Another form of the vorticity equation is derived by Chamberlain and Hunten (1987, p. 106, Eq. 2.6.4) in isobaric coordinates for steady, slow motions in a frictionless atmosphere with acceleration terms and quadratic terms being negligible:
where Vp is the gradient along isobars; and V =(u, v, 0). Again, applying the continuity equation, this relation implies that du/dp = 0, where u = dp/dt is the equivalent of vertical velocity in pressure coordinates. This implies that u = dp/dt equals a constant and the only physical solution is that u = 0 and thus that, to a first approximation, all steady motions in a rotating atmosphere with zero viscosity must be two-dimensional (i.e., barotropic). This is a fundamental conclusion and is equivalent to the Taylor-Proudman theorem in hydrodynamics that we shall come across again later.
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