Oceanic mixing by breaking internal waves

Deep-Sea Research, 1972, Vol. 19, pp. 823 to 832. Pergamon Press. Printed in Great Britain.

Oceanic mixing by breaking internal waves

CHRISTOPHER GARRETT* and WALTER MUNK t

(Received 13 March 1972; in revised form 21 June 1972; accepted 26 June 1972)

Abstract--Based on our recently proposed horizontally isotropic internal wave spectrum (with it, many uncertainties) we have developed a formalism for mixing due to internal wave shearing. Following an initial instability, the vertical shear is spread into a layer growing at just the rate required to main- tain the layer Richardson number at a critical value. For instabilities in the gross density profile, the layer thickness is determined by the vertical scale of the shear maximum; fine-structure concentrates the instability along thin sheets and leads to more frequent breaking, but of smaller vertical extent (the scale now being determined by the fine-structure). The resultant buoyancy flux can be interpreted in terms of an eddy diffusivity, but the numerical evaluation is extremely uncertain. Properties of the internal wave field will cause the horizontal scale of the mixed layer so formed to exceed by several orders the vertical scale and similarly the horizontal diffusivity to exceed the vertical diffusivity. Both diffusivities decrease rapidly with increasing depth.

1. INTRODUCTION

WE (GARRETT and MUNK, 1972; henceforth GM) have recently attempted to fit all internal wave observations in the deep sea with a universal horizontally isotropic spectrum in wavenumber, frequency-space. The assumption of universality was based on (i) the similarity of spectral shape and energy levels for measurements of the same type at different places and times, and (ii) the lack of any other way of relating different observations. There are many uncertainties in our final model. Indeed, the main result of our work should be to clarify the most important areas of ignorance.

However, we cannot resist the temptation to explore some of the consequences of the GM spectrum, with particular regard to vertical and horizontal mixing in the oceans.

2. SUMMARY OF THE PROPOSED INTERNAL WAVE SPECTRUM

The dimensionless GM spectrum is (with numbers in square brackets referring to GM equation numbers)

E(ct, to) = 2n - t Eto, l z - t to - t (to2 _ to~)-~ (2.1) [6.18]

extending from the inertial frequency to t to the local V~iis~il~i frequency n(y), and limited in wavenumber to the range 0 < ,t < #, where

# = jl~(to2 _ to2.)~, (2.2) [6.19]

j~ being the equivalent number of wave modes. Observations were fitted with

j~ = 20, E = 2~ x 10 -s . (2.3) [6.24]

*Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada. tInstitute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University

of California, La Jolla, Calif., U.S.A.

823

824 CHRISTOPHER GARRETT and WALTER MUNK

The effective local vertical wavenumber is

fl = ~n(co 2 - co 2) - * (2.4) [1.31

(see G M following [5.15] for the reason for taking n rather than V'n 2 " co 2 here). The foregoing quantities are non-dimensionalized with respect to a wave n u m b e r

and frequency = 1.22 x 10- 6 cycles c m - 1, ~¢ = 0-833 x 10- 3 cycles sec- 1 = 3 cycles h o u r - ~ (2.5)

based on the buoyancy scale depth b = (2zc_~)-1 = 1.3 km and on the Vaisala frequency at the top of the thermocline (extrapolated f rom abyssal depths). The inertial frequency co~ was taken as 0.0133 (cb~ = 0.04 cycles hour -1 ) for convenience, though it does, o f course, vary with latitude. Internal wave energy per unit surface area (integrated over depth) is

/~/~ = /3 (2n ) -1 E ~ - a ~/-1 = 0 .4 joules cm -2. (2.6) [6.24]

The mean square horizontal (not given by G M ) and vertical displacements are

[~2, ~2] = b2 ~ [X 2, Z 2] E(c~, co) d~ do) (2.7) [4.4]

= 2 2 (co2 _ co2.). (2.8) [2.10] w i t h X 2 nco-4(co 2 + c o i ) , Z = n - l c o -2

Hence ~_~ 7 - 2 En b 2, ~ = -~ oo i = ½ En -1 b 2, (2.9)

giving 720 n ~ and 7.3 n - ~ metres for the corresponding r.m.s, values. Finally, horizontal and vertical coherency scales are given by

R ~ = 2.8 # - 1 b = 58(co 2 - co2)-+ metres, ~ = 1.9 (zion) -1 b = 39n - l metres. (2.10) [6.26, 6.16]

Our estimate of the reliability of the G M spectrum, and hence of the results based on it, is as follows:

(i) Energy levels. Our value of E (and hence ~2, ~2) is within a factor o f about 3 of most deep-sea observations. The dependence on depth through n(y) is p robab ly t rustworthy to abou t the same extent.

(ii) Frequency dependence. The spectral shape of the frequency spectrum, co-2 away f rom co~, is fairly well-established, though the choice of the exponent of(co 2 - co z) in an analytical representat ion of the observed inertial cusp is somewhat arbitrary. We have chosen - I /2; a different, but still realistic, value (in the range f rom - 1/4 to -3/4, say) does not produce significant changes.

(iii) Wave number dependence. The linear dependence of the bandwidth /t on frequency (away f rom co~) is required by the fo rm of moored (frequency) and towed (wavenumber) spectra. The actual choice o f j i depends on rather limited coherence data. An error (or variability in t ime or space) by a factor of 3 is conceivable. More serious is our choice of a spectrum that is flat f rom 0 to p [or more precisely, f rom the first mode wavenumber ~(1) (09) to /t] and zero for ~ > It. The same bandwidth could be achieved in many other ways; we chose the simplest for lack of any data that would enable us to resolve the ambiguity. Our choice may well underest imate (but p robably not overestimate) such quantities as mean-square shear.

(iv) Horizontal isotropy. Here the evidence is conflicting, and we chose an isotropic spectrum for the sake of simplicity. Most quantities will be affected very little, if at all, by this assumption.

Oceanic mixing by breaking internal waves 825

3. SLOPE AND SHEAR

The mean square slope in the xl-direction is obtained by multiplying the integrand for ~ in (2.7) with (2x ~l ~)2. Similarly in the xz-direction we multiply by (2~ ~2 ~)2. For horizontal isotropy the two are the same and, with ~2 = 0~ + ~ ,

m --~ = m 2 + m~ = ~ 2 Z'---2 E((x, co) dec d o == 2co, E n - 1 ~i ~2c°-3 d~ do (3.1) j~x2

= -~ r~j~ coj E = 7 .0 x 10 -+, (3.2)

neglecting terms of order codn.

The mean square shear is ~2 = (Ofq/O~)2 + (0a2/0j))2, where (ill, fi2) = 0(~1,~2)/0l are the components of horizontal flow; it is found by multiplying the integrand for ~z in (2.7) with (2~colql) 2 (2~fli~r)2:

~2 = ~ + ~ = 8ca,ji- t E n 3 ~r2 ~ c~z co-a (co2 + co2) (co2 _ co~)-2 dec dco (3.3)

= 2~+j~ E n a FI z = 3.4 x 10 -6 n a sec -2, (3.4)

giving an r.m.s, shear of about 0.2 cm sec- 1 per metre for n = 1. The associated Richardson number is

R-i = (2~n~r)2__ = 2(~2j~ n E) -1 = 8.1 n -1. (3.5) ~2

Too much importance should not be attached to the precise values calculated here, but we do consider the relative magnitudes of slope and shear, and their depth

dependence through n, to be significant. The dependence of ( R i ) - ~ on n(y) indicates a decrease of shear instability with depth.

It takes 38 r.m.s, slopes to produce unit slope, compared with only 5.7 n -~ r.m.s. shears to reduce the Richardson number to I/4. Thus, except at great depth where n is very small, shear instability is very much more likely than overturning. This is in conflict with the suggestion of ORLANSKI and BRYAN (1969), the reason being that in the ocean much of the internal wave energy is near the inertial frequency, where the waves have a much greater shear to slope ratio than in the absence of rotation.

Shear instability rather than direct overturning is made even more likely by the p/t presence of fine-structure. Let s', refer to the fine-structured distributions of shear

and Viiis~il~i frequency, with local averages s, n, the gross quantities considered so far. From internal wave theory, vertical gradients of velocity and density are approximately proportional, so that s ' /s = n'2/n 2, leading to a Richardson number in the fine-structure

4g z n'Z 4~z n 2 (n'Z/n 2) n 2 Ri ' = s,-- T - - s2(n,+/n+) = 2 ~ i - ~ . (3.6)

Thus the probability of shear instability is greatly enhanced in a region of high density

gradient. The ratio n'Z/n 2 ofr.m.s, to mean density gradient may be of order 20 (GREGG

and Cox, 1972), and so it takes only order (n -+) r.m.s, shears to reduce Ri ' to 1/4. Unlike gross instability, breaking in a region of high density gradient must be quite common, and this has implications concerning the maintenance of fine-structure. On the other hand, it is not obvious whether frequent but small-scale breaking on fine- structure or rare gross breaking contributes more to the vertical mixing.


4. GROSS V E R T I C A L M I X I N G

If we plot the shear s ( y , t ) at any depth y as a function of time t, there will be occasions (due to random superposition of waves) when the shear exceeds the critical value sc for a Richardson number of 1/4 (Fig. 1). We assume that each of these events leads to breaking (with subsequent vertical mixing), our viewpoint thus being rather similar to that adopted by LONGUET-HIGGINS (1969) in his study of breaking surface waves. The laboratory experiments of THORVE (1971) and meteorological observations of BROWNING and WATKINS (1970) suggest that the thickness of the resulting mixed layer is such that the layer Richardson number, based on the maximum shear experi- enced, is 0-4 ___ 0.1. We shall call this the Thorpe number, denoted Th.

Of course a given depth may be involved in a mixing event due to a Richardson number < 1/4 just above or below it, even though the Richardson number at that depth is always > 1/4. A better way of visualizing the mixing events is in the y, t-plane (Fig. 1). An event starts when and where s reaches a critical value

.s. c = Ric -½ (21rn) = 2(2nn),

then quickly spreads to a thickness within which the average shear is reduced to STh = T h -½ (2rrn) = 1.6 (2nn). It then spreads slowly to some maximum value A at the time the shear is largest, the vertical extent of the shaded region at any one time indicating the depths over which the layer Richardson number is Th. THORPE (1971) has emphasized that, for mixing to occur, a Richardson number < 1/4 must persist over a long enough interval for the instability to grow large. We assume this to be the case; a point in favour of the assumption is that the time scale of the instability is of the order of the local V~iisiilii period, while the time scale of the duration of the shear will depend also on the generally much larger inertial period.

s (Y0 ,t

Sc

$Th

Y

Sc

t

Fig. 1. Schematic presentations of mixing events at a fixed depth yo (top) and in the y, t-plane (bottom). The first event is chosen to commence at Yo, the second above Yo. In the shaded areas s > sc if there were no mixing. As a result of mixing, the shear is rapidly reduced to Srh and the mixed areas extended to the Sr~-contours, which reach their maximum dimension, A, at the time

of maximum shear.


The change in potential energy per unit surface area associated with perfect mixing over a depth A in a density gradient dp/dy is E, = 1/12 g(dp/dy) A 3 = 1/12pnZA 3. (The measurements by BROWNING and WATKINS (1970) and laboratory experiments of THORPE (1973) suggest imperfect mixing, but this presumably only changes the factor 1/12 somewhat.) Let Q(A) dA designate the expected number of events between A + ½dA per unit t, y-area. S Ep(A) Q(A) dA is the average change in potential energy per unit time per unit volume, i.e. the buoyancy flux A pn z, where pn z = d(pg)/dy is the buoyancy gradient, and

A = I ~ : A 3 Q ( A ) d A (4.1)

is by definition then the appropriate eddy diffusivity. It is convenient to change from A-distributions to the statistics of S, the shear

magnitude at the time and depth of peak shear. We write s = S(1 - ½y2/y2 + . . . ) for the distribution just above and beneath the peak, where an appropriate value for

1 ~½4 s dy, we have Y will be chosen shortly. Then with Srh = A J_i~

S < so: A = 0

S = so: A = A,~, = 2x/6 Y[I - (RiJTh)*]* = 2"24Y

S = so(1 + ~): A = A,~ n (1 + 1.9c).

In nearly all events for which S exceeds s~ it will only just do so, and it is adequate to write

A = 1/12 Amian X number of peaks, with S > so, per unit t, y-area. (4.2)

If the shear components were normally distributed, which is unlikely in view of the intermittency of inertial waves (WEBSTER, 1968), then the density of peaks of s = (st 2 + s2) ½ could in principle be evaluated. The job is formidable and we have not attempted it. However, if we assume that all the shear is in one component (an incorrect assumption even for a uni-directional spectrum) and that it is normally distributed, then we may use the results of RICE (1944) to proceed further. Denote

S~ g 'F (~, co) dxdco = M,(A),

where ~. is a frequency or wavenumber and F(cz, co)is the spectrum of the shear [proportional to the integrand in (3.3)]. Then

TI = re[Me (~b)/M2 (cb)] ~ = 1~4(3re~co#)* 3[-1 = 2.2n-¢ hours (4.3)

Lt = 7t[Mo (I~)/M~ (~)]½ = (5/3) ~ (j~n)-l~ = 84n- 1 metres

designate the mean intervals in time and vertical distance between shear zeros. (Note that Lt and the vertical coherence scale I~, are of the same form and comparable in magnitude, as one would expect.) Rice statistics give the number of zero crossings times exp 2 2 (-½s,/sp) for the expected number of peaks (maxima and minima) with

2 2 2 IS[ > s,, provided that s~ > sp. s~; denotes the mean-square shear at the vertical position under consideration, namely the position in the vertical with the maximum shear amplitude. We shall take sp 2 equal to the overall mean square shear s 2, though it could


be slightly greater. There is typically one extremum in a vertical distance L1, so that

there are exp ( - ½ sc2/s 2) 7"1-1 L I - 1 events per hour-metre.* For our spectrum this is 0.0054n 3/2 exp ( -16 . 2n -~) events per hour metre,

= 5.0 x 10 -1° for n = 1, much less for smaller n. Doubling the r.m.s, shear increases this number by a factor 1.9 x 105.

An appropriate choice for the scale Y in an expansion of the shear near its vertical maximum is L 1 / ~ , so that Aml n = 60n- ~ metres and

A = 97n -3/2 exp ( - 16.2n -1) metresZhr -~ = 270n -3/2 exp ( - 16.2n -1) cm2sec -1 (4.4)

or 2.5 x 10- 5 cm 2 sec- ~ for n = 1, 4-7 cm 2 sec- ~ for double the r.m.s, shear. The eddy diffusivity estimated on this basis is clearly very sensitive to the r.m.s, shear, and also to our assumption of a Gaussian distribution of shear, and thus impossible to calculate with any confidence. The only significant feature of the result is the rapid decrease of A with increasing depth (decreasing n).

Alternatively, with Aml n = 2.24Y and Y = Ll/z~, we may write

A = 0.03L12 TZ a (4.5)

where T~ is the interval between each event in the vertical distance L~. This is similar to the result of STOMMEL and FEDOROV (1967), as is inevitable for what is, after all, a mixing length theory. In order for A to be the traditional 1 cm2sec - a (e.g. MtrNK, 1966) we require (with L 1 = 84n- a metres) an event every 25n -2 days, a result that is very sensitive to the value of L x.

5. VERTICAL MIXING RELATED TO FINE-STRUCTURE

I f both temperature and salt gradients are stable, so that layer formation by double diffusive effects (STERN and TURNER, 1969) cannot occur, internal wave breaking of the sort discussed above seems to be the most likely source of fine-structure. It may provide sharp density gradients at the top and bot tom of the mixed layer, as alleged by WOODS and WILEY (1972), and also possibly between mixed billows of slightly different composition when they collapse and are sheared over each other (THORPE, 1971). Once the fine-structure exists, the regions of high gradient provide a preferential site for shear instability.

Imagine a vertical stack of fairly homogeneous layers separated by thin sheets with high gradients.t We visualize the following sequence following initial instability (Fig. 2): the sheet (a) quickly thickens (as R i -~ Th) and then grows till the time of maximum shear (b), the sheet Richardson number always remaining equal to the Thorpe number. Residual turbulence eventually turns the thickened sheet into a new layer (c). Conditions at the times of initial instability, and of maximum gross shear, are given by

(a) s ' s ( n ' / n ) 2 "-½ ' = = Rtc (2rcn), (b) S ' S (n} /n ) 2 _~ , (5.1) = = Th (27rn.0,

*One of us prefers to argue that the distributions in time and vertical distance are independent, so that the number of events per hour-metre is exp ( - s c2 / s 2) T1 - t L1-t. For our spectrum and n = 1 the authors thus disagree by a factor 107, and beg the reader's indulgence.

tThis idealization was introduced by WOODS and FOSBERRY (1966) and is useful for discussion purposes. In fact, the resolved free-structure looks more like a vertical random process.


p(y)j~qf~ ~'J'~P'(Y)

A A s AL

Fig. 2.

\\ The density depth curve at three successive stages of an instability.

respectively. Dividing gives (s/S) (n'/n~) = (RiJTh)-L The ratio between the thickness of the new layer so formed, and that of the initial sheet, is then

A n 'z T h S 2 T h S 2 / n ' \ 2 ~ s s = n - ~ = Ri~s 2 - 4~ 2n 2 I n ) " (5.2)

Prior to the event the sheets are separated by homogeneous layers of thickness AL = (n'Z/n z) A s, and if the newly formed layer is to be of the same thickness it follows that S = (Th) -~ 2~n, which is just the condition for gross instability. I f the new layer is thicker than the old, then the gross profile is also unstable, mixing is limited in vertical extent by the thickness of the shear peak (as we have seen) and the presence of fine-structure is not crucial. Alternatively, if the new layers are thinner, then the breaking waves on the fine-structure are a small-scale diffusive process, and the scale will continue to decrease until molecular processes take over, or a new layer is formed by some means, possibly gross breaking. We shall now estimate vertical mixing by wave breaking on fine-structure.

Set v z = n'Z/n z, so that v z is the ratio of r.m.s, to mean density gradient. The change of potential energy per unit surface area for a sheet with initial density jump ~p is approximately Ep = 1/8 g~pA 2 = 1/8 pn'ZAsA z. Let P(S)dS denote the number of (gross) shear peaks in S + ½dS per unit t. An event requires

S > Ric-~v -1 (2~n) = 4~n/v, so proceeding as previously

f° A = < ~ v 6 A~ a (2~n)-4Th z S4P(S)dS> / <v2As> (5.3) ~n]v


where < > represents an average over all sheets, with different A, and v. This formula emphasizes the contributions from thick sheets with large gradients (though v 6 and the lower bound of the integral). < v2A,> is the average spacing between sheets.

We now proceed in the coarsest possible way, assuming As and v constant and again using the peak probability for a single Gaussian variable,

P ( S ) = T I- l (S/s2)exp ( _½S2/s 2)

(valid if $2~ s2). Then

A = 2Th 2 A ] T 1 t (1 + 27 -1 + 27 -2) e-~, i' = 2v -2 Ri (5.4)

which becomes exponentially small for large 7 , and varies in proportion to

~-2 = l/4v 4 (s2)2 (2zrn)-a for small y. Setting Th = 0.4, A~ = 20 cm, /'1 = 8000n -½ seconds, v 2 = 20, we have y = 0 . 8 1 n -~, and A = 0.046 cm 2 sec -~ for n = 1, (0.80 cm 2 sec -~ if we double r.m.s, shear or v).

6. H O R I Z O N T A L M I X I N G

The horizontal displacement associated with the GM spectrum is 0.72 n ½ km [equation (2.9)], so that in a large event the horizontal excursion of a particle may be several kilometres. This suggests a role for internal wave-induced horizontal mixing processes.

Denote by a the displacement shear (the vertical derivative of horizontal displace-

ment). The integrand for a 2 is (2t r i te)- 2 times that for ~2. Hence for the G M spectrum

tT2 = -~.lito7rZ2 -2 -i 2 n 3 E = 4 0 0 n 3, (6.1)

i.e. the r.m.s, displacement shear is about 20 metres per metre. The displacement shear corresponding to a velocity shear sufficient to give a Richardson number 1/4 is 120 n, a value which depends only on the ~o-dependence of the internal wave spectrum. Although maximum velocity shear is achieved on the average when the displacement shear is zero, the r.m.s, displacement shear at a time of maximum velocity shear in a mixing event will be comparable with the above figure. This suggests that if vertical mixing occurs over a distance Y, the water that is mixed comes from points with typical horizontal separation of the order of 10 2 nY. This scale will occur squared in any formula for a horizontal diffusivity, so that we expect a horizontal eddy diffusivity about l04 n 2 times the vertical diffusivity.

This feature of breaking internal waves may be related to the cusps in the T - S curves of microstructure. These cusps are oriented along lines of constant a r, showing that in these instances the microstructures of temperature and salinity nearly cancel in their effect on density. PINGR~E 0969) suggests horizontal intrusion of a well mixed region or some mechanism based on horizontal shear. The horizontal displacement required is of the order of 1 km, and so could be produced by vertical mixing over a distance of order 10n-1 metres.

A large ratio between horizontal and vertical scales is found also for the coherences [equation (2.9)]. Most of the energy is near inertial frequencies, and for to = 2co i, say, the ratio ~' , to fr~ is 2500 metres to 40 n - 1 metres. This may have a bearing on the relatively large horizontal extent (and aspect ratios of about 103 :l) of tbe homogeneous laminae, usually attributed to the collapse of a well-mixed region of smaller aspect


ratio. We expect the shear maximum to have a horizontal extent of order ,~, and to retain its identity as it sweeps out and destabilizes a large horizontal area.

7. DISSIPATION

One final vague number is the dissipation time of the GM spectrum. THORPE (1973) suggests that about 2 5 ~ of the kinetic energy lost by the shear flow appears as potential energy (the exact percentage depends on the degree of mixing and the velocity profile after mixing; neither is yet well-established). Thus the rate of loss of internal wave kinetic energy is about 4 times the rate of gain of potential energy of the basic density gradient, i.e.

dE - - = 4p .4(27tn~) 2 (7.1) ~t

at any depth. Integrating this over all depths for the GM model (n oc e-Y), assuming A constant, the rate of loss of energy per unit surface area is approximately 41tp A.~2~ t - 1. For A = 1 cm 2 sec- 1 (though we do not believe a constant value to be possible) that is about 7 ergs cm- 2 sec- ~. Thus, the time required to dissipate the total internal wave energy of approximately 4 x 10 6 e r g s c m - 2 is 6 days. It would be interesting to compare this time with the time scale of energy transfer to higher wavenumbers by resonant interactions, which may also play a vital role in determining the shape of the spectrum.

8. D I S C U S S I O N

We have visualized the mixing events to occur at the time and place of maximum (gross) shear. In the presence of fine-structure the event consists of a cloud of micro- events, each centred on a sheet. Micro-breaking leads to a continual decrease in the scale of fine-structure and ultimately its removal by molecular processes, unless it is occasionally renewed by the formation of thick layers. It seems possible that gross breaking is the dominant mixing mechanism, and micro-breaking merely a smoothing process that occurs between gross events. Our work is premature in that some of the essential phenomenology has yet to be established by observation, and that some atten- dant statistical problems have to be solved.

Due to the almost inevitable sensitivity of an eddy diffusivity to r.m.s, shear and to the tail of the probability distribution of shear, we suspect that a useful prediction of a value for the eddy diffusivity is unlikely to come from a statistical theory of wave breaking based on an observed internal wave spectrum. We do attach some weight, however, to the following rather qualitative results:

(i) Mixing is much more likely to be caused by shear instability than by direct overturning.

(ii) Vertical mixing decreases rapidly with V~iis~il~i frequency, and is possibly several orders of magnitude less near the bottom of the ocean than in the main thermocline.

(iii) The horizontal eddy diffusivity due to breaking internal waves is about 10 4 n 2 times the vertical eddy diffusivity.

We make no predictions about eddy viscosities.


Acknowledegments--S. A. THORPE has made helpful comments. Much of our work has been supported by the National Science Foundation and the Office of Naval Research. The International Centre for Theoretical Physics, Trieste, Italy assisted in the preparation of this manuscript.

REFERENCES

BgOWNING K. A. and C. D. WATglNS (1970) Observations of clear air turbulence by high power radar. Nature, Lond., 227, 260-263.

GARRETT C. J. R. and W. H. MtsNg (1972) Space-time scales of internal waves. Geophys. Fluid Dynamics, 2, 225-264.

GgEc, o M. C. and C. S. Cox (1972) The vertical microstructure of temperature and salinity. Deep-Sea Res., 19, 355-376.

LONGtYET-HIC,~INS M. S. (1969) On wave breaking and the equilibrium spectrum of wind- generated waves. Proc. R. Soc. (A), 310, 151-159.

MUNK W. H. (1966) Abyssal recipes. Deep-Sea Res., 13, 707-730. ORLANSKI I. and K. BRYAN (1969) Formation of the thermocline step structure by large

amplitude internal waves. J. geophys. Res., 74, 6975-6983. P1NGPa:.E R. D. (1969) Small-scale structure of temperature and salinity near Station Cavall.

Deep-Sea Res., 16, 275-295. RICE S. O. (1944) The mathematical analysis of random noise. Bell System tech. J., 23,

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STERN M. E. and J. S. Ttn~NER (1969) Salt fingers and convecting layers. Deep-Sea Res., 16, 497-511.

STOMMEL H. and K. N. FEDOROV (1967) Small scale structure in temperature and salinity. Tellus, 19, 306-325.

THORPE S. A. (1971) Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech., 46, 299-319.

THORPE S. A. (1973) Turbulence in stably stratified fluids; a review of laboratory experiments. Boundary Layer Meteorology (in press).

WEBSTER T. F. (1968) Observation of inertial period motions in the deep sea. Rev. Geophys., 6, 473-490.

WOODS J. D. and G. G. FOSBERRY (1966) Observations of the thermocline and transient stratifications made visible by dye. Proc. 1965 Malta Syrup. Underwater Assoc., London, Underwater Assoc., p. 31.

WOODS J. D. and R. L. WlLEY (1972) Billow turbulence and ocean microstructure. Deep-Sea Res., 19, 87-121.

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Oceanic mixing by breaking internal waves