Phenomenology for the decay of energy-containing eddies in homogeneous MHD turbulence

We evaluate a number of simple, one‐point phenomenological models for the decay of energy‐containing eddies in magnetohydrodynamic(MHD) and hydrodynamicturbulence. The MHDmodels include effects of cross helicity and Alfvénic couplings associated with a constant mean magnetic field, based on physical effects well‐described in the literature. The analytic structure of three separate MHDmodels is discussed. The single hydrodynamic model and several MHDmodels are compared against results from spectral‐method simulations. The hydrodynamic model phenomenology has been previously verified against experiments in wind tunnels, and certain experimentally determined parameters in the model are satisfactorily reproduced by the present simulation. This agreement supports the suitability of our numerical calculations for examining MHDturbulence, where practical difficulties make it more difficult to study physical examples. When the triple‐decorrelation time and effects of spectral anisotropy are properly taken into account, particular MHDmodels give decay rates that remain correct to within a factor of 2 for several energy‐halving times. A simple model of this type is likely to be useful in a number of applications in space physics, astrophysics, and laboratory plasma physics where the approximate effects of turbulence need to be included

Magnetic helicity in magnetohydrodynamic turbulence with a mean magnetic field

A computational investigation of magnetic helicity of the fluctuatingmagnetic fieldHm in ideal and freely decaying three‐dimensional (3‐D) magnetohydrodynamics (MHD) in the presence of a uniform mean magnetic field is performed. It is shown that for ideal 3‐D MHDHm, which is a rugged invariant in the absence of a mean magnetic field [Frisch et al., J. Fluid Mech. 77, 796 (1975)], decays from its initial value and proceeds to oscillate about zero. The decay of Hm is shown to result from the presence of a new ‘‘generalized’’ helicity invariant, which includes contributions from the uniform magnetic field. The loss of invariance of Hm will diminish the effects of inverse transfer of Hm on freely decaying turbulence. This is demonstrated in a discussion of the selective decay relaxation process

Rapid directional alignment of velocity and magnetic field in magnetohydrodynamic turbulence

We show that local directional alignment of the velocity and magnetic field fluctuations occurs rapidly in magnetohydrodynamics for a variety of parameters. This is observed both in direct numerical simulations and in solar wind data. The phenomenon is due to an alignment between the magnetic field and either pressure gradients or shear-associated kinetic energy gradients. A similar alignment, of velocity and vorticity, occurs in the Navier Stokes fluid case. This may be the most rapid and robust relaxation process in turbulent flows, and leads to a local weakening of the nonlinear terms in the small scale vorticity and current structures where alignment takes place.Comment: 4 pages, 6 figure

Statistical anisotropy of magnetohydrodynamic turbulence

Direct numerical simulations of decaying and forced magnetohydrodynamic (MHD) turbulence without and with mean magnetic field are analyzed by higher-order two-point statistics. The turbulence exhibits statistical anisotropy with respect to the direction of the local magnetic field even in the case of global isotropy. A mean magnetic field reduces the parallel-field dynamics while in the perpendicular direction a gradual transition towards two-dimensional MHD turbulence is observed with $k^{-3/2}$ inertial-range scaling of the perpendicular energy spectrum. An intermittency model based on the Log-Poisson approach, $\zeta_p=p/g^2 +1 -(1/g)^{p/g}$, is able to describe the observed structure function scalings.Comment: 4 pages, 3 figures. To appear in Phys.Rev.

Spectral Modeling of Magnetohydrodynamic Turbulent Flows

We present a dynamical spectral model for Large Eddy Simulation of the incompressible magnetohydrodynamic (MHD) equations based on the Eddy Damped Quasi Normal Markovian approximation. This model extends classical spectral Large Eddy Simulations for the Navier-Stokes equations to incorporate general (non Kolmogorovian) spectra as well as eddy noise. We derive the model for MHD and show that introducing a new eddy-damping time for the dynamics of spectral tensors in the absence of equipartition between the velocity and magnetic fields leads to better agreement with direct numerical simulations, an important point for dynamo computations.Comment: 10 pages, 13 figure

Simulations of MHD Turbulence in a Strongly Magnetized Medium

We analyze 3D numerical simulations of driven incompressible magnetohydrodynamic (MHD) turbulence in a periodic box threaded by a moderately strong external magnetic field. We sum over nonlinear interactions within Fourier wavebands and find that the time scale for the energy cascade is consistent with the Goldreich-Sridhar model of strong MHD turbulence. Using higher order longitudinal structure functions we show that the turbulent motions in the plane perpendicular to the local mean magnetic field are similar to ordinary hydrodynamic turbulence while motions parallel to the field are consistent with a scaling correction which arises from the eddy anisotropy. We present the structure tensor describing velocity statistics of Alfvenic and pseudo-Alfvenic turbulence. Finally, we confirm that an imbalance of energy moving up and down magnetic field lines leads to a slow decay of turbulent motions and speculate that this imbalance is common in the interstellar medium where injection of energy is intermittent both in time and space.Comment: ApJ accepted, 29 pages, 10 figures; some revisions, new figure

Dynamical age of solar wind turbulence in the outer heliosphere

In an evolving turbulent medium, a natural timescale can be defined in terms of the energy decay time. The time evolution may be complicated by other effects such as energy supply due to driving, and spatial inhomogeneity. In the solar wind the turbulence appears not to be simply engaging in free decay, but rather the energy level observed at a particular position in the heliosphere is affected by expansion, “mixing,” and driving by stream shear. Here we discuss a new approach for estimating the “age” of solar wind turbulence as a function of heliocentric distance, using the local turbulent decay rate as the natural clock, but taking into account expansion and driving effects. The simplified formalism presented here is appropriate to low cross helicity (non-Alfvénic) turbulence in the outer heliosphere especially at low helio-latitudes. We employ Voyager data to illustrate our method, which improves upon the familiar estimates in terms of local eddy turnover times

The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics

Invariance properties of physical systems govern their behavior: energy conservation in turbulence drives a wide distribution of energy among modes, observed in geophysical or astrophysical flows. In ideal hydrodynamics, the role of helicity conservation (correlation between velocity and its curl, measuring departures from mirror symmetry) remains unclear since it does not alter the energy spectrum. However, with solid body rotation, significant differences emerge between helical and non-helical flows. We first outline several results, like the energy and helicity spectral distribution and the breaking of strict universality for the individual spectra. Using massive numerical simulations, we then show that small-scale structures and their intermittency properties differ according to whether helicity is present or not, in particular with respect to the emergence of Beltrami-core vortices (BCV) that are laminar helical vertical updrafts. These results point to the discovery of a small parameter besides the Rossby number; this could relate the problem of rotating helical turbulence to that of critical phenomena, through renormalization group and weak turbulence theory. This parameter can be associated with the adimensionalized ratio of the energy to helicity flux to small scales, the three-dimensional energy cascade being weak and self-similar

Energy spectrum of turbulent fluctuations in boundary driven reduced magnetohydrodynamics

The nonlinear dynamics of a bundle of magnetic flux ropes driven by stationary fluid motions at their endpoints is studied, by performing numerical simulations of the magnetohydrodynamic (MHD) equations. The development of MHD turbulence is shown, where the system reaches a state that is characterized by the ratio between the Alfven time (the time for incompressible MHD waves to travel along the field lines) and the convective time scale of the driving motions. This ratio of time scales determines the energy spectra and the relaxation toward different regimes ranging from weak to strong turbulence. A connection is made with phenomenological theories for the energy spectra in MHD turbulence.Comment: Published in Physics of Plasma

Hydrodynamic and magnetohydrodynamic computations inside a rotating sphere

Numerical solutions of the incompressible magnetohydrodynamic (MHD) equations are reported for the interior of a rotating, perfectly-conducting, rigid spherical shell that is insulator-coated on the inside. A previously-reported spectral method is used which relies on a Galerkin expansion in Chandrasekhar-Kendall vector eigenfunctions of the curl. The new ingredient in this set of computations is the rigid rotation of the sphere. After a few purely hydrodynamic examples are sampled (spin down, Ekman pumping, inertial waves), attention is focused on selective decay and the MHD dynamo problem. In dynamo runs, prescribed mechanical forcing excites a persistent velocity field, usually turbulent at modest Reynolds numbers, which in turn amplifies a small seed magnetic field that is introduced. A wide variety of dynamo activity is observed, all at unit magnetic Prandtl number. The code lacks the resolution to probe high Reynolds numbers, but nevertheless interesting dynamo regimes turn out to be plentiful in those parts of parameter space in which the code is accurate. The key control parameters seem to be mechanical and magnetic Reynolds numbers, the Rossby and Ekman numbers (which in our computations are varied mostly by varying the rate of rotation of the sphere) and the amount of mechanical helicity injected. Magnetic energy levels and magnetic dipole behavior are exhibited which fluctuate strongly on a time scale of a few eddy turnover times. These seem to stabilize as the rotation rate is increased until the limit of the code resolution is reached.Comment: 26 pages, 17 figures, submitted to New Journal of Physic