Notizen 928
On the Spectra of Turbulent Fluids at Large k
H . Tasso
Max-Planck-Institut für Plasmaphysik,
Euratom association, Garching bei München, F.R.G.
Z. Naturforsch. 45a, 928 (1990); received May 4, 1990 The turbulence spectra of continuous fluids at large k are discussed. A basic difference in behaviour appears between conservative and dissipative systems. Only in the latter case can one eliminate higher order ultraviolet divergences.
In previous w o r k on the equilibrium statistics of the K o r t e w e g de Vries e q u a t i o n (see [1, 2]) a n d o t h e r 1-d n o n - i n t e g r a b l e b u t dispersive e q u a t i o n s [3], it was possible to p r o v e t h a t the space correlation f u n c t i o n h a s an e x p o n e n t i a l decay like This was the result of the s i m u l t a n e o u s presence of nonlinearity a n d dispersion in the e q u a t i o n . O t h e r w i s e the correla- tion f u n c t i o n w o u l d have a ö (x) b e h a v i o u r .
T h e k-spectrum being the F o u r i e r t r a n s f o r m of the c o r r e l a t i o n f u n c t i o n , the b e h a v i o u r at large k of the s p e c t r u m c o r r e s p o n d s to the b e h a v i o u r at small x of the c o r r e l a t i o n f u n c t i o n . T h e e- a | x | b e h a v i o u r sup- presses the c r u d e ultraviolet divergence in the energy s p e c t r u m a n d leads to a Lorenz s h a p e as k~2 (see [4]).
T h i s c r u d e divergence occurs in case of a Ö (x) be- h a v i o u r of the correlation function, which corre- s p o n d s to e q u i p a r t i t i o n . Since the derivatives of e~311x1
[1] H. Tasso, Phys. Lett. 96 A, 33 (1983).
[2] H. Tasso and K. Lerbinger, Phys. Let. 97 A, 384 (1983).
[3] H. Tasso, Phys. Lett. 120 A, 464 (1987).
[4] H. Tasso, Z. Naturforsch. 41a, 1258 (1986).
[5] C. Foias and G. Prodi, Rend. Semin. Univ. Padova 39, 1 (1967).
are n o t c o n t i n u o u s o r even defined, divergences can a p p e a r in the s p e c t r a of the derivatives of the fluid variables. This s i t u a t i o n seems to be u n a v o i d a b l e for c o n s e r v a t i v e c o n t i n u o u s systems.
Dissipative fluids h a v e a f u n d a m e n t a l l y different be- h a v i o u r . It h a s been p r o v e d for 2-d Navier-Stokes e q u a t i o n s t h a t the t i m e - i n d e p e n d e n t space correlation f u n c t i o n is s m o o t h for all x (see [5]). T h i s has been used recently to e s t i m a t e the large k b e h a v i o u r of the en- ergy s p e c t r u m , which t u r n s o u t to be exponentially d e c a y i n g (see [6]). In [6] the s m o o t h n e s s (C00) of the c o r r e l a t i o n f u n c t i o n h a s been a s s u m e d for the 3-d case w i t h o u t proof. T h i s a s s u m p t i o n seems plausible, h o w - ever, for dissipative fluids which usually have a t t r a c - t o r s with finite (see [7]) H a u s d o r f f dimension, t h o u g h p r o o f s in the 3-d case are missing. T h i s suggests t h a t they b e h a v e like a system with a finite n u m b e r of degrees of f r e e d o m for which n o ultraviolet-like diver- gences of a n y type c a n occur.
In t h e dissipative case, the divergences are avoided w h a t e v e r the e n s e m b l e average is, as l o n g as ergodic- ity is a s s u m e d , while in the c o n s e r v a t i v e case the kind of a v e r a g i n g is i m p o r t a n t because suitable weighting a m o n g the infinite n u m b e r of degrees of freedom pe- nalizes, for instance, the large k (see [2]). T h e penaliza- tion is, however, suitable for certain quantities b u t n o t for others, especially those c o n t a i n i n g higher deriva- tives, so it c a n n o t be g u a r a n t e e d t h a t all ultraviolet divergences are suppressed. Viscous-like dissipation seems to be the way t o eliminate completely the ultra- violet p r o b l e m .
[6] C. Foias, O. Manley, and L. Sirovich, Phys. Fluids A 2, 464 (1990).
[7] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York
1988.
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