The budget of kinetic energy is of fundamental interest in ocean modeling. The manners in which various processes contribute to this budget are considered in this chapter. In particular, a breakdown of the budget for the total kinetic energy, the external mode kinetic energy, and the internal mode kinetic energy are derived for the continuum equations in the case of a free surface. The rigid lid results are also indicated. After doing so, two aspects of the discrete model energetics are considered. First, a proof that the work done by the discrete pressure terms is equal to the work done by buoyancy is given. In addition, the arguments from Bryan (1969) and Semtner(1974) are summarized concerning the conservation of first and second moments for velocity. This result holds for the case of zero forcing, zero dissipation, and when employing centered-differenced advection of momentum. The conservation of these two moments for momentum prevent systematic errors that accumulate in time (i.e., spurious growth or decay of kinetic energy). Additionally, the preservation of the second moment eliminates the problems with Phillip's (1959) non-linear instability (Arakawa 1966, Bryan 1969). These two points have provided the strong motivation for employing centered advection of momentum in the GFDL ocean model. Note that there are alternative advection schemes for tracers, as discussed in Chapter 31. The use of these schemes for tracers does not compromise the numerical stability maintained by centered differenced momentum advection.
In general, it is important for an ocean model to provide a diagnostic of the kinetic energy density. Pragmatically, the implementation of numerous algorithms in MOM have been debugged through an analysis of the kinetic energy budget. The diagnostic option energy_analysis (Section 39.4) provides the domain averaged budget for the kinetic energy density for MOM.