Low Rossby limiting dynamics for stably stratified flow with finite Froude number
In this paper, we explore the strong rotation limit of the rotating and stratified Boussinesq equations with periodic boundary conditions when the stratification is order 1 ([Rossby number] Ro = ε, [Froude number] Fr = O(1), as ε → 0). Using the same framework of Embid & Majda (Geophys. Astrophys. Fluid Dyn., vol. 87, 1998, p. 1), we show that the slow dynamics decouples from the fast. Furthermore, we derive equations for the slow dynamics and their conservation laws. The horizontal momentum equations reduce to the two-dimensional Navier–Stokes equations. The equation for the vertically averaged vertical velocity includes a term due to the vertical average of the buoyancy. The buoyancy equation, the only variable to retain its three-dimensionality, is advected by all three two-dimensional slow velocity components. The conservation laws for the slow dynamics include those for the two-dimensional Navier–Stokes equations and a new conserved quantity that describes dynamics between the vertical kinetic energy and the buoyancy. The leading order potential enstrophy is slow while the leading order total energy retains both fast and slow dynamics. We also perform forced numerical simulations of the rotating Boussinesq equations to demonstrate support for three aspects of the theory in the limit Ro → 0: (i) we find the formation and persistence of large-scale columnar Taylor–Proudman flows in the presence of O(1) Froude number; after a spin-up time, (ii) the ratio of the slow total energy to the total energy approaches a constant and that at the smallest Rossby numbers that constant approaches 1 and (iii) the ratio of the slow potential enstrophy to the total potential enstrophy also approaches a constant and that at the lowest Rossby numbers that constant is 1. The results of the numerical simulations indicate that even in the presence of the low wavenumber white noise forcing the dynamics exhibit characteristics of the theory.