In the previous sections, we summarized an explicit scheme as: where could be any prognostic variable ( , , or ) and is an explicit estimate of and would be exact if not for implicitintime terms. The parenthesis about indicates that the term is explicit and extrapolated forward in time and for this we use the quasisecond order AdamsBashforth method: This is a linear extrapolation, forward in time, to . An extrapolation to the midpoint in time, , corresponding to , would be second order accurate but is weakly unstable for oscillatory terms. A small but finite value for stabilizes the method. Strictly speaking, damping terms such as diffusion and dissipation, and fixed terms (forcing), do not need to be inside the AdamsBashforth extrapolation. However, in the current code, it is simpler to include these terms and this can be justified if the flow and forcing evolves smoothly. Problems can, and do, arise when forcing or motions are high frequency and this corresponds to a reduced stability compared to a simple forward timestepping of such terms. The model offers the possibility to leave the tracer and momentum forcing terms and the dissipation terms outside the AdamsBashforth extrapolation, by turning off the logical flags forcing_In_AB (parameter file data, namelist PARM01, default value = True). (tracForcingOutAB, default=0, momForcingOutAB, default=0) and momDissip_In_AB (parameter file data, namelist PARM01, default value = True). respectively. A stability analysis for an oscillation equation should be given at this point. A stability analysis for a relaxation equation should be given at this point.
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