DP Pzi nfdT

p\di we obtain

In hydrostatic approximation the integral (over the atmospheric mass) internal energy Jo pi dz is proportional to the integral potential energy Jo p® dz with a proportionality factor cJR. Really, for the unsaturated moist fPs p<b dz = — zdp = 0 Jo where ps is the surface atmospheric pressure.

pdz= pRTdz,

This suggests pi dz I

Considtering this fact we can combine these two forms of energy into one, referred to as the total potential energy II = I + <b = cpT, where cp = cv + R. The corresponding budget equation takes the form an , _ l ^ vp — + (uV)n = ft - - Vpu + v—. dt p p

The kinetic energy budget equation is derived by means of scalar multiplication of (2.5.1) by v, and taking into account that in hydrostatic approximation dw/dt = 0, we have

Equations (2.5.7)-(2.5.10) include similar (but opposite in sign) terms describing the conversion from one form of energy into the other. Thus, the term gw appearing in Equations (2.5.7) and (2.5.8) and describing the work performed by the buoyancy force features the conversion of internal energy into potential energy and back, and the term vVp/p, appearing in Equations (2.5.9) and (2.5.10) and describing the work performed by the pressure force describes the conversion of total potential energy into kinetic energy and back. Not all total potential energy can be converted into kinetic energy, but only the part A = J p(Yl — n^) dV designated as available potential energy. Here, J pfl^ d V is the unavailable potential energy, that is, that part of the total potential energy which remains if the atmosphere is adjusted to a state with constant pressure over any isentropic surface (surface with equal values of specific entropy t] = cp In T — R In p + const) retaining the stable stratification; d V = a2 cos q> dX dcp dz is an element of the volume; the integration extends to the whole volume of the atmosphere.

To represent A in terms of the parameters to be measured, let us carry out the following chain of transformations:

pIldV =

Tdp p-Rlcp ds edp

Here, it is anticipated that the potential temperature 6 is equal to zero at P = Po (Po = 1000 HPa is the standard atmospheric pressure at the sea surface); the integration over s extends to the whole surface of the sphere.

Let us take into account the fact that in adiabatic processes the air mass over any isentropic surface is constant. Then the mean pressure and the value pFLdF = ^

K/Cp ds pl+RIC*d6

also remain constant above this surface. Substituting these expressions for J pYl dV and J pll^ dV in definition of A we have

But inasmuch as p > 0 and p^, by definition, is the mean value of p, then, according to Lorenz (1967), the integral of the difference (p1+R/cp — p*+B/Cp) will be positive for each isentropic surface, and therefore it can be expressed in terms of a pressure dispersion at this surface. Further, when it is considered (see Lorenz, 1967) that inclination of the isentropic surface with respect to the horizontal plane is small, then the pressure dispersion at the isentropic surface can be approximated in terms of the potential temperature dispersion or of absolute temperature dispersion at the isobaric surface. At present, there are several approximations to the relationship (2.5.11). They are all based on the approximated Lorenz formula (1967):

a2 cos <p d<p dA dp/g, where T is the global mean air temperature on the fixed isobaric surface, ya = 9/cP and y = —df/dz are adiabatic and real vertical gradients of temperature in the atmosphere.

Turning back to the derivation of the energy budget equation, we multiply (2.5.5) by L and add it to (2.5.9) and (2.5.10). As a result, we have the total energy budget equation

or, in divergent form, dt

Conversion from one form of energy to the other has already been mentioned above but nothing was said about the mechanism of this conversion. To analyse it we use the quasi-zonality of the atmospheric circulation resulting from the zonality of the insolation, and extract zonal average and eddy (related to deviations of zonal averages) components in the fields of climatic characteristics. Then any climatic characteristics (say, components u, v and w of wind velocity or air temperature T) can be presented as u

T= [r] + T* + T, where, as before, square brackets signify zonal averaging, the asterisks signify departure from zonal average values and the tilde signifies a global average value.

Accordingly, the kinetic energy K referring to the unit mass can be expressed as the sum of the kinetic energy KM of the zonal mean motion and the kinetic energy KE of eddy disturbances, that is,

K = KM + KE, KM = KM2 + M2), KE = !([u*2] + [>*2]).

Similarly, the available potential energy is defined as

0 0

Post a comment