70 research outputs found
Anomalous dissipation and regularization in isotropic Gaussian turbulence
No full textIn this work we rigorously establish a number of properties of "turbulent" solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002]. The advecting velocity field, not necessarily incompressible, is Gaussian and white-in-time, space-homogeneous and isotropic, with -Hölder regularity in space, . We cover the full range of compressibility ratios giving spontaneous stochasticity of particle trajectories. For the stochastic transport equation, we prove that generic data experience anomalous dissipation of the mean energy, and study basic properties of the resulting anomalous dissipation measure. Moreover, we show that starting from such irregular data, the solution immediately gains regularity and enters into a fractional Sobolev space . The proof of the latter is obtained as a consequence of a new sharp regularity result for the degenerate parabolic PDE satisfied by the associated two-point self-correlation function, which is of independent interest. In the incompressible case, a Duchon-Robert-type formula for the anomalous dissipation measure is derived, making a precise connection between this self-regularizing effect and a limit on the flux of energy in the turbulent cascade. Finally, for the stochastic continuity equation, we prove that solutions starting from a Dirac delta initial condition undergo an average squared dispersion growing with respect to time as , rigorously establishing the analogue of Richardson's law of particle separations in fluid dynamics
The statistical geometry of material loops in turbulence
No full textMaterial elements - which are lines, surfaces, or volumes behaving as
passive, non-diffusive markers - provide an inherently geometric window into
the intricate dynamics of chaotic flows. Their stretching and folding dynamics
has immediate implications for mixing in the oceans or the atmosphere, as well
as the emergence of self-sustained dynamos in astrophysical settings. Here, we
uncover robust statistical properties of an ensemble of material loops in a
turbulent environment. Our approach combines high-resolution direct numerical
simulations of Navier-Stokes turbulence, stochastic models, and dynamical
systems techniques to reveal predictable, universal features of these complex
objects. We show that the loop curvature statistics become stationary through a
dynamical formation process of high-curvature folds, leading to distributions
with power-law tails whose exponents are determined by the large-deviations
statistics of finite-time Lyapunov exponents of the flow. This prediction
applies to advected material lines in a broad range of chaotic flows. To
complement this dynamical picture, we confirm our theory in the analytically
tractable Kraichnan model with an exact Fokker-Planck approach
Self-Regularization in turbulence from the Kolmogorov 4/5-Law and Alignment
No full textA defining feature of 3D hydrodynamic turbulence is that the rate of energy
dissipation is bounded away from zero as viscosity is decreased (Reynolds
number increased). This phenomenon - anomalous dissipation - is sometimes
called the `zeroth law of turbulence' as it underpins many celebrated
theoretical predictions. Another robust feature observed in turbulence is that
velocity structure functions
exhibit persistent power-law scaling in the inertial range, namely for exponents over an ever-increasing (with
Reynolds) range of scales. This behavior indicates that the velocity field
retains some fractional differentiability uniformly in the Reynolds number. The
Kolmogorov 1941 theory of turbulence predicts that for all
and Onsager's 1949 theory establishes the requirement that
for for consistency with the zeroth law. Empirically, and , suggesting that turbulent
Navier-Stokes solutions approximate dissipative weak solutions of the Euler
equations possessing (nearly) the minimal degree of singularity required to
sustain anomalous dissipation. In this note, we adopt an experimentally
supported hypothesis on the anti-alignment of velocity increments with their
separation vectors and demonstrate that the inertial dissipation provides a
regularization mechanism via the Kolmogorov 4/5-law.Comment: 14 pages, 4 figure
Islands in stable fluid equilibria
No full textWe prove that stable fluid equilibria with trivial homology on curved,
reflection-symmetric periodic channels must posses "islands", or cat's eye
vortices. In this way, arbitrarily small disturbances of a flat boundary cause
a change of streamline topology of stable steady states.Comment: 6 pages, 2 figure
Circulation and energy theorem preserving stochastic fluids
No full textSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem [1]. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin’s theorem, introduced by Constantin–Iyer (2008) [3]. In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincare´ and stochastic Navier-Stokes–Poincare´ equations respectively. The stochastic Euler–Poincare equations were previously derived from a stochastic ´ variational principle by Holm (2015) [20], which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems
Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence
No full textWe develop a first-principles theory of relativistic fluid turbulence at high Reynolds and Péclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a nonperturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for nonrelativistic 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 not vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find to be broken by our coarse-graining regularization but which is restored in the limit where 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 nonrelativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of nonrelativistic kinetic energy
Flexibility and rigidity of free boundary MHD equilibria
No full textWe study stationary free boundary configurations of an ideal incompressible
magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply
connected domains, we prove that if the magnetic field and velocity field are
never commensurate, the only possible domain for any such equilibria is a disk,
and the velocity and magnetic field are circular. We give examples of
non-symmetric equilibria occupying a domain of any shape by imposing an
external magnetic field generated by a singular current sheet charge
distribution (external coils). Some results carry over to 3D axisymmetric
solutions. These results highlight the importance of external magnetic fields
for the existence of asymmetric equilibria.Comment: revised version. 18 pages, 3 figure
- …
