What Sets the Spatial Size of Coherent Objects Produced in a Reverse Cascade of Turbulent Energy?

Reverse energy cascades occur in rapidly rotating and/or highly stratified turbulence. Earlier, we studied reverse cascades with the 2D quasigeostrophic (QG) equations used in geophysical fluid dynamics. Randomly forcing at high wave numbers, we created statistically steady, turbulent, coherent objects (in this case, we created an alternating band of β-plane jets that formed a “staircase” of potential vorticity and qualitatively looked like the multiple jet streams of Jupiter, Saturn, and Neptune). The width of the jets was the length where the reverse cascade of energy stopped. What determines that length? At each wave number k in a 2D QG fluid, the ratio of the kinetic energy (KE) to potential energy (PE) is (kL_R)², where L_R is the Rossby deformation radius, so at large k, KE dominates; while at small k, PE dominates. Our 2D QG simulations suggested that the KE injected at small spatial scales, reverse cascades to smaller k until the PE equals the KE, so that the cascade stops at k = 1/L_R. In a continuously-stratified 3D fluid, L_R depends on the Coriolis parameter and the stratification. Here, we examine whether our earlier 2D QG findings about what sets the scale of the largest coherent objects are valid in real 3D stratified flows.