Baroclinic Eady wave and fronts. Part I: Viscous semigeostrophy and the impact of boundary condition

A two-dimensional viscous semigeostrophic model is developed to study the evolution of the baroclinic Eady wave and fronts with two types (free-slip and nonslip) of boundary conditions. With the free-slip boundary condition, the solution is very similar to the inviscid one but the frontal collapse is prevented by the diffusive effect. When the fronts become sharp in the mature stage, strong horizontal diffusions of momentum and potential temperature cause strong inward fluxes of geostrophic potential vorticity (GPV) at the surface fronts, so high GPV anomalies are generated at the surface fronts and advected into the interior, forming two backward-tilted plumes along the upper and lower fronts. The wave and front development can be interpreted by the interaction between the lower- and upper-level GPV anomalies in terms of GPV thinking similarly tb that in the inviscid case. When the boundary condition is nonslip, the initial growth and subsequent nonlinear evolution of the solution are significantly slower than the inviscid one, but the associated boundary layer processes allow the model to produce realistic features in the vicinity of the front. Diffusive GPV fluxes at the boundaries are caused mainly by vertical diffusions of momentum and potential temperature, so GPV anomalies are produced over broad regions behind and ahead of the front. As the GPV anomalies are transported from the boundary layer into the interior, they evolve into two mushroom clouds. The shallow boundary layer circulation, driven by the inverted geostrophic flow through Ekman pumping, produces a positive feedback to the horizontal spreading of the interior GPV anomalies. This explains why and how the GPV anomalies grow into two mushroom clouds.