Abstract
The behaviour of quadratic invariants of the velocity gradient tensor is explored when the time evolution is governed by semigeostrophic forms of the shallow water equations. The evolution equation of a certain Jacobian involving the geostrophic flow is formally similar to its counterpart under the primitive shallow water equations. The resultant deformation and the Frobenius norm do not behave in this symmetrical way. A product of the study is a straightforward derivation of the semigeostrophic potential vorticity conservation property. Results are extended to 3D baroclinic flow by using isentropic coordinates.