EXPERIMENTAL STUDY OF INTERNAL
GRAVITY WAVES OVER A SLOPE

by

David A. Cacchione
A.B., Princeton University
(1962)

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION

September, 1970

Signature of Author

Joint Program in Oceanography, Woods Hole
Oceanographic Institution and Massachusetts
Institute of Technology, Department of Earth
and Planetary Sciences

Certified by

Thesis Supervisor

Thesis Supervisor

Thesis Accepted by

C15 Chairman, Joint Oceanography Committee in the

Earth Sciences, Massachusetts Institute of Technology
- Woods Hole Oceanographic Institution

EXPERIMENTAL STUDY OF INTERNAL
GRAVITY WAVES OVER A SLOPE

by

David A. Cacchione
Ph. D. Thesis

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

and

WOODS HOLE OCEANOGRAPHIC INSTITUTION

Experimental Sedimentology Laboratory-
Department of Earth and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA

Sponsored by the U. S. Office of Naval Research

Nonr 1841 (74)

Contract Authority NR 083-157

Reproduction in whole or in part is permitted
for any purpose of the United States Government.

Distribution of this document is unlimited.

Report 70-6 November 1970

ft*? b 2-1

. -CIS"

ABSTRACT

., CALIF, 93940

Experimental Study of Internal
Gravity Waves Over a Slope

by

David A. Cacchione

Submitted to the Department of Earth and Planetary Sciences
in partial fulfillment of the requirement for the
degree of Doctor of Philosophy.

A series of laboratory experiments were conducted in a glass wave
tank to investigate the propagation of internal gravity waves up a sloping
bottom in a fluid with constant Brunt- Vaisala frequency. Measurements of
the wave motion in the fluid interior were primarily taken with electrical
conductivity probes; measurements in the boundary layer were made with
dye streaks and neutrally buoyant particles. The results indicate that,
outside of the breaking zone, the amplitude and horizontal wave number of
the high-frequency waves increase linearly with decreasing depth; this is
shown to agree with existing linear, inviscid solutions. A zone of breaking
or runup is induced by these high-frequency waves well upslope. Shadow-
graph observations show that, if the wave characteristics are coincident,
or nearly so, with the bottom slope, the upslope propagation of the low-
frequency waves causes a line of regularly spaced vortices to form along
the slope. Subsequent mixing in the vortex cells creates thin horizontal
laminae that are more homogeneous than the adjacent layers. These lam-
inae slowly penetrate the fluid interior, creating a step-like vertical density
structure.

Available linear theoretical solutions for the velocity in the viscous
boundary layer, determined to be valid for certain experimental conditions,
are used to develop a criterion for incipient motion of bottom sediment in-
duced by shoaling internal waves. The maximum sediment sizes that can be
placed into motion, according to this criterion, are larger than certain
mean sediment sizes on the continental margin off New England. This sug-
gests that internal waves might induce initial sediment movement. Specula-
tion about the geological effects of breaking and vortex instabilities is also
given. These processes, not definitely measured in the field as yet, might
also be conducive to sediment movement.

Thesis Supervisors: John Southard

Assistant Professor of Geology

Carl Wunsch

Associate Professor of Oceanography

111

TABLE OF CONTENTS

ABSTRACT iii

LIST OF SYMBOLS vii

LIST OF FIGURES

ACKNOWLEDGEMENTS xv

1. INTRODUCTION ±

GENERAL x

BACKGROUND

Internal Wave Theory and Experiment 3

Field Observations 7

Wave-Induced Bottom Sediment Motion ,Q

2. EXPERIMENTAL SYSTEM 13

APPARATUS 13

Wave Tank and Mechanical Components .... 13

Wave Generating Apparatus 15

Power unit 15

Wave maker 17

DATA ACQUISITION 17

Detection of Wave Motion 17

Temperature 19

Specific Gravity 21

Wave- Maker Motion 2i

Photographic Measurement 23

Data Recording and Display 26

EXPERIMENTAL PROCEDURE 2 8

3. DATA ANALYSIS 31

TEMPERATURE 31

MEAN DENSITY GRADIENT 31

DENSITY PERTURBATIONS 34

VELOCITIES AND BOTTOM SHEAR STRESS. ... 3?

MISCELLANEOUS 41

TABLE OF CONTENTS (Cont)

4. EXPERIMENTAL RESULTS 45

GENERAL DISCUSSION 45

THEORETICAL DISCUSSION 49

Interior Wave Field 50

Boundary Layer 53

INTERIOR WAVE FIELD - RESULTS 55

Subcritical Case 55

Wave number 56

Wave amplitude 64

Critical and Supercritical Cases 77

Wave number 80

Wave amplitude 88

BOUNDARY ACTIVITY 9 5

Case A: y < c 95

Qualitative results 95

Quantitative results 107

Cases B and C: y = c and y > c 128

Qualitative 128

Quantitative 134

DISCUSSION 150

5. SEDIMENT MOVEMENT BY INTERNAL WAVES ... 151

GENERAL DISCUSSION 151

Oceanic Conditions 152

INCIPIENT MOTION CRITERIA 165

MODEL ANALYSIS AND SPECULATION 183

REFERENCES 197

APPENDIX A 208

APPENDIX B 213

BIOGRAPHY 226

vi

LIST OF SYMBOLS

A amplitude of the stream function

a wave amplitude

a input wave amplitude

a measured wave amplitude
m

a fluid acceleration

af effective fluid acceleration at level of effective fluid velocity

c slope of internal wave characteristic

D sediment diameter in Chapter 5

D mean sediment diameter
m

F_, fluid resistance force
F

F^ hydrodynamic drag

F„ gravity force

FT hydrodynamic lift

Fp pressure force

F M virtual or apparent mass force

g gravitational acceleration

h local fluid depth over slope

h fluid depth over horizontal bottom

k horizontal wave number

k average local horizontal wave number, theoretical

k measured horizontal wave number
m

N Brunt-Vaisala frequency

n mode number in text; index of refraction in Appendix A

p pressure

R Reynolds number

Rn particle Reynolds number

Rw wave Reynolds number

Rc boundary-layer Reynolds number

s specific gravity when used with a subscript; otherwise, specific
gravity ratio

Vll

T internal wave period

t time

C^ drag coefficient

C lift coefficient

C„M coefficient of virtual mass

U average phase function, theoretical

u magnitude of the instantaneous horizontal fluid velocity, Chapter 4;

magnitude of the instantaneous velocity in boundary layer, parallel

to the bottom, Chapter 5
v magnitude of the velocity of sediment particle

w magnitude of the instantaneous vertical fluid velocity, Chapter 4;

magnitude of the instanteous fluid velocity in the boundary layer,

normal to the bottom
x,z horizontal and vertical coordinates, respectively
x , z absolute values of x,z, respectively
x total horizontal projection of the slope

a slope angle, degrees

|3 angle between fluid resistance forces F„ and a line parallel to the

slope

bottom slope in radians

y

6 boundary layer thickness

6 measured estimate of boundary layer thickness

e ak; Stokes parameter

e a k ; measured estimate of Stokes parameter
m m m ^

6 phase angle

k bed roughness expressed as a particle diameter

ix molecular viscosity

v viscosity

i/ kinematic viscosity

7, "rj coordinates along and normal to the slope

p density

Vlll

p mean density; also amplitude of density over the flat bottom,

theoretical
p' perturbation density, theoretical

p perturbation density computed from linear, inviscid solutions for

progressive internal waves on a slope

a electrical conductivity

t shear stress

t shear stress at the bed

o

t critical shear stress; i.e., that shear stress necessary to initiate

sediment movement

<f> angle of repose of sediment particle

ip stream function

ip total stream function in boundary layer

i/jT interior stream function

ip. boundary -layer stream function

co internal wave frequency

to) critical internal wave frequency

co-, semi-diurnal frequency

b. d . subscript referring to boundary layer

f subscript "f,T referring to effective kinematic fluid property or to
physical property of fluid

m quantity determined by experimental measurements

max maximum value of quantity

o subscript usually denotes those quantities pertinent to the input
region; i. e., the portion of the wave tank with horizontal bottom

s subscript Ms" referring to physical property of sediment

T subscript referring to total value; i.e., interior value plus boundary
layer value
complex conjugate
absolute value of quantity

IX

Fig.

1

Fig.

2

Fig.

3

Fig.

4

Fig.

5

Fig.

6

Fig.

7

Fig.

8

Fig.

9

Fig.

10

Fig.

11

Fig.

12

Fig.

13a

Fig.

13b

Fig.

14

Fig.

15

Fig.

16

Fig.

17

Fig.

18

Fig.

19

Fig.

20

Fig.

21

Fig.

22

Fig.

23

Fig.

24

LIST OF FIGURES Page

Wave tank and filling system 14

Wave generating apparatus 16

Wave maker in actual location 18

Conductivity probe and probe assembly . . . . 2 0

Refractometer and mini-siphon 2 2

Shadowgraph system 2 5

Data acquisition system 27

Typical temperature - depth profile 32

Typical specific gravity - depth profile .... 33

Conductivity and wave-maker data 3 6

Sample per iodogram 38

Sample work sheet of dye-streak displacements . . 4 0

Motion of neutrally buoyant particles over the flat bottom 42

Motion of neutrally buoyant particles over the slope . 44

Classification of internal waves on a slope . . 4 6

Geometry and coordinate system 48

Upslope variation of wave number, y < c . . . . 57

Upslope variation of wave number, y < c . . . . 58

Upslope variation of wave number, y < c . . . . 6 0
Measured versus computed values of wave number,

y < C 61

Measured versus computed values of wave number,

y < C 62

Measured versus computed values of wave number,

y < C 63

Measured versus computed values of wave number,

y < C 65

Measured versus computed values of wave amplitude,

y < C 67

Measured versus computed values of wave amplitude,

y < C 68

XI

LIST OF FIGURES (Cont)

Page

Fig.

25

Fig.

26

Fig.

27

Fig.

28

Fig.

29

Fig.

30

Fig.

31

Fig.

32

Fig.

33

Fig.

34

Fig.

35

Fig.

36

Fig.

37

Fig.

38

Fig.

39

Fig.

40

Fig.

41

Fig. 42
through
Fig. 49

Fig.

50

Fig.

51

Fig.

52

Fig.

53

Fig.

54

Fig.

55

Fig.

56

> r

<

c

> r

<

c

> r

<

c

> r

<

c

> r

<

c

> r

<

c

Upslope variation of wave amplitude

Upslope variation of wave amplitude

Upslope variation of wave amplitude

Upslope variation of wave amplitude

Upslope variation of wave amplitude

Upslope variation of wave amplitude

Reflection of characteristics from a slope, y > c

Upslope variation of wave number, y > c .

Upslope variation of wave number, y > c .

Upslope variation of wave number on log-log plot, y > c

Upslope variation of wave number on log-log plot, y > c

Wave number versus magnitude of zc/cxc, y > c

Upslope variations of wave number, y = c

Wave amplitude versus magnitude of zc/cxc, y> c .

Wave amplitude versus magnitude of zc/cxc, y > c

Upslope variation of wave amplitude, y > c .

Upslope variation of wave amplitude, y = c

Photographs of shadowgraph images of internal waves
over a slope, y < c

Diagram of initial vortex advancing upslope
Position of breaking

Theoretical increase in velocity components along the
slope

Estimates of boundary layer thickness and Reynolds
numbers, y < c

Vertical distribution of velocity near the bottom at
station l,y<c

Vertical distribution of velocity near the bottom at
station 2, y < c

Vertical distribution of velocity near the bottom at
station 3,y<c

69
70
71
72
73
74
78
81
82
84
85
86
87
90
91
93
94

98

106

107

110
111
113
114
115

xii

LIST OF FIGURES (Cont) Page

Fig. 57 Vertical distribution of velocity near the bottom at

station 4, y <c 116

Fig. 58 Vertical distribution of velocity near the bottom at

station 5, y < c 117

Fig. 59 Velocity and bed shear stress at selected vertical levels

near the bottom, y < c 120

Fig. 60 Measured values of mean streaming 122

Fig. 61a Experimental and theoretical values of bed shear stress

at station 1 124

Fig. 61b Experimental and theoretical values of bed shear stress

at station 2 12 5

Fig. 61c Experimental and theoretical values of bed shear stress

at station 3 126

Fig. 6 Id Experimental and theoretical values of bed shear stress

at station 4 127

Fig 62

,, &* . Photographs of shadowgraph images of internal waves

Fig°U64 over a slope, y > c 129

Fig. 65 Onset of vortex motions as a function of wave frequency 13 5

Fig. 66 Maximum size of vortex diameters as a function of

frequency 136

Fig. 67 Density-depth profile through streamers and layers . 138

Fig. 68 Average net velocities in streamers and intervening

layers 139

Fig. 69 Estimates of boundary layer thickness and Reynolds

numbers, y = c 141

Fig. 70 Estimates of boundary layer thickness and Reynolds

numbers, y>c 142

Fig. 71 Velocity-time plots at selected level near the bottom,

y = C 144

Fig. 72 Velocity and bed shear stress at five stations, y - c . 14 5

Fig. 73a Vertical distribution of velocity near the bottom at

station 1, y > c 147

Fig. 73b Vertical distribution of velocity near the bottom at

station 4, y>c 148

xm

LIST OF FIGURES (Cont) Page

Fig. 74 Velocity and bed shear stress at selected levels near

the bottom, y > c 149

Fig. 75 Frequency spectra of horizontal kinetic energy density

at site "D" 154

Fig. 76 Bathymetry off southern New England . . . . 155

Fig. 77 Internal wave measurements on continental margins . 156

Fig. 78 Idealized cross-section of continental margin . . . 158

Fig. 79 Characteristic slope and average bottom slope compared

for various a> and N 16 0

Fig. 80 Definition sketch of sediment particles . . . . 166

Fig. 81 Resistance coefficients Cn for spheres .... 175

Fig. 82 Roughness effects on the resistance coefficients for

spheres 176

Fig. 83 D./6 versus h/h 178

Fig. 84 Shields diagram 181

Fig. 85 Hypothetical size- frequency distribution of sediments on

an oceanic slope 185

Fig. 86 Continental shelf model; high frequency waves . . 18 8

Fig. 87 Continental shelf model; tidal frequency waves . . 19 0

Fig. 88 Continental slope model; high frequency waves . . 192

Fig. 89 Continental rise model; tidal frequency waves . . . 193

Fig. Al Circuit schematic for conductivity sensor. . . . 210

XIV

ACKNOWLEDGEMENTS

I am deeply indebted to Professor John Southard and to Professor
Carl Wunsch for their expert guidance throughout this study. Without their
continuing technical and moral support this work would not have been pos-
sible. Their patience to listen and their willingness to comment will not
be easily forgotten. I would also like to thank Dr. Nicholas Fofonoff and
Dr. Charles Hollister for their critical reviews of portions of this work;
their comments aided the preparation of the final product. I gratefully
acknowledge the assistance of Dr. Seelye Martin who provided the original
circuitry for the conductivity probes and gave several helpful suggestions
concerning the experimental techniques.

Several persons provided invaluable assistance during the several
phases of this study. I greatly appreciate the skilled technical assistance
given by Mr. John Annese of the departmental machine shop during the con-
struction of the experimental apparatus. The variety of computations and
other tasks that were accomplished expeditiously by Mr. David Drummond
during the analysis of data, and the fine illustrations which were provided
by Mr. Valentin Livada were indispensable to the preparation of this work.
I thank Mrs. Caroline Peterson for her careful typing of the final draft.
Of course, without my wife, Virginia, there would not have been a final
product.

I am grateful to the Education Development Center in Newton,
Massachusetts, for donating the glass tank used during the experiments.
This work was supported by the Office of Naval Research under Contract

No. Nonr 1841 (74).

xv

1. INTRODUCTION

GENERAL

The purpose of this research is threefold:

(1) to provide experimental results that describe the changing
character of single-frequency, small -amplitude internal gravity waves as
these waves propagate over a sloping bottom;

(2) to compare these results with recent theoretical solutions for
those conditions which are theoretically tractable, and to examine the phy-
sical characteristics of those conditions which are difficult to treat theo-
retically;

(3) to apply the theoretical and experimental results toward the
development of a simple analytical model that prescribes the conditions for
the initiation of bottom sediment motion induced by shoaling internal
gravity waves.

The motivation for this work was rooted in speculation - a starting
point not uncommon in experimental research. The speculation was stimu-
lated by the results of previous work that had indicated the possible signifi-
cance of internal waves as geological agents and by recent theoretical
solutions that predict the intensification of internal wave motion along
sloping bottoms.

It was decided early in this research that a clearer physical under-
standing of the shoaling process for internal waves was needed - particuarly
knowledge of the nature of the bottom boundary layer - before the two-phase
problem could be considered. The effects of internal waves on sediment
particles of various shapes, sizes, and densities were investigated during
a preliminary set of experiments. It was concluded from these kinds of
experiments that the reliability of the results and the range of test condi-
tions were limited by the experimental difficulties, mainly the difficulty

of attaining large velocities in the small experimental system. The approach
to this problem was changed to investigate the movement of sediment by in-
ternal waves analytically, based on the results of a detailed laboratory study
of shoaling internal waves over a smooth sloping bottom. Recent theoretical
analyses had cast a framework for the experimental design (Wunsch, 1969;
Keller and Mow, 1969); preliminary laboratory work had established the
experimental techniques. Earlier experiments by other researchers had
demonstrated the feasibility of some of the techniques (Mowbray and Rarity,
1967; Martin, Simmons, and Wunsch, 1969; Gibson and Schwarz, 1963),
and had shown several stimulating results (Thorpe, 1966).

The subsequent work is divided into three sections:

(1) the experimental design, instruments, and techniques are dis-
cussed in Chapters 2 and 3;

(2) the results of the experiments and their comparison with theo-
retical solutions of Wunsch (1969) are presented in Chapter 4. The results
are separated into two basic types: experiments with high-frequency waves
and experiments with low-frequency waves. The interior wave field and
the boundary layer are discussed separately for each;

(3) in Chapter 5 the balance of forces acting on a bed particle is
used to derive a simple analytical model for the stability of the particle
beneath a train of high-frequency, shoaling internal waves. The approach
closely follows that used by Ippen and Eagleson (1955) and Eagleson and
Dean (1959) in the development of an analytical model for sediment motion

*
The distinction between high and low frequency waves is discussed in

more detail in Chapter 4. Basically the incident energy of the high fre-
quency waves propagates upslope to the corner; i.e., toward the intersec-
tion of the slope and the free surface, with no reflection. By contrast, the
incident energy of the low frequency waves theoretically is back reflected
from the slope.

induced by surface waves. The applicability and limitations of the analysis
of internal waves and sediment movement are discussed in Chapter 5, and
some speculations are offered concerning the effects of breaking of internal
waves and the action of boundary layer instabilities on bottom sediment in
the ocean.

BACKGROUND

Internal Wave Theory and Experiment

In two recent papers, Wunsch (1968, 1969) has developed normal-
mode solutions for both standing and progressive internal waves in a wedge
geometry. The solutions of interest here are for progressive waves pro-
pagating into the wedge, i.e., over a linearly shoaling bottom. This theory
assumes two-dimensional motion in a stably stratified, inviscid, Boussinesq

,2 e dp0
fluid with constant Brunt- Vaisala frequency N (N = — — - — ). The geometry

and coordinate system that Wunsch considered is given in Figure 1.
Wunsch's solutions are discussed in more detail in Chapter 4; these solu-
tions are presented for comparison with the experimental results. One of
the principal features in the solutions is the suggestion of a strong intensi-
fication of velocity components along the sloping boundary. Wunsch (1969)
conducted a simple experiment in which internal waves were propagated
over a linearly sloping, rigid bottom. From photographic determinations
of the changes in wave length up the slope, he showed that the results are
in agreement with his solutions. The wave length was approximately pro-
portional to the depth. Magaard (1962) presented solutions for a similar
problem of standing internal waves except for the density distribution and
shape of the bottom profile. He considered a density distribution that can
be expressed analytically as:

p(z) = c2 (2z + Cl)1/2 (1-1)

where c-, c„ are constants. The bottom profile that he chose for

computational facility and some realism was:

h(x) = i

(2bx)'

(1-2)

where c1 and b are constants. He developed a linearized characteristic
equation for the velocity field and found that the equation was separable in
the spatial coordinates only for the case of constant depth. He also showed
that for a bottom slope y less than the slope of the characteristics c, a
solution is possible which gives waves that have increasing amplitudes and
decreasing wave lengths with depth. Magaard (1962) also found an appre-
ciable intensification of the velocity components along the sloping bottom.

Robinson (1970) has recently considered the effects of a corner
(such as the point of transition between slope and flat bottom) upon an in-
ternal wave train propagating toward the slope in an infinite medium (i.e.,
with free surface infinitely far from the horizontal bottom). Robinson
provided solutions that consist of incident and reflected waves such that
the reflected waves do not violate the radiation condition; i.e., they do
not introduce energy from the far field (at x = +*>).

Keller and Mow (1969) obtained an asymptotic solution to the
problem of internal wave propagation in a horizontally stratified fluid of
nonuniform depth by application of the principles of geometrical optics.
Their general solutions apply to the linearized equations of motion, and
the wave amplitude and fluid depth are assumed to vary very little over
horizontal distances small compared to a wave length. They gave a partic-
ular example in which N was assumed constant and the bottom was a small,
linear slope. The solutions in this case show that the wave length is
proportional to the depth, and the wave amplitude increases linearly with
decreasing depth. Keller and Mow noted that their solutions also are
in agreement with the experimental results given by Wunsch (1969).

Hogg and Wunsch (1970; also personal communication) derived
similar asymptotic solutions using a WKB approach with a two parameter

expansion scheme. Their first order, linear solutions for progressive
internal waves over a small, linear slope are equivalent to those of Keller
and Mow (1969); both solutions are compatible with the normal mode
solutions of Wunsch (1969) for small slopes (y«c). Hogg and Wunsch ex-
tended their analysis to higher-order terms in the stream function and
developed expressions for the wave-induced radiation stress. They showed
that a significant set-up of the isopycnals induced by internal waves is pos-
sible and demonstrated theoretically the plausibility of "longslope" currents
for a line of breaking internal waves oblique to the bottom contours. From
the higher order terms Hogg and Wunsch (1970) were also able to demon-
strate that in the absence of mixing the net Lagrangian motion of water
particles in a two-dimensional stably stratified fluid is necessarily zero
for internal wave motion in the proximity of a sloping, rigid boundary.

Fofonoff (1969) also examined analytically the unsteady motion
in a horizontally stratified, uniformly rotating ocean. He considered
the seaward propagation of an internal baroclinic wave that was
generated at the continental shelf edge by the diurnal tide. By exam-
ining the characteristics of the solution, he presented a numerical
example in which the values of density, latitude, and depth were those
measured at site "D" (39° 20'N, 70° W), located about 37 miles south of
the 200 meter contour. Fofonoff (1966) showed that north of this location
the bottom profile of the continental slope and the path of the characteris-
tic were approximately coincident. Again there is implication of increased
bottom shear along the slope.

Sandstrom (1966) derived the reflection properties of internal
waves from a sloping, rigid boundary. Both he and Robinson (1970) have
noted that the normal- mode solutions derived by Sandstrom from his ray
theory solutions for progressive internal waves propagating over a slope
violate the radiation condition (i.e., the complete solution requires an
anomalous set of reflected waves that introduce energy from the far field).
Sandstrom predicted that upon reflection the waves change their wave

number and amplitude; he found that the magnitude of the changes is
dependent on a ratio equivalent to y /c. Sandstrom (1966) also conducted a
set of experiments in which internal waves were generated by a flap-type
wave maker and allowed to propagate over a planar, inclined bottom. His
experimental results for a gently sloping bottom (y «c) showed qualitatively
two features that were also found in the present study, and are discussed
quantitively in Chapter 4: (1) amplification of internal waves traveling
from "deep" into "shallow" water, and (2) intensification of motion near
the slope. Sandstrom (1966) noted, however, that unfortunately the partic-
ular experiment for which these features were observed was complicated
by the presence of a surface mode that was also generated by the wave
maker. Although his apparatus was apparently capable of generating the
first internal mode, there are no experimental results in his work which
indicate that he did this. Sandstrom discussed solutions for the case y >c
(steep bottom slope), but did not present experimental results for this
case. He concluded from the linear theory that in the case y ^ c, the motion
close to the bottom becomes very large.

Longuet-Higgins (1968) extended the discussion of reflection prop-
erties of internal waves from rigid surfaces to account for various types
of bottom roughnesses. He has shown that the important parameter is not
the ratio of the scale of bottom roughness to wavelength but instead the
ratio of the scale of roughness to the thickness of the oscillatory boundary
layer. He pointed out that this suggests that small-scale irregularities
can severly affect the transmittance and reflection conditions of the bottom.

In a very thorough theoretical and experimental treatment of finite-
amplitude effects on interfacial and internal waves in a horizontal channel,
Thorpe (1968) also reported some qualitative observations of progressive
internal waves on a slope. In his experimental arrangement, a linearly
sloping, smooth bottom was mainly employed as a wave absorber. How-
ever, during the course of the experimental runs for internal waves in a
linearly stratified salt solution he noted several interesting features over

the slope. These observations are summarized here as a preface to the
results shown later in this study:

(1) the wave length decreased upslope;

(2) a large runup was apparent along the upper slope region, and the
larger amplitude waves produced a larger runup;

(3) the free surface remained undisturbed;

(4) a "rotor" was observed near the free surface on one occasion;

(5) breaking ("overturning") was observed well upslope on one
occasion;

(6) no reflections could be detected.

Thorpe (1968) showed three photographs of the waves over the
slope. His method of flow visualization, involving alternating dyed and
undyed layers, did not permit detailed observations of the breaking or
mixing, but his qualitative observations are quite instructive and essentially
agree with the results shown later.

It should also be mentioned here that excellent experimental studies
of instabilities and turbulence in a stably stratified fluid, including breaking
of interfacial waves, have been carried out by Pao (1968).

Field Observations

Two kinds of field observations are pertinent to this study: (1)
measurements that suggest shoreward propagation of internal waves over
the continental margin, and (2) observations that indicate, directly or in-
directly, the motion of bottom sediment due to passage of internal waves.

Measurements of the first kind have been presented by Lee (1961),
Gaul (1961) and Ufford (1947), among others. Their investigations presented
time-series measurements of temperature fluctuations taken simultaneously
at three stations oriented in various triangular arrays in shallow water
(depth <200 feet). The wide geographical distribution of these studies illus-
trates the ubiquity of internal motions on the continental shelves. Common

internal-wave periods were on the order of 5 to 20 minutes; wave propaga-
tion was shoreward with speeds of 0.3 to 1.2 knots. Lafond (1962) has
provided similar values. Longer period internal oscillations are described
by Lee (1961), Boston (1964), and Summers and Emery (1963). The last
study illustrated the refraction of internal waves of semidiurnal period as
they propagated shoreward over the continental slope and shelf off Southern
California. The estimated wave speed was 7 knots in deep water and slightly
less than 1 knot over the shelf.

Isolated asymmetrical internal temperature disturbances resembling
the idealized solitary wave profile of surface waves (Ippen, 1966) have also
been detected by Lee (1961), Gaul (1961), and Cairns (1967). The common
period of this sort of disturbance seems to be approximately 10 minutes,
and the propagation direction is again shoreward.

Curves of spectral kinetic energy density versus frequency are
available for velocity measurements of internal motion taken with vertical
arrays of current meters at site "D" (Fofonoff, 1968). An example of these
spectra is shown in Chapter 5.

Direct observations which reliably link the two motions of internal
waves and sediment movement are limited to a few isolated instances.
This lack of evidence might suggest both the difficulty and the insufficiency
of this kind of field measurement. Ideally, one would like to measure simul-
taneously the internal wave velocity structure and the movement of bottom
sediment beneath the wave motion. Either measurement alone is a challenge
in the field. Lafond (1965), using underwater television to observe the sedi-
ment motion and a vertical array of temperature sensors to measure the
internal wave motion, found that the movement of sediment ripples in shal-
low water (depth about 60 m) was correlated with large fluctuations in the
temperature structure. He interpreted this as evidence for sediment trans-
port by internal waves.

Indirect field data that connect the two motions are similarly limited.
Revelle (1939) inferred the connection in an attempt to explain an observed

pattern of sediment distribution. Munk (1941) calculated the theoretical
standing-wave characteristics in the Gulf of California and found that the
nodal-antinodal separation coincided favorably with Revelle's (1939) ob-
served distribution of sediment sizes. That is, the smaller sizes were
noticeably lacking beneath the nodes, where larger instantaneous velocities
are expected. However, in a subsequent detailed study of the sediment dis-
tribution in the Gulf of California Van Andel showed that the sediment pat-
tern described earlier by Revelle was not apparent in the later data.
Emery (1956; 1960) suggested that deep, standing internal waves in certain
California basins stir the bottom sediments and might produce the observed
local concentrations of coarse sediments at the basin sills.

More recent investigations of the large submarine sand waves (av-
erage height =rf 8 meters) near the edge of the continental shelf southwest of
Great Britain (water depths 180 meters) by Stride and Cartwright (1958),
Cartwright (1959), Stride and Tucker (1960), and Carruthers (1963) indicate
the possibility of interaction between internal waves and bottom sediment.
Cartwright (1959) formulated a theory which suggests that these sand waves
are maintained by a stationary internal- wave system of tidal period that
exists during the presence of the local seasonal thermocline. Measure-
ments of oscillations of the scattering layer (Stride and Tucker, 1960) and
towed thermistor records (Carruthers, 1963) support this theory. Car-
ruthers (1963) also measured the currents close to the crest of the sand
waves at one-hour intervals over a seventeen- hour period. A plot of cur-
rent vectors indicated that the predominant flow direction was normal to
the crests, with a maximum current of 0.5 knot in this direction. Inter-
estingly, the crests of the sand waves are parallel and adjacent to the
shelf edge.

Many investigators have studied the observed sediment distribution
on the continental shelf and slope off the East Coast of the United States.
Uchupi (1963), Emery (1966), and Schlee (unpublished manuscript) have
summarized much of this work. Sand waves and local topographic highs

10

are abundant on the continental shelf. The type and size distribution of
sediment varies both across and along the bottom contours. The seaward
variations are generally sharper. For instance, Uchupi (1963) delineated
four broad zones parallel to the shelf edge from Cape Cod to Hudson Can-
yon. He noted that the most seaward zone (about 70 kilometers wide) con-
sists of silty sand, sandy silt, and silt (water depth 60 to 135 meters).
Uchupi (1963) claimed that from Hudson Canyon to Cape Hatteras the outer-
most zone is composed of fine sand of approximately uniform grain size.
The surface of the continental slope in these regions consists mainly of
silt and clay; however, sand, silty sand, and gravel line the floors of the
submarine canyons that cut the slope.

Wave-Induced Bottom Sediment Motion

The writer could find no published results on wave-induced bottom
sediment motion in a stratified fluid. However, there are many published
studies of sediment motion beneath surface waves in a homogeneous fluid.
Most of these studies are empirical or semi-empirical. The obvious lack
of theoretical treatment probably stems from the complex form of the gov-
erning equations of motion, which are not well established.

One appealing approach to a laboratory and analytical study of the
problem was undertaken by Ippen and Eagleson (1955) and later extended by
Eagleson, et al (1958), and Eagleson and Dean (1959). A statistical treat-
ment of laboratory data that involved the motion of discrete spherical sedi-
ment particles due to shoaling surface waves supplemented their theoretical
presentation. Many of the complicating secondary factors, such as (1)
nonuniform bed roughness, (2) shape variations of the particles, (3) mu-
tual particle interactions, (4) nonuniform incident wave trains, (5) local
channelization and fluidization of the bed, and (6) bed permeability were
not included in their work. They were able to derive a force-balance
equation for the individual particles. Ippen and Eagleson (1955) described
two possible types of particle motion: incipient and equilibrium. Incipient
sediment motion was defined by Eagleson et al (1958) as "an instantaneous

11

condition reached when the resultant of all active forces on the particle
intersects the line connecting the bed particle contact points. " They also
defined established sediment motion as "an oscillatory or quasi-oscillatory
condition of motion reached when for some portion of each wave cycle the
sum of the instantaneous active forces is greater than that value necessary
to initiate motion. " The major difficulties in working with these definitions
are: (1) the point of application of some of the individual forces is usually
not known; (2) the various coefficients, such as those for drag, lift, and
virtual mass, are not well defined for unsteady motion; (3) usually the
effective fluid velocities acting on the bed particle must be estimated from
an approximate theoretical equation, since the exact velocity profile near
the rough bed in unsteady motion is not known. Despite these difficulties,
Eagleson and Dean (1959) were able to arrive at a semi-empirical relation-
ship for particle size, beach slope, and local wave characteristics. Eagle-
son et al (1958) compared the analytical results with field data from eight
stations on the Atlantic coast of the United States. As these authors put it,
the agreement was "fortuitously" remarkable.

The concept of an equilibrium diameter for given wave conditions
was first proposed by Cornaglia (Miller and Zeigler, 1964) and has since
been labeled the null-point theory. His arguments, as well as those of
Ippen and Eagleson (1955) and Miller and Zeigler (1964), are basically
simple. In an area of shoaling waves, every bottom sediment particle is
affected by the existing sea state. For any given sea state and bottom
slope, there will be one size class of particles in a state of equilibrium
motion. That is, this size class will move back and forth along the bottom
as successive troughs and crests pass, but will oscillate about some mean
position. In general, particles larger than the null-point size will move
offshore and those smaller will move onshore. Miller and Zeigler (1964)
noted that in the field the null point should be regarded statistically, where-
as in the laboratory it is convenient to regard it as single-valued. Their
field study also indicated the limitations and the modified applicability of
this theory in nature. However, at the same time their work showed that

12

the theory gives a satisfactory prediction of the gross sediment distribution
of a nearshore sand bottom.

Other laboratory investigations have produced relationships between
the initiation of motion of a given particle size and the associated wave con-
ditions (Manohar, 1956; Goddet, 1960; Vincent, 1958). These investigations
were all conducted in a channel with a level bottom. Vincent (1958) observed
that the fluid velocity just outside the boundary layer was almost constant
at the onset of movement of equal- diameter grains. Abou-Seida (1965)
noted that field measurements of radioactive sand tracers conducted by
Sato et al (1963) compared favorably with the laboratory relationships de-
duced by Manohar (1956) and Goddet (1960).

A second approach to this problem is relatively new. Arguing from
basic physical principles and certain assumptions, Bagnold (1963, 1966)
derived a relationship between bed load transport and the dissipation of
energy in the fluid motion. His argument rests on the assumption that a
distribution of dispersive and tangential shear stresses exists in a bed of
granular material that is subjected to shear deformation. Several sum-
maries of the topic of wave-induced sediment motion are available (Inman,
1963; Abou-Seida, 1965; Raudkivi, 1967). Such presentations of incipient
sediment motion usually include a discussion of the various criteria developed
for open channel flow. In particular, the work of Shields (1936), White
(1940), and Kalinske (1947) is often cited.

13

2. EXPERIMENTAL SYSTEM

APPARATUS

Wave Tank and Mechanical Components

The rectangular wave tank illustrated in Figure 1 has an open top
and 3/8-inch glass side walls and bottom. It is 4. 9 m long, 20. 5 cm wide,
and 38. 4 cm deep and consists of two sections of similar dimensions joined
end to end by an O-ring seal contained between plastic inserts attached to
the sections with epoxy cement. A thin layer of clear silicone sealant (Dow
Corning 701 Building Sealant) was applied along the inside seam to form a
smooth transition between the two sections. All other joints were also
formed with silicone sealant. The ends of the channel are acrylic plastic
plates sealed by Neoprene gaskets. A 14-foot length of aluminum angle
attached to five vertical aluminum angle supports behind the tank and 3/4 of
an inch above the side wall served as an instrument mount.

A filling technique similar to that described by others (Fortuin,
1960; Oster, 1965; Jaffee, 1968) was designed to establish the desired sal-
inity gradients in the wave channel. Figure 1 shows a diagram of the filling
system. Owing to the corrosive properties of salt water, the insides of the
two 55-gallon containers were coated with epoxy paint, and all seams were
caulked with silicone sealant. Wherever the liquid came into contact with
bare metal, only stainless steel or brass was used. Flexible Tygon tubing
is used throughout the filling system wherever tubing was required.

The filling rate was controlled by a brass needle valve. Each drum
was equipped with a Tygon sight tube for visual determination of the level
in each barrel. A motor driven stainless stirring rod with two brass stir-
ring vanes was mounted above the fresh water barrel. To prevent solid-
body rotation of the water, the stirring rod lay off the axis of the barrel.

The bottom slopes, one of which is shown in position in Figure 1,
were cut from Plexiglas sheets. Each slope is beveled at one end to form

14

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a smooth junction with the bottom of the tank. Four slope angles were used,
7, 15, 30, and 45 degrees. During the experiments the slopes were supported
by two parallel Plexiglas struts. These struts were positioned at the proper
angle along the inner walls of the tank. Four Lexan plastic rods with threaded
ends and plastic nuts were used to separate the struts and to maintain their
position by forcing them against the channel walls. Neoprene strips were
cemented to the outer surfaces of each strut to form a continuous and flexible
seal with the side walls of the channel. All materials forming the beach
slopes and supports were noncorrodible.

Wave Generating Apparatus

The wave generator had to meet five primary requirements:

(1) smooth and sufficiently powerful motion;

(2) desired frequency range and control;

(3) variable amplitude;

(4) compact size with components stably mounted on a plate above
one end of the channel;

(5) long running with low vibration level.

In an effort to meet these requirements the apparatus was modified several
times. Availability of various components such as the variable speed drive
often aided the choice of parts. The end product was an outgrowth of the
usual experimenter's dilemna: getting optimum performance at reasonable
cost.

Power unit. Figure 2 is a diagram of the power drive for the wave maker.
The entire drive mechanism is rigidly supported above one end of the chan-
nel by four 1-1/2-inch pipe legs with foot flanges. The base is a 1/4-inch
steel plate. The assembly consists of a Graham variable-speed drive
coupled to a 1/4-hp motor. The output shaft is joined to a fixed- ratio gear
reducer (9.5:1) via a flexible coupling. This second speed reducer is con-
nected to a Scotch yoke device that converts the rotary shaft movement into
rectilinear sinusoidal motion of a vertical shaft.

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Wave maker. Figure 3 shows the wave maker in actual location. Several
additional considerations were involved in the design of this device. First,
a unit was needed that could be inserted into position easily without appre-
ciably disturbing the stratification after the tank had been filled. Secondly,
the driving motion had to generate, to a good approximation, first- mode
internal waves while minimizing mixing at the wave maker. In addition,
the water motion in the section behind the wave maker had to be prevented
from interacting with the fluid in the working section of the channel.

Basically, the device consists of a plastic flapper plate free to pivot
about its midpoint. Hinges anchor this plate to a supporting frame. A thin
flexible rubber gasket covers the entire face of the plate and frame, thus
preventing liquid from flowing between the two sections. This gasket is
held in position by thin plastic covers cut to the same shape and dimemensions
as the frame and flapper plate. The flapper plate is driven by a stainless -
steel push rod connected to the vertical shaft of the Scotch yoke by a universal
coupling that allows for minor misalignments between shaft and flapper plate.

A closed- cell Neoprene strip was cemented to the sides and bottom
of the frame. This afforded an adequate seal between the wave flapper
assembly and the glass tank. A test showed that the wave maker could suc-
cessfully maintain a head difference of at least 6 cm of water between the
working section of the channel and the unused portion.

DATA ACQUISITION

This section describes the instruments used to record the experi-
mental data. A more detailed discussion of the quality of the measurements,
including calibration curves, error, and reliability estimates, is presented
in the next chapter and in Appendix A.

Detection of Wave Motion

The changing properties of the internal waves as they propagated
upslope were measured in the Eulerian sense with conductivity probes.

18

Fig. 3. Wave maker in actual location. Tank
depth is 3 8.2 cm.

19

The basic feature of the water medium was its strong salinity stratification.
Also, over the range of salinity values used during this study, salinity is
approximately linearly proportional to conductivity. The temperature vari-
ations in the water were found to be small (usually less than 0. 5°C over the
total depth) except in the upper two centimeters, so that below this layer the
total change of conductivity at a point is here a valid estimate of the total
change of salinity:

S = SQ + £ (a - aQ)

(2-1)

Da 1 DS
Dt " v Dt

E

where

Yj in %o/(ohm -cm)"

S in%0

a in ohm cm

Because the wave-induced fluctuations of conductivity are measured relative
to some basic state of rest, only relative changes in conductivity at a point
are needed to decode the waves.

The conductivity sensor is a spherical platinum electrode that is the
active arm of an A. C. Wheatstone bridge. Design and construction of the
probe was inspired by Gibson and Schwartz (1963). Figure 4 shows a probe
assembly, including electrode, glass housing, and connecting cable. A de-
tailed description of its construction and an analysis of its performance are
presented in Appendix A.

Temperature

Temperature of the room air and the water in the filling barrels
was measured with a mercury-in-glass thermometer which can be read to
0.05°C. Vertical and horizontal temperature profiles of the stratified

20

5mm

inner wire of coax
spot weld

3 mm

////////////////r

5.3 cm

glass annealed to Pt wire

0.3mm

Fig. 4. Upper: Photograph of conductivity probe

assembly and circuit box.
Lower: Construction of conductivity probe
tip.

21

water column were taken with a glass -coated thermistor bead that is the
active arm in a D.C. Wheatstone bridge.

Specific Gravity

Specific gravity measurements of the water in the filling barrels
were made with a lead- weighted glass bulb hydrometer. The hydrometer
can be read to 0.0005 for any specific gravity in the range from 1. 000 to
1. 070. Its accuracy is about 0. 2 percent of the maximum value in the range.

The basic specific gravity gradient was determined from measure-
ments of the vertical distribution of index of refraction. Figure 5 shows
the refractometer and mini- siphon that were used to obtain index of re-
fraction at various depths in the stratified water. These measurements
were converted to specific gravity values from calibration curves. The
siphon consists of a 0.5 mm diameter stainless steel tube supported verti-
cally by a brass bushing and two locking screws. The bushing is press -
fitted into an aluminum bar. Centering pins allow repeatable positioning of
the aluminum bar athwart the channel. Markings that are evenly spaced one
centimeter apart are inscribed on the steel siphon for vertical positioning.
Small-diameter Tygon tubing fits over one end of the siphon tube. A brass
pin that can be inserted into the open end of the Tygon tubing is used to stop
the flow. Flow rate through the siphon at mid- depth (about 16 cm of water)
is about 0.2m? /sec.

Wave -Maker Motion

A rotary potentiometer in a D.C. voltage -divider circuit transduces
the oscillating, rectilinear motion of the Scotch yoke into a fluctuating D.C.
voltage signal. A precision gear meshes with a linear rack that is cut into
the forward part of the upper vertical shaft of the Scotch yoke (Figure 2).
The spur gear is fastened to the shaft of the rotary potentiometer. Vertical
movement of the Scotch yoke causes a change in the voltage output of the
sensing circuit. The sensitivity is approximately 0. 1 volt /cm. This tech-
nique establishes a convenient and accurate frequency reference and time

22

Fig. 5. Refractometer and mini-siphon

23

base for the conductivity probe measurements. The accuracy of the circuit
as a frequency reference is primarily dependent upon the linearity of the
potentiometer. In this case, the linearity of the rotary pot is 0. 1 percent
(Giannini Control Corporation). Amplitude calibration of the signal also
provides a measure of the forcing amplitude of the wave flapper.

The period of the motion is sensed by a microswitch mounted on
the lower part of the vertical support plate for the Scotch yoke (Figure 3).
This switch is closed once per cycle by a protrusion on the rotating arm,
and provides a convenient trigger for a period counter.

Photographic Measurement

Photographic measurements of the wave motion were obtained by
recording on film the behavior of neutrally buoyant particles and various
dyes. Expandable polystyrene particles (Sinclair -Koppers Company) were
chosen because their density was adjustable to the proper range by a tech-
nique described by Bohlen (1969). In addition, their milky white color pro-
vides an excellent contrast with the surrounding water when used with a
black background and overhead lighting. After processing to achieve the
desired density range, Bohlen found that their sizes varied over a consider-
able range. Sieving produced a more uniform size distribution, with median
size about 0.2 mm. The range of specific gravity for the lot was 0. 990 to
1. 040, with an obvious skew toward the heavier limit. The mean specific
gravity was estimated at 1.020. These values were adequate for neutral
buoyancy in the range of fluid densities used during this study. The particles
were used to study orbital velocities and modal structure.

Potassium permanganate (KMnO.) was used to make the vertical dye
streaks to determine near-bottom velocities and bottom shear stress.
When dropped into a water column, crystals of this substance leave a dis-
tinct dark- red dye trace from surface to bottom. The vertical trace was
then tracked photographically as it was deformed by wave motion.

24

Small quantities (one to two cc) of common liquid food coloring or a
highly nondiffusive blue dye (Blue Dextran 2000) were injected into the fill-
ing tube by means of a 16 -gauge hypodermic needle and syringe through a
self- sealing rubber membrane at infrequent intervals. The membrane was
fitted over one end of a glass T-joint in the Tygon filling line (Figure 1).
These injections resulted in thin horizontal dye layers at various levels in
the stratified water column. This enabled qualitative measurements of the
internal waves by observation of the deformation of the dye layers by the
propagating waves. The layers were typically 3 to 5 mm thick for the food
dyes and only about 1 to 2 mm thick for the Blue Dextran 2000. The tech-
nique also aided in observing flow conditions at the entrance to the main
channel during filling. Excessive mixing at this point was indicated by rapid
diffusion of the dye, as might be expected in an area of strong turbulence.
In such cases, filling rate was decreased. Finally, the dye layers added
some aesthetic value to the final stratification.

Shadowgraph images were used to display instabilities in the boundary
layer and the violent breaking of waves in the upslope turbulent zone. The
shadowgraph system is shown diagrammatically in Figure 6. The two-way
diffusive screen is sold by Edmund Scientific Company under the trade name
Lenscreen. Its excellent dispersive properties and good transmittance pro-
duced clear and evenly lighted images. The merits and techniques of shadow-
graph flow visualization are discussed by several authors (Barnes, 1954;
Goldstein, 1965; Edgerton, 1958). A Bolex 16 mm camera was used to take
motion pictures of the shadowgraph images .

The shadowgraph lighting was provided by a tungsten filament lamp
(Beck Company) that contains a dial rheostat for light-intensity adjustments.
It also has a circular portal lens that can be moved toward or away from
the light source. This lens and an adjustable diaphragm are located in a
collimating tube through which the light rays pass. A 500-watt combination
flood- spot lamp provided illumination for the dye streak photographs.

25

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Data Recording and Display

Figure 7 shows some of the instruments used in this study. The
oscilloscope at the left of the picture has provisions for dual-channel opera-
tion (Tektronix Type 564 Storage Oscilloscope), and was used mainly to
display the output of each conductivity bridge when null was set (Appendix A).
The instrument at the right is a Brush dual-pen strip-chart recorder. The
two charts operate on the same time base but with independent voltage scales.
This recorder provided a real-time display of the transduced conductivity
fluctuations during the internal-wave runs. Thermistor temperature mea-
surements were also recorded in this way.

The heart of the data recording system, a Wang model 2300 digital
readout and recording system (Wang Laboratories), is at the center of the
photograph in Figure 7. The fastest available sampling rate (1 cycle per
second) was chosen for all phases of this investigation. The system was
programmed such that each tape record was in card-image format (< 80
characters per record) and represented one second of real time. Included
in each record were digital clock time to the nearest second and six sequen-
tial channels of data. The data characters contained five measurement
digits, decimal point location, and sign; each data point was recorded to the
nearest millivolt. The recording format simplified use of the data in the
digital computation routines on the IBM 360/67 system and in listing the
raw data in decimal format on the IBM 1401.

Several cameras were used to photograph the neutrally buoyant par-
ticles and dye lines. General photographic work was done with a Nikon F
single-lens 35 mm reflex camera. Another Nikon F camera with an auto-
matic film advance attachment was used to photograph deformations of the
dye streaks, and, in some cases, the motions of neutrally buoyant particles.
The film advance was controlled by a synchronous timing device that provides
continuous one-second trigger pulses to the camera at an accuracy of ±0. 1
percent of the timing interval. A Calumet view camera with fast Polaroid
film was normally used to photograph the neutrally buoyant particles, which

27

Fig. 7. Data acquisition system

Left: Tektronix type 564 stroage oscilloscope
Center: Wang model 2300 digital data system
Right: Brush dual-pen strip chart recorder

28

were illuminated by a flashing stroboscope. The strobe flashes were triggered
externally by a signal generator (Wave-Tek).

EXPERIMENTAL PROCEDURE

Although the procedure for each experiment varied depending upon the
purpose, the following basic procedure was employed throughout the prepara-
tion and execution of all experiments. Several days (usually three) prior to
the experiment, the two 55 -gallon barrels were filled to a predetermined level
with hot tap water to which 7 to 10 teaspoons of Alconox laboratory detergent
were added. Liberation of dissolved air accompanied the subsequent cooling
to room temperature. The detergent inhibited formation of air bubbles along
the walls of the drums. The water was also stirred manually as often as pos-
sible. On the next day, a predetermined amount of salt was added to one drum.
A submersible pump was used to mix the salt into solution and to minimize
entrainment of air at the free surface. After the wave tank had been cleaned,
the wave flapper assembled and tested, and the slope supports positioned at
the desired angle, the siphon connecting the two barrels (Figure 2) was primed,
and the stirrer in the fresh water barrel was started. The water temperature
in each barrel and the room temperature were then recorded. Readings of
specific gravity and index of refraction were taken in both barrels. During
filling, the water level was recorded as a function of time, and dye was in-
jected into the filling tube at selected intervals. Room and water temperatures
were monitored at one -hour intervals. After filling was completed, about 10
hours later, the final temperature and specific gravity of the water in each
barrel were measured. The slope was then inserted very slowly and smoothly
by sliding it down the supporting struts until its beveled edge lay flush with
the tank bottom. Very little mixing was observed, except at the edge, as the
slope moved through the stratified fluid. The coherence of the dyed layers
near the slope attested to the small degree of disturbance. The wave flapper,
whose bottom had been positioned at mid- depth during the filling, was then
inserted through the remainder of the fluid. Its driving rod had been fastened
to the vertical shaft of the Scotch yoke prior to filling. This procedure also

29

caused very little noticeable mixing. The water depth was recorded, and
the top of the tank was covered with a thin plastic sheet (Saran Wrap) to
minimize evaporation losses at the free surface. (Prior experience showed
that these evaporative losses could be quite substantial, 1-2 mm of water
per day.) In addition, the conductivity probes were cleaned and platinized
as necessary (Appendix A). Depth markings were checked and refurbished
on all probes.

On the next day, vertical profiles of temperature and index of re-
fraction were taken over the flat bottom and at other stations along the slope.
The depth was recorded, and the presence of an interface near the surface
was investigated with the shadowgraph apparatus . The plastic covering was
removed from the slope section of the tank, and the pertinent photographic
apparatus was made ready.

Prior to wave generation, the conductivity probes were given static
calibration by raising and lowering them vertically by known increments
while recording their output at each level; this allowed a comparison with
previous calibration values and provided an updated sensitivity coefficient
for each probe. Normally, the next group of measurements was taken
with the probes all at the same depth over the flat-bottom region. Adjacent
probes were separated by equal distances usually 5 to 10 cm, with the probe
farthest from the slope at a distance of at least 200 cm from the wave maker.
Runs were then conducted in which the forcing amplitude and frequency of
the wave maker were varied to include most of the values to be investigated.
These data provided a set of reference values with which the later measure-
ments over the slope were compared. The procedure used to take the dye-
streak measurements is summarized below.

(1) Prior to the initiation of wave motion, a grid marked in milli-
meters was gently lowered to the slope in the middle and parallel to the side
walls of the tank, photographed in place, and gently removed. The position
of the camera relative to the front wall of the tank was carefully measured;
this position could be repeated to within 1 percent.

30

(2) After the disturbances introduced by the grid removal had sub-
sided (usually 10 - 15 minutes), and several minutes after the wave-maker
had been turned on, several dye pellets were dropped at various locations
over the slope (normally four) and at one position over the flat bottom. The
pellets were dropped through a small glass tube at the previous position of
the grid. By choosing dye pellets that were small (diameter <0.3 mm) and
nearly spherical, irregular motion during descent was minimized. During
the rest of the experiment, photographic and conductivity measurements
were taken at various locations and times. Each experiment usually lasted
from two to four days, depending on the nature of the particular experiment
and the physical endurance of the experimenter.

31

3. DATA ANALYSIS

Several kinds of experimental data were obtained; these included
(1) temperature, (2) mean density gradient, (3) density perturbations,
(4) boundary layer velocities and bed shear stress, and (5) miscellaneous
data including mode number, net particle drift, and measurements of boun-
dary layer instabilities. The instruments and some of the procedures that
were used to obtain the various kinds of data have already been described;
this section presents a summary of the data reduction techniques and the
methods of analysis .

TEMPERATURE

Vertical temperature profiles at several horizontal locations were
made at least once before and after each experiment, and usually several
times in between. Typically the maximum temperature variations were
relatively small (<0.5°C) over the fluid interior; more irregular changes
often occurred in the near-surface layer. This layer will be discussed in
more detail in the next section. Figure 8 is a representative temperature-
depth profile.

MEAN DENSITY GRADIENT

A typical density gradient is shown in Figure 9. Measurements of
index of refraction were made at about 2 -cm intervals in the vertical and
at closer spacing near the free surface. Index of refraction was converted
directly to specific gravity by applying a calibration constant that included
a correction for the local temperature. Several times during the course of
the experiments the specific gravity values obtained in this way were checked
by carefully weighing a known volume and determining density in g/cc di-
rectly. The maximum difference in the two methods was always found to be

3
less than 1 part in 10 (or about 0. 1 percent).

32

23.5

Temperature ,

Fig. 8. Typical temperature variation with depth
Data from exp. group 4.

33

Z , cm

10 15 20

(Specific gravity — l.o)

25
I0~3

Fig. 9. Typical specific gravity variation with depth
Data from exp. group 2; computed vertical
distribution of Brunt Vaisala frequency is
also shown.

34

The linearity of the curve in Figure 9 is quite good over the fluid
interior but nonlinear tails are evident near the bottom and the free surface.
Evaporative cooling at the free surface, when the plastic wrap was removed
from the slope section of the tank during the conductivity runs, tended to
increase the density of the water there, countering the effect of lower den-
sity due to low salt concentrations. The overall result was a more homo-
geneous density layer near the free surface. Weak convective motions were
observed in this layer in the absence of wave motion by dropping potassium
permanganate crystals through the water and tracking the motions of the dye
traces in this layer. The amount of evaporation was reduced significantly
(to < 1 mm per day) when the plastic cover was used.

The variation of salt concentration with depth as a function of time
for a similar experimental arrangement has been derived theoretically
from the one- dimensional diffusion equation for a constant coefficient of
diffusion and conditions of no flux of salt across the boundaries (Mowbray
and Rarity, 1967). The theoretical concentration-depth curves were shown
to have a linear interior and nonlinear tails that thickened in time. The
point of maximum curvature located between the nonlinear and linear por-
tions of the curves might explain the sharp shadowgraph image that was
usually observed a small distance below the free surface and was inter-
preted as a change in the density gradient (i.e., ap /az #0 at this point).
An example of the vertical distribution of Brunt-Vaisala frequency N is
shown in Figure 9. The effects of the nonlinear tails cause N to decrease
sharply in the small layers. Photographic measurements of the modal
structure and of the particle displacements indicate that the basic wave
motion was not measureably affected by the small, nonlinear portions of
the density distribution.

DENSITY PERTURBATIONS

Perturbations in the density field induced by internal wave motion
were sensed with conductivity probes. Voltage variations relative to some

35

mean or null value and proportional to the conductivity changes were recorded
digitally on magnetic tape. The quality of the data was checked by using a
real-time display on a strip-chart recorder.

Normally, measurements from all four probes were taken at least
one minute before and after the wave motion to establish an instrument drift
correction. Each of six data channels was sampled sequentially once per
second (At = 1 sec); the channel allocations were four channels for probe
data, one for the wave -maker signal, and one for a reference voltage.
Samples of the raw data signals from one probe and wave- maker are shown
in Figure 10. The length of each data record depended on the particular ex-
perimental run. Record durations of least 50 wave periods, often up to 100,
were obtained for quasi- steady conditions as determined from the strip-
chart output.

The data records on the magnetic tape were then processed on the
IBM 360/65 as follows:

(1) average and trend were computed, listed, and removed;

(2) mean square of the data values was computed and listed;

(3) the periodogram was computed for 2N data points using a Fast

Fourier Transform algorithm (IBM Manual H20-0205-3) to obtain squares

of the real Fourier coefficients a, , b. such that:

k' k

N-l

v = l/2a0 + I {akcos(^) + bksin(f )} + £ a^-1)*
J k=l

3 =0, 1, , 2N-1 (3'1}

Normally, the starting point for each computation of the periodogram was
selected when it was determined that the data were quasi- steady (transients
and other irregular motions were no longer evident on the strip chart).

Output listings and selected plots of the squares of the periodogram esti-
mates a (in
m v

were obtained.

2 2

mates a (in cm ) and of the phase $ at each frequency f (in cps)

36

en

en

-en

^

i£j

cn

L±J

-cn
fT1

-CD

>
•H
4J
O

d co

O 4J

o co
o

d
CO

jj *d*

^ >

ctj

6

0)

J-i
o

d

co

<u
-d X
cu

4-1

4J . ^-v

O 4J ITS

r-« d

Cu Oi Ou

4J d
co d
■p o
co
•d <u

£ o

co u

& aw

o

o
u
o

a

00

iy i

or ti

W]

N

n

37

2,2. 2.1/2

6 = tan —
rm a,

k

m = k = 0, 1, , N-l

A sample periodogram computed from the data record represented in Fig-
ure 10 is shown in Figure 11. A calibration constant has been applied so

2 2

that the ordinate is cm instead of (volts) . The strong peak at the input

frequency and the sharp fall-off at adjacent frequencies are typical of these

results. The cut-off frequency was lower than the Brunt- Vaisala frequency

for all periodogram computations (cut-off frequency = 0.5 cps).

2 *
Estimates of the periodogram a were computed for probe data

taken over the flat bottom region. These estimates were used as reference

2
values for subsequent estimates a for data taken over the slope at similar

m 2 2

frequencies. The results are shown in a later chapter in the form a /a

to indicate the change in wave amplitude from flat bottom to slope regions.
This normalization also conveniently removes certain calibration constants.

VELOCITIES AND BOTTOM SHEAR STRESS

Time- sequenced photographs of the deformation of potassium per-
manganate streaks were used to obtain one- second averages of the Lagran-
gian velocities of these streaks at several levels parallel to the bottom in
and near the bottom boundary layer. Reliable measurements of streak dis-
placements were obtained at one-second intervals over at least one full
wave period, often down to distances of 0. 05 cm above the bottom. Tracks

*

The subscript "o" signifies a value pertinent to the flat-bottom section of
the tank, not to be confused with the general Fourier coefficient a in
Equation (3-1). °

38

0-

Fig. 11.

Sample Periodograms .

Upper: Periodogram of wave maker signal.

Lower: Periodogram of conductivity probe
output.

-2

-4-

— I-
0.1

— T—
0-2

0.3

— r~

04

Frequency , cps

-2-

-4-

-6-

~i

0.1

1

0.2

1 ■■

0.3

r

0.4

Frequency

, cps

39

of neutrally buoyant floats near the bottom generally showed very little, if
any, motion normal to the bottom, except in the zone of breaking. The dye
traces usually remained coherent to the slope, but if instabilities and tur-
bulence were present the streak traces usually developed a wavy structure
or dissipated rapidly near the bottom.

The following summarizes the methods used to obtain velocity and
bed shear stress data from the streaks.

(1) Photographic negatives which contained a series of one- second
positions of the streaks for a particular input wave amplitude and frequency/
and at each station were enlarged and transferred onto a sheet of tracing
paper by using a microfilm reader. A single sheet was produced containing
the traces of the successive positions of the dye streak at each station for
one-second sampling intervals. Successive displacements of neutrally
buoyant particles were recorded simultaneously on the same sheet. Figure
12 is an example of a worksheet that includes the dye streak and particle
positions. The numbers represent successive seconds in time relative to

t = 1 . The streaks and particles were also color-coded; this does not show
in Figure 12.

(2) The coordinates of each streak relative to its position at the slope
were punched onto IBM cards by a special purpose X-Y digitizer. The streak
positions were usually sampled at 0.02 mm increments normal to the bottom
in the boundary layer and at larger increments (0.05 mm to 0. 10 mm) above
the boundary layer. Each neutrally buoyant particle position was digitized.

(3) Successive displacements of the streaks (or particles) at a partic-
ular level above the bottom gave approximately one- second averages of the
Lagrangian velocities u. at that level;

*
The thickness of the boundary layer is tentatively defined as the distance

of the inflection point in the streak trace to the bottom.

40

a)

TS

u

c

>i

fd

0

co

CD

M

>

id

(0

CD

s

•

M

rH

P

fd

CO

Cn

II

CD

C

P

>i-H

H3

M

0

3

P

14-1 T3

0

CD

n3

>

W

G

•H

d

0

-P

0

CJ

fd

•H

CD iH

+J

W

CD

t0

5-1

O jG

O

O

CD

H

ItJ

U

CD

rd

rC

P

-P

^-s

•H

(3

CO

5

u

•fl

CD

-P

CD .Q

(D

-P

e

0)

-P

3

A

O

c

w

iH

— -

M

D^

u

CO

0

03 A

>

CD

5-1

rH

fd

d)

u

e

rH

•H

Qj-P

CD

S

U

e

r3

fd

■H

CO

D^fH

CM

•H
&4

41

„ 3>t ],t-l

j,t " At

t = 1, 2, , T

J = 1, 2, , L

T is the number of streaks (usually equal to the nearest integer
greater than the wave period).

L is the number of vertical levels.

j = 1 represents the lowest level for which the data of any particular
set are considered reliable.

As stated above, the spacing between the various vertical levels
from j = 1 was constant in the boundary layer.

(4) Estimates of bottom shear stress (one-second averages) were
computed from the velocity values at j = 1:

ul t

T = »

M Aif

t = 1, 2, , T as before.

A 77 is the distance between the bed and the level j = 1 measured normal to
the bottom.

MISCELLANEOUS

To ascertain that the wave maker was in fact generating primarily
the first mode, several photographs of the orbital motion of neutrally buoy-
ant particles over an entire vertical section were taken at various positions
along the tank. Figure 13a is a typical section over the flat bottom. The
predominantly vertical motion near the center of the water column and the
horizontal pattern near the free surface and at the bottom are characteris-
tic of internal wave motion of the first mode. The photograph was taken
with Polaroid film that was exposed by flashing a strobe through a thin slit

42

Fig. 13 a.

Time sequence of neutrally buoyant
particle positions over horizontal bottom.
Modal structure approximately equal to the
first internal mode (n = 1) .
Pointer depth: 16.0 cm.

Bright horizontal section on pointer: 1.0 cm,
Brunt-Vaisala period: 6.5 sec.
Wave period: 12.0 sec.

(Strobe: 1 flash/sec; Polaroid 4x5 film,
ASA 3000)

43

from above the tank at 2 cps. The illuminating flash was a rectangular
beam about 1. 5 cm wide and approximately centered between the side walls
of the tank. Similar photographs were taken at various positions over the
slope. Figure 13b shows a close-up view of successive displacements of
neutrally buoyant beads near the bottom of a relatively low slope (a^ 15
degrees). Note the amplification in the particle displacements nearer the
bottom.

Shadowgraph images were used to visualize the boundary layer in-
stabilities and breaking of the high frequency waves. Both of these phenomena
are discussed in Chapter 4; several photographs of the shadowgraph images
are shown in that chapter.

The net drift patterns of neutrally buoyant particles were determined
by two methods:

(1) The positions of selected particles were marked on the glass side
walls of the tank over long durations (up to 30 minutes).

(2) The positions of the particles were traced from 16 mm movies
of the shadowgraph images that were projected onto a screen of large grid
paper. By counting the frames between position marks, the approximate
time that had elapsed was estimated and a net velocity computed.

44

Fig. 13 b.

Time sequence of neutrally buoyant

particle positions over slope (a=30 deg) .

Note intensification of motion near bottom

Brunt-Vaisala period: 6.0 sec.

Wave period: 9.6 sec.

Input wave amplitude: 0.2 cm.

(Strobe: 1 flash/sec; Tri-x 35mm film.)

45

4. EXPERIMENTAL RESULTS

GENERAL DISCUSSION

This section describes the organization of the results and presents
an overview of the hydrodynamics. Subsequent sections discuss the results
in more detail and compare them with the predictions of recent theories.
The results are separated into two basic categories: (1) one that describes
the interior wave field (wave number and wave amplitude); (2) another that
describes the activity at the boundaries (wave breaking and boundary layers).

It is useful to define two distinct hydrodynamic regimes on the basis
of the reflection conditions for internal gravity waves from a smooth, rigid
slope (Phillips, 1966; Sandstrom, 1966). This subdivision is based on the
ratio of bottom slope y to the slope of the input wave characteristics c.
Figure 14 illustrates the classification and shows that regime n can be fur-
ther subdivided into two cases. Phillips (1966), among others, has shown
the slope of the characteristics to be a function only of the stability fre-
quency N and the frequency of the input waves u>, provided that N is con-
stant with depth:

c = l/(N2/w2-l)1/2 (4-1)

Since N and y were essentially constant during each experiment, the ratio
y/c could be altered by changing co. In this way, both hydrodynamic regimes
y/c<l andy/c>l were investigated during several experiments.

Tables Bl and B2 in Appendix B summarize the experimental param-
eters for the various measurements that were made with the conductivity
probes. Table Bl is divided into experimental groups that each have a simi-
lar Brunt-Vaisala frequency and bottom slope. This table is a reference for
Table B2, which lists the various input wave frequencies that were generated
during the experiments. The experimental conditions for each frequency can
be found by matching the second number in the first column of Table B2 with

46

Reflection

Regime

Case

Designation

r/c

Properties

I

A

subcritical

< l

transmissive; all
energy propagates
into corner

n

B

critical

= 1

marginally trans-
missive; energy
propagates parallel
to bottom slope

C

supercritical

> 1

nontransmissive;
energy is back-
reflected

Fig. 14. Classification of internal waves based on reflection
from a smooth, rigid slope. Slope of wave charac-
teristics = c; bottom slope = y.

47

the number of the experimental group in Table Bl. Those frequencies
which belong to hydrodynamic regime II are identified by a dash in column 7.
An asterisk is located above those frequencies for which more than one ex-
perimental run were made.

The emphasis on two hydrodynamic regimes is supported by the re-
sults. The subcritical case was examined extensively, since the results for
this case are pertinent to the later analysis of sediment motion and since
this case is important for certain oceanic conditions (Chapter 5). The
data for this case suggest a further classification into two wave types
based on the frequency of the input waves. This distinction is more justified
for the measurements that were taken over the low slope angle (15 degrees)
than those measurements taken over steeper slopes. The experimental re-
sults indicate that the high-frequency input waves decrease in wave length
and increase in wave amplitude as they propagate upslope until at some lo-
cation near the corner and below the free surface they dissipate by breaking.
The low-frequency waves in this case (y<c) initially steepen like the high-
frequency waves, but the effects are less pronounced. The steepening
diminishes upslope as these waves form an internal surge that produces a
large runup with little breaking. The water motion near the bottom is am-
plified for both wave types. The bottom shear stress is increased over the
slope and is greatest in and near the zones of breaking and runup. In addi-
tion, there is considerable vertical motion near the bottom in the breakers.

The experimental results for case B and case C (y/c>l) indicate
that the interior wave motion is very complicated. The conductivity-probe
measurements show that during most experimental runs the wave amplitude
over the slope was a maximum at or near a depth z = -ex. (The geometry
is defined in Figure 15.) In the critical case, a line of small, regularly
spaced vortices forms along the slope. The vortices have diameters on the
order of 1 cm and axes that are normal to the side walls of the tank. They
oscillate along the slope with the wave motion, collapsing and reforming
over a half-cycle. The distance from the vortex core to the slope is about

48

g
<D
4->
W
>1

in

0>

-p
id

■H
T3

H

O

O

O

>i
M

-P
(D
g
O
<D
O

IT)

-H

49

7 to 10 mm. Several wave cycles after the formation of the line of vortices,
thin horizontal layers appear in the fluid. Each layer is associated with a
vortex near the slope and represents a modification to the density structure
over the slope. Wave breaking near the corner accompanies the vortex
activity. These features are discussed in more detail later in the chapter.

THEORETICAL DISCUSSION

The experimental results presented in this chapter are compared to
the theoretical solutions of Wunsch (1969). Preliminary measurements in-
dicated that the exact linear solutions for the interior wave field fitted the
experimental data. The experimental conditions and the assumptions im-
plicit in the theory were compatible except for the boundaries. Wunsch (1969)
solved the problem for a wedge geometry bounded above by a rigid horizon-
tal top and below by a rigid bottom slope that extends indefinitely far from
the corner (the term corner refers to x = z = 0 in Figure 15). The rigid
bottom slope used in the experiments terminated a finite distance from the
corner at the transition to a horizontal bottom. The possible effects of
this discrepancy between theoretical model and experimental conditions
are mentioned in the discussion of results for cases B and C and in the
boundary layer section for case A. Although a free surface was permitted
in the experiments, the observed vertical motions of this surface were
small. In the sense that w « 0 at z = 0, the use of a free surface was
compatible with the rigid top in the theoretical model.

For a viscous Boussinesq fluid of mean density p0(z) the two-dimen-
sional perturbation equations (after Wunsch, 1969) are:

ut = ~Px/po + yy2u (4"2)

wt = "pz/po " gp'/po + yv2w (4_3)

ux + wz = 0 (4-4)

50
p't + Wpo = ° (4-5)

— La

where p' is the perturbation density, and a bar denotes partial differen-
tiation.

Interior Wave Field

The interior equations are given by the inviscid forms of Equations
(4-2) through (4-5) above. The pertinent linear solution (Wunsch, 1969)
for small -amplitude internal waves of frequency o> traveling toward the
corner is composed of two progressive waves:

ip = A

-iq£n(cx - z) -iq£n(cx + z)

e"lcot (4-6)

2n7T C + y /. „\

where n is the mode number. This solution is singular at x = z = 0 for

case A (y <c). The phases of the two waves inEquation (4-6) are arith-
metically averaged to give an average phase function U:

U = q/2 [£n(cx - z) + £n(cx + z)] (4-8)

It follows that an average local horizontal wave number is defined by k :

k
w

= 8U/3X = qf-2-^-2) (4-9)

xc x - z /

The corresponding wave number k that was determined from the conduc-
tivity probe measurements is A0 /Ax (Chapter 3).

For a time dependence of e ~ w , the incompressibility
condition (Equation (4-5)) yields:

P ■ -i/u> apQ/az w (4-10)

(the prime on p has been dropped for convenience).

51

The vertical velocity w is obtained from the stream function in Equation (4 -6),

w = a i///ax:

-icjt

w

Aiqc

1 -iqCnM 1 -iq£nN
"Me + N6

(4-11)

M = ex - z

N

ex + z

Applying Equation (4-11) into Equation (4-10) and taking the time average of
the product of p and its complex conjugate p*, we obtain an expression for
the square of the amplitude variation with coordinate position:

Pf, = \pp* = iA2(S£)2(8p,yaz>2

W

U)

(4-12)

X

+

ivr

N

2 r , /%

cos |qCn(^)}

NM

The comparative experimental value is the square of the spectral amplitude,

2
a , as determined from the periodogram of the conductivity-probe measure-

m 2 2

ments. The method of comparison between a and p was discussed in

m rw

Chapter 3.

It is useful to point out here that for y<<c, Wunsch's solutions are
the asymptotic solutions for internal waves propagating over a small slope
(also given by Keller and Mow, 1969). The dependence on local depth h
for local values of horizontal wave number k and amplitude a is

k =
x

n^c

a h
o o

(4-13)

where the subscript "o" signifies "deep" water values (i. e., constant values

of wave amplitude a and depth h ). Hogg and Wunsch (1970) have shown

o o

that the nonlinear terms are significant when e becomes order one,

a h
, o o

e = ak = — r— cn7r
x h

(4-14)

That is, the amplitudes of the next higher order term in the velocity ex-

o
pansion grows as e . This suggests that the linear solutions are no longer

52

valid over the small slope at some depth h given by e of order one in
Equation (4-14).

For the case y>c, Wunsch (1969) showed that the apparent singu-
larity at z = -ex in Equation (4-6) could be removed. Wunsch rewrote q
in the form

q = r + is (4-15)

and showed that the expression for the stream function for waves propagat-
ing upslope could be rewritten

-icot

if) = A
where

(s - ir)£nM (s - ir)#nN
e - e

(4-16)

r =

2n7r£n|A'

[(£n|A'|)2 +tt2]

-2m

s =

[Un|A|)2 +7T2]

c -

A' - ^-^ = 1/A

C + y '

and n is a negative integer. (Note that £nN is complex for ex + z < 0.)

The notation here does not conform to that of Wunsch (1969) in order to
avoid later confusion. He wrote the complex form of q as q = r\ + is,
and gave an analogous expression for Equation (4-16) in which the real part
of the exponential group was written in a power of s(i. e., 5 in his notation).

53

In order to have the velocities and their first derivatives continuous,
Wunsch derived the requirement that s > 2 or

.„ > fialfJli + ! (4.17)

7T

-n represents the lowest, nonsingular inviscid mode permissible for
these solutions. The vertical velocity is obtained from Equation (4-16):

w = Ac(s - ir)

J_ (s - ir)0nM _1 (s - ir)£nN
M e N e

e"ia)t (4-18)

2

An expression for p analogous with that in Equation (4-12) can be obtained

w

in principle from Equation (4-10) and Equation (4-18) and compared to the

2

measured amplitude a for specific coordinate positions by the method

described in Chapter 3. However, it is not simple in this case to obtain a
physically meaningful phase function that is analogous to Equation (4-8)
owing to the complicated wave forms in Equation (4-18). The amplitude
dependence on the spatial coordinates indicates that these are not simple
plane waves. By writing

w = f(x, z)eig(x' Z' t} (4-19)

where f and g are complex, a phase function U can be defined as

U = tan"1 gg (4-20)

where I(w) and R(w) are the imaginary and real parts of w, respectively.
This form of U will be used in the later discussion of theoretical and ex-
perimental results for cases B and C.

Boundary Layer

The full set of perturbation Equations (4-2) through (4-5) were used
by Wunsch (1969) to obtain linear boundary-layer solutions for case A

54

(y < c). The solutions match those for the inviscid interior just outside
the boundary layer. The stream function can be written:

^ = h+ *b.i. (4'21)

where ^T and $ are the interior and boundary layer contributions, re-

spectively. In terms of the coordinates J, rf shown in Figure 15, the
expression for <v given by Wunsch (1969) can be rewritten

Kl.-i?*

Fe-iq£n(B£) _ G e~iqU(DF)

x e-U + 1) v/b e-iwt

where

2 gov

,V2

i - 1 2 XT2 . 2

\W - N sin

;)

6 = (i - 1) P
B

F =

c cos a + sin a,

c sin a - cos a

D = c cos a - sin a

G =

c sin a + cos a

(4-22)

B D

The component of velocity parallel to the slope (in the J direction) is
given by

uT = Uj + ufe = -dipT/drf (4-23)

or from Equation (4-6), after it has been expressed in J, 77" coordinates,
and Equation (4-22),

uT = V(I)|l-e-<l+1>^

55

-icot
e

iAq

V(?)

(4-24)
Fe-iqCn(BT) _ Q e-iqfin(D|)

The behavior of the velocity component u parallel to the bottom with
variations in J and rj" is obvious: (1) u grows to infinitely large values
as { — 0; in fact, at rf = j = 0, the solution is singular, as might be ex-
pected since the interior solution is singular at the origin for v < c; (2)
for if = 0, J * 0, the solution reduces to that of the interior field; and (3)
as "rj -* + °°, the boundary-layer part (u, ) becomes infinitesimally small
and u -* u.. The solutions for w can be derived in principle from
wT = a ip/dT, but are not presented here since the measured values of wT
were very small in the boundary layer (this is discussed later in this
chapter) .

For cases B and C (y > c), Wunsch (1969) found that the boundary -
layer equation for \\j, was nonseparable in the spatial coordinates. In

fact, Wunsch showed that when the slope is near critical the scale of the

-1/3 -1/2

boundary-layer thickness is proportional to R ' , in contrast to R J for

<c, where R is a local wedge Reynolds number and h is the local depth:
R = Nh2/v

INTERIOR WAVE FIELD - RESULTS

Subcritical Case

Table B3 (Appendix B) summarizes the periodogram amplitudes
and wave numbers that were determined from the conductivity-probe mea-
surements as described in Chapter 3. The notation in the column headings
is discussed in the following sections. The experimentally determined
values are organized into columns by input wave amplitude.

7

56

Wave number. As was described in Chapter 3, an experimental wave
number k was computed from the measured phase differences between
adjacent probes (k = A0 /Ax). Columns 9, 10, and 11 of Table A3 con-
tain the appropriate values of k divided by q, i. e., - A0 /Ax (q is de-

m J n' q m' - ^

fined in Equation (4-7)). This parameter is convenient since -k corres-
ponds to the theoretical value - k given by Equation (4-9):

Ik _ I III

q w q 3x

x

x2 - (z/c)2

(4-25)

The right-hand side of this equation is evaluated for the appropriate values
of c, and the average is computed for the two probe positions. The value
listed in column 7 is

q w

1,2

x - (z/c)

(4-26)

1,2

in which the subscripts 1 and 2 indicate that the function is evaluated at the
coordinate positions for each probe and then the average is taken.

The theoretical curves in Figures 16 and 17 are straight lines with
slope of 1. The data in Figure 16 are taken from the values for relatively
high-frequency waves in Table B3. The upper right portion of the diagram
corresponds to the corner region of the slope. Figure 17 is a similar plot

for the data of the low-frequency waves. Both plots show that the wave

*
number decreases with increasing distance from the corner. The agree-
ment of the results with the theoretical curve is best for waves over the
lower section of the slope, well away from the corner. Toward the corner,

The behavior of the abscissa,

x2 - (z/c)2

, is approximately like 1/x for

constant z. (This is obviously true for x >> z/c.)

57

High Frequency Waves

EXP group

ao = 0 2 cm

1

0

2

□

3

A

5

O

X

x2-

10'

Fig. 16.
Upslope variation of wave number, y < c

58

A©

0

Low Frequency Waves

EXP group

aQ= 0.2 cm

1

O

2

□

5

0

6

0

3 4
X

1 — i — * — r

6 7 8

X2-

z
c

10'

Fig. 17.
Upslope variation of wave number, y < c

59

the data points in Figure 16 depart significantly from the theoretical curve.
It is also evident that the value farthest to the right for each experimental
group in Figure 16 is displaced toward higher wave number than that pre-
dicted by theory. This trend is illustrated in Figure 18, which includes
data for the high-frequency waves (u) = 0.094 cps) in Group 1 (Table B3).
The upslope increase in wave number is similar for both input waves
(a = 0. 2 cm and a = 0. 1 cm). However, the greater increase in wave
number for the larger waves (a =0.2 cm) at the most cornerward position
(i. e., closer to x = 0) suggests that the upslope variation in wave number
might depend on amplitude. The experimentally determined value of e at
this most cornerward position is approximately 0. 15; this might indicate
that nonlinear effects are beginning to have a significant effect on the motion.

In Figure 17 there is good agreement between theory and measure-
ment for the low -frequency waves, although most of the data values tend
to be lower than the theoretical curve in the right-hand side of the diagram
(near the corner). This suggests that the wave length of these lower fre-
quency waves decreases more slowly with decreasing distance to the corner
than theory predicts.

Another type of comparison between theoretical and experimental

values of wave number is shown in Figures 19 through 21. The data are

taken from columns 12 through 14 in Table B3 (Appendix B). Experimental

wave numbers k and k were determined for each input wave from probe
mo ^

measurements taken over the slope and over the flat bottom, respectively.

The ratio k /k is the ordinate in Figures 19 through 21 (k = A0 /Ax).
m/ o & & m m/

A comparative theoretical ratio is used as the abscissa in these figures;

i. e., kw/kw where k is the theoretical wave number computed from

Equation (4-9) for coordinates established by the probe locations over the

slope, and k is similarly computed for coordinates at the entrance to

wo
the slope region (k = A0 /Ax). (The depth z used in the computation of

k is equal to the probe depth during the determination of k and k .)
o

60

1 A©

q ax

•I02

3 -

m

2 -

I -

EXP

group 1

oj -

0 094 cps

do

, cm

0

0.2

0

0. 1

*w

10'

Fig. 18.
Upslope variation of wave number, y < c

61

_L A®m

Ko ' AX

6 -

5-

2 -

High Frequency Waves

EXP group

do = 0.2 cm

1

O

2

□

3

A

4

♦

5

O

1 r

0

~ r

i

4

_L A®w
Kw0' AX

Fig. 19.
Measured versus computed values of wave number, y < c

62

A©m

Ko AX

3-

2 -

I -

Low Frequency Waves

EXP group

ao = 0.2 cm

1

O

2

□

5

0

6

<3>

Kw,

A©w
AX

Fig. 20.
Measured versus computed values of wave number, y < c

63

_L A®.

Ko AX

m

4-i

3-

2 -

I -

EXP g

roup 1

(X) = 0.

094 cps

Oo

, cm

O

0.4

G

0.-2

I A®

w

Kwo A X

Fig. 21.
Measured versus computed values of wave number, y < c

64

Figure 19 and 20 also show better agreement between experiment
and theory for the smaller wave numbers over the lower slope sections.
Farther upslope the data for the higher frequency waves in Figure 19 tend
toward larger values of wave number relative to the theoretical values; by
contrast, the lower frequency waves in Figure 20 generally tend toward
smaller wave number. The effect of input wave amplitude on wave number
is indicated in Figures 21 and 22, for high and low frequency waves, re-
spectively. The values for the larger amplitude waves in Figure 21 appear
to deviate from theory more significantly than the smaller waves. Signifi-
cant departures from theoretical estimates are also shown for the larger
input waves of low frequency (Figure 22). In this case, however, the
measured wave numbers are smaller than the predicted values at locations
farthest to the right in Figure 22. This behavior corresponds to the pre-
vious observations (Figure 17) that the wave lengths of the lower frequency
waves apparently decrease less than theory predicts at positions well upslope.

Wave amplitude. The results of the conductivity-probe measurements for
case A, concerning the effects of shoaling on wave amplitude, are sum-
marized in Table B4 (Appendix B). The experimental group numbers cor-
respond to those in Table Bl. Experimental determinations of the period-

2 2

ogram amplitudes a and a for each input wave were made from probe

measurements over the slope and over the flat bottom, respectively (for

2
more detail see Chapter 3). The value a was determined for each of the

2 2 °
four probes so that each ratio a /a represented data taken with a par-
ticular probe. This procedure was convenient since it removed the
necessity of introducing calibration constants relating one probe to another.
That is, the measured amplitude ratios could be compared to one another
without applying relative gain factors. This method of normalization also

simplified the detection of measured changes in wave amplitude relative to

2
the input wave amplitude a . The comparative theoretical values p and

p were computed from Equation (4-12) for coordinates of the probe loca-
tions over the slope and at the entrance to the slope region, respectively.

65

A©

E

O

EXP

group

2

oj = 0

.062

cps

ao

, cm

o

0.2

□

0. 1

— I
50

0

0

2.0

30

4.0

A©

w

K

wc

AX

Fig. 22.
Measured versus computed values of wave number, y < c.

66

2 2
The ratio p fp was equally convenient, since its use circumvented a

conversion from density p to the units of a and permitted direct com-

9 9 ^

parison with the experimental ratio a /a .

nr o

The results shown in Figures 23 and 24, for the high-frequency and

low-frequency waves, respectively, suggest that the agreement between

2

theory and experiment is best for the lower values of (a /a ) , i. e., for

the wave motion on the lower and middle sections of the slope. This is in
agreement with the results for wave number. The solid lines in these fig-
ures were drawn for reference and represent points where measurement
and theory coincide. The deviations of the points from the straight line in
Figure 23 are significant for (p /p ) > 20, but no consistent pattern is
apparent in the variations. The values for lower frequency waves show
earlier departures from theoretical values than do the higher frequency

waves. For example, the measured and theoretical values shown for low-

2
frequency waves in Figure 24 are significantly different for (p /p ) > 10.

w o

There is strong indication in the trend of the points that the actual ampli-
tudes are much lower than predicted on the middle and upper sections of
the slope. At these slope sections it was also found that the measured

values of steepness a k (or e ) are considerably less than those esti-

m m v nr

mated from theory.

Measured amplitude growth is further compared with theoretical
estimates in Figures 25 through 30. In these figures the variation of wave
amplitude is examined as a function of horizontal distance x to the corner

at various fixed depths z . The data are again taken from Table B4 (Appen-

_. * c

dix B). In each of these figures the theoretical and experimental ratios of

*
The actual wave amplitude in centimeters at a point along the slope can

easily be found from the graphs. If the correct wave amplitude (in centi-
meters) for the flat bottom region is multiplied by the square root of the
measured value ((a /a ) ), the measured estimate of the actual amplitude
is obtained.

67

^2

Qm

ao

6-i

5-

4-

3 -

10

-I

El a

High Frequency Waves

EXP group

Oo = 0.2 cm

1

O

2

□

3

A

5

©

2
rO-,2

P

1 w

P.

10"

L' 0

Fig. 23.

Measured versus computed values of wave
amplitude, y < c.

68

- I2
Qm

ao

4 -l

3-

io-

Low Frequency Waves

EXP group

Qo = 0.2 cm

2

□

5

O

6

0

l_ ' 0-J

Fig. 24.

Measured versus computed values
of wave amplitude, y < c.

69

r ^-,2

or

L'OJ

8-

7 -
6-

4
3
2-

I -

r -.2

am
a~o"

©

EXP group 1

ai =

0.094

cps

G

00=0.2

cm

□

^

0

1 1

3 4

r

5

1
6

i

7

1
8

Xc
Xo

I01

Fig. 25.
Upslope variation of wave amplitude, y < c

70

'R,

-, 2

or

o-

Qm

ao

-.2

4-

3 -

EXP gi

oup

2

ai =

0.102

cps

*

ao =

0.2

cm

A

ao =

0.4

cm

O

ft,

3 4
Xc
Xo

I01

i

7

8

Fig. 26.
Upslope variation of wave amplitude, y < c

71

P

n2 ,- ^2

P,

L'oJ

or

ao

10'

5-1

4 -

2 -

EXP group

2

oj = 0.062

cps

0 ao = 02

cm

□ Pu,

AO

8

Fig. 27.
Upslope variation of wave amplitude, y < c

72

2 r -i2

EXP group

3

CJ =

0.094

c ps

D

do =0.2

cm

A

0 o = 0.4

cm

O

A,

** I01

Xo

Fig. 28.
Upslope variation of wave amplitude, y < c

73

La

or

-.2

ao

10"

EXP group 5

a; = 0 098 cps

a) =0 082 cps

-O- ao = 02 cm

O ao=0.2 cm

*> Pu

□ ft,

5-

4 -

2 -

0

&-'°'

Fig. 29.
Upslope variation of wave amplitude, y < c

74

r- ~-,2

Po

or

o J
20-,

18-
16-
14-

12-
10-
8 -
6 -
4 -
2 -

0

r- -.2
Om
do

EXP group 6

cu , cp s

Oo

P

1 U)

0.062

O

i

0 047

Q

o

0.043

A

0

3 4
Xc
Xo

i
9

0'

Fig. 30.
Upslope variation of wave amplitude, y < c

75

2 2

amplitude variations (p /p ) and (a /a ) , respectively, are plotted

along the ordinate; the ratio of the horizontal distance between the probe
and the corner x to the total horizontal projection of the slope x is plotted
along the abscissa. The depth of the probe measurements is constant for
any one of the figures, but as shown in Appendix B, Figure B4, it generally
varies for each experimental group. In Figures 25, 26, 27 and 30 the re-
sults pertain to the slope angle of 15 degrees, for which the distinction be-
tween high and low frequency waves appears most applicable. Two effects
of the method used to display the results must be mentioned here:

(1) the deviations between the theoretical and measured estimates
of amplitude are exaggerated in the graphs, since squares are plotted as
the ordinate;

(2) the peaking of the waves is a function of the vertical position z
of the probes as well as distance from the corner x . This latter effect is
subtle; since the internal waves are of the first mode, the amplitude should
be largest near the center of the water column at any position over the slope
if the modal structure remains intact. Consequently, the amplitude is
expected to vary along a horizontal line (constant depth) due to the modal
structure as well as to the amplification during shoaling.

To the right of the peak values (i. e., away from the corner) in
Figures 25 and 26 (high-frequency waves), the variations are essentially
parallel, although the experimental values are consistently higher than the
predicted. The maximum measured values are slightly higher than pre-
dicted, with an indication that the larger input waves (aQ = 0.4 cm,
Figure 26) peak sooner than the smaller waves. There is also evidence

*For this case (y < c), photographic measurements indicate that the mode
is preserved at least over the lower and middle slope areas.

76

that the measured values fall off to lower values than the theoretical values
to the left of the zone of peaking (i. e., toward the corner).

The high-frequency waves over the steeper slope (Figure 28) con-
form closely to the predicted behavior for x /x > 0. 35, but nearer the
corner the amplitudes of these waves grow less than predicted. This is
probably a result of the anomalously large growth of the theoretical solu-
tions near the corner (recall that for y < c Equation (4-6) is singular for
x = z = 0). The larger input waves (a =0.4 cm, Figure 28) show a
tendency to peak earlier than predicted. The measured value at
x /x =0. 35 for a = 0. 2 cm is suspiciously low.

Figures 27 and 30 illustrate the behavior of wave amplitude during
shoaling of the lower frequency waves for a low slope angle (15 degrees).
The measured values obviously do not achieve the high predicted values
well upslope (x /x < 0.4 in Figure 27 and x /x < 0. 5 in Figure 30).
The departure from theoretical estimates is most noticeable for the very
low frequencies (o> = 0. 047 cps and w = 0. 043 cps) in Figure 27. This sug-
gests that the waves increase in amplitude as they propagate upslope until
at some point near the half-distance of the horizontal projection of the slope
the rate of growth decreases. It was indicated earlier that these waves
tend to surge as they approach the corner. Possibly a transformation from
the sinusoidal wave form to a different wave structure (such as a solitary
wave) that is continually losing energy to viscous dissipation along the slope
accounts for the nature of the experimental data for these kinds of waves.
In any event, the amplitude of these lower frequency waves is not as large
as predicted, and there seems to be evidence of a more gradual dissipa-
tion of these low frequency waves with the formation of runup.

77

Critical and Supercritical Cases

The experimental runs for the critical and supercritical cases are
indicated by a dash in column 6, Table B2 (Appendix B). A comparison of the
input wave frequency with the critical frequency for each of these runs shows
that o)< ct> , a necessary condition for this regime. A brief description of
the reflection properties that define this regime was presented in the first
section of this chapter. In particular, for case C (supercritical) two fea-
tures are suggested by the reflection properties: (1) the presence of
back reflected wave energy; (2) concentration of this back-reflected energy
in a zone bounded by the bottom slope and the characteristic that intersects
the corner. Figure 31 further illustrates these features. This figure
shows that wave characteristics which are incident to the slope from above,
i.e., those with a downward-pointing vertical component, upon reflection
from the slope will remain below the characteristic (defined by z = -ex)
that intersects the corner. The energy flux associated with these charac-
teristics enters the slope region in a relatively broad band and exits in a
confined, triangular zone (for example, the triangular area AOB in Figure
17). As Figure 31 indicates, there is a smaller influx of energy which,
upon reflection from the flat bottom, propagates upslope within the narrow
bottom zone (for example, the path of the characteristic "b" in triangle
AOB). The characteristics associated with this latter energy flux have
upward-pointing vertical components as they enter their slope regions;
upon reflection they preserve the sense of the upward-directed vertical
components. In both cases (characteristics "a" and "b" in Figure 31) once
the characteristics reflect from the slope the horizontal direction of energy
is away from the corner.

Critical conditions (case B) were difficult to achieve experimentally
for two reasons: (1) y - c required that the wave-maker frequency be
tuned precisely; (2) it was difficult to determine the Brunt-Vaisala fre-
quency accurately during an experiment. In addition, the experimental
groups which involved the 15 degree slope angle had critical frequencies

78

0

Zc > CXc

( I)

Zc >CXc

(2)

FIG. 31. Reflection of characteristics for case C, y > c where y = tan a,
z = IzL x = Ixl and characteristics are labeled a, b, c. Char-

acteristic c is "critical" i. e., passes directly to the corner at
O in (1) and at F in (2). The increased concentration of charac-
teristics in the triangular zones (hatched areas) is illustrated by
the path of characteristic a.

79

that corresponded to long horizontal wave lengths over the flat bottom (on
the order of the tank length). Data for only six runs at the critical fre-
quency are considered reliable. During each of these runs the input wave
frequency was held to within two percent of the calculated critical frequency.

Table B5 (Appendix B) summarizes the experimental results from
the conductivity-probe measurements for this regime (y > c). Like the
organization in Table B3, the frequency, period, and probe coordinates
(x , z )* are given for each experimental run. The experimental runs
having the same N and y are again collected into groups; as before, the
experimental parameters for each experimental group are listed in Table
Bl. An asterisk in column 2 of Table B5 again indicates that the experi-
mental run was repeated at least once. The local depth h is shown in
column 6 of Table B5, and the parameter — ~ ~ in column 10 is

x - (z cr

analogous to the parameter in column 7 of Table B3.

The experimental method of determining wave number and wave am-
plitude was discussed in Chapter 3. These quantities are plotted as functions
of x /x and z /ex in this section to examine their dependence on (1) hori-
zontal distance from the slope at a fixed depth, and (2) nearness to z = ex
(i. e., distance to the theoretical position of the critical characteristic).
The pertinent theoretical curves were computed from Equations (4-18) and
(4-20) and are presented in several of the figures for comparison with the
data.

It must be pointed out here that qualitative considerations of the
linear, inviscid solution for y >c, given by Equation (4-16) suggest a priori
that these solutions might fail to fit the experimental data. It can be shown
that the velocity components derived from Equation (4-16) become anoma-
lously large for large coordinate values (x, z). This behavior was not

*

In this discussion and in Table B5 x = |x| and z = |z| for x,z as shown
in Figure 15. c c

80

observed in the experiments. In addition it is shown later that the theory
(in particular, Equation (4-17)) selects |n| = 2 as the lowest nonsingular
inviscid mode permissible over the slope region for each of the experimental
conditions tested. In the particular experimental geometry, it was not ap-
parent that the second mode was present over the slope; it was found most
often that the modal structure was approximately that of the input mode
(|n| = 1). In spite of these qualitative objections, the following discussion
of the experimental results is done quantitatively by comparison with the
theoretical predictions and by consideration of the reflection properties
(Figure 31). It is felt that this method of presentation aids the interpreta-
tion of the results.

Wave number. Figures 32 and 33 illustrate the complicated behavior of the
experimentally determined wave numbers for this case. Data from three
experimental groups and for frequencies co= 0.051 cps and co= 0.066 cps
are represented in these figures. The calculated positions of the critical
characteristic (determined by z = ex ) for two of the experimental groups
are shown along the horizontal axes. The two theoretical curves shown in
each figure were computed for two different values of the mode number (see
qualitative discussion at end of previous section). Equation (4-20) was used
to determine the minimum permissible nonsingular mode number for the
curves labeled n = 2. The mode number n = 1 was preselected for the theo-
retical curves labeled n = 1. Although this latter choice violated the con-
dition necessary for finite values of the velocity components and their first
derivatives at z = -ex (Equation (4-20)), it nevertheless forces the theoretical
mode number over the slope to be equal to the input mode of the experiments.

It is apparent in the figures that there is no straightforward relation-
ship between linear theory (n = 2) and the data. There is only qualitative
indication that the curves for n = 1 have the same trend as the experimental

results for x /x < 0.035 in Figure 32 and for x /x < 0.45 in Figure 33.
c/ o to c' o

It is obvious that the theoretical estimates for n = 1 grow toward very large
values at the critical points (for example, x /x = 0.26, Figure 33). The

81

Km
Ko

22i

20-
18-

16-
14-

12-

10-

8-
6-

4

2-

0

n =

O

n = 2

EXP

group

3

OJ

=

0.051

0

Qo = 0.2

cm

— I T 1 1 1 1 1 1 1

0.2 0.3 0 4 0.5 0.6 0.7 0 8 0.9 10

Xo

Fig. 32.
Upslope variation of wave number, y > c.

82

Km
Ko

a; = 0.

066

ao = 0.

2 cm

-O EXP

group 4

-O- EXP

group 5

0.1 0.2 0.3 0.4 05 0.6 07 0.8 09 1.0

Xo

Fig. 33.
Upslope variation of wave number, y > c

83

experimental values near the calculated critical points are apparently
bounded. The dip in the data in Figure 32 for x /x ^ 0.30 is not apparent
in Figure 33. Possibly the absence of the dip in the data from experimental
group 4 (Figure 33) can be explained by considering nonlinear effects. The
measured value of e (Table B5) for this particular data set is approximately
0.5 at x /x = 0.36. This suggests that nonlinear amplitude growth might
be significant near these positions. By comparison, the measured value of
e for the data in Figure 32 is only 0. 13 for x /x ^0. 2 (Table B5). It
should also be noted that there is a significant shift in the peak value to the
right of the calculated critical point in Figure 32. Possibly this shift indi-
cates that the singularity is removed by nonlinear or viscous processes and
that the linear inviscid solutions are not valid in this particular physical
situation.

Data from experimental groups 3 and 4 are plotted on log- log scales
in Figures 34 and 35, respectively. Theoretical curves for n = 1 are again
shown in these figures. The qualitatively similar trend between the theo-
retical curve and the data is apparent, particularly in Figure 35. Values of
wave number for the larger amplitude input waves (a =0.4 cm) tend to be
higher than those for the smaller waves (a =0.2 cm) in experimental
group 3 (Figure 34).

Figure 36 shows the results of experimental group 3 plotted as a

function of z /ex . It is obvious that the theoretical curve for n = 1 grows
& c

toward infinite values of k /k at z = ex ' the theoretical curve for n = 2

nv o c c7

(obtained from Equation (4-17)) is finite at this point. The trend in the

data values reverses at about z /cx„ * 0.80, and again at z /ex =, 1. 0.

c c ^ ^

No simple relationship between the data and the theoretical curves is
apparent from this figure.

Figure 37 summarizes the experimental results of wave number for
case B (r = c). The log-log plot emphasizes the steep rise toward larger
values of wave number for decreasing distance to the slope. It is interesting
that the magnitudes of wave number for the larger input waves (a =0.4 cm)

84

0

9"

8-

2 Ko

6-

5-

4-

3-

□

O

~ET
O

O
B

CD
O

□

o

n =

EXP

g

roup

3

CJ =

0.

051

cps

do

i

cm

0

0.2

□

0.4

0.2

0.3

0.4

0.5

Xc
Xo

Fig. 34.
Upslope variation of wave number on log-log plot, y > c

85

io-

9
8

7-
6-

1 Km

2 Ko

4-

0

n =

3-

2-

EXP

g

roup

4

U) -

0.

066

cps

ao

•

cm

0

0.2

o

0.15

0.2

03

0.4

05

06 07

Xc

Xo

Fig. 35.
Upslope variation of wave number on log-log plot, y > c

86

2.0

Fig. 36.
Wave number versus magnitude of zc/cxc, y > c.

87

9-

1 Km

2 Ko

8-

7-

Q

6-

0

4-

3-

2-

□
O

0

□

□

A

O

□

EXP group 3

EXP

group 4

oi - 0. 07 3 c p s

w = 0

.079 c ps

Zc = 5.5 cm

Zc =

II. 1 cm

a o , cm

do

, cm

0 0.2

A

0 2

B 0.4

A

□

0.2

0.3

0.4

0.5

0.6 0.7

Xc
Xo

Fig. 37.
Upslope variations of wave number, y = c.

88

in experimental group 3 are larger than the corresponding values of the
smaller input waves for x /x < 0. 3, but for x /x > 0. 3 the wave numbers

CO CO

of the larger waves are consistently less than those of the smaller waves.
In general, the wave number variations with distance to the slope for this
case are similar to the variations shown in Figure 34 for the supercritical
case. No theoretical comparison is shown, since for y - c the minimum
permissible nonsingular inviscid mode over the slope, given by Equation
(4 -17), is infinite.

In summary, the results show a complicated relationship between
wave number and position over the slope. There is evidence in Figures
34 and 35 that the wave number increases rapidly as the waves pass over
the lower and middle sections of the slope for the supercritical case, but
there does not appear to be a systematic pattern to the data near z = ex .
The variations in wave number for the critical case are qualitatively simi-
lar to those for the supercritical case; a large increase in the wave number

for smaller x /x is apparent. The trends of the theoretical values of wave
& o

number computed for n = 1 are similar to the measured values for z /ex < 1.

It is obvious that the unbounded theoretical values at z = ex for n = 1 are

c c

removed when the condition given by Equation (4-17) is satisfied. For both
cases, the measured values of e (Table B5) are greater than 0. 1 for
most of the measurements taken closest to the corner. This suggests that
nonlinear effects might be important near these positions.

Wave amplitude. The experimental estimates of wave amplitude were ob-
tained from the periodograms of the conductivity-probe measurements
(Chapter 3). The results are presented as ratios of the square of the wave
amplitude over the slope to the square of the wave amplitude over the flat
bottom for various frequencies of the input waves. Like the earlier presen-
tation for wave number, these results are illustrated graphically with

z /ex and x /x used as the horizontal axes,
c c & o

89

The results for wave amplitude are easier to interpret in terms of

the reflection properties of the slope than were the results for wave number.

Earlier theoretical discussion indicated that near the characteristic defined

by z = ex and in a zone between this characteristic and the bottom slope

(Figure 31) there is a possible region of concentrated energy flux. Figure

38 shows that the measured values of wave amplitude are highest near

z /ex =1.0 for each frequency tested in this group except u)= 0.039 cps.

However, it is not conclusively shown in this figure that the amplitudes peak

exactly at z = ex . This figure also shows that the ordinate values for the

various frequencies at equal values of z /ex generally increase with de-

creasing frequency. In the light of Figure 31, back-reflected wave energy m

might contribute smaller amounts to the total energy at locations above the

critical characteristic (i. e., at positions given by z < ex ) for frequencies

nearer to critical. This in turn might account for the lower measured

2
levels of (a /a ) for the frequencies nearer to critical in Figure 38.

Figure 39 is a similar plot for data from experimental groups 3 and 4;

two input wave amplitudes are represented in the values from experimental

2
group 3. Also shown in Figure 39 are theoretical curves for (p /p ) de-
rived from Equations (4-10) and (4-18) by the methods described earlier.

The mode number |n| was determined from the condition in Equation (4-17)

2
for curve n = 2 in this figure. The theoretical values (a /a ) decrease

b m/ o

toward the slope and undergo a smooth transition through the critical point
z /ex = 1.0, where they then begin to increase toward the slope. The
theoretical curve with n = 1 has increasing values toward the slope until the
critical point (z = ex ) is approached; a sharp dip in the theoretical values
at this point followed by a rapid rise suggests that the solutions are not well
behaved at this critical point. The ordinate values for the two input ampli-
tudes of group 3 in Figure 39 agree within 10 percent of one another for
values of z /ex < 1.0; in general, the larger input waves have slightly
higher measured amplitudes in this range. It is also apparent that the
larger input waves grow more rapidly as the value z /ex = 1.0 is approached.
Owing to the larger values of e (Table B5) for the waves of larger input

90

a

m

2

©

24-

a

0

22-

20-

o

18-

-0-

16-

14-

12-

Ar

io-

A

<>

8"

ta

0

6-

A
El

4-

A
A-

A

O

0

A

2-

i

s

— 9—

-El-
El
0
I

— P"

0

i

0

1

1

E
i

1

Zc

cXc

1

02

0.4

0.6 08

.0

1.8

20

EXP group 4

EXP group 5

Oo = 0.2 cm

Oo = 0.4 cm

cj , cps

ai , cps

O 0.073
O 0.066
A 0055
□ 0.051
0 0.039

-Er- 0.0 6 6
-A- 005 5

Fig. 38.
Wave amplitude versus magnitude of zc/cxc, y > c

91

Cl» =

0 051

cps

EXP group

do

, cm

3

O
A

0.2
0.4

4

□

0.2

Fig. 39.
Wave amplitude versus magnitude of zc/cxc, y > c

92

amplitude (a =0.4 cm), the increased disparity in the measured amplitude
values for input waves of 0.4 and 0. 2 cm, respectively, at values of
z /ex > 1.0 in Figure 39 might be the result of nonlinear effects.

Figure 40 shows the experimental results and theoretical curves for
a) = 0.051 cps (group 3) on a log-log scale; the horizontal axis is distance x
to the corner normalized by the total horizontal projection of the slope x .
The rapid increase in amplitude upslope for 0. 28 < x /x < 0. 60 is readily
apparent in the data; however, the theoretical values for n = 2 show an op-
posite trend, toward lower values for decreasing distance to the slope. The
measured amplitudes below the indicated position of the critical point remain
relatively high; this suggests amplification of the motion near the bottom.
Nonlinear effects also might be important near the corner (x /x = 0. 19)

The results for the probe measurements of wave amplitude at critical
frequencies are shown in Figure 41. The increase in amplitude over the
range 0. 28 < x /x < 0. 60 is similar to the increase shown in Figure 38 for
the same range. However, the maximum amplitude measured for these
waves is considerably less than the corresponding value for the supercritical
case (Figure 40). Larger viscous dissipation along the slope in the critical
case might explain the smaller peak amplitude levels.

In summary, the data indicate that the wave amplitudes generally

increase toward the slope; the higher frequency waves (i. e., those closer

to a? ) for these cases show rapid growth during shoaling over the middle

sections of the slope. Peak levels of amplitude were generally found near

z /ex m 1.0, with lower measured peak values for waves of critical fre-
c' c ' *

quency. In both cases the motion is relatively large near the slope. In
general, the conductivity-probe measurements of amplitude agree qualita-
tively with the reflection properties of internal waves from a rigid, sloping

bottom. There is some indication from the measured values of e that

m

nonlinear amplification of the amplitude occurs for small values of x /x .

93

10.0

0m

ao

12

A A

O

5.0

A

0

n = i — -

A

O

EXP

group 3

u) =

0.051 cps

ao

, cm

o

0.2

A

0.4

A
O

20-

.0

0.5-

0.2

n = 2

i — i — r

0.1

02

0.5

Xc
Xo

Fig. 40.
Upslope variation of wave amplitude, y > c

94

-|2

5,0—

0m

ao

0.5-

A

2.0-

1.0"

A

O

A
A

O
0

%
I

A
0

$

0.2-

EXP

group

3

a» =

0.073

cps

ao

, cm

O

0.2

A

0.4

T

i — r

"TT

0.2

0.5

Xo

1.0

Fig. 41.
Upslope variation of wave amplitude, y > c

95

It might be added here that the bounded experimental values at
z = ex suggest that viscous dissipation and nonlinear processes might act
to remove the singularity at the critical characteristic. For example, the
shift in the measured peak wave number in Figure 32, mentioned earlier in
this section, and the formation of vortices near the bottom for these cases,
discussed in the next section, suggest that the linear inviscid model is not
properly tested by these experimental results. Nonlinear and viscous effects
in the interior, particularly in the vicinity of the critical characteristic,
should be examined in future analytical treatments of this problem.

BOUNDARY ACTIVITY

The discussion of the bottom boundary layer and the zone of breaking
follows the procedure of separating the experimental results into two hydro-
dynamic regimes (Iy < c and II y > c). The distinction between these two
regimes was defined in Figure 14. Qualitative features observed during the
experiments and illustrated by several photographs are presented at the
outset as an introduction to the later discussion of quantitative results. In
the discussion of quantitative results, experimentally determined velocities
in the bottom boundary layer at several positions along the slope are shown
to agree with theoretical values computed from the linear solutions in
Equation (4-24) for case A. On the basis of the good agreement between
experiment and theory for the conditions tested in case A, Equation (4-24)
is used in Chapter 5 to develop a model for sediment movement induced by
shoaling internal waves.

Case A: y <c

Qualitative results. The shadowgraph images and the motion of neutrally
buoyant floats permitted excellent real-time observations of the oscillatory
flow in the boundary layer and breaking zone. One obvious major feature
of this flow was its intensification near the bottom along the slope as com-
pared with the flow along the horizontal bottom. The maximum velocities
generally increased at positions farther upslope, with a sharp growth in the

96

velocities near the zone of breaking. The oscillatory flow remained laminar
in the boundary layer up to the zone of breaking.

The breaking of waves on the relatively steep slopes (30 and 45
degrees) is characterized by considerable turbulence and vertical motion
below the free surface and by the generation of spatially irregular fine
structure in the density field. This kind of breaking is also observed for
relatively high-frequency waves on the smaller slopes (7 and 15 degrees).
In contrast, the lower frequency waves (i. e., those waves with frequencies
closer to co ) break much less violently, and typically form surges that run
up the slope for relatively long distances.

More detailed observations of the higher frequency waves indicate
that shortly after the first arrival of the waves near the top of the smaller
slopes (a = 7 and 15 degrees), a small vortex forms along the slope be-
neath the crest of the advancing wave form. Figures 42 and 43 illustrate
the onset of this instability beneath the wave crest. In Figure 43 the maxi-
mum amplitude of the incoming wave at the entrance to the slope region
was approximately 0.4 cm, whereas the corresponding amplitude in Fig-
ure 42 was 0. 2 cm. The time that had elapsed from the start of the wave
maker is shown in the guide sheet preceding Figure 42. Subsequent devel-
opment of the instability in Figure 42 is shown in Figures 44a and 44b, and
later stages of the instability in Figure 43 are shown in Figure 45a and 45b.
The lamellar structures in these photographs represent small-scale depar-
tures from the constant vertical index-of- refraction gradient characteristic
of the stratified fluid with no wave motion. A constant index-of-refraction
gradient (or equivalently, constant density gradient) produces a uniform
uninteresting shadowgraph image (see center of field in Figure 42). In
spite of the apparently chaotic density structure, the basic wave-form can
still be discerned in the photographs. Qualitatively, the larger amplitude
waves show a higher intensity of breaking, as evidenced by the contrast be-
tween Figure 44b and Figure 45b. Not shown in this series of photographs
is the protrusion of thin horizontal layers back into the interior region

97

over the slope. The protrusions lose their intensity (as measured by
photographic contrast) over very short horizontal distances (5 to 10 cm)
and are confined to the upper vertical levels, where the breaking occurs.
The protrusions, thin tongues of mixed fluid that are weakly converted back
into the interior, are probably caused by mixing in the zone of breaking.

In contrast to the higher frequency waves, the waves of lower fre-
quency on a small slope (7 degrees or 15 degrees) show less mixing and
fewer laminae in the zone of breaking. Each wave forms a surge in which
the velocities of neutrally buoyant particles within the surge become approxi-
mately equal to the wave celerity as the wave progresses upslope, somewhat
analogous to solitary surface waves. The surging motion eventually produces
a runup and backwash zone along the bottom very near the corner. The runup
appears as a thin streamer in the shadowgraph image and represents a pro-
trusion of fluid of higher density into the near-surface layer. Figures 46a
through 46c illustrate the development of this phenomenon for waves whose
initial amplitude is 0.4 cm, Figures 47a through 47c show waves whose
initial amplitude is 0. 2 cm. Vortex instabilities initially form beneath the
wave crests and propagate upslope with the wave form. The quasi- steady
development for the 0.4 cm and 0. 2 cm waves is shown in Figures 46c and
47c, respectively. In these figures, the surging motion essentially fills
the water column in the left portion of the photographs. Farther upslope a
vortex forms and eventually dissipates into runup.

Figures 48a through 49c show a corresponding development of break-
ing for waves of 10. 7 sec period and 0.4 cm initial amplitude over a 30-
degree slope. The mixing becomes very intense after the collapse of the
initial vortex (Figure 48a). It is interesting to note the reversal in the cir-
culation of the lead vortex as it propagates upslope and collapses. The re-
versal is a consequence of the shear between the surging motion near the
bottom and the opposing flow above the vortex. The following sketch in
Figure 50 illustrates this pattern.

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Fig. 43.

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Fig. 44 a

Fig. 44 b

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Fig. 45 a

Fig. 45 b

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Fig. 46 a

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Fig. 47 a.

Fig. 47 b

Fig. 47 c

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Fig. 48 a.

Fig. 48 b.

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Fig. 49 a

Fig. 49 b,

Fig. 49 c.

106

FIG. 50. Advancing initial vortex at successive locations A, B, and C
Note observed reversal in circulation at position C.

The outflow above the vortex balances the flow of water that is advected
toward the corner along the bottom by the wave motion.

As indicated by the well developed lamellar structures in Figures
49b and 49c, the steeper slopes appear to produce a high intensity of breaking.
The formation of a surge with associated runup was not observed for the
steeper slopes (30 degrees and 45 degrees) for any wave frequency tested.

Figure 51 presents a summary of the observed positions of breaking
on slopes of 30 degrees. The accuracy of the distance L, is low, owing to
the difficulty in defining the point of breaking; however, the relative order
of the values of L, for the waves of different frequencies shown in the figure
is reliable. The general trend indicates that for a given N, the higher fre-
quency waves break first, i. e., in deeper water.

107

FIG. 51. Position of breaking

a = 30°, N = 0.97

-1 rr. Li, cm
u), sec T, sec b?

a = 0. 2 cm a =0.4 cm
o o

0.722 8.7 13.4 14.0

0.668 9.4 12.5 12.9

0.593 10.6 10.6 11.1

0.561 11.2 9.9 10.5

0.537 11.7 9.0 9.5

0.515 12.2 8.2 8.6

0.495 12.7 7.7 8.2

Quantitative results. The experimental techniques that were used to mea-
sure the motion in the boundary layer are discussed in Chapter 2. The
results of these measurements are compared with the predictions of the
linear, viscous boundary layer solutions in order to check the validity of
these solutions for certain conditions and provide some empirical estimates
of the motion where the theory does not apply.

Since successive displacements of the dye streaks and neutrally buoy-
ant particles were measured at one-second time intervals, the velocity
values computed from these measurements are essentially one- second time
averages in the Lagrangian sense. Equation (4-24) provides a linear,
Eulerian solution for the velocity field parallel to the bottom in the viscous
boundary layer; this solution is identical with the Lagrangian one to first
order in e. The Lagrangian velocity uT can be determined in principle
from the Eulerian velocity u„:

uL (a, t) = uE (a,t) + Jto u(a, t') dt' ■ V^ u£ (4-27)

where a is a position of the dye streak, say, at time to; t - t is small com-
pared to the wave period T, and V is the gradient operator (Phillips, 1966).

108

— 2 —

This expression for ul is correct to 0(e ) if u„ is correct to this order.

Li Hi

Let

(D ■ A? (2) (4_28)

UE = eUE + e UE

where u is the linear solution for the velocity field. It is easy to show

that there are two second order contributions to uT in Equation (4-27):

(1) u ^2' and (2) (f* u dt') • Vu£. The first contribution (1) is due to the

o
nonlinear terms in the equations of motion; the second (2) is commonly re-
ferred to as the Stokes velocity and can be computed in principal from Equa-
tion (4-21). Hogg and Wunsch (1970) have shown that to order e the time
average of u for this problem is zero. This implies that there is no net
transport of water particles induced by internal waves near a sloping boun-

dary. It also implies that since u„(l) = 0, then

uL = -(£ uE(1)dt') • VuE(1) (4-29)

o

where the bar denotes a time average over one wave period.

A complete analysis of the data was carried out for three wave fre-
quencies in regime I (y<c). Two of these analyses involved waves on a
30 degree slope (co = 0. 105 cps and co = 0.094 cps, group 7, Figure Bl,
Appendix B) and one involved waves on a 15 degree slope (w = 0. 105 cps
group 2). For the sake of brevity, and without any appreciable loss of de-
tail, only the results from w = 0. 105 cps will be discussed at length.
Instances where measurements from the other runs showed significant
features of deviations from this one will be mentioned in the discussion.
The results for regime n (y > c) are discussed in a later section.

109

A time-sequenced series of streak and particle positions from
which velocity information was obtained is shown in Figure 12. The streaks
are numbered according to consecutive, one-second time marks relative to
the initial one. The numbers on the particles correspond to those for the
streaks. The increase in displacement of the streaks nearer to the bottom
agrees qualitatively with the amplification predicted by Wunsch (1969) for
y <c (Figure 52).

Figure 53 summarizes some of the relevant parameters that were
estimated from the measurements. The five station marks in the diagram
show the locations of dye pellets. The coordinates for these locations are
given in columns 2 and 3. The definitions of the various Reynolds numbers
and estimates of boundary -layer thickness are given in the legend. The
approximate maximum error in the measured estimates of the boundary-
layer thickness (6 ) is 10 percent of the listed value.

The boundary- layer behavior for steep slopes (30 degrees) is anom-
alous on the lower portion of the slope. The data at station 4 for co = 0. 105
cps (Figure 57) and go = 0.094 cps both reveal erratic streak displacement.
The estimates of boundary-layer thickness at this location were slightly
larger or approximately equal to those at station 3. The measured thick-
ness of the boundary layer at corresponding positions on the smaller slope
(15 degrees) generally increased in an upslope direction to the zone of
breaking. The anomalous behavior for steeper slopes was probably associ-
ated with an adjustment in the modal structure as the waves progressed
from the flat-bottom region to one of appreciable slope. The irregularities
in the motion at station 4 are discussed later in this section.

The larger values of 6 as compared with s_ at stations 1, 2, and
3 represent significant departures of the measured estimates from the
theoretical values. In addition, the theoretical value is constant over the
entire slope, whereas the measurements indicate that 6 varies slightly

with slope positions. Over the flat bottom, 6 is slightly larger than the

1/2 m

value given by (2i//co) ' , although this difference was not significant for

110

n =

0.6 0.8 1.0

Fig. 52. Theoretical prediction of increase in

velocity components along the slope. Units
are arbitrary (after Wunsch, 1969) .

Ill

tJ =

0.66 rad/sec, T = 9.5 sec

N =

0.97 rad/sec, C = 0.93

a =

30 deg 7 = 0.577

o :
:o =

0.2 cm

0.097 cm"1, XQ = 64.7 cm

Ko

hQ = 30.0 cm

1

2

3

4

5

6

7

8

9

10

11

12

Station
No.

X

c
(cm)

h
(cm)

RL

MO"3

*W

RB

61
cm

62
cm

5
~m

cm

6s
cm

a
m

cm

f

1

8.7

5.0

2.4

209.0

41.2

0.17

0.26

0.35

0. 17

1.78

0.52

2

23.0

13.3

17.2

32.4

21.7

0.17

0.26

0.47

1.16

0.70

0.08

3

36.0

20.8

42.0

19.3

13.0

0.17

0.26

0.39

2. 17

0.54

0.04

4

51.0

29.4

83.8

62.2

25.0

0. 17

0.26

0.39

1.74

0.97

0.05

5

100.0

30.0

87.3

4. 1

3.6

0.17

0.26

0.22

8.70

0.25

0.01

R = Nh /v - local wedge Reynolds number

L

R

B

= a w/v = local wave Reynolds number

= a u> 6 fv = local boundary layer Reynolds number

6+ = (2v/co)

1/2

= (2y/N)1/2 (o-/(o-2-sin2a)1/2 , a = u>/N

= «iA

a k sin a
m m

FIG. 53. Estimated of boundary layer thickness and
Reynolds numbers, y < c.

112

the experiments in which oj = 0. 105 cps, a = 15 degrees. It appears that
the boundary layer thickness given by linear theory gives the correct
order, mm rather than cm, but the measured values tend to be significantly
higher. However, it must be recalled that the definition of boundary layer
thickness for the purposes of measurement, i. e., the distance between the
bottom and the point of inflection in the velocity profiles, is somewhat
arbitrary, and might account for the discrepancies with the theoretical
values.

It should also be mentioned here that at station 1, the computed
value of e is relatively high (e a 0. 5); nonlinear effects are probably im-
portant there. It was mentioned earlier that Wunsch (1970) recently showed
theoretically that there can be a mean upwelling velocity independent of wave
motion in a layer whose thickness is 6

w

aw~(,*)1/4/N1/2

where v and k are the vertical diffusivities of momentum and heat, re-
spectively. In terms of molecular processes, i/~102cm ' and

2 / -1

k ~ 10-3 cm /sec# Using the value of N given in Figure 53, N ~ 1 sec ,

we can estimate 8 ~ 0o 06 cm for this particular experiment.

The curves in Figures 54 through 58 show velocity versus depth for
all five stations. Each curve represents a one-second time average of the
Lagrangian velocity parallel to the bottom., The consecutive numbers on the
curves indicate the time history of this velocity over one wave cycle. The
curves were computed only to the vertical level nearest the bottom for
which the streak data was considered reliable. The horizontal velocity
scales are uniform in each of the figures except for the expanded scale of
the flat bottom results (Figure 58) and the condensed scale at station 1; the
vertical scales are the same for all figures.

It is important to note that at each of these stations the observed
motion in and near the boundary layer was essentially parallel to the bottom.

113

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118

Only very small velocities normal to the bottom were detected from the
motion of the plastic beads, and this vertical motion was usually confined
to regions adjacent to and in the zone of breaking.

A reason for the anomalous behavior of the velocities at station 4
(Figure 57) was hypothesized earlier in this section. An alternative ex-
planation is that the abrupt change in bottom geometry acts as a source of
diffracted waves which locally affect the pattern of motion. However,
since the density did not vary linearly with depth at the bottom (for example,
Figure 9), but instead gradually became more homogeneous in a small
layer near the bottom, the dynamics in this transition region are probably
quite complicated. Visual observations of the motion of neutrally buoyant
particles and dye patches that were placed at the transition in bottom geom-
etry during several experiments (for 15 and 30 degree slope angles) did not
disclose any significant patterns other than those already mentioned.

Figure 59 shows a comparison between the theoretical and experi-
mental RMS values of the velocities at selected vertical levels shown in
Figures 54 through 58. The experimental RMS value at each level was
computed as:

RMS " lUMS;

where

\\

MS

I (up/r , i = l, 2,

and T was chosen as the smallest integer greater than the wave period.
For this particular set of data, T = 10 (wave period a 9. 5 sec). For
stations on the slope, theoretical RMS values were obtained from the Eulerian
expression for the boundary layer velocity component parallel to the slope
(Equation (4-24)):

URMS(the0ry) = (1/2»E * ue)1/2'

119

These RMS values are listed in Figure 59 as first order theoretical estimates
of the RMS Lagrangian velocities.

Qualitatively, we might expect that the measured values are more
accurately represented by the sum of the first order Eulerian values and
the second order values consisting of the Stokes velocity component and the
nonlinear component, particularly farther upslope where e become larger.

When the various sources of uncertainty in the measurements of
velocity and experimental parameters (AQ, N, w, h, etc.) are considered,
a rather good comparison would require that the theoretical and experimental
values fall within 15 percent of one another at each level tj • The results in
Figure 59 show that excluding station 4 there is excellent agreement (with-
in 10 percent) between the measured RMS values and the corresponding
first order, theoretical values. The excellent comparison at station 1 is
particularly remarkable, considering that the estimated value of e is approx-
imately 0. 5 there. The high measured values at station 4 are a consequence
of the anomalous motion that has already been discussed.

The results for the maximum velocity |u| max parallel to the bottom
at the selected levels (columns 6, 7, 8, Figure 59) also show good agreement
between experiment and theory. Theoretical estimates of the maximum
values of this velocity component were again obtained from Equation (4-24).
The experimental values are generally higher than the theoretical first order
values. The large deviations between experiment and theory at station 4 are
again obvious. The results for the other analyses of this kind show similar good

120

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121

agreement between theory and experiment. The velocity values for station
4 with the 15 degree slope are more in agreement with the theoretical
values (within 25 percent) than are the results in Figure 59 and did not show
anomalously large velocity values for this station.

The asymmetry in the velocity profiles at station 1 (Figure 54)
contrasts with the more symmetrical curves for the other stations.
Station 1 is located well upslope and only a few centimeters from the zone
of breaking. The skewness toward positive velocities suggests that there
might be a measurable net upslope motion of the water particles over one
wave cycle. Although the maximum velocity is directed downslope (curve
1, Figure 54), the magnitudes of the upslope component are quite high
(approximately 1 cm/ sec) for a considerable fraction of the wave cycle.
The local measured value of e is about 0. 5 at this station, which suggests
that nonlinear effects might be important. It is also interesting to note
that <5 at this station shows a decrease from that at station 2 (Figure 53).

Experimental estimates of net drift at the outer edge of the boundary
layer were computed at each station by algebraically averaging the mea-
sured velocity values at the estimated level of 6 in Figures 54 through 58,
respectively. The average is for one wave cycle (sampling interval of one
second) and represents a single estimate of mean streaming at each station.
No statistical significance can be inferred from this method, but the mea-
sured results in Figure 60 show that the streaming is insignificant at all
slope stations except station 1. Here the net drift is approximately 1.3 mm/
sec. This drift is probably induced by mixing in the nearby (3-5 cm) zone
of breaking. As discussed earlier, protrusions of thin fluid laminae slowly
advect into the interior over the slope from the zone of breaking. The direc-
tion of the drift measured at station 1 is toward the breaker, possibly indi-
cating that this flow compensates in part for the mass efflux in the protru-
sions. The observed direction net drift near the free surface above this
station is also toward the zone of breaking.

122

FIG. 60. Measured values of mean streaming at ij = 6
stations 1 through 4 on slope.

Station No. x , cm |", cm u , cm/sec

1 8,7 10.05 +0.13

2 23.0 26.6 -0.07

3 36.2 41.8 -0.003

4 51.0 58.9 -0.023

123

Owing to the poor reliability of the raw streak data very near the

wall (<0.07 cm), the best experimental estimate of bed shear stress, t ,

' o'

was obtained by extending the reliable portion of the velocity-depth curves

to the zero point at the bottom. The shear stress was then computed from

a simple straight-line fit between the zero point and the lowest point on the

velocity curve. This gives a low estimate of the bed shear stress, since

the actual shape of the velocity curves in the boundary layer is more

exponential. The maximum bed shear stress A- /max measured at each

location is listed in column 7 of Figure 59; the time variations in r over

o

one wave cycle corresponding to the time variations in the velocities in
Figures 54 through 58 are shown in Figure 61(a - d). The theoretical values
of bed shear stress were computed from Equation (4-24):

tq = m au/a^ , rf = 0

where

" = 0.01 —62^.
cm-sec

The maximum theoretical value is shown in column 8 , Figure 59,
for comparison with the corresponding measured value. Since the theo-
retical estimates were made from the linear Eulerian expression of velocity
at each station, they correspond to the experimental Lagrangian estimates
only to first order in e. It can be shown that \i varies by only 4 percent over
the density range for the particular experimental conditions shown in Fig-
ure 53. In spite of the poor comparison of the estimates of boundary layer
thickness given by linear theory and experiment, the theoretical estimates
of maximum bed shear stress at stations 1-3 compare reasonably well with
the measured values. This is apparently the consequence of the higher ex-
perimental values of maximum velocity as compared to the first-order
theoretical values for levels near the boundary. Figure 61 also shows the
excellent fit of the theoretical curve computed from first order solutions to
the experimental values at stations 1, 2, and 3.

124

Shear stress, dynes/cm'
7.0-
6.0-

5.0-

4.0-

3.0

2.0

1.0-

-1.0
-2X)

-3i0
-4.0
-5.0
-6.0
-7.0

O measured

— theoretical curve

Fig. 61 a.

Experimental and theoretical values of bed
shear stress t0 at station 1. The experi-
mental values were obtained for each
second of time t during one wave cycle.

125

Shear stress, dynes/ cm
3.0-1

© me a sured

— theoretical curve

Fig. 61 b. Experimental and theoretical values of bed
shear stress tq at station 2. The experi-
mental values were obtained for each second
of time t during one wave cycle.

126

Shear stress, dynes /cm
2.5-1

© measured
— theoretical curve

Fig. 61 c.

Experimental and theoretical values of bed
shear stress x0 at station 3. The experi-
mental values were obtained for each second
of time t during one wave cycle.

127

Shear stress , dynes /cm
2 5t

time , sec

© measured

— theoretical curve

Fig. 61 d. Experimental and theoretical values of bed
shear stress tq at station 4. The experi-
mental values were obtained for each second
of time t during one wave cycle.

128

Cases B and C: y = c and y > c

The description of the boundary -layer motion in this section follows
a format similar to that presented in the discussion of boundary activity for
case A (y < c). The qualitative remarks deal mainly with the observations
and photographs of shadowgraph images, whereas the quantitative results
were derived from the measurements of the motion of dye streaks and
neutrally buoyant particles.

Qualitative During those runs for the critical case (y = c) with moder-
ately steep slopes (30 and 45 degrees), the zone of breaking near the cor-
ner was characterized by intensive mixing as evident in the shadowgraph
images, similar to that described for the high-frequency waves on a 30
degree slope (see earlier discussion for y <c in this chapter). For runs
with a slope of 15 degrees, each wave, for both the critical and supercriti-
cal cases, formed a surge that usually ended with a short runup. This
runup usually occurred at the bottom of the more homogeneous near- surface
layer, if present, and appeared like a gentle, lapping motion with much
smaller upslope penetration than the runup for the subcritical case, which
persisted over a much longer distance. In summary, the intensity of
breaking and runup for y > c is decreased as compared with the experi-
mental observations of these processes for the subcritical case.

Figures 62 through 64 illustrate in a very striking way that the in-
ternal wave motion for these cases (y > c) is associated with the growth of
an instability along the slope. The shadowgraph observations of the insta-
bility illustrated in these figures indicate that a line of regularly spaced
vortices forms near the bottom. These vortices grow and decay over each
half cycle as they oscillate along the slope at the frequency of the input
waves. It appeared that on the downslope propagation during the wave
cycle, the vortices would "flatten", indicating that they form only during
the upslope movement of the water particles. This is indicated in Figures
63a and 63b. The circulation of adjacent vortices has the same sense;

129

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130

Fig. 62 a

Fig. 62 b

131

Fig. 63 a

Fig. 63 b

132

Fig. 64 a

Fig. 64 b.

133

this circulation is indicated diagrammatically below. The axis of each
vortex is oriented approximately normal to the side walls of the tank,
and the maximum distance observed between the core of adjacent vortices was
was about 2 cm. Unlike the vortices which form beneath the shoaling,
high-frequency waves well upslope and which precede the violent breaking
(Figure 45), these vortices retain their cyclic growth and decay pattern
for the duration of the input wave motion. Their regular spacing was much
shorter than the wave lengths of the input waves, particularly at the middle
and lower slope regions.

Circulation in Vortices

Shortly after the formation of the line of vortices, thin streamers or
wisps appear near the cusp of each vortex and penetrate horizontally back
into the interior of the fluid (note the thin lighter bands that alternate with
the darker bands in Figures 62 through 64). Qualitatively, it appears that
fluid of slightly different densities from vertical levels adjacent to each
vortex cell is mixed in the cell and returns to the interior as a thin streamer
at a new equilibrium level. The result of this process is alternate layers of
thin, nearly homogenous streamers and thicker, linearly stratified laminae.

134

Quantitative (1.) Vortices and Layering. The onset of vortex activity for
any particular run normally occurs within 10 to 20 wave periods following
the arrival of the first wave motion at the corner region of the slope. The
initial vortex generally appears somewhere along the middle section of the
slope, and the instability spreads gradually toward the corners. The ini-
tiation of vortex activity was observed as a function of input frequency for
experimental groups 4 and 5; the results are shown in Figure 65. These
results indicate that, for this set of experiments, (1) the vortices occur
earliest (in time measured from the start of the wave maker) for a> =* u>
even though the lower frequency waves have a higher group velocity; (2)
there is some w < a; for which the vortices do not appear, and (3) there
is a small range of co slightly greater than a> (i. e., waves in the sub-
critical range) for which vortices form.

The diameters of the vortices were difficult to measure accurately
from the shadowgraph images because of the constant process of either
growth or decay during each half cycle. From data taken from various
photographs, the size of the vortex diameters is largest for co =* co and
generally decreases for lower and higher frequencies. Figure 66 shows
the size variation of maximum vortex diameters measured at midslope
positions during experimental group 4. The corresponding input wave
frequencies are listed in column 1.

The position of the vortices relative to the bottom also varies over
each half cycle of the wave motion. As a vortex grows in diameter it also
appears to move away from the boundary; conversely, as its diameter
diminishes its distance from the bottom decreases. The maximum distance
measured between the center of a vortex and the bottom was approximately
1.2 cm. *

It is worth noting here that the alternate motion toward and away from the
boundary by a diminishing and enlarging vortex cell gives the impression
of a periodic pumping in a direction normal to the bottom.

135

tc

t

1.0 1

0.9-
0.8-
0.7-
0.6-
0.5-
0.4-
0.3-
0.2-
0.1

0.8

O

o

o

o

o

o

o

o

o

o

09

1.0 I.I

OJc

c^- = O. ^s sec

tc = 250 sec.

delimit the range
of observed vortex
formation.

FIG. 65. Onset of vortex motion as a function of input
wave frequency.

136

-1

it), sec

T, sec

d, cm

0.546

11.5

0.32

0.547

11.7

0.39

0.519

12.1

0.43

0.495

12.7

0.52

0.487

12.9

0.51

0.480

13.1

0.52

0.468

13.4

0.47

0.458

13.7

0.47

0.430

14.6

0.43

FIG. 66. Maximum size of vortex diameters
as a function of frequency

a) = 0.495 sec
c

N =0.99 sec

-1
-1

137

Shadowgraph images of the thin, horizontal streamers (for example,
Figure 62b) usually intensify with time if the wave motion continues. Their
initial appearance generally varies with frequency much in the same way as
the onset of the vortices. The streamers are usually sharply outlined as
shadowgraph images near the vortices, but within short horizontal distances
from the slope (order of 2 -4 cm) their images weaken. Although they can
be detected out to distances of 20 cm from the bottom from the shading
variations on the shadowgraph, the contrast between the images of the
streamers and adjacent layers becomes very indistinct at distances greater
than 20 cm from the slope. Their deepest observed penetration of the
interior was approximately 25 cm, measured horizontally away from the
slope. This particular measurement was made after one hour of continu-
ous wave motion at the critical frequency for a run during experimental
group 4 {a = 30 degrees). The thickness of the streamers during this
particular run was 2 mm near the boundary, and the vertical distance
separating two streamers was typically 5 mm. Normally the vertical
separation between streamers is approximately equal to the diameter of
the nearest vortex. The shadowgraph images of the streamers are most

distinct for wave frequencies near oj .

c

Several conductivity probe traverses through short vertical sections
(3 and 4 cm) were made in order to detect changes in the linear density-
depth profiles that might be associated with the streamers. Figure 67 shows
the measured structure for a typical vertical traverse of a probe after the
formation of the streamers. Small scale irregularities in the curve (after
the wave motion) are typical of these traverses. The values along the
horizontal axes represent changes in output voltage relative to the null po-
sition that was chosen approximately at the center of the traverse; the
vertical axis is actual tank depth in cm. Changes in equivalent density de-
termined from the calibration curves are also shown along the horizontal
axis. Although sharp changes in density at the edges of the streamer are
apparent from the photographs, some conductivity probe and specific gravity

138

-1.0 "0.5

0.5 1.0

Ayo- I03, g/cc

Z.Cm -0.5 -0-4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

8-

9-

10-

12-

I L

J 1 I L

J I I L

AV, volts

Fig. 67.

Density-depth profile through a region of
streamers (numbered) and intervening layers

139

measurements suggest that the density is not homogeneous within the
streamer, but has a small, nonzero gradient there. This latter remark is
based on only six measurements within two streamers (not shown in Figure
67), and these were taken near the associated vortex cell close to the slope,
where the streamer could easily be identified from shadowgraph observa-
tions. The apparent density inversion at layer "3" in Figure 67 is not
significant.

The net movements of neutrally buoyant particles in the streamers
and in the invervening layers were observed for several runs. The typical
net motion (away from the slope in the streamers and toward the slope in
the intervening layers) is indicated diagrammatically in Figure 68. Aver-
ages of the net velocity measurements in the streamers and layers are
listed in the table in Figure 68. Each individual determination of net mo-
tion that went into the computation of this average involved tracking particles
for a minimum of 10 wave periods. The differences among the various num-
bers are not significant; however, a magnitude of about 0. 1 mm/sec in each
zone is reliable.

Streamer

o .^Intervening
Layer

Case

Average net
velocities in mm/sec

Streamer

Intervening
Layer

Critical
(y = c)

0.14

0.21

Super
Critical

y > C

0.17

0.13

Fig. 68. Average net velocities in streamers and intervening layers.

140

(2) Boundary- layer description. Owing to the complicated near-
bottom motion caused by waves of frequencies at or near critical, it was
difficult to choose a level that accurately described the physical extent of
the boundary layer. The height above the bottom of the inflection point in
the streak curves was selected as an approximate measure of the boundary-
layer thickness. Admittedly, the choice is somewhat arbitrary; however,
the change in curvature in the streaks was generally well defined at this
point during at least the total time of sampling. In any event, this estimate
of 5 should be regarded as only approximate. In addition to the values of
6 , other estimates of the boundary -layer thickness are presented in
Figure 69 for comparison.

The values of s„ and 6„ represent two estimates of the boundary-
layer thickness that result from scaling considerations. Wunsch (1969)

found that for w^w there is a balance in the governing boundary- layer

C -1/3

equation over a distance proportional to R ' . He also showed that the

equation is nonseparable in the spatial coordinates. This type of balance

-1/3
(R ' ) is represented by <5 0 in column 8. The value of s„ represents a

-1/2
boundary-layer thickness of Stokes type (<5 aR ' ) and is presented for

comparison in column 9. Firm conclusions based on quantitative com-
parisons between the scaling estimates and s are not justified here since
6 is only approximate for each slope station.

Similar results for a representative run of the supercritical cases
are shown in Figure 70. For these measurements, the boundary- layer
motion at each station was very regular, and the dye- streak profiles gen-
erally resembled those described for the subcritical case. There was no
evidence of vortex activity with the associated generation of thin streamers

The estimates of 6j and <5o differ approximately by a factor of 2; this is
obvious from their definitions.

141

<jO =

N =
a =

i:

0.495 rad/sec, T = 12.7 sec

0.97 rad/sec, C = 0.593

30 deg 7 = 0.577

0.2 cm

0.056 cm'1, X = 113.0 cm

33.5 cm

1

2

3

4

5

6

7

8

9

10

11

12

Station
No.

X

c
(cm)

h
(cm)

RL

■ io-3

*W

RB

(cm)

"2
(cm)

'3

(cm)

5
m

(cm)

a
m

(cm)

f

1

15.6

9.0

7.86

68.98

21.14

0.201

0.45

0. 10

0.362

1. 18

0. 12

2

24.2

14.0

19.03

92.91

26.51

0.201

0.52

0.10

0.391

1.37

0.09

3

43.3

25.0

60.63

102.64

29.08

0.201

0.64

0. 10

0.408

1.44

0.05

4

53.7

31.0

93.2

25.66

14.65

0.201

0.68

0. 10

0.411

0.72

0.04

5

110.0

33.5

109.2

7.15

5.27

0.201

0.70

0. 10

0.283

0.38

0.02

R^ = Nh2/y

RB =amw6m^
o, = (2v/io)1/2

local wedge Reynolds number

local wave Reynolds number

local boundary layer Reynolds number

= hR,

= hR

-1/3

'-1/2

= (iz/N)

1/2

a k sin a
m m

FIG. 69. Estimates of boundary layer thickness
and Reynolds numbers, y = c.

142

io -
N =
a =

Si

0.405 rad/sec, T = 15.5 sec
0.97 rad/sec, C = 0.460
30 deg 7 = 0.577
0.2 cm
0.043 cm"1
33.5 cm

XQ = 145.7 cm

1

2

3

4

5

6

7

8

9

10

11

12

Station
No.

X

c

cm

h
cm

RL

.io-3

*W

RB

61
cm

52
cm

63
cm

6
m

cm

a
m

cm

c

1

12.1

7.0

4.75

15.1

11.4

0.22

0.42

0. 10

0.46

0.61

0.06

2

22.5

13.0

16.39

61.3

22.4

0.22

0.51

0.10

0.45

1.23

0.07

3

38,1

22.0

46.95

30.7

17.3

0.22

0.61

0. 10

0.49

0.87

0.03

4

48.5

28.0

76.05

31.4

11.8

0.22

0.66

0.10

0.33

0.88

0.02

5

100.0

33.5

108.86

1.3

1.5

0.22

0.70

0.10

0.21

0.18

0.004

RL

Nh /v = local wedge Reynolds number

a 2u>/y = local wave Reynolds number

m

R

B

a u) 6 Iv = local boundary layer Reynolds number

hi ¥

(2y/u>)1/2
-1/3

= h R

= h R,

-1/2 _

(v/N)

1/2

= a k sin a
m m

FIG. 70. Estimates of boundary layer thickness and
Reynolds numbers, y > c.

143

or breaking along the entire slope. Unlike the subcritical case, with slopes
of 30 degrees, the displacements of dye streaks and neutrally buoyant par-
ticles at station 4 did not show any irregularities or anomalous behavior.
The measurements of the boudary-layer thicknesses at each station are
considered to be as reliable as those given for the subcritical case
(Figure 53).

(3) Velocity and Shear Stress. One-second velocity averages were
again computed from the streak displacements; for the critical case these
values are shown only for one vertical level: that described above for the

measurements of s . This level was chosen since it normally was the

m J

vertical position closest to the slope for which the streak curves remained
coherent over the entire wave cycle during sampling. The dye streaks
usually became quite distorted above this level (probably due to the motion
in the wisps and streamers), and generally the streaks were very faint
adjacent to the slope (probably due to mixing). These velocity measure-
ments for the various stations on the slope are presented in Figure 71;
the experimental conditions at the time these values were determined are
listed in the upper right-hand section of Figure 69.

The values of velocity in Figure 71 are plotted along the vertical
axis in cm/sec, and the successive seconds of time during the wave cycle
are shown along the horizontal. The phase has been adjusted artificially
so at time t = 1 the velocity is nearly zero. The average distance above
the bottom for which these measurements were taken is indicated for each
station. This distance is approximately equal to <5 (Figure 69). A sum-
mary of the maximum and RMS values of velocity is given in Figure 72 for
each station (including the velocity variations on the flat bottom not shown
in Figure 71).

144

velocity , |£c

- 1.0
-0.5

o-^

--0.5

--I.0
1.0

05

o_

^-0.5

-- 1.0
1.0

0.5

Station I

O

G

O

O

-©-

o

~0~

G

G

G o

G

Station 2

G

G

G

G

G

-e-

G

G

G

G O

G © G

Station 3

G

G

G

G

G

--05

-■10

- 1.0

- 0 5

G

G

G

G

Station 4

G

0 G G

G

^o — e-

--0.5

-©-

G

G

G G

G

G

~i 1 1 1 1-

3 4 5 6 7

time , sec
FIG. 71

1 1 1 1 1 1 T"

8 9 10 II 12 13 14

145

The last column of Figure 72 gives rough estimates of the maximum
bed shear stress,

, I _ u max

t max =* a J—L-

1 o1 r 6

where ~ju = 0. 01 gm /cm-sec, that were computed at the level of the inflec-
tion point (rf = 6 ) described above. This kind of estimate assumes a
laminar velocity distribution that remains essentially linear with distance
above the bottom over the wave cycle. This assumption is not accurate
for two reasons: (1) the actual profiles of velocity are probably not linear,
and (2) the observed dissipation of the dye streaks below 6 indicates pos-
sible turbulence in this region. The estimates of It are intended

& ' o ' max

only as approximate lower bounds on the local maximum bed shear stress;
the actual values are probably larger due to a nonlinear velocity distribu-
tion with superimposed turbulent fluctuations near the bed.

Fig. 72. Summary of velocity and bed shear stress

values for critical case (y^ c) measurements.

Station
(see Fig.

11)

1 u 1 max
cm/sec

URMS
cm/sec

rolmax
dynes /cm 2
(XlO+2)

cm
<5m

1

0.65

0.38

1.80

0.36

2

0.68

0.44

1.74

0.39

3

0.72

0.45

1.76

0.41

4

0.35

0.25

0.85

0.41

5 0.18 0.13 0.64 0.28

The velocity measurements show that the magnitude of the maximum
oscillatory motion at the selected level of measurement is nearly uniform at

the middle and upper slope stations. The value of lul . on the lower slope

rc r ■ ' max

(station 4) is smaller than that farther upslope (stations 2, 3, and 4) by about

146

a factor of 2. The increase in maximum velocity between the flat bottom
and station 1 is approximately a factor of 4. This is considerably lower
than that found between similar locations for the subcritical-case measure-
ments shown in Figure 59. The measured bed shear stress (as defined
above) at station 1 is greater by only a factor of 3 than that for the flat
bottom. A striking feature in the results is the smooth sinusoidal appear-
ance of the velocity- time curves in Figure 71. The apparent lack of irreg-
ularities in these curves suggests that in spite of the vortex- streamer
production, the basic harmonic motion of the input waves is still quite
recognizable.

The Lagrangian velocity measurements for the supercritical case
are represented by the results in Figures 73a and 73b. These velocity-
depth profiles are analogous to those shown in Figures 54 through 58 for
y <c. Only the results at two stations (1 and 4) are shown here to illustrate
the regularity of the motion. The absence of the vortices and thin streamers
for this frequency (go < co ) was also accompanied by a return to an apparent
periodic motion of the dye streaks and particles near the bottom. No wave
breaking was observed near the corner. The maximum and RMS velocity
values that were taken from selected levels in the velocity -depth curves at
each station are listed in Figure 74. The maximum value of bed shear
stress is also shown in the last column of this figure.

The velocity curves at all stations are quite symmetrical and do not
indicate any measureable net motion; the measured net displacement of
neutrally buoyant particles over several wave periods corroborates this

finding. It is also evident that u at the selected levels increases

" ' 'max

only slightly from station 4 to station 2, and then decreases at station 1,
which is farthest upslope. This behavior contrasts sharply with the large
increase in |u| at station 1 for the high-frequency waves (y <c, case A),

The estimated bed shear stress at the slope locations increases by a factor
of 4 or 5 from that over the flat bottom. In this case, 5 in Equation (4 -7a)
was chosen as the smallest t\ for which the dye streak data were considered
to be reliable (see first value of rf for each station in Figure 74).

147

u

to

A

o

<-»

„

t-i

c

04

o

o

4-1

cd

4-J

U)

■U

^_

«J

6

CO

<D

-—i

•H

<4-l

O

u

>

a

x

4-J

CL

a)

X)

6

i

i

5>i

u

•H

O

O

CM

i— 1

O

<u

i

>

ctf

«

ro

t-

O

i

O

M

fa

148

b

ro
d

CVJ

d

o

i

CO

d
i

ro
d

d

I

o
A

c
o

4-i

td

4-i
0)

4-i
Cfl

CO

4-1
O
S-t

a

4->

a
a>

i

4J
•H
O
O
r-H

at
>

,0
CO

O

149

FIG. 74. Velocity and bed shear stress at selected levels

t max

1 o1

Station
No.

V
cm

1 'max
cm/sec

Iu'rms

cm/sec

(xlO2)
dynes/cm

1

0.11

0.18

0.08

1.64

0.26

0.26

0.14

0.38

0.29

0.17

0.45

0.28

0.17

2

0.16

0.32

0.22

2.00

0.30

0.44

0.30

0.45

0.48

0.33

0.66

0.50

0.33

3

0.13

0.20

0.13

1.54

0.24

0.30

0.20

0.38

0.35

0.24

0.53

0.36

0.24

4

0.13

0.25

0.17

1.92

0.24

0.35

0.23

0.37

0.38

0.25

0.51

0.35

0.23

5

0.12

0.05

0.03

0.42

0.21

0.07

0.04

0.32

0.11

0.06

0.46

0.11

0.07

150

DISCUSSION

The experimental results can be summarized briefly. The measure-
ments of the wave motion in the subcritical range y <c indicate that the
linear theoretical solutions (Wunsch, 1969) accurately represent the data
for the higher frequency waves in the interior and in the boundary layer
over the slope for e less than approximately 0. 5. The data for the lower
frequency waves in the subcritical range do not agree with these solutions
as well as the data for the higher frequency waves. For both of these wave
types, no significant reflections were measured; the wave energy dissipates
primarily in a narrow upslope zone. The higher frequency waves break
turbulently in this zone, where mixing generates laminae of various thick-
nesses that protrude back into the interior. By contrast, the lower fre-
quency waves form surges that dissipate in runup near the corner. Large
intensification in the velocities parallel to the slope were measured for all
waves in this range.

The critical case, y ~ c, is characterized by less violent breaking
near the corner. Regularly spaced line vortices form along the slope dur-
ing the upslope motion of the water particles. The mixing in these vortices
generates thin streamers of fluid that penetrate horizontally back into the
interior over most of the depth. Intensification of the near-bottom oscillatory
velocities was also evident; however, the flow pattern in the vortices com-
plicated the kinematics near the bottom.

In the supercritical case, y > c, no breaking was seen near the
corner. The vortices and streamers persisted until the wave frequency
reached low values, below which the vortex production subsided. The
motion in the boundary layer was very regular in the absence of the insta-
bilities. In the fluid interior the measurements show that the wave motion
intensifies near the critical characteristics, whose position is given by
z = -ex.

151

5. SEDIMENT MOVEMENT BY INTERNAL WAVES

GENERAL DISCUSSION

This chapter attempts to assess the role of shoaling internal gravity
waves in the initiation of bottom -sediment movement in the ocean, in the
light of the theoretical and experimental results discussed in the previous
chapters. Although the concept of internal waves as an effective geological
agent in the ocean is not new, very few published studies have dealt speci-
fically with the erosion, transportation, or deposition of sediments on sub-
marine slopes. Chapter 1 has provided some historical background on this
subject. Several explanations for the apparent lack of attention are possible:
(1) our knowledge of the mechanics of sediment movement beneath the more
familiar oceanic surface waves is still incomplete; (2) the wide spatial
separation of available data on bottom sediment types on the continental
margins hampers understanding of local sedimentary processes; (3) reli-
able measurements of oceanic internal waves, particularly those taken in
conjunction with a sediment- sampling program, are few; (4) until very
recently, theoretical and experimental studies of internal waves passing
over a bottom with variable depth were virtually nonexistent. In spite of
these problems, there have been inferences to the potential importance of
this process (Lafond, 1962; Hulsemann, 1968). There exists a body of
internal-wave measurements (see Chapter 1) which indicate that internal
waves propagate shoreward normal to the bottom contours. (Wunsch, 1969,
has also shown theoretically that these waves can refract, in the sense that
their crests tend to align parallel to the bottom contours.) These measure-
ments, together with the results of the previous chapters, promote the fol-
lowing discussion about the interaction of these waves with bottom sediment.

gfc

Defant (1961) gives two illustrations of the experimental work of Zeilon
(done in 1934) in which interfacial waves incident onto a model shelf- slope
configuration were studied.

152

The first section in the following discussion presents various
examples of internal-wave measurements in the ocean and discusses quali-
tatively some features of sediment distribution on the continental margins,
particularly off the northeastern United States. The purpose of this section
is to provide some background material on ocean conditions relevant to the
later discussion and to demonstrate the wide range of values of the various
hydrodynamic and sediment properties. The second section develops a
criterion that describes the initiation of bottom -sediment motion. This
criterion is basically an extension of the analysis developed by Eagleson,
et al (1958) for incipient motion of discrete bottom- sediment particles in-
duced by surface waves. The analysis for shoaling internal waves over a
constant bottom slope is then specialized for the case of small slopes
(y <<c)> which is shown to be particularly appropriate for high-frequency
internal waves in the ocean. Another criterion (the Shields curve, with a
later extension by Vanoni, 1964), which represents the more traditional
approach to the problem of incipient motion of bottom sediment and is based
primarily on dimensional analysis and empirical results, is also discussed.
This latter criterion is shown to be approximately equivalent to the first.

The analysis is then applied to specific oceanic measurements, and
the results are discussed in a final section, in light of the various limita-
tions. Qualitative remarks concerning the effect on sediment movement by
breaking internal waves and critical slope conditions (y^c) are also given
in the last section. Throughout the subsequent discussion, consideration of
the interaction between internal waves and sediment particles is only an
initial attempt to evaluate the possibilities of incipient motion induced by
this interaction; the actual importance, or even existence, of this process
under real conditions must await future field studies.

Oceanic Conditions

Chapter 1 provides several references to actual measurements of
internal wave motion in the ocean, including studies of the directional

153

properties of these waves (for example, Summers and Emery, 1963). In

most cases the measurements were made either by recording fluctuations

of temperature or horizontal velocity at fixed positions from a moored buoy

or anchored ship or by sampling the vertical distribution of temperature

with devices like the bathythermograph. Since the frequency content of the

internal wave field is not known a priori, a common means of distinguishing

the contributions to the measured changes by the various frequency compo-

*
nents is the energy spectrum. Figure 75 illustrates kinetic energy spectra

at several levels computed from horizontal velocity data taken at site TT>
(location shown in Figure 76) and described by Fofonoff (1969). The large
values of spectral energy density at the inertial and tidal frequencies are
typical of this location. Frequencies and amplitudes of vertical tempera-
ture fluctuations have been measured by others (Chapter 1). Figure 77
demonstrates the wide range of amplitudes and frequencies represented in
previous measurements of internal waves in the ocean. The measurements
by Gaul (1961) and Ufford (1947) taken over the outer continental shelf and
continental slope suggest that wave amplitudes of 1 meter to 5 meters are
not unreasonable for the higher frequency internal waves.

*

Time series of measurements of vertical velocity or temperature fluctua-
tions can be used to compute frequency spectra of potential energy. Mea-
surements of horizontal velocities like those taken at side MD" (location
given in the following text) are the basis for kinetic energy spectra.
Fofonoff (1969) discussed the spectral ratio of potential energy to kinetic
energy in terms of a linear internal wave model and found that this ratio is
approximately unity for low frequencies (near inertial frequency), but ap-
proaches infinity for co -* N. He noted that this latter behavior is probably
caused by the exclusion of the advective terms in the linear model. Voor-
his (1968) measured vertical internal motions near site "D" with neutrally
buoyant floats. After comparing the frequency spectra of potential energy
and vertical kinetic energy (obtained from the measurements) with the spec-
tra of horizontal kinetic energy computed from the site "D" records,
Voorhis found that there is an approximate equipartition of potential and
kinetic energies for frequencies between inertial and Brunt- Vaisala. He
interpreted this equipartitioning to indicate the predominance of internal
wave motion in the measurements.

154

10

SITE "D" (39'20'N, 70* W)
2143 1004m 0CC.1966 (Temp.)

2204
2203

51 1 m
101 Jrn

#*•.

Feb, i96r
reb.i9S7

4500

(velocity)
(velocity)

cm /sec

CO2

2204

I

1

!:| jfs 2 20 3

2143

10

-2

10

-4

10*2 10(

FREQUENCY ( CyC^ S /hour )

Fig. 75. Frequency spectra of horizontal kinetic energy
density at site "D".

155

T3
C
(0

-H

tr>
C

w

0)

c

<D

-p

o
to

4-1
M-l
O

-P

<D

6
>i

■C

-P

id

CQ

V£>

0>
•H
Cm

156

FIGURE 77
INTERNAL WAVE MEASUREMENTS ON CONTINENTAL MARGINS

Source

Water Depth
(meters)

Wave Height
(meters)

Wave Period

Celerity*
(knots)

1.

Lafond
(1963)

20

7 (max.)

7 min. to
12 hr.

0.11-0.6

2.

Boston
(1964)

20, 33

8

12 hr.

3.

Gaul
(1961)

62

1 - 3

10 min.

0.5 - 1.5

4.

Ufford
(1947)

40, 100

1 - 30

9 min. -
2hr.

0.08-0.68

5.

Emery
(1956)

1350

130 - 200

12 hr.

(standing)

6.

Lee
(1961)

20

3 - 7

5-15 min.

7.

Summers and

Emery

(1963)

10 - 1500

30

12 hr.

1 (shallow)
7 (deep)

*In all cases, the dominant direction of propagation was shoreward.

157

In order to describe internal wave motions in the ocean, an estimate
of the vertical distribution of Brunt-Vaisala frequency is needed. The aver-
age vertical distribution of Brunt-Vaisala frequency for locations seaward
of the continental slope (near site "D", for example) is typically character-
ized by relatively large values of N in the seasonal thermocline, if present,
and in the deeper main thermocline. Figure 78 shows measured values of
Brunt-Vaisala frequency N versus depth for stations near site "D"; the hydro-
graphic data were provided by Volkmann (personal communication). An
idealized distribution of Brunt-Vaisala frequency with depth is also shown in
this figure. The idealized case consists of three distinct vertical layers; in
each layer the density gradient is assumed to be constant (i. e., N ^ constant).
This hypothetical vertical distribution of N is later used with the idealized
topography also shown in Figure 78 as a basis for various models of sedi-
ment movement induced by internal waves.

The generalized topographic cross-section in Figure 78 is not con-
strued to be representative of any particular area, but is taken as an
idealized case in which the slopes are linear and the boundaries between
the three provinces (shelf, slope and rise) are well defined. Each idealized
spatial province in this figure is similar to the experimental conditions
described in previous chapters in the sense that N and y are constant in
each.

The actual bottom topography of the various provinces comprising
the continental margins of the oceans not only varies markedly with geo-
graphic position but also can be quite complicated locally. Featureless
and even bottom slopes are probably rare. In his summary of the continental
shelves and slopes of the world, Shepard(1963) remarks that it is exceptional
to find a shelf that has a constant slope out to the shelf edge; constancy of
slope usually implies widely spaced sample points. The values for bottom
slopes shown in Figure 78 are recommended averages taken from Shepard
(1963) and Heezen, et al (1959). More detailed descriptions of the physiog-
raphy, structure, and sedimentology of the continental margin off the east

158

i

1

o rtf

^ o O

00 O "-H

r- C C

C P

+J • «J g

(fl &ig E

i-3 C ^ 0

53

0 iH U

g J 0

fQ

0 > -H

(D

^ «■— ' fd

N

4-J S C

■H

5 0

^— »

H

id - - 03

c

fd

-P c r- u

0

0)

KJ . . Q)

•H

T3

T3 <H "^ D- 4-)

•H

- o

o

o

o

o

o

o

o

o

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o

1

fO

1

1

i

10

1

i

00

1

1

o o
o o
o o

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in

c

c

-H

-p

3

Xi

•H

u

+J

W

•H

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T5

0)

N

•H

H

fd

<L

>C

•H

"C

C

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•

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•

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r

»

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£

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w

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<u

w

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(0

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0

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c

<u

0

3

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w

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159

coast of the United States is presented by Uchupi (1963, 1968), Emery (1966)
Pratt (1968), Schlee (unpublished manuscript), and Emery, e_t aj. (1969).

At this point it is instructive to compare average values of the
oceanic slopes y with hypothetical estimates of the slopes of the internal
wave characteristics c in order to determine which case (y <c, y^ c, or
y >c) in Figure 1, if any, is predominant. Figure 79 summarizes the
relative values of c and y for typical values of Brunt- Vaisala frequency N,
wave frequency a>, and a constant value of Coriolis frequency (f = 0.042 cph).
The results show that the continental shelves are approximately critical
(i. e., y a c) for internal wave frequencies near inertial for each of the values
of N shown; for frequencies greater than the frequency of the semi-diurnal
tide u) — , the shelf is subcritical.

The assumption that the average gradient of the continental shelf is
small compared to the slope of characteristics c for normally incident
internal waves (i. e., y <<c) appears to be reasonable for high-frequency
internal waves. For example, if co>0. 1 cph and N > 1. 0 cph, orifa)>0.5
cph and N = 6. 0 cph, then y <<c is approximately valid according to
Figure 79. The average gradient of the continental slope might be critical
for a relatively wide range of internal wave frequencies depending on the
local values of N and f. For example, in Figure 79 for N in the range
considered (0. 5 cph <N <6. 0 cph), the range of critical frequencies is
approximately 0. 045 cph < co <0. 03 cph; this range includes the semi-
diurnal tide. The average gradient of the continental rise is approximately
critical for frequencies between inertial and tidal (semi-diurnal) for
0.5 cph < N < 3.0 cph.

It is tempting to speculate about the consequences of y > c for
oceanic conditions; this is done later and only touched upon here. The ex-
perimental results show that for internal wave frequencies equal to or
slightly less than critical, vortex instabilities form along the slope. If
this occurs along submarine slopes, the mixing in these vortices, with
consequent formation of thin streamers that might advect fluid horizontally

160

.001

Characteristic slope c (ordinate) computed for various
frequencies w and selected Brunt Vaisala frequencies /V
Average oceanic slopes Y are also shown.

161

back into the interior, might provide a means of stirring the fluid near the
bottom and moving the very fine suspended matter seaward. This agitation
might inhibit or prevent deposition of fine suspended matter, and thus main-
tain a relatively turbid and well-mixed fluid layer near the bottom. How-
ever, it is dangerous to extrapolate the laboratory observations to unobserved
natural phenomena.

The intention of the above discussion of average gradients of the
oceanic slopes was to present the gross features of these slopes in relation
to an idealized hydrodynamic structure (Figure 79); local bottom features
such as animal markings, sediment ripples, depressions, and rock outcrops
were intentionally left out of consideration. It is realized that these additional
features might locally affect the internal wave motion by causing complicated
reflection patterns of the internal wave characteristics (Longuet-Higgins,
1969) and by influencing the flow in the boundary layer.

The actual scales of bottom roughness on oceanic slopes are quite
variable. Spatial variations of these scales have been illustrated by several
excellent photographic and echo-sounding studies of submarine topography.
A series of camera stations spaced two miles apart on a line extending south
of Martha's Vineyard and lying approximately normal to the bottom contours
was described by Northrup (1951). Sediment samples were taken at most of
the camera stations. Northrup noted that there was good evidence for the
action of bottom currents on the outer shelf. Others (for example, Uchupi,
1963) have described topographic features such as sand waves with heights
of 5 to 25 meters on the continental shelf. Owen and Emery (1967) cited
photographic evidence for intensive erosion of the continental slope south of
of Martha's Vineyard by strong bottom currents. They found several un-
usual features in their photographs including small masses of sediment that
protruded a centimeter or so above the general level of the bottom; in this
instance they described the overall appearance as "sand-blasted". Bowin,
et al (1967) made an extensive photographic and sediment survey of the
western and southern slopes of Plantagenet Bank (near Bermuda) in water

162

depths of 60 meters to 2000 meters. The height scales of the various sand
ripples that are clearly shown in some of their stereo-photographs appear
to be on the order of 10 cm. (The ripples were observed at various depths,
but especially between 950 and 1200 meters.)

The effects these local variations in bottom roughness actually have
on the internal wave motion have not been determined. In fact, what role
internal waves might have in forming or maintaining features such as ripples
or sand waves is uncertain, and must await field studies of this problem.
The analysis and discussion in the subsequent sections deals only with the
simpler problem of a linearly sloping bottom, mainly for reasons of analy-
tical tractability. It is probably true that the wide variations in the hydro-
dynamic and topographic features of the ocean include this case; its consider-
ation might provide some insight into more complex situations.

A general discussion of the distribution of sediments on the various
continental shelves and slopes is given by Shepard (1963). For the purposes
of this discussion, only a brief summary of the sediment types in the area
shown in Figure 76 will be given. Uchupi (1963) and Schlee (unpublished
manuscript) presented detailed descriptions of the bottom sediments in
this region. In his report Schlee noted that although the sediments on the
shelf are dominantly quartzose sands that are moderately sorted, many of
the sediment samples from the continental shelf (about 20 percent) showed
sizable silt-clay fractions. Schlee also noted that there is some evidence
for a seaward reversal in the median sand sizes (i. e., fine to coarse in a
seaward direction) at limited sections of the shelf break southeast of Cape
Cod. This seaward reversal has been found by others.

Several studies have indicated that the sediments on the outer shelf
shown in Figure 76 are predominantly Tertiary, and that modern sediments
are either lacking or the result of reworking of the older material (Donahue
et al 1966). The dynamics of the by-passing of the shelf by much of the silt
and clay of modern sediments, particularly on that part of the shelf below
surface wave base, are not well known. Emery (1966) states that waves

163

and currents on the continental shelf produce enough motion to cause much
of the silt and clay of modern sediments to travel across the outer shelf
and come to rest in deeper waters. It is suggested later in this chapter
that near-bottom velocities induced by internal waves might be partly re-
sponsible for the sediment transport in this region.

Pratt (1968) described the continental slope off the eastern coast of the
United States as a complex feature whose surfac e is generally more irregular on
the upper parts . In the area under consideration the m edian diameter of the sur -
face sediments is in the silt range; quartzose sands are locally predominant on the
upper slope (Schlee, unpublished manuscript). The continental rise has been de-
scribed as a depositional apron sloping seaward from depths of about 2000 m to
about 5000m at a gradient of between 1:100 and 1:700 (Heezen, etal 1959). The
surface is covered with fine well-sorted sands and coarse silts interbeddedwith
clays. Emery (1966) suggests that these sediments were deposited by turbidity
currents; however, Heezen, etal (1966) maintain that ocean bottom currents
flowing parallel to the contours are responsible for most, if not all, continental

rise sedimentation.

Actual sediment data that are later compared with the analytical

results were obtained from the work of Hollister (personal communication),
Stetson (1939), Pratt (1968), and Emery and Ross (1968). The geographic
locations of the sampling sites are shown in Figure 76. These results are
particularly useful, since the sample locations are on lines extending off-
shore near site "D". The hydrodynamic data at and near site "D" provide
convenient inputs to the models considered in a later section in which
incipient- motion criteria are discussed for the various sediment data.

Before proceeding to the analysis of sediment movement by internal
waves, some consideration of the boundary layer along oceanic slopes would
be worthwhile. Studies of sediment movement in rivers and beneath surface
waves have indicated the importance of the flow characteristics (laminar or
turbulent) in this layer in determining the motion of bottom sediment. Linear
solutions due to Wunsch (1969) for the boundary layer beneath internal waves
on a sloping bottom were discussed in Chapter 4. Wunsch (1970) also

164

considered theoretically oceanic boundary mixing for quasi- steady conditions
in a stratified fluid adjacent to a sloping boundary, and showed that a mean
vertical velocity is induced at the sloping boundary to satisfy the no-flux
condition. He found that the effect is confined to a boundary layer of thick-
ness

, vl/4ml/2

8 = {v K) ' /N '

where

v is the eddy momentum coefficient

k is the eddy heat coefficient

N is the Brunt-Vaisala frequency, as before.

He also pointed out that the various estimates of the mixing coefficients

2

v , k in the ocean are unreliable, and showed that for v , k ~ 1 cm /sec,

-3 -1
and N = 2 X 10 sec , then 6 ~ 20 cm.

Attempts to define v for various oceanic processes are abundant

(Defant, 1961); however, near a sediment- strewn bottom that is disturbed

by periodic and quasi-steady currents, such a definition is elusive. Based

on measurements of suspended matter near the bottom of the continental

slope east of Chesapeake Bay (Ewing and Thorndike, 1965), Ichiye (1966)

computed an eddy diffusivity from a linearized version of the Fickian dif-

2
fusion equation. He arrived at a figure of 0. 12 cm /sec for the eddy

viscosity. Others have shown photographically that the bottom along the
submarine slopes can be quite turbid; high concentrations of suspended mat-
ter in this near-bottom zone might increase both the viscosity and the den-
sity of the fluid. By contrast, many photographs show remarkable clarity
and suggest low concentrations of suspended material near the bottom. It
therefore appears that the local conditions along oceanic boundaries are
sufficiently variable to preclude a single, all-inclusive value of v • The
above discussion suggests that many processes are operative along the
oceanic slopes (currents, animal stirrings, etc.); internal waves are only

165

one possible kind. The actual viscosity near the bottom must be the result
of the composite effects of the processes.

INCIPIENT MOTION CRITERIA

The various approaches to the formulation of analytical criteria
that adequately describe the initiation of motion of bottom sediment due to
water waves are limited owing to the complexity of the process. Several
authors (Inman, 1963; Raudkivi, 1967; Abou-Seida, 1965) have summarized
these approaches and discussed some of the limitations of each method.

Prior to the development of an analytical model, it might be helpful
to present a dimensional analysis of the problem of initiation of sediment
motion induced by internal waves, in order to show the large number of
independent variables and dimensionless groupings of these variables that
are possible. The specific problem is to relate functionally the incipient
diameter of the movable sediment particles to the pertinent hydrodynamic,
geometric, and sediment variables. It is assumed that internal gravity
waves of a given mode number in a continuously stratified fluid propagate
over a bottom slope that has a sedimentary bed. The density stratification
is approximately linear and stable, and the bottom slope is constant. The
bed is assumed to consist of noncohesive particles with mean diameter D ;
the roughness is arbitrarily described by a length scale k . Figure 80
shows the idealized geometry and particle relationships; this figure is
similar to that given by Eagleson and Dean (1959).

Incipient diameter is the largest size within a local size-frequency distri-
bution of particles that can be placed into motion by the action of the hydro -
dynamic and gravitational forces. The condition of incipient motion induced
by surface waves on beaches has been described as follows. At some point
on the offshore slope, the instantaneous hydrodynamic forces may become
sufficiently large to cause static instability of bed- sediment particles of a
given size. The diameter of this size is referred to as the incipient diam-
eter (Johnson and Eagleson, 1966).

166

V

°r

FIG. 80. Definition sketch of sediment particles; forces and
geometry.

symbol

name

definition

point of
application

F
rG

submerged

3

ttD /6 p g (s -sf)

center of

weight

gravity - buoyancy

mass

2,2

F
F

fluid

CF pf ttD /8 uf

J.

resistance

dynamic pressure and

top

VM

pressure
force

virtual mass
force

viscous shear acting
over surface of particle

ttD /6 p. a
r — o

instantaneous pressure center
gradient force present in mass
fluid whether or not
particle is present. Equal
to the inertial force of
the displaced fluid.

CM ttD /6 pf af

force required to accel- top
erate fluid due to
presence of particle

of

167

Eleven independent variables are selected a priori as being important
in determining the incipient diameter D. for this problem. The functional re-
lationship can then be expressed as

D. = f(a, u), N, n, Pf, v, h, r, Dm, Pg, K) (5-1)

(The symbols are defined in the front section of this paper.) By dimensional
analysis the functional relationship can be expressed equivalently in terms of
eight independent dimensionless groups:

VDm= f(a/h' W/N' n' ^c' awDm/y ' aw/[(s-DgDm]1/2,
V) (2) (3) (4) (5) (6)

(5-2)

^[(s-Dg/Dj1/2, DmA)

(7) (8) '

where s = s /s„.
s/ f

The first four expressions on the right in Equation (5-2) relate to the
*
local wave field ; the remaining groups relate either to the wave-sediment

interaction or to the sediment properties alone. Dimensionless groups (5)
through (8) resemble those developed by Carstens, et al (1969) in his analy-
sis of incipient motion due to surface waves. The large number of dimension-
less groups indicates the complexity of the process and the potential difficulties
in developing accurate dynamic models of this process. Several other vari-
ables have purposefully been omitted; these include measures of sorting and
packing. These factors would increase the complexity of the process.

If the problem is simplified to consideration of the balance of forces
acting on a single, protruding bed particle such as shown in Figure 80 , an

*
a/h is proportional to the local Stokes parameter e , since k =* cmr /k and

e = ak (Chapter 4).

168

expression can be derived for the incipient diameter in terms of the local,
wave- induced velocity field in the viscous boundary layer, certain resistance
coefficients, bottom slope, and sediment properties. Furthermore it is
postulated that incipient conditions hold when the sum of all hydrodynamic
and gravitational moments acting on the particle at any instant is equal to
zero. Such a condition can be represented (similar to Eagleson and Dean,
1959) by

Y ! M = 0 = (F sin ^ sin 0 ± FF cos |3 (1 + cos <f>) /5„v

± FyM (1 + cos (f>) ± Fp cos <f> - FQ sin (</> +a)) ^

Brief definitions of each force and its assumed point of application
for this model are given in Figure 80. The points of application follow from
the discussion in Ippen and Eagleson (1.955) and Eagleson and Dean 0959).
Ippen and Eagleson (1 955) give a more detailed discussion of the various
forces and their representation. The fluid resistance can be rewritten in
terms of components normal to and parallel to the bed; these components
are commonly referred to as lift and drag, respectively (Schlichting, 1955):

FL = Fp sin 0 = CL Pf ttD2/8 uf2

FD = FF cos 0 = CD Pf ttD2/8 uf2

To obtain an expression for the incipient diameter from Equation
(5-3), the following assumptions are made.

(1) The magnitude of the acceleration is small compared to the
magnitude of the effective velocity (|a |, |af| << |uf |). If the advective
accelerations are neglected (this is a linearized model), then the instanta-
neous accelerations are approximately equal to u> times the instantaneous
velocity. Their neglect is reasonable for internal waves in the ocean (say
o)^ 10 sec ).

169

(2) The lift coefficient is approximately equal to the drag coefficient.
The actual relationship used in this model (CT « 0. 85 C„) was obtained from

Li U

Chepil (1958). It is recognized that this relationship was found for particles
in a turbulent boundary layer; its applicability to laminar conditions has not
been verified.

Applying assumptions (1) and (2) above and the lift and drag expres-
sions for the fluid resistance in Equation (5-3) and solving for D, the incipi-
ent diameter is expressible as

u2C
3 f n

Di-f-V5,!1-! (5"4)

where

r =

± (1 + cos <f>) + 0.85 sin <f>
(sg/sf - 1) sin (0 Tff)

The sign convention necessitates the use of absolute value for r; if u. is
directed upslope the choice of the bottom signs would give a negative value
of r. The value of D. represents the largest diameter that can be moved
for particular sediment and internal-wave conditions.

The quantity r is dependent on<£, s /sf, and a- Since emphasis is

S J

placed here on noncohesive particles in sea water, the choice of s -2. 65
(quartz) and sf = 1.027 appears reasonable. The density variation in the
water column over the continental margins off the northeastern United
States is < 1 percent. The angle of repose <£ for both well-rounded and
very angular grains of sizes > 0. 3 mm was shown graphically by Albertson,
et al (1963). There is strong evidence from the nature of the curves for
rounded and angular grains of small diameters (< 0.3 mm) that <£ approaches
30 degrees (± 3 degrees). Eagleson, et al (1958) showed that for the idealized
situation similar to Figure 80 this angle can be derived geometrically:

tan ^ = s : t-/9 (5-5)

[(D/Kr + DA- 1/3] l/l

170

If the diameters of the bed grains are approximately equal to the equivalent

diameters of the roughness (D/k m 1), tan § = 0. 53 or </> <* 28 degrees. If

these values of s , sr, and 6 are used and a is assumed to be small, such
s f

that a << <j), then | r | =* 2. 01 for upslope motion and | r | =* 3. 08 for down-
slope motion.

Consequently, on small slopes {a << </>),

Uf CD
D. = 2.31 (downslope incipient motion) (5-6a)

Uf CD
D. = 1.51 (upslope incipient motion). (5-6b)

The problem now remains to obtain a value uf in Equation (5-6a) or
(5 -6b) that is the maximum velocity which effectively acts at the uppermost
point of the grain surface (77" = D.). The boundary-layer velocity at this
point can be expressed as the real part of Equation (4-24)

and

uf = 4^ [{F sine1 - G sine2} - e"V6

X {F sin^ + D./5) - G sin (0 + D^)}]

/ 2v» \1/2

2 M2 . 2

\a; -N sin of/

(5-7)

(5-7a)

where

4 cnA ' c - y

61 = qjfn(By) + wt , e2 = qen(Pf) + wt

For the case of small slopes in the sense y << c, Wunsch (1969) has shown
that q « — — . Furthermore, for distances well away from the corner
(cx>> |z |) the coefficient A can be written in terms of the input wave
motion

171

a co
o

2k

also k =

CriTT

(5-7b)

o

where a and k are the amplitude and wave number of the internal waves
oo

over the flat bottom, respectively. This expression is true for an input
area whose top and bottom are horizontal and rigid. If the condition that
the vertical velocity of the wave motion at the top and bottom of the input
channel is approximately equal to zero, then Equation (5-7b) is approximately
true for an input channel with nonrigid upper and lower boundaries. It is
important to note that these approximations of q and A imply a smooth
transition in flow conditions from the input region to that over the slope
(i. e., in two dimensions this implies a smooth matching of stream lines at
the transition between the two regions).

For conditions of small slope (y << c) and a smooth bottom transi-
tion, several simplifications are possible in Equation (5-7). For small
slope, y « a « sina and for high enough wave frequencies, co/N>> sin ex.
In addition,

M a c:

N =* c

c'

C

so that

u„ <* -

a omTT

-D./6
sin03 - e smie^ + D^/b)

(5-8)

where

0„ = q£n(c£) + cot ;

6 - (^)
CO

V2

Since k = -r — and h
o h
o

— y X

r

£ for small y,

then

a co
uf " cTh/h0)

-D./5
sine,, - e sin (0„ + 0^/5)

(5-9)

with n = 1.

172

After a short amount of manipulation it can be shown that the maximum
value of the terms inside the brackets on the right side of Equation (5-9)

can be approximated as ^2 D./6 to order (D./6) . For waves of 1-hour

-3 l l -2 2

period, co =* 1.7 x 10 rad/sec, and for y ^ 10 cm /sec, 6 ^ 10 cm.

Typical sediment diameters on the outer shelf are on the order of 10 cm,

indicating that the above approximation is reasonable to about one percent.

Since for incipient motion conditions it is the maximum value of the velocity

at "rj = D. that is the important quantity, then from Equation (5-9)

a a)

|u,| a ?u/u c \2 D./6 (5-10)

1 f'max c(h/h ) * v

for the following assumptions:

(1) small slopes such that y << c and a <<<£;

(2) D.<<6.

It is also worth recalling here that the results of the laboratory experiments
described in the earlier chapters for the subcritical case (y < c) suggest that,
at least for the conditions tested, the velocity field in the boundary layer was
adequately described by the linear theoretical solutions along the slope to the
point of breaking. For any particular case in which y <<c is not valid, but
y <c still holds, the complete expression for u (Equation (5-7)) must be
used in Equation (5-4). However, it has been shown (Figure 79) that y <<c
is reasonable for several average oceanic conditions.

If Equation (5-10) is applied to Equation (5-6a), then

D. = 0.216
l

c (h/hQ)

n2

aQu)

2
g 5 (downslope incipient /,. --»

C~ motion)

This equation shows that in order to determine D. some estimate of boundary
layer thickness s (or equivalently, viscosity y) must be made. Owing to the
difficulties in obtaining oceanic measurements of this quantity, this choice is
not simple. However, a lower bound can be estimated: the value of -5 obtained

173

from Equation (5-7a) with v - y,, where y, is the kinematic viscosity

-9 9

(y, w 10 cm /sec). The variation in y, for sea water is shown by

Higgins (1962); typically y, varies from about 0.008 cm /sec at 30°C to

2

about 0.015 cm /sec at 0°C (salinity about 35%o). Although coefficients

of viscosity have been suggested by other authors to describe mixing
processes that have scales larger than molecular, the current state of
knowledge of the oceanic boundary layer induced by internal waves does not
justify an a priori assumption of y >vk- The problem of selecting a repre-
sentative value of y was discussed at the end of the previous section in this
chapter. The basis for other choices of viscosity here lies in an "intuitive
feeling" that natural oceanic boundary layers are turbulent. The actual
viscosity is probably a result of several processes at work, including in-
ternal waves. It is worth recalling, however, that the experimental runs
with linear, smooth slopes showed no appreciable net transport of fluid in
the boundary layer outside of the breaking zone for y <c. In the discussion
of net Lagrangian motion in Chapter 4, this observed absence of net trans-
port was interpreted to mean that there was no measurable mixing induced
by the internal waves along the slope up to positions near the breaking zone.
The turbulent dissipation of the high-frequency waves for y <c in the break-
ing zone and the mixing induced by the vortex instability for y> c are two
possible instances for which a choice of y >y, might be made, at least for
the laboratory conditions.

The other apparent difficulty in the direct application of Equation
(5-6a) or Equation (5-6b) to oceanic conditions is the choice of the drag co-
efficient C . Considerable experimental evidence shows that Cn is largely
dependent on a suitable Reynolds number R-. for solid bodies immersed in a
moving fluid (Prandtl, 1952; Batchelor, 1967). Usually the relationship be-
tween Cn and Rn is given for a steady flow past a smooth and regular body
such as a sphere in an unbounded fluid. In the case of an irregularly shaped
particle resting on a rough bed in an oscillatory flow, the relationship be-
tween C and Rn is not well known. Eagleson et al (1958) showed

174

measurements from which the drag coefficients for spheres rolling down a
smooth, inclined boundary immersed in water or oil were evaluated from
the relation

CD pf— Vs = FG Sln a

where v is the measured translational velocity of the sphere of diameter D.
Their results are shown in Figure 81. Line A in this diagram is the con-
ventional curve for C~ °c r for spherical particles moving in an un-
bounded fluid (Batchelor, 1967). Eagleson et al (1958) later extended these
results experimentally to account for the presence of a rough bottom.
Figure 82 shows that for constant values of particle diameter D, decreasing
roughness size k (i.e., increasing ratio D/k) causes a reduction in the drag
coefficient determined for a smooth bottom from Figure 81. In their studies
of incipient motion of bed particles due to surface waves, Eagleson and
Dean (1959) chose CD = 19. 2/RD with D/k =* 1 and RD = uf D/y. On the
basis of a large number of experiments in which C~ was computed from an
equation similar to Equation (5-4) for various measured values of uf, D.,
<£, a, and s /sf, Eagleson and Dean (1959) concluded that there was no
qualitative disagreement between the values of Cn obtained in this way with
those values obtained from Figures 81 and 82. It is not obvious, however,
that their results show conclusively that the two methods of computing C
are equivalent for incipient motion of the particles. On the other hand,
their results are quite convincing that the two methods are in agreement
for discrete particles that are already in motion.

In view of the apparent importance of the local particle Reynolds
number Rn on the value of C , it is useful to establish some upper bound

on R for the oceanic problem. Sediments on the outer shelf have median

-2

diameters on the order of 10 cm. Lacking specific measurements of

175

io

O

m

C^

O

m

cr.

•-{

c

fd

<u

*

D

O

T3

C

tC

C

0

w

V

ro

.H

O

(T.

(0

K

U

<D

-P

4-1

<\J

<d

O

in

<D

V-i

0)

x:

Q

g

Q
>

CO

0

ii ^

cr c

u

en

4J

C

~~~

Q)

•H

o

-H

4-1

14-1

<D

i

0

o

O

0)

U

C

id

4->

U)

00

•H

W

b

*o

ro

CsJ

O

O

C\Jw

U_°

>

<|C\J

Q^

ii

a

C

J>

00

M

Cm

176

1.0-

08-

0.6-

smooth

R = const

0.4-

0.2-

o.i-

.08"
.06-

incipient motion

.04

T
2

4

6

8 10

D/k

FIG. 82. Effect of bed-particle geometry on the resistance

coefficients for spheres (after Eagleson and Dean, 1959)

177

internal-wave velocities in the bottom boundary layer, a value of 100 cm/sec

*
is selected as a crude upper estimate.

-2 2

If v = vk * 10 cm /sec, then RD a 100. For larger values of y

or smaller uf, RD becomes less than 100. This upper bound suggests that,
according to Figure 81, CD » 192/RD for a hydraulically smooth bottom.
If the additonal assumption that D/k ~ 1 is made, then C = 192/R
(Figure 82). If this value of CD is applied to Equation (5-11) with

D

u D.

Rn = and v ^ 8 co/2, then

D/*=31-36iSTE7y (5"12)

for incipient downslope motion on small slopes.

It is obvious from Equation (5-12) - and not unexpected - that the
local incipient diameter is a function of the boundary layer thickness (or
alternatively, the viscosity v) as well as the local depth and input wave
conditions. The value of D. increases linearly with decreasing depth, sug-
gesting progressive sorting of sediment. Equation (5-12) predicts that sed-
iment particles having diameters smaller than D. will initially move down-
slope so that there should be a trend toward smaller mean diameters in the
seaward direction. Since the model does not describe the subsequent history
of bottom particles once they are initially moved from their roughness hol-
lows, it is not possible from this analysis to trace their net motion after
initial movement. However, one might speculate that once the particles
are set into motion they might be advected to different locations as bed load

*A few values of instantaneous near-bottom velocities on the continental
margins are available. For example, Emery and Ross (1968) reported a
maximum value of 70 cm/sec on the continental slope south of Martha's
Vineyard; the flow direction was parallel to the bottom contours.

178

if there is net motion of the surrounding fluid. The net motion of the fluid
need not depend on mixing induced by internal waves, but can be caused by
other currents. It is emphasized that this analysis pertains to initiation of
motion of noncohesive bed particles for the conditions assumed; subsequent
motion as established bed load or suspended load is only speculative.

The inverse relationship between the ratios D./6 and h/h can be
conveniently illustrated for different choices of a , N, and oj; Figure 8?
is an example of this relationship. Each straight line in the diagram
represents a particular input wave amplitude a for internal waves that

o

are assumed to be normally incident to the slope. The larger values of a
are permitted only for h large in the sense a /h << 1. (The linear in-
ternal wave theory used in this analysis is valid only for small -amplitude
waves.) Finite-amplitude effects have not been considered explicitly. Con-
sequently, the results are valid upslope to a point where the Stokes param-
eter e becomes order one; the exact value of e for this limiting condition
is not precisely defined. A discussion of e for the small-slope approxi-
mation is given in Chapter 4. The experiments show that for y < c, the
high-frequency waves induce a zone of breaking for values of e > 0. 5. A
value of e > 0. 5 might be used to establish a limiting depth ratio h/h for
this model; i. e., at depth ratios shallower than h/h the linear analysis is
invalid. If a = mh where m << 1 and positive, then from Equation (4-14)

h/hQ = (f ctt)1/2 (5-13)

where

mode n = 1.

For example, for relatively high-frequency, first-mode internal waves, such

that

that

that w* N/2 and a = 0.01 h (m = 0.01), it follows from Equation (5-13)

/ -2\1/2

179

o.ooi-

0.000

0.00001

QQ- 0.002

FIG. 83. Ratio of incipient diameter D. to boundary-layer

thickness S for various positions on the slope, qiven
by h/h , and for selected amplitudes a.f of :he

o

incident internal waves.

180

If e = 0. 1, h/hQ * 0.4; if e = 0.5, h/hQ a 0.2. Figure 83, shows the
limiting depth ratios derived in these examples.

Before discussing the results of this model in terms of oceanic
conditions it is instructive to show that the form of Equation (5-12) is in
some sense compatible with the familiar Shields parameter S,

S " (*sVl>Vs (5"14)

where D is equivalent to D. and r is the shear stress necessary to pro-

S 1 c

duce initial motion of noncohesive bed particles in a steady flow. Shields
(1936) presented experimental results relating this parameter to a partic-
ular Reynolds number. Others (for example, Vanoni, 1964) provided sub-
sequent experimental results that established a single curve relating this
Shields parameter S to the particle Reynolds number R*n, where

R* = S, withu* =\Jr/pf

D v

The diagram in Figure 84 shows the classical Shields curve with some
later experimental results plotted on it; the Shields parameter is the ordi-
nate, and the particle Reynolds number is the abscissa. Two basic prob-
lems arise in the application of this curve to the present problem: (1)
the experimental measurements represented by the curve were made for
boundary layers that were fully turbulent; and (2) in most cases the mean
flow conditions were steady (channel flow). White (1940) derived a similar
semi-empirical result for the critical stress at the bed for steady flow
condit ions and a laminar boundary layer:

S = 0.18 tan 0 (5-15)

where S is given by Equation (5-14). According to White, tan <j> is about
unity which gives S « 0. 18, as compared to about 0.03 to 0.06 for S ob-
tained by Shields (Figure 84). White found, however, that the value of S

181

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</-

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WHITE)

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/ o! : i 1 1 1 !

i

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I

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<J |tS x ^ ln ' " (." (.^ ^ -.- tl .",

T

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■

0

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o o

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TURBULENT
VELOCITY PROFILE

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182

for turbulent flow in the boundary layer was about one-half that for laminar
conditions. The remaining discrepancies in the results of White for turbu-
lent flow with those of Shields were attributed to the more intense turbulence
in the flows with fully developed velocity profiles which Shields used (Vanoni,
1964). The difference between the two results according to Vanoni was that
for both turbulent and laminar flow conditions in the boundary layer, White
found that t was apparently independent of R*n- By contrast, Figure 84
shows that t is dependent on R* to values of R* near 10 where the
Shields parameter approaches a constant value of about 0.06.

The mean conditions of steady flow that are implicit in Figure 84
are not unreasonable for low-frequency internal wave phenomena, since
the advective accelerations are small, as was assumed in the earlier model.
The compatibility of the earlier model with White's equation for laminar
flow conditions (Equation (5-15)) can be demonstrated simply. If the bed
shear stress is written as

t = n au/a^T

and if au/a rj" is approximated by a linear fit:

au/aii" ~ Au/6 at au>/c6

1/2
where Au * aco/c, then for 6 = (2y/u)) ; (i. e., small slopes) and ju = v pf

the bed shear stress can be rewritten approximately

o

t = ypf act)/c6)=(6/2)aco /c . (5-16)

On small slopes it was shown earlier that

a = a h /h (correct to first order in e ). (5-17)

Combining Equations (5-16) and (5-17) it follows that

2

183

Applying this expression for bed shear stress into White's result for
incipient motion in a laminar boundary layer (Equation (5-15)) an expres-
sion is obtained for D./s:

2

aoa>

V6 = K gc(hAo)

(5-19)
K = 2. 78

" (Pg - Pf) tan 0

The functional form of this result is equivalent to Equation (5-12). A closer
comparison between this equation and Equation (5-12) is not justified because:
(1) Equation (5-17) does not include a Reynolds number dependence; (2)
Equation (5-19) assumes a linear velocity profile in the boundary layer; and
(3) Equation (5-19) does not specify the point of application of the effective
velocity acting on the bed grains.

A brief summary of the preceding analysis might be helpful here.
The foregoing model deals with conditions of incipient motion of noncohesive
bed particles induced by internal gravity waves shoaling over a small linear
slope. The static equilibrium of these particles on the bed was defined in
terms of an incipient diameter D.. The analysis predicts that the net in-
stantaneous hydrodynamic and gravitational moments acting on the particle
about the contact points with other bed particles will cause those particles
with diameters less than D. to move downslope initially. Equation (5-12)
relates D. to the internal wave parameters (a , qj, c), a local depth ratio
(h/h ), and the boundary layer thickness (s ). It is also shown that Equa-
tion (5-12) is compatible with the results of Shields (1936) and White (1940).

MODEL ANALYSIS AND SPECULATION

As defined above, the criterion in Equation (5-12) essentially
defines the condition of neutral equilibrium set up by instantaneous hydro-
dynamic forces of internal waves and that of gravity acting on discrete bed
particles that are resting on a rough bed. A different dynamic balance is

184

needed to define a condition of equilibrium for particles that have already
been placed in motion. This balance must depend on the net transport cur-
rents influencing the particle motion, including any net currents that might
be induced by the internal waves, and the downslope pull of gravity. How-
ever, as was indicated in Chapter 4, steady transport of water particles
induced by internal waves, near a sloping boundary in a stably stratified
fluid, other than that caused by diffusion, theoretically cannot occur in a
plane normal to the depth contours unless other processes of mixing are
active. Consequently, it is not possible to predict a type of oscillating
equilibrium for onshore-offshore motion of sedimentary particles for which
D< D.. In the case of surface waves, Ippen and Eagleson (1955) and Eagle-
son and Dean (1959) have shown that a balance between the forces on a par-
ticle due to the mass transport current in the bottom boundary layer, directed
upslope, and the downslope component of gravity creates a situation where
sediment particles of increasing size in the onshore direction are in oscil-
lating equilibrium. The diameter of the local equilibrium size is often
referred to as the "null" or "equilibrium" diameter D . The location of
this type of equilibrium for a particular sediment size is called the "null"
point. Miller and Zeigler (1964) and Eagleson and Johnson (1966) discuss
sediment sorting by surface waves and give some comparisons with field
data. The only available direct evidence for net fluid transport induced by
internal waves on a slope are the laboratory results that were presented in
Chapter 4.

In view of the absence of other evidence for the existence of mass
transport by shoaling internal waves, only the effects of the incipient condi-
tion which was derived in the preceding section are considered here. Curves
similar to that shown in Figure 83 can be used to determine the hypothetical
distribution of D. along a linear bottom slope if 6 is known. Suppose that
the initial size-frequency distribution of noncohesive bed particles is given
by the solid curve in Figure 85 at each position along the slope. After the
internal waves have acted to produce incipient motion of sediment having

185

Frequency

nit ia I

distribution

sma

Sediment size

large

FIG. 85. Hypothetical size-frequency distribution of
sediments on an oceanic slope.

186

diameters less than D. at each depth according to Equation (5-12), the final
shape of the size-frequency distribution at each location might eventually
have a spike at D. and appear as shown by the broken line in Figure 85.
The sediment size at which the spike occurs decreases in the offshore
direction.

It is unreasonable to require this simple analysis, with Equation
(5-12) as its tidy result, to explain the complicated sedimentary pattern
found on the continental margins. Instead, it is instructive to postulate
various simple models to determine at what locations and for which hydro-
dynamic conditions, if any, the predicted values of D. according to Equa-
tion (5-12) might exceed the observed natural sediment diameters.

Models are constructed for three provinces defined by the idealized
bottom geometry and the simplified distribution of Brunt-Vaisala frequency
shown in Figure 78. Each layer (i. e., the three fluid layers intersecting
the shelf A, slope B, and rise C) with constant N is assumed to support
small-amplitude, single- frequency internal waves of the first mode that
propagate shoreward at approximately normal incidence to the slopes.
Furthermore, only examples that satisfy the condition y << c were con-
sidered, except for the hypothetical instance of waves with semi-diurnal
frequency over the continental shelf and continental rise. In these cases,
it is shown that the ratio y/c is on the order of one-tenth. Values of the
pertinent hydrodynamic parameters are listed on each diagram. In addi-
tion, since the depth range over which the analysis is valid is limited to
small values of e (as discussed in the initial section of this chapter), the
depth ratios h/h at which e = 0. 1 and e = 0.5 are indicated on each diagram.
It should be recalled that the experiments for y < c showed breaking of in-
ternal waves at e > 0. 5. On this basis, it was suggested that the linear
solutions might be invalid if local values of e exceed 0.5. The sources
for the sediment data are listed in each diagram; the approximate locations
of the sampling areas for each source are shown in Figure 76.

187

Figure 86 treats a continental- shelf model with relatively high-
frequency internal waves (co m N/2, N = 6 cph); the pertinent assumptions
in the model are specified in the figure. The ordinate is mean diameter
Dm (for actual sediment data) or incipient diameter D. (computed from
Equation (5-12)); the horizontal axis is the depth ratio h/h . As mentioned
above, values of h/hQ computed from Equation (5-13) for two selected values
of e and for the particular wave conditions (a , w, c) of the model are

marked long the horizontal axis. The variations of D. with h/h are shown

i ' o

as theoretical stright-line curves for selected values of H:
H =-2-

A >/2
o k

where a is input wave amplitude in meters,

v is viscosity in cm /sec,

A = 1 m,
o

-2 2

v, = 10 cm /sec.

(Ao„k1/2 = 10 cm2 • sec"1/2.)

For example, the curve in Figure 86 labeled H = 5 represents

1/2 2 -1/2

a v =50 cm • sec ' , or if v - v^, then a = 5 meters. This kind

of log-log plot simplifies presentation of the analytical results; the theoreti-
cal curves are straight lines (Equation (5-12)), and D. can readily be obtained

for various combinations of a and v (or 5). The field data of mean sedi-

o

ment diameters in Figure 86 show a general trend toward decreasing sizes
in an offshore direction. This trend reverses for h/h > 0.4 in Stetson's
data and for h/h > 0. 5 in Hollister's data.

Interpretation of the relationship between the actual mean values D

and the theoretical distributions of D. was described earlier and will be

1

summarized here. D. represents the maximum sediment diameter that is in

188

h/ho

Model pqrometers

N = 6 cph

u) - 3 cph

c = 0 577

f = 0.062 cph

A0 = I m

uK - I0~2 cm2 /sec

H =

a0 v

1/2

W2

u,2-f2

-i 1/2

LN2-a,2

Sediment data
O Ho Mister
A Stetson # 4

FIG. 86. Continental shelf model; relatively high frequency
waves .

189

neutral equilibrium at slope positions given by h/h for hypothetical internal-
wave types and idealized bottom conditions. The analytical model predicts

initiation of motion of bed particles in a downslope direction if D < D .

m ^ i

In Figure 86 incipient motion of measured sizes D is predicted for

H > 50 (for example, a = 5m,!/ = 1 cm /sec) and h/h > 0.48 (for the

cut-off condition imposed by e = 0. 5). If H = 5 (a = 5m, v= v-J) and if

O K.

e = 0. 5 is the limiting value for the linear analysis, then all values of D

m

from Stetson's data for 0. 25 < h/h < 0. 45 could be placed into motion by
the waves. It might be noted here that the nonlinear effects which are sig-
nificant farther upslope (where e is order one, say) could make the case for
possible sediment movement even stronger. For instance, the experiments
for y < c showed that the measured velocities in the boundary layer near
the zone of breaking were significantly larger than corresponding measure-
ments taken farther downslope. Consequently, one might expect that the
presence of nonlinearities in the velocity field might increase the values of
D. for positions well upslope. The consideration here of a strictly linear
analysis is in this sense a "worst" case: if initiation of motion of measured
diameters is predictable from the linear model, then the inclusion of non-
linear effects strengthens the argument. We might also recall that values of
v >v* are included mainly on the supposition that the oceanic boundary -layer
dynamics might involve an eddy viscosity.

Figure 87 is another shelf model similar to Figure 86 except that
the internal-wave frequency is assumed to be semi-diurnal. In this model,
the requirement y/c <<1 is necessarily relaxed to y/c<l (y/c u 0.01).
The case for incipient motion of the measured sediment sizes is not as con-
vincing as that given in Figure 86. The theoretical curves indicate that D

is less than D. for relatively small- amplitude waves (a < 5m) only if

2 1
v > 1 cm /sec (see H = 50, for example). The values of D^ along theoreti-
cal curves H = 1 and H = 5 are below the mean diameters; this suggests that
waves of 5-meter amplitudes with v = v, will not move the measured mean
sizes.

190

Model parameters
N = 6 cph

H =

aQ v

cu= 0. 083 cph

c = 0.01

f = 0.062 cph

c =

~u>2- f2~

1/2

Ao= 1 m

_N2-u,2

i/R = I0~2 cm2/sec

h/ho

Sediment data
O Hollister

A Stetson # 4

FIG. R7. Continental shelf model ; wave of tidal frequency
(semi-diurnal) .

191

The continental slope model in Figure 88 assumes relatively high-
frequency waves (a = N/2; N = 3 cph) because critical conditions (r * c)
are approximately achieved along the slope for waves of lower frequencies,
such as the frequency of the semi-diurnal tide. The data shown in this
figure were obtained from various sources; the lines of sample locations are
shown in Figure 76. Initial movement is predicted for sediment along the

lower extremity of the slope (h/h > 0. 8) for H = 10. For H = 50 (a = 5m,

2

v = 1 cm /sec), initiation of sediment motion is possible for each of the

measured values along the slope.

A smaller number of data points are shown in the continental rise
model in Figure 89. The available values of mean diameter tend to be in
the range of fine to very fine silt; the ordinate (D. or D ) has been scaled
accordingly. It is worth recalling the earlier discussion of sedimentary
types on the continental rise. Several previous workers (Heezen, et al
1966, for example) noted that clean quartzose sands and coarse silts inter-
bedded with the clays can be locally dominant. These larger sizes are not
represented in the data plotted in Figure 89. It is obvious from Pratt's
data, however, that the mean diameters found on the lower parts of the rise
are generally larger than farther upslope. The computed incipient diameters
exceed the mean diameters for H = 5 (a = 5m, y - yA on the upper rise.

O K.

Initial movement of the larger measured sizes on the lower rise requires
larger wave amplitudes (or larger viscosity), such that H > 50. The effects
of cohesive forces acting on the smaller sizes (very fine silts and clays)
have not been considered; these forces might inhibit initial movement and
thereby require larger values of H for incipient motion.

The results of this section can be summarized briefly. It has been

shown that, within the constraints of the models, the predicted values of

incipient diameters D. can exceed the values of mean diameters D mea-
r i m

sured at various positions on the continental margin southeast of New England.
The condition for incipient motion of bed particles induced by internal waves
shoaling over a small slope whose average gradient is linear is given by

192

Model parometers

N= 3 cph
oj= 1.5 cph
c = 0.5768
f = 0.062 cph

Ao= I m

v*- I0-2 cm2/sec

H =

QoV

C

1/2

AoV*

1/2

C =

a,2-f2

nl/2

LN2- W2J

e

h/ho

Sed

iment dc

ita

0

Stetson

# 5

□

S t etson

# 6

A

Pratt

O

Emery

and Ross

(averac,

e of 3

samples )

FIG. 88. Continental slope model; relatively high-frequency
waves .

193

\

\s

\
\

■

\

Di

\ \H=50

■

\ \

\

\ \

\

\H=I0 \

\

\
\

\ \
\

\ \

\

\ \

\

\ H=5 \

- 0.1

\
\

\ \

o \

-

0 \

©\ 0 \

.

\ H=l \ ©

\

-

\ \

\ \

\

\

_ 0.05

© \ \

\

-

\ \

\

-

\

\ \

symbol

h/ho

€

a0 , m

\
\
\
\

a

0.0 1 1

0.5

I

b

0.025

0. 1

I

c

0.025

0.5

5

d

0.057

0- 1

5

\

\

0

05

1 , ■ ,

01

1

0.5 .

1

1

b,c

Model parameters

N =

1 cph

a> =

0.083

cph

c • =

0.056

f =

0.062

cph

A0=

Im

H =

c =

a0z/

1/2

A0i/,

1/2

h/ho

Sediment data
© Pratf

i/ = 10 2cm2 /sec

a;2-f2

-l 1/2

N2-oj2

FIG. 89. Continental rise model; waves of tidal frequency
(semi-diurnal) .

194

Equation (5-12). It is obvious from Equation (5-12) that larger values of
wave amplitude and viscosity produce larger incipient diameters at specified
values of h/h . Incipient motion of the measured sediment is predicted for
specific instances in the models considered above:

(1) on the outer continental shelf (h > 50 m, say) with N = 6 cph,
to = 3 cph, and H > 5;

(2) on the outer continental shelf with N = 6 cph, semi-diurnal
frequencies, and H > 50;

(3) on the lower continental slope (h/h > 0. 8), for N = 3 cph,

a; = 1. 5 cph and H = 10, or on the middle sections of the continental slope
withH= 50;

(4) on the upper and middle sections of the continental rise for
N = 1 cph, oj = 0. 083 cph, and H > 5, or on the lower rise with H > 50.
The internal wave amplitudes suggested in the discussion of the various
models are representative in that they fall within the range measured in
the ocean. The various proposed values of viscosity v (or boundary layer
thicknesses 6) are speculative.

The available data for specific sediment distributions on the conti-
nental margins do not correspond in a simple way to the predicted distri-
butions; the natural setting is too complex to fit these idealized models.
However, what is not answerable at this time is whether or not the internal
waves do in fact initiate motion of bottom sediment: for certain conditions,
the models predict that they can. It must be noted that an additional attri-
bute of the sediments on the slope and rise that is probably important in
increasing the local forces necessary to dislodge the grains is the existence
of cohesive forces acting between particles of clay and very fine silt sizes.
The incipient-diameter criterion in Equation (5-12) assumed noncohesive
bed particles. Other factors, such as bed roughness scales greater than
granular, stirring by bottom organisms, bed fluidization, grain angularity,
and packing of bed grains, are probably significant locally in the transport

195

of bed particles. Their effects are difficult to treat analytically; future
field studies might establish their actual importance.

In the light of the experimental results presented above, the effects
of two other phenomena induced by internal waves must be considered.
Breaking internal waves of high frequency were observed to produce turbu-
lent conditions and increased instantaneous velocities near the bottom. The
breaking often generated filimentous horizontal layers which penetrated the
adjacent flu id interior. Secondly, experimental runs with y> c showed a
vortex instability along the slope; the mixing in the vortices caused a ver-
tical structure of alternating horizontal streamers and layers within which
the net fluid transport was away from and toward the slope, respectively.
If these processes occur in nature (there are no definitive field measure-
ments of either process) speculation about their possible impact for sedi-
ment processes on the continental margins raises some interesting questions.

Wherever the breaking might occur on the slope, sediment that is
well-sorted and coarse relative to nearby material probably will be associ-
ated with this zone. The finer materials might be winnowed out and trans-
ported from the area by currents. Hogg and Wunsch (1970) have shown that
for sufficiently large values of eddy viscosity and slightly oblique incidence,
in theory the breaking internal waves can generate "longslope" currents.
Hence, a littoral drift of fine materials similar to that observed for surface
waves and supplementing the effects of the contour currents (Heezen, et al
1966) might be possible. If the position of breaking along the slope is a
function of time, i. e., if the stratified layer containing the waves shifts its
vertical position over a time scale of days or longer, as seasonal or local

if.

The laboratory measurements suggest that the breaking occurs well up-
slope (for r < c). In fact, when the stratified section was beneath a less
strongly stratified or homogeneous layer, the breaking occurred near or at
the base of this upper layer.

196

thermoclines might, then a time- sequenced winnowing out of fine materials
might occur along the bottom over a wide offshore traverse. Deposition of
recent sediments entering the area from the continental sources might be
prevented, and local size-frequency distributions might be skewed toward
the coarser sizes. A dynamic process like this might help to explain the
observed scarcity of recent sediments on the outer shelf.

Finally, speculation is ventured for the effects of breaking internal
waves at the edge of the continental shelf. Suppose that relatively large-
amplitude, high-frequency internal waves were generated at infrequent
intervals (for example, by the passage of severe surface storms) in the
seasonal thermocline when the upper extent of the thermocline was at the
approximate depth of the shelf edge. Sudden substantial increases in bot-
tom turbidity due to the suspension of bed materials by breaking of these
large-amplitude waves might increase the density of the water along the
shelf-slope transition to values high enough to generate density currents.
This mechanism might provide a means of continuing erosion of the slope
face at irregular times and places by relatively weak small-scale turbidity
currents.

The vortex instabilities and streamers that were created in the lab-
oratory, generally for y > c, might also provide a means of increasing
suspension of bottom materials, by increased turbulence within the eddies
near the bed, and by transporting fine suspended materials away from the
slope. The net transport in the streamers might move the finer sediment
(clay and very fine silt) away from the continental slope to deeper sites of
deposition on the rise or abyssal plain.

None of these mechanisms has been observed in the ocean. The
case for increased mixing along the oceanic slopes is not new; Munk (1966)
and Wunsch (1970) have called attention to this process. It is hoped that
these experiments have illuminated some of the hydrodynamic features of
shoaling internal waves, and strengthened the case for internal waves as
geological agents on the oceanic slopes.

197

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207

REFERENCES
(ERRATA)

American Optical Company, informal table 53. Sodium Chloride, 1969.

Ives, David J. G. , and Janz, George J. , 1961, Reference Electrodes,
Academic Press, New York.

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insulated platinum microelectrodes, Science, 131, 1309-1310.

208

APPENDIX A

The use of conductivity cells in A. C. circuits to detect fluctuations
of conductivity in electrolyte solutions has been briefly summarized by
Gibson and Schwarz (1963). In this section we describe only the particular
sensors and circuitry that were used during the set of experiments discussed
in the text.

DESCRIPTION

The sensor is a single- electrode, platinum conductivity probe whose
construction is shown diagrammatically in Figure 4. The rounded platinum
tip and the L shaped glass housing minimized flow disturbances induced by
the probes. The probe assembly was oriented so that the horizontal portion
of the glass housing containing the probe tip was approximately parallel to
the crests of the internal waves (i.e., perpendicular to the side walls of the
tank). This prevented the wake of the vertical shaft from washing by the
electrode and introducing spurious values. The electrical path was between
this electrode and the solution ground (a large copper strip placed along the
tank bottom beneath the slope). The platinum tip was normally platinized
by a method similar to that described by Wolbarscht et al 0 960): (1) the
probe was immersed in a 1 percent solution of Chloroplatinic acid to which
lead acetate (80 percent by weight) had been added; (2) current from a 10 v
D. C. source in series with a 10, 000 ohm resistor was passed between the
probe (cathode) and a platinum wire (anode) either for about 15 seconds or
until a thin stream of bubbles was emitted from the probe tip. The result
was a thin grayish-black coating on the platinum probe. The effects of
this coating on probe performance are discussed in Ives and Janz (1961);
essentially, the platinization minimizes drift in the output of the sensor.

The probe is the active arm in an A.C. Wheatstone Bridge (Figure
Al). Potential imbalances across the bridge caused by fluctuations of con-
ductivity at the probe are amplified, detected (diode rectification), and

209

recorded by the A-D conversion described in Chapter 2. The high excitation
frequency (60 kc) and low excitation voltage (0. 5 v rms) inhibited polariza-
tion of the electrolyte at the probe tip. This was essential in order to mini-
mize signal drift and capacitance problems.

PERFORMANCE

Four complete circuits like that shown in Figure AT were used. It
was found that the levels of signal output changed by as much as 15 percent
when probes were interchanged, owing to nonuniform it ies and irregular
platinization of the probe tip; this usually necessitated a calibration check
before each experiment. Normally this calibration consisted of recording
output voltage as a function of probe depth before each run. This provided
an updated sensitivity value that could be checked with previous values.
If it was observed that sensitivity changed markedly (> 10 percent) between
runs, the probe tip was checked for fouling and cleaned as necessary. The
cleaning process is like that described by Ives and Johns (1961). Prior to
initial platinization, or when the platinized coating was cleaned off with a
soft tissue, the following procedure was used:

(1) wash in warm concentrated HNO„ for 30 seconds,

(2) wash in distilled water,

(3) polish platinum tip with glass rod,

(4) treat polished electrode with warm 50 percent solution

of Aqua Regia (4 Volumes HO, 3 Volumes HCp , 1 Volume
HNOJ for 15 seconds,

(5) wash in distilled water,

(6) wash in warm HNO„ for 30 seconds,

(7) wash in distilled water,

(8) dry in air,

(9) place in distilled water when not in use.

Normally, the output signal was acceptable up to three hours of continuous
use; however, shortly thereafter degradation of the signal to noise ratio

210

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and erratic output values necessitated cleaning of the probe tip (steps (5)
through (7) above).

The following list summarizes the average performance of the
probes determined from experimental data and calibration tests. The
length units are presented for convenience and are based on:

where

P = P0 (l-0z)

3 = 0. 001 g/cc/cm

(1) sensitivity: 0. 3 volt/0. 001 g/cc (0. 3 v/cm)

(2) resolution: 0. 5 x 10~ g/cc (0. 5 mm)

(3) accuracy: 5 percent of maximum vertical displacement

of probe relative to stratified fluid at rest
for displacements < 0. 5 cm.

(4) linearity: 2 percent of maximum vertical displacement

of probe relative to stratified fluid at rest
for displacements < 0. 5 cm.

(5) drift: variable, depending on quality of platiniza-

tion. (Typically 20 millivolts/hour.)

It might be useful to present functional relationships between some
of the physical variables for sodium chloride solutions that were assembled
during this study. Several sources were used to obtain the relationships
(L.L. Higgins, 1962; International Critical Tables, 1929; American Opti-
cal Company, 1969).

(1) n = 1.3330 + 0.244 (p-1); T = 20°C
(accurate to approximately 1 percent)

(2) S = 1.39 (p-1) x 103; T = 20°C
(accurate to approximately 1 percent)

(3) a = 1.902 (p-1); T = 20°C
(accurate to approximately 5 percent)

212

where

n is index of refraction,

3
p is density on g/cm

S is salinity in °/oo,

cr is electrical conductivity in (ohm-cm)

-1

213

APPENDIX B

214

TABLE Bl

Group N N <jqc u)c Tc hQ a Y

sec" cps sec" cps sec cm deg (tana)

1 0.776 0.123 0.199 0.038 31.51 31.55 14.9 0.266

2 0.990 0.157 0.248 0.039 25.35 31.60 14.5 0.259

3 0.970 0.154 0.456 0.073 13.80 31.80 28.0 0.532

4 0.988 0.156 0.494 0.079 12.72 31.65 30.0 0.577

5 0.945 0.151 0.473 0.075 13.29 33.35 30.0 0.577

6 0.952 0.152 0.235 0.038 26.69 33.45 14.3 0.255

7 0.972 0.154 0.497 0.079 12.7 33.5 30.0 0.577

215

TABLE

B2

Ref. No.

(j)

T

ko.

c

q

cps

sec

cm"1

1-2

0.129*

7.76

0.142

1.426

17.105

2-3

0.129

7.76

0.150

1.522

8.614

3-4

0.109

9.14

0.095

0.962

4.532

4-2

0.102*

9.85

0.084

0.843

9.913

5-5

0.102*

9.85

0.083

0.883

4.020

6-3

0.098*

10.24

0.081

0.822

4.075

7-4

0.098

10.24

0.079

0.793

3.627

8-5

0.098*

10.24

0.080

0.853

3.818

9-6

0.098

10.24

0.080

0.848

10.124

10-1

0.094*

10.67

0.116

1.168

13.546

11-2

0.094

10.67

0.074

0.743

8.625

12-3

0.094*

10.67

0.075

0.764

3.656

13-1

0.094*

10.67

0.116

1.168

13.546

14-4

0.090*

11.13

0.070

0.697

2.897

15-4

0.086

11.64

0.065

0.653

2.506

16-5

0.086

11.64

0.046

0.486

—

17-4

0.082*

12.19

0.061

0.612

2.085

18-5

0.082*

12.19

0.061

0.650

2.227

19-4

0.078*

12.80

0.057

0.573

1.537

20-4

0.072*

13.86

0.051

0.515

--

21-4

0.066*

15.06

0.046

0.462

—

22-5

0.066*

15.06

0.046

0.488

--

23-6

0.062

16.00

0.043

0.453

4.922

24-2

0.062*

16.00

0.042

0.420

4.366

25-1

0.055*

18.28

0.051

0.517

5.518

26-4

0.055*

18.28

0.037

0.373

--

27-5

0.055

18.28

0.037

0.393

--

28-3

0.051

19.69

0.035

0.350

--

29-4

0.051*

19.69

0.034

0.343

--

30-3

0.049

20.51

0.033

0.335

--

31-6

0.047*

21.33

0.031

0.325

2.971

32-6

0.043*

23.26

0.028

0.296

2.407

33-4

0.039*

25.60

0.025

0.256

--

216

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TABLE

B4

Expt.
Group (j)

T

xc

zc xc/xo

K/fh) knA?)

*nfij

fe/a(

cps

sec

cm

cm

ao=0ilcmao=0.Zcmao=0,

1 0.094

10.67

130.6

9.5 1.101

0.716

_ _

2.642

100.6

0.848

1.716

3.016

3.242

80.6

0.680

3.261

3.666

4.474

70.3

0.593

4.488

3.045

5.123

70.9

0.598

4.410

3.038

5.202

55.9

0.471

6.060

4.246

7.996

45.6

0.385

4.791

3.782

3.412

40.6

0.342

2.544

1.412

2.679

0.055

18.28

130.6

1.101

0.704

0.884

100.6

0.848

1.780

0.723

0.867

80.6

0.680

3.650

1.268

1.602

70.3

0.593

5.380

2.712

6.550

70.9

0.598

5.260

--

—

55.9

0.471

8.735

1.633

3.687

45.6

0.385

9.758

0.802

1.357

40.6

0.342

8.678

0.837

1.221

2 0.102

9.85

132.6

6.6 1.085

0.735

_ _

_ _

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102.6

0.840

1.901

--

1.614

1.723

82.6

0.676

4.081

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4.202

3.782

72.6

0.594

6.258

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8.062

11.820

72.6

0.594

6.258

4.410

8.108

11.238

57.6

0.471

12.428

6.083

15.490

14.774

47.6

0.389

19.295

9.378

21.441

25.845

41.6

0.340

23.432

--

25.005

24.242

51.0

0.417

16.479

14.210

--

24.370

43.3

0.354

22.419

10.942

14.205

23.332

37.1

0.304

24.437

10.138

25.256

19.539

32.3

0.264

19.940

6.214

14. 178

10.839

0.062

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132.6

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1.715

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4.354

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5.083

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7.272

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57.9

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42.9

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1.471

26.616

51.0

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21.520

6.122

19.504

43.3

0.355

32.771

12.645

13.248

37.1

0.304

43.212

4.571

12.368

32.3

0.265

47.519

2.082

11.488

221

TABLE B4 (Cont)

Group

00

T

xc

z

c

XCA0

2 2

(Pw/Po) (am/ao)

(an/ao)

(am/ao)

cps

sec

cm

cm

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aQ=0.2cm

ao=0. 4cm

3

0.094

10.67

54.1

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5

0.905

1.427

2.728

3.034

33.8

0.565

8.702

11.192

6.322

25.5

0.426

24.331

23.630

22.024

21.0

0.351

46.975

33.112

71.648

26.3

0.440

21.816

21.608

25.424

21.7

0.363

42.216

42.892

44.450

16.7

0.279

91.863

52.984

60.728

11.6

0.194

175.545*

73.624*

86.540

0.129

7.76

26.3
21.7
16.7
11.6

0.440
0.363
0.279
0.194

17.754
30.243
48.736
26.475

16.136
42.114
56.270
20.924

4

0.098

10.24

48.3
32.3
25.5
20.5

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1

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0.571
0.450
0.362

1.770

6.238

10.014

17.103

0.086

11.64

48.3
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25.5
20.5

0.853
0.571
0.450
0.362

1.877

8.990

22.063

83.440

0.090

11.13

46.8
30.8
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10,

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0.827
0.545
0.425
0.336

2.071

9.424
20.394
57.874

0.078

12.80

46.8
30.8
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0.827
0.545
0.425

2.279
16.582
74.749

19.0

0.336X 1946.208

5

0.098

10.24

47.4
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11.

0

0.427
0.335
0.257
0.207

2.038

4.535

9.121

12.364

1.485

3.221

11.402

15.100

0.082

12.19

47.4
37.2
28.5
23.0

0.427
0.335
0.257
0.207

2.241

6.078

18.656

50.672

1.881

8.563

19.068

37.403

222

TABLE B4 (Cont)

Expt.
Group

00

cps

T

sec

xc

cm

zc
cm

xc/xo

(Pw/Po) (am/ao)

aQ = 0. 1cm

(a
ao

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6

0.062

16. 00

177.9

112.9

82. 9

62.9

9. 5

1. 351
0.861
0. 632
0.480

0.318
1.713
4.777
9.839

2.227
4. 292
7. 008

0.047

21. 33

177.9

112. 9

82. 9

62.9

1. 357
0.861
0.632
0.480

0.318

1.784

5.729

15.146

1.747
4.472
4.090

0.043

23.26

177.9

112. 9

82.9

62. 5

1. 357
0. 861
0. 632
0.477

0.295

1.820

6.260

18.788

1.364
7. 346
6.042

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BIOGRAPHICAL SKETCH

The author was born, August 4, 1940, and raised in Erie, Pennsyl-
vania. After receiving an A. B. degree in geology from Princeton University
in June, 1962, he entered the Naval Officers Candidate School in Newport,
Rhode Island. Following commissioning into the U.S. Navy in December,
1962, he spent three years aboard the U.S.S. Saint Paul where he served
as an engineering and a weapons officer. During this time he married the
former Virginia Louise Beardsley in San Diego, California. He did grad-
uate study in oceanography at the U. S. Naval Postgraduate School in
Monterey, California for one year before transferring to M.I.T. in Septem-
ber 1966.

The author is a Lieutenant on active duty in the U.S. Navy, a
member of the American Geophysical Union, and a member of the Society
of Sigma Xi. He has two children, David Scott (5) and John Daniel (3).

Unclassified

Security Classification

DOCUMENT CONTROL DATA -R&D

(Security classification of title, body of abstract and indexing annotation must be entered when the overall report is clattll

1. ORIGINATING activity (Corporate author)

Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge/ Massachusetts 02139

2a. REPORT SECURITY CLASSIFICATION

Unclassified

2b. GROUP

3. REPORT TITL

EXPERIMENTAL STUDY OF INTERNAL GRAVITY WAVES OVER A SLOPE

4 DESCRIPTIVE NOTES (Type of report and inclusive dates)

Doctoral thesis

5 AUTHORiSi (First name, middle initial, last name)

David A. Cacchione

6. REPOR T D A TE

November 197 0

la, TOTAL NO. OF PAGES

226

76. NO OF RE FS
111

Ba. CONTRACT OR GRANT NO.

Nonr 1841 (74)

b. PRO J EC T NO .

Contract Authority NR 083-157

9a. ORIGINATOR'S REPORT NUMBER(S)

70-6

9fc. OTHER REPORT NOiS) (Any other numbers that may be assigned
this report)

DISTRIBUTION STATEMENT

Distribution of this document is unlimited.

11. SUPPLEMENTARY NOTES

12. SPONSO RING Ml LI T AR Y ACTIVITY

Office of Naval Research

ABSTRACT

Propagation of internal gravity waves over a slope in a fluid with
constant Brunt-Vaisala frequency was studied in a wave tank,
motion in the fluid interior was measured with conductivity probes, and
in the boundary layer, with dye streaks and neutrally buoyant particles
Outside the breaking zone, the amplitude and horizontal wave number of
high-frequency waves increase linearly with decreasing depth; this is
in agreement with existing linear, inviscid solutions. Well upslope, a
zone of breaking or runup is produced by the high-frequency waves,
the wave characteristics are closely parallel to the bottom slope,
upslope propagation of low-frequency waves produces a line of regularly
spaced vortices along the slope, which slowly penetrate into the fluid
interior; mixing in these vortices sets up thin horizontal laminae that
are more nearly homogeneous than adjacent layers.

A criterion for incipient motion of bottom sediment under shoaling
internal waves was developed using available theoretical solutions
velocity in the viscous boundary layer, found to be valid for certain
experimental conditions. The predicted maximum sediment sizes that can
be moved by typical oceanic internal waves are larger than typical
sediment sizes on the continental margin off New England,
breaking and vortex formation, not yet observed in the ocean, might
also be conducive to sediment movement.

DD

FORM

I NO V 61

1473

Unclassified

So i ' :-

Unclassified

Internal gravity waves
incipient movement of sediment

LINK A

Unclassified

Security Classification

Thesis

C15

131803

Cacchione

Experimental study
of internal gravity
waves over a slope.

Thesis

CIS

131803

Cacchione

Experimental study
of internal gravity
waves over a slope.

thesC 15

E,^,mental study of/

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