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where p is the density value averaged over the ocean area; the integration extends to the whole ocean volume and we then represent the climatic characteristics as the sum of average (in the above-mentioned sense) values and departures from them. We designate the former values by tildes, and the latter values by asterisks. After substitution of these expressions into (2.5.46) we obtain dp*/dt + (uV)p* + w* dp/dz = F*, (2.5.50)

where we have taken into account the fact that w = 0 and Fp = 0, and, hence, dp/dt = 0.

We multiply Equation (2.5.50) by —gp*(dp/dz)~1 and then integrate it over the ocean volume and use the appropriate boundary conditions. Then, discarding negligible terms describing the effect of river and underground run-off, we arrive at the following equation:

dp/dz

This is the budget equation for the available gravitational potential energy in the ocean. The first term on the right-hand side in this equation represents the mutual conversions of the kinetic and available gravitational potential energy; the second term represents the production and degeneration of the available gravitational potential energy. Let us note that, because w = 0, p*p* = pw, and therefore not all potential energy but, rather, only that part defined by Equation (2.5.49) is involved in the exchange with the kinetic energy (see (2.5.48)).

Separation of K and A into zonal and eddy components and the derivation of the corresponding budget equations are carried out formally in the same manner as for the atmosphere. Without repeating this procedure we turn to a discussion of empirical estimates of ocean energy cycle characteristics. According to Oort et al. (1989) the kinetic energy density of the zonally averaged circulation and eddy disturbances in the World Ocean for the annual mean conditions amounts to, on average, 0.006 and 0.075 J/m2, and the available gravitational potential energy density is 4.40 and 1.68 J/m2 respectively. For comparison: in the atmosphere these are equal to 4.5 and 7.3 J/m2, and 33.3 and 11.1 J/m2, respectively.

The above suggests that the kinetic energy in the ocean is much less than the available gravitational potential energy. This, in turn, means that the conversion of the available gravitational potential energy into kinetic energy of large-scale ocean circulation is small, and hence the time scales of processes in the ocean are much larger than in the atmosphere. Indeed, the rates of generation and dissipation of kinetic energy in the ocean are of the order of 0.1 W/m2 and the rate of generation of available gravitational potential energy and its conversion into kinetic energy is of the order of 2 x 10" 4 W/m2 (see Lueck and Reid, 1984). In other words, the ocean is characterized by the approximate parity between the production of available gravitational potential energy and its conversion into kinetic energy, on the one hand, and the generation and dissipation of kinetic energy, on the other.

Moreover, the kinetic and available gravitational potential energy in the ocean are much less than in the atmosphere, which results from the fact that in the ocean the thickness of the layer in which the main density disturbances are concentrated is no more than several hundred metres, while in the atmosphere the temperature disturbances extend to the limits of its total thickness. The fact that the ocean stratification is more stable than that of the atmosphere is of no small importance. Thus, in interaction with the atmosphere the ocean behaves as a passive, inertly responding partner.

To complete this discussion, let us enumerate the most distinguishing features of the energetics of the climatic system as a whole. According to empirical data classified by Monin (1982) the incoming short-wave solar radiation flux per unit area of the upper atmospheric boundary amounts to 1356 W/m2. Part of this radiation (the planetary albedo) which is reflected back to space is equal, on the average, to 0.30, so that the shortwave solar radiation flux assimilated by the climatic system amounts to

244 W/m2. The latter is redistributed in the following way: from the 70% of the incoming solar radiation remaining after reflection, 20% is absorbed by the atmosphere and 50% is absorbed by the ocean and the land. In its turn, the short-wave solar radiation absorbed by the ocean and land is expended on evaporation from the underlying surface (24%) and on sensible heat exchange with the atmosphere (8%). The remaining part (20%) is expended on ocean and land heating and hence on the conversion of short-wave solar radiation into long-wave thermal radiation from which 14% is absorbed by the atmosphere and 6% is emitted into space. Only a very small part (about 4 W/m2, or slightly more than 1%) of the assimilated solar radiation is converted into the kinetic energy of atmospheric and ocean motions. The rate of kinetic energy dissipation must be of the same order. Finally, the heat energy, absorbed by the atmosphere and equal to 64% of incoming short-wave solar radiation (20% of this is determined by the absorption of short-wave radiation, 16% is due to the absorption of long-wave radiation and 30% is determined by the exchange of sensible and latent heat with the underlying surface), is returned to space by means of long-wave radiation, thus completing the energy cycle. This is illustrated in Figure 2.15.

### 2.6 Angular momentum budget

The absolute angular momentum of a unit mass relative to rotation of the Earth's axis is defined by the expression where the first term on the right-hand side is called the planetary momentum (it describes the angular momentum of the unit mass rotating together with the Earth as a solid body); the second term is called the relative momentum (it has its origin in motion relative to the rotating Earth). The relative momentum is thought to be positive when moving from west to east, and negative when moving backwards.

We apply the individual derivative operator for (2.6.1) and write down the expression for dM/dt, which takes the form

Combining this relationship with the equation of motion for the zonal velocity component:

Combining this relationship with the equation of motion for the zonal velocity component:

we obtain

from which it follows that a change in the absolute angular momentum is determined only by the torque of pressure and friction forces.

We integrate (2.6.2) over the longitude and then over the vertical within the limits of the atmospheric thickness. Then, instead of (2.6.2), we obtain