Nesting is used for local refinement to resolve eddy dynamics that would not be accessible otherwise. Unstructured meshes offer this functionality too by adjusting their resolution according to some goal function. However, by locally refining the mesh one does not necessarily achieves the goal resolution, because the eddy dynamics and the ability of eddies to release the available potential energy in particular, also depend on the dynamics on the upstream part of the coarse mesh. It is shown through a suite of simple experiments with a zonally re-entrant channel that baroclinic turbulence can be away from equilibrium in wide (compared to a typical eddy size) zones downstream into the refined area. The effect depends on whether or not the coarse part is eddy resolving, being much stronger if it is not, and almost disappearing if it is. Biharmonic viscosity scaled as cube of mesh element size is generally sufficient to control the smoothness of solutions on the variable mesh. However, noise in the vertical velocity field may be present at locations where the mesh is varied if momentum advection is implemented in the vector invariant form, which points to a variant of the Hollingsworth instability. Smoothness of vertical velocity is recovered for the flux form of momentum advection.