Fryderyk Wilczyński

Magnetic buoyancy instability, which is of astrophysical importance, results from the influence of magnetic pressure variations on the density of a fluid in a gravitational field. It is inherently a compressible phenomenon and is, as such, fully described by the equations of compressible magnetohydrodynamics (MHD). For analytical and computational reasons, it is often convenient to study compressible MHD within simpler, asymptotically consistent reduced systems; the two most widely used result from the Boussinesq and anelastic approximations. Within the standard Boussinesq approximation of MHD, leading to the equations of Boussinesq magnetoconvection, magnetic buoyancy is excluded. It can, however, be included by a rescaling of the basic-state variables and by making further assumptions about the perturbation length scales. Within the anelastic approximation, no special measures are taken to incorporate magnetic buoyancy. It is, however, a priori unclear as to whether this neglect is justified, particularly in the light of the Boussinesq results. Our aims here are thus twofold. The first is to formulate the relationship between descriptions of magnetic buoyancy in the compressible, anelastic and Boussinesq systems. In so doing, we show that, under both the anelastic and Boussinesq approximations, magnetic buoyancy can be included either through a combination of a weak field and strong gradient, or, conversely, a strong field and weak gradient. Each has its own asymptotically consistent reduction, with dedicated governing equations. Our second aim is to address, through a linear stability analysis, under which conditions the standard anelastic system provides a faithful representation of magnetic buoyancy instability. For completeness, we also formulate the energy principle of ideal MHD within the anelastic framework, and demonstrate the relation with its fully compressible counterpart.

The dynamics at the edge of fusion confinement devices is driven by interchange instabilities and involves the motion of plasma across two regions—the “core region” and the scrape-off layer (SOL)—distinguished by whether field lines are, respectively, closed or connected to the wall. Motivated by this phenomenon, we present an extensive linear stability analysis of a two-layer plasma model encompassing the coupled interactions between the region with closed field lines and the SOL. We focus on the effect of varying the particle diffusivity and ion viscosity, revealing the significant variation in the spatial structure of the critical modes. In addition, we have investigated the dependence of the stability threshold on the ratio of the width of the region with closed field lines to that of the SOL; this dependence is strong when the ratio is sufficiently small, but becomes insignificant once the ratio is of order unity.

The dynamics at the edge of fusion confinement devices involves the motion of plasma across two regions—the core and the scrape-off layer—that may have very different properties. Motivated by this problem, and exploiting the analogy between the equations governing plasma interchange dynamics and those of classical Rayleigh-Bénard convection, we consider the linear stability of two-dimensional, two-layer miscible convection. We focus specifically on the influence of three particular parameters: the ratio of the viscosities in the two layers, the ratio of the thermal diffusivities, and the ratio of the depths of the two layers. The key result is that, depending on the parameters of the problem, the most unstable mode can take one of three quite distinct forms: whole layer solutions, in which the eigenfunctions of the stream function and temperature extend over both layers of fluid; localized solutions, with the velocity cells or the temperature perturbation (or both) confined to just one of the layers; and segregated solutions, in which the fluid motion and temperature perturbation are confined to different fluid layers.

We present an extensive linear stability analysis of a two-dimensional fluid model used to study the plasma dynamics in the scrape-off layer of tokamaks. The model equations are based on the Braginskii fluid equations under the assumptions of drift ordering and electrostatic plasma. The model also employs the commonly used slab geometry approximation, whereby the magnetic field is assumed constant and straight, with the effects of curvature reintroduced as effective gravitational terms. We study the linear instability in the system by solving a boundary value problem, thereby extending previous studies, which focused on a local analysis. Furthermore, we demonstrate that the governing plasma equations for the scrape-off layer can be viewed as describing a thermal convection problem with additional effects. The new features include a non-uniform basic state gradient, linear damping terms, and additional advective terms. We characterize the conditions at the onset of instability and perform an extensive parameter scan to describe how the stability threshold varies as a function of plasma parameters.