D. P. Brownjohn

Flux Separation in Stellar Magnetoconvection

The effect of a strong magnetic field on photospheric convection in a cool star like the Sun can be established by relating high-resolution solar observations to results from nonlinear models that rely on computation. The patterns of motion in numerical experiments on three-dimensional, compressible magnetoconvection depend not only on the strength of the imposed vertical magnetic field but also on the aspect ratio λ of the computational box. In a wide box (λ=8) with a moderately strong field, the flow organizes itself so that magnetic flux is separated from the motion. There are regions with strong fields and small-scale oscillatory convection next to almost field-free regions with clusters of broad and vigorous convective plumes. In the solar photosphere, this corresponds to the difference between the patterns of granulation in plage regions (with fields greater than 100 G) and in the adjacent quiet Sun.

Compressible magnetoconvection in three dimensions : pattern formation in a strongly stratified layer

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur.

Compressible magnetoconvection in oblique fields: linearized theory and simple nonlinear models

The linear stability of a layer of compressible fluid, permeated by an oblique magnetic field, is discussed. It is shown that regardless of the system parameters, all bifurcations generically lead to travelling waves. Wave speeds and direction of the wave propagation are investigated. Symmetry arguments are used to show that when the field is almost vertical, waves with a wave vector aligned with the tilt are preferred over those with a wave vector perpendicular to the tilt. The nonlinear development of the travelling waves is explored using simple model equations.

Nonlinear compressible magnetoconvection Part 2. Streaming instabilities in two dimensions

We have conducted further numerical experiments on two-dimensional fully compressible convection in an imposed vertical magnetic field and interpreted the results by reference to appropriate low-order models. Here we focus on streaming instabilities, involving horizontal shear flows, which form an important mechanism for the breakdown of steady convection in relatively weak fields for boxes of sufficiently small aspect ratio. While these shearing modes can arise even in the absence of a field, they will typically lead only to travelling and modulated waves unless there is a field to provide a restoring force. For magnetoconvection a new and dramatic form of pulsating wave appears after a complex sequence of secondary bifurcations

Oscillations and secondary bifurcations in nonlinear magnetoconvection

Complicated bifurcation structures that appear in nonlinear systems governed by partial differential equations (PDEs) can be explained by studying appropriate low-order amplitude equations. We demonstrate the power of this approach by considering compressible magnetoconvection. Numerical experiments reveal a transition from a regime with a subcritical Hopf bifurcation from the static solution, to one where finite-amplitude oscillations persist although there is no Hopf bifurcation from the static solution. This transition is associated with a codimension-two bifurcation with a pair of zero eigenvalues. We show that the bifurcation pattern found for the PDEs is indeed predicted by the second-order normal form equation (with cubic nonlinearities) for a Takens-Bogdanov bifurcation with Z2 symmetry. We then extend this equation by adding quintic nonlinearities and analyse the resulting system. Its predictions provide a qualitatively accurate description of solutions of the full PDEs over a wider range of parameter values. Replacing the reflecting (Z2) lateral boundary conditions with periodic [O(2)] boundaries allows stable travelling wave and modulated wave solutions to appear; they could be described by a third-order system.

Nonlinear compressible magnetoconvection. Part 3. Travelling waves in a horizontal field

We present results of numerical experiments on two-dimensional compressible convection in a polytropic layer with an imposed horizontal magnetic field. Our aim is to determine how far this geometry favours the occurrence of travelling waves. We therefore delineate the region of parameter space where travelling waves are stable, explore the ways in which they lose stability and investigate the physical mechanisms that are involved. In the magnetically dominated regime (with the plasma beta,

PHOTOSPHERIC CONVECTION IN STRONG MAGNETIC FIELDS

\hat{\beta}

Oscillatory convection in sunspot umbrae

= 8), convection sets in at an oscillatory bifurcation and travelling waves are preferred to standing waves. Standing waves are stable in the strong-field regime (

Magnetic flux separation in photospheric convection

\hat{\beta}

Nonlinear compressible magnetoconvection Part 1. Travelling waves and oscillations

= 32) but travelling waves are again preferred in the intermediate region (