  The variational free-Lagrange (VFL) method for shallow water is a
free-Lagrange method with the additional property that it preserves the
variational structure of shallow water. The VFL method was first derived in
this context by \cite{AUG84} who discretized Hamilton's action principle with a
free-Lagrange data structure. The purpose of this article is to assess the
long-time conservation properties of the VFL method for regularized shallow
water which are useful for climate simulation. Long-time regularized shallow
water simulations show that the VFL method exhibits no secular drift in the (i)
energy error through the application of symplectic integrators; and (ii) the
potential vorticity error through the construction of discrete curl, divergence
and gradient operators which satisfy semi-discrete divergence and potential
vorticity conservation laws. These diagnostic semi-discrete equations augment
the description of the VFL method by characterizing the evolution of its
respective irrotational and solenoidal components in the Lagrangian frame. Like
the continuum equations, the former exhibits a $\text{div}^2\mathbf{U}$ term
which indicates that the flow has a very strong tendency towards a purely
rotational state.
  Numerical results show (i) the preservation of shape and strength of an
initially radially symmetric vortex pair in purely rotational regularized
shallow water and (ii) how the Voronoi diagram retains the history of the flow
field and (iii) that energy is conserved to $\mathcal{O}(\Delta^2)$ and
potential vorticity error to within 5% with no secular growth over a 50 year
period.
