Transient Growth of Submesoscale Instabilities

Submesoscale instabilities are analyzed using a transient growth approach to determine the optimal perturbation for a rotating Boussinesq fluid subject to baroclinic and symmetric instabilities. We consider a base flow with uniform shear and stratification and compute the non-normal evolution of linear perturbations over finite-time in an ageostrophic, non-hydrostatic regime. Stone (1966, 1971) showed that the stability of the base flow to normal modes depends on the Rossby and Richardson numbers, with instabilities ranging from baroclinic modes (Ri >1) to symmetric (Ri < 1) and Kelvin-Helmholtz (Ri < 1/4) modes. Non-normal transient growth at short time represents a faster mechanism for the energy growth of perturbations and may provide an energetic link between large-scale geostrophic flows and dissipation via submesoscale instabilities. Here we consider two- and three-dimensional optimal perturbations by means of direct-adjoint iterations of the linearized Boussinesq Navier-Stokes equations to determine the form of the optimal perturbation and the optimal energy gain, and explore the mechanisms that contribute to the difference in energy transfer (horizontal buoyancy flux vs Reynolds stress) between short-term transient growth and long-term modal growth.