Small-scale Current Helicity Research Articles

Magnetic Helicity Dissipation and Production in an Ideal MHD Code

Abstract We study a turbulent helical dynamo in a periodic domain by solving the ideal magnetohydrodynamic (MHD) equations with the FLASH code using the divergence-cleaning eight-wave method and compare our results with direct numerical simulations (DNS) using the Pencil Code. At low resolution, FLASH reproduces the DNS results qualitatively by developing the large-scale magnetic field expected from DNS, but at higher resolution, no large-scale magnetic field is obtained. In all those cases in which a large-scale magnetic field is generated, the ideal MHD results yield too little power at small scales. As a consequence, the small-scale current helicity is too small compared with that of the DNS. The resulting net current helicity has then always the wrong sign, and its statistical average also does not approach zero at late times, as expected from the DNS. Our results have implications for astrophysical dynamo simulations of stellar and galactic magnetism using ideal MHD codes.

The influence of helical background fields on current helicity and electromotive force of magnetoconvection

Motivated by the empirical finding that the known hemispheric rules for the current helicity at the solar surface are not strict, the excitation of small-scale current helicity by the influence of a large-scale helical magnetic background fields on nonrotating magnetoconvection is demonstrated. It is shown within a quasilinear analytic theory of driven turbulence and by nonlinear simulations of magnetoconvection that the resulting small-scale current helicity has the same sign as the large-scale current helicity while the ratio of both pseudo-scalars is of the order of the magnetic Reynolds number of the convection. The same models do not provide finite values of the small-scale kinetic helicity. On the other hand, a turbulence-induced electromotive force is produced including the diamagnetic pumping term as well as the eddy diffusivity but no alpha effect. It is thus argued that the relations by Pouquet & Patterson (1978) and Keinigs (1983) for the simultaneous existence of small-scale current helicity and alpha effect do not hold for the considered model of nonrotating magnetoconvection. Calculations for various values of the magnetic Prandtl number demonstrate that for the considered diffusivities the current helicity grows for growing magnetic Reynolds number which is not true for the velocity of the diamagnetic pumping -- in agreement with the results of the quasilinear analytical approximation.

Ribbons characterize magnetohydrodynamic magnetic fields better than lines: a lesson from dynamo theory

Blackman and Brandenburg argued that magnetic helicity conservation in dynamo theory can in principle be captured by diagrams of mean field dynamos when the magnetic fields are represented by ribbons or tubes, but not by lines. Here, we present such a schematic ribbon diagram for the α2 dynamo that tracks magnetic helicity and provides distinct scales of large-scale magnetic helicity, small-scale magnetic helicity, and kinetic helicity involved in the process. This also motivates our construction of a new ‘2.5 scale’ minimalist generalization of the helicity-evolving equations for the α2 dynamo that separately allows for these three distinct length-scales while keeping only two dynamical equations. We solve these equations and, as in previous studies, find that the large-scale field first grows at a rate independent of the magnetic Reynolds number RM before quenching to an RM-dependent regime. But we also show that the larger the ratio of the wavenumber where the small-scale current helicity resides to that of the forcing scale, the earlier the non-linear dynamo quenching occurs, and the weaker the large-scale field is at the turnoff from linear growth. The harmony between the theory and the schematic diagram exemplifies a general lesson that magnetic fields in magnetohydrodynamic are better visualized as two-dimensional ribbons (or pairs of lines) rather than single lines.

Current helicity constraints in solar dynamo models

AbstractWe investigate to what extent the current helicity distribution observed in solar active regions is compatible with solar dynamo models. We use an advanced 2D mean-field dynamo model with dynamo action largely concentrated near the bottom of the convective zone, and dynamo saturation based on the evolution of the magnetic helicity and algebraic quenching. For comparison, we also studied a more basic 2D mean-field dynamo model with simple algebraic alpha quenching only. Using these numerical models we obtain butterfly diagrams for both the small-scale current helicity and the large-scale magnetic helicity, and compare them with the butterfly diagram for the current helicity in active regions obtained from observations. This comparison shows that the current helicity of active regions, as estimated by −A·B evaluated at the depth from which the active region arises, resembles the observational data much better than the small-scale current helicity calculated directly from the helicity evolution equation. Here B and A are respectively the dynamo generated mean magnetic field and its vector potential.

CURRENT HELICITY OF ACTIVE REGIONS AS A TRACER OF LARGE-SCALE SOLAR MAGNETIC HELICITY

We demonstrate that the current helicity observed in solar active regions traces the magnetic helicity of the large-scale dynamo generated field. We use an advanced 2D mean-field dynamo model with dynamo saturation based on the evolution of the magnetic helicity and algebraic quenching. For comparison, we also studied a more basic 2D mean-field dynamo model with simple algebraic alpha quenching only. Using these numerical models we obtained butterfly diagrams both for the small-scale current helicity and also for the large-scale magnetic helicity, and compared them with the butterfly diagram for the current helicity in active regions obtained from observations. This comparison shows that the current helicity of active regions, as estimated by $-{\bf A \cdot B}$ evaluated at the depth from which the active region arises, resembles the observational data much better than the small-scale current helicity calculated directly from the helicity evolution equation. Here ${\bf B}$ and ${\bf A}$ are respectively the dynamo generated mean magnetic field and its vector potential. A theoretical interpretation of these results is given.

Comparison between turbulent helical dynamo simulations and a non-linear three-scale theory

Progress towards understanding principles of non-linear growth and saturation of large-scale magnetic fields has emerged from comparison of theoretical models that incorporate the evolution of magnetic helicity with numerical simulations for problems that are more idealized than expected in astrophysical circumstances, but still fully non-linear. We carry out a new comparison of this sort for the magnetic field growth from forced isotropic helical turbulence in a periodic box. Previous comparisons between analytic theory and simulations of this problem have shown that a two-scale model compares well with the simulations in agreeing that the driver of large-scale field growth is the difference between kinetic and current helicities associated with the small-scale field, and that the backreaction slows the growth of the large-scale field as the small-scale current helicity grows. However, the two-scale model artificially restricts the small-scale current helicity to reside at the same scale as the driving kinetic helicity. In addition, previous comparisons have focused on the late-time saturation regime and less on the early-time growth regime. Here, we address these issues by comparing a three-scale model to new simulations for both early- and late-time growth regimes. We find that the minimalist extension to three scales provides a better model for the field evolution than the two-scale model because the simulations show that the small-scale current helicity does not reside at the same scale as that of the driving kinetic helicity. The simulations also show that the peak of the small-scale current helicity migrates towards lower wavenumbers as the growth evolves from the fast to saturated growth regimes.

A mean-field approach to the propagation of field patterns in stratified magnetorotational turbulence

Local shearing box simulations of stratified magneto rotational turbulence invariably exhibit cyclic field patterns which propagate away from the disc midplane. A common explanation for this is magnetic buoyancy. The recent analysis by Shi et al. however shows that the flow is buoyantly stable below one disc scale height H, necessitating an alternative explanation in this region. We here conduct and analyse direct numerical simulations to explain the observed behaviour by means of a mean-field description. Apart from the mean radial and azimuthal field, we monitor the small-scale current helicity, which we propose as a key indicator for saturation. Reconstructing the horizontally averaged field, we demonstrate that the problem can be reduced to a one-dimensional induction equation. By means of the so-called test field method, we then determine the underlying closure parameters. Our analysis shows that, apart from a possible direct MRI dynamo, two distinct indirect dynamo mechanisms operate in the disc. This resolves the issue of the "wrong" sign of the MRI dynamo effect. Finally, we use the obtained closure parameters to run a dynamically quenched dynamo model. This model approximately recovers the observed field patterns in the mean fields. Moreover, the model reproduces the prevailing parity and the distinct phase pattern in the small-scale current helicity. The latter property might open a potential route to understand the saturation of MRI induced turbulence.

Nonlinear Current Helicity Fluxes in Turbulent Dynamos and Alpha Quenching

Large scale dynamos produce small scale current helicity as a waste product that quenches the large scale dynamo process (alpha effect). This quenching can be catastrophic (i.e., intensify with magnetic Reynolds number) unless one has fluxes of small scale magnetic (or current) helicity out of the system. We derive the form of helicity fluxes in turbulent dynamos, taking also into account the nonlinear effects of Lorentz forces due to fluctuating fields. We confirm the form of an earlier derived magnetic helicity flux term, and also show that it is not renormalized by the small scale magnetic field, just like turbulent diffusion. Additional nonlinear fluxes are identified, which are driven by the anisotropic and antisymmetric parts of the magnetic correlations. These could provide further ways for turbulent dynamos to transport out small scale magnetic helicity, so as to avoid catastrophic quenching.

The α-Effect and Current Helicity for Fast Sheared Rotators

We explore the f -effect and the small-scale current helicity, , for the case of weakly compressible magnetically driven turbulence that is subjected to the differential rotation. No restriction is applied to the amplitude of angular velocity, i.e., the derivations presented are valid for an arbitrary Coriolis number, z * = 2 z cor , though the differential rotation itself is assumed to be weak. The expressions obtained are used to explore the possible distributions of f -effect and h c in convection zones (CZ) of the solar-type stars. Generally, our theory gives f { { > 0 in the northern hemisphere of the Sun and the opposite case in the southern hemisphere. In most cases the h c has the opposite sign to f { { . However, we show that in the depth of CZ where the influence of rotation upon turbulence (associated with z *) and the radial shear of angular velocity are strong, the distribution of f { { might be drastically different from a classical cos è -dependence, where è is colatitude. It is shown that f { { has a negative sign at the bottom and below of CZ at mid latitudes. There, the distribution of h c is also different from cos è , but it does not change its sign with the depth. Further, we briefly consider these quantities in the disk geometry. The application of the developed theory to dynamos in the accretion disk is more restrictive because they usually have a strong differential rotation, | ‘ log z / ‘ log r | > 1.

Magnetoconvection and dynamo coefficients:

We present numerical simulations of three-dimensional compressible magnetoconvection in a rotating rectangular box that represents a section of the solar convection zone. The box contains a convectively unstable layer, surrounded by stably stratified layers with overshooting convection. The magnetic Reynolds number, Rm, is chosen subcritical, thus excluding spontaneous growth of the magnetic field through dynamo action, and the magnetic energy is maintained by introducing a constant magnetic field into the box, once convection has attained a statistically stationary state. Under the influence of the Coriolis force, the advection of the magnetic field results in a non-vanishing contribution to the mean electric field, given by . From this electric field, we calculate the α-effect, separately for the stably and the unstably stratified layers, by averaging over time and over suitably defined volumes. From the variation of α we derive an error estimate, and the dependence of α on rotation and magnetic field strength is studied. Evidence is found for rotational quenching of the vertical α effect, and for a monotonic increase of the horizontal α effect with increasing rotation. For , our results for both vertical and horizontal α effect are consistent with magnetic quenching by a factor . The signs of the small-scale current helicity and of the vertical component of α are found to be opposite to those for isotropic turbulence.