Inertial oscillations and frontogenesis driven by a quadratic horizontal density variation

We consider inviscid rotating flow driven by a horizontally quadratic density variation in a horizontally unbounded slab. This configuration permits a similarity solution, removing the dependence on the horizontal coordinate from the vorticity and temperature equations, which are then solved by numerical integration along characteristics. At large values of Rossby number, the flow proceeds to a singularity in a similar manner to the non-rotating flow with the same initial conditions. At small values of Rossby number there are inertial oscillations of growing amplitude, which have been analysed using the method of multiple scales. The oscillations become desynchronised between the upper and lower parts of the domain, and static instability appears for a small fraction of each oscillation period. Eventually the oscillations give way to the rapid formation of a singularity, in contrast to geostrophic adjustment theory which predicts that a singularity will form only if the Rossby number is sufficiently large.