## 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.

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