Self-preservation of large-scale structures in Burgers' turbulence

Physics LettersA 182 (1993) 109-113 PHYSICS LETTERS A North-Holland

Self-preservation of large-scale structures in Burgers' turbulence

Erik Aurcll a,b, Sergey N. Gurbatov c,d and Igor I. Wertgeim b,e • Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden b Center for Parallel Computers, Royal Institute of Technology, 100 44 Stockholm, Sweden c Radiophysical Department, University of Nizhny Novgorod,

23 Gagarin Avenue, 603600 Nizhny Novgorod, Russian Federation a Department of Mechanics, Royal Institute of Technology, 100 44 Stockholm, Sweden e Institute of Continuous Media Mechanics, Russian Academy of Sciences,

1 Akad. Korolyov Street, 614061 Perm, Russian Federation i

Received 29 June 1993; accepted for publication 16 September 1993 Communicated by A.P. Fordy

We investibate the stability of larl~seale structures in Bursers' equation under the perturbation by high wave-number noise in the initial conditions. Analytical estimates are obtained for random initial data with spatial spectral density k n, n < 1. Numerical investigations are performed for the ease n=O, using a parallel implementation of the fast Legendre transform.

The appearance of ordered structures is possible for nonlinear media with dissipation [ 1 ]. These structures often form a set of cells with regular bc- haviour, alternating with randomly loealised zones of dissipation. One nonlinear dissipative system with such behaviour is the well-known Burgers' equation,

02/) OV + v ~ x = l Z v ( x , t = O ) = v o ( X ) (1) oS g-X X

where/z is the viscosity coefficient. The solutions of Burgers' equation with random initial conditions display two mechanisms, which are also inherent to real turbulence: nonlinear transfer of energy through the spectrum, and viscous damping in the small scale region. Equation (1) was proposed by Burgers [2 ] as a one-dimensional model of fluid turbulence. It has since been shown to arise in a large variety of non-equilibrium phenomena, when parity invari- ante holds [ 3]. Burgers' equation has applications to nonlinear acoustics, nonlinear waves in thermo- elastic media, modelling of the formation of large- scale structures in the universe, and many other sys- tems where dispersion is negligibly small compared

Permanent addresses.

with nonlinearity (see ref. [4] and references therein).

Burgers' turbulence is an example of strong turbulence, the properties of which are determined by the strong interaction of large numbers of harmonic waves. In Burgers' turbulence, due to the coherent interaction of harmonics, saw-tooth shock waves are formed, which may be treated as a gas with strong local interaction between particles [4,5]. The colli- sion of shock fronts leads to their merging, which is analogous to inelastic collisions of particles, and to an increase of the integral scale of turbulence.

In the limit of zero viscosity the solution of ( I ) for the velocity field is given by

x - y ( x , t) OS(x, t) v ( x , t ) = t , v ( x , t ) = - Ox ' (2)

where y(x , t) is where the absolute maximum is re- alised of the function

( x - y ) 2 G(x, y, t) =S0(y) 2t '

X # t

S o ( X ) = J vo(y) dy. (3)

By analogy between the solutions of Burgers' equa-

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Volume 182, number 1 PHYSICS LETTERS A 8 November 1993

tion and the flow ofpaxticles [4], we shall call So(y) the initial action, and S(x, t)=G(x, y(x, t), t) the action at time t. This solution can be computed by a Legendre transform of initial data [6]. We shall assume that the spectrum of random initial velocity field has the following form,

go(k) =ot2 knbo( k ) ,

bo(k<ko) = l, bo(k>ko) = 0 . (4)

The curvature Ks of the initial action in (3) may then be estimated as

O/2 L.n+3 ha'0 2 2

K 2 = ( [ S ~ ( y ) ] 2 ) = n+3 =so~linen,

,-u2bn+ 1 a ~ = ( v 2 ) = ~,.~_____z_o (5)

n + l

Taking into account that the curvature of the parabola in (3) is 1/t, we get that when r~t~= lmi./ao =1(,71 the curvature of the parabola is much smaller than the curvature of the initial action. This implies that the global maximum of G(x, y, t) is in the neighbourhood of the local maximum of SO(y), and that y(x, t) is a stepwise non-decreasing function of x. Thus the velocity field has universal behaviour,

v(x, t)= X--yk t ' X k _ 1 < X < X k , (6)

in each cell between shock positions Xk~ Each cell is characterized by two numbers: the inverse Lagran- gian coordinate Yk, and the value of the initial action in the cell Sk=SO(Yk). The dissipation zone is in this ease at the shock front, and hence has zero width. The positions of the shocks are determined by the equality at the point Xk of the values of two absolute maxima, G(Xk, Yk-~, t)=G(Xk, Yk, t):

Xk= ½ (Yk + Yk--~ ) + Vkt ,

Vk = SO(Yk-1) --SO(Yk) (7) Y k - - Y k - - l

It is easy to see that the rate of collisions of the shocks depends on the asymptotic behaviour of the structure function of the initial action

ds(p) = ( [ s o ( x + p ) - (So(x)]2)

~a~o l-n, i f n < l ,

~ 2 a 2, if n> 1 . (8)

Here ~2 = (S 2 ) . We can estimate the integral scale of the turbulence l( t)= Ix-yJ from the condition that the parabola and the initial action are of the same order:

ds[l(t) ]1/2~12/t. (9)

From ( 8 ), (9) it therefore follows that we have two different types of growth for the scale l(t):

l ( t )~ (ott) 2/(n+3), if n< 1 .

(ast) 1/2, if n> 1 . (10)

It was shown in ref. [ 4 ] (see also ref. [ 6 ] ) that the common feature, for both types of initial spectra, is the existence of self-similarity of statistical properties of solutions, which are determined by only one scale l(t). From (10) one can see that in the case n < 1, the behavior of Burgers' turbulence at time t is determined by the local large-scale behaviour of the initial spectrum. This was the motivation for a simple qualitative model of Burgers' turbulence (at n < 1 ) as a discrete infinite set of modes - the spatial harmonies km=koy -m ( y ~ 1 ), sufficiently spaced in the spatial spectrum [7,8 ]. The amplitudes A 2 of the harmonics were chosen from the condition that the mean spectral density of the harmonics in the in- terval Ar~=k,,+~-km was identical to the random noise spectral density: A 2 =go(kin)Am. Assuming that the influence of small-scale components on the large-scale one is small, it was obtained that the growth of the average scale for this discrete model is the same as in the case of the continuous spectrum, given by (10). The idea of independent evolution of large-scale structures has also been used to obtain long-time asymptotics of the cylindrical Burgers equation [ 9 ].

It is possible to show that the assumption of rel- atively independent evolution of the large-scale structures is valid for the continuous model of Burgers' turbulence as well. In ref. [10] it was obtained analytically that the interaction of a regular positive pulse with noise in Burgers' equation does' not change the evolution of the large-scale structure

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of the pulse, when the value of the initial action of the pulse is greater than the dispersion of the noise action. This result was confirmed by numerical sim- ulations based on the asymptotic solution (2), (3) of Burgers' equation. Here we shall give some simple estimates of this effect for random perturbations with initial spectrum (4), and illustrate it by results of numerical experiment for the case of truncated white noise (n -- 0 in (10) ).

Numerical experiments were performed using a parallel version of the fast Legendre transform algorithm implemented on a Connection Machine CM- 200 [ 6,11 ]. The algorithm uses the property that the inverse Lagrangian function y(x, t) is a non-decreasing function of x, and specific low-level instructions on the Connection Machine, that allow one to per- form simultaneously partial maximization operations over divisions of the range, that are only known at run-time. The most primitive computation of the maximum of (3) takes for N points O(N 2) operations on a sequential computer, The serial fast Le- gendre transform takes O(Nlog N) operations in 1D (and O(N21og2N) operations for data on an N X N grid in 2D [ 12 ] ). Our parallel fast Legendre transform takes O(log2N) operations on an ideal parallel computer with an unlimited number of physical processors, connected in a hypercube network (this is the type of connection used on the CM-2 and CM- 200 models). The Connection Machine simulates an arbitrary number of processors with a finite number (on our machine 8192). Our algorithm will therefore eventually go linearly with the ratio of initial- ized gridpoints (N) to the actual number of physical processors. For the class of one-dimensional prob- lems we have considered here, the computational time including input, output and Connection Ma- chine initialization, is much less than one minute for a total number of points up to N= 220

We shall consider the evolution of two initial random perturbations to(X) and ~o (x):

~ 0 ( X ) = / ) 0 ( X ) "t"/)h(X) . ( 11 )

The power spectra of both processes are described by (4) with n< 1, but the process Vo(X) has spectral components in the range [0,/c.], while the process b'~o (x) is in the range [0,/Co] with ko:** k.. Restricted to the range [ 0, k. ] the two processes are identical, Vh(X) being their difference in the large wave-hum-

ber range [k., ko]. The correlation coefficient between Vo and ~o is ro= (k./ko)(n+,)/2,~: 1. In par- ticular, we made calculations for ko°)/k.=22 and k(o2)/k. =2 6.

Three initial realisations, as described above, are shown in fig. 1. At t ~ tn ~- 1/ak(. n+3)/2 all three processes are transformed to a sequence of triangular pulses with universal behaviour inside the cells, ac- cording to (6). The solution v(x, t) will be stable, relative to the high wave-number perturbation Vh(X), if the fluctuations of both the inverse Lagrangian co- ordinates, Ayk=Pk--Yk, and the shock positions, AXk=Xk--Xk, are small with respect to the integral scale of turbulence l(t). While the asymptotic properties of Burgers' turbulence are determined by the behaviour of the initial action, the values of distur- bances ~ k and Ayk will be determined by the value of the variance o 2 of the perturbation action Sh(X)= yxVh(y) dy,

°t2 [ 1- - (k . /ko) t -n] , (12) °~s= <Sh~ > = (1-n)k'.-"

and its correlation scale Is~, (k.ko)-'/2. The main di- vergence between ~ and v is due to the different ve- locities of shocks, Vk and Pk, and these errors increase with time. From (7) one sees that the fluctuations of the velocity are determined by the fluctuations of the action Sh(y) in the neighbourhood of the local maximum of So(y). This problem is similar to the analysis of statistical properties of

t ' 'T --[~ 40 '3' --W

20

Vo 0

-20

-4O

560 600 640 680 x

Fill. 1. Initial velocity realizations with cutoff wave-numbers k. (l),ko ¢1) (2) andk¢o 2) (3).

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Burgers' turbulence with stationary initial action [4 ]. Thus we get that the relative deviation of shock positions is equal to

e ( t ) = <(A(x2)>I/2 ~--OhSt (13) l(t) 12(t) '

which means that the behaviour of ~(t) depends on the growth law of the integral scale l(t). From (10), (12), (13) we obtain that in the case n < 1 and

1 1 ~( t )~ [ /~ / ( t ) ]o_ . ) /2 l n ( ~ / / ~ )

1 t(l_.)/(n+3). (14)

Here we have taken into account that for ko ~ k., the value of the absolute maximum of the random process Sh(y) in the vicinity of a local maximum of So (y) has a double logarithmic distribution with average value ~Ohs[ln(ko/k.)] t/2, and variance decreasing with correlation scale Is proportional to ln(ko/k.) [4,13 ]. It may be that one shock in v cor- responds to a cluster of shocks in ~, but they are spread out over a distance no more than ~ (14). Therefore one can say that the large-scale structures of Burgers' turbulence are self-preserving due to multiple merging of shocks. The stability of large- scale structures is illustrated by results of numerical calculations, presented in fig. 2. Three realisations of

a velocity field are shown, corresponding to the initial conditions in fig. 1 with cutoff wave-numbers equal, respectively, to k., ko ° ) and ko (2). One can see that the large-scale behaviour of all those realizations is similar, and only the fine structure depends weakly on the cutoff wave-number. The correlation coefficient between the velocity fields with cutoff wave-numbers k. and ko (2) increases from the value 0.035 for the initial perturbation (fig. 1 ) up to the value 0.92 at the stage of developed shocks (fig. 2) . This becomes even clearer when comparing the cor-

150

i00

50

~IJ, ffJo

0

-50

-1000 2000 4000 6000 8000 x

Fig. 3. Realizations of the action fields with cutoffwave~numbers k. ( I ), k~ s> (2) and k~ 2) (3) at t ~ t, and initial action for the wave-number kfk. (4).

0.08 1 "i' -- 0.06 '2' --

O.04 ~ / / /

0.02 I

0 , V -0.02

-0.04 / I -0.06 -

-0.08

-0.10 2000 4000 6000 8000 x

Fig. 2. Realizations of the velocity field with cutoff wave-numbers k. (l),k~ ') (2) andk~ ") (3) att:*,t,.

80

60

40

20

0

560 600 640 680 x

Fig. 4. Realizations of the action fields with cutoffwave-numbers k. (1) and k~ 2~ (2) at t:~ t, and initial actions for the same wave- numbers (3), (4).

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Volume 182, number I PHYSICS LETTERS A 8 November 1993

responding initial and final actions, presented in figs. 3 and 4. There one sees that the action for higher cutoff values can be obtained from the smaller one ap- proximately by a shift along the vertical axis. This is in accordance with estimates following from the double logarithmic distribution o f Sh(y).

In conclusion, we would like to point out that the stability o f large-scale structures relative to small- scale perturbations is similar to the independence o f the asymptotic evolution o f Burgers' turbulence (when n < 1 ) of the value o f the viscosity coefficient ~u [4 ] .

This work is supported by the Swedish Natural Research Council under contract S-FO 1778-302 (E.A.), the G6ran Gustavsson Foundation (S.N.G.) and the Swedish Institute (I .W.). The authors are grateful to the Center for Parallel Computers and the Department o f Mechanics o f the Royal Institute o f Technology for hospitality.

References

[ 1 ] P.O. Davis and A.J. Yule, J. Fluid Mech. 69 (1975) 513.

[2] J.M. Burgers, The nonlinear diffusion equation (Reidel, Dordrecht, 1974).

[ 3 ] Y. Kuramoto, Chemical oscillations, waves and turbulence (Springer, Berlin, 1983).

[4 ] S.N. Gurbatov, A.N. Malakhov and A.L Saichev, Nonlinear waves and turbulence in nondispersive media: waves, rays and particles (Manchester Univ. Press, Manchester, 1991 ).

[5] T. Tatsumi and S. Kida, J. Fluid Mech. 55 (1972) 659. [6] Z.-S. She, E. AureU and U. Frisch, Commun. Math. Phys.

148 (1992) 623. [7] S.N. Gurbatov, I.Yu. Detain and A.I. Saichev, Soy. Phys.

JETP 60 (1984) 284. [8 ] S.N. Gurbatov and D.G. Crighton, to appear in Chaos. [ 9 ] B.O. Enflo, in: Proc. 12th Int. Syrup. on Nonlinear aoaustics

Austin, TX (1990) p. 131, to appear in Soy. Phys. Radiophys. Quant. Electr. (1993).

[ 10 ] S.N. Gurbatov, I.Yu. Dentin and N.V. Pronchatov-Pubtsov, Soy. Phys. JETP 64 (1986) 797.

[ 11 ] E. Aurell, in: Annual report, Center for Parallel Computers (1993) p. 68; E. Aurell and I.I. Wertgeim, submitted to Second European CM Users Meeting (Pads, October 1993).

[12] A. Noullez, A fast algorithm for discrete Legendre transforms, preprint, Observatoire de Nice (1992).

[ 13 ] I.G. Yakushkln, Soy. Phys. JETP 54 ( 1981 ) 513. [ 14 ] S.N. Gurbatov, A.I. Saichev and I.G. Yakushkin, Soy. Phys.

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Self-preservation of large-scale structures in Burgers' turbulence