During an adjustment, the slope of the interface declines to a new value, Sn, and AGPE is converted into KE. Under the conditions of geostrophy and stagnancy of the lower layer, velocity and total kinetic energy are:
where g' = gAp/p is the reduced gravity, and f is the Coriolis parameter. The total amount of GPE of the system is reduced:
Assuming that the process is reversible and adiabatic, the total energy is therefore conserved:
From Eqns. (3.151), (3.152), and (3.153), we obtain
where X = ^g'H/f is the deformation radius. The fraction of APE converted into KE is thus
For a circum-global current system, with L = 1,000 km, H = 1,000 m, and g' —
0.01 m/s2, at mid latitudes f — 10-4/s, r2 — 167, n — 1/168 — 0.006; while for an equatorial channel, f — 10-5, r2 — 10/6, n — 3/8. Thus, the APE of such a system can be quite efficiently converted into KE. However, the conversion rate for a mid- or high-latitude channel is much smaller. This result is consistent with the scaling of the wind-driven gyre. As discussed in Section 4.1.3, the ratio between kinetic energy of the mean flow, kinetic energy of eddies, and APE of the mean state is Kmean : Keddies : n = 1 : 100 : 1000. Therefore, owing to the geostrophic constraint, only a small portion of the APE can be converted into kinetic energy. We should also mention that the velocity field is associated with huge amount of APE.
Redefinition of meso-scale AGPE Although problems do occur when the MS AGPE is applied to the study of basin-scale dynamics, such a concept remains a very powerful tool in the study of meso-scale dynamics, in particular for understanding the essence of baroclinic instability. It is well known that the GPE of the mean state is the quantity that is released and converted into KE and potential energy of meso-scale eddies; therefore, it is highly desirable to be able to find a way to keep using this commonly accepted concept of AGPE. One way is to limit its usage to problems with a horizontal scale on the order of the deformation radius. In the following discussion, Eqn. (3.144) is applied to the world's oceans with a grid based on different horizontal resolutions.
First (Case A), we apply Eqn. (3.144) to each 1° x 1° grid cell in the world's oceans,
1.e., density averaged over the four corners of each 1° x 1° grid cell is used as the reference density and stratification in Eqn. (3.144). Assuming that the density distribution within this grid cell is a bi-linear function of the local coordinates, Eqn. (3.144) is reduced to a simple finite difference form. The results of this calculation will be referenced as MS_1 AGPE.
Second (Case B), Eqn. (3.144) is applied to the world's oceans, with a 2° x 2° resolution and similar finite difference scheme as for MS_1 AGPE. This case is termed MS_2 AGPE.
Case A (MS_1)
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