Book Review of Accretion Power in Astrophysics (3rd Edition) by Juhan Frank,Andrew King,Derek Raine Cambridge University Press, 2002, XIV+384 pp., £27.95, hardback (ISBN 0 521 62053 8), paperback (ISBN 0 521 62957 8).

The velocity, pressure, perturbation magnetic field, helicity and electromotive force driven by an isolated buoyant parcel in an unbounded, rapidly rotating, electrically conducting fluid in the limit of small Elsasser number and very small Ekman number are calculated, visualized and analyzed. On the scale of the parcel, the solution is identical to that obtained in the limit of small Ekman number and zero Elsasser number. On the scale of the Taylor-column, it is elongated in the direction of the applied magnetic field and compressed in the direction perpendicular to it. The α-effect calculated by averaging the electromotive force on planes normal to rotation is strongly anisotropic: near the parcel and in the inner part of the Taylor-column it is strongest when the applied magnetic field is perpendicular to rotation and gravity; in the outer part of the Taylor-column it is strongest when the applied magnetic field is in the same plane as rotation and gravity.     

Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell

Abstract

Dynamic interaction between magnetic field and fluid motion is studied through a numerical experiment of nonlinear three-dimensional magnetoconvection in a rapidly rotating spherical fluid shell to which a uniform magnetic field parallel to its spin axis is applied. The fluid shell is heated by internal heat sources to maintain thermal convection. The mean value of the magnetic Reynolds number in the fluid shell is 22.4 and 10 pairs of axially aligned vortex rolls are stably developed. We found that confinement of magnetic flux into anti-cyclonic vortex rolls was crucial on an abrupt change of the mode of magnetoconvection which occurred at Δ = 1 ～ 2, where A is the Elsasser number. After the mode change, the fluid shell can store a large amount of magnetic flux in itself by changing its convection style, and the magnetostrophic balance among the Coriolis, Lorentz and pressure forces is established. Furthermore, the toroidal/poloidal ratio of the induced magnetic energy becomes less than unity, and the magnetized anti-cyclones are enlarged due to the effect of the magnetic force. Using these key ideas, we investigated the causes of the mode change of magnetoconvection. Considering relatively large magnetic Reynolds number and a rapid rotation rate of this model, we believe that these basic ideas used to interpret the present numerical experiment can be applied to the dynamics in the Earth's and other planetary cores.     

Large wavenumber convection in the rotating annulus

Abstract

The annulus model considers convection between concentric cylinders with sloping endwalls. It is used as a simplified model of convection in a rapidly rotating sphere. Large azimuthal wavenumbers are preferred in this problem, and this has been exploited to develop an asymptotic approach to nonlinear convection in the annulus. The problem is further reduced because the Taylor-Proudman constraint simplifies the dependence in the direction of the rotation vector, so that a nonlinear system dependent only on the radial variable and time results. As Rayleigh number is increased a sequence of bifurcations is found, from steady solutions to periodic solutions and 2-tori, typically ending in chaotic behaviour. Both the magnetic (MHD convection) and non-magnetic problem has been considered, and in the non-magnetic case our bifurcation sequence can be compared with those found by previous two-dimensional numerical simulations.     

Nonlinear Magnetoconvection and the Geostrophic Flow - Subcritical Solutions

The magnetoconvection problem under the magnetostrophic approximation is investigated as the nonlinear regime is entered. The model consists of a fluid filled sphere, internally heated, and rapidly rotating in the presence of a prescribed, axisymmetric, toroidal magnetic field. For simplicity only a dipole parity and a single azimuthal wavenumber (m = 2) is considered here. The leading order nonlinearity at small amplitude is the geostrophic flow U _g which is introduced to the previously linear model (Walker and Barenghi, 1997a, b). Walker and Barenghi (1997c) considered parameter space above critical and found that U _g acts as an equilibration mechanism for moderately supercritical solutions. However, for solutions well above critical a Taylor state is approached and the system can no longer equilibrate. More importantly though, in the context of this paper, is that subcritical solutions were found. Here subcritical solutions are considered in more detail. It was found that, at is strongly dependent on . ( is the critical value of the modified Rayleigh number is a measure of the maximum amplitude of the generated geostrophic flow while , the Elsasser number, defines the strength of the prescribed toroidal field.) R_m at proves to be the key measure in determining how far into the subcritical regime the system can advance.     

Formation of magnetic flux tubes in cylindrical wedge geometry

Three-dimensional (3D) MHD numerical simulations have not been able to demonstrate convincingly the spontaneous formation of large vertical flux tubes. Two-dimensional (2D) magnetoconvection in axisymmetric cylinders forms a central magnetic flux tube surrounded by annular convection rings. To study the robustness of this type of solution in three dimensions, the nonlinear resistive MHD equations are solved numerically in a 3D cylindrical wedge from an initially uniform vertical magnetic field. It is shown that the 2D result is retrieved for small domain radii. However, for larger radii the central axis loses its importance and in this case many convection cells form in the numerical domain. Magnetic flux is captured between cells where flow converges and the reduced amount of flux that congregates at the central axis is eroded by the surrounding convection.     

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α²-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.     

Symmetries of time-dependent magnetoconvection

Abstract

In the presence of a magnetic field, convection may set in at a stationary or an oscillatory bifurcation, giving rise to branches of steady, standing wave and travelling wave solutions. Numerical experiments provide examples of nonlinear solutions with a variety of different spatiotemporal symmetries, which can be classified by establishing an appropriate group structure. For the idealized problem of two-dimensional convection in a stratified layer the system has left-right spatial symmetry and a continuous symmetry with respect to translations in time. For solutions of period P the latter can be reduced to Z ₂ symmetry by sampling solutions at intervals of ½P. Then the fundamental steady solution has the spatiotemporal symmetry D ₂ = Z ₂ ? Z ₂ and symmetry-breaking yields solutions with Z ₂ symmetry corresponding to travelling waves, standing waves and pulsating waves. A further loss of symmetry leads to modulated waves. Interactions between the fundamental and its first harmonic are described by the group D _2h = D ₂ ? Z ₂ and its invariant subgroups, which describe solutions that are either steady or periodic in a uniformly moving frame. For a Boussinesq fluid in a layer with identical top and bottom boundary conditions there is also an up-down symmetry. With fixed lateral boundaries the spatiotemporal symmetries, again described by D _2h and its invariant subgroups, can be related to results obtained in numerical experiments and analysed by Nagata et al. (1990). With periodic boundary conditions, the full symmetry group, D _2h ?Z ₂, is of order 16. Its invariant subgroups describe pure and mixed-mode solutions, which may be steady states, standing waves, travelling waves, pulsating waves or modulated waves.     

Effects of boundary conditions on the onset of convection with tilted magnetic field and rotation vectors

The problem of the onset of thermal convection is considered, firstly when a uniform tilted magnetic field is present, and secondly in a frame rotating about an oblique axis. If up–down symmetry is broken we expect to find only bifurcations that lead to travelling waves. Numerical studies show, however, that in a Boussinesq fluid the spectrum of eigenvalues can be symmetrical about the real axis, even when the boundary conditions are asymmetrical. Here we show analytically that this symmetry property indeed holds for a wide range of boundary conditions and hence that both steady solutions and standing waves are allowed.     

Buoyancy-driven perturbations in a rapidly rotating,electrically conducting fluid: part I – flow and magnetic field

The steady velocity, perturbation pressure and perturbation magnetic field, driven by an isolated buoyant parcel of Gaussian shape in a rapidly rotating, unconfined, incompressible electrically conducting fluid in the presence of an imposed uniform magnetic field, are obtained by means of the Fourier transform in the limit of small Ekman number. Lorentz and inertial forces are neglected. The solution requires at most evaluation of a single integral and is found in closed form in some spatial regions. The solution has structure on two disparate scales: on the scale of the buoyant parcel and on the scale of the Taylor column, which is elongated in the direction of the rotation axis. The detailed structures of the flow and pressure depend linearly on the relative orientation of gravity and rotation, with the solution for arbitrary orientation being a linear combination of two limiting cases in which these vectors are colinear (polar case) and perpendicular (equatorial case). The perturbation magnetic field depends additionally on the relative orientation of the imposed magnetic field, and three limiting cases of interest are presented in which gravity and rotation are colinear (polar–toroidal case), gravity and imposed field are colinear (equatorial–radial case) and all three are mutually perpendicular (equatorial–toroidal case). Visualization and analysis of the velocity and perturbation magnetic field vectors are facilitated by dividing these vector fields into geostrophic and ageostrophic protions. In all cases, the geostrophic and ageostrophic portions have different structure on the Taylor-column scale. The buoyancy force is balanced by a pressure force in the polar case and by a flux of momentum in the equatorial case. The pressure force and momentum flux do not decay in strength with increasing axial distance. Far from the parcel, the axial mass flux varies as the inverse one-third power of distance from the parcel. The velocity has a single geostrophic vortex in the polar case and two vortices in the equatorial case. The perturbation magnetic field has two, four and one geostrophic vortices in the polar–toroidal, equatorial–radial and equatorial–toroidal cases, respectively. To facilitate comparison of the present results with numerical simulations carried out in a finite domain, a set of boundary conditions are developed, with may be applied at a finite distance from the parcel.