  We consider the problem of coarse-graining in the context of finite-volume
fluid models. If a variable is defined on a high-resolution grid it may be
coarse-grained so that it is defined on a grid of lower resolution. In general
this will cause some information about the variable to be lost. In particular,
horizontal divergences, gradients or other operators calculated on the coarse
grid after projecting may differ from those calculated on the fine grid. In
some cases we are able to choose averaging weights for coarse-graining such
that the coarse-grid operators will give a result approximating that of the
corresponding fine-grid operators applied on the fine grid. In this work we
derive general conditions on the averaging weights that allow the divergence
and gradient to be preserved. These conditions are applied to a regular
triangular mesh with B-grid variable placement in which the fine-grid
resolution is some integer multiple $N$ of the coarse-grid resolution. For this
case we find particular values for the averaging weights that preserve the
divergence, and a set of different averaging weights that preserve the
gradient. We observe that the vertical component of the curl is also preserved
by the same coarse-graining that preserves divergence. These coarse-grainings
are applied to data from FESOM2 simulations and we demonstrate that using the
particular coarse-grainings derived herein gives an overall reduction in $L^1$
error when compared with other methods.
