5.1.2 Motivation for separating the modes

Although there are several technical problems associated with the separation into the vertically averaged and vertically dependent modes, it is essential to build large scale ocean models using some version of this separation for the following reasons:

1.

Without a separation, the full momentum field will be subject to the CFL constraints of the external mode gravity wave speed, which is roughly $\sqrt{g \, H} = 200-250 \mbox{m}\,\mbox{s}^{-1}$ for ocean depths $H = 4000\mbox{m}-6000\mbox{m}$. When splitting, the internal modes, which are roughly 100 times slower than the external mode, can be integrated with approximatly 100 times longer time steps, thus enhancing the utility of the model for climate simulations.

2.

As vertical resolution is improved, the computation requirements for the barotropic mode will remain the same. However, for a non-separated model, adding vertical resolution adds more equations which are subject to the barotropic mode time step. Modern ocean simulations are tending towards increasing the vertical resolution in order to improve the representation of vertical physical processes such as boundary layers. Therefore, the low efficiency of the non-separated model is a greater burden for these high vertical resolution models.

There are two fundamental methods in MOM for solving the momentum equations. The traditional rigid lid method completely filters out the very fast waves associated with the external mode by fixing the ocean surface to be flat. This filtering transforms the generally hyperbolic external mode problem to an elliptic problem. The free surface, in contrast, admits the fast external waves and so care must be exercised in order to maintain numerical stability, and additional care must be exercised due to the possible linear interaction between the depth independent and depth dependent modes. It turns out that each method, and certain variants thereof, imply far reaching consequences for the numerical methods and physical content of the whole model. Much of the discussion in the remainder of this chapter elaborates on these consequences. The remainder of this section provides a general overview of these two methods, and later sections and chapters provide the full details.