Poincare Wavre Theorem

How does rotation affect cloud morphology? In the preceding section, we were able to obtain spherical structures because the supporting thermal pressure is inherently isotropic. If we now consider the cloud to be rotating about a fixed axis, the associated centrifugal force on each fluid element points away from that axis, distending the equilibrium structure accordingly. It is natural, then, to erect a cylindrical coordinate system whose z-axis lies in the direction of rotation (see Figure 9.5). We suppose that each fluid element has a certain steady velocity u$ about this axis, but has no motion in the z- or ^-directions. Here, w = r sin 0 denotes the cylindrical radius. We further suppose that the cloud is axisymmetric, so that all ^-gradients vanish, and that it has reflection symmetry about the z = 0 equatorial plane.

A physical quantity of key importance is j = wu$, the specific angular momentum about the z-axis. Written in terms of j, the centrifugal force per unit mass on each element is j2/w3. This term must be incorporated into equation (9.1) for hydrostatic equilibrium. If we again posit

Figure 9.6 Illustration of Poincare-Wavre theorem. The specific angular momentum j is constant along cylinders that are centered on the rotation axis. For rotational stability, the value of j must increase outward, as indicated.

Figure 9.6 Illustration of Poincare-Wavre theorem. The specific angular momentum j is constant along cylinders that are centered on the rotation axis. For rotational stability, the value of j must increase outward, as indicated.

an isothermal equation of state, then force balance in the w- and z-directions requires dp_ p dw dw j2

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