## M

is the mass-weighted mean height of the fluid. We can think of (z> as the ''height of the center of mass.'' It is evident from Eq. 8-7 that potential energy can be released only by lowering the center of mass of the fluid.

### 8.3.2. Available potential energy

Consider again the stratified incompressible fluid discussed in Section 4.2.2. Suppose, first, that dp/dz < 0 (stably stratified) and that there are no horizontal gradients of density, so p is a function of height only, as depicted in Fig. 4.5. In Section 4.2.3 we considered the energetic implications of adiabatically switching the position of two parcels, initially at heights zi and z2, with densities p1 and p2, respectively, with z2 > z1 and p2 < p1. Since p is conserved under the displacement (remember we are considering an incompressible fluid here), the final state has a parcel with density p1 at z2, and one with p2 at z1. Thus density has increased at z2 and decreased at z1. The center of mass has therefore been raised, and so PE has been increased by the rearrangement; none can be released into kinetic energy. This of course is one way of understanding why the stratification is stable. Even though such a fluid has nonzero potential energy (as defined by Eq. 8-7), this energy is unavailable for conversion to kinetic energy. It cannot be reduced by any adiabatic rearrangement of fluid parcels. Therefore the fluid has available potential energy only if it has nonzero horizontal gradients of density.

Consider now the density distribution sketched in Fig. 8.9 (left); a two-layer fluid is shown with light fluid over heavy with the interface between the two sloping at angle y. This can be considered to be a highly idealized representation of the radial density distribution in our tank experiment GFD Lab XI with heavier fluid (shaded) to the right (adjacent to the ice bucket) and light fluid to the left. Let us rearrange the fluid such that the interface becomes horizontal as shown. Now all fluid below the dashed horizontal line is dense, and all fluid above is light. The net effect of the rearrangement has been to exchange heavy fluid downward and light fluid upward such that, in the wedge B, heavy fluid has been replaced by light fluid, while the opposite has happened in the wedge A. (Such a rearrangement can be achieved in more than one way, as we shall see.) The center of mass has thus been lowered, and the final potential energy is therefore less than that in the initial state: potential energy has been released. Since no further energy can be released once the interface is horizontal, the difference in the potential energy between these two states defines the available potential energy (APE) of the initial state.

The important conclusions here are that available potential energy is associated with horizontal gradients of density and that release of that energy (and, by implication, conversion to kinetic energy of the motion) is effected by a reduction in those gradients.

The previous discussion can be made quantitative as follows. Consider again Fig. 8.9, in which the height of the interface between the two fluids is given by h(y) = 2 H + y y, and to keep things simple we restrict the slope of the interface Y < 1H to ensure that it does not intersect the lower and upper boundaries at z = 0, z = H respectively. Direct integration of Eq. 8-5 for the density distribution shown in Fig. 8.9, shows that the potential energy per unit length in the x-direction is given by (as derived in Appendix A.1.2)

FIGURE 8.9. Reduction in the available potential energy of a two-layer fluid moving from an initial state in which the interface is tilted (left) to the final state in which the interface is horizontal (right). Dense fluid is shaded. The net effect of the rearrangement is to exchange heavy fluid downward and light fluid upward such that, in the wedge B, heavy fluid is replaced by light fluid, whereas the opposite occurs in the wedge A.

FIGURE 8.9. Reduction in the available potential energy of a two-layer fluid moving from an initial state in which the interface is tilted (left) to the final state in which the interface is horizontal (right). Dense fluid is shaded. The net effect of the rearrangement is to exchange heavy fluid downward and light fluid upward such that, in the wedge B, heavy fluid is replaced by light fluid, whereas the opposite occurs in the wedge A.

rH rL

gpz dy dz

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