160 research outputs found

### Gibbsian Hypothesis in Turbulence

We show that Kolmogorov multipliers in turbulence cannot be statistically
independent of others at adjacent scales (or even a finite range apart) by
numerical simulation of a shell model and by theory. As the simplest
generalization of independent distributions, we suppose that the steady-state
statistics of multipliers in the shell model are given by a
translation-invariant Gibbs measure with a short-range potential, when
expressed in terms of suitable ``spin'' variables: real-valued spins that are
logarithms of multipliers and XY-spins defined by local dynamical phases.
Numerical evidence is presented in favor of the hypothesis for the shell model,
in particular novel scaling laws and derivative relations predicted by the
existence of a thermodynamic limit. The Gibbs measure appears to be in a
high-temperature, unique-phase regime with ``paramagnetic'' spin order.Comment: 19 pages, 9 figures, greatly expanded content, accepted to appear in
J. Stat. Phy

### Turbulent Cascade of Circulations

The circulation around any closed loop is a Lagrangian invariant for classical, smooth solutions of the incompressible Euler equations in any number of space dimensions. However, singular solutions relevant to turbulent flows need not preserve the classical integrals of motion. Here we generalize the Kelvin theorem on conservation of circulations to distributional solutions of Euler and give necessary conditions for the anomalous dissipation of circulations. We discuss the important role of Kelvin's theorem in turbulent vortex-stretching dynamics and conjecture a version of the theorem which may apply to suitable singular solutions

### Fluctuations in the Irreversible Decay of Turbulent Energy

A fluctuation law of the energy in freely-decaying, homogeneous and isotropic
turbulence is derived within standard closure hypotheses for 3D incompressible
flow. In particular, a fluctuation-dissipation relation is derived which
relates the strength of a stochastic backscatter term in the energy decay
equation to the mean of the energy dissipation rate. The theory is based on the
so-called ``effective action'' of the energy history and illustrates a
Rayleigh-Ritz method recently developed to evaluate the effective action
approximately within probability density-function (PDF) closures. These
effective actions generalize the Onsager-Machlup action of nonequilibrium
statistical mechanics to turbulent flow. They yield detailed, concrete
predictions for fluctuations, such as multi-time correlation functions of
arbitrary order, which cannot be obtained by direct PDF methods. They also
characterize the mean histories by a variational principle.Comment: 26 pages, Latex Version 2.09, plus seceq.sty, a stylefile for
sequential numbering of equations by section. This version includes new
discussion of the physical interpretation of the formal Rayleigh-Ritz
approximation. The title is also change

### Fluctuation-Response Relations for Multi-Time Correlations

We show that time-correlation functions of arbitrary order for any random
variable in a statistical dynamical system can be calculated as higher-order
response functions of the mean history of the variable. The response is to a
``control term'' added as a modification to the master equation for statistical
distributions. The proof of the relations is based upon a variational
characterization of the generating functional of the time-correlations. The
same fluctuation-response relations are preserved within moment-closures for
the statistical dynamical system, when these are constructed via the
variational Rayleigh-Ritz procedure. For the 2-time correlations of the
moment-variables themselves, the fluctuation-response relation is equivalent to
an ``Onsager regression hypothesis'' for the small fluctuations. For
correlations of higher-order, there is a new effect in addition to such linear
propagation of fluctuations present instantaneously: the dynamical generation
of correlations by nonlinear interaction of fluctuations. In general, we
discuss some physical and mathematical aspects of the {\it Ans\"{a}tze}
required for an accurate calculation of the time correlations. We also comment
briefly upon the computational use of these relations, which is well-suited for
automatic differentiation tools. An example will be given of a simple closure
for turbulent energy decay, which illustrates the numerical application of the
relations.Comment: 28 pages, 1 figure, submitted to Phys. Rev.

### Resonant Interactions in Rotating Homogeneous Three-dimensional Turbulence

Direct numerical simulations of three-dimensional (3D) homogeneous turbulence
under rapid rigid rotation are conducted to examine the predictions of resonant
wave theory for both small Rossby number and large Reynolds number. The
simulation results reveal that there is a clear inverse energy cascade to the
large scales, as predicted by 2D Navier-Stokes equations for resonant
interactions of slow modes. As the rotation rate increases, the
vertically-averaged horizontal velocity field from 3D Navier-Stokes converges
to the velocity field from 2D Navier-Stokes, as measured by the energy in their
difference field. Likewise, the vertically-averaged vertical velocity from 3D
Navier-Stokes converges to a solution of the 2D passive scalar equation. The
energy flux directly into small wave numbers in the $k_z=0$ plane from
non-resonant interactions decreases, while fast-mode energy concentrates closer
to that plane. The simulations are consistent with an increasingly dominant
role of resonant triads for more rapid rotation

### Equation-free implementation of statistical moment closures

We present a general numerical scheme for the practical implementation of
statistical moment closures suitable for modeling complex, large-scale,
nonlinear systems. Building on recently developed equation-free methods, this
approach numerically integrates the closure dynamics, the equations of which
may not even be available in closed form. Although closure dynamics introduce
statistical assumptions of unknown validity, they can have significant
computational advantages as they typically have fewer degrees of freedom and
may be much less stiff than the original detailed model. The closure method can
in principle be applied to a wide class of nonlinear problems, including
strongly-coupled systems (either deterministic or stochastic) for which there
may be no scale separation. We demonstrate the equation-free approach for
implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic
partial differential equation.Comment: 7 pages, 2 figure

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