  Adaptive meshes have the potential to improve the accuracy and efficiency of
atmospheric modelling by increasing resolution where it is most needed. Mesh
re-distribution, or r-adaptivity, adapts by moving the mesh without changing
the connectivity. This avoids some of the challenges with h-adaptivity (adding
and removing points): the solution does not need to be mapped between meshes,
which can be expensive and introduces errors, and there are no load balancing
problems on parallel computers. A long standing problem with both forms of
adaptivity has been changes in volume of the domain as resolution changes at an
uneven boundary. We propose a solution to exact local conservation and
maintenance of uniform fields while the mesh changes volume as it moves over
orography. This is solved by introducing a volume adjustment parameter which
tracks the true cell volumes without using expensive conservative mapping.
  A finite volume solution of the advection equation over orography on moving
meshes is described and results are presented demonstrating improved accuracy
for cost using moving meshes. Exact local conservation and maintenance of
uniform fields is demonstrated and the corrected mesh volume is preserved.
  We use optimal transport to generate meshes which are guaranteed not to
tangle and are equidistributed with respect to a monitor function. This leads
to a Monge-Amp\`{e}re equation which is solved with a Newton solver. The
superiority of the Newton solver over other techniques is demonstrated in the
appendix. However the Newton solver is only efficient if it is applied to the
left hand side of the Monge-Amp\`{e}re equation with fixed point iterations for
the right hand side.
