51 research outputs found
Burgers Turbulence with Large-scale Forcing
Burgers turbulence supported by white-in-time random forcing at low
wavenumbers is studied analytically and by computer simulation. It is concluded
that the probability density Q of velocity gradient displays four asymptotic
regimes at very large Reynolds number: (A) a region of large positive gradient
where Q decays rapidly (reduction of gradient by stretching); (B) an
intermediate region of negative gradient where Q falls off as the inverse third
power of gradient (transient inviscid steepening of negative gradient); (C) an
outer power-law region of negative gradient where Q falls off as the reciprocal
of gradient (shoulders of mature shocks); (D) a final region of large gradient
where Q decays very rapidly (interior of mature shocks). The probability
density of velocity difference across an interval r, divided by r, lies on Q
throughout regions A and B and into the middle of C, for small enough
inertial-range r.Comment: Revtex (8 pages) with 11 postscript figures (separate file
Passive Scalar: Scaling Exponents and Realizability
An isotropic passive scalar field advected by a rapidly-varying velocity
field is studied. The tail of the probability distribution for
the difference in across an inertial-range distance is found
to be Gaussian. Scaling exponents of moments of increase as
or faster at large order , if a mean dissipation conditioned on is
a nondecreasing function of . The computed numerically
under the so-called linear ansatz is found to be realizable. Some classes of
gentle modifications of the linear ansatz are not realizable.Comment: Substantially revised to conform with published version. Revtex (4
pages) with 2 postscript figures. Send email to [email protected]
Statistical mechanics of Fofonoff flows in an oceanic basin
We study the minimization of potential enstrophy at fixed circulation and
energy in an oceanic basin with arbitrary topography. For illustration, we
consider a rectangular basin and a linear topography h=by which represents
either a real bottom topography or the beta-effect appropriate to oceanic
situations. Our minimum enstrophy principle is motivated by different arguments
of statistical mechanics reviewed in the article. It leads to steady states of
the quasigeostrophic (QG) equations characterized by a linear relationship
between potential vorticity q and stream function psi. For low values of the
energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a
strong westward jet. For large values of the energy, we obtain geometry induced
phase transitions between monopoles and dipoles similar to those found by
Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of
topography. In the presence of topography, we recover and confirm the results
obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a
different formalism. In addition, we introduce relaxation equations towards
minimum potential enstrophy states and perform numerical simulations to
illustrate the phase transitions in a rectangular oceanic basin with linear
topography (or beta-effect).Comment: 26 pages, 28 figure
Invariant measures of the 2D Euler and Vlasov equations
We discuss invariant measures of partial differential equations such as the
2D Euler or Vlasov equations. For the 2D Euler equations, starting from the
Liouville theorem, valid for N-dimensional approximations of the dynamics, we
define the microcanonical measure as a limit measure where N goes to infinity.
When only the energy and enstrophy invariants are taken into account, we give
an explicit computation to prove the following result: the microcanonical
measure is actually a Young measure corresponding to the maximization of a
mean-field entropy. We explain why this result remains true for more general
microcanonical measures, when all the dynamical invariants are taken into
account. We give an explicit proof that these microcanonical measures are
invariant measures for the dynamics of the 2D Euler equations. We describe a
more general set of invariant measures, and discuss briefly their stability and
their consequence for the ergodicity of the 2D Euler equations. The extension
of these results to the Vlasov equations is also discussed, together with a
proof of the uniqueness of statistical equilibria, for Vlasov equations with
repulsive convex potentials. Even if we consider, in this paper, invariant
measures only for Hamiltonian equations, with no fluxes of conserved
quantities, we think this work is an important step towards the description of
non-equilibrium invariant measures with fluxes.Comment: 40 page
Statistics of Dissipation and Enstrophy Induced by a Set of Burgers Vortices
Dissipation and enstropy statistics are calculated for an ensemble of
modified Burgers vortices in equilibrium under uniform straining. Different
best-fit, finite-range scaling exponents are found for locally-averaged
dissipation and enstrophy, in agreement with existing numerical simulations and
experiments. However, the ratios of dissipation and enstropy moments supported
by axisymmetric vortices of any profile are finite. Therefore the asymptotic
scaling exponents for dissipation and enstrophy induced by such vortices are
equal in the limit of infinite Reynolds number.Comment: Revtex (4 pages) with 4 postscript figures included via psfi
Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution
Using a Maximum Entropy Production Principle (MEPP), we derive a new type of
relaxation equations for two-dimensional turbulent flows in the case where a
prior vorticity distribution is prescribed instead of the Casimir constraints
[Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a
Gaussian prior is specifically treated in connection to minimum enstrophy
states and Fofonoff flows. These relaxation equations are compared with other
relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776
(1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a
small-scale parametrization of 2D turbulence or serve as numerical algorithms
to compute maximum entropy states with appropriate constraints. We perform
numerical simulations of these relaxation equations in order to illustrate
geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure
Quasi-stationary States of Two-Dimensional Electron Plasma Trapped in Magnetic Field
We have performed numerical simulations on a pure electron plasma system
under a strong magnetic field, in order to examine quasi-stationary states that
the system eventually evolves into. We use ring states as the initial states,
changing the width, and find that the system evolves into a vortex crystal
state from a thinner-ring state while a state with a single-peaked density
distribution is obtained from a thicker-ring initial state. For those
quasi-stationary states, density distribution and macroscopic observables are
defined on the basis of a coarse-grained density field. We compare our results
with experiments and some statistical theories, which include the
Gibbs-Boltzmann statistics, Tsallis statistics, the fluid entropy theory, and
the minimum enstrophy state. From some of those initial states, we obtain the
quasi-stationary states which are close to the minimum enstrophy state, but we
also find that the quasi-stationary states depend upon initial states, even if
the initial states have the same energy and angular momentum, which means the
ergodicity does not hold.Comment: 9 pages, 7 figure
Vortices in (2+1)d Conformal Fluids
We study isolated, stationary, axially symmetric vortex solutions in
(2+1)-dimensional viscous conformal fluids. The equations describing them can
be brought to the form of three coupled first order ODEs for the radial and
rotational velocities and the temperature. They have a rich space of solutions
characterized by the radial energy and angular momentum fluxes. We do a
detailed study of the phases in the one-parameter family of solutions with no
energy flux. This parameter is the product of the asymptotic vorticity and
temperature. When it is large, the radial fluid velocity reaches the speed of
light at a finite inner radius. When it is below a critical value, the velocity
is everywhere bounded, but at the origin there is a discontinuity. We comment
on turbulence, potential gravity duals, non-viscous limits and non-relativistic
limits.Comment: 39 pages, 10 eps figures, v2: Minor changes, refs, preprint numbe
Universally Coupled Massive Gravity, II: Densitized Tetrad and Cotetrad Theories
Einstein's equations in a tetrad formulation are derived from a linear theory
in flat spacetime with an asymmetric potential using free field gauge
invariance, local Lorentz invariance and universal coupling. The gravitational
potential can be either covariant or contravariant and of almost any density
weight. These results are adapted to produce universally coupled massive
variants of Einstein's equations, yielding two one-parameter families of
distinct theories with spin 2 and spin 0. The theories derived, upon fixing the
local Lorentz gauge freedom, are seen to be a subset of those found by
Ogievetsky and Polubarinov some time ago using a spin limitation principle. In
view of the stability question for massive gravities, the proven non-necessity
of positive energy for stability in applied mathematics in some contexts is
recalled. Massive tetrad gravities permit the mass of the spin 0 to be heavier
than that of the spin 2, as well as lighter than or equal to it, and so provide
phenomenological flexibility that might be of astrophysical or cosmological
use.Comment: 2 figures. Forthcoming in General Relativity and Gravitatio
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