The Geostrophic Turbulence of Boundary Buoyancy Anomalies

Abstract

Quasigeostrophic flows are induced by spatial variations in interior potential vorticity and boundary buoyancy. In the first part of this dissertation, we develop the geostrophic turbulence theory of boundary buoyancy anomalies in a quasigeostrophic fluid with vanishing potential vorticity. We find that the vertical stratification controls both the interaction range of boundary buoyancy anomalies and the dispersion of boundary-trapped Rossby waves. Buoyancy anomalies generate longer range velocity fields and more dispersive Rossby waves over decreasing stratification [$\mathrm{d}N(z)/\mathrm{d}{z} \leq 0$, where $N(z)$ is the buoyancy frequency] than over increasing stratification [$\mathrm{d}N(z)/\mathrm{d}{z} \geq 0$]. Consequently, the surface kinetic energy spectrum is steeper over decreasing (mixed-layer like) stratification than in the classical uniformly stratified model. We therefore suggest that this steepening of the spectrum over mixed-layer like stratification accounts for the $k^{-2}$ spectrum found in the wintertime upper ocean. This suggestion is consistent with numerical and observational evidence indicating that surface geostrophic velocities over wintertime extratropical currents are largely induced by surface buoyancy anomalies. We also find that, under certain conditions, the nonlinear interplay of boundary-trapped Rossby waves with the turbulence spontaneously reorganizes the flow into homogenized zones of surface buoyancy separated by surface buoyancy discontinuities, with sharp eastward jets centered at the discontinuities and weaker westward flows in between. Jet dynamics then depend on the vertical stratification. Over decreasing stratification, we obtain straight jets perturbed by dispersive eastward propagating waves. Over increasing stratification, we obtain meandering jets whose shape evolves in time due to westward propagating weakly dispersive waves. In the second part of this dissertation, we investigate normal modes in the presence of boundary-confined restoring forces, with the ultimate aim of obtaining an energy-conserving modal truncation of the quasigeostrophic equations. Such a modal truncation would generalize classical $N$-layer models to account for non-isentropic boundaries. Although we obtain orthogonal sets of vertical modes that diagonalize the energy and potential enstrophy in the presence of non-isentropic boundaries, we find that the loss of a crucial symmetry in the vertical coupling between the modes prevents modal truncations from conserving energy. Consequently, energy conserving modal truncations are not possible in the presence of non-isentropic boundaries.