Geostrophic turbulence near rapid changes in stratification

Geostrophic turbulence near rigid vertical boundaries is controlled by the forward cascade of buoyancy variance, exhibiting a shallow energy spectrum, secondary roll-up of filaments into small vortices, and a corresponding vorticity PDF with fat tails. These effects, well-described by the surface quasigeostrophic (SQG) model, also arise near step-function jumps in stratification. Here we investigate geostrophic turbulence near rapid but smooth jumps in stratification, modeled by N(z) = N₀ + N_d tanh(z/δ). The rapidity of change is controlled by the length scale δ, and the profile approaches a step function as δ → 0. The Green's function for the PV-streamfunction relationship, determined using a WKB approximation, is used to predict the structure of the spectrum, under various assumptions about the vertical and wavenumber distribution of PV.

Numerical simulations of freely-evolving turbulence generated by baroclinic instability, in both quasigeostrophic and Boussinesq models, verify the predictions and reveal that the jump in stratification has two effects: it alters the Green's function in the region of the jump, and it produces a peak in PV near the jump, approaching a Dirac delta-function as δ → 0. When the Green's function is integrated against this sharp PV distribution, contributions far from the jump (|z| ≫ δ) are suppressed, and the kinetic energy spectrum flattens. This occurs for a range of wavenumbers above the deformation wavenumber associated with the vertical extent H₀ of the domain, f/(N₀H₀), but smaller than a wavenumber associated with the vertical scale of the jump, K_δ = f/(N_d δ). The vertical distribution of the flattened spectrum decays over a distance proportional to δ. Implications of these results for observations of geostrophic turbulence in the upper ocean will be discussed.