A two-level time-stepping method for layered ocean circulation models: further development and testing

In [R.L. Higdon, A two-level time-stepping method for layered ocean circulation models, J. Comput. Phys. 177 (2002) 59] a two-level time-stepping method was developed for layered ocean circulation models. The method is designed to be used with a barotropic-baroclinic splitting that separates the fast and slow motions into subsystems that are solved by different techniques. The discussion in Higdon (2002) includes the development of the scheme, a linearized stability analysis, a description of some techniques for practical implementation in a nonlinear model, and some numerical testing. Subsequent additional testing revealed a need for further development of the techniques for nonlinear implementation. The purpose of the present paper is to describe these algorithmic improvements and to develop and report some additional numerical experiments. The algorithmic issues involve the relation between velocity and momentum density as layer thicknesses tend to zero, limiting mass and momentum fluxes between thick and thin cells near variable bottom topography, solving the barotropic equations that describe the fast motions in the system, and conserving mass within individual layers. This paper also develops a test problem involving external and internal Rossby waves in a two-layer fluid; the separation into modes makes it possible to test the time-stepping schemes for the barotropic and baroclinic systems independently. The paper concludes with some numerical tests that include the Rossby wave problem, an upwelling/downwelling problem that involves fluid interfaces moving upward and downward along sloping bottom topography, and a double-gyre circulation that displays meanders and eddies.