5.2.1.3 Fresh water forcing in the rigid lid

A constant surface height, $\eta =0$, does not imply a vanishing vertical velocity component at the ocean surface. The basic reason is that the surface height is a Lagrangian coordinate and the vertical velocity at the surface is Eulerian. Relatedly, as seen by the equation (4.29) for the surface kinematic boundary condition, $\eta =0$ does not imply w(z=0)=0 if there is a surface fresh water flux. This is the fundamental point raised in the work of Huang (1993). As can be seen through the kinematic boundary condition, allowing w to fluctuate at the surface, while still keeping $\eta =0$, will enable a more physical means to force a rigid lid model with fresh water $q_{w} \ne 0$ while at the same time filtering out the external mode gravity waves. Setting both $\eta =0$and w(z=0)=0 precludes a direct use of fresh water forcing. As such, the upper boundary is effectively closed to fresh water in the traditional rigid lid method. Instead, the effects of fresh water must be introduced through a virtual salt flux added to the salinity equation. The implementation of other surface boundary conditions likewise may involve certain unphysical assumptions.

In summary, the tradeoff that Bryan (1969) made was to eliminate the need for computing a velocity potential while sacrificing a physically based fresh water forcing. Huang (1993) argues that this choice is not satisfactory for large scale modeling since it eliminates some fundamental modes of the ocean circulation. Additionally, it possibly affects the sensitivity of the oceanic variability which might be realized in a coupled ocean-atmosphere model. Currently, the choice made in the development of MOM is to not implement the method from Huang. Rather, it is to focus on the free surface method for the reasons explained below.