On Ertel's potential vorticity theorem. On the impermeability theorem for potential vorticity

摘要

The different expressions for Ertel's potential vorticity theory are presented using the spatial as well as the material description of the fields. The kinematical foundations in the material description are especially simple as they only involve derivative commutations with respect to the material variables. In addition, a precise mathematical expression for the so-called impermeability theorem is provided, clarifying the sense in which such a theorem can be understood. This is done by introducing a suitable transformation of the fluid velocity, making the impermeability theorem an expression of mass conservation and of the conservation of the particles identity in the transformed fluid.