Showing papers on "Magnetic Prandtl number published in 2009"

Simple mechanism for reversals of earth's magnetic field.

Abstract: We show that a model, recently used to describe all the dynamical regimes of the magnetic field generated by the dynamo effect in the von Karman sodium experiment, also provides a simple explanation of the reversals of Earth's magnetic field, despite strong differences between both systems. The validity of the model relies on the smallness of the magnetic Prandtl number.

The turbulent magnetic prandtl number of mhd turbulence in disks

Abstract: The magnetic Prandtl number PrM is the ratio of viscosity to resistivity. In astrophysical disks the diffusion of angular momentum (viscosity) and magnetic fields (resistivity) are controlled by turbulence. Phenomenological models of the evolution of large-scale poloidal magnetic fields in disks suggest that the turbulent magnetic Prandtl number PrM,T controls the rate of escape of vertical field from the disk; for PrM,T ≤ R/H vertical field diffuses outward before it can be advected inward by accretion. Here we measure field diffusion and angular momentum transport due to MHD turbulence in a shearing box, and thus PrM,T, by studying the evolution of a sinusoidal perturbation in the magnetic field that is injected into a turbulent background. We show that the perturbation is always stable, decays approximately exponentially, has decay rate k 2, and that the implied PrM,T ~ 1.

Turbulent resistivity driven by the magnetorotational instability

Abstract: Aims We measure the turbulent resistivity in the nonlinear regime of the MRI, and evaluate the turbulent magnetic Prandtl number Methods We perform a set of numerical simulations with the Eulerian finite volume codes Athena and Ramses in the framework of the shearing box model We consider models including explicit dissipation coefficients and magnetic field topologies such that the net magnetic flux threading the box in both the vertical and azimuthal directions vanishes Results We first demonstrate good agreement between the two codes by comparing the properties of the turbulent states in simulations having identical microscopic diffusion coefficients (viscosity and resistivity) We find the properties of the turbulence do not change when the box size is increased in the radial direction, provided it is elongated in the azimuthal direction To measure the turbulent resistivity in the disk, we impose a fixed electromotive force on the flow and measure the amplitude of the saturated magnetic field that results We obtain a turbulent resistivity that is in rough agreement with mean field theories like the Second Order Smoothing Approximation The numerical value translates into a turbulent magnetic Prandtl number Pmt of order unity Pmt appears to be an increasing function of the forcing we impose It also becomes smaller as the box size is increased in the radial direction, in good agreement with previous results obtained in very large boxes Conclusions Our results are in general agreement with other recently published papers studying the same problem but using different methodology Thus, our conclusion that Pmt is of order unity appears robust

Simulations of magnetorotational turbulence with a higher-order godunov scheme

Abstract: We apply a new, second-order Godunov code, Athena, to studies of the magnetorotational instability (MRI) using unstratified shearing box simulations with a uniform net vertical field and a sinusoidally varying zero net vertical field. The Athena results agree well with similar studies that used different numerical algorithms, including the observation that the turbulent energy decreases with increasing resolution in the zero net field model. We conduct analyses to study the flow of energy from differential rotation to turbulent fluctuations to thermalization. A study of the time correlation between the rates of change of different volume-averaged energy components shows that energy injected into turbulent fluctuations dissipates on a timescale of Ω–1, where Ω is the orbital frequency of the local domain. Magnetic dissipation dominates over kinetic dissipation, although not by as great a factor as the ratio of magnetic to kinetic energy. We Fourier-transform the magnetic and kinetic energy evolution equations and, using the assumption that the time-averaged energies are constant, determine the level of numerical dissipation as a function of length scale and resolution. By modeling numerical dissipation as if it were physical in origin, we characterize numerical resistivity and viscosity in terms of effective Reynolds and Prandtl numbers. The resulting effective magnetic Prandtl number is ~2, independent of resolution or initial field geometry. MRI simulations with effective Reynolds and Prandtl numbers determined by numerical dissipation are not equivalent to those where these numbers are set by physical resistivity and viscosity. These results serve, then, as a baseline for future shearing box studies where dissipation is controlled by the inclusion of explicit viscosity and resistivity.

Prandtl number dependence of convection driven dynamos in rotating spherical fluid shells

Abstract: The value of the Prandtl number $P$ exerts a strong influence on convection-driven dynamos in rotating spherical shells filled with electrically conducting fluids. Low Prandtl numbers promote dynamo action through the shear provided by differential rotation, while the generation of magnetic fields is more difficult to sustain in high-Prandtl-number fluids where higher values of the magnetic Prandtl number $P_m$ are required. The magnetostrophic approximation often used in dynamo theory appears to be valid only for relatively high values of $P$ and $P_m$. Dynamos with a minimum value of $P_m$ seem to be most readily realizable in the presence of convection columns at moderately low values of $P$. The structure of the magnetic field varies strongly with $P$ in that dynamos with a strong axial dipole field are found for high values of $P$ while the energy of this component is exceeded by that of the axisymmetric toroidal field and by that of the non-axisymmetric components at low values of $P$. Some conclusions are discussed in relation to the problem of the generation of planetary magnetic fields by motions in their electrically conducting liquid cores.

Viscous and Resistive Effects on the Magnetorotational Instability with a Net Toroidal Field

Abstract: Resistivity and viscosity have a significant role in establishing the energy levels in turbulence driven by the magnetorotational instability (MRI) in local astrophysical disk models. This study uses the Athena code to characterize the effects of a constant shear viscosity nu and Ohmic resistivity eta in unstratified shearing box simulations with a net toroidal magnetic flux. A previous study of shearing boxes with zero net magnetic field performed with the ZEUS code found that turbulence dies out for values of the magnetic Prandtl number, P {sub m} = nu/eta, below P {sub m} approx 1; for P {sub m} approx> 1, time- and volume-averaged stress levels increase with P {sub m}. We repeat these experiments with Athena and obtain consistent results. Next, the influence of viscosity and resistivity on the toroidal field MRI is investigated both for linear growth and for fully developed turbulence. In the linear regime, a sufficiently large nu or eta can prevent MRI growth; P {sub m} itself has little direct influence on growth from linear perturbations. By applying a range of values for nu and eta to an initial state consisting of fully developed turbulence in the presence of a background toroidal field, we investigatemore » their effects in the fully nonlinear system. Here, increased viscosity enhances the turbulence, and the turbulence decays only if the resistivity is above a critical value; turbulence can be sustained even when P {sub m} < 1, in contrast to the zero net field model. While we find preliminary evidence that the stress converges to a small range of values when nu and eta become small enough, the influence of dissipation terms on MRI-driven turbulence for relatively large eta and nu is significant, independent of field geometry.« less

Large-scale dynamos at low magnetic prandtl numbers

Abstract: Using direct simulations of hydromagnetic turbulence driven by random polarized waves it is shown that dynamo action is possible over a wide range of magnetic Prandtl numbers from 10–3 to 1. Triply periodic boundary conditions are being used. In the final saturated state the resulting magnetic field has a large-scale component of Beltrami type. For the kinematic phase, growth rates have been determined for magnetic Prandtl numbers between 0.01 and 1, but only the case with the smallest magnetic Prandtl number shows large-scale magnetic fields. It is less organized than in the nonlinear stage. For small magnetic Prandtl numbers the growth rates are comparable to those calculated from an alpha squared mean-field dynamo. In the linear regime the magnetic helicity spectrum has a short inertial range compatible with a –5/3 power law, while in the nonlinear regime it is the current helicity whose spectrum may be compatible with such a law. In the saturated case, the spectral magnetic energy in the inertial range is in slight excess over the spectral kinetic energy, although for small magnetic Prandtl numbers the magnetic energy spectrum reaches its resistive cut off wavenumber more quickly. The viscous energy dissipation declines with the square root of the magnetic Prandtl number, which implies that most of the energy is dissipated via Joule heat.

Parameter dependences of convection driven dynamos in rotating spherical fluid shells

Abstract: For the understanding of planetary and stellar dynamos an overview of the major parameter dependences of convection driven dynamos in rotating spherical fluid shells is desirable. Although the computationally accessible parameter space is limited, earlier work is extended with emphasis on higher Prandtl numbers and uniform heat flux condition at the outer boundary. The transition from dynamos dominated by non-axisymmetric components of the magnetic field to those dominated by the axisymmetric components depends on the magnetic Prandtl number as well as on the ordinary Prandtl number for higher values of the rotation parameter $\tau$. The dependence of the transition on the latter parameter is also discussed. A variety of oscillating dynamos is presented and interpreted in terms of dynamo waves, standing oscillation or modified relaxation oscillations.

The Dynamo Bifurcation in Rotating Spherical Shells

Abstract: We investigate the nature of the dynamo bifurcation in a configuration applicable to the Earth's liquid outer core, i.e. in a rotating spherical shell with thermally driven motions. We show that the nature of the bifurcation, which can be either supercritical or subcritical or even take the form of isola (or detached lobes) strongly depends on the parameters. This dependence is described in a range of parameters numerically accessible (which unfortunately remains remote from geophysical application), and we show how the magnetic Prandtl number and the Ekman number control these transitions.

The saturation limit of the magnetorotational instability

Abstract: Simulations of the magnetorotational instability (MRI) in a homogeneous shearing box have shown that the asymptotic strength of the magnetic field declines steeply with increasing resolution. Here I model the MRI-driven dynamo as a large-scale dynamo driven by the vertical magnetic helicity flux. This growth is balanced by large-scale mixing driven by a secondary instability. The saturated magnetic energy density depends almost linearly on the vertical height of the typical eddies. The MRI can drive eddies with arbitrarily large vertical wavenumber, so the eddy thickness is either set by diffusive effects, by the magnetic tension of a large-scale vertical field component, or by magnetic buoyancy effects. In homogeneous, zero magnetic flux simulations, only the first effect applies and the saturated limit of the dynamo is determined by explicit or numerical diffusion. The exact result depends on the numerical details, but is consistent with previous work, including the claim that the saturated field energy scales as the gas pressure to the one quarter power (which we interpret as an artifact of numerical dissipation). The magnetic energy density in a homogeneous shearing box will tend to zero as the resolution of the simulation increases, but this has no consequences for the dynamo or for angular momentum transport in real accretion disks. The claim that the saturated state depends on the magnetic Prandtl number may also be an artifact of simulations in which microphysical transport coefficients set the MRI eddy thickness. Finally, the efficiency of the MRI dynamo is a function of the ratio of the Alfven velocity to the product of the pressure scale height and the local shear. As this approaches unity from below, the dynamo reaches maximum efficiency. Farther from the disk midplane, the Parker instability will dominate the local dynamics and the dynamo process.

Alpha effect and diffusivity in helical turbulence with shear

Abstract: Aims. We study the dependence of turbulent transport coefficients, such as the components of the α tensor (αij) and the turbulent magnetic diffusivity tensor (ηij), on shear and magnetic Reynolds number in the presence of helical forcing. Methods. We use three-dimensional direct numerical simulations with periodic boundary conditions and measure the turbulent transport coefficients using the kinematic test field method. In all cases the magnetic Prandtl number is taken as unity. Results. We find that with increasing shear the diagonal components of αij quench, whereas those of ηij increase. The antisymmetric parts of both tensors increase with increasing shear. We also propose a simple expression for the turbulent pumping velocity (or γ effect). This pumping velocity is proportional to the kinetic helicity of the turbulence and the vorticity of the mean flow. For negative helicity, i.e. for a positive trace of αij, it points in the direction of the mean vorticity, i.e. perpendicular to the plane of the shear flow. Our simulations support this expression for low shear and magnetic Reynolds number. The transport coefficients depend on the wavenumber of the mean flow in a Lorentzian fashion, just as for non-shearing turbulence.

Constraints on the Magnetic Buoyancy Instabilities of a Shear-Generated Magnetic Layer

Abstract: In a recent paper addressing the solar Ω-effect, we found that the action of forced vertical (radial) shear on the vertical component of poloidal magnetic field could induce magnetic buoyancy that produced rising, undulating tubular magnetic structures, as often envisaged arising from the solar tachocline. However, such dynamics were only found to exist under extreme circumstances (extremely large forcing). Here, we analytically examine the reasons underpinning the difficulties in obtaining magnetic buoyancy in this type of system. Specifically, under various assumptions about the maintenance of the shear, we derive some analytic limits on the ability of localized radial shear to produce a toroidal magnetic field from a poloidal field. First, we consider a local time-dependent context, where an unmaintained shear builds a toroidal magnetic layer over time by shearing the poloidal field. Second, we consider the possibility that maintenance is necessary to obtain stationary buoyant dynamics, and examine a local time-independent context, where the shear is maintained by a weak forcing. In both situations, we derive estimates or mathematical bounds for the toroidal magnetic energy that can be realized and its gradients, and thus evaluate the likelihood of magnetic buoyancy instabilities. We find that the results can be expressed as conditions on the imposed shear-flow Richardson number, Ri, and the magnetic Prandtl number, σM, requiring either low Ri or high σM for instability. We found the former in earlier work by Vasil & Brummell, and perform a new simulation here confirming the latter. These results suggest that, for the case of the solar tachocline where σM is small and Ri is large, the assumptions of our models must be violated, and a more comprehensive model is likely required.

Chaotic magnetic field reversals in turbulent dynamos

Abstract: We present direct numerical simulations of reversals of the magnetic field generated by swirling flows in a spherical domain. In agreement with a recent model, we observe that coupling dipolar and quadrupolar magnetic modes by an asymmetric forcing of the flow generates field reversals. In addition, we show that this mechanism strongly depends on the value of the magnetic Prandtl number. The generation of magnetic field by the flow of an electrically conducting fluid, i.e., the dynamo effect, has been mostly studied to understand the magnetic fields of planets and stars [1]. The Earth and the Sun provide the best documented examples: they both involve a spatially coherent large scale component of magnetic field with well characterized dynamics. Earth’s dipole is nearly stationary on time scales much larger than the ones related to the flow in the liquid core, but displays random reversals. Reversals also occur for the Sun but nearly periodically. The magnetic field changes polarity roughly every 11 years. Reversals have been displayed by direct simulations of the equations of magnetohydrodynamics (MHD) [2] or of mean field MHD [3] and have been modeled using low dimensional dynamical systems [4, 5]. It has been observed recently that the magnetic field generated by a von Karman flow of liquid sodium (VKS experiment) can display either periodic or random reversals [6] as well as several other dynamo regimes, all located in a small parameter range [7]. The ability of all these very different dynamos to reverse polarity is their most striking property. This is obviously related to the B → −B symmetry of the MHD equations, implying that if a magnetic field B is a solution, −B is another solution. However, this does not explain how these two solutions can be connected as time evolves. The VKS experiment has provided an interesting observation. In this experiment, the flow is driven in a cylindrical container by two counter-rotating coaxial propellers. When they rotate at roughly the same frequency, a magnetic field with a dominant dipolar component aligned with the axis of rotation is generated. Time dependent magnetic field with periodic or random reversals are observed only when the difference between the two rotation frequencies is large enough [6]. We have shown that this can be re

Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

Abstract: We prove that smooth solutions of nonideal (viscous and resistive) incompressible magnetohydrodynamic (MHD) equations satisfy a stochastic law of flux conservation. This property implies that the magnetic flux through a surface is equal to the average of the magnetic fluxes through an ensemble of surfaces advected backward in time by the plasma velocity perturbed with a random white noise. Our result is an analog of the well-known Alfven theorem of ideal MHD and is valid for any value of the magnetic Prandtl number. A second stochastic conservation law is shown to hold at unit Prandtl number, a random version of the generalized Kelvin theorem derived by Bekenstein and Oron for ideal MHD. These stochastic conservation laws are not only shown to be consequences of the nonideal MHD equations but are proved in fact to be equivalent to those equations. We derive similar results for two more refined hydromagnetic models, Hall MHD and the two-fluid plasma model, still assuming incompressible velocities and isotropic transport coefficients. Finally, we use these results to discuss briefly the infinite-Reynolds-number limit of hydromagnetic turbulence and to support the conjecture that flux conservation remains stochastic in that limit.

The Saturation Limit of the Magnetorotational Instability

Abstract: Simulations of the magnetorotational instability (MRI) in a homogeneous shearing box have shown that the asymptotic strength of the magnetic field declines steeply with increasing resolution. Here I model the MRI driven dynamo as a large scale dynamo driven by the vertical magnetic helicity flux. This growth is balanced by large scale mixing driven by a secondary instability. The saturated magnetic energy density depends almost linearly on the vertical height of the typical eddies. The MRI can drive eddies with arbitrarily large vertical wavenumber, so the eddy thickness is either set by diffusive effects, by the magnetic tension of a large scale vertical field component, or by magnetic buoyancy effects. In homogeneous, zero magnetic flux, simulations only the first effect applies and the saturated limit of the dynamo is determined by explicit or numerical diffusion. The exact result depends on the numerical details, but is consistent with previous work, including the claim that the saturated field energy scales as the gas pressure to the one quarter power (which we interpret as an artifact of numerical dissipation). The magnetic energy density in a homogeneous shearing box will tend to zero as the resolution of the simulation increases, but this has no consequences for the dynamo or for angular momentum transport in real accretion disks. The claim that the saturated state depends on the magnetic Prandtl number may also be an artifact of simulations in which microphysical transport coefficients set the MRI eddy thickness. Finally, the efficiency of the MRI dynamo is a function of the ratio of the Alfv\'en velocity to the product of the pressure scale height and the local shear. As this approaches unity from below the dynamo reaches maximum efficiency.

Absolute versus convective helical magnetorotational instability in a Taylor-Couette flow.

Abstract: We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field helical magnetorotational instability HMRI using the inductionless approximation defined by a zero magnetic Prandtl number Prm=0. The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small-amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a relatively narrow range beyond its purely hydrodynamic limit defined by the Rayleigh line. There is not only a lower critical threshold at which HMRI appears but also an upper one at which it disappears again. The latter distinguishes the HMRI from a magnetically modified Taylor vortex flow. Second, we find an absolute instability threshold as well. In the hydrodynamically unstable regime before the Rayleigh line, the threshold of absolute instability is just slightly above the convective one although the critical wavelength of the former is noticeably shorter than that of the latter. Beyond the Rayleigh line the lower threshold of absolute instability rises significantly above the corresponding convective one while the upper one descends significantly below its convective counterpart. As a result, the extension of the absolute HMRI beyond the Rayleigh line is considerably shorter than that of the convective instability. The absolute HMRI is supposed to be self-sustained and, thus, experimentally observable without any external excitation in a system of sufficiently large axial extension.

Tayler instability of toroidal magnetic fields in MHD Taylor-Couette flows

Abstract: The nonaxisymmetric 'kink-type' Tayler instability (TI) of toroidal magnetic fields is studied for conducting incompressible fluids of uniform density between two infinitely long cylinders rotating around the same axis. It is shown that for resting cylinders the critical Hartmann number for the unstable modes does not depend on Pm. By rigid rotation the instability is suppressed where the critical ratio of the rotation velocity and the Alfven velocity of the field (only) slightly depends on the magnetic Prandtl number Pm. For Pm=1 the rotational quenching of TI takes its maximum. Rotation laws with negative shear (i.e. d\Omega/dR<0) strongly destabilize the toroidal field if the rotation is not too fast. For sufficiently high Reynolds numbers of rotation the suppression of the nonaxisymmetric magnetic instability always dominates. The angular momentum transport of the instability is anticorrelated with the shear so that an eddy viscosity can be defined which proves to be positive. For negative shear the Maxwell stress of the perturbations remarkably contributes to the angular momentum transport. We have also shown the possibility of laboratory TI experiments with a wide-gap container filled with fluid metals like sodium or gallium. Even the effect of the rotational stabilization can be reproduced in the laboratory with electric currents of only a few kAmp.

Dissipative and nonaxisymmetric standard-MRI in Kepler disks

Abstract: Deviations from axial symmetry are necessary to maintain self-sustained MRI-turbulence. We define the parameters region where nonaxisymmetric MRI is excited and study dependence of the unstable modes structure and growth rates on the relevant parameters. We solve numerically the linear eigenvalue problem for global axisymmetric and nonaxisymmetric modes of standard-MRI in Keplerian disks with finite diffusion. For small magnetic Prandtl number the microscopic viscosity completely drops out from the analysis so that the stability maps and the growth rates expressed in terms of the magnetic Reynolds number Rm and the Lundquist number S do not depend on the magnetic Prandtl number Pm. The minimum magnetic field for onset of nonaxisymmetric MRI grows with Rm. For given S all nonaxisymmetric modes disappear for sufficiently high Rm. This behavior is a consequence of the radial fine-structure of the nonaxisymmetric modes resulting from the winding effect of differential rotation. It is this fine-structure which presents severe resolution problems for the numerical simulation of MRI at large Rm. For weak supercritical magnetic fields only axisymmetric modes are unstable. Nonaxisymmetric modes need stronger fields and not too fast rotation. If Pm is small its real value does not play any role in MRI.

Oscillatory plate temperature effects of free convection flow of dissipative fluid between long vertical parallel plates

Abstract: Unsteady free convection flow of dissipative viscous fluid between two long vertical parallel plates is studied under the temperature of one of the plates is oscillating about a constant nonzero mean temperature. Approximate solutions to coupled non-linear partial differential equations governing the flow have been carried out for the transient velocity, the transient temperature, the amplitude and phase of the skin friction, and the rate of heat transfer. The velocity and temperature fields are shown graphically whereas the numerical values of the amplitude and phase are listed in tables. The effects of the Prandtl number, the Eckert number and frequency on the flow field are discussed. It is found that the transient velocity increases with increasing Prandtl number for fluids with Prandtl number less than unity but it decreases with increasing Prandtl number for fluids with Prandtl number greater than unity.

Hydromagnetic flow of viscous incompressible fluid past a wedge with permeable surface

Abstract: Flow of a viscous, electrically conducting fluid past a wedge having permeable surface is analyzed. A constant transpiration through the wedge surface is assumed. The equations governing the flow and the magnetic field being reduced to local non-similarity equations are solved numerically. The implicit finite difference method, as well as the local non-similarity method is being used in finding the solutions of the reduced equations against the transpiration parameter, ξ. Perturbation solutions for small and large ξ values are also obtained. Effect of the physical parameters, such as, the magnetic force parameter, S, the magnetic Prandtl number, Pm and free stream velocity gradient, n, on the local skin-friction coefficient, f'' (0, ξ), and the local current density coefficient, g'' (0, ξ ), are shown graphically. It is found that the perturbation solutions agreed excellently with other solutions at the two extreme ranges of ξ values. From the present investigation we further observe that, incase of withdrawal of fluid both the momentum and magnetic boundary layers decrease with the increase of ξ. On the other hand these layers increase with ξ value when fluid is being injected trough the surface. Further we notice that there is an onset of reverse flow in the down-stream region in case of blowing of fluid and the starting point of this flow, approximately, is ξ = -0.6.

Stratorotational instability in MHD Taylor-Couette flows

Abstract: Aims. The stability of the dissipative Taylor-Couette flow with a stable axial density stratification and a prescribed azimuthal magnetic field is considered. Methods. Global nonaxisymmetric solutions of the linearized MHD equations with toroidal magnetic field, density stratification, and differential rotation are found for both insulating and conducting cylinders. Results. Hydrodynamic calculations for various gap widths show that flat rotation laws such as the Kepler rotation are always unstable against SRI. Quasigalactic rotation laws, however, are stable for wide gaps. The influence of a current-free toroidal magnetic field on SRI strongly depends on the magnetic Prandtl number Pm: SRI is supported by Pm > 1 and it is suppressed by Pm < 1. For rotation laws that are too flat, when the hydrodynamic SRI ceases, a smooth transition exists to the instability that the toroidal magnetic field produces in combination with the differential rotation. For the first time this nonaxisymmetric azimuthal magnetorotational instability (AMRI) has been computed in the presence of an axial density gradient. If the magnetic field between the cylinders is not current-free, then the Tayler instability occurs, too. The transition from the nonmagnetic centrifugal instability to the magnetic Tayler instability in the presence of differential rotation and density stratification proves to be complex. Most spectacular is the “ballooning” of the stability domain by the density stratification: already a small rotation stabilizes magnetic fields against the Tayler instability.

Magnetohydrodynamic dissipation range spectra for isotropic viscosity and resistivity

Abstract: Dissipation range spectra for incompressible magnetohydrodynamic turbulence are derived for isotropic viscosity μ and resistivity η. The spectra are obtained from heuristic closures of spectral transfer correlations for cases with Pm=μ/η≤1, where Pm is the magnetic Prandtl number. Familiar inertial range power laws are modified by exponential factors that dominate spectral falloff in the dissipation range. Spectral forms are sensitive to alignment between flow and magnetic field. There are as many as four Kolmogorov wavenumbers that parametrize the transition between inertial and dissipative behavior and enter corresponding spectral forms. They depend on the values of the viscosity and resistivity and on the nature of alignment in inertial and dissipation ranges.

Eddy diffusivity from hydromagnetic Taylor-Couette flow experiments.

Abstract: The stability problem of hydromagnetic Taylor-Couette flows with toroidal magnetic fields is considered for various magnetic Prandtl numbers. Only the most uniform (but not current-free) field has been treated. For high enough Hartmann numbers, the toroidal field is always unstable due to the magnetic kink-type instability, which is stabilized by rigid basic rotation. The axial electric current, which drives the instability, is reduced by the electromotive force induced by the instability itself. Numerical simulations show that this electromotive force only depends on the molecular magnetic diffusivity rather than the viscosity. The resulting eddy diffusivity should be on the order of the molecular diffusivity for all the considered magnetic Prandtl numbers. If this is true also for very small magnetic Prandtl numbers (not possible to simulate) then one can use this effect to measure the eddy diffusivity eta(T) in a laboratory. In a sodium experiment (without rotation), a detectable potential difference of approximately 16 mV between top and bottom will result for a container of 1 m length and a gap width of 10 cm.

Well-posedness of the infinite prandtl number model for convection with temperature-dependent viscosity

Abstract: We establish the well-posedness of the infinite Prandtl number model for convection with temperature-dependent viscosity, free-slip boundary condition and zero horizontal fluxes.