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

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Abstract
Because of its possible applications to the solar differential rotation, hydromagnetic dynamo theory, and the geomagnetic secular variation, the dynamics of baroclinic, geostrophic β-plane flow in a zonal (toroidal), magnetic field are studied. The equations describing the flow in the absence of a magnetic field are the same as those formulated by Charney and Stern and by Pedlosky. Due to the horizontal magnetic field, potential vorticity is no longer conserved. Vertical magnetic fields can be produced, but the scaling excludes their feedback on the motions and horizontal fields. Thus, the system in its present simplest form can not complete a dynamo cycle, but suitable relaxation of some scaling restrictions may overcome this difficulty. The scaled equations are perturbed about a steady axially symmetric zonal flow and zonal (toroidal) magnetic field. Changes in the initial state are inferred from products of perturbation quantities. These include the growth in the meridional plane of axially symmetric circulations and magnetic fields (i.e., poloidal fields). The energetics of the system are examined. It is shown that the potential vorticity theorem of Charney and Stern no longer holds due to the magnetic field. For short wavelength disturbances, the field should dominate in determining the stability properties. Bounds are placed on the complex phase velocities of normal mode disturbances. For flows with vertical and horizontal shear, these are the same as found by Pedlosky for the nonmagnetic case. The bounds on growth rates of disturbances in barotropic flows are tightened by the magnetic field. Such flows will be stable to all wavelengths if the field is large enough.

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