On the Ertel and impermeability theorems for slightly viscous currents with oceanographic applications

In this note we make a theoretical analysis of how a mild fluid viscosity c an affect the potential vorticity for stratified fluids in a rotating syste m. The classical Ertel (1942) Theorem is applied to slightly viscous fluids to obtain the law of conservation corresponding to novel invariants. These invariants do not have a classical form: indeed one example is a simple cl assical potential vorticity multiplied by a function of time. It has to be stressed that similar relations hold for a large class of conserved quantit ies, such as tracers, entropy, etc. Our results are compared with the recen t Impermeability Theorem by Haynes and McIntyre (1987). Among other results this comparison provides an effective way of estimating the importance of frictional effects on the potential vorticity evolution along streamlines, also using a generalization of the classical Margules equation. In general these ideas can be fruitfully applied to various cases of oceanic interest, in an analysis of superficial phenomena as observed in satellite imagery o r systematic analysis of deep currents. So here we analyse some interesting aspects of equatorial currents, small space-scale motions, as well as stea dy or quasi-steady fronts and cold filaments, namely the long patches that are often observed in thermal satellite images.