8.3.3. Release of available potential energy in baroclinic instability

In a nonrotating fluid, the release of APE is straightforward and intuitive. If, at some instant in time, the interface of our two-component fluid were tilted as shown in Fig. 8.10 (left), we would expect the interface to collapse in the manner depicted by the heavy arrows in the figure. In reality the interface would overshoot the horizontal position, and the fluid would slosh around until friction brought about a motionless steady state with the interface horizontal. The APE released by this process would first have been converted to kinetic energy of the motion, and ultimately lost to fric-tional dissipation. In this case there would be no circulation in the long term, but if external factors such as heating at the left of Fig. 8.10 (left) and cooling at the right were to sustain the tilt in the interface, there would be a steady circulation (shown by the light arrows in the figure) in which the interface-flattening effects of the circulation balance the heating and cooling.

In a rotating fluid, things are not so obvious. There is no necessity for a circulation of the type shown in Fig. 8.10 (left) to develop since, as we saw in Section 7.3, an equilibrium state can be achieved in which forces associated with the horizontal density gradient are in thermal wind balance with a vertical shear of the horizontal flow. In fact, rotation actively suppresses such circulations. We have seen that if Q is sufficiently small, as in the Hadley circulation experiment GFD Lab VIII (Section 8.2.1), such a circulation is indeed established. At high rotation rates, however, a Hadley regime is not observed; nevertheless, APE can still be released, albeit by a very different mechanism. As we discussed in Section 8.2.2, the zonal state we have described is unstable to baroclinic instability. Through this instability, azimuthally asymmetric motions are generated, within which fluid parcels are exchanged along sloping surfaces as they oscillate from side to side, as sketched in Fig. 8.8. The azimuthal current is therefore not purely zonal, but wavy, as observed in the laboratory experiment shown in Fig. 8.7. The only way light fluid can be moved upward in exchange for downward motion of heavy fluid is for the exchange to take place at an angle that is steeper than the horizontal, but less steep than the density surfaces, or within the wedge of instability, as shown in Fig. 8.10 (right). Thus APE is reduced, not by overturning in the radial plane, but by fluid parcels oscillating back and forth within this wedge; light (warm) fluid moves upward and radially inward at one azimuth, while, simultaneously heavy (cold) fluid moves downward and

FIGURE 8.10. Release of available potential energy in a two-component fluid. Left: Nonrotating (or very slowly rotating) case: azimuthally uniform overturning. Right: Rapidly rotating case: sloping exchange in the wedge of instability by baroclinic eddies.

outward at another, as sketched in Fig. 8.8. This is the mechanism at work in GFD Lab XI.9

8.3.4. Energetics in a compressible atmosphere

So far we have discussed energetics in the context of an incompressible fluid, such as water. However, it is not immediately clear how we can apply the concept of available potential energy to a compressible fluid, such as the atmosphere. Parcels of incompressible fluid can do no work on their surroundings and T is conserved in adiabatic rearrangement of the fluid; therefore potential energy is all one needs to consider. However, air parcels expand and contract during the redistribution of mass, doing work on their surroundings and changing their T and hence internal energy. Thus we must also consider changes in internal energy. It turns out that for an atmosphere in hydrostatic balance and with a flat lower boundary it is very easy to extend the definition Eq. 85 to include the internal energy. The internal energy (IE) of the entire atmosphere (assuming it to be a perfect gas) is, if we neglect surface topography (see Appendix A.1.3)

Note the similarity to Eq. 8-5. In fact, for a diatomic gas like the atmosphere, cv/R = (1 - k) /k = 5/2, so its internal energy (neglecting topographic effects) is 2.5 times greater than its potential energy. The sum of internal and potential energy is conventionally (if somewhat loosely)

referred to as total potential energy, and given by

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