7.2.5 Global water budget

For the computation of global budgets, it is useful to consider the global integral of the free surface height tendency given in equation (7.18). Due to the no-normal flow condition, the global integral of the convergence $- \nabla_{h} \cdot {\bf U}$vanishes. Additionally, for a closed fresh water budget, the global integral of q_w vanishes. Hence, the global integral of $\eta_{t}$ vanishes. This result means that the volume of water in the global ocean is constant, assuming no net water flux into or out of the ocean. Constant total ocean volume results for such a closed system when assuming an incompressible fluid. Assuming the total surface area of the ocean is constant, the global integral of $\eta_{t}$ vanishing implies that the global integral of $\eta$ is a constant. This constant can conveniently be set to zero. Conservation of volume is an important constraint to satisfy with a discretization of the free surface method.

For regional models, the total model volume may vary due to fresh water flux or due to fluxes through open model boundaries. For these cases, the global integral of the free surface height need not be constant in time. Also, for coupled models, there can be source and sink terms on land and in the atmosphere which may render the ocean volume non-constant.