An estimate of turbulent velocities in the ocean

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An Estimate
Of ‘Vurbulent Velocities
In the Ocean

by

T. E. Pochapsky

MARCH 1, 1959

Contract

N6-ONR-27135

e F2er?00 ToED 0

MAO A

IOHM/TEI

_ ASTIA No, AD-214881 CU-87-59-ONR-271-Phys.

Columbia University
Hudson Laboratories
Dobbs Ferry, N. Y.

R. A. Frosch
Director

Technical Report No. 67
AN ESTIMATE OF TURBULENT VELOCITIES IN THE OCEAN

by
T, E, Pochapsky

March 1, 1959

This report consists of Copy No. 24

23 pages of 75 copies

Research sponsored by
Office of Naval Research
Contract N6-ONR-27135

Further distribution of this report, or of an abstract or
reproductions, may be made only with the approval of Chief
of Naval Research (Code 466). Reproduction in whole or in

part is permitted for any purpose of the United States
government.

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Abstract

Past eddy diffusion measurements have suggested that turbulent
velocity fluctuations occur in the deep ocean. An attempt has been
made to determine the magnitude and scale of the fluctuations from
estimates of shear velocities in the stably stratified ocean. The rate
at which the kinetic energy of these fluctuations is converted into
heat by viscosity is compared with an estimate of the rate at which
kinetic energy is supplied to the water by forces acting on the boundaries
of the ocean.

The primary purpose of this report is to obtain an order of
magnitude estimate of turbulent fluctuations which would be useful in
designing experiments for turbulence measurements at sea. It is shown
that vertical velocity fluctuations of less than 0.2 cm sec! can be
expected in the deep ocean and that characteristic fluctuating vertical
displacements would be about 50 cm. Corresponding magnitudes for the

horizontal fluctuations are 2 cm sec! and 5 km ,

Introduction

The general motion of ocean waters has features which are simi-
lar to those of atmospheric circulation. Details of this motion, however,
are not known and predictions based on hypothetical models do not have the
easy verification possible for a weather forecast.

The purpose of this report is to assemble a brief description of
the mechanisms relating to oceanic turbulence so as to obtain a better
picture of conditions to be dealt with experimentally. Ideally, one would
wish to measure the random turbulent velocities as distinguished from
steady or periodic motions and then determine the cause of the turbulence.
The arguments used here suggest, however, that the turbulent motions in
the ocean are so slow and weak that it may be impractical to obtain accu-
rate direct velocity measurements. There also seems to be no suitable
model of the ocean which can serve as a basis for deriving turbulent
velocities from, say, mean flow or transport measurements. Nevertheless,
turbulent fields can be specified in terms of velocity correlations
(Batchelor, 1953), and measurements of this type are feasible. They
could be done, for instance, by measuring the pair correlation separation
between free neutrally-buoyant containers. It is hoped that the eddy
lengths and time scales derived in this report on the basis of rather primi-

tive arguments will be useful in the design of such experiments.

The procedure adopted to estimate the turbulent structure is
first to determine in a general way the mean motion field associated
with the ocean. This is done so that the extent of turbulence-producing
shear flows can be appreciated and so that an upper limit can be set on
the velocity fluctuations.

The mean motion would be damped out unless it were maintained by
external forces. Because an equilibrium, of sorts exists, the dissipation
rate associated with the motion of the water must equal the rate at which
energy is supplied to the water in mechanical form from the exterior so
as to Cause it to move. Although this energy eventually appears as heat,
it first contributes to the kinetic energy of large scale motions and is
transferred to a viscous loss mechanism by means of intermediate turbulent
motions. Consequently this dissipation rate is an important index of the
intensity of the turbulence and calculations are made to establish an
upper limit for this quantity.

Finally, expected turbulent velocities and dissipation rates are
incorporated into models which consider shear flow, stability criteria, etc.,
so as to try to determine eddy sizes and time scales which are, hopefully,
of an order of magnitude of those that actually occur in the ocean.

For simplicity, all data will refer to the North Atlantic between

10°N and 60°N.

Energy of the Ocean
If the velocity distribution in the mean flow throughout the ocean

were known, the problem of estimating the degree of turbulence would be

considerably simplified. Sufficient data of this type does not exist.

Consequently, isolated measurements and somewhat more extensive measure-
ments across interesting areas, such as the Gulf Stream, must serve as
guides to guesses for the kinetic energy in any desired region.

The energy that must be supplied continuously from outside the
ocean to maintain the existing general circulation is ultimately of lunar
or solar origin. Much of it, however, enters indirectly through the
atmosphere. In the calculations below, only an upper limit is established
for this rate of doing mechanical work on the ocean which in turn is
equated to the rate at which the general motion produces viscously gener-
ated heat.

It will be necessary to consider the effect of the inhomogeneous
properties of the ocean on the flow primarily because work must be done
to overcome adverse density gradients. Fortunately, oceanographers
have always attached importance to a knowledge of the inhomogeneous structure
and adequate data is available for present purposes. This subject will be

considered in the subsequent section dealing with the stability of the ocean.

Total Energy
There is a large, almost steady, reservoir of kinetic energy in
ocean movements which range from the high velocities in the Gulf Stream
to the almost stationary deep ocean, An analysis was made of the im-
pressive Gulf Stream flow to determine the extent of the movement of water
in that flow relative to surrounding waters. Flow profiles obtained by

oe
Wust (S, 674) were used for the Florida Straits and data obtained by

ES
Available information on the ocean can often be found in The Oceans by
Sverdrup, Johnson, and Fleming (1942). References to that book in this
report will be abbreviated to read "(S, page number)".

Hae

Worthington (1953) for flow off Montauk Point. The results of these

calculations showed

Stream Average Flow of Volume Flow
Area Velocity Kinetic Energy
2 -1 -1 3. -1_,,-6
cm cm sec ergs sec meter sec x10

Florida Strait

360x10? 66 12x101® 24
Montauk Point
870x10" 69 38x10! 60

Flow maps suggested that the Gulf Stream circuit is about 15,000 km long,

20

so that roughly 13x10” cc of water is involved in the flow.

The area of water of the North Atlantic from the equator to 60°N,

including the Mediterranean, is about 4x10 “km” , and the total volume

is about 1.6x1072 ce » Consequently, the Gulf Stream circuit involves
about 1 per cent of the North Atlantic water and has an average kinetic
energy of approximately 6. 3x10° ergs, as deduced from the Montauk data.
The velocity associated with this kinetic energy is 110 cm sec! . Peak
velocities in the swiftest part of the current reach about 250 cm Sec.
(Worthington, 1953).

Turbulent velocities are random and have statistical features
which are relatively stationary in time. The large streaming motions can
be considered to be a shear flow which generates turbulence. Meanderings
of this flow, however, may be difficult to separate from turbulent fluctu-
ations. If the results for usuai jet flows are an indication, (Townsend

1956) the fluctuating turbulent velocities within the stream should not

exceed 5 per cent of the stream velocity and so are probably below 10cm sec !,

On the west side of the Atlantic, this high level of motion would take
place in a strip about 50 miles wide and 800 meters deep. The velocity
gradients in this strip would generally not exceed 0.0002 sec | ina
horizontal direction and 0.005 sec vertically.

Additional energy concentrations are found in "eddies" outside
the general course of the Gulf Stream as well as in the counter current
flowing beneath it. In both cases, measurements by Swallow and Worthing-
ton (1957) showed that mean velocities of about 10 cm sec! were not the
exception. These eddies are not considered to be included in the random
background turbulence. On the other hand, Swallow (1957) has obtained
deep current velocities of the order of 1 cm sec | near Salvage Island.

Outside the relatively small volume of strong streaming motion,
the highest velocities are probably tidally induced. These velocities,
of the order of 10 cm sec! (Bowden 1954), have the additional feature
of being periodic.

The above considerations do not give a very firm basis for esti-
mating the turbulent velocity fluctuations in deep ocean. For want of a

: ; -l
better guess, it will be assumed that they do not exceed about 1 cm sec .

Energy Dissipation
An estimate of the maximum rate at which mechanical energy is
supplied to the ocean by the moon can be made on the basis of the
secular acceleration of the moon. Jeffries (1952) estimates that lunar
forces do work on the earth at the rate of 1.4x102? ergs BCCu If

the assumption is made that this energy is delivered uniformly over the

oceans alone, 3.9 ergs sec! would be supplied to each square centimeter

Om. (GS, 15)

of ocean surface, The total area of ocean involved is 361x10
When discussing Taylor’s method of computing tidal dissipation, Jeffries

shows that practically all of this energy is lost in the form of heat in
shallow seas. Thus, very little of it could be converted into turbulent
motion in deep ocean.

The sun produces motion in the sea by a variety of processes. Direct
heating of the water produces mechanical effects as does heating of the atmos-
phere with its concomitant winds.

The energy supplied by the winds was obtained from the product of

the stress t exerted by wind on the surface of water and the associated

water velocity. Stresses were calculated from the equation

= 3.2x10°H" ,

where W is the wind velocity, and Wand <q are inc.g.s. units. A marine
atlas (U.S. Navy 1955) was used to estimate average winds for each month in
the North Atlantic. An average wind stress of 1:1 dynes em” was obtained.
The surface velocity imparted to the water was then assumed to be 3 per cent
of the wind velocity (Hughes, 1957) and the shear stress-velocity products
for each month were obtained. The average rate of energy input to the water
during a year was found to be 32 ergs me sec !, Knauss (1956) used a
value of 15 ergs em sec ~! which he obtained from averaged surface currents.
Most of this energy is probably converted to motion in the layer of water
above the thermocline.

An approximate value for the kinetic energy obtained from the sun
can be obtained by not questioning details of the process. Although the

yearly average incidence of radiation on the North Atlantic is approximately

-004 cal sec tem" (S, 103), only a small fraction of that energy can be

26-

transformed to a motional form other than heat. In fact, motion is
produced only as a Goushouancetee heating or cooling which supplies
mechanical energy by 1) evaporating water over large areas of the ocean
so that a return flow from other areas is induced, 2) increasing the
salinity or density of surface waters which then tend to sink, and

3) heating surface waters so that they expand above the equilibrium
water level and gain potential oiprie

The importance of a given process depends on the season (S, 122),
At 47°N, 12°W, for instance, most evaporation takes place from September
to November and the latent heat is supplied more from the thermal energy
stored during the summer months than from solar heating at the time.

To estimate the magnitude of the mechanical energy that could be
obtained as a result of evaporation, assume that steady evaporation has
lowered the depth of the ocean over a large area by the amount h and
raised the level by precipitation in another area by a simi lay amount,
Water flowing from the high areas to the low would then lose potential

or gain kinetic energy at the rate

2hpg os = 4,8x10~° h ergs cmisisect. a:

The quantities p and g are the density of sea water and the acceler-
ation of gravity respectively. Substitution of these values and the value
for the evaporation rate “ , of 2.9x107° cm sec 2 per unit area of ocean
into the above expression yields the quantity on the right of the equation.
The evaporation rate was determined from the annual rate of evaporation

Sim?

over all oceans, 334x10 km yr CS, 120). The kinetic energy gained in

aie

this process would be comparable to the maximum spplied by the moon only
if h exceeded, say, 10° cm, but the time required to evaporate this
quantity of water without its being replaced is much too long to be
realistic.

In a similar way it can be shown that the motion associated with
the density changes produced by heating or cooling are very small.

Finally, it is necessary to consider the potential energy of
the 8x10~% grams of salt left behind at the surface each second by the
evaporating water. If this salt fell 4000 meters to the bottom of the
ocean without mixing and transformed all its potential into kinetic energy,
it would release 28 ergs sec om - Obviously, such complete penetration
cannot occur, but this mechanism could induce some small stirring of deep
waters.

Mention was not made of the effect of returning rain waters and a
multitude of other possible processes. In any event, it would be hard to
imagine that more energy could be obtained than in a process in which the
sun served as a means of pumping distilled water from the surface of the
ocean and returning it to the bottom of the ocean from which it eventually
returns to the surface. Then, the same result would be obtained as was
calculated for the falling salt precipitate. Thus the maximum kinetic
energy realizable from solar heating of the water is below 56 ergs sec !om
of ocean surface, as obtained by adding the hypothetical salt and fresh
water contributions.

It should be noted that the maximum value of 56 ergs sec Jem -

for the kinetic energy imparted to the water by solar heat is very small

compared to the incident radiation of approximately 10° ergs ice an .
Conversion is thus very inefficient and there remains the possibility
that some subtle and unrecognized mechanism supplies more motion to the
ocean than calculated above. Lacking this evidence of another process,
the present value for the rate of supply of motional energy can certainly
be considered an overestimate, probably by at least one order of magnitude.
Heat flows from the ocean floor at the rate of approximately
107° cal fii secu , (Bullard, 1954), Mechanical effects associated with
this flow rate can be ignored except possibly in parts of the deep ocean
which may be practically unaffected by motion imparted by the other sources
mentioned.
Summarizing, the maximum average kinetic energy possible produced
below a unit area of ocean surface by the various sources is:

Tide 3.9 ergs sec 1 emi

Wind 32 y

Direct Sun 56 "

Most of this energy might only be associated with water in the layer
approximately 100 meters deep over the thermocline. In that event, steady
turbulence could not dissipate energy faster than about 100 ergs sec!
per unit area of ocean surface of 107° ergs sec! per unit volume of
water above the thermocline. Some of the energy must appear in deep ocean,
but it is hard to believe that the fraction could exceed 1/10. In that
event, 2x107> ergs ay sea! coma tha the Gedo supplied to maintain

turbulence in a 4000-meter ocean, ‘i.e., 107x107! x¢4x10°) 7: 6

gow

Homogeneous Isotropic Turbulence

For turbulent motion to be homogeneous in a medium, it is neces-
sary tt the statistical properties of the velocity fluctuations be
identical at all points of the medium. Because the ocean has boundaries,
the turbulence could not be homogeneous unless most of the eddies had a
size that is small compared to, say, the depth of the ocean. If, in ad-
dition, the ocean were a homogeneous fluid, with no temperature or density
gradients, it is possible that these smaller eddies would produce velocity
fluctuations which are similar in all directions; i.e., the turbulence
would be isotropic as well as homogeneous.

The ocean is not a homogeneous liquid. Nevertheless, if the turbu-
lent eddies are small enough, the effect of the density gradients in turn
might prove to be so small that the existing theory of homogeneous isotropic
turbulence can be applied to obtain a useful first approximation of the
turbulent motion,

Instead of trying to determine whether homogeneous isotropic turbu-
lence takes place in the ocean, or not, it would be more convenient to
assume that it does and then determine whether the associated size of the
eddies is consistent with known conditions in the ocean. This could be
done if the total energy, E , of the turbulent motion were known together
with the rate, ¢ , at which heat is produced as a result of the viscous
motion. Upper limits for these two quantities were established in the
preceding section.

The theory of homogeneous turbulence (Batchelor, 1953) shows the

energy density, E(k,t) , associated with particular eddy sizes at high eddy

=10=

Reynolds number is
BCL) enue

where E(k,t) and e are ona unit weight basis, a7~0.3, and = on

is a measure of the eddy size. Because most of the energy is concen-
trated in the larger eddies, this expression can be integrated over all
k > La to give the total energy without particular concern for the
small amount of energy attributed to those smaller eddies having a low
Reynolds number. Then this may be written
Be OA eS

In deep water, it was shown that e < 2x107° ergs gm sec” and
E < 4 ergs gm (for velocities below 1 cm/sec). Substitution shows a
rN of 3, 4x10" cm to be representative of the size of the larger eddies
if the equality signs hold for ¢« and E . The fact that ) turns out
to be almost equal to the depth of the ocean is completely fortuitous,
and these calculations do no more than indicate that a homogeneous ocean
would be continuously well stirred. This conclusion would also hold for
reasonable reductions in the values of ¢ and E used. Inasmuch as the
ocean is not well stirred, the theory of homogeneous isotropic turbulence
is generally inapplicable. If water above the thermocline is treated
similarly, now, with E <4 and ¢€ = 107° + \ becomes 675 cm, This
layer is actually about 104 cm deep and can be considered fairly well
stirred by the wind so that the theory may have possible general appli-
cation in this case.

Energy is lost from the larger eddies by transfer to smaller ones.

Eventually eddies become small enough for velocity gradients to be large

=11Ts

enough to permit normal viscous loss and little further diminution in
size takes place. Dissipation by fluid viscosity occurs at lengths

characterized by (Batchelor, 1953):

ne
27°
in| Ee f )

where y , the kinematic viscosity, is .015 ( in c.g.s. units) for water.

(ss)

For ¢ = 2x107> ergs, 1 = 0.7 cm. The velocities associated with these

eddies would be extremely smali.

The Stably Stratified Ocean

In the previous section it was shown that if turbulent velocities
are approximately lcm sec! in homogeneous water, eddy sizes would be
almost equal in magnitude to the depth of the ocean. However, if a unit
volume of water could be moved isothermally along a vertical eddy from
the cold bottom of the ocean, the density difference between it and the
surrounding water would necessitate doing work against gravity in the amount
of 70,000 ergs in order to reach the surface. Because the kinetic energy
at depth is only of the order of an erg, the sizes of the vertical eddies
must be several orders of magnitude less than the depth of the ocean.

The ability of turbulent stresses to move a fluid against gravi-
tational forces can be expressed quantitatively by Richardson's number
(Proudman 1953):

a bbe

i es 2

Where the denominator represents work supplied by large scale motion

working against the Reynolds stresses of turbulent motion. In this equation,

B=

g is the gravitational force; p , the fluid density; z _, the height
of the layer considered; U , the velocity of mean, or non-turbulent,
flow. The implication of this equation is that a horizontal shearing
= ' generates turbulent energy at a rate which must be larger

than some critical rate at which energy is.lost by mixing at the higher

motion,

gravitational potential near the top of an eddy. In order to have turbu-
lence, R; must not exceed some critical value. Some type of horizontal
shearing motion is needed for continuation of the turbulent motion and
this may be the result of tidally induced boundary flow over the bottom
of the ocean, wind driven surface currents, or even gradients in the
large scale horizontal eddy motions.

Values of the stability, - a , were calculated for the North
Atlantic from data obtained on the Crawford cruise 16, 1 Oct. -11 Dec.,

1957 (Metcalf, 1957). The averages were

depth, meters stability —

1/p = - em?

1000 - 2000 2.5 x10"?
2000 - 3000 4x10 19
3000 - 4000 2x10 19

Townsend (1958) showed that turbulence in the developed shear
flow of a stably stratified medium will collapse if R; is more than

about 0.1. For a stability of 2x10°19 em7! , the gradient required

=13-

“1 Multi-

to maintain turbulence must be greater than 1.4107? sec
plying this by the depth of the ocean, 4x10°em, yields a value of

560 cm/sec for the change in velocity of the top relative to the bottom,
exclusive of the sharp drops occurring in the boundary layers. Tidal
currents are of the order of 10 cm/sec and wind induced velocities
rarely exceed 30 cm/sec. Consequently it is doubtful that wind or tide
could produce vertical turbulent fluctuations in the deep ocean, if Town-
send's criterion is applicable. ‘It should be noted that the largest

gradients in the Gulf Stream are only about BOR” seer

Eddy Diffusivity

A turbulent fluid can. transport properties such as momentum or
temperature at a much higher rate than a non-turbulent liquid.as a result
of the eddy motion. Measured values of the eddy viscosity for momentum
transport or eddy diffusion for the transport of impurities can be used
to get a measure of the associated turbulent velocities.

Eddy diffusion coefficients have been obtained at various locations
in the ocean (S, 481) by measurements of changes in temperature, salinity,
concentration of dissolved gases, etc; eddy viscosity measurements, which
represent momentum transport, are generally based on relations assumed to
exist in the velocity field being studied. In a horizontal direction
both types of coefficient are approximately equal, but in the vertical
direction the viscosity coefficient is ten or more times larger than the
diffusion coefficient. A given coefficient changes in going from the
vertical to the horizontal direction by the staggering factor of about one

million.

%
Both coefficients result from a mass transport and they can have equal C.g.S.
numerical values because water conveniently has unit specific heat (S92).

-14-

Recent determinations of deep ocean vertical eddy diffusion
coefficients, A, » were made by Koczy (1956) and were based on measure-
ments of the concentration of radium. Values of about 10 gm cm sec
obtained near the bottom fell to values below 1 gm Cmissecis at.a
height of 3 km. The viscosity of non-turbulent water is about
-01 gm cm! sec! . Therefore, the presence of turbulent motion in deep
ocean seems certain.

These measurements allow an estimate of the energy lost by the
velocity gradient previousiy calculated to be necessary to maintain turbu-
lence. This energy is of the order of (Proudman, 1953)

2

cals? ~

AL dz

= 10x(1.4x10 ~)

or 2x107> ergs cn Again, the agreement obtained with the same value
established for the maximum rate of energy supplied from outside the ocean
to maintain the turbulence is fortuitous. Nevertheless, the high gradient
is not inconsistent with the maximum rate of energy dissipation even though
it is inconsistent with the expected wind-induced or tidal gradients. If
the turbulence is not induced by a mean horizontal shear flow, two other
mechanisms come to mind: 1) large and small scale horizontal motions inter-
act to produce vertical fluctuations, and 2) internal waves in the strat-
ified medium interact with the horizontal eddies to produce an effective
vertical turbulent component of motion (Townsend, 1958).

In shear flows, the eddy viscosity is equal to (Townsend, 1956)

=5=

where Hand W are the root-mean-square turbulent components in the
horizontal and vertical directions. In such flows uw= 0.4 u", (Towncend,
1956), so that the r.m.s. velocity in the minimum velocity gradient re-
quired to maintain turbulence is about 0.2 cm sec! for A= gm em !sec?.
Further insight into the turbulent diffusion can be obtained by
noting that particles moving in a one-dimensional random walk have a distri-
bution after a sufficient time which is described by the diffusion equation.
The diffusion coefficient is replaced by 3nt? ,» Where n is the jump rate
and 2? is the constant jump distance. If is associated with a
characteristic eddy size, 3nt has a value which approximates the velocity.
For an A, of 10 gm cn. sec! for a velocity of 0.2 cm deen! , & must
be 50 cm , the order of size of the eddies needed in the vertical di-
rection. The time required for a particle to circulate around an eddy
is then about 250 seconds.
An attempt was made to apply the random walk method to horizontal
diffusion for the case where the horizontal motion consists primarily
of a shear flow. Vertical fluctuations in position would move the dif-
fusing particles in random steps through layers of differing horizontal
velocities. Any particle leaving a reference plane as a result of
vertical motion would be replaced by another particle having a horizontal
velocity characteristic of a plane separated from the reference plane

by a distance equal to the vertical eddy scale. Then many particles in

the reference plane would have velocities differing from that of the

ay oe

mean velocity of the plane by eda

- As an approximation, it was

assumed that all particles in this plane were engaged in a random walk

Stee

with a step length of t, a tT where T= = is the vertical time scale.
The jump rate for conditions described in the previous paragraph, .008 secu.
then requires that a velocity gradient of 4 sec! be present to obtain
diffusion coefficients of 10° gm em? sec! - Such a gradient obviously
cannot be present, and so diffusion probably takes place as a result of
existing horizontal eddies having a large scale.

Some idea of the characteristics of the horizontal eddies can be

obtained from the horizontal diffusion coefficient, Ay , and from the

maximum rate of energy dissipation, 2x107° ergs gm”! sec! . Thus,

eh 23 Bea On ae
5 ny, = Uy ty = 10> = Au
dU, 2 AL? -5
Ay Ge ~ Ay G) < 2x10

H
lead to
u,, < 2 cm secs
H

ty > 5x10" cm

and a time scale of the order of several days. The value of these results
should be tempered by noting that 1) the expression for the energy dissi-
pation applies to shear flows and may be appreciably in error when applied
to the ocean, 2) the resulting length scale may be too large to have a
meaningful random walk approximation, and 3) the horizontal diffusion
coefficient used is based on experiments which were probably obtained over
a smaller geographical area than required to obtain good statistical data

for the large eddies expected.

Se

Discussion

The preceding computations suggest that the vertical component
of the background turbulent motion in deep ocean has a typical velocity
less than 0.2 cm sec! , a displacement of more than 50 cm and a fluctu-
ation time greater than 250 sec. Corresponding figures for the hori-
zontal fluctuations are 2 cm Bees > okm , and 2x10" seconds.

These figures were based on the assumption that the turbulence is gener-
ated by comparatively steady shear flows in the ocean for which the
Richardson's number does not exceed 0.1. They are consistent with
observed diffusion rates and the maximum possible rate at which external
forces at the boundaries of the ocean could supply energy to maintain the
motion.

The requirement on the Richardson's number is a stringent one.

It can be criticized as applying to a developed shear flow and not to
uncertain oceanic conditions. In fact, other observers have reported
numbers ranging from 10 to 100. Reference should be made to Townsend
(1958) for a discussion of this point.

If the Richardson's number is assumed to be 10 instead of 0.1,
the shear velocity gradients would be one-tenth as large and the vertical
velocities would be reduced by a factor of about 3 . Both the length
and time scales would increase. The smaller Richardson's number was used
in the preceding computations so as to obtain an upper limit for the
velocity fluctuations.

It was necessary to attribute horizontal diffusion to a horizontal

eddy structure. The origin of this motion remains unknown. It is possible

ie=

that alternating shear stresses, caused by wind or moon, in a Coriolis
field may build up such a velocity field. On the other hand, it may
somehow be induced by major currents such as the Gulf Stream. These
possibilities, however, must remain speculative until more attention
is given the problem.

The structure of the field of horizontal motion is important
because it probably determines the vertical fluctuating velocities.

So far, these vertical turbulent motions were postulated to explain
vertical diffusion, and gradients in the horizontal motion were then
postulated as necessary to maintain them. There remains the possibility
that the relation between horizontal motion and vertical diffusion is
not quite so indirect.

In effect, ideas derived from studies of turbulent motion in
shear flow have led to a picture of an ocean in which the role of shear
flow is uncertain. Despite this, it is probable that the approach used
was simple enough for the limits established on the motion to be valid
to within an order of magnitude.

Much continued experimental work on "ambient" oceanic turbulence
is obviously needed, but, if the present results are any indication,
this research will be difficult. Diffusion measurements on a large
scale are particularly convenient because they can be made on material
introduced naturally, or artificially well in advance of the gradient
measurements. The outlook for making direct measurements of the turbu-
lent velocities or associated pressure fluctuations is discouraging.
Fluctuations in the horizontal velocity of the order of a centimeter

per second over a day would be difficult to separate from periodic tidal

=19=

or other large-scale motions. Variations on the vertical motion are
also small enough to pose severe measuring problems. Pressure fluctu-
ations are almost numerically equal to the square of the velocity and
so are of the order of 1 dyne on . These pressure changes would be
difficult to detect at the expected frequencies of 0.004 sec or lower.

In most cases the detection equipment could not be tied to a
ship. Free floating devices, such as Swallow's neutrally buoyant
cylinder, offer exceptional opportunities for quiet measurements and
invaluable data could be collected if telemetering devices were developed
to send up information directly from such cylinders.

A promising method of studying the turbulent motion is to follow
the motion of two or more neutral cylinders at depth. Measurements of
changes in their separation with time should give detailed information

on the velocity structure of the water.

EJ0=

References

Batchelor, G. K,
The Theory of Homogeneous Turbulence, Cambridge, University Press (1953).

Bowden, K. F.
Deep-Sea Res. 2, 33-47 (1954).

Bullard, E. C.
Deep-Sea Res. 1, 65-66 (1954).
Hughes, P.
Quart. J. R. Met. Soc. 82, 494-502 (1956).

Jeffries, H.
The Earth, Cambridge, University Press (1952).

Knauss, J. A.
J. Acoust. Soc. Am., 28, 443-446 (1956).

Koczy, F. F.
Nature, Lond., 178, 585-586, (1956).

Metcalf, W. G.
Woods Hole Ocean. Inst., Ref. 58-31 (1957).

Proudman, J.
Dynamical Oceanography, John Wiley, New York (1953),

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Swallow, J. C.
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Townsend, A. A.
The Structure of Turbulent Shear Flow, Cambridge, University Press (1956).

Townsend, A. A.
J. Fluid Mech. 3, 361-372 (1958).

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Marine Climatic Atlas of the World, Vol. 1, North Atlantic Ocean,
Pub. by Direction of the Chief of Naval Operations, Washington, D. C. (1955).

Worthington, L. V.
Woods Hole Ocean. Inst., Ref. 53-85 (1953).

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Office of Naval Research (Code 466)
Navy Department

Washington 25, D. C.

copies 1-2

Commanding Officer and Director

U. S. Navy Underwater Sound Laboratory
Fort Trumbull

New London, Connecticut

copies 3-4

Director

U. S. Naval Ordnance Laboratory
White Oak

Silver Springs 19, Maryland
copies 5-6

Commander

U. S. Naval Air Ordnance Test Station
China Lake, California

copy 7

Director, Marine Physical Laboratory
University of California

Scripps Institute of Oceanography
San Diego, California

copy 8

Director

Bell Telephone Laboratories
Whippany, New Jersey

copies 9-10

Committee on Undersea Warfare
Naval Research Council
National Academy of Sciences
2101 Constitution Avenue
Washington 25, D. C.

copies 11-i2

Chief, Bureau of Ships (Code 688)
Navy Department

Washington 25, D. C.

copies 13-14

The Hydrographer

U. S. Navy Hydrographic Office
Washington 25, D. C.

copies 15-16

Director

Naval Research Laboratory
Washington 20, D. C.

Attn: Code 5500, Dr. H. L. Saxton
copies 17-18

Commanding Officer and Director

U. S. Naval Electronics Laboratory
Point Loma

San Diego 52, California

copies 19-20

Commander

U. S. Naval Air Development Center
Johnsville, Pennsylvania

copies 21-22

Dee b> Vo Hunt

Harvard University
Cambridge, Massachusetts
copy 23

Director

Woods Hole Oceanographic Institution
Woods Hole, Massachusetts

copy 24

Director

Lamont Geological Observatory
Torrey Cliff

Palisades, New York

copies 25-26

Chief, Bureau of Aeronautics (AV-43)
Navy Department

Washington 25, D. C.

copies 27-28

Chief, Bureau of Ordnance
Navy Department
Washington 25, D. C.
copies 29-30

Chief of Naval Operations

Navy Department

Washington 25, D. C.
(Op-001) copy 31
(Op-91) copies 32-33
(Op-312) copies 34-35
(Op-316) copies 36-37

Commanding Officer
ONR Branch Office

Navy No. 100

Fleet Post Office

New York, New York
copies 38-39

British Joint Services Mission
Main Navy Building, Room 4924
Washington 25, D. C.

via

Chief of Naval Operations (Op-316d)
Navy Department

Washington 25, D. C.

copies 40-42

Admiralty Research Laboratory
Teddington-Middlesex, England

via

Chief of Naval Operations (Op-316)
Navy Department

Washington 25, D. C.

copy 43

Commander

Armed Services Technical
Information Agency

Arlington Hall Station

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copies 44-53

Commander

Submarine Development Group II
U. S. Naval Submarine Base
Box 70

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copies 54-55

Weapons Systems Evaluation Group
Office of Sec'y of Defense

Room 2E1006- Pentagon Building
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copy 56

Commanding Officer
ONR Branch Office

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New York 13, New York
copy 57

Canadian Joint Staff

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via

Chief of Naval Operations
(Op-316)

Navy Department

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copies 58-60