1 Tidal Theory of the Thermal Wind

The baroclinic instability problem is considered in the framework of Laplacian tidal theory. The Hilbert space of the quasigeostrophic vorticity budget is spanned by spheroidal functions. The fluid is linearly stable against quasigeostrophic disturbances. As the essential source of irregular ocean-atmosphere motions, baroclinic instability is ruled out by tidal theory. The midlatitude β-plane budget of vorticity fluxes is inconsistent with basic laws of motion on the rotating spherical surface. Realistic numerical simulations of global wave dynamics and dynamical circulation instabilities require a covariant account of fluid motion on the spherical planet. I. Introduction Major difficulties in numerical climate simulations result from the fact that the potential energy in the ocean-atmosphere system is closely associated with large-scale features of its density field while a considerable fraction of its kinetic energy resides on fairly small eddy scales [1]. State-of-the-art models for weather prediction and climate simulation capture large-scale features of the global circulation with some degree of realism if they are determined by large-scale features of topography and external forcing. However, simulation of the dynamical control of density by transfer processes between the small scales of kinetic energy and the large potential energy scales is less satisfactory [2]. The key to these energy-and vorticity-fluxes is fluid instability. Of the numerous instabilities in the climate system none is considered as fundamental as baroclinic instability [3]. This concept refers to the growth of weakly divergent Rossby waves in a stably stratified and vertically sheared fluid on the rotating spherical surface. The contemporary understanding of cyclogenesis, predictability-limits and the transition to chaos, turbulence and stochasticity in the climate system revolves essentially around this process. Hence, it also provides the paradigm for the design and interpretation of irregular fluid motion in numerical circulation models, ranging from resolved scales down to the parameterization of subscale processes. Baroclinic instability theory invokes four sets of approximations. First: shallow water theory with constant mean layer-thicknesses and-velocities [3]. Second: quasigeostrophy. For tidal theory, Longuet-Higgins [4] has shown that this approximate closure of the vorticity budget yields a meaningful and elegant Rossby wave filter of Laplace's tidal equation. In baroclinic fluids this approach governs the thermal wind. Third: the midlatitude β-plane. To avoid formal difficulties with spherical coordinates, a tangential cartesian plane is pinned to the sphere at some extratropical latitude in analogy to Kelvin's well-established equatorial β-plane. Fourth: a number of scaling assumptions which neglect fluid velocities relative to planet rotation …