It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.
Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.
We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.
More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.