Besides regular, spatially periodic solutions, our systems show also spatio-temporal complex patterns. They appear in some cases as competing attractors to the periodic patterns or they show up as transients, when the systems are approaching a new state after having passed a secondary bifurcation. The complex patterns are often characterized by topological defects (dislocations), which present spatial variations on a length scale considerably smaller than the basic wavelengthλc = 2π/qc. On the other hand, it is sometimes important to describe large- aspect ratio systems with a lateral extensionL >> λc to capture properly modulational instabilities of rolls. Furthermore, in particular at moderate Prandtl numbers, meanfloweffects come into play, which are slowly decaying in space. They are for instance responsible for spiral defect chaos in RBC [5, 26].
One option is to solve the hydrodynamic equations directly by discretization in space. The codes run then on supercomputers and L < 30 λc is the practical limit.
Alternatively one can switch to q-space where the horizontal derivatives become mul-tiplications with the components of q. Not too far from onset the vertical variations are not too strong, such that they can be resolved by few Galerkin modes. However, a wide range of q-vectors in the Galerkin ansatz is required to cover the different length scales in complex pattern. The calculations are done on a grid q- space on discrete interpolation points qm,n = ∆q (m, n) with −N2 ≤ m, n ≤ N2. We chose typically
∆q = qc/k, which corresponds to L = k λc. Furthermore k ≤ N/6 is required to include at least the higher harmonics with wave vector 3 qc. A reasonable choice is N = 128, k = 12 but N = 256 can easily be handled as well. It is obvious that the direct solution of theODE’s for the Galerkin expansion coefficients is only possible for smallN (≤10), which is not sufficient to describe reliably complex patterns. However,
the system of ODE’s in Eq. (1.13) (N ×N ×3M coupled equations) has turned to be very useful to generate numerical solutions of the hydrodynamic equations for large systems. In contrast to standard discretization schemes it allows to concentrate on the most important active modes (<(λ) not too small) which carry the dynamics of the system, whereas the passive modes can be adiabatically eliminated. It is crucial that the most time consuming manipulations can be based on the fast Fourier transforma-tions (F F T). With respect to time, the linear operator is diagonal and can be treated fully implicit whereas the nonlinear part is treated with an explicit slightly modified Adams- Bashforth scheme (for further details, see App. B) The scheme is very ro-bust and allows large time steps. Due to a one- to- one correspondence to the Galerkin scheme the solutions can be tested by comparison. In this way, for the first time typical modern convection experiments could be reproduced in numerical simulations. It has been even demonstrated that to some extent two- dimensional experimental snapshots of patterns (temperature field at the upper plane) can be used to construct reliably the full three dimensional convection structure (see [18, 5]).
Chapter 2
Convection in a Rotating Annulus with an Azimuthal Magnetic Field
Introduction
Convection driven by thermal buoyancy in rotating fluid layers heated from below in the presence of a magnetic field is a typical problem encountered in planetary and stellar fluid dynamics. The most commonly treated version of this problem corresponds to the case when both, the axis of rotation and the direction of the imposed homogenous magnetic field, are parallel to the gravity vector. The onset of convection in this case was considered by Chandrasekhar [27]. He found the surprising result that Lorentz force and Coriolis force may counteract each other such that the critical Rayleigh number for onset of convection is lower than in cases when either the magnetic field or the rotation rate vanishes.
For planetary and stellar application the configuration of rotation axis and magnetic field direction perpendicular to each other and perpendicular to the gravity vector could be more important. It corresponds to the case of convection in the equatorial regions of rotating spherical fluid shells when an azimuthal magnetic field is imposed. Toroidal magnetic fields in the electrically conducting cores of planets or in the solar atmosphere are believed to be often much stronger than the poloidal components which can be measured from the outside. It is thus of interest to study the properties of convection in this situation which can also be realized in laboratory experiments through the use of the rotating annulus configuration (see Fig. 2.1a). In this case the centrifugal force is used as an effective gravity. Although the motivation for studying the rotating magnetic annulus convection has originally arisen in the geophysical context, the problem is also of interest from a more general point of view since it is a good example for pattern formation in the presence of two competing directional effects. In fact, as will be demonstrated in this thesis a large variety of convection patterns is found already at small values of the rotation and magnetic field parameters for moderate values of the Rayleigh number. In this respect the present study can be regarded as an extension of the paper by Auer et al. [24] to the case when a homogeneous magnetic field is added.
A rotating annulus experiment corresponding to the paper of Auer et al. [24] has been 17
performed by Jaletzky and Busse [28].
In Sec. (2.1) the mathematical formulation of the problem is discussed. In the small gap approximation, the problem reduces to the case of a horizontal fluid layer heated from below with magnetic field, axis of rotation, and the vertical direction corresponding to the x−, y− and z− axis of a Cartesian system of coordinates. A sketch of the geometrical configuration to be considered in this thesis is shown in Fig.
2.1b.
The results of the linear theory for the onset of convection which has first been considered by Eltayeb [8] more than 30 years ago are discussed in Sec. (2.2). In order to obtain simple expressions, Eltayeb has used idealized boundary conditions, namely stress-free conditions for the velocity field and electrically infinitely conducting boundaries for the magnetic field. In the present study the more realistic case of rigid, electrically insulating boundaries is considered. The weakly nonlinear analysis and a stability analysis of convection rolls is described in Sec. (2.3). Some of the instabilities of convection rolls can be understood on the basis of analytical results obtained in earlier work on the related problem of convection in the absence of a magnetic field [24, 25]. But new mechanisms of instability are introduced by the Lorentz force. Future research and potential application are discussed in the concluding section. Some results of this chapter have been already accepted for publication [29].