Stable Split Time Stepping Schemes for Large-Scale Ocean Modeling

from millennial variations of the thermohaline circulation to the rapid timescales of surface gravity waves. LargeAn explicit time integration of the primitive equations, which are often used for numerical ocean simulations, would be subject to a scale ocean circulation models typically eliminate the fastshort time step limit imposed by the rapidly varying external gravity est oscillations through use of the primitive equations. waves. One way to make this time step limit less onerous is to split Sound waves are removed by assuming that flow is incomthe primitive equations into a simplified two-dimensional set of pressible, while the hydrostatic approximation to the vertiequations that describes the evolution of the external gravity waves cal momentum equation eliminates the need to solve a and a much more slowly evolving three-dimensional remainder. The two-dimensional barotropic equations can be rapidly integrated three-dimensional elliptic equation for pressure or resolve over a large number of short time steps, while a much longer time the timescales associated with vertically propagating gravstep can be used with the much more complicated remainder. Unfority waves. The fastest remaining timescales are associated tunately, it has recently been demonstrated that an inexact splitting with horizontal propagation of external gravity waves, with into the fast and slow equations can lead to instability in the explicit a speed of about ÏgD, where g is the gravitational accelerintegration of the slow equations. Here a more exact splitting of the equations is proposed. The proposed split time stepping scheme ation and D is the depth of the ocean. In the deep ocean, is demonstrated to be stable for linear inertia–gravity waves, subject this speed is typically on the order of 225 m/s. The next to a time step limit based on the inertial frequency and internal fastest timescales are associated with internal gravity wave gravity wave speeds. Q 1997 Academic Press propagation or horizontal advection, both with speeds of order a few meters per second. There is a strong incentive to use a time stepping scheme with time steps that are