The size of the discrete time step used to integrate the equations is limited by the fastest motion that has to be resolved, in this case the external gravity waves. One way to get around this limitation is to separate the fast and slow motions into separate subproblems. The fast external motions are essentially 2-D due to approximate independence from depth, which leads to the common option of obtaining the 2-D velocity field from a vertical average of the horizontal velocity field in the original 3-D equations.
This procedure can give rise to computational instabilities since the operator splitting method is inexact except for the case of a linearized flow with a horizontal bottom and a rigid lid. In this case one solution is exactly independent of depth and the horizontal velocity field obtained corresponds exactly to the vertically averaged velocity. However, if any of the restrictions are relaxed the fast and slow motions can be mixed by variable bottom topography or nonlinearities and can result in numerical instabilities if an explicit method with a long time step is used to advance the slow motion component in time. See Higdon and Bennett (1996).
where g' is the reduced gravity and L and U are, respectively, length and velocity scales imposed by the boundary conditions of the problem. The name comes from the fact that this is an overall parameter describing a whole flow as opposed to the gradient and flux Richardson numbers. See Turner (1973).
A major use of this method is to gauge the waxing and waning of glacial periods since the deposition of large amounts of water on land in the form of glaciers leaves the water enriched in and the water depleted of the heavy isotopes.