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Quasigeostrophic flows are induced by spatial variations in interior potential vorticity and boundary buoyancy. We begin by developing the geostrophic turbulence theory of boundary buoyancy anomalies in a 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 also find that the nonlinear interplay of Rossby waves with the turbulence spontaneously reorganizes the flow into homogenized zones of surface buoyancy separated by buoyancy discontinuities, with sharp eastward jets centered at the discontinuities. 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. Finally, 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. However, we find that the loss of a crucial symmetry in the vertical coupling between the modes prevents modal truncations from conserving energy, and so no such modal truncation is possible.