A. Pouquet

Strong MHD helical turbulence and the nonlinear dynamo effect

To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971~~)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number. Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘AlfvBn effect ’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of largescale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a - power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are - 2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field willgrow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frisch et aZ. (1975), results from a competition between the helicity and Alfvh effects and yields an inertial range with approximately - 1 and - 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale Zinj and the rate k (per unit mass), the time of build-up of magnetic energy with scale L 9 Zinl is t % L( prp;nj)-k 21 FLM 77

Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence

Some of the consequences of the conservation of magnetic helicity

A weak turbulence theory for incompressible mhd

\int \rm{a.b}\it{d}^{\rm{3}}\rm{r\qquad (a\; =\; vector\; potential\; of\; magnetic\; field\; b)}

KOLMOGOROV-LIKE SPECTRA IN DECAYING THREE-DIMENSIONAL SUPERSONIC FLOWS

for incompressible three-dimensional turbulent MHD flows are investigated. Absolute equilibrium spectra for inviscid infinitely conducting flows truncated at lower and upper wavenumbers k min and k max are obtained. When the total magnetic helicity approaches an upper limit given by the total energy (kinetic plus magnetic) divided by k min , the spectra of magnetic energy and helicity are strongly peaked near k min ; in addition, when the cross-correlations between the velocity and magnetic fields are small, the magnetic energy density near k min greatly exceeds the kinetic energy density. Several arguments are presented in favour of the existence of inverse cascades of magnetic helicity towards small wavenumbers leading to the generation of large-scale magnetic energy.

A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible MHD. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. We find energy spectra solution of the kinetic equations. The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

A TURBULENT MODEL FOR THE INTERSTELLAR MEDIUM. II. MAGNETIC FIELDS AND ROTATION

A numerical simulation of decaying supersonic turbulence using the piecewise parabolic method (PPM) algorithm on a computational mesh of 5123 zones indicates that, once the solenoidal part of the velocity field, representing vortical motions, is fully developed and has reached a self‐similar regime, a velocity spectrum compatible with that predicted by the classical theory of Kolmogorov develops. It is followed by a domain with a shallower spectrum. A convergence study is presented to support these assertions. The formation, structure, and evolution of slip surfaces and vortex tubes are presented in terms of perspective volume renderings of fields in physical space.

Current and vorticity dynamics in three‐dimensional magnetohydrodynamic turbulence

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0 ê‖. Numerically and analytically, we find energy spectra E± ∼ k± ⊥ , such that n+ +n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k‖ = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Influence of Cooling-Induced Compressibility on the Structure of Turbulent Flows and Gravitational Collapse

We present results from two-dimensional numerical simulations of a supersonic turbulent flow in the plane of the galactic disk, incorporating shear, thresholded and discrete star formation (SF), self-gravity, rotation and magnetic fields. A test of the model in the linear regime supports the results of the linear theory of Elmegreen (1991). In the fully nonlinear turbulent regime, while some results of the linear theory persist, new effects also emerge. Some exclusively nonlinear effects are: a) Even though there is no dynamo in 2D, the simulations are able to maintain or increase their net magnetic energy in the presence of a seed uniform azimuthal component. b) A well-defined power-law magnetic spectrum and an inverse magnetic cascade are observed in the simulations, indicating full MHD turbulence. Thus, magnetic field energy is generated in regions of SF and cascades up to the largest scales. c) The field has a slight but noticeable tendency to be aligned with density features. d) The magnetic field prevents HII regions from expanding freely, as in the recent results of Slavin \& Cox (1993). e) A tendency to exhibit {\it less} filamentary structures at stronger values of the uniform component of the magnetic field is present in several magnetic runs. f) For fiducial values of the parameters, the flow in general appears to be in rough equipartition between magnetic and kinetic energy. There is no clear domination of either the magnetic or the inertial forces. g) A median value of the magnetic field strength within clouds is

Numerical study of dynamo action at low magnetic Prandtl numbers.

\sim 12\mu

Dynamical length scales for turbulent magnetized flows

G, while for the intercloud medium a value of