Braginskii magnetohydrodynamics for arbitrary magnetic topologies: coronal applications

Abstract

We investigate single-fluid magnetohydrodynamics (MHD) with anisotropic viscosity, often referred to as Braginskii MHD, with a particular eye to solar coronal applications. First, we examine the full Braginskii viscous tensor in the single-fluid limit. We pay particular attention to how the Braginskii tensor behaves as the magnetic field strength vanishes. The solar corona contains a magnetic field with a complex and evolving topology, so the viscosity must revert to its isotropic form when the field strength is zero, e.g. at null points. We highlight that the standard form in which the Braginskii tensor is written is not suitable for inclusion in simulations as singularities in the individual terms can develop. Instead, an altered form, where the parallel and perpendicular tensors are combined, provides the required asymptotic behaviour in the weak-field limit. We implement this combined form of the tensor into the Lare3D code, which is widely used for coronal simulations. Since our main focus is the viscous heating of the solar corona, we drop the drift terms of the Braginskii tensor. In a stressed null point simulation, we discover that small-scale structures, which develop very close to the null, lead to anisotropic viscous heating at the null itself (that is, heating due to the anisotropic terms in the viscosity tensor). The null point simulation we present has a much higher resolution than many other simulations containing null points so this excess heating is a practical concern in coronal simulations. To remedy this unwanted heating at the null point, we develop a model for the viscosity tensor that captures the most important physics of viscosity in the corona: parallel viscosity for strong field and isotropic viscosity at null points. We derive a continuum model of viscosity where momentum transport, described by this viscosity model, has the magnetic field as its preferred orientation. When the field strength is zero, there is no preferred direction for momentum transport and viscosity reverts to the standard isotropic form. The most general viscous stress tensor of a (single-fluid) plasma satisfying these conditions is found. It is shown that the Braginskii model, without the drift terms, is a specialization of the general model. Performing the stressed null point simulation with this simplified model of viscosity reveals very similar heating profiles compared to the full Braginskii model. The new model, however, does not produce anisotropic heating at the null point, as required. Since the vast majority of coronal simulations use only isotropic viscosity, we perform the stressed null point simulation with isotropic viscosity and compare the heating profiles to those of the anisotropic models. It is shown than the fully isotropic viscosity can over-estimate the viscous heating by an order of magnitude.

The effect of anisotropic viscosity on the nonlinear MHD kink instability

“…The interpolation function s(|B|) is derived in [17] by considering how the distribution of momentum transport depends on the magnetic field strength. The interpolating function is given by…”

“…In practial terms, the parameter a 0 varies the size of the region where isotropic viscosity dominates in the numerical simulations. To align with earlier work in [17], we choose a 0 = 150. The application of the switching model in simulations of a stressed null point in [17] provides a fuller exploration of the isotropic feature of the model.…”

“…Although the Braginskii tensor theoretically achieves this smooth transition, in practice the implementation of the full tensor in finite-difference MHD codes leads to numerical errors in the viscosity at locations where the magnetic field strength is very weak, due to a lack of sufficient resolution. Further exploration of this numerical issue can be found in [17]. One approach to solving this problem is to design a numerical scheme specifically to treat the full Braginskii tensor.…”

“…Another approach is to devise a model that captures the main physics of the Braginskii tensor suitable for the solar corona and can be implemented in existing MHD codes [8]. The second option was taken up by MacTaggart, Vergori and Quinn [17], who developed a phenomenological model of anisotropic viscosity in the solar corona that captures the main physics for viscosity in the corona, namely parallel viscosity in regions of strong field strength and isotropic viscosity in regions of very weak or zero field strength. In this paper, we will refer to this model of viscosity as the switching model, for short.…”

“…weak field) regions, to compensate for resolution issues in simulations. In numerical tests of stressed null points, comparing the isotropic and anisotropic models showed that the use of isotropic viscosity overestimates the viscous heating by an order of magnitude [17].…”