By virtue of Eq. (2.34), one always has div u = 0. (2.36)
Hence, absolute-vorticity lines cannot begin or end in the fluid; they are either closed curves or terminate on the boundary. By making use of Eq. (2.22), one can also write u = curl (u + n x r) = Z + 2«, (2.37)
where Z is the relative vorticity, that is, the curl of the velocity measured in the rotating frame of reference.
Let us now derive the equation expressing the rate of change of vorticity in a continuous motion. Using a formula well known in vector analysis,
we can take the curl of Eq. (2.27) to obtain d Z 1 ( 1 \ --+ curl (u x u) = — grad p x grad p + curl ( — f) . (2.39)
By making use of Eq. (2.36), one finds that curl (u x u) = u ■ grad u — u ■ grad u + u div u. (2.40)
Since n is a constant vector, one also has dZ/dt = du/dt. Hence, Eq. (2.39) becomes
-= u ■ grad u — u div u +—- grad p x grad p + curl — f . (2.41)
Dt p2 p
Combining Eqs. (2.6) and (2.41), one obtains the vorticity equation
— — = — ■ grad u +—- grad p x grad p +— curl — f . (2.42)
The first term on the right-hand side of this equation represents the action of velocity variations on the ratio u/p. The second term, the so-called baroclinic vector, modifies this ratio whenever the surfaces of constant pressure and constant density do not coincide in the fluid. The third term represents the rate of change of the ratio u/p due to diffusion of vorticity by viscous friction.
Since the vector (u/p) ■ grad u has no counterpart in the momentum equation, it warrants further discussion. Thus neglecting the baroclinic vector and the curl of the frictional force, we obtain
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