  We outline how discrete analogues of the conservation of potential vorticity
may be achieved in Finite Element numerical schemes for a variational system
which has the particle relabelling symmetry, typically shallow water equations.
  We show that the discrete analogue of the conservation law for potential
vorticity converges to the smooth law for potential vorticity, and moreover,
for a strong solution, is the weak version of the potential vorticity law. This
result rests on recent results by the author with T. Pryer concerning discrete
analogues of conservation laws in Finite Element variational problems, together
with an observation by P. Hydon concerning how the conservation of potential
vorticity in smooth systems arises as a consequence of the linear momenta.
  The purpose of this paper is to provide all the necessary information for the
implementation of the schemes and the necessary numerical tests. A brief
tutorial on Noether's theorem is included to demonstrate the origin of the laws
and to demonstrate that the numerical method follows the same basic principle,
which is that the law follows directly from the Lie group invariance of the
Lagrangian.
