Stability of Baroclinic Flows in a Zonal Magnetic Field: Part III

Abstract

The study of the stability of two-layer baroclinic flows in a zonal magnetic field (Part II) is generalized to include a parabolic flow profile in the upper layer. Contrary to the nonmagnetic case, angularities may be present in the equations even when the potential vorticity gradient does not change sign, if the magnetic field is increased beyond a certain strength. Below this strength, power series solutions show that, just as in the nonmagnetic use, horizontal shear renders shorter waves unstable. Reynolds stresses are seen to transport momentum up the gradient, while smaller Maxwell stresses oppose them. Properties of the solutions in this paper and in Part II are compared to solar observations. At gas densities ?10?4 gm cm?3, a ?35K equator-pole temperature difference would be enough to give baroclinically unstable disturbances in a zonal field of 100 gauss. The perturbation vertical fields produced from this magnitude zonal field will have magnitude ?1 gauss, as well as a zonal wave number of 6?8, ?e folding? times of a few solar rotations, and a pronounced tilt upstream away from the maximum of zonal flow. All these characteristics are qualitatively consistent with the solar observations. The Reynolds and Maxwell stresses are seen to act in such a way as to be capable of maintaining the solar differential rotation in the manner proposed by Starr and Gilman. It is suggested that the same process could be used to maintain differential rotation in the dynamo theory. The production of perturbation vertical and mean poloidal fields, and the changes in the toroidal field by the disturbances are also seen to correspond to dynamo processes. Suitable generalizations of the equations may allow a complete dynamo cycle to take place. If so, the system would have the important computational advantage of putting all the dynamics on essentially the same time and space scales. Finally, the ?magnetic modes? recently suggested by Hide to be responsible for the slow westward drift of the geomagnetic field are very unlikely to be baroclinically unstable. They could, however, be fed energy from baroclinically unstable ?inertial modes.? Suggestions are made for further studies of the scaled equations presented.