LONG WAVE STUDY OF MONTEREY BAY

by

Thomas John Lynch

I United States

Naval Postgraduate School

THESIS

LONG

WAVE

STUDY OF

MONTEREY

BAY

-

by

Th(

?mas

John

Lynch

September 1970

TRTs ctocumerrE Kas Been approved "for pubTTc
release and sale; its distribution Is unlimited.

Tl 35743

IBRARY
NAVAL POSTGRADUATE SCHOOD
MONTEREY, CALIF. 93940 .

Long Wave Study of Monterey Bay

by

Thomas John ^Lynch
Lieutenant Commander, United States Navy
B.S. in Commerce, St. Louis University, 1961

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL
September 1970

LIBRARY

HAVAL POSTGRADUATE SCHOOO

KONTEREY, CALIF. 9394Q y

ABSTRACT

Monterey Bay, on the west coast of the United States, is unique in
that it is a large, symmetric, semi-elliptical bay divided north and
south by the deep Monterey Canyon. The effect of the canyon on seiching
within the bay and on long wave oscillations within the bay was studied
by analyzing synchronized wave records at each end of the bay. Power
spectra and cross spectra calculated for five periods selected from six
months continuous data indicate the Monterey Canyon has a profound effect
on the bay's oscillating characteristics. The canyon appears to act as
an impedence barrier dividing the bay into two independent oscillating
basins each having recurring long-period waves which persist during
significant long-wave activity.

TABLE OF CONTENTS

I

I. INTRODUCTION 11

II. INSTRUMENTATION 18

III. ANALYSIS PROCEDURE 23

A. MONTEREY DATA 23

B. SANTA CRUZ DATA 25

C. CROSS SPECTRA CALCULATIONS 30

IV. ANALYSIS OF RESULTS 36

A. IDENTIFICATION OF LONG-WAVES PRESENT 36

B. INTERPRETATION OF INDIVIDUAL SPECTRA 43

C. INTERPRETATION OF CROSS SPECTRA 56

D. COMPARISON WITH WILSON'S MODEL 64

V. ERROR ANALYSIS 75

VI. SUMMARY 77

APPENDIX A - Digitizing Graphical Analog Records Using

the Calma Company Model 480 78

APPENDIX B - Computation of the Fast Fourier Transform

Using the IBM/360 Subroutine RHARM 84

APPENDIX C - Least Squares Curve Fitting . 86

LITERATURE CITED 88

INITIAL DISTRIBUTION LIST . 89

FORM DD 1473 91

f

LIST OF TABLES

Table Page

I Long-Period Waves in Monterey Harbor 15

II Marine Advisors Data for Monterey Harbor 15

III Marine Advisors Data for Santa Cruz Harbor 15

IV Periods of Oscillation from Residuation 16
Analyses

V Periods of Oscillation from Spectral Analysis 17

VI Computed Response Characteristics of Santa Cruz 22
Stilling Well

VII Summary of Significant Spectral Peaks in 42
Monterey Bay

VIII Summary of Significant Cross Spectral Peaks in 58
Monterey Bay

IX Comparison of Results with Wilson's Numerical 72
Model

LIST OF FIGURES

Figure Page

1 Monterey Bay 13

2 The Bubbler Tide Gage 19

3 Least Squares Curve for Santa Cruz and Monterey, 24
6 December 1969

4 Bristol Company Chart 6112 27

5 Geometric Relations for Conversion from 28
Curvilinear Coordinates (R,9) to Rectilinear
Coordinates (Y,t)

6 Linear Interpolation 30

7 Fundamental Longitudinal Oscillation of Monterey 36
Bay

8 Fundamental Transverse Oscillation of Monterey Bay 38

9 Two-step Shelf Topography 40

10 Sea-surface Heights, Santa Cruz/Monterey, 44
0730, 6 November 1969 to 0330, 7 November 1969

11 Sepctral Wave Analysis, 6 November 1969 45

12 Sea-surface Heights, Santa Cruz/Monterey, 46
0130-2130, 3 December 1969

13 Spectral Wave Analysis, 3 December 1969 47

14 Sea-surface Heights, Santa Cruz/Monterey, 49
2348-1848, 5-6 December 1969

15 Spectral Wave Analysis, 6 December 1969 50

16 Sea-surface Heights, Santa Cruz/Monterey, 51
0810-0410, 14-15 January 1970

17 Spectral Wave Analysis, 14 January 1970 52

18 Sea-surface Heights, Santa Cruz /Monterey, 54
0215-2215, 20 January 1970

19 Spectral Wave Analysis, 20 January 1970 55

Figure Page

20 Cross Spectra and Phase Relationships, Monterey/ 59
Santa Cruz, 6 November 1969

21 Cross Spectra and Phase Relationships, Monterey/ 60
Santa Cruz, 3 December 1969

22 Cross Spectra and Phase Relationships, Monterey/ 61
Santa Cruz , 6 December 1969

23 Cross Spectra and Phase Relationships, Monterey/ 62
Santa Cruz, 14 January 1970

24 Cross Spectra and Phase Relationships, Monterey/ 63
Santa Cruz, 20 January 1970

25 Numerical Calculations of Modes of Oscillation 65
of Monterey Bay

26 Numerical Calculations of Modes of Oscillation 66
of Monterey Bay

27 Numerical Calculations of Modes of Oscillation 67
of Monterey Bay

28 Numerical Calculations of Modes of Oscillation 68
of Monterey Bay

29 Profiles of Water-surface Elevation Along Axis 70
of Bay for Four Lowest Modes of Transverse
Oscillation in Monterey Bay

ACKNOWLEDGEMENTS

!
The author wishes to express his sincere appreciation to Dr.

Edward B. Thornton of the Department of Oceanography, Naval Post-
graduate School, under whose direction this paper was written. His
suggestions, recommendations, and farsightedness were extremely
helpful in the successful completion of this project.

In addition the writer is indebted to Mr. Howard Bethel of the
City of Santa Cruz, for the care and maintenance of the Santa Cruz
tide gage and to Mr. Paul Stevens of Fleet Numerical Weather Central
for his assistance in the use of the Calma Company Model 480 digitizer
and in programming the CDC 6500 computer.

to

I. INTRODUCTION

Monterey Bay (Figure 1) lies on the west coast of the United States,
located about sixty nautical miles south of San Francisco. The Bay is
a large semi-elliptical bay which has some topographical features which
are unique. A deep trough of the North Pacific basin, bounded on the
north by the Mendocino Seascarp and on the south by the Murray Sea-
scarp, approaches closer to the coast at Monterey Bay than at any other
point along the North American coastline. As a result, long waves propa-
gated across the Pacific Ocean in an easterly direction may be somewhat
contained. The continental shelf within the bay extends out to approxi-
mately the 600 foot depth contour and is cut by the deep Monterey Canyon
which has a volume of over 50 cubic miles. Monterey Canyon will be seen
to exert significant influences on the oscillating characteristics of
the Bay.

There have been several studies concerned with the effects of long-
period waves in Monterey Bay (a long-period wave for this study is
defined as a wave having a period in excess of 1 minute), but these
studies have not been able to define the causes of the long wave
activity and have left many areas of inquiry. Hudson [1947] proposed
a surge model of Monterey Harbor for the U. S. Army Corps of Engineers.
He used six months data (October 1946 - April 1947) from three elec-
tronically synchronized sea-surface recorders. These results are
tabulated in Table I. The presence of the long-period waves was
observed. However, the cause of the surge mechanism was unknown.

Wilson [1965] conducted a long -wave study of Monterey Bay for
the U. S. Army Corps of Engineers to determine the feasibility of

11

construction of an engineering model of the surge phenomenon that
occurred at various times within Monterey Bay. Statistical data were
collected from sensors located at Monterey Harbor, and Santa Cruz
(Figure 1). Three long-period wave recorders were also installed
within Monterey Harbor and were in continuous operation from October
1963 to April 1964. Two were arranged so that tides and sea-swell
were filtered out, and a third recorded swell approaching the harbor.
Wilson concluded that the surge within Monterey Harbor was not a result
of incoming swell as no correlation was found to exist between long-
period waves within Monterey Harbor and the incoming sea-swell. The
Santa Cruz sensor was designed so that it functioned as a long-wave
recorder. The recorder operated continuously from October 1963 to
February 1964. Wilson's results for Monterey Harbor are summarized
in Tables II and III.

Wilson performed a residuation analyses of different records from
sensors located around the bay in order to determine a local evaluation
of the oscillations within the bay. Residuation analyses is accomplished
by successively subtracting "apparent" periods from the wave record
until a smooth trace remains. The subjectivity of which this method is
accomplished can lead to ficticious results. These results are listed
in Table IV. Table V is a synopsis of spectrum analyses for three days
record of the Monterey sensors. Wilson concluded, based on Tables IV
and V along with calculated modes of oscillation, Figures 24-29, that
Monterey Canyon functions as an impedence between north and south
portions of the bay, and therefore, free oscillations are to a large
extent uncoupled. Wilson further states that the effect of any sharp
discontinuity in submarine topography, equivalent to the edge of a

12

continental shelf, is to serve as a nodal position for any shelf
oscillation to which the shelf is susceptible.

iRobinson [1969] analyzed long-wave activity in Monterey Bay on 23
January and 20 April 1969. Tide recorders were located at Monterey
Wharf #2 and Santa Cruz Municipal Wharf, however, the records were not
synchronized. Thus, cross spectra and phase information was not
obtainable. Robinson concluded that Monterey Canyon affected the
seiching motions within the bay, however, it did not appear to divide
the bay into two independent oscillating basins, as concluded by Wilson
[1965]. Similar periods and amplitudes were found at Monterey and
Santa Cruz concurrently, but correlation between the two locations
could not be computed due to non-synchronization of records.

In the present study simultaneous tidal records from tide gages
located at Monterey Wharf #2 and Santa Cruz Municipal Wharf were
analyzed during periods of significant long-wave activity in Monterey
Bay. Continuous analog records were maintained from 3 November 1969
to 30 April 1970 and five days (6 November, 3 December, 6 December,
14 January, 20 January) were selected for analysis and comparison.

The objectives were to perform an overall long-wave study of
Monterey Bay, and to examine various modes of free oscillation of the
bay to determine if in fact seiching does exist in the bay. This was
accomplished through examination of the individual power spectra at
Monterey and Santa Cruz, computation of the cross spectra and deter-
mination of phase differences of long-period waves present.

14

TABLE I
Long-Period Waves in Monterey Harbor (after Hudson, 1949)

PERIOD

AVERAGE HEIGHT

PERCENTAGE OF

(min)

(ft)

TIME

PRESENT

1-2

0.4

20

2-4

0.5

30

4-15

not given

15

TABLE II
Marine Advisor's Data for Monterey Harbor (after Wilson, 1965)

SENSOR

PERIOD
(min)

AVERAGE HEIGHT
(ft)

PERCENT OF
TIME PRESENT

1
2

1.7-14
1-14

0.1-2.5
0.1-3.0

0-50
0-55

TABLE III
Marine Advisor's Data for Santa Cruz Harbor (after Wilson, 1965)

PERIOD

AVERAGE HEIGHT

PERCENT OF

(min)

(ft)

TIME PRESENT

1-2.3

n

0-18

2.3-4

> 0.2-2.0

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10-14

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17

II. INSTRUMENTATION

I

A. MONTEREY

The data for Monterey was obtained using a standard Coast and
Geodetic Survey automatic tide gage. (Manual of Tide Observations,
1965), located on Monterey Municipal Wharf #2, Monterey Harbor. The
tide gage is maintained daily by NPS personnel.

The Monterey tide gage senses changes in water level by means of
a float/pulley arrangement, which is entirely mechanical. The recording
drum is advanced by a clock mechanism. The drum speed is designed for
1 in/hr , however, the hourly feed was measured to be 1.03 in/hr . The
instrument is time checked daily with time marks being accurate within
+ 5 seconds. The marigram is recorded in rectilinear coordinates on
plain white paper. A stilling well, which is a 12 inch diameter steel
pipe with a 1 inch orfice in the bottom, serves as low pass filter for
the majority of wind waves (periods of one minute and below).

B. SANTA CRUZ

The Santa Cruz data were obtained with a Bristol Model 28 gas-
purging pressure (bubbler), portable tide gage, (Figure 2), located
on Santa Cruz Municipal Wharf. This instrument senses changes in
water level by means of a nitrogen-filled tube which is connected to
a bellows system (Manual of Tide Observation, 1965).

A bubbler orfice chamber was connected to the end of the sensing
tube to reduce sensitivity to short period wave action. There is also
a bellows inlet needle valve which could be throttled to filter add-
itional wave action from the record.

18

(P

®r

r

$*?&*&&**}

©

t

TO

s

*®

— <§)

1. NITROGEN BOTTLE

2. PRESSURE REDUCING VALVE

3. PRESSURE DIFFERENTIAL REGULATOR

4. ADJUSTABLE NEEDLE VALVE

5. TRANSPARENT BUBBLER CHAMBER

6. PULSATION DAMPER

7 STRIP CHART RECORDER WITH TRANSDUCER

8. TELEMETERING PRESSURE TRANSMITTER

9. TUBING

10. BUBBLER ORIFICE CHAMBER

11. STILLING WELL

FIGURE 2 THE BUBBLER TIDE GAGE.

19

The pressure reduction valve is regulated to allow a constant
pressure greater than that due to the maximum head anticipated over
the orfice. The pressure differential regulator together with the
shunt line provide for a constant pressure difference of about 3
pounds per square inch across the flow regulator. Thus, the rate of
flow can be adjusted to a desired constant value and is not dependent
on the tide stage. Flow is through a silicone oil-filled bowl so that
it can be monitored visually. A rate of 30-60 bubbles per minute was
used for this study.

Advantages of this scheme are that the sensitive elements of the
gage need not be designed to withstand the underwater environment; and
fluctuations in barometric pressure are not directly reflected on the
record.

A significant portion of the system capacity is at the submerged
orfice, thus providing some high frequency filtering and protecting
the small diameter supply line and orfice from marine fouling.

A Bristol Company clock recorder provides an analog record. The
record is on a 6 inch (15.2 cm) wide strip chart and is recorded in
curvilinear coordinates. Paper advance is adjustable and 6 in/hr was
utilized. The clock drive has an eight-day spring, and the record is
sufficient for about seven days. Manufacturer's claims are that
hysteresis and/or non-linearity are limited to 1$ of the full-scale
value. For sea water use, the instrument is calibrated to correspond
to a specific gravity of 1.025.

In addition to the above high-frequency filtering mechanisms, a
stilling well was designed and installed to attain the desired
filtering of high frequency and waves [Robinson, 1966]. The well was

20

constructed of a 20 foot section of polyvinyl chloride (pvc) 6 inch

inside diameter pipe. It was capped on 16, 1/4 inch inside diameter

holes drilled in the side. Copper sleeves were inserted to eliminate
I

fouling. The 16 holes provided the capability of increasing the orfice
from a 1/4 inch to a 1 inch diameter opening.

The response characteristics of the well were determined theoretic-
ally for two orfice sizes and three different wave frequencies using
the equation for the rate of water rise in a well [Robinson, 1969] .

dhVdt = 0.6 a/A V2g(h - h )

where ,

a = orfice area
A = well area
0.6 = empirical orfice flow coefficient
g = acceleration due to gravity
h_ = water height outside the well, i.e., the forcing function
h. = water height insdie the well.
The forcing function, hn, was chosen to be a simple sine function
of unit amplitude, and of frequency equal to the wave frequency of

interest. The initial conditions were h. = 0 at t = 0. The results

i

are summarized in Table VI.

The response characteristics for the Monterey stilling well were
not calculated since the orfice area to well ratio is larger than the
Santa Cruz well, providing response characteristics better than those
of the Santa Cruz. A 1/4 inch orfice was used in the Santa Cruz well.

21

TABLE VI

Computed Response Characteristics of Santa Cruz Stilling Well

(from Robinson, 1965)

PERIOD

ORFICE DIAMETER

PHASE IAG

AMPLITUDE

(in.)

(deg)

REDUCTION
(#)

20 sec

0.25

180

95

20 sec

1.00

72

45

60 sec

0.25

75

91

60 sec

1.00

30

5

25 min

0.25

6

1

25 min

1.00

0

0

22

III. ANALYSIS PROCEDURE

A . MONTEREY

The Monterey tidal records were digitized at 100 points per inch
giving a sampling interval of 34.95 seconds. The discrete Fast Fourier
Transform utilized requires 2x2 data points, and for the records
analyzed, 2048 data points were used which gave a record length of
71545 seconds, or approximately 20 hours. In order to remove the tidal
influence and reduce "leakage", the records were high pass filtered by
fitting a least-squares polynomial curve to the raw data and the curve
was then subtracted from each ordinate point (Appendix C). An example
of the fit to sample data points is shown in Figure 3. It was found
that the 6th-degree polynomial provided the best fit for this particular
tidal cycle.

Once the tidal effects were removed, the data is subjected to the
Fast Fourier Transform to obtain the energy density and phase estimates
as a function of frequency. The IBM/360 standard subroutines HARM and
RHARM were used to calculate the Fourier coefficients, from which the
one-dimensional spectra are derived. This method and procedure is
explained in Appendix B.

Subroutine RHARM gives the raw Fourier's coefficients

AQ/2, BQ = 0, A^B^ A2,B2,...A1j/2, BN = 0

which are then combined as

A. iB.

21 + "21 J - 1,2, ...N

to give the one-sided spectral estimate

23

Santa Cruz
6 Dec 69

■u

0)

60
1-4
<U

0)

u

CO

4-1

J-l

CO

to

01
CO

333

Time (10 seconds)

Monterey
6 Dec 69

Time (10 seconds)

Figure 3
Least Squares Curve for Santa Cruz and Monterey, 6 Dec 69

24

V(V*"/)©

where

T = record length (seconds)

, 2
E. = spectral estimate (m -sec)

The associated frequencies are

f n f If-1 p N-l
1 "' ' 2 '" T' 3 " T ••*' N " T

The raw spectral estimates were then smoothed over five frequencies,
giving each new estimate 10-degrees of freedom and increased confidence
in the spectral peaks. Smoothing over five spectral estimates removes
spurious peaks, but does not smear the actual peaks. The 90$> con-
fidence limits for 10 degrees of freedom are interpreted as meaning
that one can state with 90$ accuracy that the actual spectral estimate
is greater than .49E. but less than 1.60E., as the power spectrum is
distributed according to a CHI-SQUARE distribution.

B. SANTA CRUZ

The analysis procedure of the Santa Cruz data was more complex than
the Monterey data due to two considerations:

(1) The Santa Cruz tidal trace is recorded on curvilinear
coordinates, but is accepted as rectilinear coordinates
by the CAIMA 480 digitizer.

(2) The sampling interval of Santa Cruz is 6 seconds which
must be interpolated and matched point for point in time
with the Monterey data, in order to compute the cross
spectra.

25

The unadjusted amplitudes and time increments were obtained from use
of the CALMA 480 digitizer, and computer program CONVERT. The analog
recotd was recorded on Bristol Company chart 6112, shown in Figure 4,
which has curvilinear coordinates (t,R0). The digitizer then recorded
incremental pairs, in the order X and Y (of the cartesian coordinates
of the tidal curves), every .01 inch of stylus travel. It was there-
fore necessary to form the cartesian coordinates by summation within
the computer. These cartesian coordinates were then converted to the
approximately correct T, R0 coordinates by employing geometric
relationships .

The computation of the correct value of t. (time axis) is critical
to the conversion. On the time scale, 0.1 inch of record was equivalent
to a sample interval of 6-seconds. The summation of X, X., representing
the horizontal travel was converted to the t, R9 coordinates by

t. = x. + r - vr2

R - Y.

i

where ,

t. = time in curvilinear coordinates

l

R = radius of curvature for arcs of constant t.

i

Y. = the ordinate on the cartesian scale

l

X. = time scale in equally spaced increments of t on the
cartesian scale.

These geometric relations are shown in Figure 5 where

-1 r~2 2

0. = TAN Y./VR - Y.
l 11

The adjusted heights are then computed as:

Y. = R0

l

where Y. is now the adjusted sea-surface elevation value. Since the

26

«M

vD

U

(0

X!

<f

O

<D

>>

M

c

0

«0

bO

Cu

•r-i

B

fe

o

u

r-l

O

■u

(0

i-i

U

«

27

m

<D

u

3

60
•H

co

CU
•U

d

O
O

u

>-l 4J

ctj •»

CU >H
C ^-'

•H

t— I w

•r-l <U

> *->

M CO

3 C

-0
E J-l

o o
u o

4-1 CJ

c u

O CO
•r-l <u

co c:
cu i— i

> -H
C 4-J
O CJ

CJ CU

o o

M-l 4J
CO

C

o

■U
CO
^
CU

erf

-u

i

CU

o

28

space increment is .01 inch, the adjusted time, t. is

t. - 600 X.
1 l

The adjusted sea surface elevation (with reference to an arbitrary
level) and adjusted times, were not in general equally spaced. There-
fore, an interpolation scheme was used to obtain data points at a
sampling interval of 34.95 seconds in order to match the Monterey data,
point for point (Figure 6) .

A linear interpolation was performed between time increments greater
than 34.95 seconds. The slope of the line is

a gi - Yi-P
^ " (t. - tul)

The horizontal spatial scale for the interpolated values is,

X. = (i-l)A where A = 34.95 sec.

l

The resultant interpolated amplitude is,

A. - tJ (X. - t. .) + Y. .

l At x l-l l-l

Effectively, the computer checks each successive time value until
the desired interval of 34.95 seconds is exceeded, then interpolates
between the value greater than 34.95 seconds and the prior value which
has an increment less than 34.95 seconds. Santa Cruz data were then
aligned with the Monterey data point for point.

The first 2048 points of the Santa Cruz data were handled in the
same manner as the Monterey data. The tidal influences were removed
by the least-squares scheme, and the adjusted data were analyzed in
the same manner as the Monterey data.

29

> t,x

Figure 6
Linear Interpolation

C. CROSS SPECTRA

Given two random, stationary time-series, f (t) and f9(t) the
cross-spectra is defined as:

*12 (a) = h { *12 (T>e"iaTdT

(1)

where ,

T ■ time lag

O = angular frequency
cp12 = the cross-covariance between f7(t) and f_(t) given by

T/2
cp12 (t) = lim f fx(t) f2(t+T) dt (2)

T-«»

■T/2

30

where

T = time lag

T = record length

t = t ime
If the time lag T is replaced by -T and substitution of the dummy
variable 6 = t - t we have,

T/2

cp (-t) = lim f f (6 + t) t (6)d6
T-«> *
-T/2

then, the cross-covariance is an even function such that

cp12 (~t) = cp21 (t) (3)

The cross -spectrum (1) can be written (using Euler's formula)

e"1QT = COS oT-i SIN or
as

f on oo

$12 ^ = 2n { [ ^12^^ C0S 0TdT"i J ^U^7^ SIN 0Td'

[ -OO -00

since cp19(T) is not, in general, an even function, therefore, the cross
spectrum is a complex function of a.

The cross spectrum combines both the cosine and sine transforms of
cp1?(T). From this, if the cross-spectrum can be resolved into odd and
even complnents , it can then be written as,

Too oo •»

$ -(a) = 2~"< [<$ even (t) COS QTdT-i f cp odd (t) SIN oTdr\ (4)

t -00 -00

since the cosine (sine) transform of an odd (even) function is zero.
For convenience we define two functions A(t) and B(t) where

31

A12(T)

B12(T) =

cp12CO

-

cp2i(T)

2

<P12(t)

+

cp21(T)

(5)

If T is replaced by -T in (5), and using (3) gives A J-t) = -A (t)
and B19(-t) = B19(t) where A is an odd function and B even. Since
cp19(T) is resolved as

cp12(T) = A12(t) + B12(t)

we see A and B are the odd and even parts respectively of cp19.
Using this equation, (4) can be written as

(6)

$12(a) = 2^1 / B12(T) C0S OTdT~i / A12(T) SIN ^^

I. -00 -00

The co-spectrum is defined as the real part of (6)

00

C12(Q) » ~ J B12(T) SIN QTdT

while the negative of this imaginary part defines the quadrature-
spectrum

00

Q12(o) - - |^ A12(t) SIN QTdT.
-co

The cross spectrum is then written in complex form

*12(<J) = C12(a) - iQ12(a) (7)

If a time series is correlated with itself the resulting co-
variance function is even. The cosine transform of the auto-covariance
function is the power spectrum of the record. The co-spectrum is the
cosine transform of the even part of the cross-covariance function,
and gives the amount of power due to in-phase (0 ) or out of phase +

32 )

180° components between the two records. The quadrature spectra is
similar, however, the components are out of phase by a constant amount.
|The spectral phase difference e19(a) is defined by

-1 Q12(o)

and is the argument of the complex conjugate of the spectrum.

In this study the Fast Fourier Transform method was used to determine
the power spectra vice the auto-correlation method. This resulted in a
slightly different scheme of calculating the cross-spectra, as described
below.

From subroutine RHARM the Fourier coefficients

A0 Ai Bl \

2~' B0 = °' T> T> ••' 2~' BN = °

were determined for both the Monterey and Santa Cruz data.

From the Fourier coefficients we form the modules for each station,

VJ)-? + V- |FMlfilei (Monterey)

and

iB

Fs(j) = ^ + -21 ■ |Fs|ele2 (Santa Cruz),

or

A. iB.

^ = -J. _

Fs(j) = Fg(j)e fc2 = ■£■ ^- (complex conjugate)

where j = 0,1,2,. . .N.

Then the cross-correlation function,

2N-1 . . _

(p12(T) = S F (j) FM(j)e1J0lT

j=l

where g. = 2rr/N dt

33

and the cross spectra

T/2

•l2(j)- VJ> VJ> " k / cp12(T)e-1JVdT

-T/2

form a Fourier integral pair.

The cross-spectra $ ( j ) can be expanded as

$12(j) = Vj) FS(j) = l*M<J)le ^V^'6 2
- |FM(j)||Fs(j)|ei(ere2)

where

e = TAN"1 Qty (Monterey)

J

e2 = TAN"1 (-T*). (Santa Cruz)

By use of Euler's formula,

|A|e10 = |A| (cos 0 + i sin 0)
we have, expanding $,?(j),

*12(j) = |FM(j)||Fs(j)| cos (£l - e2)

+ i|FM(j)||Fs(j)| SIN (Gl - g2),

or, $-_ - [co-spectra (in phase)] + i [quad-spectra (out of phase)].

For the co-spectra, if g. - £j, COS (0) = 1, and we have maximum

value for in phase. If £, = g„ + rr/2 in the quad spectra, SIN tt/2 = 1

and we have maximum out of phase. If we arbitrarily let e9 = 0 then

sin r, ^„»^
fcl QUAD

tan e, = = arrr

fcl cos G CO

and the phase difference,

34

From this, the co-spectra, quad-spectra, cross-spectra and phase
difference was computed for each spectral component.

35

IV. ANALYSIS RESULTS

I
A. IDENTIFICATION OF LONG WAVES PRESENT

Spectral analysis of ten tidal records indicated that the long-wave
activity in Monterey Bay is characterized by a few specific spectral
peaks which dominate the spectra. Their period identification and
possible causes are explained below.

51.8 Minute Period

According to Wilson (1965), the fundamental mode of longitudinal
oscillation of the north-south extremities of Monterey Bay is 44.2
minutes. If the bay is approximated as an enclosed basin of uniform
depth and length, (Figure 7), the periods of oscillation are roughly
calculated in the following manner

Monterey

Santa Cruz

Figure 7
Fundamental Longitudinal Oscillation of Monterey Bay

36

The shallow-water wave celerity is approximated by,

C = Vgh
where ,

C = wave celerity
g = acceleration due to gravity
h = average depth of bay
The wave length of the fundamental period is

L - 2w
letting C = L/T, and substituting gives,

2w
T, =

1 Vgh

or in general

1 2w

T - iTr n = 1, 2, ...

n n vgh

for the harmonics of the fundamental mode of oscillation.

Substituting,

w = 20nm

h = 240 feet

g = 32.2 ft/sec2
gives the fundamental period of oscillation

T, = 46.5 minutes

The 51.8 minute long -wave appears throughout the different spectra,
and is evaluated as the fundamental period of longitudinal oscillation.

36.1 Minute Period

The fundamental mode of transverse oscillation of the east-west
extremities of Monterey Bay is computed by Wilson [1965] to be 32.3
minutes. Again, as in all Wilson's calculations, he assumed a nodal

37

line from Pt . Santa Cruz to Pt . Pinos . From Figure 8 the fundamental
transverse oscillation is calculated.

Assumed Node

Moss Landing

Figure 8
Fundamental Transverse Oscillation of Monterey Bay
The fundamental wave length is

L = 4b
Using the shallow water wave equation

and substituting gives,

C =V/ih =

1 4b

n n Vgh

For Monterey Bay,
b = 8nm
h = 240 ft
the fundamental period is calculated as

n = 1, 2,

38

T. = 36.3 minutes .

The calculated fundamental transverse mode, which is in agreement

I
with! Wilson is present in Monterey four out of the five days analyzed,

however, it only appears in Santa Cruz on 3 December 1969.

In addition to the above, one can also consider the possible pre-
sence of significant edge-wave activity [Munk, Snodgrass and Carrier,
1956]. The edge-wave is a wave form in shallow water on sloping
shelves and beaches. It is called an edge-wave because the waves
approach with the crest perpendicular to the beach and travel along
the coast. The waves are sinusoidal, and it is assumed the shelf
slope is constant. The celerity of the edge-wave as a function of
period and shelf slope is given by,

C = fi sin B

2tt

where ,

C = wave celerity

T = period of edge -wave

3 = shelf slope

g = acceleration due to gravity.
Substituting C = L/T we have the fundamental period and harmonics
of the edge -wave

T = /— -r^-r (2n^T) n = 1, 2, ...

n Vg sin 3 ' »

where L is the wave length equal to twice the longitudinal distance
across the bay for the fundamental mode.
Substituting,

g = 32.2 ft/sec

39

0 = 600 ft/8nm

•z = 20nm

the fundamental period of oscillation and associated harmonics of edge-
waves for Monterey Bay are computed as ,

T = 35.8, 25.3, 20.2, 17.8, 15.7, ... minutes

These modes are also called "trapped modes" from wave-guide theory
as the wave energy is trapped into a continental wave guide where in
theory the energy remains trapped indefinitely under idealized conditions.

Although the 35.8 minute long-wave is significantly present as
indicated above, it is not evaluated as an edge-wave as phase computations
do not support any pronounced edge wave activity.

27.7 Minute Period

The 27.7 minute period appears to be the most persistent long-wave
in the bay. It appears in the spectrum having significant energy
density on four out of five days analyzed, and is present at both Santa
Cruz and Monterey on 6 December 1969 and 14 January 1970. The 27.7
minute wave is also the dominant long-wave in Robinson's [1965] analyses.
Such waves can be interpreted as shelf-waves [Munk, 1962]. The formula
for the case of normal incidence on a two-step topography (Figure 9)
consisting of the continental shelf of width A and constant slope S1,
and the continental slope of width B with slope S is

Figure 9
Two-Step Shelf Topography

40

T..2TTA 1

where ,

T_ = fundamental period of the shelf wave

3 = constant which is dependent upon ratios S'/S and h'/h

where h'/h is the ratio of depth of break in continental
shelf to depth of sea floor at the bottom of continental
slope .
g = acceleration due to gravity.
For the case of Monterey Bay, choosing the 100 fathom curve as h'
and average distance A equal to 8.0nm and letting 4(3/n = 1.54 when
S ' < < S and h' < < h, the formula for the she If -wave fundamental
period becomes

T = ^— -=L

x0 (60)1.54 Vgh7 '

Substituting,

A = 48000 ft (average distance to 100 fathom curve)
g = 32.2 ft/sec2
h' =600 ft
we have ,

Tn ■ 29.6 minutes .
Modification of A and h' in the shelf-wave formula for different
locations around the bay changes the fundamental period only slightly.

The remaining long-waves which are present during significant long-
wave activity are those of periods 22.5, 18.9, 16.3, 13.6, 10.1, and
9.3 minutes. These are considered to be harmonics of the basic funda-
mental period 51.8, 36.1 and 27.7 minutes, and are in general weak in
spectral density. The significant spectral peaks found on various dates

are summarized in Table VII.

41

en

X X

X

X

X

I

ON

i

.—1

•

X X

X X

X X

X

X

o

I— 1

vO

CO

X X

X

X

X X

i-H

CO

vD

X X

X

X

X

.—1

On

oo

X

X

X X

X

i—i

•"-\

CO

m

W

•

H

CM

X

X

§

CM

M

s

>«•

r^»

co

•

Q

r^

X

X X

X X

X

X

X X

O

CM

M

1*

w

S

.—I

•

X

X X

X

X

X

vO

CO

00

•

X X

X

X

X X

X

X

i— 1

m

N

N

N

N

N

N

N

3

3

S3

3

3

3

3

3

>> u

>> ^

o

>, ^

>> ^

>s ^

>. >-l

>^ U

CU O

CU u

M

0) CD

0) c_>

<u u

CU U

<u u

h

u

H

h

J-l

u

u

h

CU eg

0) CO

<C

<U CO

<U CO

CO CO

CU CO

CU CO

4J 4J

4J 4J

4-) 4J

J-J 4-1

■U -u

•U 4J

•U 4J

c c

C 3

Q

c c

e c

c 3

c c

3 C

i— i

O CO

O cO

>— 1

O CO

O CO

O CO

O CO

O CO

ON

X co

X CO

2 co

X co

S CO

2 CO

S co

vO

ON

.—1

o

o

o

O

o\

ON

ON

r^

r*.

c

t^

r^

vO

vO

vO

O

W

c

3

CO

3

U

H

P>

o

O

CO

CO

c

CO

a.

3

o

ai

CU

->

•-5

•r-l

-i

<

23

Q

o,

XI

<t

O

o

CO

o

sO

CO

vO

i-{

CM

ai

CM

CM

3

PQ
<3
H

>>
CO
PQ

>>
cu
U
CU

4-1

c
o
2

CO

CO

CU

P-,

CO

S-l
4-1

o

CU

ex

CO

c

CO
CJ
1-1

H-t

•H
3
W)
•H
CO

M-l

O

>.

u

3
CO

42

B. INTERPRETATION OF INDIVIDUAL SPECTRA

Figures 10, 12, 14, 16, and 18 show the synchronized tidal traces
of Santa Cruz and Monterey on dates indicated. The traces are plotted
with a sampling interval of one minute which is approximately equal to
the Nyquist period of 70 seconds, which is the lowest period which can
be analyzed using the 34.495 sampling rate. The tidal traces are repro-
duced here for the first 16 hours of each period analyzed.

6 November 1969

The individual spectra for 6 November 1969 are plotted in Figure 11,
The record tends to be a "noisy" record with considerable long-wave
activity. The weather conditions in Monterey Bay were calm winds and
seas in the early morning, increasing to 15 kts of wind from the south-
west about 2000. The surface air temperature was 55 F and skies were
overcast for most of the day. The 42.5 minute non-recurring long
wave appears to be significant (the term significant peak used in this
study is a peak that has significant spectral density when compared
with other peaks in the same record vice having a specific value) at
both Monterey and Santa Cruz only on this date. Most noticable in the
records are waves of periods 51.8, 42.5, 16.3 and 10.1 minutes which
appear at both stations and have significant energy-density.

3 December 1969

Figure 13 shows the individual power spectra for Monterey and Santa
Cruz from 0130-2130 3 December 1969. Examination of Figure 12 shows
that significant wave activity in the bay commenced about 0950.
Initially the surface winds were from the north at 2 kts, shifting to
south at 15 kts by 1200. Skies were clear and surface air temperature
increased from 45 F in the early morning to 58 F in the afternoon. The

43

Santa Cruz
6 Nov 69

Time (.5x10" seconds)

Figure 10

Sea Surface Heights, Santa Cruz/Monterey
0730 6 November 1969 to 0330 7 November 1969

44

eg

U

<u

05

I
I

u

0)

■u
<u
B

W

c
Q

00
0)

c
u

CM

o

CO

I
I

w

•U

E

i-i

CO

c

Q

00

S-i
<D

c

hi

Frequency (millihertz)
Santa Cruz 0730 - 0330, 6 Nov 69-7 Nov 69

Frequency (millihertz)
Monterey 0730 - 0330, 6 Nov 69-7 Nov 69

Figure 11
Spectral Wave Analysis, 6 Nov 69

45

(0

sx
to

W

4)

O
03
14-1

S-l

co

CO

Santa Cruz
3 Dec 69

Time (.5x10 seconds)

Monterey
3 Dec 69

Time (.5x10 seconds)

Figure 12

Sea Surface Heights, Santa Cruz/Monterey
0130 - 2130, 3 Dec 69

46

C\l

o

CD

w
I

I

05

J-i
01

■u

0)

E

c
a)
D

>>

00

)-.

0)

c
w

Frequency (millihertz)
Santa Cruz, 0130 - 2130, 3 Dec 69

Frequency (millihertz)
Monterey, 0130 - 2130, 3 Dec 69

Figure 13
Spectral Wave Analysis, 3 Dec 69

47

spectra shows intense long-wave activity at Santa Cruz and moderate
activity at Monterey. The effect of persistant southerly winds may
account for this difference, however, no correlation is drawn. The
36.1, 13.6 and 10.1 minute waves have pronounced energy at both
locations, while the 51.8, 27.7 and 16.3 minute waves are significant
at Santa Cruz alone. Note, that at Santa Cruz the 13.6 minute wave
can very well be a harmonic of the 27.7 minute shelf wave while the
16.3 and 10.1 minute waves could be a harmonics of the fundamental
transverse mode of 36.2 minutes.

6 December 1969

The tidal traces for 6 December 1969, Figure 14, indicate heavy
long-wave activity at Santa Cruz and only light to moderate wave
action at Monterey. The winds in Monterey Bay were northwest at 16 kts
in the early morning decreasing to 2 kts in the evening. The surface
air temperature averaged 55 F and clear skies were observed during the
day. Although in contrast to the situation observed on 3 December 1969
no relation of wind to long -wave activity can be drawn during this
period. The 27.7 minute she If -wave is readily apparent in both records
with the 13.6 minute harmonic present at Santa Cruz alone. The Santa
Cruz picture is quite similar to that experienced earlier on 3 December
1969 at Santa Cruz. Monterey is generally weak in energy density and
the shelf-wave appears to be the only active wave of any significance.

14 January 1970

The analyses of wave records on 14 January 1970 are the most dis-
tinctive of the study. The sea surface traces, Figure 16, show moderate
to heavy wave activity at both stations. During the period analyzed,
winds were southerly at 35 kts and heavy rain was falling in the bay.

48

4J i »

0)

w

•u

jr
too

•H
CU

ac

o>
o

CO

m

CO

CO

cu
c/>

o

!

Santa Cruz
6 Dec 69

DJ>

Time (.5x10 seconds)

Monterey
6 Dec 69

Du!

Time (.5x10 seconds)

Figure 14

Sea Surface Heights, Santa Cruz/Monterey
2348 - 1848, 5-6 Dec 69

49

Csl

o
u

CO

I
I

CO

u

<v

4-J

CD

E

T-l

CO

£
0)
O

S^

0)

c

CO

C

<u

Q

d
w

Frequency (millihertz)
Santa Cruz, 2348 - 1848, 5-6 Dec 69

<M

-36^

"v^v-

^%-v_

og:

J24

ajb

Frequency (millihertz)
Monterey, 2348 - 1848, 5-6 Dec 69

Figure 15
Spectral Wave Analysis, 6 Dec 69

50

Santa Cruz
14 Jan 70

DJ.i

-4

Time (.5x10 seconds)

CO

01
4J

JC
60
tH

<U

0)

o
«o

CO

«

Monterey
14 Jan 70

oj;

-4

Time (.5x10 seconds)

Figure 16

Sea Surface Heights, Santa Cruz/Monterey
0810 - 0410, 14 Jan 70 - 15 Jan 70

51

Frequency (millihertz)
Santa Cruz, 0810 - 0410, 14 Jan 70 - 15 Jan 70

DC,

a:s

Frequency (millihertz)
Monterey, 0810 - 0410, 14 Jan 70 - 15 Jan 70

Figure 17
Spectral Wave Analysis, 14 Jan 70

52

Winds decreased to 2 kts in the late evening, however, surface air
temperature remained near 50 F during the entire period. The 27.7
minute wave is the primary long-wave present at both locations. In
addition the 18.9 minute wave is present at Santa Cruz but extremely
weak at Monterey. Again the relationship between weather conditions
and long-waves present is not characterized by any definite pattern.
The 51.8 minute wave is present at both locations, but is weak.

20 January 1970

Wave activity at Santa Cruz was again more intense than that at
Monterey. Winds were light and variable in the morning increasing to
southerly at 15 kts with heavy rain in the early evening. The wave
activity was similar to that experienced on 3 December 1969 showing
moderate southerly winds, heavy wave activity at Santa Cruz and light
activity at Monterey. However, with only two examples of this situ-
ation, a clear pattern is not observed. The Santa Cruz spectrum shows
some long -wave activity present at many discrete frequencies, however,
the 51.8 and 13.6 minute periods are the only recurring wave periods
present. In this record the 27.7 minute she If -wave was weak at Santa
Cruz. The 36.1 and 27.7 minute waves were observed at Monterey, how-
ever, overall wave activity at Monterey was weak.

Looking at the overall picture, including Robinson's [1969] data,
it was seen that the 27.7 minute wave evaluated as a shelf-wave, was
the dominant long-wave in the bay. This 27.7 minute period is averaged
over 5 frequency band widths therefore appears within the range of 26.3
to 29.4 minutes. Various values of average depth of the shelf and
distance to the shelf, in general, yield periods which fall within this
range. The fundamental transverse period of oscillation was observed

53

M-l

CO

J-J

s:

60
•H

<D
X

9)

u

cit

<4-l

C3
O
W

CD
CD
M-l

CO

4-J

J3
00
•H

0)

u

CO
«4-l

3
C/>

(0

<u
en

Santa Cruz
20 Jan 70

a j 4

Time (.5x10 seconds)

Monterey
20 Jan 70

^ ^2

-4
Time (.5x10 seconds)

Si)<

Figure 18

Sea Surface Heights, Santa Cruz/Monterey
0215 - 2215, 20 Jan 70

54

<N

u

1

CD

I

9

1

o

CO

u

CD

CD

E

>»

4->

— i

1-1

o

CO

C

cu

O

>>

GO

l-i

0)

C

w

CI

1 )

o

Frequency (millihertz)
Santa Cruz, 0215 - 2215, 20 Jan 70

tN

o

CJ

<D
CO

I

I

CO

u

<u
•u
CD

e

>> 3

4J a
•H

CO

a

a)
O

00

h

? CJ

Frequency (millihertz)
Monterey, 0215 - 2215, 20 Jan 70

Figure 19
Spectral Wave Analysis, 20 Jan 70

55

regularly at Monterey, but only once at Santa Cruz. The fundamental
longitudinal period was significant at both locations most of the time.
No definite relationship was drawn between atmospheric conditions and
long-wave activity, however, a strong southerly wind seemed to lead to
heavy wave activity in Santa Cruz. Also it was noted that most of the
time, long-wave activity when present in the bay, was much more intense
at Santa Cruz than at Monterey.

C. INTERPRETATION OF CROSS SPECTRA AND PHASE

Table VIII gives a summary of selected recurring long-period waves
which have significant cross power when compared with other peaks in
the same record. Some high energy-density long-waves which only appear
once or twice in the entire study are excluded from the analyses. The
cross spectra is a measure of the combined energy-density resulting
from contributions of each power spectra but gives no information as to
the forcing function of the wave or to whether the waves at the two
locations are correlated with each other. Two isolated waves of the
same period but operating independently, each having considerable
energy, would yield a high cross power. Phase relationships determine
whether or not the waves appearing at two locations are coupled.

Figures 20 through 24 show the cross spectrum and phase differences
of each long-wave between Monterey and Santa Cruz. Both raw spectra
were smoothed over five frequency band widths. However, considerable
difficulties arose when attempting to smooth the phase relationships.

The process of smoothing over band widths assumes that the data is
the result of a stationary random process. The long-waves analyzed in
this study may be viewed as deterministic functions and therefore what
is considered in the record is a number of deterministic functions

56

superimposed on so called "white noise". The ideal way to smooth the
phase information would be to smooth over ensembles, however, this could
not be done because the long-waves are, in general, a transient phenomenon
and usually not a stationary process. The Fast Fourier Transform (FFT)
method of calculating the power spectrum is relatively new and, while
the raw phase information is considered correct, the correct method to
smooth relationships when using the FFT have not been completely worked
out [Enochson and Otmes , 1968].

In this study the raw phase information was calculated utilizing
the cross spectra as a weight function such that

. e. $.
e ■ 1 ^ i = 1,2,3,4,5

i

where

e ■ smooth phase difference

£ . = raw phase

$. = raw cross power.
1

This method was adopted after investigating several alternatives. The
calculation resulted in those raw phase estimates associated with a high
cross power to be dominant in the smoothing process. Those relation-
ships are shown in Figures 20 through 24 and summarized in Table IX for
selected periods. There are reasons to doubt the accuracy of calcu-
lating the smoothed phase using FFT in this manner which circumvents
computing the correlation function. Hence, the smoothed phase calcu-
lation must be viewed with reservations.

Table VIII is a summary of significant cross spectral peaks for
selected long-waves which recur throughout the five days analyzed.
Figures 20 through 24 show the raw and smoothed cross spectra for the

57

PERIOD

DATE

(minutes)

6 Nov 69

3 Dec 69

6 Dec 69

14 Jan 70

20 Jan 70

51.8

X

X

X

42.5

X

36.1

X

X

27.7

X

X

X

22.5

18.9

X

X

16.3

X

13.6

X

10.1

X

X

X

9.3

X

TABLE VIII
Summary of Significant Cross Spectral Peaks in Monterey Bay

58

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63

dates analyzed. As expected the cross spectra is dominated by the 27.7
minute shelf-wave and its associated harmonics. The cross spectral

summary is in general agreement with Table VII, the summary of indivi-

i

dual spectral peaks. The phase differences, between a particular wave
at Santa Cruz and Monterey appear to be random, which implies the
Monterey Canyon acts as a barrier dividing the bay into separate north-
south basins which oscillate independently. This is in general agreement
with Wilson [1965] .

D. COMPARISON WITH WILSON'S THREE-DIMENSIONAL NUMERICAL MODEL

Wilson [1965] assumed a boundary nodal line for his solution of his
three-dimensional model of the oscillating characteristics of Monterey
Bay. This node is a line drawn from Pt . Santa Cruz to Pt . Pinos on the
Monterey Pens inula. This assumption tends to be the greatest objection
to Wilson's results, although an enclosed bay will usually have a node
across the seaward opening for particular modes of oscillation. The
oscillating characteristics are depicted in Figures 25 through 28 where
Figure 25(a) gives the general bathemetry of the bay. These depths
were used as grid points in order to generate the numerical solution.
The other figures show the increasing complex longitudinal modes of
oscillation. The contour lines are the amplitudes of the water level
normalized to the highest anti-node for the mode. Each complex mode
of oscillation is drawn in a simplified version to the left of the
figure. The nodal line assumption is particularly constraining to the
longitudinal mode of oscillation which can be seen in Figure 25 where
mode 1 (i.e., the fundamental) tends to oscillate transversely due to
the nodal constraint. The lowest modes would be most affected by the
nodal assumption and would have less constraint on higher modes.

64

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68

The periods of oscillation of the longitudinal mode were calculated

as,

Tn =44.2, 29.6, 28.2, 23.3, 21.6, 20.4,

19.4, 18.7, 17.6, mins.

Mode 1 has a fundamental period of 44.2 minutes and indicates strong
oscillation at Santa Cruz and weak at Monterey. Mode 2 is similar to
mode 1 while mode 3 gives the only strong oscillation at Monterey and
weak oscillations at Santa Cruz. Mode 4 is similar to modes 1 and 2 ,
but is somewhat more complex as are successive modes.

Wilson does state that his assumed node may very well be incorrect
and possible should be moved further seaward. The boundary conditions
of the model would then be changed, resulting in new oscillating
periods and characteristics. This would result in a lengthening of
the fundamental period.

Wilson also computed two-dimensional modes of oscillation of
Monterey Bay. This is the transverse oscillation of the east-west
extremeties bounded at Moss Landing on the east and the assumed node on
the west. The sequence of modal periods of oscillation were calculated

as

Tn =32.3, 14.3, 9.5, 7.0, ... mins.
The profiles of water-surface elevation for the transverse mode are
shown in Figure 29.

Considering both the longitudinal and transverse modes of oscillation,
Wilson notes that decreased solution confidence is placed on each „
cessive harmonic, however, the highest mode in which he has confide,
is not stated. This increasing error in successive modal solutions can
be decreased by choosing a finer grid. Wilson's model probably is most
valid for modes greater than mode 2 and less than mode 7.

5UC-

:nce

69

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70

As stated earlier, Wilson concluded that the deep Monterey Canyon
does exert a profound influence on Monterey Bay oscillations as it

effeptively sets a barrier to free oscillations, between the north and

!

south portions of the bay. The result is that wave activity acting in
both locations would remain largely uncoupled and would give rise to a
random phase .

Table IX summarizes some of the results of this study with that of
Wilson's model. It is apparent that the long-wave activity at Santa
Cruz is more intense than that at Monterey. This tends to support the
conclusion that the two basins act independently. Robinson [1965]
found similar magnitudes at both stations and concluded that the
oscillations were not uncoupled. The fundamental longitudinal mode of
51.8 minutes was found to be weak at Monterey and stronger at Santa
Cruz thus supporting Wilson's findings. The 20.5 minute wave corres-
ponding to Wilson's mode 6 is in general weak at both stations. This
is in agreement with Wilson's results. The greatest contrast between
the results of this study and that of Wilson's is the effect and identi-
fication of the 27.3 minute wave, evaluated as a shelf-wave here, but
as the third mode of oscillation by Wilson. This wave, according to
Wilson, should be strong at Monterey and weak at Santa Cruz. In general,
this was not found to be the case. The ratios of energy-density at
Santa Cruz to energy-density at Monterey is greater than 1 on 6 November,
3 December and 6 December indicating the opposite of Wilson's findings.
The last two periods studied show a ratio of .94 on 14 January 1970 and
.49 on 20 January 1970, neither of which support Wilson's calculated
results. It is considered that the 27.3 minute wave predicted by Wilson
is in fact present during long-wave oscillations, but is evaluated as a

71

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73

shelf-wave vice a harmonic of the fundamental frequency of oscillation
of the bay. Most of the other periods predicted by Wilson were found to
exist. However, it must be remembered that a discrete Fourier analysis
was performed and the data smoothed. This has the result of "forcing"
some energy into discrete band widths allowing only certain periods to
appear in the spectrum.

As a result of this comparison the present study tends to verify
many of Wilson's conclusions. The observations that wave amplitudes
are generally higher at Santa Cruz and that the phase relationships tend
to be random lead to the conclusion that oscillations of Monterey Bay
at Santa Cruz and Monterey tend to operate independently of each other
due to the impedence barrier formed by Monterey Canyon.

74

V. ERROR ANALYSIS

Errors encountered during the course of the study were basically
of three types :

(1) Time errors can be introduced when recording raw data on an
analog trace.

(2) Errors may be generated in matching the Santa Cruz raw
data to Monterey data.

(3) The effect of the numerical methods of calculating the
power-spectrum and the cross spectrum can lead to mis-
interpretation of results.

Each of these effects are examined below in order to determine, in a
qualitative sense, their effect on the calculated results.

Time Induced Errors

The tide gages at Santa Cruz and Monterey are both mechanical
devices which must be wound at required intervals. Both were maintained
within the required periods. The sampling interval of 34.495 seconds
for Monterey and 6.00 seconds for Santa Cruz were computed by inches/
time relationships between two successive time checks on each record
for each period analyzed. The times at Santa Cruz were recorded from
radio station WW (Naval Observatory Time) and are correct to the second.
The times at Monterey were obtained from the telephone company and are
accurate within + 5 seconds. Time checks were maintained daily for the
entire period. The recording speed at Monterey was found to be 1.03
+ .01 inches/hour and 6.00 + .02 inches/hour for Santa Cruz. Synchronized
records are critical to phase computations. However, the resulting
error in obtaining the raw data was considered insignificant.

75

Errors Generated in Aligning Records

The Santa Cruz record was aligned with the Monterey record by
converting the coordinate system from curvilinear coordinates to
rectilinear coordinates. Linear interpolation was then used to match
data points.

The Santa Cruz data yielded a data point every six seconds giving
high resolution in digitizing. The original 12,000 plus data points
were reduced to 2048 points. Reduction and conversion of data in this
manner resulted in very minor raw data errors.

Numerical Calculation Methods

The effect of approximating a continuous tidal trace with a step-
function and transforming the step-function by numerical methods through
use of a digital computer has several effects on the final analyses, two
of which could result in misleading results of interpretation.

When a continuous curve is approximated by a step function, a
secondary wave form of high frequency energy is superimposed on the
recorded wave form. To evaluate this possible error the sampling rate
was reduced and records were analyzed and checked for aliasing. It was
found the aliased power was quite small and in no case greater than 10$.

The second source of error is the effect of transforming the record
into discrete values of energy-density as a function of discrete periods.
Thus, the energy is forced into certain band widths and centered about
the middle frequency. As a result of smoothing over five frequencies,
the frequency interval was .00007 cps ranging in periods from 397.4
minutes to 1.16 minutes. Comparison of raw and smooth power and cross
spectra show that the raw spectra was not smeared due to smoothing the
original 1025 spectral estimates as "spikes" in the raw spectra were
still present after the smoothing process was completed.

76

VI. SUMMARY

Power spectra analyses and cross spectral analyses of wave records
in Monterey Bay indicate the presence of persistent unique periods
with the phase differences between Monterey and Santa Cruz tending to
be random. Coupled with this and the fact that wave energy at Santa
Cruz is in general greater than that at Monterey, it is concluded that
the Monterey Submarine Canyon has a profound effect on seiching motions
within the bay. This effect appears to divide the Monterey Bay into
north and south basins where oscillations occur essentially independent
of each other.

77

APPENDIX A

DIGITIZING GRAPHICAL ANALOG RECORDS USING THE CALMA COMPANY MODEL 480

The Calma digitizer reduces analog graphical data to digital
magnetic tape for computer processing and analysis. It consists of a
freestanding tracing bed and a separate electronics/recorder module.
The tracing bed is motor-driven for adjustment to the most comfortable
digitizing position.

To digitize analog graphical data directly on magnetic tape, the
operator traces the analog curve with a moveable stylus/carriage assembly,
As the curve is traced, movements of the stylus in either the X or Y
direction causes pulsed to be generated by the direct digital pickoff
mechanism. Incremental motions are recorded as characters on magnetic
tape, leaving the task of summing the increments for whole-value coordi-
nates to the final processing computer. This method of digitizing is
both faster and more accurate than template methods.

The digitizer reads and records data in the X and Y direction, with
a sampling interval which can be set to 0.01, 0.02, 0.04, 0.08, or 0.15
inches. The maximum absolute sampling error for the machine is 0.012
inches. The output is external BCD, stored on 556 bpi, 7 channel, tape.
The tape can be made compatible with the IBM/360 system, however, it is
easier to use the CDC/6500 computer.

A FORTRAN IV program wirtten for the CDC/6500 can be utilized for
interpreting the digitized record.

The program accomplished the following:

(1) Converts the data from external BCD to display code.

(2) Arranges the data in column matrices of 80 character length.

78

(3) Interrogated each character in the matrix to determine if
the character is a

(a) Point flag - a designator to label a specific data
point of interest

(b) Plus or minus travel in the X-direction.

(c) Plus or minus travel in the Y-direction.

(4) Sums the Y direction travel.

(5) Records incremental travel in X direction (plus or minus).
This allows for negative travel while digitizing. Negative travel

is common if digitizing a curvilinear trace of if variations in data are
closely spaced.

(6) Prints the Y value V(N), for each designated increment in
the X direction.

(7) Prints the X increment value U(N), showing the incremental
travel (plus or minus) for which the Y value was obtained.

(8) Punch cards for V(N) and U(N) which are used as inputs for
other computer programs as desired.

Headings may be entered on the tape with keyboard control, but
CONVERT is not designed to read tape headings nor does the program have
the capability to search for a particular set of data on the tape. Con-
sequently, all data on the tape will be analyzed each time CONVERT is
used.

The following sequence of operations can be used as a guide for the
digitizing procedure:

(1) Mount chart on tracing bed so that horizontal axis of chart
is aligned with X axis of digitizer.

(2) Turn system power on.

79

(3) Load tape.

(4) Initiate "Load Forward" on tape module.

(5) Press RECORD ERROR. (If recorder is properly loaded, RECORD
ERROR indicator will go out when push button is pressed,...
digitizer will automatically shift to KEYBOARD MODE.)

(6) Enter required identifiers from keyboard if necessary.
(Not required with CONVERT program.)

(7) Position stylus at beginning of curve. Lock Y-axis, and
check alignment.

(8) Initiate TRACER Mode, de-activate SKIP (SKIP button out,
light off), and unlock stylus.

(9) FLAG the tape once.

(10) Trace the record as desired. The first point which will not
be computed, or printed, will be X = 0, Y = 0. The speed at
which the stylus can be moved without destroying data
accumulated is well in excess of the speed normally used to
digitize the record. However, if exceeded, the RECORD ERROR
will light and record will be deleted.

(11) When tracing is complete, lock the stylus, initiate FLAG
PEDAL twice and IRG button once.

(12) Press rewind button on tape module and remove tape.

The user of the Calma digitizer can reap the benefits of the instru-
ment's speed and accuracy only if the digitizing operator understands
and applies the system's basic rules of operation. It is strongly
suggested that new digitizing operators read, study and understand the
textual instructions (CAIMA MODEL 480 DIGITIZER, Instruction and Main-
tenance Manual) before attempting to operate the digitizer.

80

(13) The magnetic tape from the digitizing process is used as INPUT
to program CONVERT. The approximate number of words on the
tape should be estimated so that CONVERT may be dimensioned
accordingly.

In addition, if the tape contains any parity errors, or was not
deguassed properly prior to usage, the program will only interpret a
portion of the data or possibly not run at all. These problems have
been encountered in the past.

(14) The program CONVERT is included and is self-explanatory.
Output consists of (a) sampling rate, (b) length of record
(sec), (c) amplitude values (inch) V(N), and (d) incremental
travel in X-direction U(N). These may be summed in another
program to determine proper time or spatial frame.

(15) It must be remembered that CONVERT is FORTRAN IV written for
the CDC/6500 located at FNWC Monterey, and must be adapted
for other computers.

81

PROGRAM CONVERT ( I NPUT, OUTPUT, PUNCH)
C THIS PROGRAM CONVERTS THE DIGITIZED RECORD OF THE
C MONTEREY TIDE GUAGE FROM EXTERNAL BCD TO AMPLITUDE
C VALUES FOR A CONSTANT SAMPLING PATE DELT, AVGX IS THE
C AVERAGE DISTANCE THE RECORDING DRUM ADVANCES PER ONE
C HOUR PEAL TIME, I* E, THE DATA SAMPLING »ATE.
C REFERENCE-CALMA CO, MODEL 3 03 DIGITIZER INSTRUCTION
C MANUAL. DIMENSION STATEMENT MAY BE MODIFIED TO
C COMPUTE AS MANY DATA PCINTS AS REQUIRED.

DIMENSION U( 17000) ,V( 17000) ,N(80) , I BUFF ( 17000 ) ,NK(80J
C PROGRAM READS L SETS OF DATA FROM TAPE

READ 97, L
97 F0RMATU2)

DO 116 JJ=1,L

READ 98, AVGX
93 FORMAT(F10.3)
C COMPUTE SAMPLING RATE (SECONDS)

DELT=36.0/AVGX

PRINT 99, DELT
99 FORMAT( 1H0,2X,25H SAMPLING INTERVAL EQUALS , IX , E 15 .7 , IX
1,7H?EC0NDS)

PRI.T 700, JJ
700 FORMAT! 1H0,9HJJ EQUALS, 2X,I2)
C ZERO FILL AMPLITUDE ARRAYS

DO 100 1=1,17000

U( I ) = 0.0

V( I )=C.C

100 CONTINUE

C ZEROIZE INPUT/OUTPUT BUFFER

101 DO 202 1=1,17000
IBUFF( I )=0

202 CONTINUE

COUNTX=C.O

COUNTY=0.0

NUM=17000
C COUNT NUMBER OF DATA POINTS ACCUMULATED

M=l

K8 = 0
C CALL SUBROUTINE LIOF TO READ INPUT TAPE WHERE NPAR IS
C A PARITY CHECK, AND NEOF CHECKS FOR END OF FILE

CALL LIOF( 5LRBC01,IBUFF,NUM,NPAR,NE0F)

IF(NEOF) 602,200

200 K=-7
KF = 0

201 K=K+8
KA=K+8
KC = 0

00 102 KB=K,KA
KC = KC + 1

NK(KC)=IBUFF(KB)
IF(NMKC).EO.O) KF = 1

102 CONTINUE

C DECODE TAPE 80 WORDS AT A TIME
DECODE( 80,103,NK) ( N< I ) , I =1 ,80 )

103 F0RMAT(80R1)

C INTERPRET BCD ON TAPE AND COMPUTE AMPLITUDES

DO 104 19=1,80

IF(N( I9).EQ.50B) GO TO 106

IF(N( I9).E0.55BJ GO TO 107

IF(N( I9).EQ.34B) GO TO 103

IF(N( I9).EQ.A7B> GO TO 105
C SYMBOL / (50B) REPRESENTS AN INCREMENT TRAVEL IN THE
C MINUS X OR Y DIRECTION BY THE DIGITIZER.
C SYMBOL 0#(55B) , REPRESENTS A ZE"0 INCREMENT TRAVEL IN
C THE X OP Y DIRECTION BY THE DIGITIZER.
C SYMBOL 1,( 3AB) , REPRESENTS AN INCREMENT TRAVEL IN THE
C POSITIVE X OR Y DIRECTION BY THE DIGITIZER.
C SYMBOL *V(47B), IS A FLAG INSERTED IN THE RECORD BY
C THE PERSON DIGITIZING.

GO TO 10^
105 PRINT 205, M, 19
205 F0RMAT(1H0,2I10)

82

c
c

c
c

1106

107

108
109

110
111

112

104

115

215
213
116

600
601
602

THIS S
IDENTI
IF(M.G
GO TO
COMPUT
RX=-0.
K8=K8+
GO TO
COMPUT
RX=0.0
Kd=K8+
GO TO
COMPUT
RX=0.0
K8=K8+
K3=K8/
K3=2*K
IFCK3.
COUNTX
IF(CCU
GO TO
COMPUT
U(M)=C
GO TO
COUNTY
IF(CCU
GO TO
COMPUT
V(M)=C
COUNTX
M=M+1
STOP I
IF(M.G
CONTIN
IF(KF.
GO TO
TOTAL
POINTS
TIME=(
PRINT
FORMAT
17,2X,7
PRINT
PRINT
PUNCH
PRINT
PRINT
PUNCH
FORMAT
FORMAT
CONTIN
STOP
PRINT
FORMAT
STOP
END

CN ALLOWS FIRST TEN WCRDS OF TAPE TO BE
TION DATA IF DESIRED.
) GO TO 113

ECTI

FICA

T.10

104

E NEGATIVE TRAVEL

CI

1

109

a ZERO TRAVEL

1

1C9

E POSITIVE TRAVEL

1

1

2

3

E0.K8) GO TO 111

=CCLNTX+RX

NTX.NE.O.O) GO TC 110

104

E X INCREMENT TRAVEL

CLNTX

104

=CCUNTY+RX

NTX.NE.O.O) GO TC 112

104

E AMPLITUDE VALUE

CLNTY

= 0.0

F DA
T.17
LE

EO.l
201
TIME
TIM
M*DE
115,
( 1H
H HO
AND
215,
213,
AND
215,
215,
( 1H
( 14F
UE

TA POINTS EXCEED
000) GO TO 600

) GO TO 113

ARRAY

OF
ES T
LT)/
TIME
,10X
URS,
PUNC
<V( I
( V( I
PUNC
<U( I
(U( I
,10X
5.2)

THE RECORD EQUALS THE NUMBER
HE SAMPLING I NTE RVAL ,DELT .
3600.0

OF DATA

,28H TOTAL TIME OF RECORD EQUALS , 2X , E 15 .

)

H AMPLITUDE VALUES

) ,1=1, V)

) ,I=1,V)

H X INCREMENTS

) ,1=1, Kl

),I=1,M)

,14F7.2)

601

( 1H1,20X,35H****

U AND V SPACE INADEQUATE *****)

83

APPENDIX B

COMPUTATION OF FAST FOURIER TRANSFORM USING IBM/360 SUBROUTINE RHARM
This subroutine is a one -dimensional Fast Fourier Transform (FFT)
analysis based on the Cooley and Tukey [1965] algorithm. The program
is designed to analyze I data points where

I = 2M M - 3, 4, 5, .... 20.

The FFT not only greatly reduces the number of calculations from
earlier analysis schemes, but also reduces the round-off errors in the
Fourier coefficients. The computations and round-off errors are
reduced by log?(I)/l.

The subroutine HARM is then called by RHARM to compute the complex
coefficients :

*K " i £ <X2j-1X2J+l> ¥ JK

K = 0,1,2,.. .N-l.

N N
Then for K = l,2,...,-r - 1, j (with the bar denoting conjugation)

AK = 2 (AK + \-x) ' AK = 2 (AN-K_AK}

and,

CK = i<AK+A£ei(""'jK)>

* v i o N.N
for K = 1,2,... ^ - 1, 2

i _ • r n H\

CN-K = 1 (A-' " A" fil

(AK ' AK 6 N"K 2 }

for K = 1,2,..., | - 1

84

If we let C_ = 2 (ReA0 " ImA(P

CN = 2 (ReA0 + ImA0>

We finally have :

AQ = 2 Re (C0), BR = -21m (CK) ,

B0 = 0, ^ = 2 Re (CN),

The Fourier coefficients

V B0 = °' V Bi""V BN = °are

obtained for input X., j = 0,1,2 ,.. .2N-1, where N = 1/2, for the

following equation.

N-l
X. = | AQ + E (^ COS (njK/N) + BR SIN (TTjk/N) +
K=l

85

APPENDIX C

LEAST SQUARES CURVE FITTING
The scheme used to accomplish the least squares fit for a sixth-
degree polynomial is described below. Given a set of data pairs (X., Y.)
(i = 0,N) which can be interpreted as measured values of the coordinates
of the points on the graph of y = f(x), assume that the unknown
function f(x) can be approximated by a linear combination of suitably

chosen functions, f_(x), f..(x), ... f, (x) of the form F(x) = A-f^fx) +

U 1 o U U

A1f1(x) + A2f2(x) + A3f3(x) + A4f4(x) + A5f5(x) + A^Cx) where the
unknown coefficients A , A., . . . , A, are independent parameters to be
determined and the degree of least squares polynomial, M is such that
M = 6 < N. The difference between the approximating function value
F(xj) and the corresponding data value Y., is called the residual r.
and is defined by the relation

rj = F(X.) - Y. (i = 0, N).

The function F(x) that best approximates the given set of data in

a least squares sense is that linear combination A fn(x) + A.f , (x) +

... + A,f,(x) of functions f (x) that produces the minimum value of
bo k

the sum Q of the squared residuals where

Q = S r.2 = S [f(X.) - Y.2].

For the case of curve-smoothing by polynomial least squares, we
approximate the function y = f(x) over the range of data points
(X. , Y.) ( i = 0, N). The parameters A , A- , ... A, are then deter-
mined such that

86

" ■ S ri2 ■ E [P6<Xi> " Yi2]

is £ minimum. That is, a 6th degree polynomial curve is fitted to the
data points in a least squares sense as defined earlier. The normal
equations for the least-squares polynomial can be written as

N+1ZX. EX.

1 1

2 3

EX. EX. E X.
ill

E X.

i

E X.'

i

EX.6 EX.7 E X.8.
ill

E X

12

E Y.

l

E X.Y.
i l

E X.6Y.
i i

where :

N = number of data pairs

X. = time increment (seconds)
l '

Y. = amplitude values (inches)

The solution of the matrix is the sixth-degree polynomial,

P,(x) = A_ + A.X1^ A0X20 + A_X3_ + A.X4, + A_X5_ + A,X6,
o 0 1122 33 44 55 66

which is then subtracted from each data point to remove tidal effects,

acting as a high-pass filter. Choosing a sample interval of 90

increments, 23 data pairs were utilized for the least squares fit.

87

LITERATURE CITED

1. Cooley, J. W. and J. W. Tukey, 1965. An Algorithm for the Machine
Calculation of Complex Fourier Series, Mathematics of Computation,
Vol. 19, No. 90, April, 297-301.

2. Hudson, R. Y., 1949. Wave and Surge Action, Monterey Harbor,
Monterey, California, Technical Memorandum No. 2-301, Waterways
Experiment Station, Vicksburg, Mississippi, U. S. Army Corps of
Engineers.

3. Manual of Tide Observations, 1965. U. S. Coast and Geodetic Survey
Publication 30-1, U. S. Government Printing Office, Washington, D. C

4. Munk, W. H., 1962. Long Ocean Waves, The Sea , Volume One, M. N.
Hill, ed., Interscience Publishers, New York.

5. Wilson, B. W. , J. A. Hendrickson, and R. E. Kilmer, 1965.
Feasibility Study for a Surge - Action Model of Monterey Harbor,
California , Waterways Experiment Station, Vicksburg, Mississippi,
U. S. Army Corps of Engineers.

6. Enochson, L. D. and Otmes , R. K. , Programming and Analysis for
Digital Time Series, Data 1968.

7. Munk, Snodgrass , and Carrier, 1956. Edgewaves on the Continental
Shelf, Science, 27 January 1956, Vol. 123, No. 3187.

88

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1. Defense Documentation Center 2
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2. Library, Code 0212 2
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3. Oceanographer of the Navy . 1
The Madison Building

732 N. Washington Street
Alexandria, Virginia 22314

4. Department of Oceanography, Code 58 3
Naval Postgraduate School

Monterey, California 93940

5. Dr. E. B. Thornton, Code 58Tm 3
Department of Oceanography

Naval Postgraduate School
Monterey, California 93940

6. Dr. W. C. Thompson, Code 58Th 1
Department of Oceanography

Naval Postgraduate School
Monterey, California 93940

7. U. S. Army Corps of Engineers 1
Waterways Experiment Station

Vicksburg, Mississippi 39180

8. Dr. B. W. Wilson 1
Science Engineering Associates

San Marino, California 91108

9. LCDR Thomas J. Lynch 3
USS Vigor (MSO 473)

Fleet Post Office

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10. V. G. Grauzinis 1

Northwest Oceanographers , Inc.
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89

11. Mr. Thorndike Saville, Jr., Director
Research Division

Coastal Engineering Research Center
520 Little Falls Road, N. W.
Washington, D. C. 20016

12. Mr. Orville T. Magoon

U. S. Army Engineer District

Corps of Engineers

100 McAllister Street

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13. Dr. Taivo Laevastu, Code Ro
Fleet Numerical Weather Central
Monterey, California 93940

14. Dr. Joseph J. von Schwind, Code 58Vs
Department of Oceanography

Naval Postgraduate School
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{Security classi tic ation of title, body of abstract and indexing annotation must be entered when the overall report is classltied)

1 ORIGINATING AC Tl VI T Y (Corpora te author)

Naval Postgraduate School
Monterey, California 93940

2«. REPORT SECURITY CLASSIFICATION

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2b. GROUP

3 REPORT TITLE!

Long Wave Study of Monterey Bay

4 DESCRIPTIVE NOTES (Type ol report and, inclusi ve dates)

Master's Thesis; Sppt-Pmhpr 1Q7D

? r.

5- Au THORIS) (First name, middle initial, last name)

Thomas John Lynch

6 REPORT DATE

September 1970

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This document has been approved for public release and sale;
its distribution is unlimited.

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12. SPONSORING MILI TAR Y ACTIVITY

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13. ABSTRAC T

Monterey Bay, on the west coast of the United States, is unique in
that it is a large, symmetric, semi-elliptical bay divided north and
south by the deep Monterey Canyon. The effect of the canyon on
seiching within the bay and on long wave oscillations within the
bay was studied by analyzing sychronized wave records at each end
of the bay. Power spectra and cross spectra calculated for five
periods selected from six months continuous data indicate the Monterey
Canyon has a profound effect on the bay's oscillating characteristics.
The canyon appears to act as an impedence barrier dividing the bay
into two independent oscillating basins each having recurring long-
period waves which persist during significant long -wave activity.

DD ,F°Rv"..1473 (PAGE "

<ov es
S/N 0101 -807-681 1

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key wo BO!

Monterey Bay
Long wave
Seiching
Power spectra
Cross spectra

FORM

I NOV

101 -807-682 1

1 .'£?.. 14 73 (back,

ROLE

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Security Classification

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'\

Thesis

L965

e.1

wave study of
Bay.

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