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THESIS

THE ONE-DIMENSIONALITY OF THE UPPER OCEAN MIXING AND
THE ROLE OF ADVECTION DURING THE POLE EXPERIMENT

by

Ricky Eugene Shook

September 1980

Thesis

Advisor: R. Vi. Garwood, Jr.

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4 TITLE CaMrf Suk(«f(a)

The One-Dimensionality of the Upper Ocean Mixing
and the Role of Advection During the POLE Experi-
ment

7. AUTHOIt<«>

Ricky Eugene Shook

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Mathematical Models, Mixed Layers, Turbulent Boundary Layer, Mixing, Advection
Ocean Thermal Structure; 1-Dimensional Numerical Model Turbulence

20

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one-dimensional mixed layer model (barwood, 1976, 1977) is used to simulate
xed layer depth and temperature observed during POLEX, a component of the
Pacific Experiment. POLEX occurred during January-February 1974 with mid-
observations (35°N, 155°W) of temperature and salinity made from R/P FLIP,
FLIP was under free-drift conditions. The results of this simulation show
pparent horizontal advection, due to the drift of R/P FLIP, was important in
at and salinity budgets, but that vertical mixing and mixed layer depth
s were controlled primarily by the one-dimensional response of the turbulent
c energy budget to the local atmospheric forcing. Occasionally, surface

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order of tens of meters) the thickness of ocean surface turbulent boundary layer,
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Approved for public release; distribution unlimited,

The One-Dimensionality of the Upper Ocean Mixing and the
Role of Advection During the POLE Experiment

by

Ricky Eugene Shook

Lieutenant, United States Navy

B.S., Capital University, Columbus Ohio, 1973

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL

September 1980

ABSTRACT

A one-dimensional mixed layer model (Garwood, 1976, 1977) is used to
simulate the mixed layer depth and temperature observed during POLEX, a
component of the North Pacific Experiment. POLEX occurred during Janu-
ary-February 1974 with mid-ocean observations (35°N, 155°W) of tempera-
ture and salinity made from R/P FLIP, while FLIP was under free-drift
conditions. The results of this simulation show that apparent horizon-
tal advection, due to the drift of R/P FLIP, was important in the heat
and salinity budgets, but that vertical mixing and mixed layer depth
changes were controlled primarily by the one-dimensional response of the
turbulent kinetic energy budget to the local atmospheric forcing. Oc-
casionally, surface salinity flux, due to large precipitation rates, can
significantly alter (on the order of tens of meters) the thickness of
the ocean surface turbulent boundary layer, with a demonstrated decrease
in thickness during a single period of strong precipitation.

TABLE OF CONTENTS

I. INTRODUCTION 9

A. PURPOSE OF THE STUDY — 9

B. MILITARY IMPORTANCE - - - — 11

C. LITERATURE REVIEW - - Jl

II. DATA ANALYSIS 18

A. METHODS - - 18

B. SURFACE FLUXES - - - 18

1. Surface Wind Stress 21

2. Surface Heat Flux - 21

3. Salinity Flux - --- 22

4. Buoyancy Flux 22

C. ADVECTION 23

1. Heat and Salinity Budgets 23

2. Calculation of the Rates of Advection 26

D. MIXED LAYER DEPTH — 28

1. Observed Mixed Layer Depth 28

2. Comparison of the Model -Computed Mixed Layer

Depth with Observed Mixed' Layer Depth 29

3. Salinity Effects on the Mixed Layer Depth 31

III. SUMMARY - - - 33

FIGURES - - 35

LIST OF REFERENCES 67

INITIAL DISTRIBUTION LIST — 70

NOTATION

b Buoyancy

2 -3
. , , Turbulent buoyancy flux (cm sec )

C Degrees Centigrade

Cj Non-dimensional drag coefficient

c Heat capacity of sea water (cal gm~ C' )

-2 -1
E Rate of evaporation (gm cm sec )

E- Vapor pressure of air

a

E Saturated vapor pressure of marine air

f Coriolis parameter

g Acceleration due to gravity

h Turbulent boundary layer depth (modeled)

h. Mixed layer depth (modeled)

h« Observed mixed layer depth

m Meters

-2 -1

P Rate of precipitation (gm cm sec )

-2 -1
Q^ Upward surface heat flux (cal cm sec )

a

-2
Q. Back Radiation (cal cm sec

-2
Q Latent heat flux (cal cm sec

-2
,Q. Sensible heat flux (cal cm sec

-2
Qq Clear sky radiation (cal cm sec

-2
Q-j- Net surface heat flux (cal cm sec

S Salinity (0/00)

-rrrp- Turbulent salinity flux (0/00 cm sec' )

T Temperature

T Temperature in the air
a

T Sea-surface temperature

Y^ — r Turbulent temperature flux (mwatts cm )

t Time (hours)

u^ Frictional velocity at ocean surface

Tj- Mean wind speed (m sec" )
a

, , Vertical turbulent velocity flux
u w -^

V Velocity vector (cm sec" )

V^ Velocity vector due to ocean currents

c -^

V^ Velocity vector due to Ekman flow
e "^

V Velocity of current due to quasi-geostropic flow

V Velocity due to the drift of R/P FLIP

w Vertical component of turbulent velocity

z Depth (meters)

dz Depth increment

a Expansion coefficient for heat

6 Expansion coefficient for salt

p Density of sea water (gm cm )

p Density of air (gm cm )

T Surface stress (dynes cm )

G Temperature

-v Represent the total instantaneous value

< > Denotes the vertical average

A Increment of change in a variable

ACKNOWLEDGEMENTS

The author wishes to express appreciation to Dr. R. W. Garwood,
Department of Oceanography, Naval Postgraduate School, for his time,
interest and guidance throughout this study, and to Dr. R. L. Elsberry
for his helpful discussions and review of this thesis.

Finally, I would like to express my gratitude to my wife, Darlene,
and to my son and daughter for their patience, understanding, and sup-
port during the last year.

I. INTRODUCTION

A. PURPOSE OF STUDY

The objective of this research was to investigate the conclusions
which were reached by Simpson and Paulson [1979] from ocean data ob-
tained as part of the Pole Experiment (POLEX), a component of the North
Pacific Experiment (NORPAX). POLEX was part of the first process ori-
ented NORPAX experiment. The experiment was named POLE to indicate that
the horizontal extent of the sampling was small compared to the largest
scales investigated in NORPAX. Simpson and Paulson concluded: "One-
dimensional mixed-layer deepening models failed to predict the mixed-
layer depth and temperatures observed during POLE. Horizontal advec-
tion, as evidenced from the salinity maximum frequently occurring at the
bottom of the mixed layer and other near-surface changes in salinity and
temperature not associated with local surface forcing, are responsible
for the failure".

This study will investigate the one-dimensionality of the POLE
temperature and salinity data and determine if the mixed layer depth
changes in response to the local atmospheric conditions (one-dimensional)
or if these changes are caused by two and/or three dimensional advective
processes. To accomplish this task, a non-stationary, one-dimensional
bulk model of the mixed layer originally proposed by Garwood [1976, 1977]

Simpson, J. J. and Paulson, C.A., 1979: Observation of Upper Ocean
Temperature and Salinity Structure During POLE Experiment, J. Phys.
Oceanoqr. , 9, page 869.

is expanded by including a salinity budget. A comparison of the re-
sults of a strictly one-dimensional model of the upper ocean with obser-
vations can be used to make an assessment of the one-dimensionality of
the different variables which are involved in the upper ocean boundary
layer response to the local atmospheric forcing.

One hypothesis to be examined is that horizontal advection may be
important for the heat and salt budgets, and at the same time vertical
mixing and mixed layer depth may be controlled primarily by one-
dimensional or local forcing alone. One objective of this study will
be to determine how responsive are the variables of heat and salt in
the upper ocean boundary layer to one-dimensional effects. A second
objective of this study is to use the data obtained as part of POLEX to
see if the one-dimensional mixed layer model proposed by Garwood will
predict the observed mixed layer depth, even if advection dominates the
local changes in heat and salt content in the upper ocean.

The data used in this report contain observations made from R/P FLIP
while under free-drift conditions. As a result of the free-drift condi-
tion of FLIP during the data collection period, advection as seen by R/P
FLIP will be a combination of the actual advection in an Eulerian coor-
dinate system for the area and advection due to the drift of FLIP.
Henceforth, the term advection will be used to mean a combination of the
actual advection term and apparent horizontal advection due to the drift
of FLIP.

The data were collected during the period 28 January 74 through
14 February 74. During that time, FLIP occupied a station approxi-
mately 800 miles north of the Hawaiian Island Chain under free drift
conditions. The position of FLIP ranged from SS^SQ'N, ISS^OS'W to

10

SA^Se'N, 155 25'W. The experimental site is hydrodynamically complex,
as shown in Figure 1. The Subtropical Front is known to meander between
32" and 35°N (Roden, 1974; Barnett,1976) . The region of the trade winds

northeast of Hawaii has air-sea fluxes of latent heat in excess of 35

-2 -1
cal cm day (Wyrtki, 1965). The Subtropical water mass formed in this

region contrasts markedly with the less saline Eastern North Pacific
Central Water characteristically encountered north of 35''N. The Horse
Latitudes are located only 3" of latitude to the south of the observa-
tion area and the North Pacific Current is expected to affect the gen-
eral hydrography of the region.

B. MILITARY IMPORTANCE

Mixed layer depth is an important factor when dealing with sound
propagation in the ocean. It is used to determine such sonar detection
variables as convergence zone distances, source level depth, and cutoff
frequency for surface duct propagation. A model which could predict the
change in mixed layer depth in response to the local atmospheric forcing
could be incorporated in the existing acoustic models, for example the
Fast Asymptotic Coherent Transmission (FACT) and the Naval Interim Sur-
face Ship Model (NISSIM), presently being used at Fleet Numerical Ocean-
ography Center (FNOC). A one dimensional mixed layer model would help
the naval tactician to directly correlate the variability of the ocean
with his weapon system and determine the most efficient deployment of
the naval Anti-Submarine Warfare (ASW) assets.

C. LITERATURE REVIEW

The vertical fluxes of heat, salt, and momentum across the sea-air
interface are the sources of almost all oceanic motions. In the fully

n

turbulent oceanic mixed layer bounded by the sea-air interface above and
by the dynamically stable water mass below, the vertical fluxes are
large. Below the mixed layer, they are usually negligibly small so that
one can decouple the mixed layer from the underlying stable, quiescent
water mass. This homogeneity is the root of the term "slab", which is
often used to describe the layer as depicted in Figures 2 and 3.

When dealing with sound propagation in the ocean, the term mixed
layer depth is sometimes taken to be that area with a uniform tempera-
ture structure, but this is not always the case. In Figure 4 the
bottom of the mixed layer depth (Region I) appears to be at about 80
meters, and below there is a strong temperature gradient. However, ex-
amination of the 0^ profile shows that the mixed layer depth is about 30
meters. This shallow halocline was probably caused by an earlier rain-
fall which decreased the near-surface salinity. The assumption of ver-
tical homogeneity in a bulk model, because of the problem associated
with solving for the interior fluxes of buoyancy and momentum, reduces
to the need to know only the surface and entrainment fluxes. However,
only small vertical gradients in these mean variables may be associated
with large turbulent fluxes. Therefore, the slab assumption should not
be as automatically applied to the turbulent kinetic energy budget.
The immediate local reaction of this mixed layer to those fluxes re-
sults in a homogeneous water column, i.e., vertical uniformity in the
mean velocity and density. This adjustment to the density and velocity
structure of the surface layers of the ocean to variable fluxes has been
the subject of a large number of studies since Ekman's [1905] treatise
where he originated the concept of a depth of frictional resistance for

12

the upper section of a wind stressed ocean. He further suggested that
the wind-driven current will have a similar depth, the Ekman depth.

Much of the one-dimensional theory for the ocean surface turbulent
boundary layer or mixed layer seems to be dependent upon the validity
of two basic hypotheses. The first of these is that vertical mixing
within the turbulent boundary layer and entrainment mixing at its base
occur in response to local atmospheric forcing, i.e., the surface_wind
stress and net heat radiated or fluxed across the sea-air interface. The
second hypothesis is that the mechanical energy budget is the key to the
understanding and prediction of the mixed layer dynamics.

Most observations show that local atmospheric forcing such as an
increase in wind speed increases the mixed layer depth and low wind speed
and high solar radiation decreases mixed layer depth. This is less true
with increased depth below the mixed layer. Observations by Pollard and
Tarbel [l975] showed that horizontal coherence between the fluctuating
wind-driven currents near the surface was large and consistent with the
local amplitude and phase differences of the wind history. With in-
creased depth, the horizontal coherence decreased monotonically as did
the vertical coherence of the deeper current measurements with those
near the surface. Kroll [1975] pointed out that the time scale of these
strongly inertial currents is on the order of a couple of days, depend-
ing upon the stability of the underlying water column as well as the
difference between the driving frequency and the local inertial fre-
quency. The turbulent motions within the mixed layer are expected to be
even more responsive to the local atmospheric conditions because of the
small dissipation 'time scale for the turbulent kinetic energy — minutes

13

to hours, which depends upon the layer depth and turbulence intensity.
Both the mean and turbulent kinetic energies in the mixed layer are ex-
pected to reflect the local boundary conditions for wind stress and sur-
face buoyancy flux. Measurements of turbulent velocity fluctuations,
which were made by Powell et al [l975] and Jones and Kenney [1977], show
that the root-mean-square turbulent velocity does scale with the water
surface friction velocity, u^. Camp and Elsberry [l978] and Elsberry
and Camp [1978] demonstrated the role of strong synoptic scale wind
events for deepening. Garwood and Hal pern [l976] and Elsberry and Raney
[1978] indicated the importance of diurnal heating, especially at times
between storms. Hindcasting simulations by Thompson [1976], Mellor and
Durbin [l975] and Camp [1976] support the importance of surface boundary
fluxes.

Kraus and Turner 1967 improved the one-dimensional model originally
developed by Kraus and Rooth [i961J . This was accomplished by consider-
ing the turbulent kinetic energy budget utilizing the heat equation and
a mechanical energy equation. Because the frictional generation of heat
is negligible, the vertically integrated heat equation provides a rela-
tionship far the conservation of potential energy. They parameterized
the mechanical production rate in terms of the friction velocity, but
neglected the viscous rate of dissipation and the effects of entrainment
shear production.

The application of mechanical energy principles to the problem of
mixed layer dynamics has caused some disagreement. An important contro-
versy in mixed layer modeling has been centered around whether mean kin-
etic energy or turbulent kinetic energy leads to the downward entrainment

14

buoyancy flux at the base of the turbulent boundary layer. The mean
kinetic energy advocates, including Thompson [j976j , Pollard p9773 and
Price [1977] , argue that this mean flow energy is directly converted to
potential energy by a mean flow instability that causes the bulk
Richardson number to be less than some critical constant of the order
one. Turbulent kinetic energy advocates do not disagree with the pos-
sibility of this mechanism, but they argue it is not the dominant mech-
anism because turbulence-generated instabilities will usually suffi-
ciently erode the interface to preclude the occurrence of the mean flow
instability. De Szoeke and Rhines [1975] and Yun Q978] demonstrated
that wind-shear production of turbulent energy is ultimately the most
important source of energy for deepening the mixed layer, if the surface
buoyancy flux is neglected. Even when the mean flow energy is important,
they find that the rate of conversion of this energy to potential energy
via entrainment shear production is controlled by the constraints of the
turbulent kinetic energy budget. The Kraus-Turner turbulent energy model
has not proven to be entirely satisfactory, and a number of changes have
been suggested. Stevenson [1979] examined a number of alternative para-
meterizations for dissipation. Zeman and Tennekes [1975], Elsberry et al
1^976], Alexander and Kim jj976] and Kim [l976] added depth-dependent
dissipation terms that helped to reduce unrealistic deepening-

Garwood [1976, 1977J developed an ocean mixed-layer model using the
Navier-Stokes equation of motion with the geostrophic component eliminated,
the continuity equation in incompressible water, the heat equation from
the first law of thermodynamics, the conservation of salt equation, and
a linearized equation of state. The entrainment hypothesis depends upon

15

the relative distribution of turbulent energy between horizontal and
vertical components and is offered as a plausible mechanism for govern-
ing both entrainment and layer retreat.

Garwood suggested that planetary rotation influences the dissipation
of turbulence for deeper mixed layers and enables a cyclical steady state
to exist on an annual basis. Furthermore, the rate of entrainment for the
stable regime can not be a simple linear extrapolation of the unstable
situations. Unlike the atomospheric boundary case, most of the solar
radiation does not penetrate the layer. Therefore, downward turbulent
heat flux in the oceanic boundary layer is as important as the upward
flux during the course of both diurnal and annual cycles. The non-
linearity of the interface entrainment tendency parameter, which is
greatest for stable surface boundary conditions, results in a modulation
of the long-term trend of the mixed-layer depth by the diurnal component
of surface heat flux. In this model, buoyant production is somewhat
more efficient than shear production as a source of energy for vertical
mixing because of its unique effect on the vertical component of the
turbulent velocity. The buoyancy equation is generated from the heat
and salt equations together with an equation of state.

p = Pq 1 - a (0 -

■ e^) + 6

and the definition for buoyancy,

b = g (pQ -

■ P'/PQ

a,

%

.s - Sq

)], (1)

(2)

In equations 1 and 2 9 is temperature, S salinity and p density
while a and 3 are the expansion coefficients for heat and salt, respec-
tively, and g is gravity. The tilde represents the total instantaneous

16

value and the subscript zero denotes a representative but arbitrary con-
stant value. The generalization of using b rather than 9 will cast the
model equations in a form equally applicable to those situations where
evaporation and precipitation contribute significantly to the surface
buoyancy flux and the structure of the evolving pycnocline. The buoyancy
equation also has a more obvious and direct role in the mechanical energy
budget as shown in Figure 5.

Miller |j976j included salinity in the mixed-layer model of Kraus and
Turner |j967] . He concluded that salinity is important in determining
the density structure when applying a mixed-layer model to the entire
ocean, but the relative importance of salinity effect on the short term
evolution of the density profile may not be significant. Paulus lj978j
included salinity in the mixed layer model of Camp [19761. ^^ concluded
that the effects of salinity structure were noticeable only in deepening
regimes.

17

II. DATA ANALYSIS

A. METHODS

The upper ocean temperature and salinity data and the meterological
data during the NORPAX POLE experiment were obtained from Simpson and
Paulson n977j. The parameters required for surface boundary condition
computations, which are used in Garwood's model, include wind speed, wind
direction, cloud cover, sea surface temperature, air temperature (dry
bulb), dew point and rain code. These parameters were recorded at ir-
regular time intervals. A linear interpolation technique was used to
obtain values for these input parameters at regular time intervals. The
resulting interpolated data were compared with the original data. Fig-
ures 6 through 12 show that no noticeable error was introduced into the
parameters by the use of a linear interpolation.

Temperature and salinity profiles as in Figures 3 and 4 were digi-
tized by using a Houston Instrument Hi Pad Digitizer and a Tektronix
4052 computer. The digitized profiles of temperature and salinity were
linearly interpolated into hourly intervals. This provided for both
initial conditions in the one-dimensional model and for fields to be
used to compute observed mixed layer depth, h (t), and to compare with
subsequent model computations of T(z,t) and S(z,t).

B. SURFACE FLUXES

The mixed layer model uses the total surface heat flux (Q-r(O)), in-
cident solar radiation, wind speed at a height d, rate of evaporation
(E), and rate of precipitation (P) to calculate surface fluxes of

18

buoyancy (heat and salt) and momentum. The friction velocity, in air,
is calculated using the following formulas:

^s = Pa^d^a

-^c = P.c.u? (3)

where u"^ is the mean wind speet (m/sec),
a

c. is the non-dimensional drag coefficient (1.4x10 ),

-3 3
p. is the density of air (1.25x10 gm/cm ), and
a

T is the surface stress (dynes/cm ).
The turbulent fluxes of latent heat (Q ) and sensible heat (Q. ) are
estimated using the following bulk aerodynamic formulas:

% - =d(-38Es - E,)u, (5)

The net back radiation (Q, ) is estimated from the following empirical

formula reported by Husby and Seckel |j975j

Q. = 1.14 x 10"''(273.16 + Tj^ (.39 - .05Ey^) (1. - .6C^) (7)
D s a

where E is the saturated vapor pressure of the marine air directly in
contact with the sea surface (.98 corrects for salt defects),
T, is the air temperature (degrees Centigrade),

a

E, is the vapor pressure of air at approximately 10 m based on dew

a

point temperature,
T is the sea-surface temperature (degrees Centigrade), and
C is the fractional cloud cover.
The upward heat flux is then:

Qa = Qe ^ Qh " Qb (8)

19

The solar insolation, Q , was estimated by

s

Q3 = (1. - a-h (1. - .66C^)Qq (9)

The constants a and b are adapted from Tabata 1_1964] and the cubic
cloud cover correction from Laevast IT960J . The coefficient C is the
fractional cloud cover and « is the mid-day elevation angle of the sun.
The clear sky radiation, Qq, is given by

Qq=Aq+Ai cos(J)+Bi si n({)+A2COs2c{)+B2S i n2(i) (10)

from the formula developed by Seckel and Beauday 11973]. The coeffi-
cients (Aq, etc.) were calculated by harmonic representation of the values
presented in the Smithsonian Meteorological Tables (List, 1958) with

(})=(2Tr/365)(t-21 ) where t is the Julian day of the year. Turbulent
temperature flux.

T'w',QX ='Qa_ (11)

pcp

turbulent velocity flux.

.2

u'w'(Q) = Ui (12)

turbulent salinity flux.

S'w'(Q^ = (P-E)S(o) (13)

and turbulent buoyancy flux,
(0)

b'w^.^ = g [ctT'w'^Q) - 6S'w'(qJ (14)

where a is the coefficient of thermal expansion,

3 is the coefficient of contraction of sea water due to salinity

variations, and
S/qx is the surface salinity value.

20

Figures 13, 14, 15 and 16 are graphs which show the hourly values of

wind stress, t, total heat flux at the surface, Qj, salinity flux, S'w'

and buoyancy flux, b'w', respectively, as calculated by the one-dimensional
model. These figures can be compared with Figures 17 and 18 which show the
interpolated hourly values of wind stress, i, total heat flux at the sur-

face, Qj, buoyancy flux, b'w', and various other components which were re-
ported by Simpson and Paulson [l977l. The values of these fluxes cal-
culated here are nearly identical to the values reported by Simpson and
Paulson. In Simpson and Paulson's graph on buoyancy flux (Figure 17), a
positive buoyancy flux results when Q . > 0. In the one-dimensional
model (Figure 16) a positive buoyancy flux results when Q . < 0 and E > P,

1 . Surface Wind Stress

The wind stress calculated using the bulk approximation with a

-3 -2

drag coefficient of 1.4x10 has a mean value of about 0.66 dynes cm

for the experimental period. Figure 13 shows the output for the surface

wind stress as calculated by the one-dimensional model. Analysis of

Figure 13 shows that for periods up to as long as 4 days the wind stress

is considerably less than 1 dyne cm , and it is evident that only two

periods of relatively high winds were encountered during the experiment.

One period of high wind near the middle of the experiment (time = 200-260

hours) with a maximum wind stress of 2.5 dynes cm , and the other at the

-2
end of the experiment with a maximum wind stress of 3.3 dynes cm .

2. Surface Heat Flux

Comparison of the total heat flux with net all-wave flux (Figures
17 and 19) shows that the net all-wave flux is generally the dominant
term in the surface heat balance. However, a two-day period of enhanced

21

latent heat flux, due largely to a significant increase in wind speed
midway through the experiment (time = 250 hours), is evident when com-
paring Figures 13 and 14.

Simpson's |j977j calculation of the total heat budget for the
experiment suggested that the ocean gained 201 cal cm for the time per-
iod from 3 Feb 74 - 12 Feb 74. Simpson's heat budgets calculations indi-
cated that ocean gained heat for 6 days, was in near thermal equilibrium
with the atmosphere for 2 days and lost heat to the atmosphere for 2-day
period. These results, coupled with the low wind stress, imply that the
upper dynamics might have been dominated by net surface heating for a
significant part of the experiment.

3. Salinity Flux

Figure 15 shows the salinity term as computed by the one-
dimensional model (equation 13) without including the precipitation term
in the computation. The precipitation term was omitted in the calculations
because no reasonable correlation could be made between a rain code given
in the experiment data report (Simpson and Paulson 1977) and an amount of
precipitation. The results of model runs which included precipitation
(estimated by the rain code) were compared to model runs which neglected
precipitation altogether. These model comparison runs showed no signifi-
cant difference except during one period when precipitation was much
larger than evaporation. Simpson and Paulson JJ979J observed that over-
all evaporation and precipitation were in near equilibrium, with precip-

_2
itation exceeding evaporation by 23 mg cm ,

4. Buoyancy Flux

Stratification in the ocean exerts a strong dynamical influence

22

on mixing since a stably stratified fluid requires work against buoyancy
forces if mixing is to occur. For a stably stratified fluid, the density
increases with depth and the buoyancy flux is directed downward, while
the converse is true for an unstable, stratified fluid. In the one-
dimensional model, surface buoyancy flux, b'w' /qn , is determined by the
air-sea transfers (equation 14). Figure 16 shows the model computed
buoyancy flux with Q . < 0 and E > P resulting in a positive buoyancy
flux. The net heat flux. Figure 14, dominates the surface values of
buoyancy flux during the experiment. During daytime periods, negative
buoyancy flux results from the domination of the solar heating at the
surface, while during the night periods a positive buoyancy flux is
produced by the combined effect of net long-wave radiation and the up-
ward turbulent fluxes of heat and moisture.

C. ADVECTION

The determination of the importance of advection to the heat and
salt budget is necessary before an assessment can be made on how one-
dimensional is the response of the variables of heat and salt are in
the upper boundary layer. The velocity vector V of the advection term
(V'(VT,VS)), is separated into two components in order to determine
which component dominates the advection term.

1 . Heat and Salinity Budgets

Neglecting internal sources and sinks, the equation for conser-
vation of heat or salt is:
0

-d L

9(T,S)+r7(T,S) + 9(T,$)'w' + 8(T,S)'u' + 3(T,S)'v'

8t d z d X 9y

(a) (b) (c) (d) (e)

23

dz=0 (15)

where T is heat, S is salinity, t is time, d is depth and V is the velo-
city vector in a coordinate system moving with the drifting platform.
The meanings of these terms are:

(a) Heat or salinity change with time,

(b) Heat or salinity change due to advection in the moving coordinate
system, assuming that when averaged over a day such features such
as internal waves would not contribute to advection,

(c) Vertical heat or saltflux divergence, and

(d) and (e) Horizontal heat or salinity flux divergence.

Terms (d) and (e) are assumed to be small because they can not be
evaluated. Therefore, equation (15) can be rewritten in terms of the

variable temperature:
0.

./[

|I.t-VT

dz = - (T'w'^qP .T'w'(^)

(16)

(f)

(g)

Term (f) is equal to the surface heat flux Qj/pc , where Qy is the net

-2 -1
surface heat (QT=Qs~QK-Qe'Qh"'"^c^ ' ^^'' ^^ ^^^ ' ^ ""^ ^^^ density of sea

-3 -1 -1

water, 1 gm cm , and c is the specific heat of sea water, 1 cal gm C

Term (g) is assumed to be negligible if d is well below the mixed layer,

i.e., d > h . The vertical integrated mean advection of temperature is

therefore.

./.

-d

(VVT)dz =

[/TdzJ,{/TdzJ,
-d -d

t

At

PC,

(17)

24

The same procedure used to derive (17) may be followed to find the
average advective change in salinity. The surface salinity flux is
given in equation (13). Thus the vertically integrated mean advection
of salinity is:

P

•VS)dz =

-d

r 0

Sdz
•--d

n u -1 p u

r 0 n
Sdz

At

/

S(E-P)dt

(18)

The magnitude of actual advection (in an Eulerian coordinate system)

can be checked by estimating V and VT or VS. The term V'(VT, VS), is the
advection quantity averaged over a time interval. At. The velocity vector
V can be separated into two components.

V = V + V^
c p

(19)

where V^ is the velocity vector due to the ocean currents and V^ is the
c -^ p

velocity vector due to the drift of R/P FLIP. The velocity vector due to
the ocean currents is itself composed of two components:

c g e

(20)

where V is the velocity of the current due to the quasi-geostrophic flow
and is assumed to be parallel to the surface temperature contour. There-
fore V -VT is assumed to be negligible. The order of magnitude of V ,
the velocity current due to Ekman flow, is a function of the wind stress
and can be estimated by,

|Vg| = [Tl = 1.57 cm sec'"' = 0.0565 km hour"'' (21)

phf

-2

where t is the average wind stress, 0.66 dyn cm , p is the density of
sea water, h is average mixed layer depth, 5000 cm and f is the Coriolis

25

parameter, Zcosinc}), where oo is the angular velocity of the earth and (j)
is the latitude.

The velocity calculated due to Ekman flow is insignificant when com-
pared with the drift rate velocity calculated for R/P FLIP (V =0.3 km hour" )
from 28 Jan 1974 to 14 Feb 1974. The calculated Ekman velocity is not suf-
ficient enough to explain the drift of POLE. Hence, R/P FLIP was by no
means flowing with the same mixed layer water mass throughout the experi-
ment. Therefore, the velocity vector V will depend on the drift rate of
R/P FLIP, V = 7 . What Simpson and Paulson [j979] call advection was not
actual advection for the area but apparent advection due to the drift of
R/P FLIP.

2. Calculation of the Rates of Advection

The distance R/P FLIP traveled from 0700 hours 28 Jan 1974 to
0700 hours 14 Feb 1974 (408 hours), was calculated to be 121 km, using
a rhumbline distance calculation and a position of SS^Se'N, 155°05'W to
34''36N, 155''25'W. Vertical profiles of temperature and salinity were
taken from R/P FLIP from 0000 hours 30 Jan 1974 through 0700 hours Feb
1974 (total of 367 hours) which covered a distance of approximately
109 km. Figures 19 and 20 show the observed changes in heat and salt
fluxes. The amounts of advection in a coordinate system moving hori-
zontally with R/P FLIP were then estimated by subtracting the accumu-
lated surface heat and salt fluxes from the changes in heat and salt
content, respectively. Figures 21 and 22 show the results of this cal-
culation. Least square fits were performed on the results, giving a

-2 -1
slope of 2.0 cal cm hour for heat content gain and a slope of

0.45 /OO cm hour" for salinity gain. These values can be used to

26

compute the average horizontal temperature and salinity gradients by
multiplying by the total hours of observation (367 hours) and dividing
by the average mixed layer depth (approximately 40m) and the distance
FLIP traveled during the observational period (109 km),

|V^T| = 0.0017 C km"^ (22)

|V|^S| = 0.0004 °/00 km'^ (23)

where |VmT| is the horizontal temperature gradient and IVmSI is the hori-
zontal salinity gradient.

For comparison purposes, gradients were obtained from a NORPAC
Atlas ri960j by dividing the atlas temperature change (AT = 0.2 "C) and
the atlas salinity change (AS = 0.05 /OO) between FLIP starting and
final positions (35''39'N, 155''05'W to 34''36'N, 155°25'W) by the total dis-
tance (121 km) which FLIP traveled,

IV^Tj^ = 0.0017°C km"^
|V^T|^ = 0.0004 ^/OO km'^
where IVmTL and |VmSL are the horizontal temperature and salinity
gradients that were calculated from the NORPAC Atlas. Although these
values agree with the earlier calculation, equations (22) and (23), they
were only obtained to show agreement in order of magnitude with recorded
atlas values. These results further illustrate the fact that what
Simpson and Paulson |l979] call advection was probably the result of the
drift of R/P FLIP.

Figures 19 and 21 show that the temperature field, which is a
function of both time and depth, has large fluctuations which can only
be explained by advection, or the drift of FLIP which is a non one-
dimensional effect. These large fluctuations dominate the heat budget

27

on the time scales shorter than 100 hours, but over a longer time scale,
surface flux is equally important (one-dimensional effects). Figure 23
is a plot of observed sea surface temperature and a plot of the model-
computed mixed layer temperature, with the model -computed temperature
slightly offset on the temperature scale for comparison. Comparing the
two plots, a one-dimensional component can be detected in the observed
sea surface temperature time series but the stronger component due to
the drift of FLIP is evident. Therefore, this apparent advection (as
seen by FLIP) of temperature is important on all time scales, but domi-
nates the short time scale. Figures 20 and 22 show that advection of
salinity dominates the salinity budget over both the short and long time
scales up to at least 400 hours, which was the period of observations.

D. MIXED LAYER DEPTH

1 . Observed Mixed Layer Depth

The observed mixed layer depth, shown in Figure 17 as used by
Simpson and Paulson h979j, is the shallowest depth at which the den-
sity is not more than 0.02 a. units greater than the density at a depth
of 5 m. However, salinity and temperature changes can occur above
z = 5m. In such cases, the above definition will result in a misleading
computed time series for mixed layer depth. A more practical definition
for observed mixed layer depth, which would be more consistent with the
precision of the available POLEX observational data, would be the shal-
lowest depth at which the observed value of o^ is 0.02 a. units greater
than the observed surface value. For comparison with the Garwood model,
the observed mixed layer depth (hp) will be defined according to this
latter definition.

28

Contours of temperature for the observation period are shown in
Figures 24 and 25. Figure 24, which is based on a contour interval of
0.5 C, shows a mixed layer depth as varying from near the surface to ap-
proximately 90 meters. Figure 25, which has a contour interval of 0.05 C

2

after having been smoothed twice, shows a mixed layer from near the sur-
face to approximately 80 meters. This illustrates the importance of
resolution when determining mixed layer depth from temperature contours.
A higher resolution frequently results in a shallower mixed layer depth.
Therefore, a temperature contour interval of 0.5 C is not sufficient
resolution to accurately determine mixed layer depth from the contour
plot. Although density is a function of both temperature and salinity,
during this time series the salinity influence was not significant, so
that the density was mainly dependent upon temperature. A mixed layer
depth definition could then be defined as the shallowest depth at which
the temperature is not more than 0.1 C less than the temperature at the
surface which would compare well with the similar definition based on the
observed density structure, as is evident by comparing Figure 25 with the
observed mixed layer depth in Figure 26.

2. Comparison of the Model -Computed Mixed Layer Depth with Observed
Mixed Layer Depth

Figure 26 is a time series of the depth of the model computed
turbulent boundary layer, -h(t), and of the observed mixed layer depth,
-hgCt). The value of h should be less than or equal to the mixed layer
depth as determined from observed 3p/az if the model simulation of the

2

Smoothing was accomplished by giving the center data point a weight

of 2 and all. of the surrounding data points a weight of 1.

29

vertical density structure changes are accurate. Figure 27 is a plot of
the same functions, h(t) and hQ(t) between time = 200 hours and time = 400
hours. Notice that the model results compare favorably with the observa-
tions except for a period of about one day starting at hour 285. Figure
28 is the model computed mixed layer depth, h,(t). The value of -h-jCt)
is taken to be the depth at which the model a;^^ is 0.02a. units greater than
the value of <o^> where < > denotes the vertical average for the turbu-
lent boundary layer. This definition is equivalent to that for hQ(t) if
the turbulent boundary layer is homogeneous. Figure 29 shows both hi(t)
and the observed mixed layer depth, h (t) for the same time period as
Figure 28. Notice that h,<_ h for all t. Although h^(t) compares well
with h(t), there are times during which hQ(t) is more comparable to h, (t).
For example, for the period between t = 285 hours and t = 310 hours,
hr.{t) is most similar to h-ilt). This happens because, although the tur-
bulent boundary layer may be quite shallow, the associated transient
thermocline is not sufficiently strong to influence the values for hi(t)
and hglt).

Only during one time period, at about time = 100, is the observed
mixed layer depth found to be significantly deeper than either the tur-
bulent boundary layer, h(t), or the mixed layer depth, hi(t), computed
by the model. This difference between the predicted and observed mixed
layer depth could be the result of R/P FLIP drifting through a warm core
eddy or filament of water mass having more tropical characteristics.
Figures 21 and 22, which are the graphs of the advective components of
the heat and salt budgets, show a large increase in the magnitudes of
both the advection of heat and salt at about time = 100 hours. This

30

large increase in advection of both heat and salt is consistent with the
hypothesis that R/P FLIP drifted across either an eddy or a filament of
water from the south of the Subtropical front. The Subtropical water
mass which lies to the south is warmer and more saline than the Eastern
North Pacific Central water.

3. Salinity Effects on the Mixed Layer Depth

An additional model simulation was conducted in order to deter-
mine what effect precipitation has on the mixed layer depth. A steady

-5 -2 -1
rate of precipitation of 5 x 10 gm cm sec was prescribed over a 12-
hour period starting at t = 200 hours. At all other times precipitation
was assumed to be negligible. This event corresponds in magnitude, at
least, with the rate of precipitation reported by Simpson [1977] for
5 February 1974. Figure 30 shows a rapid decrease in the surface salin-
ity with the injection of the precipitation. It drops from about 34.11 /OO
to 33.99 /OO. This drop in surface salinity causes a rapid rise in the
surface salinity flux as shown in Figure 31. Figure 32 shows the model
prediction of the mixed layer depth with and without the precipitation

event. The associated increase in downward buoyancy flux, b'w'/QS due
to the increase in salinity flux (Figure 31) caused the model mixed
layer depth to deepen much less rapidly than for the case without pre-
cipitation. The effect would have been even more apparent if the pre-
cipitation period had not coincided with strong winds.

Analysis of Figure 29 shows that the turbulent boundary layer
depth predicted by the model was approximately 20 meters deeper than the
observed mixed layer depth at t = 210 hours. This time period coincides
with that reported by Simpson and Paulson [19791 as the period of most

31

intense precipitation. These results suggest that, if the amount of
precipitation were accurately known and included in the model boundary
conditions, there would be an even better correlation between the model
mixed layer depth and the observed mixed layer depth.

32

III. SUMMARY

This analysis of 15 days of salinity, temperature and density pro-
files from R/P FLIP as part of the NORPAX POLE experiment and the ac-
companying model study suggest that, non one-dimensional effects were
y/ery important for both the heat and salt budgets. Vertical mixing and
mixed layer depth are controlled primarily by the one-dimensional response
of the turbulent kinetic energy budget to the local atmospheric forcing.
It is apparent that the non one-dimensional effects may be largely ex-
plained by the drift of R/P FLIP. Results from POLE site showed that this
apparent horizontal temperature advection (as observed by the moving plat-
form) was important on all time scales and dominated the small time scale.
Nevertheless, the one-dimensional model computation of mixed layer depth
compared favorably with the observed mixed layer depth. It is interest-
ing that even though this apparent advection usually dominated the salt
budget over both the long and short time scales, the one-dimensional mixed
layer model response to the surface salinity flux can be significant.
During the single period of strong precipitation, the mixed layer depth
was shallower (on the order of tens of meters) than it would have been
without precipitation. This is consistent with the results of Miller
[l976J, showing that salinity flux can at least occasionally play an
important part in mixed layer dynamics in the subtropics. Further
studies would have to be conducted with observational data that included
the time, duration, and amount of precipitation before a firm conclusion
can be made on the longer-term importance of salinity effects on the

33

mixed layer depth. Qualitatively, model performance was improved when
the influence of precipitation was taken into account.

Simpson and Paulson jj979j concluded that the one-dimensional models
failed to predict the mixed layer depth and temperature observed during
POLE. Why did the models of Pollard et al |2973j and Niiler [1975] , which
were used by Simpson and Paulson, fail to predict the observed mixed layer
depth? One part of the explanation for the model's apparent failure could
be the definition of mixed layer depth used by Simpson and Paulson. An
accurate determination of the mixed layer depth from a vertical profile
can only be made by considering the entire profile, an erroneous analysis
of the mixed layer depth is probable whenever there is a transient
pycnocline above z = 5 meters. Therefore, contrary to the results of
Simpson and Paulson [.1979], it has been shown that the one-dimensional
turbulence closure model used here does seem to explain most of the ob-
served mixed layer depth changes and that the mixed layer depth is more
influenced by the local surface fluxes than by horizontal advection.

34

45*

40'

P •• < * I ^ff^»^P^^PW^«^l

^■1 Fi|> y y^"n»»ii ■■■w^^

K^ mm ^»»

REGION OF SUBARCTIC FRONT

^PREVAiZlNG V/ESTERLIE5

'^^.

REGION OF SUBTROPICAL FRONT

%'s'//^Jvwoes

TRADE WIND REGION

^ ''^'^^f^^

160'

50*

Figure 1. The location of the R/P FLIP during the POLE

experiment (35^N, ISS^W) in relation to general
oceanic features. (Simpson and Paulson 1977)

35

z

h--
■h 5"

P-P^

/

>•

<U>

-> tz

^

1
1 /

^

" 1

1

5 II

y^"^

III

u(v)

Figure 2. Idealized density and mean velocity profiles of the ocean

mixed layer ( ). Mixed layer depth is (h), (5) is the

thickness of the interface or entrainment zone. Region I
is the fully turbulent mixed layer depth. Region II is the
slightly stable, intermittently-turbulent entrainment zone.
Region III is the stable underlying watermass having negli-
gible vertical fluxes in comparison to those of Region I.
(Garwood 1977)

36

SIGHR-T

^ 1 H

+

25.4 25.6 25.8 26-0 26-2

SRLINITY in PPT

I • 1 • 1 1 1 1 1 1 1

34.05 34.10 34.15 34.20 34.25 34-30

TEMPERRTURE IN DEG C
I • 1 • f--H 1 1 1 . 1

02/09/74

Figure 3. Typical POLEX data temperature, salinity, and sigma-t pro-
files taken from Simpson and Paulson [1979]. Added to the
profile is Region I the fully-turbulent mixed layer of
depth h. Region II the slightly stable, intermittently-
turbulent entrainment zone of thickness 5. Region III is
the stable underlying water mass having negligible fluxes
in comparison to those of Region I.

37

I — H

SIGrii^-T
-) » h

H \-

25.4 25.6 25.8 26.0 26.2 25.4
SRLIMITY IN PPT

+

34.05 34.10 34.15 34.20 34.25

TEMPEi^nTURE IN DEO C
! 1 1 • ; i 1 ! ! ^■

I

POLE 046 TI^tE 0604 (W) 02/03/74

Figure 4. Typical POLEX data temperature, salinity, and sigma-t pro-
files taken from Simpson and Paulson [1979]. Added to the
profile is Region I the fully-turbulent mixed layer of
depth h. Region II the slightly stable, intermittently-
turbulent entrainment zone of thickness 6. Region III is
the stable underlying water mass having negligible fluxes
in comparison to those of Region I.

38

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TtrC IN HOURS xio'

40

Interpolated hourly values of mass and buoyancy fluxes,
surface salinity, sea surface temperature, surface
sigma-t, and the various components of the heat flux
involved in the determination of the buoyancy flux.
(Simpson and Paulson 1977)

51

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Figure 18. Interpolated hourly values of the various components
of the surface momentum and heat balance are shown.
(Simpson and Paulson 1977)

52

o
o

0.00

80.00

160.00 240

TIME

00

IN

320.00

HOURS

Figure 19. The observed changes in heat content in the upper
40 meters (solid line). Model computed accumulated
surface heat flux (dotted line). The origin of the
abscissa is 0000 hour (local) 30 January 74.

53

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Figure 20. The observed changes in salt content in upper 40
meters (solid line). Model -computed accumulated
surface salinity flux (TOT), dashed line, which
is negligible in comparison with the observed
changes in salt content. The origin of the
abscissa is 0000 hour (local) 30 January 74.

54

o
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Figure 21

160.00 240.00

TIME IN

320.00

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Advection of heat (solid line) determined by '
subtracting the accumulated heat flux from the
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cm

hour

55

0.00

80.00

160.00 240.00 320

TIME IN HOURS

00

Figure 22. Advection of salt (solid line) determined by
subtracting the accumulated surface salinity
flux from the changes in salt content (Figure
20). Least square fit to data resulted in a
slope 0.45 0/00 cm hour-1.

56

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66

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190975

Shook

The one-dimension-
ality of the upper
ocean mixing and the
i7F£6cd» of adveGtienB^UF-S
ing the POLE experiment.

nfcHQ^ 2 82 03

Thesis 190975

Sii83 Shook

The one-dimension-
al ity of the upper
ocean mixing and the
role of advection dur-
ing the POLE experiment.