Magnetic Prandtl number

About: Magnetic Prandtl number is a research topic. Over the lifetime, 1035 publications have been published within this topic receiving 27903 citations. The topic is also known as: Prm & Prandtl magnetic number.

A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution

Abstract: A broad class of astronomical accretion disks is presently shown to be dynamically unstable to axisymmetric disturbances in the presence of a weak magnetic field, an insight with consequently broad applicability to gaseous, differentially-rotating systems. In the first part of this work, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate, which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field. Fluid motions associated with the instability directly generate both poloidal and toroidal field components. In the second part of this investigation, the scaling relation between the instability's wavenumber and the Alfven velocity is demonstrated, and the independence of the maximum growth rate from magnetic field strength is confirmed.

Strong MHD helical turbulence and the nonlinear dynamo effect

Abstract: 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

Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields

Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence

Abstract: A numerical model of isotropic homogeneous turbulence with helical forcing is investigated. The resulting flow, which is essentially the prototype of the α2 dynamo of mean field dynamo theory, produces strong dynamo action with an additional large-scale field on the scale of the box (at wavenumber k = 1; forcing is at k = 5). This large-scale field is nearly force free and exceeds the equipartition value. As the magnetic Reynolds number Rm increases, the saturation field strength and the growth rate of the dynamo increase. However, the time it takes to build up the large-scale field from equipartition to its final superequipartition value increases with magnetic Reynolds number. The large-scale field generation can be identified as being due to nonlocal interactions originating from the forcing scale, which is characteristic of the α-effect. Both α and turbulent magnetic diffusivity ηt are determined simultaneously using numerical experiments where the mean field is modified artificially. Both quantities are quenched in an Rm-dependent fashion. The evolution of the energy of the mean field matches that predicted by an α2 dynamo model with similar α and ηt quenchings. For this model an analytic solution is given that matches the results of the simulations. The simulations are numerically robust in that the shape of the spectrum at large scales is unchanged when changing the resolution from 303 to 1203 mesh points, or when increasing the magnetic Prandtl number (viscosity/magnetic diffusivity) from 1 to 100. Increasing the forcing wavenumber to 30 (i.e., increasing the scale separation) makes the inverse cascade effect more pronounced, although it remains otherwise qualitatively unchanged.

Simulations of nonhelical hydromagnetic turbulence.

Abstract: Nonhelical hydromagnetic forced turbulence is investigated using large scale simulations on up to $256$ processors and ${1024}^{3}$ mesh points. The magnetic Prandtl number is varied between 1∕8 and 30, although in most cases it is unity. When the magnetic Reynolds number is based on the inverse forcing wave number, the critical value for dynamo action is shown to be around 35 for magnetic Prandtl number of unity. For small magnetic Prandtl numbers we find the critical magnetic Reynolds number to increase with decreasing magnetic Prandtl number. The Kazantsev ${k}^{3∕2}$ spectrum for magnetic energy is confirmed for the kinematic regime, i.e., when nonlinear effects are still unimportant and when the magnetic Prandtl number is unity. In the nonlinear regime, the energy budget converges for large Reynolds numbers (around 1000) such that for our parameters about $70%$ is in kinetic energy and about $30%$ is in magnetic energy. The energy dissipation rates are converged to $30%$ viscous dissipation and $70%$ resistive dissipation. Second-order structure functions of the Elsasser variables give evidence for a ${k}^{\ensuremath{-}5∕3}$ spectrum. Nevertheless, the three-dimensional spectrum is close to ${k}^{\ensuremath{-}3∕2}$, but we argue that this is due to the bottleneck effect. The bottleneck effect is shown to be equally strong both for magnetic and nonmagnetic turbulence, but it is far weaker in one-dimensional spectra that are normally studied in laboratory turbulence. Structure function exponents for other orders are well described by the She-Leveque formula, but the velocity field is significantly less intermittent and the magnetic field is more intermittent than the Elsasser variables.