Abstract:
Baroclinic instability drives synoptic-scale extratropical weather and mesoscale ocean dynamics. Linear instability theory is a classical tool for understanding baroclinic instability, but its fundamental assumption of infinitesimal perturbations about a mean flow leaves a large gap between linear theory and fully nonlinear baroclinic dynamics. DelSole (J. Atmos. Sci. 2004) pointed out a connection between ‘instantaneous optimals’ from non-normal linear instability theory and the energetics of fully-nonlinear turbulence, which we pursue here. We begin by finding exact analytical expressions for instantaneous optimals in the ageostrophic Eady problem, followed by a similar analysis (without exact solutions) for the general quasigeostrophic baroclinic instability problem. The solutions are paradoxical in light of DelSole’s connection to the fully nonlinear problem: They indicate that eddies of any horizontal scale can extract energy from the mean flow equally efficiently. We make a conjecture, whose details are specific to the quasigeostrophic problem, that sustained energy extraction from the mean should occur at horizontal length scales where there is an unstable normal mode, i.e. where the linear problem has an exponentially-growing mode. Finally, we examine the impact of horizontal discretization on the normal modes and instantaneous optimals of the Eady problem. We find that standard second-order discretizations on the Arakawa C grid (but not the B grid) introduce spurious unstable modes near the grid scale, while conversely the C grid does a better job of representing instantaneous optimals.