We may interpret the terms on the right-hand side of (18.4.1) as follows. The first term represents the Laplacian of the advection of the perturbation thickness by the basic state vertically-averaged mean wind. The second term is proportional to the Laplacian of the advection of the basic state thickness by the vertically-averaged perturbation meridional wind. The third term represents the differential advection of the perturbation vorticity by the basic state wind. Thus, it appears that three distinct physical processes force vertical motion in this model. However, it can be shown that the first and third terms can be combined to give an expression identical to the second term so that (18.4.1) may be written

(d2/dx2 - 2X2)®2' = -(2fo/aAp)d2{Ux(d^i'/dx + dy3'/dx)}/dx2 (18.4.2)

Since (d2/dx2 - 2X2)©2' — — ^27, and the thermal wind Ux — — dX/dy, where X is mean temperature, we have from (18.4.2)

Alternatively, since the right-hand side of (18.4.2) can be expressed in terms of perturbation vorticity, we may write (18.4.2) in the form w27 - UTdZ27/dx (18.4.4)

where Z2' = (Z1' + Z3')/2 is the vertically-averaged perturbation vorticity.

Thus, in the linearized two-level model, the net forcing of vertical motion is proportional to: (1) the advection of the basic state temperature field by the vertically-averaged perturbation meridional wind, or (2) the advection of the vertically-averaged perturbation vorticity by the basic state thermal wind.

A schematic in Fig. 18.2 depicts the phase relationships between the geopotential field and the divergent secondary motion field for a developing unstable baroclinic wave in the two-level model over the midlatitudes where UT is usually positive. Summarizing the above results, one may state that

(a) Cold (warm) advection forces sinking (rising) motion, or

(b) Negative (positive) vorticity advection by the thermal wind forces sinking (rising) motion.

Fig. 18.2 Schematic showing (a) vertical motion forced by basic state horizontal thermal advection, and (b) vertical motion forced by divergence and vorticity changes in the field of a developing unstable baroclinic wave in the two-level model. Symbols used: C-cold, W-warm, U-upward, D-downward, Tr-trough, R-ridge, DV-divergence, CV-convergence. The circles with arrows show the directions of the secondary circulations

Fig. 18.2 Schematic showing (a) vertical motion forced by basic state horizontal thermal advection, and (b) vertical motion forced by divergence and vorticity changes in the field of a developing unstable baroclinic wave in the two-level model. Symbols used: C-cold, W-warm, U-upward, D-downward, Tr-trough, R-ridge, DV-divergence, CV-convergence. The circles with arrows show the directions of the secondary circulations

18.5 Energetics and Energy Conversions in Barodinic Instability 18.5.1 Definitions 18.5.1.1 Internal Energy

Let the internal energy of a vertical section dz of a column of air of unit cross-section be denoted by dl. Then, by definition, dl = p cv T dz, where p is the density, cv is the specific heat at constant volume, and T is the absolute temperature of the air in the vertical section. Integrating fron the earth's surface to the top of the atmosphere, the total internal energy of the column of air is given by

18.5.1.2 Potential Energy

On the other hand, the gravitational potential energy, dP, of the same vertical section at a height z above the earth's surface is given by dP = p g z dz.

Integrating through the atmosphere, the total potential energy of the column is given by

0 po

Integrating (18.5.2) by parts and using the ideal gas law, we obtain,

p0 J

Thus we find that I and P are related to each other as

and the total potential energy of the atmosphere may be expressed as

18.5.1.3 Available Potential Energy and Kinetic Energy

The total potential energy as defined above (18.5.4), however, is not a suitable measure of energy in the atmosphere, because the bulk of it is unavailable for useful work for a variety of reasons. Only a tiny part is available for atmospheric circulation, as discussed below, for a model atmosphere. Let us consider two equal masses of dry air of uniform potential temperatures, 91 and 92, with 91 < 92, separated by a vertical partition (Fig. 18.3).

Horizontal dashed lines indicate approximate isobaric surfaces. Arrows show the direction of motion when the vertical partition is withdrawn. The approximate rest position of the surface of discontinuity after re-arrangement of the airmasses is indicated by an inclined dashed line.

Let the ground level pressure on each side of the partition be 1000 mb. It is obvious that when the vertical partition is removed, there will be an adiabatic rearrangement of the air masses, the warmer air moving towards the colder air aloft, and the colder air undercutting the warmer air near the ground in opposite directions. The horizontal movement will generate a certain degree of vertical motion in the two airmasses with downward motion in the colder airmass and upward motion in the warmer airmass till a mass balance is reached along a surface of airmass discontinuity, indicated by a dashed line inclined to the vertical, as shown schematically in

We may get an idea of the kinetic energy generated by the re-arrangement of the masses within the same volume by considering the total energy of the system before and after removal of the vertical partition. Since the total energy is conserved during an adiabatic process, we have where K denotes kinetic energy.

If the air masses are initially at rest, K = 0. Thus, if we use primed quantities to denote the final state, we have

Fig. 18.3 Two airmasses of different potential temperatures separated by a vertical partition

With the aid of (18.5.4), it can be shown that the kinetic energy that is generated by the re-arrangement of the masses is given by the relation

According to the above relation, K' increases with decrease in the value of I' and becomes maximum when I' attains its absolute minimum value, I'', i.e., when the warmer air lies entirely over the colder air and the surface of separation between the two air masses becomes horizontal. In this extreme case, the total potential energy that becomes unavailable amounts to (cp/cv) I''. No further reduction of potential energy is possible beyond this stage. The maximum possible kinetic energy corresponding to maximum available potential energy in the atmosphere that can be realised by adiabatic re-arrangement of two air masses is, therefore, given by the expression

Lorenz (196o) has shown that the available potential energy (abbreviated to A.P.E.) in the earth's atmosphere is given by the volume integral of the variance of the potential temperature on isobaric surfaces over the entire atmosphere. Thus, if 0 is the average potential temperature at a given isobaric surface and 0' the local deviation from the average, the average A.P.E. per unit volume is to satisfy the proportionality

where V denotes the volume and the average value is denoted by underlining.

Observations indicate that for the atmosphere as a whole, the mean A.P.E. is only about 1/2oo of the mean total potential energy and that of what is available, only about 1/1o can be converted into mean kinetic energy. This is equivalent to saying that almost 99.95% of the mean total potential energy of the atmosphere is unavailable for any useful work. Thus, from the point of view of energy conversions, the atmosphere is highly inefficient as a heat engine.

18.5.2 Energy Equations for the Two-Level Quasi-Geostrophic Model

In the two-level quasi-geostrophic model (18.3.1-18.3.3), the perturbation temperature field is proportional to - ^3), the 25o-75o mb thickness. Thus, in conformity with the discussion in the previous sub-section, the available potential energy in this model is proportional to - y'3)2. To show that this is, indeed, the case, we derive the energy equations for the two-level model system as follows. We first multiply (18.3.5) by - (18.3.6) by - and (18.3.7) by (^ 1 - y'3). Next, we integrate the resulting equations over one wavelength of the perturbation in the zonal direction, noting that the average of any term over one wavelength will be indicated by an angle bracket as shown below.

where L is the wavelength of the perturbation.

Thus, for the first term involving differentiation with respect to time in (18.3.5), the average after multiplication by - ^ yields

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