5.2.1.2 Vanishing velocity at the ocean surface

A related assumption made by Bryan (1969) is to set the vertical velocity at the surface to zero

w(z=0) = 0. (5.6)

Setting w to zero at the ocean surface is not necessary for eliminating the external mode gravity waves. Again, $\eta =0$ is sufficient. As discussed in Section 6.1, Bryan's choice to set w(z=0) = 0 is based on the consequent ability of a single streamfunction to specify the two components of the barotropic velocity. If w is allowed to fluctuate at the surface, and $\eta =0$ is still imposed, then a velocity potential must be used in addition to the streamfunction (see Section 7.6). Both the streamfunction and velocity potential have separate elliptical problems which must be solved. In large scale ocean models, especially those with realistic geography and topography, the elliptic problems are quite expensive computationally. Therefore, Bryan's choice to set w(z=0) = 0 removed a large amount of computational burden from the model.