By Melvin E. Stern (Eds.)
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3b) can be linearized. 7. 9) and where i denotes an eastward pointing unit vector. - admissible for motions having large lateral scale L because w 4 IV1. 6) is Ik 252 x VI - lg Vhl This vertical component of the Coriolis force tends to contribute a surface pressure of order Jdz k * 2C2 x V Igh Vhl - and the horizontal gradient of this is of order L-'lgh Vhl. 5) is the asymptotic shallow water equation for a sphere. 4. 9) cannot be neglected. 60 111. DENSITY C U R R E N T S References Benjamin, T.
Let us first examine the behavior of a rotating fluid when disturbed by an infinitesimal velocity field V . 1) by replacing d V / d t with aV/at. Let @‘(x,y , z, t ) denote the perturbed value of @ in a Cartesian system with the z axis pointing in the direction of w, and let f = 21wl denote the Coriolis parameter. The x, y , z components of relative velocity are again denoted by u, u, w, and the corresponding components of 2w x V are then given by -fu, fu, 0. 2. 5) by taking the x derivative of the first equation and t h e y derivative of the second equation, and then adding the results.
Another limiting case is provided by a liquid having a free surface and resting on the spherical surface of a nonrotating planet. Here the direction of the plumb line will obviously determine the dynamical vertical, for those motions whose horizontal scale R is large compared to the mean layer thckness H. But in the more general case, we have t o consider the “competition” between the rotation rate f2 and the local force of gravity gk. In all previous considerations of this chapter, f2 is parallel to gk, and thus the two forces “cooperate” in defining the dynamical vertical.