4.4.2 Mass conservation

Now consider the mass of the infinitesimal column of water

$$\delta m \int_{z_1}^{\eta} dz \, \rho \, \delta A.$$ = (4.42)

In this expression, $\rho$ is the mass density. The column mass changes when either the volume or the density is changed,

$$\partial_{t} (\delta m) \rho(\eta) \, \delta A \, \partial_{t} \eta + \int_{z_1}^{\eta} dz \, \partial_{t}\rho \, \delta A.$$ = (4.43)

Mass conservation implies that this change is due to mass flux through the box surface, i.e., from the convergence of the horizontal mass flux and from the water coming through the bottom and through the free surface

$$\partial_{t} (\delta m) \left(Q_{w} + w_{1}\, \rho_{1} -\nabla_{h} \cdot\int_{z_1}^\eta dz \, \rho {\bf u}_{h} \right)\,\delta A,$$ = (4.44)

where $\rho_{1}$ is the density of water entering from the bottom of the layer, and Q_w is the mass flux density of water entering through the free surface. This result leads to the mass balance equation for the surface layer

 

$$\partial_{t} \int_{z_1}^{\eta} dz \, \rho + \nabla_{h} \cdot\int_{z_1}^\eta \rho {\bf u}_{h} Q_{w} + w_{1}\, \rho_{1}.$$ = (4.45)

A more transparent form emerges from the assumption of a vertically uniform density in the surface layer, and $\delta m \approx \rho_s \, h \, \delta A$, which leads to

$$\partial_{t}(\rho_s \, h) + \nabla_{h} \cdot (h \, \rho_s \, {\bf u}_{h}) = Q_{w} + w_{1} \, \rho_{1},$$ (4.46)

or in the alternate form

 

$$\partial_{t} \, h + \nabla_{h} \cdot (h \, {\bf u}_{h}) \frac{ w_{1} \rho_{1} + Q_{w} - h \,D_h\rho_s/Dt } {\rho_s}.$$ = (4.47)

Comparison with the volume conservation equation (4.41) reveals three differences. The first is the presence of the density ratio weighting the vertical velocitiy w₁. To leading order, this ratio is close to unity. The second is the occurence of the fresh water mass flux instead of the volume flux. The third difference is the fundamentally new term

$$-\frac{h} {\rho_s}\, \frac{D_h\rho_s}{Dt} -\frac{h} {\rho_s} \left(\partial_{t} \rho_s +{\bf u}_{h} \cdot\nabla_{h} \rho_s\right)$$ = (4.48)

This term acts to increase the surface height whenever the density of the surface layer is reduced, such as occurs when the layer is heated. It is this effect which is absent in the current formulation of MOM. A general way to incorporate this effect is to reformulate the model's equations in their non-Boussinesq form.

Differences between a volume conserving and mass conserving ocean model are discussed more thoroughly in the papers by Greatbatch (1994) and Mellor and Ezer (1995). Both papers argue that the difference in sea level height is a spatially independent, time dependent height.