where , and are the tendencies due to advection, diffusion and forcing, respectively, namely:
and the forcing can be some arbitrary function of state, time and space. The term, , is required to retain local conservation in conjunction with the linear implicit freesurface. It only affects the surface layer since the flow is nondivergent everywhere else. This term is therefore referred to as the surface correction term. Global conservation is not possible using the fluxform (as here) and a linearized freesurface (Campin et al. [2004]; Griffies and Hallberg [2000]). The continuity equation can be recovered by setting and . The driver routine that calls the routines to calculate tendencies are S/R CALC_GT and S/R CALC_GS for temperature and salt (moisture), respectively. These in turn call a generic advection diffusion routine S/R GAD_CALC_RHS that is called with the flow field and relevant tracer as arguments and returns the collective tendency due to advection and diffusion. Forcing is add subsequently in S/R CALC_GT or S/R CALC_GS to the same tendency array.
The space and time discretization are treated separately (method of lines). Tendencies are calculated at time levels and and extrapolated to using the AdamsBashforth method:
where at time step . The tendency at is not recalculated but rather the tendency at is stored in a global array for later reuse.
The tracers are stepped forward in time using the extrapolated tendency:
Strictly speaking the ABII scheme should be applied only to the advection terms. However, this scheme is only used in conjunction with the standard second, third and fourth order advection schemes. Selection of any other advection scheme disables AdamsBashforth for tracers so that explicit diffusion and forcing use the forward method.
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