: PEfinal

V dz

if z2 - zi is small, where (dp/dz)r = ——P1) V 'E (Z2 - Z1) is the mean density gradient of the environmental state. Note that the factor g (z2 - Z1)2 is always positive and so the sign of APE depends on that of (dp/dz)E. Hence, if (dp/dz)E> 0, rearrangement leads to a decrease in APE and thus to the growth of the kinetic energy of the parcels; therefore a disturbance is able to grow, and the system will be unstable. But if (dp/dz)E < 0, then APE > 0, and potential energy cannot be released by exchanging parcels. So we again arrive at the stability criterion, Eq. 4-6. This energetic approach is simple but very powerful. It should be emphasized, however, that we have only demonstrated the possibility of instability. To show that instability is a fact, one must carry out a stability analysis analogous to that carried out in Section 4.1.2 for the ball on the curved surface (a simple example is given in Section 4.4) in which the details of the perturbation are worked out. However, whenever energetic considerations point to the possibility of convective instability, exact solutions of the governing dynamical equations almost invariably show that instability is a fact, provided diffusion and viscosity are sufficiently weak.

4.2.4. GFD Lab II: Convection

We can study convection in the laboratory using the apparatus shown in Fig. 4.6.

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