P. L. Sulem

A simple dynamical model of intermittent fully developed turbulence

We present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales N &2-” only a fraction /3n of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974)’ is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation. The P-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot’s (1974, 1976) ‘fractal dimension’. For threedimensional turbulence, the correction B to the Q exponent of the energy spectrum is equal to +( 3 - D) and is related to the exponent p of the dissipation correlation function by B = Qp (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality B < Qp. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off. The predictions of the P-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function (8v6(Z)) and the dissipation correlation function (e(r) e(r + 1)) are related by

Tracing complex singularities with spectral methods

Abstract A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the space variable(s), the solutions have for any t > 0 complex-space singularities at the edge of an analyticity strip of width δ ( t ) Loss of regularity corresponds to the vanishing of δ ( t ). Numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the Fourier transform of the solutions, from which δ ( t ) is readily obtained. Its time evolution can be traced down to about one mesh length. By extrapolation of δ ( t ), such numerical experiments provide evidence suggesting finite-time blow-up or all-time regularity. The method is tested on the inviscid and viscous Burgers equations and is applied to the one-dimensional nonlinear Schrodinger equation with quartic potential and to the two-dimensional incompressible Euler equation, all with periodic boundary conditions. In the latter case evidence is found suggesting that existing all-time regularity results can be substantially sharpened.

Current and vorticity dynamics in three‐dimensional magnetohydrodynamic turbulence

Spectral numerical simulations of homogeneous incompressible magnetohydrodynamic turbulence at Reynolds mumbers up to about 500, are performed using a uniform grid of 1803 collocation points. Strong vorticity and current sheets obtain both in the presence and in the absence of magnetic nulls. Contrary to vortex sheets in hydrodynamics, these structures do not destabilize into filaments, but are locally disrupted. They are the main loci of kinetic and magnetic dissipations.

Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability

The well-posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by Birkhoff in two dimensions and is shown here to hold both in two and three dimensions. Incompressible, inviscid and irrotational flow with a velocity jump across an interface is assumed. In two dimensions, global existence of a weak solution to the Euler equation with such initial conditions is established. In three dimensions, a Lagrangian representation of the vortex sheet analogous to the Birkhoff equation in two dimensions is presented.

Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows

Direct numerical simulations of the three‐dimensional Euler equations at resolutions up to 2563 for general periodic flows and 8643 for the symmetric Taylor–Green vortex are presented. The spontaneous emergence of flat pancakelike structures that shrink exponentially in time is observed. A simple self‐similar model that fits these observations is discussed. Focusing instabilities similar to those leading to streamwise vortices in the context of free shear layers [J. Fluid Mech. 143, 253 (1984)], are expected to subsequently concentrate the vorticity and produce isolated vortex filaments. A finite time singularity for the Euler equation is not excluded as the result of interactions among these filaments.

Transverse dynamics of dispersive Alfvén waves. I. Direct numerical evidence of filamentation

The three-dimensional dynamics of a small-amplitude monochromatic Alfven wave propagating along an ambient magnetic field is simulated by direct numerical integration of the Hall-magnetohydrodynamics equations. As predicted by the two-dimensional nonlinear Schrodinger equation or by more general amplitude equations retaining the coupling to low-frequency magnetosonic waves, the transverse instability of the pump leads to wave collapse and formation of intense magnetic filaments, in spite of the presence of competing, possibly linearly dominant, instabilities that in some instances distort the above structures. In computational boxes, including a large number of pump wavelengths, an early arrest of the collapse is possible under the effect of quasi-transverse instabilities that drive magnetosonic waves and also prescribe the directions of the filaments.

Inertial ranges and resistive instabilities in two‐dimensional magnetohydrodynamic turbulence

Direct numerical simulations of decaying two‐dimensional magnetohydrodynamic flows at Reynolds numbers of several thousand are performed, using resolutions of 10242 collocation points. An inertial range extending to about one decade is observed, with spectral properties depending on the velocity–magnetic field correlation. At very small scales, resistive tearing destabilizes current sheets generated by the inertial dynamics and leads to the formation of small‐scale magnetic islands, which may then grow and reach the size of inertial scales.

Collisionless magnetohydrodynamics with gyrokinetic effects

Anisotropic magnetohydrodynamics equations, which also capture the dynamics of quasi-transverse small scales obeying the gyrokinetic ordering, are derived using fourth-rank moment closures, based on a refined description of linear Landau damping and finite Larmor radius (FLR) corrections. This “FLR-Landau fluid model” reproduces the dispersion relation of low-frequency waves, up to scales that, in the case of quasi-transverse kinetic Alfven waves, can be much smaller than the ion gyroradius. The mirror instability, which requires temperature anisotropy, is also captured, together with its quenching at small scales. This model that accurately reproduces the collisionless dissipation of low-frequency modes, should provide an efficient tool to simulate mesoscale turbulence in a magnetized collisionless plasma.

Alfvén-wave filamentation

Transverse focusing or filamentation of weakly nonlinear Alfven waves propagating in a dispersive medium is studied using an amplitude-equation formalism. Special attention is devoted to the small-β, regime, where kinetic effects are weak and Alfven-wave trains are unstable to convective filamentation. It is shown that, according to their initial duration, focusing wave packets can collapse in a finite distance or, conversely, the focusing can be arrested by the development of magnetosonic waves, which in both cases may lead to the formation of sharp acoustic fronts. This effect, which dominates the usual longitudinal steepening, provides an efficient mechanism to heat the plasma without requiring large-amplitude waves. It can significantly contribute to the acceleration of the solar wind.

A Landau fluid model for warm collisionless plasmas

A Landau fluid model for a collisionless electron-proton magnetized plasma that accurately reproduces the dispersion relation and the Landau damping rate of all the magnetohydrodynamic waves is presented. It is obtained by an accurate closure of the hydrodynamic hierarchy at the level of the fourth-order moments, based on linear kinetic theory. It retains nongyrotropic corrections to the pressure and heat flux tensors up to the second order in the ratio between the considered frequencies and the ion cyclotron frequency.