Abstract
© 2013 Royal Meteorological Society.Divergence and vorticity are well known to be geometrically invariant quantities in that their mathematical forms are independent of the orientation of the coordinate axes. Various other functions of the elements of the horizontal velocity gradient tensor are invariants in the same sense: examples are the resultant deformation and the determinant and Frobenius norm of the tensor. A brief account of these quadratic invariants is given, including expressions relating them to divergence and vorticity and to one another, and noting their occurrence in the divergence equation. Assuming shallow-water dynamics with background rotation, time-evolution equations for the resultant deformation and the other quadratic invariants are derived and compared. None rivals the vorticity and potential vorticity equations for compactness, but each may be written quite concisely in terms of familiar quantities. Corresponding time-evolution equations under quasi-geostrophic shallow-water dynamics are also derived, and lead to a simple prognostic equation for the ageostrophic vorticity.