Abstract
We develop first-principles theory of relativistic fluid turbulence at high Reynolds and P\'eclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for non-relativistic turbulence, with hydrodynamic fields in the inertial-range described as distributional or "coarse-grained" solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation-laws of smooth Euler solutions but are afflicted with dissipative anomalies in the balance equations of internal energy and entropy. The anomalies are shown to be possible by exactly two mechanisms, local cascade and pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required for their non-vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find is broken by our coarse-graining regularization but which is restored in the limit that the regularization is removed, similar to relativistic lattice quantum field theory. In the formal limit as speed of light goes to infinity, we recover the results of previous non-relativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of non-relativistic kinetic energy.
Highlights
Relativistic hydrodynamics has a growing range of applications in current physics research, including energetic astrophysical objects such as gamma-ray bursts [1] and pulsars [2], high-energy physics of the early Universe and heavy-ion collisions [3], condensed matter physics of graphene [4,5] and strange metals [6,7], and black-hole gravitational physics via the fluid-gravity correspondence in AdS/CFT [8,9,10,11]
Relativistic turbulence is observed in numerical solutions of conformal hydrodynamic models [15,16,17], and an analogous phenomenon is seen in their dual AdS black-hole solutions [18]
Geroch [72] and Lindblom [73] have argued that this close agreement with the LandauLifshitz or Eckart constitutive relations will hold in the energy or particle frame, respectively, for a wide set of extended dissipative relativistic fluid models that are hyperbolic, causal, and well posed
Summary
Relativistic hydrodynamics has a growing range of applications in current physics research, including energetic astrophysical objects such as gamma-ray bursts [1] and pulsars [2], high-energy physics of the early Universe and heavy-ion collisions [3], condensed matter physics of graphene [4,5] and strange metals [6,7], and black-hole gravitational physics via the fluid-gravity correspondence in AdS/CFT [8,9,10,11]. Onsager’s theory of dissipative Euler solutions and its application to fluid turbulence is still essentially unknown to the wider physics community, This is unfortunate because it is the most comprehensive theoretical framework for high Reynolds turbulence and generally applicable to kinetic-energy dissipation in incompressible fluid turbulence and to cascades in magnetohydrodynamic turbulence [41,42], to dissipative anomalies of Lagrangian invariants such as circulations [43] and magnetic fluxes [44], and to cascades in compressible Navier-Stokes turbulence [45,46,47,48]. High Reynolds turbulence is characterized by ultraviolet divergences of gradients of the velocity and other thermodynamic fields, referred to as a “violet catastrophe” by Onsager [27] Regularizing these divergences introduces a new arbitrary length scale l upon which objective physics cannot depend, and exploiting this invariance yields the main conclusions of the theory on fluid singularities, inertial range, local cascades, etc.
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