Quasigeostrophic dynamics of a finite-thickness tropopause

A model of tropopause dynamics is derived that is of intermediate complexity between the three-dimensional quasigeostrophic model and the surface quasigeostrophic (SQG) model. The model assumes that a sharp transition in stratification occurs over a small but finite tropopause region separating regions of uniform potential vorticity (PV). The model is derived using a matched-asymptotics technique, with the ratio of the thickness of the tropopause region to the typical vertical scale of perturbations outside as a small parameter. It reduces to SQG to leading order in this parameter but takes into account the next-order correction. As a result it remains three-dimensional, although with a PV inversion relation that is greatly simplified compared to the Laplacian inversion of quasigeostrophic theory. The model is applied to examine the linear dynamics of perturbations at the tropopause. Edge waves, described in the SQG approximation, are recovered, and explicit expressions are obtained for the corrections to their frequency and structure that result from the finiteness of the tropopause region. The sensitivity of these corrections to the stratification and shear profiles across the tropopause is investigated. In addition, the evolution of perturbations with near-zero vertically integrated PV is discussed. These perturbations, which are filtered out by the SQG approximation, are represented by a continuous spectrum of singular modes and evolve as sheared disturbances. The decomposition of arbitrary perturbations into edge-wave and continuous-spectrum contributions is discussed.