LJBflARY

TECHNICAL REPORT SCCTKXI
NAVAL POSTGRADUATE SCHOOL
MONTEREY. GAUFOOMIA S&940

NPS-61Md73111A

NAVAL POSTGRADUATE SCHOOL

//

Monterey, California

15

Nov

1973

NPS-61Md73111A

Pred

icting Sound ]

Dhai

se and Amplitude Fluctuations

due

to

Micros tructure

in

the

s Upper Ocean

Physics & Chemistry

H.

Medwin

Department

Approved for public release; distribution unlimited.

FEDDOCS

D 208.14/2:NPS-61MD73111A

NAVAL POSTGRADUATE SCHOOL
Monterey , California

Rear Admiral Mc B. Freeman M0 U. Clauser

Superintendent Provost

TITLE: Predicting Sound Phase and Amplitude Fluctuations Due to Micro-
structure in the Upper Ocean

AUTHOR: H. Medwin

ABSTRACT:

Tne temporal and spatial variations of the index of refraction cause fluctuations
of sound phase and amplitude that can be completely understood only by defining
the index in terms of the duration, location, range and time of the acoustic
experiment. A truncated "universal" spatial correlation function of the index has
been derived from a simplified form of the Kolmogorov-Batchelor spectrum of
temperature fluctuations in a homogeneous, isotropic medium. Although this
correlation function is shown to be predictable simply from the depth of the
experiment, it is of only limited validity with respect to large spatial lags.
However, a Gaussian extrapolation of the "universal" correlation function
together with the standard deviation of the index provides simple useful predic-
tions of the sound fluctuations due to temperature micros tructure in the upper
ocean.

This task was supported by Naval Ship Systems Command (Code PMS 302).

TABLE OF CONTENTS

1. Introduction: Sources of Sound Phase and Amplitude Fluctuations

1.1 Microstructure in Mixed Water

1.2 Layered Microstructure

1.3 Other Underwater Structural Features

2. Wave Number Spectrum and Spatial Correlation of the Index of
Refraction in a Mixed Layer

2 . 1 The Wave Number Spectrum

2.2 The Spatial Correlation and Structure Functions of the
Microstructure

3. Sound Phase and Amplitude Fluctuations Due to Microstructure

3 . 1 Phase Fluctuations

3.2 Amplitude Fluctuation

3.3 Small Amplitude Fluctuations: Laboratory Experiments

3.4 Large Amplitude Fluctuations

3.5 Fluctuations at Sea

3.6 The Limits of Predictions

4. Conclusion
References

Initial Distribution List
Form DD 1473

1 . Introduction
Sources of Sound Phase and Amplitude Fluctuations

As it proceeds away from its source, a sound at sea encounters changing
values of temperature, salinity and density, as well as entrained objects and
bubbles in motion. These acoustically significant quantities are constant
neither in space nor in time. Slower than the microscale fluctuations due
to surface waves and turbulence of the medium, which show perceptible changes
during a time scale of the order of seconds , are changes due to internal waves
with periodicities of the order of minutes , tidal and diurnal variations , and
seasonal changes. Depending on the duration of the study, some of these
fluctuations may appear to be simply periodic, some show the spectral char-
acteristics of a narrow band noise, many are describable only by statistical
methods , and still others, such as the appearance of schools of fish or the
incidence of storms, are largely intermittent in character. (Weston 1969).
It is important to make the point that it is not the inhomogeneities but rather
it is the temporal change in the position or character of the inhomogeneities
that is the source of interest in this report.

The phase of a sound wave travelling through a stationary medium depends
on the speed of sound along the path. This dependence is expressed by
equations 1.1 and 1.2:

^^ ds = «• ^fiw ds

<p= sound phase
where c = c (T,S ,H, n(a)da) (1.2)

|V| = magnitude of component of material
velocity along the ray path

k = wave number along ray path

T = temperature

S = salinity

H = depth

n(a)da = number of bubbles of radius , a, in increment, da

The change of amplitude of a non-divergent, unattenuated sound beam
along a fixed path through the water is caused by the lens-like patches or
layers in the medium which create convergences or divergences of sound power.
This focusing and defocusing changes with movement of the patches, and the
sound amplitude at the receiver thereby fluctuates.

1 . 1 Micros tructure in Mixed Water

It will be many years before the turbulent, patchy, nature of the ocean
will be unscrambled by fluid dynamicists and physical oceanographers . How-
ever, a few general concepts and landmark ideas of turbulence theory will be
helpful in guiding our thinking particularly about the upper ocean.

Near the ocean surface, depending on the time of day and the meteoro-
logical history at the location, there is often a well-mixed layer of so-called
"isothermal" water. The name is a reflection of the fact that common bathyther-
mographs (BTs) , show such a region as isothermal. In fact, temperature probes
of higher resolution and shorter time constants have revealed that the crudely
isothermal region is characterized by drifting patches of water of slightly dif-
fering temperature, Fig. 1.

H — I — I — I — f-

-l — l — I — I — I — I — I — h

I I I — f— f

4-H f-

H 1 — I-

Fig. 1 Temperature fluctuations Run 6,4.3 Meters, 0354, 22 October 1971

Within such a region there are temperature fluctuations at a point whose
standard deviation, gl, may be of the order of 0.1 C. We will assume that
this value of ol may be obtained at a point by averaging over an appropriate
time, or at a given time by averaging over an appropriate local space
(ergodicity) . These values are a function of time of day and depth of measure-
ment. The short-time standard deviation of the temperature fluctuations meas-
ured with respect to the mean temperature is generally greatest near the surface,
greatest during the late afternoon, particularly during sunny periods. (Sagar 1960)

For example, measurements (Seymour 1971) made during the late afternoon
of a sunny day in October, one mile off San Diego, Calif, reveal a decrease
in gl from 0 . 15 to 0 . 14 to 0. 13 C as the measurement depth was increased
from 7 to 9 to 14 meters. The effect of night time cooling is interesting; by
early morning of the following day the rms temperature fluctuations in the
experiment had decreased to approximately 0.06C and showed no clear
dependence on depth. The water temperature gradient was only about 0.1 C/
meter, being warmer at the surface.

Although material velocity variations generally have a lesser acoustic
effect than temperature variations (Neubert 1970), there are circumstances

where the motion of the medium is significant, and may even overwhelm the
temperature influence. Two situations should be considered: Near the surface
of the ocean the underwater material velocity will be almost totally dependent
on the surface wave height. For long-crested (swell) waves in deep water
the vertical and horizontal, waveward, components of the material velocity for
any frequency component of the surface wave will be almost totally orbital
"near" the surface.

The depth dependence is given by

V (x ,Z) = V (x ) e"XwZ (1.3)

w w

where x = 2 ft/A = surface wave propagation constant or
w w ,

wave number

Z = depth

V (x ) = orbital velocity of the x wave component at the
w

surface

Let us call the above underwater motion that is completely, and directly,
ascribable to the surface wave action, the "coherent component" of the
material velocity. Superimposed on. the coherent motion of the medium there
will be the "incoherent" turbulent motion, often, and for simplicity, assumed
to be independent of direction. The region in which three dimensionally
isotropic, or two dimensional horizontally isotropic, turbulence can be assumed
to exist is a function of depth and of sea surface wave conditions. As a
second example there is evidence (Dunn 1965) of horizontally isotropic turbu-
lence in Menai Straits, 30 minutes before high tide, during low sea states at
depths of 10 feet and below. The material velocity was not isotropic at 5 ft.
depth.

Since the temperature fluctuations in the microstructure at sea are generally
very small compared to the average temperature it is appropriate to use the
linearized form of the speed dependence on temperature.

c = c + bAT

r

Ac - b(T) AT
where c = reference speed at temperature, T (1.4)

Ac = c - c

A r t1'5)

b(T) = = linear rate of change of speed with
temperature at T

1 .2 Layered Micros tructure

It is now recognized (Woods 1966) that there are large regions of the
ocean within and below the thermocline , which, for extended periods of time,
possess an ordered structure of alternate layers of turbulent and relatively
stable water. About one half of the ocean's regions below the mixed upper
layer may show this very extensive active layered character. The active regions
contain identifiable "layers", warmer than the liquid above or below them,
ranging from thickness sometimes as little as 10 cm or less and extending for
hundreds of meters, up to layers of thickness 10 to 15 meters extending for
tens of kilometers. The layers are separated by thin "sheets" of relatively
steeper temperature gradients of order 0.01 C /cm. Temperature inversions
of the same steepness with compensating salinity variations are found in these
active regions. And scattered throughout the active volume there are intermittent
and patchy regions, again, larger in extent than in thickness. It appears that
(Nasmyth 1970) turbulence microstructure in these layers is always accompanied
by temperature microstructure. On the other hand temperature microstructure
has been found that it is virtually free of turbulence microstructure , and that
may represent evidence of a former turbulent microstructure.

1 . 3 Other Underwater Structure Features

Frontal regions of moving water masses have been found, paralleling the
meteorologist's well-developed picture of air mass behavior. For example, one
might expect (Nasmyth 1970) a front with an angle of advance of 1.5 , in which
quiescent warm water is being uplifted by a mass of highly turbulent colder
lower water. It is not yet clear how common such situations are in the oceans
of the world .

Internal waves, generated within the ocean mass, are the analogue of
ocean surface waves. They are volume gravity waves which possess their
maximum vertical displacement amplitude at the interface between two water
masses of different densities, but their motion is detectable far above and
below this interface. All the characteristics of the water mass will change at
a point through which internal waves pass. For example, temperature fluctua-
tions of a few degrees with periodicities of the order of minutes or hours are
commonly found.

2. Spectrum and Spatial Correlation of the
Index of Refraction

Studies of temperature and velocity structure within the ocean concentrate
on the wave number spectral description or its transform, the spatial correlation.
It is clear that patchiness of salt concentrations or bubble content at sea could
also be described effectively in these same terms. The goal in turbulence
research has been to find a universal spectrum.

2 . 1 The Wave Number Spectrum

The underwater material velocity that is uncorrelated with the ocean
surface displacement, the turbulent velocity, cannot be completely isotropic.
It is easy to see that the very long wave lengths of the Z component, at least,
would be affected by the depth of the water and the depth of the mixed layer.
Nevertheless, it is convenient for the theorist to talk of localized isotropic
turbulence as an idealized situation.

The guiding light in turbulence studies has been the Kolmogorov
Hypothesis which deals with isotropic turbulence. The Hypothesis states that,
regardless of the mean flow and large scale motions (long wave length) from
which turbulent energy comes, the small scale components of a "fully
developed" turbulent region are in statistical equilibrium with each other. This
statistical equilibrium means that the turbulent motion in a given wave length
receives its time-independent energy density from larger motions and discharges
it to smaller scale motions. The general behavior can be observed qualitatively
in many diverse turbulent flow phenomena. Consider as examples, the smoke
from the end of a cigaret or the motion of the Gulf stream. In both cases the
large scale eddies break up into smaller ones and these, in turn, disintegrate
into still smaller motions. If energy continues to flow at a constant rate from
the largest eddies, and to be dissipated at the same rate in the smallest eddies,
the intermediate scale motions contain their fixed energy densities. Ultimately
the energy reaches an extremely small scale, from which it deteriorates by
shear viscous losses into the random molecular motion which we identify as
heat.

The band of turbulent velocity wave lengths through which the isotropic
energy "cascades" from large scale to the smallest scale is called the
"inertial sub-range". The smallest scale wave lengths within which viscous
dissipation occurs is called the "dissipation" or "viscous sub-range" .
Kolmogorov assumes that, within the inertial sub-range, the turbulence is
statistically independent of the large scale sources from which it originally
derived. In this inertial sub-range the turbulent power spectral density,
<£> (x) has been shown to be (Batchelor 195 6) uniquely defined by the two
parameters, the kinematic shear viscosity, v , and the rate of supply (or
removal) of energy, e . The equation for the one dimensional PSD is

*v(x) = bx"5/3 (2.1)

where b = b {v , e)

x = 2tt/A= wave number of a turbulent
velocity component

A = effective wave length of a turbulent
velocity component

€ = energy dissipation, per unit mass, per unit time

V = kinematic shear viscosity of water

The -5/3 rds power law has been amply verified for components of turbulent
velocity in laboratory experiments and, to a lesser extent, at sea.

We are interested in the turbulent velocity not only because it is a
significant source of sound fluctuations in special situations such as turbu-
lent channels but, more importantly, because it is a guide to the behavior of
temperature variations. And temperature variations are generally the principal
sources of sound fluctuations in the ocean.

The temperature often acts as a convected, passive contaminent which
closely follows the turbulent velocity behavior. The spectrum of temperature
fluctuations has been studied for such a case (Batchelor 1959) and again the
-5/3 law is predicted for the inertial subrange. The predictions have been
largely verified (Stewart 1962, Grant 1968) by ocean measurements.

4

o

Fig. 2 Temperature and velocity spectra at depth 2 7m in the open sea, from

Grant (1968a). The predicted transition wave number, x. /is shown,
as calculated from Eq. (2.8).

Fig. 2 plots the spectra (Grant 1968) of a component of velocity and tempera-
ture measured at 27m depth, compared with the theories described in the
references. The figure shows the extension of the Kolmogorov theory from
the inertial subrange, where the -5/3 slope is confirmed, into the higher wave
number dissipation sub-range. The boundary between the two subranges is often
assumed to be at the Kolmogorov wave number,

= H

b

= (e h3)k (2.2)

Values of € , derived from measurements of the turbulence spectrum, are
a function of depth and have been found (Grant 1968) to be as high as 0.5
cm sec atjan acean depth of 15 meters in well-mixed water and as small
as 1.7 x 10 cm in quiet water below the mixed region at 213 meter depth.

Looking again at Fig. 2 we see a rise in the temperature spectrum, also
predicted Jpy BatcheLpr (1959). At this depth, 27m, £ was determined to be
5.2 x 10 cm /sec . Assuming v = 1.4 x 10 " cm /sec, Eq. 2.2 yeilds
x =6.6 cm . The rise in the temperature spectrum occurs between x and

S _2 °

x —2.4 x 10 x . At greater depths, the dissipation constant and the Kolmogorov

wave number are generally smaller.

Having identified the high wave number end of the isotropic -5/3 spectrum,
we now seek to establish the low wave number limit. Unfortunately, at the low
wave number end of the spectrum of temperature micro structure, the PSD is
very dependent on the local conditions and history of the water mass, and only
empirical results are available to guide us.

A useful, but very crude, generalization that can provide some guidance
is that the largest temperature wave length that could conceivably be
isotropic at the depth H would be of extent 4H. The corresponding isotropic
wave number would be

2TT TT in i\

This would suggest that for Fig. 2, where H = 27m, the lower limit of the
-5/3 spectrum might be at

Hm = 5*8 x 10~ cm_1 (°r lo8io Hra =-3-2).

In fact, the data of Fig. 2 show that at values of x much smaller than 5.8 x
10~ (perhaps at x- 0.02 cm" ) both the temperature and the velocity PSD
break away from the -5/3 slope. This suggests that, realistically, the lower

10

end of the -5/3 spectrum to be expected at sea is somewhere between x and

x . We will call that transition wave number, x .
o t

The appropriate variable to describe sound phase and amplitude fluctuations
will turn out to be the speed of sound, and so it is the spectrum of fluctuations
of the speed that we must seek. The connection is direct. For simplicity we
will assume that the speed depends only on the temperature. We have previously
seen that the fluctuations in temperature micros tructure are much less than the
average temperature. The fluctuations in the speed of sound are also a very
small fraction of the average value. It therefore turns out to be most useful to
define not only the index of refraction of the sound speed, at position U

— ♦
ft(l)s _i^_ (2 < 4)

<c(R)>

— +

but also the excess index of refraction, \i (R) , which will be the particular
parameter for all future discussions of fluctuations.

H (R), c <»>" ic <*)> = n ft. k< !
<c (R)>

The mean value, ( ) , is to be calculated over the time of the experiment. It
is essential to realize that [± (R) is sensitive to position R* . However, for
typographic convenience we will drop the notation for the explicit dependence
on position so that henceforth we write p. (R) =|i. Further, we define

/ \ a * - I 2\z (2-6)

(p,) = o and a = \\t, )

The point must also be made that if the time or duration of our averaging
process is changed, the value of <|_i) and a will change. We must immediately
accept that these quantities must have their temporal limits restricted. A
logical criterion for the restriction is that for the present we are concerned
only with locally homogeneous regions of the ocean, only for times comparable
to the duration of the acoustic experiment. We will be more precise about
these limitations , shortly.

Since the excess index, \i, is proportional to the excess speed of sound,
which in turn is assumed to be proportional to the excess temperature, AT, a
spectrum of M- can be expected to resemble the spectrum of AT. It will have an
inertial subrange in which the log of the power spectral density, $ (x) will have
a slope of -5/3 when plotted against log x , as in Fig. 2. That isotropic inertial
subrange will have an upper limit x* and a lower limit which we will call x^.

11

The lower bound of the isotropic inertial subrange of temperature x
appears to be approximately a function of depth, H, alone if we define t
it as

\ = °-5 (*o Km>
- 0.3 (V/3 <

TT

2H

(2.7)

Fig. 3 is a graph of x^ calculated from Grant's (1968) values of c and H, as
well as those of Nasmyth (1970).

O GRANT etal (1968)
X NASMYTH (1970)

20

40 60 80

DEPTH, H METERS

00

Fig. 3 Transition wave numbers, calculated from Eq. (2.7) using data of
Grant (1968a) and Nasmyth (1970). The empirical fit to the data,
given by Eq. (2.8) is shown.

.The equation of the best simple approximation to this graph, is of the form

-H/h

x = B e

(2.8)

where B = 9 .0 m-1
h = 40.0m
H = depth, m.
x = lower wave number limit of isotropic
temperature spectrum, m

12

For example, for the experiment at 27M depth, shown in Fig. 2, Eq. (2.8)
allows us to diagnose that the medium was no longer isotropic for x<x ~
0.05 (log. _ x = -1 .3) . Study of Fig. 2 verifies that especially the velocity ,
but also the temperature spectrum, breaks away from the -5/3 slope at
log x — -1.3. We will use the criterion of Eq. (2.8) to guide us in our predic-
tion of acoustic fluctuations due to micros tructure.

A graph of the spectrum of the expected excess index of refraction at sea
is shown in Fig. 4 .

log K,

Fig. 4 Typical spectra $ (x) and the approximate, simplified, version to be
used here to calculate spatial correlation.

It is typified by: a large scale, (small wave number) non-isotropic, highly
variable, section at x<x that depends on location and time of day: an
isotropic universal subrange that follows the Kolmogorov Law

-5/3
$ (x) = b (x) forxt<x^x*

-5/3

a universal dissipation subrange with §> (x) >bx ' for x* ^x^x , and a

rapid exponential drop-off for x>x . We will generalize and use this spectrum

in the next section. The data available show that x «x. «x .

o t m

2 .2 The Spatial Correlation and Structure Function

The alternative to the spectral description of fluctuations of temperature
or index of refraction is either the correlation function or the structure function,

13

which is simply derived from the spatial correlation. These functions turn out
to be the most direct way to calculate the sound fluctuations , as we will see
in Section 3 .

One of the earliest determinations (Liebermann 1951) of the spatial
correlation of temperature fluctuations was conducted by mounting a therm-
istor on a submarine operating at a depth of approximately 50m. The
standard deviation of the temperature fluctuations was found to be approxi-
mately 0.4C at this depth (time of day, weather conditions unreported).
The spatial correlation could be fitted approximately by the exponential law
(shown by solid line in Fig. 5)

cT <§)

<T (x) T (x + g)>
2

= e

-S/a

(2.9)

where a = 60. cm

which thereby defined a correlation length, sometimes called a "patch size
of 60 cm.

i.o k^

\:i

Kolmogoroy

N^_ 0 ■*•••

100 120

E cm

Fig. 5 Temperature correlation function from Liebermann (1951) with data
(circles), empirical fits to exponential (solid line), and Gaussian
(dashed line, correlation functions. The correlation based on Fig. 4
with x = 0.24 cm is shown by dotted line, identified as "Kolmogorov1

Others have used the Gaussian correlation function

2

CT (4) =e

-(£/*)'

(2.10)

to describe micros tructure fluctuation. A Gaussian correlation function
(dashed line) has been drawn in Fig. 5 for comparison with the exponential
functions and the data (circles).

14

Both functions define the same correlation length, since they both reach
e at £= a. The correlation function of the index of refraction will equal
that of the temperature, if other parameters affecting the speed of sound
have negligible effect, and if the fluctuations are small. Then

cT(*)=cu («>

It is not always possible to determine the correlation function C (£) or
C (4) by measuring it directly. In any event it would be a boon if we could
use the known universal characteristics of ocean micros true ture to deduce
the spatial correlation function in the real world outside of the laboratory.
Although this turns out to be not completely feasible, there are certain
generalizations about the spatial correlation function that can be deduced; one
of these is its form at small values of £.

The spatial correlation function of the excess index of refraction can be
obtained from the wave number spectrum by the Fourier transform theorem. In
three dimensions this relation is

s(?)-"-f- ^ y*)ei*'* dH (2-n)

where f* is the vector spatial lag.

If we now assume that the changes of the index of refraction are independent of
the direction being studied we can simplify the integration. In spherical
coordinates

H»r. = urease- and dn = 2tth sin©- dn d &

Then C (r) = -*Z— fQ K2 dH j" s (h) sin@r ^nr*** rf

a ^

and the £ integration yields

We now define the one dimensional spectral function, $ (x) for the isotropic
medium

"• 2

S^ (h) dn = 4tt h S^ (h) dH = § (h) dn and change (2.12)

notation, r -» 4 # for consistency with previous work, so that

15

% (§) = —\- £ li£ri V (K) dH- <2-13> ■

£ may be in any assigned direction for the isotropic medium.

In order to integrate (2.13) we start with the spectral conclusions of the
previous section (Fig. 4). We are prepared to make four simplifying assump-
tions in order to proceed:

1) $ (H) = $ + s(H - K ) for K < K , : *

M- m my ra (2.14)

$m;-$o Source Subrange

where s is the slope constant= xm

$m = peak isotropic PSD; $0 = $ (°)

2) * 00 = constant = 4 for * < * < ** Transition Subrange

K -5/3 ra x

3) §^ (h) = $m ( t/H) for Ht < k < Kq Inertial Subrange

4) $ 00 = o for H > H0 Dissipation Subrange

The approximate spectrum that follows these assumptions is shown by the
dashed line of Fig. 4.

The effect of the simplifying assumptions is to replace the highly variable
anisotropic range and the fixed, but complicated, isotropic range by a fixed,
easily-integrable , spectrum. This will permit us to assess the relative impor-
tance of each of the sub-ranges and to determine what universal generalization
can be made, (if any) about C (£) at sea.

Using the assumed spectrum of Eq. (2.14) it is possible to immediately
determine the excess local index of refraction, a in terms of the peak spectral
density, $ , and the constants x and x . Recalling that the mean squared value
of any fluctuating quantity may be obtained by integrating the spectral density over
all values of the appropriate parameter (frequency or wave number) we find

(V00

(u ) = J $ (K) dK
^r O JJ,

m m t

Since x « x and & < & , the result of the integration simplifies to
m t o m

(j, = 5/2 k $
t m

16

It is interesting to determine the parts of the spectrum most responsible
for this variance of the fluctuations . The terms in the integration were

1/2 x <£ source subrange

m m

x, $ transition subrange

t m

3/2 xx $ inertial subrange

t m

Since x « x we conclude that the contributions to cr come almost equally
from the transition and the inertial subranges. Furthermore, our simplifying
assumption of the particular form of the source subrange can have little effect

on (j because that contribution is proportional to x , and x «x .

\i m m t

A second general conclusion can be reached if we restrict our interest to
very small values of the spatial lag, £ . Then we can substitute the expansion
for the sin x£ and integrate Eq. (2.13) with ease

— Jo $ (h) dx a__ (J H [$ + m (h - x )] dx

+ Jk A

m m

§ dx
nv m

+ jHoX2J (^)5/3dK

K m x

2
The first integration yields (\i >, as we saw above, so that the first term

is unity. The integration over the three terms in the curly bracket is also simple

so that

.2 . 5/3 4/3

C (§) = 1 - % $m Kt ^—

r2 (2.16)

8 o-

and with the aid of (2.15)

C (§) = 1 - 0.05 k 2/3 x 4/3 §2 for x 6«1. (2.17)

fJL tO O

The significance of this result appears if we consider any specific depth ^ say
50m. Then (2.3) gives x = tt/10 cm" , (2.8) predicts x = 2.4 x 10~ cm"
and therefore, at 50m depth, we can expect

C (§) = 1 - O.J468 §2

17

For small displacements this is an expansion of the Gaussian form

C^ <g) = e"(§/a)2^ 1 - (§/a)2; § « a
where & „ 0#6Q£ cm

We therefore conclude that at a depth of 50m the Gaussian form is
appropriate only for very small spatial lags, £«0.68 cm. The conclusion is
not significantly different at other depths.

We are now prepared to develop the universal correlation function based
on the spectrum of (2.15). The correlation function will be "universal" only
in so far as it is calculable from the universal part of the spectrum. To be
safe, we will calculate the correlation only for £< u /x .

For this restricted part of the correlation function it turns out that the universal
Kolmogorov subrange is dominant so that very useful results can be obtained
with a minimum of mathematical calisthenics.

Starting again with Eq. (2.13) we split the integration into the contributions
for the source, transition, and inertial subranges. We have assumed zero spec-
tral density in the dissipation subrange so that integral is zero.

auCU (?> ' V- o SlvEH § *« + S (H - H )]dH + J ^V^»'dH + J °sAlLHi* ^5/3 ,

M- U H = o H§ u in v m/J h h£ m k hZ m x

m t

The coordinate for integration is now changed to 0 = x £

to « ■ i/:l -&**\ ♦ -f - *» * h:! 5<^>v ♦ ?2i:^*/H

m 3 '

s in R R £i

The approximations are now made for J p d/S = |9- -~r-, + 5°5 , in the source

integral and in the transition integral. Because of the rapid convergence due
to the jg 8/3 in the denominator, the third integral is comfortably approximated
by using the sine expansion from zero to tt/2 and again from ir/2 to n. Using
(2.15) the result of these integrations, valid for £ £— with about 5% error, is

\

C (i) = 1.0 - 0.48(xt£)2/3 +0.28 (x^)2 +6.7 x 10~4(xt£)4 (2>18)

The curve is shown in Fig. 6

I.O-

*-*

\j<J> 0.8-

v:

3. a6-

^ 0.4-

0.2-

.

0

i

i

i

i

i

i

w

0 05

1.0

1.5

2.0

2.5

3.0

M

Fig. 6 Spatial correlation function C (x £) derived from spectrum of Fig 4

It turns out that the source subrange makes essentially no contribution to
the correlation function for 4<tt/x ; its effect would be for greater spatial lags.
The transition and inertial subranges, together, are totally responsible for
C (£) for £<tt/x . The inertial subrange is particularly dominant for small
spatial lags (the 2/3 power term).

Assuming that x, is given by Eq. (2.8), it is possible to calculate C (£)
for any depth. The calculation has been made for the depth of Liebermanfi's
experiment (50m), for which Eq. (2.8) gives x = 0.024 cm , and the predicted
correlation is plotted as the dotted line in Fig. 5. The agreement with the
experimental data is remarkably good, up to the limiting spatial lag, £ — tt/x.

Recapitulating, the universal spatial correlation function that has been
derived from the assumed isotropic ocean spectrum has a Gaussian correlation
function from £ = 0 to approximately £ — 2tt/x . it then drops off with a slope
of close to -2/3 under the control of the inertial subrange. This slope decreases
to close to zero at the limit of our study, £= tt/x . Experimental studies show
that, beyond £ = 77/x the correlation value will continue to decrease slowly,
begin to oscillate about this drop-off with a spatial periodicity and a specific
character that depends on the precise form of the spectrum in the transition and
the source subranges.

19

3. SOUND PHASE AND AMPLITUDE FLUCTUATIONS

3 . 1 Phase Fluctuations

Let us begin our study of the variable propagation of sound in the ocean
by considering plane wave sound phase fluctuations along a ray path in a
statistically homogeneous, isotropic, medium. We will assume that the
sound frequency is high enough so that the wave length is much smaller that
the correlation length of the microstructure, that is ka »1. Again a is the
correlation length, the distance in which the spatial correlation drops to e
We will first consider the ray approximation, and relatively small propagation
path, so that ka »R/a, where R= path length.

The dual condition is best written in terms of the wave parameter,

For this ray development (Bergmann 1946) we require D «1. The time to
travel the distance, R, in the x direction is given by

pR dx PR n(x,T)

t = i — ; — =\ = — ^ dx

•'o c(x,T) Jo c

o

where n = c/c = local index of refraction of the medium, T = clock time and
c = <c) . Both c and n are functions of position and time of the experiment.

The particular duration required for this traverse will depend on the type
and arrangement of inhomogeneities at the instant of the sound travel. The
time of travel is assumed to be short compared to the time for the medium to
rearrange itself. That is, the medium is assumed to be "frozen" during the
sound traverse, and to change into another configuration during the time
between traverses.

Our interest is in the duration of the travel, averaged over either many
statistically equivalent inhomogeneous paths, or averaged over many traverses
along a time-varying inhomogeneous path. We assume that these two averaging
processes are equivalent, that is, we assume that the medium is ergodic.

J <n(x,-%

<t> = — J <n(x,T)> dx = — J dx
c o c o

o o

because the average

index, (n) , is unity.

20

It is the deviation from the mean that we must calculate:
At = t - <t> =r — J n (x,T) dx — J dx

CO CO

o o

i rR

= — J0 M.(x,T)dx
where p, (x,T) = n (x,T) -1

From this we get the square of the deviation from the mean (At) =^ — „)

( Jo |jdx f^dx) which can be written (At) =— J^ J^ |j (x^T)^ °(X2'T)

o

dxldx2

2 1 R R

The mean square deviation from the mean is (At > = - dxn f " (u (xn ,T)

/m\\_i cz«Jol«Jol

[i (x2,T)> dx2 o

The integrand can be written in terms of the spatial correlation of the excess
index of refraction C (£):

<H(Xl,TMx2,T)> = (n*> C^ (x2-Xl) = a* C (VXl) .

a2 „ /xR

2 R I

In these terms (At ) = — ^ J C (x^-xjdx,

0 o ^ / 1 2

v p *. 4. (3.3)

o

We seek the mean squared phase fluctuation which is calculated by
substituting cp = ceAt where (pis the instantaneous deviation of the phase from
the value due to the mean speed, c . The units of <p are radians. Since the
medium is presumed to be statistically homogeneous, it is the separation,
£ = |x - x |, that is significant not the particular end points, x or x .
Therefore, "the mean squared phase fluctuation is

<<p2> =a2k\fc (4) d£ D«l (3.4)

2 2 2 2

Since (cp } = (cp> + Var cpand (cp> = o, by definition, (cp )= Var (p= a . (3.5)

We have only to know the spatial correlation of the index of refraction, from
the source to the receiver, in order to carry out the integration, and obtain
the variance of cp .

2 2

It should be pointed out that the quantity a = ([± > must be known at the

time, and location of experiment. The significance of this comment is that, in

the ocean, the evaluation of a can be strongly dependent on position and the

range at a given time, or a function of the time and duration of measurement.

21

For example during passage of an internal wave, at positions in the thermo-
cline there will be large variations of the temperature and the speed of sound,
over times of the order of several minutes.

It is also convenient, sometimes, to assume that the spatial correlation
function drops to zero after a few correlation lengths in either direction from
any reference point. Then for the sound path of length R »a we can extend
the integration to infinity in both directions , or double the integration from
zero to infinity, without affecting the total integration,

00

a2= <ip2) = 2 a2 k2Rf C (£) dt D« 1 (3.6)

Given a correlation function that drops rapidly and monotonically to zero,
Eq. (3.6) is more readily evaluated for the mean squared phase fluctuation
for the high frequency condition than (3.4).

When the path is long, so that phase interferences occur, a wave analysis
is necessary. We will not go through the derivation for the phase fluctuations
through the same type of medium at this other extreme of propagation parameters ,
but simply quote the result (Chernov 1967), applicable for low frequencies or
very large path lengths .

a = <tp2) =v k2 rTc (£) dp D»l (3.7)

The point to be observed at this time is that, regardless of the value of
the wave parameter, D, the variance of the fluctuations of phase are proportional
to the length of the sound path, the sound frequency squared, and the variance
of the fluctuation of the excess index of refraction, a , along the path, Further-
more, regardless of the form of the spatial correlation function, there is only
the factor, 2, relating the mean square phase fluctuation obtained for very
large or very small values of the wave parameter, D. This factor will not be
altered if we change the form of the spatial correlation function, either for
convenience in the evaluation of the integrals , or as we seek to describe
the medium more accurately.

Another point must be made at this time. Mintzer (1954) has shown that
solutions such as we are considering can be valid only for single scattering
in a weakly in homogeneous medium. That is, although we have ka »1 as a
requirement for the relative size of the "patches", and kR >?1 as a requirement
for far field propagation, the fluctuations of the medium, a must be so small that

2 2

k* a. aR « x

In practice this condition is not difficult to fulfill; field measurements
generally show q <10

22

The Gaussian correlation function

C (£) <=e "^/a) (3.8)

is attractive because of the simplicity in evaluating the integrals of Eqs . (3.6)
and (3.7). For the Gaussian correlation function the phase fluctuations are:

a2 = <cp2> = ^CT2 k2 Ra D«l (3.9)

2 . 2V y? 2 2 (3.10)

and a = <<n > =-r— a k R a D» 1

<P ? . ;M

It has been shown (Krasilnikov 1956, and Chernov 1967) that, for a
Gaussian correlation function, the variance of phase fluctuations over all
values of D is described by

a2 = JjLj2 k2 R a ( 1 +4 tan_1D) (3.11)

<P 2 u D

3 .2 Amplitude Fluctuations

The amplitude fluctuations are often put in terms of the ratio of the
change of the pressure amplitude to the original amplitude,

V = ( P"Po ) (3.12)

P
o

where P is the sound amplitude at x = R and P is the amplitude of the plane
wave at the starting position x = o. In terms of this parameter it turns out
(Bergmdnn 1946, Mintzer II 1953, Chernov 1967) that the mean squared
fluctuation of amplitude in the high frequency extreme is

for D = 4R/ka2 «1 (3.13)

and in the wave interference region (Mintzer I 1953) it is simply

o2 = <x2> = a2 K2 Rfc (£)d£ D»l (3.14)

A (_i Jo (j ^ ^

where cr is the fractional standard deviation of the pressure amplitude

o = <(P-PQ >2)% (3.14a)

A Po

23

Comparison of expressions for amplitude fluctuations with those for phase
fluctuations is revealing. For example, for the high frequency situation D «1
the mean squared amplitude fluctuations depend on an integration of the fourth
derivative of the correlation function in the transverse directions, y and z,
rather than a straightforward integration over the correlation function in the x
direction as in Eq . (3.8) for the phase shift. This means that the amplitude
fluctuation is caused by the curvature (really double curvature) of the correla-
tion of the refractive index, transverse to the sound path. The dimpled medium
in the case of amplitude fluctuations therefore behaves as a series of lenses
coverging or diverging acoustic energy along the sound path (notice the evalua-
tion at zero displacement in the y and z direction, that is, along the beam).

At the other extreme of propagation parameters, however, when the range
is larger, or the frequency is lower, such that R/a » ka (D »1), the sound
field fluctuations are caused by field distortions due to a large number of
inhomogeneities along the path in the medium; these distortions interfere,
constructively and destructively along the entire path and this interference
effect is common to both phase and amplitude. For this condition the mean
squared fluctuations of phase and amplitude are identical. Both are linearly
dependent on: a) the distance of propagation, b) the frequency squared, and
c) the integrated spatial correlation function.

It must be pointed out that all of the derivations to this point have been
based on the assumption of single scattering by weak fluctuations. The
consequent solutions are valid only in so far that very long paths and very
large fluctuations are not allowed. In practical propagation over very long
paths the magnitude of the phase and amplitude fluctuations cannot increase
with R without limit, or they would reach the absurd conditon of fluctuations
that are greater than the quantity itself. In fact, "saturation" must take
place and the increase of the fluctuation with range must reach a limit that
will be indicated, shortly.

If the Gaussian correlation function of the index is assumed (Krasilnikov
195 6) the variance, ah, is:

°l <X2> - C-^fV) k' R a <1-^-tan"1 D)1 (3.15)

al (X2> = (-^T") °l (R/a>3 D <K l 0.16)

aA = (X2> = (~T~) °l ^ (Ra) D >;> l (3'17)

24

3. 3 Small Amplitude Fluctuations: Laboratory Experiments

2
The dependence of small values of or on the parameters of the medium and

the frequency has been put on a firm experimental basis in a beautiful set of

laboratory studies (Stone and Mintzer 1962) in which the micros tructure of the

index of refraction was obtained by heating the tank of water through which

the sound beam was propagated. First the rms value and the spatial correlation

function of the temperature microstructure were obtained by direct measurement.

From these data the value a / and the correlation function, Cyi (£ ) , of the

excess index of refraction were calculated. The spatial correlation function was

fitted to the Gaussian form, and the correlation length, a, was calculated from

Eq. (3.8). The measured temperature micros tructure , fitted to the Gaussian

correlation function showed a correlation length, a = 3.5 cm.

Additional comparison showed that the rms excess index of refraction,
deduced acoustically, was approximately 25% less than the value 1.6 x 10 ,
calculated directly from temperature measurements. One flaw was revealed
in the experiment: the acoustically determined value of the assumed Gaussian
correlation length was only about one-third of the value (3.5 cm) directly
measured by thermistors. This discrepancy was possibly due to the error of
assuming that the correlation function was Gaussian. As shown in section 2,
the Gaussian form is appropriate only for very small displacements. A
combination of Gaussian and Kolmogorov correlation functions would probably
have been a better description of the medium for insertion into Eqs. (3.13) or
(3.14).

3. 4 Large Amplitude Fluctuations - Large Ranges

So far we have considered small fluctuations for two cases:

1) Change of sound pressure amplitude, P, in the presence of zero

(or negligible) phase shift. In section 3.2 we gave expressions for

2 P -P 2 2

cr. = // o_. v . The type of pressure field at x = R that results in <ja is

o

described by the phase diagram for "P" in Fig7a where P represents two

possible scattered pressures at time, t position x. It is the vector addition

of various values of P to the undisturbed field, A (x) , that results in the

statistically changing resultant pressure amplitude, P, at angle <p , with the

original wave. This P is used to calculate a* . It is to be noted that the

random amplitude P has components, a , in phase with the original wave,

A, and b , 90 out-of- phase with A. It is a rather than b that is effective

s s s

in changing the value of P.

2) Change of sound phase due to varying index of refraction. In the
derivations of section (3.1) leading to expressions for <j it was assumed that
amplitude fluctuations have a negligible effect on phase fluctuations. From
Fig. 7a it is seen that the out-of-phase component of scatter b , rather than
ag , is the principal source of phase changes.

25

(b) LARGE FLUCTUATIONS

Fig. 7

Two realizations of scattered pressure, P added to undisturbed pressure A.
P is composed of the in-phase part, a and the out-of- phase part, b . For
small fluctuations Fig. 7a the pressure Amplitude P, is principally a function
of A and a only, and the phase (pis principally determined by A and b . For
large fluctuations Fig. 7b both P and <p depend on A, a and b .

We now consider large fluctuations for which the situation of Fig. 7b is
more appropriate. We will no longer consider that the phase fluctuations have
negligible effect on the mean square amplitude or that the amplitude fluctuations
have negligible effect on the mean square phase.

The complete picture at x = R, which we have represented in Fig. 7 can be
described analytically by

p(x,t) = A(x) cos (cut -kx) + a (x,t) cos (tot - kx) + b (x,t) sin (tut - kx)

s s

(3.18)

26

where A (x) is the magnitude of the pressure of the remaining unscattered field

at x;

a (x,t) = time varying magnitude of the scattered pressure which is in phase

with the original wave at x. We assume that values of a are Gaussian distributed

and that <a > = 0;

S o

b (x,t) = time varying magnitude of the scattered pressure which is 90 out-of-

phase with the original wave at x. We assume b is Gaussian distributed and

<b > = 0. b *A.

s s

Equation (3.18) resolves the fluctuation scattered pressure, P of Fig . 7 into
components in-phase with the depleted original wave A (x) ana out-of- phase
with it. A typical situation in Fig. 7 shows that

«2 2 /I
P = a + b
s s s

To get the typical amplitude, Pf of the total wave at x as a function of A,
a , b , Eq. (3.18) is rearranged to

P = (A + a ) cos (cot - kx) + b sin (wt - kx) (3.19)

= P cos (o>t - kx - $)

where P = pressure amplitude at x (3.20)

r/» x2 u 2-,l/2

= [(A + a ) +b ]
s s

<$ = phase angle with respect to original

-1 bs
wave at x = tan ( ) (3.21)

A+a

s

It is useful to expand _P and $ for small fluctuations a — b «A.
Using the binomial theorem for _P

-A(l +-5"- +%A1N )-A+as (3-22)

and the small angle approximation for $

I = tan"1 (-^- ) -

A+a ; A (3.23)

s

27

2
It is therefore clear that for small fluctuations the <j of Eq. (3.12) and

(3.17) can be interpreted from the point of view of (3.18)and (3.22)

P-Po
Po

=. A

+

a« " AA

A
o

2

°A =

Va

'(XP

Var aq

A 2
o

(3.24)

so that

A **' A s.

o

2
As we anticipated from our study of Fig. 7a, the value of a is therefore

relatively independent of the out-of- phase components of scatter, b , because

s

their effect shows up to be relatively small in the small fluctuation approximation
of (3.22).

2
Likewise, for phase fluctuations the a of Eq. (3 .4) - (3 . 11) can be

understood from (3.18) and (3.23) as ^

2 Var b

;s

<P Az (3.25)

2
which shows that the value of o is virtually independent of the in-phase

variations, a , in the small fluctuation approximation of (3.23).

For large fluctuations we follow the method of Brownlee (1973). First
we recognize that the diversion of acoustical energy into scattered energy will
cause the average in-phase pressure amplitude to reduce from the value A
at x = o to A at x; we will assume that the rate of attenuation from the
original magnitude is proportional to the distance traveled,

dA „ ,

— = - a dx

A s

where a = attenuation of the original

wave due to scattered energy.

Integrating for a range R, A = A e s where A = P = pressure (3.2 6)

amplitude of (unscattered) wave at x=o.

The average intensity of the part of the wave in step with the original wave
will therefore be proportional to

A2 =A 2 e'2ttsR (3.27)

o

We will assume conservation of energy for the total wave system; that is , we
assume that whatever energy is taken from the original wave goes into the
scattered wave and that this is principally forward scattered (Skudrzyk 1961).
This means that the intensity is conserved, when averaged over any plane
perpendicular to the direction of propagation.

28

From (3.20)

<P02> = < P2)

UJ > = <(A4.s)2 + b/)

■= <A2> + (2ao A) + (a 2> + <b 2>

_ 9 9 9 ^ ^

Ao " * + (as > + <bs > (3.28)

To determine the attenuation rate, a , we use (3.2 7) and (3.28) and form

S3

An2 -A2 _ (as2) + (bs2> _ , -20i R

A 2 " A 2 " L "e S (3.29)

A A

o o

The evaluation can be made in the small fluctuation range where the
ingredients of (3.29) are well known. There, using the expansion of the
exponential for small argument and (3.24) and (3.2 5)

aA + a9 = 2asR (3.30)

The condition for long range is of particular interest.
Then we have, from (3.7) and (3.14)

°l = al >D>>1 <3-31>

so that the attenuation constant of the original beam due to energy going into
fluctuations , is

* =a2/R ,D» 1

(3.32)

In particular, when the scatter is due to a medium with a Gaussian correlation
function, C (£), we have, using (3.17),

a =-^-a2K2a (3.33)

s 2 |i

29

We can now proceed to consider large ranges and large fluctuations.
The quantity observed in any experiment is the pressure amplitude P. Eq. (3.2 0)
shows P to be a function of all three quantities: The steady coherent component,
A, and the fluctuating random components, a and b . The conventional way to
express the amplitude variations is in the ratio

(CAV)2= <P*> -2<P>2 (3-34)

<P>2

where CAV = coefficient of amplitude fluctuations. It might be noted that for
small fluctuations we have simply CAV = a., Eq. (3.14a). Here, for large
fluctuations we find, from the conservation of energy assumption that

<P2>= <Po2> = Ao2 (3.35)

2
But the calculation of <P> involves a more involved evaluation of the mean

value of P given by Eq. (3.2 0) when both a and b vary randomly, each with an

assumed Gaussian probability density function.

The mean value of the sum has been found by Brownlee (1973) and is

<P> =^AQ [1 -e-2asR}HexP[-2/(e2^R-i)]}^ £

^ i •>, , (3.5)b2 (3.5 7)b3 (3.36)

where £ = 1 + Jb + ^, .j. + tjj v* +

b « fcCe2** -D"1

2 2

When this is now squared and combined with P in Eq. (3.34) the A cancels and

the result is expressible solely in terms of the attenuation constant, a , and the

length of path, R.

(CAV)2 =-1 +-^exp ( ,l„ m . ) {[1 -exp (-2« RJIS2]}"1 (3.37)

77 exp(2asR)-i s

This messy expression has a simple maximum value for large fluctuation and
large ranges; we get

(CAV)2 = -1 + -4" = 0.273 for 2a R » 1

v tt s (3.38)

or (CAV) = 0.522

max

The complete behavior of the coefficient of amplitude variation is shown in
Fig. 8.

30

VM

i<?d) ,

= AVO

0}

G

O

4-1

co

(0

O

0)

3

4->

-1-1

4-J

d

o

fO

3

4->

i—4

o

3

CD

1 — 1

T3

>4-l

00

3

4->
-•-1

a)

d

a

H

£

(0

C

o

O

IS

c

-C

o

0

•r-4

CO

4->

OJ

TJ

S-4

-1-1

(0

>

N

T3

ffi

(0

-a

^

d

LO

-a

CM

c

4->

m

rd

4->

CO

r^

0)

CD

to

jC

cr>

6S

4->
4-4

r— 1

O

O

c

,*

0

£

d

-1— 1

4->

o

d

4-1

of Shvac
ure only

IO

co
C

data
truct

d

o

-r-4
4->

CO

cD o

x: u,

H O

1-4

-i-H

rd

. S

>

^o

<VJ

(D

O £

d

TJ

4->

" CD
co 3

-I-H

c -o

i— 1

o

a,

m-j cD

6

« -Q

rd

3 0

^■M

<-M

o *-

d

O

4->

— i CD

C

0)

•^H

o

fO co

■1-1

C CO

4-4

o

:>

a)

0

O

GO

•

Cn

PL,

o b

31

which is the graph of the square root of (3.37). We observe that the fluctuations
(CAV) = a for small values of a R, that is, for small fluctuations. The linear
dependence of (CAV) on range continues until approximately a = 0.35 after which
growth slows. (CAV) reaches 90% of its ultimate value at approximately <j =
0.6.

2
Knowing the micros tructure constants, g and a, Fig. 8 and Eq. (3.37)

may be used to determine the range at which The CAV due to micros tructure can

be expected to reach any value up to the maximum CAV = 0.522.

3 .5 Sound Fluctuations at Sea

In section 2 we found that the beautiful, and easily manipulated, Gaussian
correlation function is precisely correct only for very small spatial lags at sea.
In the previous section we saw that the Gaussian correlation function may yield
the wrong answers for the coefficient of amplitude variation, even when very
carefully tested in the laboratory. It is time to turn to the expression for the
limited microstructure correlation function developed in section 2 for an assumed,
reasonable, ocean spectrum, Eq. (2.18).

The standard deviation of the phase fluctuations and the coefficient of
amplitude variation for all values of the wave parameter, depend on the inte-
gration of the spatial correlation function over all values of £from |= 0 to
infinity. Let us look realistically at the upper limit of the integrations such
as in Eq. (3.7) and (3.14) which yield q and a . However, although the
correlation, defined by ^

c it) - ^(x) ^(x+£)>

VK (J2

M

can be determined for any spatial lag, the value of C (£ ) at £ >R is quite
irrelevent to a calculation of the sound fluctuations for a path of extend, R.
Therefore practical acoustical application of integrals of C (£)d£ will be
satisfied by an upper integration limit, R, rather than °° .

The microstructure study of section 2.2 leads us to have confidence in the
form of C (£) only for spatial lags over which the Kolmogorov spectrum is
dominant. We have set these limits as 0 <£<tt/x . In order to evaluate the
remainder of the integral consider Fig. 9 which is based on temperature meas-
urements by Seymour (1971) and transformation to spatial coordinates by
Haley (1972). The experiment was performed at a depth of 7.3 m, using the
stable NUC Oceanographic Tower, 1 mile off San Diego. At this depth,
Fig. 3 indicates a microstructure with transition wave number, x — 0.075 cm
Equation (2.18) then gives the Kolomogrov-dominated spatial correlation function
plotted as the dotted line curve to its expected limit £= 7t/x =42 cm. The
agreement with the experimental data is excellent until £ — 25-30 cm, after which

32

<AJ»

-■-I

CD
C
-i-i

T3
-•-I

"o

en

C

o

■I-I
•M

o
c:

p

4-1

c

o

CD

xi

>

o

o
o

x: jo

U

CD ^_,

CO

o

(D

x:

t*»

w

o g

6 c

a p

CD

CD

w
(0

en

CD fd

fd
CO

0

S-.

p

-M

u

p
ii

CO

O

S-.

O

XS
CD
+->
fO

o

c

p

+J

CD

Xi
+j

CO

•iH

CD

p

T3
CD

o
o cp

r-> CD

O
T3

CO

6 H
CD

> o

. (D

0> g

Q_ lO

^ en

C-> _e

2 *

c

CD

£

CD

3

en
fO
CD

c 6

.2 -

-t-> fO
CD CD

o

(D

a
en

>

O

cn

o .2

£ -3

r— I (0

O ^

-S o

o
o

CD

^-. a
co X
j3 (D

CO Z!

CD

C
(0

-rH

en
en

P
fO

,Q O

•■H

33

it levels off while the actual correlation function shows a continuing decrease
and then an oscillation with small amplitude around the zero correlation axis.

In order to obtain the integral of C (£) for a propagation experiment in such
a sea we have two choices: One solution would be to determine C (£) and
C (£) out to the range of the acoustic experiment; this would not only be a
formidable task but unrealistic for any long range experiment. The second, more
expedient, choice is to accept a judicious combination of theory and empiricism.
We propose to use the theoretically based microstructure of Fig. 6 to determine
the correlation length, a, and then to use the Gaussian correlation function to
approximate the true correlation curve. Judging by Fig. 9, this will result in
slightly too large a contribution for JC (£)d£ between 4 = 0 and £ = 0.5M. We
will assume that subsequent positive and negative parts of C (£) will cancel
each other and contribute nothing to J*C (£)d£ from £=0.5Mto£=R. In the
example being discussed, the value of rjhe correlation length, a = 25 cm.

Having found a suitable way to determine the equivalent correlation length,
all of our algebra for the sound fluctuations due to a Gaussian correlation func-
tion is assumed to be applicable, even though the true correlation function is
by no means Gaussian.

The general method for the determination of the correlation length is found
by referring to Fig. 6. The correlation function is down to e at

x £ = x t a = 1.8 (3.39)

therefore for our approximation

1.8

a =

xt

Furthermore, using the empirical approximation to the transition wave number
that we developed from Fig. 3, Eq. (2.8), we have the generalization for the
upper ocean.

a = 0.2eH/4°
where H = depth, m (3.40)

a = temperature (or index) correlation distance in m.

It is evident from the data scatter of Fig. 3, from which the constant
B and H are determined, that (3.40) can be only a guideline for the prediction
of the correlation distance.

34

Although a often decreases with increasing depth, a general rule describing
(j as a function of depth is not possible at this time. In fact, when such a rela-
tion is found there will certainly be other important parameters in the equation,
such as time of day, to which a is particularly sensitive at shallow depths.

When a is measured at the place and time of an acoustics experiment,
Eq. (3.40) provides a value of correlation length, a, that completes the needs
for evaluating q or a . For example, Shvachko (1967) has made measurements
of a and CAV af depMs 20, 35, 40M using sound of frequency 25 kHz. If we
take Schvachko's value of a / R/ k/ and use his depth in Eq. (3.40) to obtain
a, we can calculate q from (3.17) and predict his CAV from (3.37) or the graph
Fig. 8. This has been done in the following way:

ILLUSTRATIVE EXAMPLE

Shvachko (1967) has performed an acoustic fluctuation experiment at
frequency 25 kHz, range 20 m, in the sea of Norway. The medium had a
standard deviation of the excess index of refraction n = 17.8 x 10 . We are
to calculate the predicted cr .

At b. /depth 20 m reading from Fig. 3 x = 0.055 cm"" or using (2 . 8) x =
9.0e~ = 5.45 m~

1 8
therefore from (3.39) a = — '— = 0.33 m

t

20/40
or directly from (3.40) a = 0.2 e ' = 0.33 m

CO 2 (25.x 103) ._. e -1

k= — = -rh lnA 104.6 m

c 1.5 x 10s3

calculate D= 7—2 n i. cl, 0?2\ = 84« >>l
ka^1 = (104.6)(.33^)

therefore, using the Gaussian assumption (3.17) for D »1

A 2 u

= (0.89) (17.8 x 10"5)2(104.6)2(240.)(0.33)

= 0.0244

oA = 0.16

This is to be compared with Shvachko's CAV = 0.17

35

Since the <j is small, we are in the region of the curve (Fig. 8) where we
should expect CAV = <j . Therefore the small discrepancy between our predicted
<j and the experimenter's measured value of CAV is a good confirmation of our
generalized theory based on the truncated universal correlation function of the
temperature microstructure. All of the other data points except one, from the
Shvachko experiment, provide similar confirmation as shown in Table 3.1 and
Fig. 8.

Location, Time

TABLE 3 . 1

Depth m. R.m j x 10

r~

15

Theory
CAV aA (~CAV)

N.W. Atlantic, May 1962

N.E. Atlantic June 1962

Sea of Norway, July 1962

40

200

13.7

0.095 .136

40

665

25.4

0.264

.46

40

1150

4.7

0.141

.113

35

940

2.7

0.071

.055

35

600

5.4

0.100

.087

35

480

11.0

0.158

0.16

20

160

17.8

0.173

0.16

20

160

25.2

0.173

0.18

Table 3 . 1 Comparison of experimental data of Shvachko (1967) with predictions
based on the microstructure theory Eq. (3.40 and 3.17) of section 3. Sound
frequency, 25 kHz. The points are plotted in Fig. 8 where or is the theoretical
value and CAV is the experimental value. The symbols in the first column are
those used on the figure.

3 . 6 The Limits of Predictions

The calculation of amplitude fluctuations by the techniques of this report
has been constrained by several approximations and assumptions. The ex-post-
facto type of comparison with the experiments of Liebermann and Seymour-
Haley show that our truncated universal temperature correlation function based
on the universality of local microstructure in the upper ocean is not far from
the mark, and is usable as a function of depth, alone.

The highly variable values of cr in the ocean preclude our use of the
equations for predicting sound amplitude or phase variations unless we have
in-situ data of <j at the time of the experiment. However, if <j is available
as in the Shvachko experiments , our predictions of <ja and CAV are quite
acceptable.

We have continually emphasized that our predictions are limited to the
upper ocean. By this we mean approximately the upper 100 m in temperate or
tropical waters. The reason for this restriction is that the recent data of
Nasmyth (1972) for depths below 100 m do not fit our Fig. 3 which we use

36

to determine x and correlation distance, a. The reason is clear, at greater
depths in the temperate oceans, and even at shallow depths in the more stable
polar regions , the common existence of layers and thin sheets of highly
distinctive water masses contradicts our requirement for locally homogeneous,
isotropic water. Under these conditions it is not yet possible to generalize in
the sense of this section.

A further limitation to our development is the continuing restriction,
ka » 1 , on which the derivations for q and cr were based. At shallow depths,
say H=10 m, the correlation length is approximately a = 30 cm which constrains
the frequency to f =- — » 1 kHz. At greater depths in the mixed region larger

values of a would relax this limitation. Brownlee (1973) feels that the require-
ment ka » 1 is too strong.

We have not yet considered direction of propagation. Since a is a function
of depth, the fluctuations will vary with angle of inclination of propagation. In
principle the solution to the problem can be approximated by integration

2 JIT v2 „R 2
7A = T" K Jo ^

*-f- K [ a (z) a (z) dR

£ 'O u

as in a ray diagram calculation. In practice, although we have equation (3.40)
for a(z)

a(z) =0.2ez/4° (3.40a)

our frustration at generally not knowing how a varies with time, or location is
now compounded by the non-trivial requirement that a be known at all depths
along the ray path.

We have also assumed that C (£ ) is constant during the time of the experi-
ment. Any inconstancy will, of course, cause a variation in the signal. Some
idea of the variation in x and a is possibly indicated by the scatter in data
points in Fig. 3.

Throughout this section, the assumption has been made, for simplicity,
that cr and C (£ ) are dependent only on the temperature micros tructure. In
fact, ^studies of fluctuations of phase have shown that, within approximately
5 to 10 m of the surface (depending on wind speed), fluctuations of the speed
of sound and values of <j are more dependent on variations of bubble radius
and number than on temperature fluctuations. This statement is particularly
true at sound frequencies near 60 kHz, which corresponds to the resonance
frequency of a large population of bubbles. However, for any frequency less
than 100 kHz, the complete description of q within the upper 10 meters of
the ocean will require information about bubbles as well as the temperature
structure.

Finally, we must remind the reader that this report deals only with
micros tructure. To the extent that our study is correct we have evolved
a way to understand and to predict the sound fluctuations that are universal,
and repeatable. These fluctuations are in fact the minimum fluctuations to
be experienced at sea. Any larger scale fluctuations due to internal waves
or multipath propagation will be superimposed on the ones that we have
studied so far. A fertile field of study remains for the curious. The inverse
problem, to predict the g from a knowledge of a or cr now appears also to be
a timely subject for research. ■

4. Conclusion

A truncated "universal" spatial correlation function of the excess index of
refraction has been derived for the upper ocean. The correlation function is
predictable from a knowledge of only the depth of the acoustic experiment.
Although its applicability is appropriate only for reasonably well-mixed water,
and there only for small spatial lags, it appears to be essentially correct to
spatial correlations beyond the correlation length, that is, beyond correlation
values e . A Gaussian extrapolation of this universal correlation function
permits previously derived theoretical expressions for sound phase and ampli-
tude fluctuations to be used without further changes. The predictive technique
is simple, and appears to work well for a large range of experimental conditions

38

REFERENCES

Alexander, C. H. (1972), "Sound Phase and Amplitude Fluctuations in an
Anis tropic Medium", M.S. Thesis, Naval Postgraduate School, Monterey,
California 93940.

Batchelor, G. K. (1956), "The Theory of Homogeneous Turbulence",
Cambridge University Press, England.

Batchelor, G. K. (1959), "Small-Scale Variation of Convected Quantities
Like Temperature Parts 1 and 2 (with I.D. Howells and A. A. Townsend)
in Turbulent Fluid", J. Fluid Mech. 34, 443-448.

Bergmann, P. G. (1946), Physical Review, 7£, 486-492.

Brownlee, L. R. , (1973) "Rytov's Method and Large Fluctuations," J. Acoust.
Soc. Am. J5_3, 156-161.

Campanella, S. J. and A. G. Favret, (1969), "Time Autocorrelation of Sonic

Pulses Propagated in a Random Medium", J. Acoust. Soc. Am., 46_, 1234-1235

Chernov, L. A. (1967), "Wave Propagation in a Random Medium", Dover Pubs. ,
New York, 169 pp. Translated by R. A. Silverman.

Dunn, Do J. , (1965), "Turbulence and Its Effect Upon the Transmission of
Sound in Water", J. Sound Vib. 2_, 307-327.

Grant, H. L. , A. Moilliet and W. M. Vogel, (1968 a), "The Spectrum of

Temperature Fluctuations in Turbulent Flow", J. Fluid Mech. 34./ 423-442.

Grant, H. L. , A. Moilliet and W. M. Vogel, (1968 b) , "Some Observations
Of the Occurrence of Turbulence In and Above the Thermocline" ,
J. Fluid Mech. 34., 443-448.

Haley, M. C. , (1972), "Small Scale Interactions in the Near Surface Ocean",
M. S. Thesis, Naval Postgraduate School, Monterey, California 93940.

Kennedy, R. M., (1969), "Phase and Amplitude Fluctuations in Propagating
Through a Layered Ocean", J. Acoust. Soc. Am. 46_, 737-745.

Krasilnikov, V. A., and Obukhov, A. M. (1956), Soviet Physics Acoust., 2_,
103-109.

Liebermann, L., (1951), "The Effect of Temperature Inhomogeneities in the
Ocean on the Propagation of Sound" , J. Acoust. Soc. Am. 23, 5 63-570.

39

Mintzer, D. , (1953 a and b) , "Wave Propagation in a Randomly Inhomogeneous
Medium", J. Acoust. Soc. Am. 25_, 922-927, II. J. Acoust. Soc. Am., 26,
186-190. —

Nasmyth, P. W. , (1970), "Oceanic Turbulence" , Ph.D. Thesis, Physics
Department, University of British Columbia, (69 pp.).

Neubert, J. A. and J. L. Lumley, (1970), "Derivation of the Stochastic
Helmholtz Equation for Sound Propagation in a Turbulent Fluid" ,
J. Acoust. Soc. Am. _48, 1212-1218.

Potter, D. S. , (1957) and S. R. Murphy, "On Wave Propagation in a Random
Inhomogeneous Medium", J. Acoust. Soc. Am. 29_, 197-198.

Sagar, F. H. , (1960), "Acoustic Intensity Fluctuations and Temperature
Micros tructure in the Sea", J. Acoust. Soc. Am., _32_, 112-121.

Seymour, H. A., Jr., (1972), "Statistical Relations Between Salinity,
Temperature and Speed of Sound in the Upper Ocean", M. S. Thesis,
Naval Postgraduate School, Monterey, California 93940.

Shvachko, R. F. (1967) "Sound Fluctuations and Random Inhomogeneities
in the Ocean", Soviet Physics-Acoustics 13_, 93-97.

Skudrzyk, E. J. , (1961), "Thermal Micros tructure in the Sea and Its

Contribution to Sound Level Fluctuations", Proceedings NATO Symposium,
Lecture 12, pp. 199-233. Ed. by V. Albers , Plenum Press.

Stewart, R. W. and H. L. Grant, (1962), "Determination of the Rate of
Dissipation of Turbulent Energy Near the Sea Surface in the Presence
of Waves", J. Geophys. Res., 67_, 3177-3180.

Stone, R. G. and D. Mintzer, (1962), "Range Dependence of Acoustic

Fluctuations in a Randomly Inhomogeneous Medium", J. Acoust. Soc. Am.,
34_, 647-653.

Stone, R. G. and D. Mintzer, (1965), "Transition Regime for Acoustic

Fluctuations in a Randomly Inhomogeneous Medium", J. Acoust. Soc. Am.,
38_, 843-846.

Tatarski, V. E. , (1967), "Wave Propagation in a Turbulent Medium",
translated by R. A. Silverman, Dover Pub. , New York, (285 pp.).

Urick, H. J., (1967), "Principles of Underwater Sound for Engineers " ,
McGraw-Hill, New York.

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Predicting Sound Phase and Amplitude Fluctuations
Due to Micros tructure in the Upper Ocean

7. authors;
Herman Medwin

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Department of Physics
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19. KEY WORDS (Contlmia on ravaraa aide If neceeaary and Identify by block number)

Sound Fluctuations
Ocean Micros tructure
Upper Ocean

20. ABSTRACT (Continue on ravaraa alda II neceeeary and Identity by block number)

The temporal and spatial variations of the index of refraction cause fluctuations
of sound phase and amplitude that can be completely understood only by defining
the index in terms of the duration, location, range and time of the acoustic
experiment. A truncated "universal" spatial correlation function of the index ha:
been derived from a simplified form of the Kolmogorov-Batchelor spectrum of
temperature fluctuations in a homogeneous, isotropic medium. Although this
correlation function is shown tn he nrpHir.tahlP simnly from tho H^pth rvf tho

DD I j°NM73 1473 EDITION OF 1 NOV 68 IS OBSOLETE Unclassified

(Page 1) S/N 0102-014-6601 I

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experiment, it is of only limited validity with respect to large spatial lags.
However, a Gaussian extrapolation of the "universal" correlation function
together with the standard deviation of the index provides simple useful
predictions of the sound fluctuations due to temperature micros tructure in
the upper ocean.

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