Peter Bartello
Department of Mathematics & Statistics,
Department of Atmospheric & Oceanic Sciences, McGill University, Montréal, Québec

From Quasigeostrophic Turbulence to Stratified Turbulence

The march toward ever higher resolution in "realistic" atmospheric and oceanic models has renewed the community's interest in stratified turbulence with weak rotation, such that the flow does not engage in the decoupling of vortex motion and inertial-gravity waves, leading to quasigeostrophy in the large scales. Recent work on this limit will be presented. Following the conjecture of Lilly (1983) that there is an inverse cascade in stratified turbulence resulting from the vertical decoupling at low horizontal (U/NL) and vertical (U/NH) Froude numbers, there were several numerical studies (e.g. Herring & Métais 1989) with forcing at small scales in order to simulate an inverse cascade. These clearly demonstrated the lack of an inverse cascade in this limit. Since the energy had to go downscale for statistical stationarity to be achieved, co-worker Michael Waite and I decided to focus our attention on stratified turbulence downscale of the forcing in the energy cascade range. We examined the characteristic vertical scale as a function of stratification and viscosity (Prandtl=1). If the Reynolds number is sufficiently high, then as one increases the stratification, the characteristic vertical scale collapses such that the vertical Froude number remains of order unity. The horizontal Froude number goes to zero, as the horizontal scale is not much affected. Rescaling the equations in this way reveals that the vertical advection term does not become small. A number of anisotropic statistics (e.g. energy and energy transfer spectra with respect to k_h and k_z) were shown to support this conclusion. Continuing to increase the stratification at fixed Re, the characteristic vertical scale eventually falls below the viscous scale and becomes independent of stratification. Only in this limit is the classical layerwise decoupling obtained. We also performed a suite of simulations, all at strong stratification, but with varying rotation in order to observe the transition from stratified turbulence (vertical scale~N/U) to quasigeostrophic turbulence (vertical scale~fL/N). Throughout this work we examined the statistics of linear wave and vortical (PV) modes separately. It was shown that they can only be expected to have dynamical relevance when U/NH<<1.
While this is the case in quasigeostrophic scaling, it is only true in the buoyancy range of vertical scales in stratified turbulence. In the QG limit, care was taken to remove at least some of the higher-order corrections to balance from the wave contribution.
Finally, with atmospheric and oceanic turbulence modelling in mind, we explicitly calculated the energy drain on the linear vortical modes exerted by the linear wave modes as a very preliminary step toward a parameterisation of "wave drag" in balanced models.