Abstract
An analytic semiโgeostrophic model of frontogcnesis is described which exhibits the growth of a frontal discontinuity surface. the initial distribution of temperature on a plane horizontal boundary consists of a warm twoโdimensional strip embedded in an isothermal background; the potential vorticity in the atmosphere above is uniform and positive. A barotropic deformation field with axis of dilation along the warm strip forces a discontinuity in the velocity field as the warm air is occluded and becomes detached from the lower boundary.
The solution demonstrates some important general properties of the inviscid semiโgeostrophic equations when discontinuities are present. In particular, the vertical velocity on the lower boundary becomes nonโzero at the front. the maximum value of the vertical velocity occurs here when the front first forms, but then lifts off the boundary. the front itself constitutes a line of infinite potential vorticity, so that the global integral of massโweighted potential vorticity is conserved only if the boundary of the domain is redefined to exclude the front. the implications of these properties for numerical modelling are discussed.
For typical atmospheric rates of deformation forcing, the accelerations of air parcels extremely close to the front are large enough to violate the conditions of validity of semiโgeostrophic theory. the conditions of hydrostatic balance are, for all practical purposes, never violated.