Magnetic Eddy Diffusivity Research Articles

Large-scale dynamo action of magnetized Taylor–Couette flows

ABSTRACT A conducting Taylor–Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler–Spruit type. The flows are unstable against non-axisymmetric perturbations which form electromotive forces defining α effect and eddy diffusivity. If both degenerated modes with m = ±1 are excited with the same power then the global α effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite α effects if only an isolated mode is considered but this intrinsic helicity of the single-mode is too low for an α2 dynamo. Moreover, an αΩ dynamo model with quasi-Keplerian rotation requires a minimum magnetic Reynolds number of rotation of Rm ≃ 2000 to work. Whether it really works depends on assumptions about the turbulence energy. For a steeper-than-quadratic dependence of the turbulence intensity on the magnetic field, however, dynamos are only excited if the resulting magnetic eddy diffusivity approximates its microscopic value, ηT ≃ η. By basically lower or larger eddy diffusivities the dynamo instability is suppressed.

On Kinematic Generation of the Magnetic Modes of Bloch Type

Kinematic generation of electrically conductive fluid by space-periodic flow is considered. The generated magnetic modes have a form of a product of three-dimensional field having the same spatial periodicity and a Fourier harmonic whose wave vector q is an arbitrary constant. We present the examples of computation of the modes associated with the maximum (over q) growth rates for a generic flow velocity where the magnetic α-effect is present and for a parity-invariant flow lacking the α-effect and featuring negative magnetic eddy diffusivity. It is shown that magnetic modes associated with the maximum over q growth rates are characterized by weak separation of spatial scales, and, therefore, neither of the two noted effects can be regarded as the main mechanism responsible for their generation.

Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

We present examples of Padé approximations of the α -effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Padé approximants for the computation of tensors of magnetic α -effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity 1 / η around 1 / η = 0 . This yields the values of the dominant growth rate to satisfactory accuracy for η , several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard “double” (real*8) and extended “quadruple” (real*16) precision, and perform symbolic calculations in Mathematica.

Negative magnetic eddy diffusivity due to oscillogenic α-effect

We study large-scale kinematic dynamo action of steady mirror-antisymmetric flows of incompressible fluid, that involve small spatial scales only, by asymptotic methods of the multiscale stability theory. It turns out that, due to the magnetic $\alpha$-effect in such flows, the large-scale mean field experiences harmonic oscillations in time on the scale O($\varepsilon t$) without growth or decay. Here $\varepsilon$ is the spatial scale ratio and $t$ is the fast time of the order of the flow turnover time. The interaction of the accompanying fluctuating magnetic field with the flow gives rise to an anisotropic magnetic eddy diffusivity, whose dependence on the direction of the large-scale wave vector generically exhibits a singular behaviour, and thus to negative eddy diffusivity for whichever molecular magnetic diffusivity. Consequently, such flows always act as kinematic dynamos on the time scale O($\varepsilon^2t$); for the directions at which eddy diffusivity is infinite, the large-scale mean-field growth rate is finite on the scale O($\varepsilon^{3/2}t$). We investigate numerically this dynamo mechanism for two sample flows.

Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation.

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic α-effect or negative eddy diffusivity mechanisms, large-scale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

Plasma Motions and Turbulent Magnetic Diffusivity of Active Region AR 12158 Using a Minimum Energy Functional and Non-Force-Free Reconstructions of Vector Magnetograms

We present a generalization of the resistive minimum-energy fit (MEF-R: Tremblay and Vincent, Solar Phys. 290, 437, 2015) for non-force-free (NFF) magnetic fields. In MEF-R, an extremum principle is used to infer two-dimensional maps of plasma motions [ $\boldsymbol{v}(x,y)$ ] and magnetic eddy diffusivity [ $\eta _{\mathrm{eddy}}(x,y)$ ] at the photosphere. These reconstructions could be used as boundary conditions in data-driven simulations or in data assimilation. The algorithm is validated using the analytical model of a resistive expanding spheromak by Rakowski, Laming, and Lyutikov (Astrophys. J. 730, 30, 2011). We study the flaring Active Region AR 12158 using a series of magnetograms and Dopplergrams provided by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). The results are discussed for a non-force-free magnetic-field reconstruction [ $\boldsymbol{B}_{\mathrm{NFF}}$ ] (Hu and Dasgupta in Solar Phys. 247, 87, 2008). We found that the vertical plasma velocities [ $v_{z}(x,y)$ ] inferred using MEF-R are very similar to the observed Doppler velocities [ $v_{r}(x,y)$ ]. Finally, we study the potential spatial correlation between microturbulent velocities and significant values of $\eta_{\mathrm{eddy}}(x,y)$ .

NEGATIVE MAGNETIC EDDY DIFFUSIVITIES FROM THE TEST-FIELD METHOD AND MULTISCALE STABILITY THEORY

The generation of large-scale magnetic field in the kinematic regime in the absence of an alpha-effect is investigated by following two different approaches, namely the test-field method and multiscale stability theory relying on the homogenisation technique. We show analytically that the former, applied for the evaluation of magnetic eddy diffusivities, yields results that fully agree with the latter. Our computations of the magnetic eddy diffusivity tensor for the specific instances of the parity-invariant flow-IV of G.O. Roberts and the modified Taylor-Green flow in a suitable range of parameter values confirm the findings of previous studies, and also explain some of their apparent contradictions. The two flows have large symmetry groups; this is used to considerably simplify the eddy diffusivity tensor. Finally, a new analytic result is presented: upon expressing the eddy diffusivity tensor in terms of solutions to auxiliary problems for the adjoint operator, we derive relations between magnetic eddy diffusivity tensors that arise for opposite small-scale flows v(x) and -v(x).

Cross helicity at the solar surface by simulations and observations

The quasilinear mean-field theory for driven MHD turbulence leads to the result that the observed cross helicity <u \cdot b> may directly yield the magnetic eddy diffusivity \eta_{T} of the quiet Sun. In order to model the cross helicity at the solar surface, magnetoconvection under the presence of a vertical large-scale magnetic field is simulated with the nonlinear MHD code NIRVANA. The very robust result of the calculations is that < u_z b_z >\simeq 2 < u\cdot b> independent of the applied magnetic field amplitude. The correlation coefficient for the cross helicity is about 10%. Of similar robustness is the finding that the rms value of the magnetic perturbations exceeds the mean-field amplitude (only) by a factor of five. The characteristic helicity speed u_{\eta} as the ratio of the eddy diffusivity and the density scale height for an isothermal sound velocity of 6.6 km/s proves to be 1 km/s for weak fields. This value well coincides with empirical results obtained from the data of the HINODE satellite and the Swedish 1-m Solar Telescope (SST) providing the cross helicity component < u_z b_z >. Both simulations and observations thus lead to a numerical value of \eta_{T} \simeq 10^12 cm^2 /s as characteristic for the surface of the quiet Sun.

Cross Helicity and Turbulent Magnetic Diffusivity in the Solar Convection Zone

In a density-stratified turbulent medium, the cross helicity 〈u′⋅B′〉 is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations, we find the cross helicity and the mean vertical magnetic field to be anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface will exceed the value of 1 gauss km s−1. Its sign is anti-correlated to that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.

Convective plan-form of two-scale dynamos in a plane layer

We study generation of magnetic fields, involving large spatial scales, by convective plan-forms in a horizontal layer. Magnetic modes and their growth rates are expanded in power series in the scale ratio, and the magnetic eddy diffusivity (MED) tensor is derived for flows, symmetric about the vertical axis in a layer. For convective rolls we demonstrate that MED is never below molecular magnetic diffusivity. For cell patterns possessing the symmetries of a rectangle, critical values of molecular magnetic diffusivity for the onset of small- and large-scale magnetic field generation are the same. No instances of negative MED in hexagonal cells have been detected. A family of plan-forms has been found numerically, where MED is negative for molecular magnetic diffusivity over the threshold for the onset of small-scale magnetic field generation. However, the region in the parameter space, where large-scale dynamo action is observed, is small.

Generation of multiscale magnetic field by parity-invariant time-periodic flows

We study generation of magnetic fields involving large spatial scales by time- and space-periodic small-scale parity-invariant flows. The anisotropic magnetic eddy diffusivity tensor is calculated by the standard procedure involving expansion of magnetic modes and their growth rates in power series in the scale ratio. Our simulations, conducted for flows with random harmonic composition and exponentially decaying energy spectra, demonstrate that enlargement of the spatial scale of magnetic field is beneficial for generation by time-periodic flows. However, they turn out, in general, to be less efficient dynamos, than steady flows.

Differential rotation and meridional flow in the solar supergranulation layer: Measuring the eddy viscosity

We measure the eddy viscosity in the outermost layers of the solar convection zone by comparing the rotation law computed with the Reynolds stress resulting from f-plane simulations of the angular momentum transport in rotating convection with the observed differential rotation pattern. The simulations lead to a negative vertical and a positive horizontal angular momentum transport. The consequence is a subrotation of the outermost layers, as it is indeed indicated both by helioseismology and the observed rotation rates of sunspots. In order to reproduce the observed gradient of the rotation rate a value of about 1.5 x 10^{13} cm/s for the eddy viscosity is necessary. Comparison with the magnetic eddy diffusivity derived from the sunspot decay yields a surprisingly large magnetic Prandtl number of 150 for the supergranulation layer. The negative gradient of the rotation rate also drives a surface meridional flow towards the poles, in agreement with the results from Doppler measurements. The successful reproduction of the abnormally positive horizontal cross correlation (on the northern hemisphere) observed for bipolar groups then provides an independent test for the resulting eddy viscosity.

Dynamo effect in parity-invariant flow with large and moderate separation of scales

It is shown that non-helical (more precisely, parity-invariant) flows capable of sustaining a large-scale dynamo by the negative magnetic eddy diffusivity effect are quite common. This conclusion is based on numerical examination of a large number of randomly selected flows. A few outliers with strongly negative eddy diffusivities are also found, and they are interpreted in terms of the closeness of the control parameter to a critical value for generation of a small-scale magnetic field. Furthermore, it is shown that, for parity-invariant flows, a moderate separation of scales between the basic flow and the magnetic field often significantly reduces the critical magnetic Reynolds number for the onset of dynamo action.

Alpha-Effect, Helicity and Angular Momentum Transport for a Magnetically Driven Turbulence in the Solar Convection Zone

For a given field of magnetic fluctuations the dynamo-α, as well as the kinetic and current helicities, have been computed, assuming that turbulence is subject to magnetic buoyancy and global rotation. The sign of the dynamo-α is positive in the northern hemisphere and negative in the southern hemisphere. The kinetic helicity has just the same latitudinal distribution as the α-effect, indeed there is no minus sign between the dynamo-α and the kinetic helicity. Also the current helicity 〈j ⋅ B 〉 changes its sign at the equator. It is negative in the northern hemisphere and positive in the southern hemisphere. Our current helicities (due to fluctuations) and α-effects are thus always out of phase, this confirming a previous result of Keinigs (1983) and Radler and Seehafer (1990). The signs of the α-effect and both helicities correspond to the numerical simulations by Brandenburg and Schmitt (1998). We have also computed the turbulent angular momentum transport, which proves to be always inwards, as in hydrodynamic simulations without magnetic fields (Chan, 1999). Thus we can easily explain why, in the supergranulation zone, deeper layers appear to rotate faster than the solar surface plasma, or why in the solar tachocline at high-latitudes the angular velocity decreases outwards. The dynamo number derived from the observed current helicity reveals the α-effect to be rather small, unless the magnetic eddy diffusivity is not as low as the sunspot decay suggests (i.e., 1011 cm2 s−1).