The vertical scale of the Eady problem in the presence of a surface mixed layer

In this work we investigate how the relevant vertical scale for the Eady problem changes in the presence of a convective boundary layer at the surface. This is simply modeled as a bottom layer with prescribed depth and zero static stability.

Although the standard QGPV cannot be defined over the mixed layer, the quasi-geostrophic problem is still well-posed in that region. In fact, potential temperature plays the same role as q does in the stratified case in that: (i) it defines the geostrophic flow (via thermal wind), except for a surface contribution and (ii) it is advected by that geostrophic flow alone. The eigenvalue problem can then be formulated in the usual manner coupling the stratified and unstratified regions.

Using this framework, we address the problem of defining a relevant Rossby depth, by examining how the penetration of the modes changes as the mixed layer expands. Our motivation is to understand what determines the eddy scale in idealized model runs (described elsewhere) that are statically neutral over a large fraction of the troposphere.