Yaya dpdp J

The definition of C(AM, KM) was given above.

Equation (2.5.36) is the equation for the available potential energy budget. The terms on the right-hand side describe, respectively, the generation of available potential energy by zonal mean motion, mutual conversions of available potential and kinetic energy of zonal mean motion and mutual conversions of available potential energy of zonal mean motion and eddy disturbances. The last named process has its origin in the so-called baroclinic instability of zonal motion.

Let us turn to Equation (2.5.33) to derive the budget equation for the available potential energy AE of eddy disturbances. We rewrite it in divergent form, then multiply by 0', average over the longitude and use Equations (2.5.34). As a result we obtain

+ 2l&][9*v*D cos q> + — (i[0]2|>] + [0*2]|>] + 2[0][0*co*])

cp T dp

Subtracting (2.5.33) from (2.5.40) we obtain I. JZp + _J_ (i([0*2]M + [(TP*»*]) cos q>)

+ ™2([0*2][>] + [0][0*co*]) + ( [0*t?*] — + [0*co*] — )[0]

We turn in (2.5.41) from [0*2] to [71*2] and multiply the resulting equation by cPyJT(ya — y), integrating it over the whole mass of the atmosphere.

If we introduce designations

for the available potential energy of eddy disturbance and its generation, respectively, and take into account the definitions of mutual energy conversions (see (2.5.29) and (2.5.39)), then the budget equation for AE takes the form

The energy cycle of the atmosphere described by Equations (2.5.22), (2.5.28), (2.5.36) and (2.5.44) is shown in Figure 2.14. The arrows here signify the directions of energy transitions; the numerical estimates are based on factual data throughout the decade 1963-1973 and relate to the total years' average conditions. An analysis of this figure leads to the following conclusions. The annual mean radiational heating of the atmosphere in the tropics and cooling in high latitudes result in the generation of the available potential energy of zonal mean motion. Affected by baroclinic instability, it is converted into available potential energy of eddy disturbances with a rate of 1.27 W/m2. This conversion, together with the generation of the available

0(Am)

/

m

36.9

C(AM,KM)

4.7

D(Km)

1.12

46.0

0.15

5.9

0.18

m

40.0

0.02

4.9

0.28

0.99

0.11

0.32

C(Am,A£) C(Kc

,KJ

1.27

0.33

1.56

030

0.99

0.21

r

Ke

3(A£)t

11.5

C(Ac,Ke)

7.5

D(KE)

0.7

6.8

2.0

7.3

1.7

0.6

5.5

2.2

6.4

1.9

0.7

1.7

1.5

Figure 2.14 Energy cycle of the global atmosphere, according to Oort and Peixôto (1983). The upper numbers indicate annual mean conditions; the middle numbers indicate winter conditions; and the lower numbers indicate summer conditions. Different forms of energy are in 105 J/m2; generation, energy transitions and dissipation are in W/m2.

Figure 2.14 Energy cycle of the global atmosphere, according to Oort and Peixôto (1983). The upper numbers indicate annual mean conditions; the middle numbers indicate winter conditions; and the lower numbers indicate summer conditions. Different forms of energy are in 105 J/m2; generation, energy transitions and dissipation are in W/m2.

potential energy of eddy disturbances (0.74 W/m2), determined by release of latent heat, is balanced (due to the mechanism of barotropic instability) by the conversion of available potential energy into kinetic energy of eddy disturbances. A small part (0.33 W/m2) of the incoming kinetic energy of eddy disturbances goes towards maintaining the zonal average circulation that is equivalent to reverse energy transfer over the spectrum (i.e. the transfer from motions with small horizontal scales to large ones) - the phenomenon known as negative viscosity. The main part (1.7 W/m2) is involved in direct cascade energy transfer from the interval of energy supply to the viscous range, where the kinetic energy of eddy disturbances dissipates into heat. Finally, of the kinetic energy of the zonal mean circulation, part (0.18 W/m2) dissipates, and part (0.15 W/m2) transforms into the available potential energy of zonal mean circulation under the influence of ordered meridional motions. This is the energy cycle of the atmosphere in general terms. Its schematic representation can be given in the form AM => AE => KE =>

The above-mentioned features of the atmospheric energy cycle are qualitatively similar not only in different seasons of the year, but in different hemispheres. From the quantitative perspective, the differences reduce to amplification of the generation of available potential energy of zonal mean motions and eddy disturbances in summer, to the appearance of the powerful transport of available potential energy of the zonal mean circulation from a summer hemisphere into a winter one, to the disappearance of the reverse energy transfer over the spectrum and its replacement by direct transfer, and, finally, to the intensification of kinetic energy dissipation in winter. Let us note one more interesting detail: there is almost a doubling of amplitudes of seasonal variations for different forms of energy in the Northern Hemisphere as compared to their values in the Southern Hemisphere, which is caused by the different ocean-land area ratio in both hemispheres.

Ocean

Before embarking upon a discussion of ocean energetics, let us note that the compexity of the equation of sea water state relating density to temperature, salinity and pressure excludes the possibility of representing the total potential energy as the sum of internal and potential energies. But it does not prevent the introduction of the concept of available gravitational potential energy.

We define the potential energy $ so that it takes on zero value at a reference depth z = — zr, that is, ® = gp(z + zr)/p0, where p is the sea water density and p0 is its constant value, and apply the operator d/df to both parts of this equation. As a result we obtain d®/8t + (uV)<D = g(p/p0) w + g(z + zr)(8/8t + uV)p/p0. (2.5.45)

We find the second term on the right-hand side of (2.5.45) using the evolution equation for density dp/dt + (uV) p = Fp, (2.5.46)

where Fp are sources and sinks of mass in the ocean.

Substitution into (2.5.45) yields the budget equation for potential energy

SQ/dt + (uV)<D = g(p/p0)w + g(z + zr)Fp/Po. (2.5.47)

The budget equation for kinetic energy K = \u\2/2 can be written similarly to (2.5.10) as dK/dt + (uV)K = — (l/p0)Vpu - g(p/p0)w + v-F. (2.5.48)

We now introduce the definition of the available gravitational potential energy:

dp/dz

0 0

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