j. r. herring et al- statistical and dynamical questions in stratied turbulence

8/3/2019 J. R. Herring et al- Statistical and Dynamical Questions in Strati ed Turbulence

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Mech. Struct. & Mach., 29(1), 101 (2001)

Statistical and Dynamical Questions inStratified Turbulence

J. R. Herring1, Y Kimura2, R. James3, J. Clyne3, and P. A. Davidson4

1N.C.A.R., Boulder CO 803072Graduate School of Mathematics, Nagoya University,

Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan3N.C.A.R., Scientific Computing Division

4Department of Engineering, Cambridge, CB2 1PZU.K.

AbstractWe examine homogeneous turbulence under stably stratified and neu-

tral conditions, including decaying and randomly forced cases. Our toolsinclude direct numerical simulations (DNSs) and elements of statistical the-ory. Our DNSat 5123permit large scales to develop from the dynamicsat smaller, energy-containing scales. The above resolution permits a Taylormicroscale R 150. The size distribution of such large scales is closelyrelated to conservation principles, such as angular momentum, energy, and

scalar variance; and we relate these principles to our DNS results. Stratifiedturbulence decays more slowly than isotropic turbulence with the same ini-tial conditions. We offer a simple explanation in terms of the diminution ofenergy transfer to small scales resulting from phase-mixing of gravity waves.Enstrophy structures in stratified flows (scattered pancakes) are distinctlydifferent from those found from isotropic turbulence (vortex tubes). For theforced case, we examine the modification of the inertial range induced bystrong stratification (k5/3 k2). We note that the development of thevertically sheared horizontal flow (VSHF) mode of Smith & Waleffe (2002)is closely associated with strong gravity waves at large scales.

101

Copyright C 2000 by Marcel Dekker, Inc. www.dekker.com


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I. ISOTROPIC TURBULENCE AND RESOLUTION

ISSUES AT LARGE SCALES

Direct Numerical Simulations (DNS) are an important tool in under-

standing the dynamics of turbulent flows. Often, in an effort to reach ashigh a Reynolds number as possible, the large-scale dynamics is given shortshrift. Thus the results are box-limited. This defect is easily noticed in theshape of E(k) at small k so that the energy spectra of such simulations,E(k), are at maximum very close to the box size ( i.e., k=1). One of ourgoals here is to avoid such limits and to display the dynamics that de-velop via NavierStokes in the homogeneous context. Our simulations areof modest resolutions (5123), and so is the Reynolds number (R 250).

The equations of motion to be investigated are:

(t 2)u = p u u (1) u = 0 (2)

We first recall the scaling predicted by Loitsyansky (1939) stemming froman invariance of an integral moment of the two-point velocity correlation.This may be written as:

I=

d2u(x) u(x + ) (3)

Here u = (u,v,w). This translates into spectral language, (u(x) =

dku(k)eikx),etc. as

I= lim0

1

2

0

dxU(x)G(x), G(x) = 64x2

(1 + x2)3(4)

with U(k) E(k)/(2k2). Thus, in order for Ito exist

U(k) = C(t)k2

+ Ok3

. (5)A consequence of (3) would be that C(t) is independent of t. We remark

that the validity of both (3) and (4) is independent of viscosity. Much hasbeen written about the existence ofI, and we refer to Frisch s book (Frisch,1992, pp. 114 & 197) for discussion and references. We only mention herethat C(t) is a slowly increasing function of t according to EDQNM closures(Lesieur & Schertzer, 1978). Is this a defect of the closure or a manifesta-tion of a true time-dependence ofI? Of importance also is that E(k) k2was also predicted by Birkhoff (1953) & Saffman (1967), and as an invis-cid equipartition spectrum (Lee, 1951). Lees result is interesting in thatif a DNS has too little viscosity, the large scales can develop spuriously ask2, because of an under-resolved vorticity spectrum. Using 5123DNS we

explore the question of whether (5) is maintained during decay. In general,


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we find that it is quite easy to observe a transition k4 k2 during thedecay. This invariably happens if R(t = 0) > 40, which is quite surpris-ing. Thus it is easy to understand why the earlier study of this issue byChasnov (1993) found it necessary to employ an eddy viscosity to dampen

the high wave numbers. Fig. 1 shows such a result, for which a stronghyper-viscosity is applied, and for which (5) seems valid. Is C(t) constantfor this run? Our results are somewhat inconclusive, but not inconsistentwith Lesieur & Schertzer (1978). We mentioned inviscid equipartition forwhich k2, k 0 could be a solution. We explore this case in Fig. 2, whichhas the same initial conditions as Fig. 1. Here the k4 spectrum is main-tained for a rather long time, with the k2 inviscid spectrum invading fromlarge k. There remains the question of why, in this inviscid case, the highwave number k2 does not upset the low wave number k4 spectrum, forcingit to a k2 shape. Perhaps the answer is that there is no energy transferassociated with the inviscid k2 spectrum. This behavior is not physical inhomogeneous turbulence because, as Saffman and Birkhoff noted, the pref-actor multiplying k2 is an invariant, so if it starts out zero it must stayzero. The spurious k4 to k2 transition is probably a result of the imposedperiodic boundary conditions.

Before turning to stratified turbulence, we should note that there arerealizable initial conditions for which Loitsyanskys integral is nonexistent:for example, a spectrum (k k0), for which the correlation functions

u(x)u(x + ) sin(k0)/(ko). (6)Since we know that such sharp spectra developed into k4, k 0, we

conclude that it may be possible that I diagnoses an attractor for theturbulence.

II. Stably Stratified Turbulence

The stratified equations of motion generalize (1)(2) as:

(t 2)u = p u u gN + 2 u (7)(t 2) = N w u (8) u = 0. (9)

Here u = (u,v,w) and is the deviation of the temperature field from itsmean, whose constant vertical gradient is nondimensionalized to 1.

The Loitsyansky invariant for stratified turbulence follows from David-

sons analysis of the equivalent MHD problem as:


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I =

r2u udr (10)

with

r = r2x + r2y, u = (iux +juy, 0). (11)

(Davidson, 2001). To represent the anisotropy associated with the buoy-ancy term, we represent the velocity field u(k) by:

u

e1e2

=

e1(k)1(k) + e2(k)2(k)(k g)/|k g|

(k (k g)/|k (k g)

(12)

In what follows, we specify both r and k in polar coordinates so that r =r2 sin2 , dr = r2dr sin dd, and dk = k2dk sin dd. Where conve-nient we call = cos = cos . We may now work through the analogoussteps leading to (4) for isotropic turbulence to get

1

1

d

1

1

dF(, )

0

x2dx2

[1(x,) + 22(x,)] (13)

In this expression,

F(, ) = (1 2) G

1 + i,

(1 2)(1 2)

G(a, b) 24a4 + 9b4 72a2b2[a2 + b2]9/2

(14)

and

i(k) = |i(k)|2, i = (1, 2). (15)

Here, we have rescaled r x. Equations (13) and (14) are much morecomplicated than (4), but they nonetheless lead to the same conclusionthat U(k) k2 as k 0. The dependence of U(k) is not so restricted,but we should recall the theorem of Cambon, Jeandel & Mathieu (1981):

2(k, ) 1(k, ) (1 2), 1. (16)It is useful to record (7)(9) in the Fourier representation, and use (13)

for economy ((1, 2) instead of (u,v,w)):

t

12

= M

12

+

e1 fe2 f

f

(17)

Here,


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M =

0 2 cos 02cos 0 Nsin

0 Nsin 0

(18)

and

{f, f} (F1, F2, F3) (19)are the Fourier amplitudes of the nonlinear terms in (7)(9), and is the

nondimensional rotation. Finally, we transform (7)(9) with an expansionin terms of the eigen vectors of M, vi(k), and its eigenvalues, i. These are

(v1, v2, v3) =

10

0

, 1

2

0i

1

, 1

2

0i

1

(20)

0,1 = (0, iNsin()) (21)

and then expand

=

i(k, t)vi (22)

(t + i)i(k, t) = Fi (23)We record i and i only for the case = 0, which is the only case studied

here. The relation between and is explained by12

3

= T

12

1 0 00 i/2 i/2

0 1/

2 1/

2

12

(24)

Here F= T F, with F defined just after (16). T has for its inverse itscomplex adjoint.

Finally, we note viaclosure for the vi (i.e.,

|| = 1 and f(M) = |f()|) that

exp(M t) =

1 0 00 0 0

0 0 0

+ 1

2

0 0 00 1 i

0 i 1

eiNt sin

+1

2

0 0 00 1 i

0 i 1

eiNt sin (25)

Equation (24) describes the evolution of {1, 2, } in the absence of thenonlinear terms (the rapid distortion approximation, which has short-term

validity if the initial conditions are sufficiently random). We see from (23)


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that for the angular average of {1, 2, }, only 1 survives at long times,if the nonlinearities are suppressed and N.

The striking feature of stably stratified turbulence is that it decaysslower than isotropic three-dimensional turbulence with the same initial

conditions. Fig. 3 illustrates this. It shows the decay of enstrophy for thestandard E(k, 0) = k4 exp(k2) energy spectrum. Here, R(0) = 80, andthe resolution is 2563. Stratification (with N=10) is turned on when theskewness reaches its maximum (about 1 eddy circulation time).

We note that at late times, an approximate E(t) t1 obtains. Thefirst-order question is then how to explain the slowdown in the decay, fromapproximately t1.5 in isotropic DNS. A significant feature of such flows isits strong anisotropy, with the vorticity organized into scattered pancakes asshown in Fig. 4. Here we show the late-time organization of the enstrophy,E(r, t) | u(r, t)|2.

The flow is strongly anisotropic, but its degree of anisotropy tends tosaturate an N. This is indicated in Fig. 5, which shows the angulardistribution of the vorticity vector.

It is remarkable that although stratified flows contain waves, the evo-lution of the enstrophy patches shows little wave-like fluctuation. Perhapsthis is because pancake organization signifies strong vertical variability, forwhich the wave frequency,

=

N2 sin2() + 42 cos2() (26)

is near zero. Here cos = kz/k, and we include in a possible rotationrate, .

Perhaps the apparent lack of waves in pancakes may be explained viaan analogy to Taylor columns in rotating turbulence. Think of Taylors ex-periment as discussed in the introduction of Greenspans book (Greenspan,1968). A penny is slowly towed across the base of a rapidly rotating tank.A Taylor column is seen to move with it, spanning the fluid from the pennyto the top of the tank. But how does the fluid lying well above the penny,but within the column, know it must move with the penny? Now the Taylorcolumn (which is the analogue of pancakes) exhibits no wave-like featuresto the casual observer. However, if, as noted by Greenspan, one carefullyanalyzes the flow on the fast time-scale of inertial wave propagation (atime-scale based on group velocity/tank size, rather than frequency), thenone finds that the information telling the column to move is transmittedupward from the penny in the form of fast, low-frequency inertial wavespropagating in the vertical direction. (Recall that the fastest waves, interms of group velocity, have the lowest frequency.) In short, the quasi-

steady Taylor column is the manifestation of fast inertial waves. Without


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the waves there would be no column. However, if you look at the flow yousee no wave-like propertiesjust a moving column. In a similar way the pan-cakes could be a manifestation of the fast transmission of information bylow-frequency, horizontally-propagating gravity waves. As with the Taylor

column there is no obvious wave-like behavior, but without the waves therewould be no pancakes.

The following is a rough estimate of the decay of kinetic energy forstratified turbulence, based on closure. We recall first a simple wave-numberdiffusion equation for the evolution of the energy spectrum, E(k, t) as pro-posed by Leith (1968): EDQNM):

(t + k2)E(k, t) = k

k4k

k0

p2dpE(p)

(k)

E(k)

k2

. (27)

Here, (k) is the eddy relaxation rate, which for isotropic turbulence isk0

p2dpE(p). (Actually, Leith proposed k3/2

(k, t).) For strongly strati-

fied turbulence, we expect

N. Why, though, should waves act to dampenthe energy? An essential point here is that gravity waves phase mix thusproviding an attenuation of correlations in time. Kaneda (1998) has stressedthis point in his application of rapid-distortion theory to stratified turbu-lence. See also Hunt & Carruthers (1990). Then by integrating (5) over[0,k], with k in the inertial range there follows E(k) 2/3k5/3, if wetake (k)

k0

p2dpE(p). If, on the other hand (k) N, as for stronglystratified flow, there follows,

E(k) (N )1/2/k2. (28)Note that we have ignored anisotropic effects here, so our argument is

rough. But, according to Fig. 3 anisotropy is not overwhelming, even forstrong anisotropy. We may now estimate the decay of E(t) by the followingargument. There results,

E(t) t5/7 (29)The exponent is the average of 3D decay (10/7) and 2D (0). The exponent

in Fig. 3 is -1 instead of -5/7. This may be attributable to finite R 35in the DNS, just as our estimate of the decay of isotropic turbulence issomewhat faster than indicated by the EDQNM calculations (-1.5 insteadof 1.37). Clearly, such estimates should be replaced by more secure EDQNMor DIA calculations, such as those proposed by Godeferd & Cambon (1994)and Sanderson et al. (1991). We remark that the spectral form (28) hasbeen compared to DNS for rotating turbulence by Yeung & Zhou (1998).A similar slowdown in the cascade to small scales has been noted in MHD

turbulence by Galtier et al. (1997).


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108 Herring et al.

One problem with the brief analysis given above is its near-isotropyassumption. Davidsons extension of the invariants (equations (19)(13))suggests that only the horizontal energy u2

participates in invariants, where

our crude analysis involves the total energy. In fact, the decaying (isotopi-

cally gathered) energy has k2

, k 0. Perhaps a reanalysis of the problemin terms of (1, 2, 3) (see equations (10)(24)) would be more appropri-ate variables than 1(k), 2(k), (k)).

It is nevertheless of interest to examine the k2 prediction of equation(27). For this purpose we note that forced turbulence is able to reach furthertowards the asymptotic regime, where such power laws are expected. Hencewe force the flow with a solenoidal Gaussian random force acting on u(k)at some wave number k0. It is isotropic. We examine two cases, k0 = 5, andk0 = 10. Of course the k0 = 5 will reach the higher Reynolds number, andthe k0 = 10 will have the better statistics.

Fig. 6 presents the steady-state values of 1(k), 2(k), & (k)for k0 = 5. Here k denotes the cylindrically gathered spectra, summedover kz. We see some evidence here for a 2k2. The forcing of 1 isclearly evident, but not for 2 nor . Note that is in near lockstep with2 as would be expected from (7)(9), for strong stratification. We mustnote that power-law behavior is observed only for certain ways of formingspectra. For example, Fig. 7 shows isotropically gathered spectra ((|k|)),etc. Here there is evidence of power laws for any of the variables.

Where is evidence of wave-like behavior? We expect this in 2 and. Clearly, 1(r, t) displays a spectrum of oscillations, and has been notedby Metais & Herring (1989), but is there evidence for wave-like behavior inthe spectra? Fig. 8 presents data on this point. Here we see that after theapproach to a statically steady state only the lowest k modes of 2 and show wave-like behavior. This mode is the vertically sheared horizontalflow (VSHF) of Smith and Waleffe (2002). Its frequency is about half of N,

but notice that it emerges as a vertical response to a horizontal forcing.Finally we comment on the statistics of stratified flow. We know that

isotropic turbulence develops strong non-Gaussianity in its acceleration.For example, Fig 9. (top panel) shows the Eulerian acceleration for isotropicturbulence, according to DNS at 1283 resolution. The near exponentialdistribution is similar to experiments, of which those of Champagne et al.(1977) are the most secure. The lower panel shows the same histogramfor strong stratification (N=10). Note the conversion from exponential toGaussian, with an accompanying anisotropy (w2 much weaker than u2 orv2 but both near Gaussian).


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III. Concluding Comments

In this paper, we have invoked integral invariants of the Loitsyanskytype as a basis to discuss the issue of box-limited DNS. An adequate reso-

lution of the large scales of the flow implies a certain behavior of the energyspectrum near k = 0. For isotropic flows, our DNS results at 512 3 suggestthat if initial conditions have E(k, 0) kp, p > 4, then p 4 as t .However, we fail to confirm that C(t) (as in (5)) is constant. Rather C(t)increases in rough accord with Chasnovs finding (1993). We do not knowif this slow increase of C(t) holds up as t . The assumptions neededfor the derivation ofIinclude: (1) homogeneity, and (2) spacial reflectioncovariance, which asserts that(u(x, t) = u(x, t), and (x, t) = (x, t)may be used for quantities such as u(x, t)F and (x, t)F, for any F.

For stratified flow, the generalization of the Loitsyansky invariants ap-plies only to the horizontal dynamics. Thus, the strong vertical variability(the VSHF mode of SmithWaleffe at k = 0) seems outside its purview.It is this mode that bears the wave-like signature as far as spectra are con-cerned. It is excited directly through a 1, 2 coupling, with the scale of1 near the horizontal forcing. Finally, stratified flows are closer to Gaus-sian than unstratified flows. This is seen in the distribution of Eulerianacceleration (as shown in Fig. 9), as well as other measures of non-lineartransfer, e.g., skewness factors for velocity and the temperature field. Thismay suggest that closure approximations will have more success here thanfor unstratified turbulence.

One final comment about the more mathematical aspects of stratifiedflows setting apart inviscid stratified flows from the unstratified case. Forthe latter, a finite time singularity in vorticity is thought to develop, but noproof has yet emerged. An examination of this issue viaDNS indicates thatsuch is indeed the case, with the singularity developing after a few eddy

circulation times (Kerr, 1998). For stratified flows, similar DNS studiesreveal no singularity. Of course, such are always resolution-limited, but thefact that most turbulent atmospheric flows are stably stratified suggeststhat a focus by mathematicians on this issue may be important.

References

1. Caillol, P. & Zeitlin, V. (1999). Kinetic Equations and Stationary En-ergy Spectra of Weakly Nonlinear Internal Gravity Waves. Preprint.

2. Cambon, C., Jeandel, D., & Mathieu, J. (1981). Spectral modeling ofhomogeneous nonisotropic turbulence. J. Fluid Mech., 104: 247262.

3. Champagne, F. H., Friehe, C., A., Larue, J. C. & Wyngaard, J. C.


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(1977). Flux measurements, flux estimation techniques and fine-scaleturbulence measurements in the unstable surface layer over land. J.Atmos. Sci., 34: 515-530.

4. Chasnov, J. R. (1993). Computation of the Loitsyansky Integral in

decaying isotropic turbulence. Phys. Fluids A 5 (11): 25792581.5. Davidson, P. A. (2001). An Introduction to Magnetohydrodynamics.

Cambridge, U. K.: Cambridge Univ. Press.6. Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov, Cam-

bridge Univ. Press, Cambridge7. Galtier, S., Politano, H. & Pouquet, A. (1997). Self-similar energy de-

cay in magnetohydrodynamic turbulence. Phys. Rev. Lett., 79: 28072810.

8. Godeferd, F. S. & Cambon, C. (1994). Detailed investigation of en-ergy transfer in homogeneous stratified turbulence. Phys. Fluid, 6 (6):20842100.

9. Greenspan, H. P. (1969). Theory of Rotating Fluids. Cambridge, U.K.: Cambridge Univ. Press.

10. Herring J. R. & Kimura, Y. (2002). Structural and Statistical As-pects of Stably Stratified Turbulence. In Y. Kaneda and T. Gotoheds. Statistical Theories and Computational Approaches to Turbu-lence. Springer-Verlag, Berlin, Heidelberg, New York, pp. 1542.

11. Hunt, J. C. R. & Carruthers, D. J. (1990). Rapid distortion theoryand the problem of turbulence. J. Fluid Mech., 212: 497532.

12. Kerr, R. M. (1993). Evidence for a singularity of the three-dimensionalincompressible Euler equations. Phys. Fluids A, 5: 17251746.

13. Kimura, Y. & Herring J. R. (1996). Diffusion in stably stratified tur-bulence, J. Fluid Mech., 328: 253269.

14. Lee, T. D. (1952). On some statistical properties of hydrodynamicaland magneto hydrodynamical fields. Q. Appl. Math., 10: 6974.

15. Lesieur, M. & Schertzer, D. (1978). Amortissement auto similaritedune turbulence a grand nombre de Reynolds J. de Mecanique, 17:609646.

16. Loitsyansky, L. G. (1939). Some basic laws for isotropic turbulent flow.Trudy Tsentr. AeroGidrodin. Inst: 332.

17. Metais, O. & Herring, J. R. (1989). Numerical studies of freely decay-ing homogeneous stratified turbulence. J. Fluid Mech., 202: 117148.

18. Saffman, P. G. (1967). The large-scale structure of homogeneous tur-bulence. J. Fluid Mech.,27: 58193.

19. Smith, L. & Waleffe, F. (2002). Generation of slow large scales inforced rotating stratified turbulence. J. Fluid Mech., 451: 145168.

20. Yeung, P. K. & Zhou, Y. (1998). Numerical study of rotating turbu-

lence with external forcing. Phys. Fluids, 10 (11): 28953244.


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Received April 13, 2004

Revised May 2004


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nu Pr Bv2 Om2

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1.15E+01 2.06E-02 2.71E+00 3.43E-03

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Figure 2. Energy Spectrum, E(k,t), for inviscid run. Notice (1) an ap-proximate k4 low wave number E(k, t), which has an approximate constant

C(t); (2) at high k the development of an approximate inertial range (forshort times) with an eventual inviscid k2 spectrum.


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Figure 3. Total kinetic energy for decaying strongly stratified turbulence.After Herring & Kimura (2002).


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N2

= 0 N2

= 1

N2

= 10 N2

= 100

Figure 4. Enstrophy profiles for isotropic turbulence, N2 = 0 (panel 1),and increasingly strong stratification with N2 = 1, (panel 2), N2 = 10,(panel 3) and N2 = 100, (panel 4). After Kimura & Herring (1996). Reso-lution = 1283


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Figure 5. Angular distribution of enstrophy for stratified turbulence ofvarious degrees of stratification (N2 = 0, 1, 10, 100) for two times duringthe decay of the flow. After Kimura & Herring (1996).


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Figure 7. 1(|k|, t), 2(|k|, t), and (|k|, t) for horizontally forcing at|k| = 5 at t=27.5. Note the difference between this figure and the previous,indicating highly layered structures, as described in the text.


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10

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0 5 10 15 20 25 3010

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0 5 10 15 20 25 30

k = 1.5 k = 5.5

k = 10.5 k = 30.5

(k ,t)

(k ,t)

(k ,t)

(k ,t)

1(k ,t)

1(k ,t)

1(k ,t)

1(k ,t)

2(k ,t)

2(k ,t)2(k ,t)

2(k ,t)

Figure 8. 1(k, t), 2(k, t), and 2(k, t) at k = 1.5 (upper left),k = 5.5, (upper right), k = 10, (lower left), and k = 30.5 (lower right)as functions of time, t. In each panel, is the top curve, with 2(k, t),the lower, except for k = 30.5 , for which 1(k, t) 2(k, t).


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Figure 10. Enstrophy of a turbulent field at the time of maximum totalenstrophy. Initial field is Gaussian. Note the bent vortex tubes typical ofsuch turbulence. Flow is unstratified.


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Figure 11. Enstrophy of a stratified turbulent field at the late time.Initial field is the same as in previous figure. Note the scattered pancakearrangement of the vorticity, with frequently dipole stacking.


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Fi 12 E t h f t tifi d t b l t fi ld f th l t

j. r. herring et al- statistical and dynamical questions in stratied turbulence

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