Interactions between vortical and high-frequency modes in Boussinesq turbulence

Peter Bartello (McGill University), Michael L Waite (NCAR) and Lydia Bourouiba (McGill University)

The title of this meeting ("Spontaneous imbalance") seems to take for granted that the typical state of the atmosphere and oceans is "balanced" and that this is momentarily disrupted from time to time at certain scales. It is now generaly accepted that the slow manifold is fuzzy in the sense that its dimension is that of the entire phase space, but that unbalanced motions above a certain amplitude are extremely improbable. The work described here aims to explore how this amplitude varies as a function of the motion's Rossby and Froude numbers. Considering that there is as yet no strict definition of balance, our analysis is based on simple linear normal modes. The nonhydrostatic Boussinesq equations have been examined on an f-plane for maximum simplicity. To leading order as Fr->0, the potential vorticity is determined by the zero-frequency geostrophic modes (unless of course they have very little variance) and equilibrium statistical mechanics can be employed on the inviscid truncated set. Numerical simulations of forced and decaying turbulence support the picture that emerges.

Turbulence generated by random isotropic i.c.'s in a strongly stratified and strongly rotating environment participates in a nonlinear geostrophic adjustment via a downscale cascade of wave energy and a first-order decoupling of geostrophic modes. When the turbulence is stratified but not rotating, a strong coupling remains even as Fr->0. If the result is balanced at large scales, it surely is not below L~H~U/N where overturning and the generation of 3D turbulence occurs. Although not within the range where the PV decomposition corresponds to the linear eigenmodes, if energy is injected into the high-frequency modes only, then some energy is transferred to the vortical modes (see also Lelong 2005). Can we call this spontaneous balance? Results from rotating (but not stratified) simulations will also be discussed. Here, the approach to the limiting behaviour with respect to exchange between vortical modes and high-frequency modes is non-monotonic, with a maximum exchange occurring at an intermediate Rossby number.