E

the familiar "top-heavy" condition. It is this instability that leads to the convective motions discussed above. Using Eq. 4-4, the stability condition can also be expressed in terms of temperature as unstable neutral stable if I —

Note that Eq. 4-6 is appropriate for an incompressible fluid whose density depends only on temperature.

4.2.3. Energetics

Consider now our problem from yet another angle, in terms of energy conversion. We know that if the potential energy of a parcel can be reduced, just like the ball on the top of a hill in Fig. 4.3, the lost potential energy will be converted into kinetic energy of the parcel's motion. Unlike the case of the ball on the hill, however, when dealing with fluids we cannot discuss the potential energy of a single parcel in isolation, since any movement of the parcel requires rearrangement of the surrounding fluid; rather, we must consider the potential energy difference between two realizable states of the fluid. In the present case, the simplest way to do so is to consider the potential energy consequences when two parcels are interchanged.

Consider then two parcels of incompressible fluid of equal volume at differing heights, z1 and z2 as sketched in Fig. 4.5. They have the same density as their respective environments. Because the parcels are incompressible they do not expand or contract as p changes and so do not do work on, nor have work done on them by, the environment. This greatly simplifies consideration of energetics. The potential energy of the initial state is

Now interchange the parcels. The potential energy of the final state, after swapping, is

The change in potential energy, APE, is therefore given by

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