P. C. Matthews

Pattern formation with a conservation law

Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

New instabilities in two-dimensional rotating convection and magnetoconvection

Abstract We describe a new instability of rolls in three related convection problems. The instability results in a large-scale modulation of the amplitude of the convection (in contrast to a phase instability such as that of Eckhaus) and may be supercritical, with strongly amplitude-modulated rolls predicted at onset. Essential to the instability is the presence of a conserved quantity (whose existence is contingent upon an appropriate choice of boundary conditions), which gives rise to a corresponding slowly evolving large-scale mode. In rotating convection between stress-free boundaries, the conserved quantity is the velocity component along the axes of the rolls; in magnetoconvection it is the magnetic flux through the layer. In rotating magnetoconvection both quantities may be conserved. In each case the appropriate amplitude equations to describe convection near onset consist of a modulation equation of Ginzburg–Landau type for the amplitude of the rolls, coupled to a large-scale modulation equation for each conserved quantity. The large-scale mode(s), although damped according to linear theory, can destabilise the rolls at onset. It is found that rolls are unstable for sufficiently small Prandtl number in rotating convection, and for sufficiently small magnetic diffusivity in magnetoconvection. Throughout we consider only the onset of convection through a stationary bifurcation, but we find, remarkably, that in rotating magnetoconvection stable travelling waves may be found at onset due to this new instability.

Instability of rotating convection

Convection rolls in a rotating layer can become unstable to the Kuppers-Lortz instability. When the horizontal boundaries are stress free and the Prandtl number is finite, this instability diverges in the limit where the perturbation rolls make a small angle with the original rolls. This divergence is resolved by taking full account of the resonant mode interactions that occur in this limit: it is necessary to include two roll modes and a large-scale mean flow in the perturbation. It is found that rolls of critical wavelength whose amplitude is of order e are always unstable to rolls oriented at an angle of order e 2/5 . However, these rolls are unstable to perturbations at an infinitesimal angle if the Taylor number is greater than 4π 4 . Unlike the Kuppers-Lortz instability, this new instability at infinitesimal angles does not depend on the direction of rotation; it is driven by the flow along the axes of the rolls. It is this instability that dominates in the limit of rapid rotation. Numerical simulations confirm the analytical results and indicate that the instability is subcritical, leading to an attracting heteroclinic cycle

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasizing how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced. 76W05, 76F10

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.

Solar Magnetoconvection (Invited Review)

In recent years the study of how magnetic fields interact with thermal convection in the Sun has made significant advances. These are largely due to the rapidly increasing computer power and its application to more physically relevant parameters regimes and to more realistic physics and geometry in numerical models. Here we present a survey of recent results following one line of investigations and discuss and compare the results of these with observed phenomena.

Dynamo action in simple convective flows

Several simple steady flows generated by thermal convection are investigated, to determine which of them are able to generate a magnetic field through dynamo action. For steady convection in a non–rotating layer of fluid in the form of a square or hexagonal pattern, no dynamo action is found. However, spurious dynamo action can easily be obtained if the numerical resolution used is inadequate. Such erroneous results occur both with a fully spectral method and with a mixed pseudo–spectral and finite–difference method. For convection in a rotating layer, it is found that even the simplest form of convection, two–dimensional rolls, can act as a dynamo, provided that the nonlinearity of the flow is taken into account. The dynamo operates most efficiently for moderate values of the rotation rate and fails in the rapidly rotating limit. In the nonlinear regime a branch of steady equilibrated dynamos is found, but dynamo action ceases when the thermal forcing becomes sufficiently strong.

Instability and localisation of patterns due to a conserved quantity

Abstract We describe the influence of a conserved quantity on the stability properties of roll, square and hexagonal patterns near stationary onset. The appropriate systems of amplitude equations are analysed: they comprise equations for the amplitudes of the pattern modes, together with an additional coupled equation governing the evolution of a large-scale mode associated with the conserved quantity. We find that the presence of this large-scale mode may result in the destabilisation of all regular roll, square or hexagonal patterns, leading to amplitude modulation and strong localisation of the pattern. Our analytical results are complemented throughout by numerical simulations of a model partial differential equation, to illustrate the modulational instabilities and their nonlinear development.

Pulsating waves in nonlinear magnetoconvection

Numerical experiments on compressible magnetoconvection reveal a new type of periodic oscillation, associated with alternating streaming motion. Analogous behaviour in a Boussinesq fluid is constrained by extra symmetry. A low-order model confirms that these pulsating waves appear via a pitchfork-Hopf-gluing bifurcation sequence from the steady state.