Turbulence produced by internal waves in the oceanic thermocline at mid and low latitudes

ELSEVIER

Dynamics of Atmospheres and Oceans 24 (1996) 1-14

Ocean basins and lakes Turbulence produced by internal waves in the oceanic thermocline at mid and low latitudes M.C. G r e g g a,., D.P. W i n k e l a, T.B. S a n f o r d a, H. P e t e r s b a University of Washington, Seattle, WA 98105, USA b State University of New York at Stony Brook, Stony Brook, N~, USA Received 1 July 1994; revised 7 February 1995; accepted 24 February 1995

Abstract When mid-latitude internal waves are at the background state modeled by Garrett and Munk (J. Geophys. Res., 80: 291-297, 1975), wave-wave interactions transfer energy to dissipative scales so slowly that instabilities generate overturns not much larger than the scales at which viscosity dissipates the turbulence. Consequently, high wavenumber spectra contain only the viscous decay portion of the universal turbulent spectrum. Increases in low-wavenumber shear above background produce larger overturns and turbulence sufficiently intense to form well-developed inertial subranges. The turbulent region of the vertical spectrum is separated from the wave region by a k31 rolloff that does not vary with the amplitude of low-wavenumber shear, indicating that the wave field is saturated. At low latitudes, internal wave shear is generally more intense than at mid latitudes, but the turbulence is similar to that found with mid-latitude waves at background. Dynamically, the anomaly implies that wave-wave interactions transfer energy to dissipative scales more slowly at low latitudes than at high latitudes. Kinematically, this anomaly results from steeper slope in the rolloff range.

1. Observations and analysis To observe turbulence in the thermocline and simultaneously relate the turbulence to the larger-scale shear that produces it, we use the Multi-Scale Profiler (MSP). The MSP carries three sets of sensors--electromagnetic, acoustic, and airfoil--to measure eastward, u, and northward, v, velocities over wavelengths from 1000 m to 10 mm. Before calculating spectra, we subtract the ensemble

* Corresponding author. 0377-0265/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 0 2 6 5 ( 9 5 ) 0 0 4 0 6 - 8

2

MIC. Gregg et al./Dynamics of Atmospheres and Oceans 24 (1996) 1-14

average profiles, ~(x 3) and ~(x3), and apply the Hann filter to minimize limitations of using a finite data window. Velocity spectra, ~VEc(k3) -- ~u(k3) + ~,(k3), are taken over successive blocks of 5 or 10 kPa (0.5 or 1 m). These spectra are then converted to shear spectra and normalized to form spectra of the gradient Froude number, q~Fr(k3)- (2'n'k3)2~VEL(k3)N -2. Vertical wavenumber, k3, is the reciprocal of wavelength in cycles per meter (cpm), and N is the buoyancy frequency in radians per second (s-l). Integration gives the cumulative shear variance, k~

Fr2(k3) - fk,@Fr(m) a m

(1)

Termed the Froude function, Fr 2 = 4 when the r.m.s, gradient Richardson number is one-quarter. Profiles of the viscous dissipation rate, e, are calculated by integrating the shear spectra. For transverse velocities assumed to be isotropic, e : 7.5u~i"(ZTrk3)2dPvEt.(k3)dk 3, ( W k g

l)

(2)

where k 0 = 1 cpm or 2 cpm, and k , is determined empirically as the wavenumber where the spectrum is spatially resolved or noisy. As a reference for background internal waves, we use the Garrett and Munk (1975) modification known as GM76 and described by Gregg and Kunze (1991). Except for small changes in shape at low wavenumber, the Froude form of GM76 is independent of N. When internal waves are the only process affecting the profile, the strain spectrum is (I)st(k 3) = (2rrk3)2~(k3)/(Ocro/Ox3) 2, where ~o is GM76 = qbFr G~76 / 3 , as for a free wave with a potential density. At all wavenumbers, qbst frequency 1.4f. For high wavenumbers we use the turbulent velocity spectra of Nasmyth (published by Oakey (1982)) and of Panchev and Kesich (1969). The shear form of the velocity spectrum peaks at 0.16k~, where k~ = (2~')-1(ff//23)1/4 is the viscous, or Kolmogorov, wavenumber in cyclic units, and v is the kinematic viscosity. For the turbulent scalar spectrum we use the form derived by Batchelor (1959).

2. The mid-latitude background state

During P A T C H E X in October 1986, we observed internal waves at 34°N, 127°W in the eastern North Pacific and found their spectra to be similar to GM76 (Gregg and Sanford, 1988). Within this background field of internal waves, turbulence occurred intermittently in thin patches and had low intensity (Fig. 1). Averaging the ensemble of 28 P A T C H E X profiles gives g -= 7 × 1 0 - 1 ° N 2 N o 2 (W kg - l )

(3)

which is very close to the result of Henyey et al. (1986), who calculated the average energy flux from low to high wavenumbers in a field of random large-scale waves

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14

-0.3

24

u,v / rn s 1 0

25

26

~e ! kg m "3

3

0.3

27

10 1° 10 .7 E / W kg -1

Fig. 1. Typical PATCHEX profile. Zero velocity is arbitrary in this and other profiles except TH 2. The dissipation rate, e, is averaged over 0.05 MPa (5 m) in this and subsequent figures. having the background spectrum formulated by Munk (1981). Using ray tracing, they introduced test waves of intermediate wavelength and followed them until their wavenumber grew to 0.2 cpm, when their energy was treated as irreversibly headed for dissipation. Henyey et al. (1986) also developed an analytical model as a check on their numerical model. Both models predict e ( x N -2, as does a separate calculation by McComas and Miiller (1981). The magnitude of e obtained by Henyey et al. (1986) also agrees with (3) when referenced to GM76 instead of Munk (1981). Munk's formulation decreases the shear variance by ¢r/2 relative to GM76, a consequence that was not intended and is not called for by observations (W. Munk, personal communication, 1990). Osborn (1980) estimated diapycnal diffusivity from e and N 2 observations using Kp ~< 0.2EN -2 ( m 2 s -1)

(4)

The North Atlantic Tracer Release Experiment ( N A T R E ) seems to be confirming (4) by comparing it with diffusivity estimated from the thickening of thin tracer streaks. The most recent results are Kp = 1.1 × 10 -5 m z s -x from the tracer (Ledwell et al., 1994) and the microstructure survey (Schmitt et al., 1994). Internal waves during N A T R E were somewhat more intense than GM76, as using (3) for E gives Kp ~< 5 X 10 -6 m 2 s - t , constant with depth and 30 times Kr, the molecular diffusivity (Gregg and Sanford, 1988). The average P A T C H E X spectrum is nearly flat (k3°) for k 3 < 0.1 cpm (Fig. 2). Flat spectra result when the shear field consists of many uncorrelated waves. In spite of thermohaline intrusions dominating higher wavenumbers, strain spectra can be formed at low wavenumbers and are very close to the GM76 shear-to-strain ratio. The internal wave spectrum cuts off at k c = 0.1 cpm and then decreases as

4

M.C. Gregg et al./Dynamics of Atmospheres and Oceans 24 (1996) 1-14 10 ~ :

'

'

'

'"I

'

'

''''"I

'

'

''''"I

'

'

''''"

PATCHEX ~M76

"7

34°N 127°W

E Q. ¢o

I0°

10-1 10 .2

10 0

10 1

10 1

t0 2

ks / cpm Fig. 2. Froude spectrum for P A T C H E X .

k~ 1 until 1 cpm. The inflection near 1 cpm corresponds to the largest overturning scales and is the beginning of the turbulent range. The average • in Table 1 gives 14.1 cpm for k, = (27r)-l(E/u3) 1/4, the viscous, or Kolmogorov, wavenumber in cyclic units. The universal shear spectrum peaks at 0.16k~ = 2.3 cpm. The k 31 rolloff is close to the internal wave rolloff in the stratosphere. Rather than being broadband and random like oceanic internal waves, stratospheric waves usually have very narrow bandwidth in frequency and wavenumber, and increase in amplitude as they propagate upward into thinner air. Two explanations have been advanced for the uniform rolloff at k 3 > k c. The saturated hypothesis holds that the waves become unstable at k c and break by shear instability and convective overturning at higher wavenumbers (Dewan and Good, 1986). For a single wave, Smith et al. (1987) modeled the saturated spectrum as a~SAT TFr = 0 . 5 k 3 t, which is

Table 1 Average conditions for the pressure ranges marked in the figures Data P Pn FSI FS2 FS3 FS4 FS5 FS6 FS7 THll.4°N TH2°N THI°N TH0 ° C1.7 ° CO° C3°N

103 N (s l)

e

3.0 3.2 12.5 5.4 4.3 5.0 5.0 5.5 4.5 2.9 2.7 2.8 2.9 2.8 2.8 2.9

1 . 7 × 10 m 1.9×10 " 1 . 1 × 10 x 2 . 9 x 10 -s 5 . 8 × 1 0 '~ 5 . 0 x 10 - ~ 5 . 0 × 10- 9 3.8×10 9 1 . 0 × 10 s 8.07<10 ii 1.3 × 10 m 7 . 0 × 1 0 1~ 2 . 2 × 1 0 1o 2 . 2 x 10 m 1.7 × 10 - l0 7 . 0 × 10 - i1

(Wkg

Ko I)

(m2s-l) 3 . 9 × 10- 6 3.9×10 -s 1.4× 10-5 2 . 0 x 10 - 4 6 . 1 x 10 - 5 4 . 0 × 10 - s 4 . 0 × 10 - 5 2.6×10-5 1 . 0 × 10 - 4 1.9× 10-6 3.5 × 10 - 6 1.8×1-6 5.8 × 1 0 - 6 5.8× 10-6 4.4 × 1 0 - 6 1 . 7 × 10 - 6

M.C. Gregg et al. /Dynamics of Atmospheres and Oceans 24 (1996) 1-14

5

shown on our spectra. D.C. Fritts (personal communication, 1991) believes that the waves begin to lose energy at k o but breaking begins at higher numbers, perhaps 2~rk c. The second explanation was advanced by Hines (1991a), who argued that the waves do not become unstable at k c. Instead, energy is shifted to higher wavenumbers by strong Doppler shifting until the waves eventually break. Hines (1991b) developed a spectrum that asymptotically approaches k 3 k Plotted in Fig. 2, Fr 2= 0.52-0.64 for k c = 0.09-0.11 cpm. Thus, there is no reason to suspect widespread shear instability at the cutoff wavenumber. Forming the probability distribution of Fr 2 by first-differencing velocity profiles over 10 m and using constant N 2 gives only 0.2% with Fr 2 > 4 (Gregg et al., 1993). Owing to many thermohaline intrusions, we cannot directly compute overturning scales, I. Other observations (Dillon, 1982; Peters et al., 1988) are consistent with l-( e / N 3 ) 1/2, the scaling developed by Ozmidov (1965) for overturns limited by stratification. Using (3) gives l = O . 0 7 ( N / N o ) - 1 / 2 = 0.094 m. The e distribution is approximately lognormal and thus highly skewed. Consequently, the larger e values are produced by overturns exceeding 0.1 m, but visual inspection shows about 1 m as the largest overturning scale. Because the form of saturation producing the rolloff does not produce significant wave breaking, we conclude that Hines's approach is more likely than the instability models to explain oceanic spectra.

3. Mid-latitude anomalies

The PATCHEX north profiles were taken at 42°N, 126°W in a coastal jet off California. Although the jet was above 1 MPa, velocity and dissipation are larger

-0.3

25

u,v / m S"1 0

0.3

26 27 Oo / kg m"a

10"1° 10;' e / W kg ~

Fig. 3. TypicalPATCHEX north profile.

6

M.C. Gregg et al./Dynamics of Atmospheres and Oceans 24 (1996) 1-14 ........

i

........

.

PATCHEX north 42°N. 126°W 5.75-9.25 MPa

101 "7,

10 °

.... l

NASMYTH

10 ¸ '[

.......

10-2

i

10-1

\"" '~

........

t

10 0

........

i

,. , . . . . .

101

10 2

k3 / cpm

Fig. 4. Froude spectrum for PATCHEX north. than P A T C H E X throughout the profile (Fig. 3). For example, the average E is more than ten times P A T C H E X , giving Kp = 3.9 × 10 -5 m 2 s - 1 (Table 1). Using P A T C H E X north as the most energetic example, Gregg (1989) expressed the average dissipation rate in terms of the increased shear as • = 7

X

lO

10

(N

2

4 ~4 / N O2 )(S1,,/5,O~M76 )

(5)

$10 is the 10 m first-difference shear corrected for attenuation by the first-difference filter. The expression is adapted from Henyey et al. (1986) to use shear, which is measured well by MSP, instead of kinetic energy, which is not measured well. The Froude spectrum for P A T C H E X north is 5 GM76 at 0.01 cpm and slopes steadily downward until it intersects an extension of the k 3 ~ GM76 rolloff somewhat below 0.1 cpm (Fig. 4). Owing to the irregular shape, the beginning of the rolloff cannot be defined visually. For GM76, k c = 0.1 cpm and F r 2 ( k c ) = 0.7. For P A T C H E X north, the closest estimate to Fr 2 = 0.7 gives k c = 0.04 cpm. The turbulent spectrum begins at 0.4 cpm, where Fr 2 = 2.5, and is moderately well developed with a distinct inertial subrange. To observe internal waves in strong mean shear and near topography, in June 1990 we took stations across the Florida Strait at 27°N (Fig. 5). The Gulf Stream was centered on Station 3 near 0.5 MPa. At each station we took about ten profiles, each to within 5 m of the bottom. Most drops ended in sections of continuous strong turbulence much thicker than the homogeneous bottom boundary layer (Fig. 6). Spectra taken below the core of the stream vary by factors of 2-3 at low wavenumbers and ten at high wavenumbers (Fig. 7). All except Station 1 lie well above GM76 and cut off where they intersect the GM76 k~ 1 rolloff extended to

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14 1

2

3

4

5

6

7

7

2 t~

n

3

2

4

ffl

~

5

7 8

0

20

40 distance /km

60

80

Fig. 5. Florida Strait stations, June 1990. Contours show average northward velocity (in m s - 1). Labeled dashed lines are the maximum depth of the surface boundary layer (SBL) and the maximum heights of the homogeneous bottom layer (HBL) and the turbulent bottom layer (TBL). Vertical dashed lines show where the spectra were taken.

lower wavenumbers. Considering the different circumstances, the similarity of the rolloff to those of PATCHEX and PATCHEX north demonstrates that it is a very robust feature. The rolloff follows GM76 until the beginning of the turbulent range.

-1.2

U,V / -0.6

m s "1

0.0

co

24

26 oe / kg m "3

28 10 "1° 10 .7 e / W kg -1

Fig. 6. Profile at Florida Strait Station 4. High ~ at the bottom extends well into the stratified section above the homogenized bottom boundary layer.

8

M.C. Gregg et al./Dynamics of Atmospheres and Oceans 24 (1996) 1-14 10 2

,

i

, ii,,,

I

i

I

i l,li,

i

l

I

i

iilll[

i

I

I II114

Sin 2

~

;tn 7

"7

E Q.. ¢o 10 0 n-LI-

e

10 2

I

10 - 2

I i l l f HI

' I J,,JHI

J

,

,,,~HJ

t , ~J,t,t

10 °

10 2

k3 / c p m Fig. 7. F r o u d e s p e c t r a f r o m the Florida Strait.

Stations 2 and 7 are the most energetic at low wavenumbers and have inertial subranges spanning a decade. Their strong turbulence produces Kp = (1-2) × 10 4 m 2 s - ' . Observed E values are within a factor of two of (5) in the center of the strait, but are factors of 5-10 times higher near the sides. Correcting with the shear-to-strain ratio as proposed by Polzin (1993) does not help. It appears, therefore, that the elevated E is not simply related to the shear variance.

4. Low-latitude anomalies During T R O P I C H E A T 2 we profiled between I1.4°N and 0 ° near 140°W. Shears were several times GM76 but much less than predicted by Munk (1981). At 140°W the internal waves are superimposed on 'deep jets' below the undercurrent (Fig. 8). The jets have magnitudes of ___0.1 m s - ' and are about 150 m thick. As a function of vertical wavenumber and frequency, (I)GrM76(k3,09) is proportional to fw-3(oj 2 _f2)1/2, where f is the Coriolis parameter. Integrating over w from f to N removes the f dependence, making d')GM76(b TFr ""3"~ independent of f. To account for decreased coherences observed near the equator, Munk (1981) made qb~Sl(k3,w) proportional to ((o 2 --f2)1/2~o-3 SO that integrating makes ~MrSl(k 3) Ot f - t. Munk, therefore, predicted shear variances of 14 GM76 at 2 ° and 29 GM76 at 1 °"

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14 U,V / ITI S "1

-0.4 0o0

0.4

0,8

0 MSP

2

4

e~ 8 10 22

24 26 o e / kg m a

2 8 10 "1° "10-7 E ! W kg ~

Fig. 8. TROPIC HEAT 2 profile on the equator. The Equatorial Undercurrent is centered at 1.5 MPa.

lOS~

.......

i

....... i

........

¢ (.

1

j

•

10

i

i

?

I

i'A~,,"~,t~.l,~t~,%O"N

10"

, I

, I

ln-2 " --

~r. "-~I~ • •

o

10°

j ~;~GM76

I: .... " ' .~. ~ / b

10

t

101

".. j"11:4°N

10~

]102 q

~

103

¢')

.......

TROPIC H EAT 2 6.4-9.9 MPa

St

~

,

'Ko'

,

i I

.... "I~ U . ~ O •T

pM7611 !

%t '.~" ._~'J

i !

10 .3

: ~'~ r~.L.~ , \~,.~,. 10"1

........ J ........ i ........ t 10 .2 10 "1 10 0 10 1

k3 /

......

cpm

Fig. 9. TROPIC HEAT 2 spectra.

1 0 .4

10 2

r..)

10

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14 u,v -0.8-0.4

22

/ rn S-1 0.0

0.4

24 26 ~e / kg m 3

28 10 "1° 10 .7 ~ / W kg -1

Fig. 10. C O A R E 3 profile at 0 °. The Equatorial Undercurrent is centered at 2.5 MPa.

1°s I

~

.......

,

........

,

........

,

........

lO 2

q~

[...~. St t. " " - - ~ N

COARE leg 3 6.4-10 MPa 101

100 "7

E 10-1 o

e

~,gO°N

,,

~

'k.

1°43'S

,o,

10 .2

°iN, 100

%

10"~1

........

10-2

~

10.3

, . . . . . . . . , . . ."~...~ . . . . . . 10-4 10-1 10 0 101 10 2 k 3 / cpm

Fig. l 1. Froude spectra for C O A R E 3.

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14

11

TROPIC HEAT 2 Froude spectra from 11.4°N, 2°N, I°N, and 0°N differ less than a factor of two at all wavenumbers (Fig. 9). At low wavenumbers they are 1.5-2 GM76, much less than predicted by Munk (1981). The spectra are also more peaked than mid-latitude spectra, indicating that the wave field may have a narrower bandwidth. For k 3 > k c = 0.03 cpm, the spectra roll off along Yd'~Fr SAT o r slightly below. Near 0.2 cpm the slope steepens to approximately k 31.4. The steep rolloff produces a more distinct rise into the turbulent range. The dissipation rates give K~ within a factor of two of PATCHEX (Table 1). Because these profiles lack thermohaline intrusions, strain can be obtained from temperature. For k 3 < k c , the strain spectra are approximately flat and vary in amplitude from 1/3 GM76 at ll.4°N to 3 GM76 at 0°N (Fig. 9). If one wave frequency dominates the low wavenumbers, as seems likely owing to the peaked spectra, the shear-to-strain ratios correspond to average frequencies of 1.10f at ll.4°N, 1.27f at 2°N, and 1.7f at I°N. Eriksen (1993) obtained a similar latitudinal variation in average frequency from calculations of the response of the equatorial ocean to rapidly moving wind bursts. Changes in slope of (1)st match features in di)Fr. Between k c and 0.2 cpm, where (I)Fr follows YFrtI~SAT, the strain spectra transition to the rolloff. At 0.2 cpm, where the (l)Fr rolloff steepens, (I)st rolls off close to GM76 but with slightly steeper slope. The strain rolloff ends at 0.16 k,, the peak of the turbulent range in di)Fr. The turbulent range approximates the scalar spectrum derived by Batchelor (1959). During COARE 3 we profiled along 156°E. The profiles are roughly similar to those at 140°W (Fig. 10). At low wavenumbers the Froude spectra are more

102

;

. . . . . . . .

~ f ~

101 -

I

. . . . . . . .

I

. . . . . . . .

I

. . . . . . .

:

SAT

PATCHEX north 5.7 - 9.9 MPa

~

10 o COARE

leg 3

~

~.~,~P~ ~ ~-,.7,o-,Owkg'-; 1 0 "1

........ 10-2

i 10-1

........

..... .~

"..~..

i ........ 10 0

- I~

:,

~ . ." . . . . . 10 1 10 2

k3 / cpm Fig. 12. Froudespectra for PATCHEXnorth and COARE 3 0°N.

12

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14

J~ . . . . . . . .

w

j

' •

25

7 I ~ r

i

20

t

6

I

t 15 ¢.

%_ ii

2 -,

2

3-

Ii

t \~

:7 2/

..

~.,,/C3 0° ~

-

~ . / c 3 1 . 7 ° s --

:

~k'~'~ , , ~ C33°N

'"..:::,.\~., --. \ , , ',,, •"

TH

.'

-,.

-

-

5

'.'-~'".%.,

.,,.

-

o~ 10-2

10

- ~" "Y7,=,':': ~'--,~. . . . . . . . . 10"~

100 100

k3 / cpm Fig. 13. F r o u d e f u n c t i o n s n o r m a l i z e d by G M 7 6 d i s s i p a t i o n s p e c t r a a r e o n the right.

t01

o 102

k3 / cpm are s h o w n o n t h e left, a n d v a r i a n c e - p r e s e r v i n g

energetic than those from 140°W, and their roiloffs begin at higher wavenumbers, mSAT closer to the GM76 rolloff extended to low wavenumbers than to ~ e r (Fig. 11). Otherwise the rolloff and turbulent ranges are similar to those for T R O P I C H E A T 2, so that Kp again is no larger than P A T C H E X . The C O A R E 3 strain spectra are elevated even more than ~Fr, particularly at 0 °, where the rolloff remains twice GM76. C. Eriksen (personal communication, 1994) predicted preferential elevation of CI)st compared with (Dvr as a consequence of the/3 effect on near-equatorial responses to wind forcing.

5. Discussion

Variations in turbulent production result from changes in wave-wave interactions with variations in spectral amplitude and shape. The present procedure for mid latitudes requires assuming that E depends only on total shear variance to k c , perhaps corrected for shifts in the average frequency of the wave field. These are empirical results that break clown when extended from mid to low latitudes, as shown vividly in Figs. 12 and 13. In the former, C O A R E 3 drops from 2 - 3 times P A T C H E X north for k 3 < 0.1 cpm to 1/5 in the turbulent range. One possibility is that the C O A R E 3 internal wave field may contain only a few dominant waves

M.C. Gregg et al. / Dynamics of Atmospheres and Oceans 24 (1996) 1-14

13

rather than continuous distributions in wavenumber and frequency. If so, the steep rolloff may simply be the envelope of the narrow band of wavenumbers forming the spectrum, and E may be lower because there are fewer and weaker interactions. In Fig. 13 several other curves are closer to unity at 0.01 cpm and peak between 0.02 and 0.03 cpm, indicating that this may be a general condition at low latitudes. In any event if shear variance is the sole factor responsible for producing turbulence, it does not work in the same way everywhere or there would not be essentially two dissipation spectra on the right for such a varied range of shears on the left. These anomalous results also reveal the limitations of statistical approaches to wave-wave interactions. Our ensemble-averages were obtained by sampling sites for 1-12 days. The data are relatively stationary because at most places the internal wave field evolves slowly, allowing time for adequate sampling. In addition to lacking the three-dimensional structure of the wave field, we have not observed the evolution of the internal wave field. Only then will we really be able to understand the dynamics producing turbulence. In the interim, calculations of interaction rates with altered spectra would be helpful. Henyey et al. (1986) simulated variations in wave energy level by changing the dimensionless energy density of Munk (1981) while retaining the canonical shape. A good start would be to do similar calculations with varied shapes, including narrow band fields that are not separable, i.e. ones with wavenumber distributions varying with frequency.

Acknowledgments The Office of Naval Research and the National Science Foundation funded the collection and analysis of these observations.

References Batchelor, G.K., 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. J. Fluid Mech., 5: 113-139. Dewan, E.M. and Good, R.E., 1986. Saturation and the 'universal' spectrum for vertical profiles of horizontal scalar winds in the atmosphere. J. Geophys. Res., 91: 2742-2748. Dillon, T.M., 1982. Vertical overturns: a comparison of Thorpe and Ozmidov length scales. J. Geophys. Res., 87: 9601-9613. Eriksen, C.C., 1993. Equatorial ocean response to rapidly translating wind bursts. J. Phys. Oceanogr., 23: 1208-1230. Garrett, C.J.R. and Munk, W.H., 1975. Space-time scales of internal waves: a progress report. J. Geophys. Res., 80: 291-297. Gregg, M.C., 1989. Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94: 9686-9698. Gregg, M.C. and Kunze, E., 1991. Shear and strain in Santa Monica Basin. J. Geophys. Res., 96: 16709-16719. Gregg, M.C. and Sanford, T.B., 1988. The dependence of turbulent dissipation on stratification in a diffusively stable thermocline. J. Geophys. Res., 93: 12381-12392. Gregg, M.C., Seim, H.E. and Percival, D.B., 1993. Statistics of shear and turbulent dissipation profiles in random internal wave fields. J. Phys. Oceanogr., 23: 1777-1799.

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