On Divergence- and Gradient-Preserving Coarse-Graining for Finite Volume Primitive Equation Ocean Models

We consider the problem of coarse-graining in finite-volume fluid models. Projecting a variable on a fine grid onto a coarser grid will in general cause some information about the solution to be lost. In particular, horizontal divergences and gradients will not be the same when calculated on the coarse grid from a projected solution as when they are calculated on the fine grid. Therefore, it is necessary to choose the method of coarse-graining carefully via a weighted average so that these properties will be conserved in the resulting field. We derive general conditions on the averaging weights that allow these properties to be preserved. We then take the particular case of a regular triangular mesh in which the fine-grid resolution is some integer multiple $N$ of the coarse-grid resolution. For this case we particular values for the averaging weights preserve the divergence for general $N$, and different weights that preserve the gradient for the case $N = 2$. These coarse-grainings are applied to data from FESOM2 simulations and we demonstrate that using this coarse-graining gives a significant improvement over other methods.