640 GROEN AND GROVES [CHAP. 17 



The numerical techniques to be used here approximate the differential 

 equations by difference equations, the space and time intervals of which have 

 to be chosen in such a way as to guarantee sufficient computational stability. 

 We shall not enter into further technical details of these methods here. 



C. Surges along an Open Coast 



Let us first consider the nature of surges in the open ocean far from a coast. 

 It is instructive to take an idealized region of the ocean the mean surface of 

 which is a plane. Except in the very shallowest regions of the ocean the bottom 

 stress Tb is very small and can be neglected, and the surface displacement t, is 

 very small compared to the depth h. Then, linearizing equations (3), (4) and (5), 

 and neglecting the tidal potential, gives 



^+f\ixV + ghV:,,yt, = l{-hV:c,yP0 + Ta) (50) 



ct p 



Vx,2/-U + ^ = 0. 



8t 



Now let us first consider the case of constant depth. Either U or ^ can be 

 eliminated from these equations. Eliminating the former, we obtain 



^/^V..,2__/2 ___._, (51) 



where F is the forcing function, given by 

 1 



F = 

 P 



h yx,y^P0 + ^x,y''Ca +f I CUrle Ta dt 

 



(52) 



A solution for free barotropic waves is obtained by setting F equal to zero. 

 One such solution is a progressive wave form 



t, = A cos [k{x cos 6 + y sin 6) — at], 



where 6 is an arbitrary direction toward which the wave form is propagated, 

 and k and a are related according to 



a2 = ghk^+f^. (53) 



It is seen from (53) that |a| is always greater than/; i.e. this type of wave 

 motion can occur only if its period is less than half a pendulum day. The 

 orbital motion associated with this surface-wave form does not lie in a vertical 

 plane perpendicular to the crests, as it would on a rotationless earth. 



For forced waves it is interesting to compare the effects of atmospheric 

 pressure and horizontal wind stress. The horizontal forces arising from the 

 gradient of the atmospheric pressure are conservative ; i.e. there is no curl. 

 The wind stress, on the other hand, does have a curl, which exerts an integrated 

 effect on the water motion. If steady winds were to blow, the forcing function 

 would increase without limit, and so would the elevation and volume transport. 

 But this results from having neglected dissipation. In the real ocean, friction 



