A. M. Soward

The onset of thermal convection in a rapidly rotating sphere

The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than the true global value by an O (1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex s -plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the global criterion on a related two-parameter family of models, which includes the spherical convection problem for particular O (1) values of his parameters. Since he used one of them as the basis of a small-parameter expansion, his results are necessarily approximate for our problem. Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Yanos relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the leading-order value of the critical Rayleigh number together with its first-order correction for various values of the Prandtl number. Numerical solutions of the relevant PDEs have also been found, for values of the Ekman number down to 10 −6 ; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.

The onset of thermal convection in rotating spherical shells

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number

Fast dynamo action in a steady flow

E

A convection-driven dynamo I. The weak field case

is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius

On the Finite amplitude thermal instability of a rapidly rotating fluid sphere

r_o

Mathematical aspects of natural dynamos

and have short azimuthal length scale

Thermal and magnetically driven convection in a rapidly rotating fluid layer

O(E^{1/3}r_o)

A Kinematic Theory of Large Magnetic Reynolds Number Dynamos

. They also confirmed the localization of the convection about some cylinder radius

Scalar transport and alpha-effect for a family of cat's-eye flows

s\,{=}\,s_M

Convection driven dynamos

roughly