Surface Quasigeostrophic Turbulence in Variable Stratification

Abstract

Numerical and observational evidence indicates that, in regions where mixed-layer instability is active, the surface geostrophic velocity is largely induced by surface buoyancy anomalies. Yet, in these regions, the observed surface kinetic energy spectrum is steeper than predicted by uniformly stratified surface quasigeostrophic theory. By generalizing surface quasigeostrophic theory to account for variable stratification, we show that surface buoyancy anomalies can generate a variety of dynamical regimes depending on the stratification’s vertical structure. Buoyancy anomalies generate longer range velocity fields over decreasing stratification and shorter range velocity fields over increasing stratification. As a result, the surface kinetic energy spectrum is steeper over decreasing stratification than over increasing stratification. An exception occurs if the near surface stratification is much larger than the deep ocean stratification. In this case, we find an extremely local turbulent regime with surface buoyancy homogenization and a steep surface kinetic energy spectrum, similar to equivalent barotropic turbulence. By applying the variable stratification theory to the wintertime North Atlantic, and assuming that mixed-layer instability acts as a narrowband small-scale surface buoyancy forcing, we obtain a predicted surface kinetic energy spectrum between k−4/3 and k−7/3, which is consistent with the observed wintertime k−2 spectrum. We conclude by suggesting a method of measuring the buoyancy frequency’s vertical structure using satellite observations.

The buoyancy staircase limit in surface quasigeostrophic turbulence

“…and hence (from the inversion relation (2.16)) a linear-in-wavenumber inversion relation θk = −(k/σ 0 ) ψk . We also recover a linear inversion function (2.18) in the small-scale limit; Yassin & Griffies (2022) show that m(k) → k/σ 0 as k → ∞ regardless of the functional form of σ (z).…”

“…This in contrast to the equivalent barotropic model where the staircase limit is reached at smaller k ε /k r for more local flows (with a smaller deformation radius) (Scott et al 2022). This difference is likely due to the presence of a distinguished length scale (the deformation radius) in the equivalent barotropic model, which, in the absence of β, encourages the formation of plateaus of potential vorticity surrounded by kinetic energy ribbons (Arbic & Flierl 2003;Yassin & Griffies 2022).…”

“…(2.10) where x = (x, y) is the horizontal position vector, z is the vertical coordinate, k = (k x , k y ) is the horizontal wavevector, k = |k| is the horizontal wavenumber and t is the time coordinate. The non-dimensional wavenumber-dependent vertical structure, Ψ k (z), is determined by the boundary value problem (Yassin & Griffies 2022)…”

“…Because these inversion functions can be approximated by power laws m(k) ≈ k 0.49 and m(k) ≈ k 1.50 at small wavenumbers, the induced velocity fields nearly coincide with the velocity fields computed from power law inversion functions with α = 0.5 and α = 1.5. However, note that not all inversion functions can be approximated by power laws (Yassin & Griffies 2022).…”

“…We also investigate how surface quasigeostrophic jets depend on the underlying vertical stratification. Yassin & Griffies (2022) show that the vertical stratification modifies the interaction range of vortices in the surface quasigeostrophic model. Suppose we have an infinitely deep fluid governed by the time-evolution of geostrophic buoyancy anomalies at its upper boundary.…”