1,080 research outputs found
Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence
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
Structure preserving numerical methods for the ideal compressible MHD system
We introduce a novel structure-preserving method in order to approximate the
compressible ideal Magnetohydrodynamics (MHD) equations. This technique
addresses the MHD equations using a non-divergence formulation, where the
contributions of the magnetic field to the momentum and total mechanical energy
are treated as source terms. Our approach uses the Marchuk-Strang splitting
technique and involves three distinct components: a compressible Euler solver,
a source-system solver, and an update procedure for the total mechanical
energy. The scheme allows for significant freedom on the choice of Euler's
equation solver, while the magnetic field is discretized using a
curl-conforming finite element space, yielding exact preservation of the
involution constraints. We prove that the method preserves invariant domain
properties, including positivity of density, positivity of internal energy, and
the minimum principle of the specific entropy. If the scheme used to solve
Euler's equation conserves total energy, then the resulting MHD scheme can be
proven to preserve total energy. Similarly, if the scheme used to solve Euler's
equation is entropy-stable, then the resulting MHD scheme is entropy stable as
well. In our approach, the CFL condition does not depend on magnetosonic
wave-speeds, but only on the usual maximum wave speed from Euler's system. To
validate the effectiveness of our method, we solve a variety of ideal MHD
problems, showing that the method is capable of delivering high-order accuracy
in space for smooth problems, while also offering unconditional robustness in
the shock hydrodynamics regime as well
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