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The Kolmogorov theory of turbulence indicates an energy cascade in turbulence. Is there a corresponding version of relativistic fluid?

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It would appear that there is, based on a quick internet search. But I don't know enough about relativity to answer. Obviously if turbulence exists, there will be some cascade of energy... Are you asking if similar hypothesis can be made? For instance, are you asking if the cascade inviscid and locally isotropic? –  tpg2114 Dec 11 '12 at 15:48
    
Thanks. What I would like to know is how we can formulate same argument for relativistic fluid. The problem about relativistic fluid is, first time and space coordinate are mixed where "energy" is not naturally defined, second density can not be assumed invariant any more. So is there any similar thing like cascade of energy-momentum? –  Xinyu Li Dec 12 '12 at 17:46
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2 Answers

The answer is yes. Despite the importance of relativistic hydrodynamics and the reasonable expectation that turbulence is likely to play an important role in many astrophysical systems, extremely little is known about turbulence in a relativistic regime. However, your particular question has been answered and studied using numerical modelling; see this great 2012 paper for details of such a study. In fact, the Kolmogorov spectra appears unchanged from the classical theory for ideal relativistic hydrodynamics. The Kolmogorov spectra is directly applicable to relativistic magnetohydrodynamic flows as well. In driven relativistic plasmas, the velocity field is often found to be highly intermittent, but its power-spectrum is found to be in good agreement with the predictions of the classical theory of Kolmogorov (independent from the Lorentz factor), and in good agreement with the classical Kolmogorov theory.

Note however, that most flows studies have been with mildy relativistic flows with Lorentz factors of 1-10.

For me, the reason for Kolmogorov's direct applicability to driven relativistic flows is the relative length scale on which the turbulence occurs compared to the bulk motion. This is directly related to the fact that the cascading eddies move relative to the bulk flow itself.

I hope this helps.

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There are special and general versions of relativistic fluid. Two nice book of the formula can be found in S. Weinberg's Gravitation and Cosmology and Landau's book vol. 6. You can also find few other special books of this topic.

As I know, the applications of these formula are still rare. Someone are trying to use them in plasma cosmology, which is still a developing field.

Another problem is that, many formula of relativistic fluid haven't been confirmed yet. People just extended them from non-relativistic version.

For cascade, of course! The relativistic version should give same results in non-relativistic limit. But, I don't know whether anyone give a detailed calculations of that.

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This is completely wrong. Relativistic fluid dynamics is well understood, and applied extensively to a wide range of phenomena. This includes astrophysics (supernova explosions, jets, disks, etc) and relativistic heavy ion collisions. We now know that the formalism developed in Landau and Weinberg can also be derived from the AdS/CFT correspondence. To the best of my knowledge, there is no fundamental difference between rel and non-rel turbulence, in particular the energy cascade is the same. –  Thomas Feb 8 '13 at 13:31
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@Thomas: And we know that the formalism developed in Landau and Lifshitz fails for dissipative fluids, right? See Section 14.4 in relativity.livingreviews.org/Articles/lrr-2007-1. I think there still remain some open questions... –  UwF Feb 13 '13 at 15:04
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