Sound-proof models for magnetic buoyancy

Abstract

Magnetic buoyancy is a phenomenon by which regions of strong magnetic field in a plasma can rise. In the Sun, magnetic buoyancy transports field from the deep interior to the surface, where it produces sunspots. This field is believed to originate in the solar tachocline, a thin layer of rotational shear beneath the convective envelope, wherein field lines are stretched azimuthally to produce a layer of strong horizontal field. A description of magnetic buoyancy and its role in the solar interior is provided in Chapters 1 - 2. To model the magnetic buoyancy phenomenon, one typically solves the fully compressible fluid equations of magneto-hydrodynamics (MHD). This is because the presence of strong magnetic field affects both the density and the pressure of the fluid. However, the presence of fast acoustic waves in the fully compressible equations can present difficulties for analytical and numerical analyses. Therefore, several methods have been developed to filter out these waves, leading to various “sound-proof” models, including the Boussinesq, anelastic and pseudo-incompressible models. These sound-proof models are presented in Chapter 3, along with the assumptions under which they are derived. In Chapter 4 we assess the validity of each of these approximate models for describing magnetic buoyancy in the context of the solar interior. A general sound-proof model is introduced and compared to the fully compressible system in a number of asymptotic regimes, including both non-rotating and rotating cases. We obtain specific constraints that must be satisfied in order that the model captures the leading-order behaviour of the fully compressible system. We then discuss which of the existing sound-proof models satisfy these constraints, and in what parameter regimes. We find that the pseudo-incompressible model and a formulation of the anelastic model both reproduce the leading order behaviour of the fully compressible system in the most general parameter regimes we consider. An alternative, and complementary, way to assess the validity of any model is to consider its mathematical properties, particularly conservation laws. An ideal magnetohydrodynamic fluid is a Hamiltonian system, and conserves energy, momentum, and magnetic flux. However, sound-proof models are derived using approximations that may violate the Hamiltonian structure of the system. In Chapter 5, we compare the sound-proof models to the compressible system by considering the mathematical properties of the linearised equations. For a Hamiltonian system, the equations describing perturbations to any static state are guaranteed to be self-adjoint, a fact that is useful in obtaining stability criteria. We derive constraints under which our general linearised sound-proof system is self-adjoint. We show that there is a unique set of self-adjoint sound-proof equations, namely the pseudo-incompressible equations, that conserves the same energy as the fully compressible system. The results presented in Chapters 4 and 5 neglect the effects of viscosity and magnetic diffusion. However, the presence of these diffusivities, as well as thermal diffusion, can lead to so-called double-diffusive instability, even in situations where the thermal and magnetic stratifications are individually stable. To date, this instability has only been extensively studied in the magneto-Boussinesq regime, which assumes very small vertical scales. In Chapter 6 we determine the behaviour of the double-diffusive instability outside of the Boussinesq regime by numerically solving the linearised fully compressible equations. We conclude in Chapter 7 by summarising the key results as well as detailing some potential avenues for future work.