386 General Theory of Ocean Currents in a Homogeneous Sea 



from the surface of the figure backwards gives rise to a uniform current from left to 

 right; at first there will be an equilibrium in it between the gradient and Coriolis forces. 

 If the depth of the sea increases in the current direction (bottom slopes downward) 

 then for a constant flow amount, since the current cross-section becomes larger, there 

 must be a decrease in velocity. The equilibrium between the two forces will be disturbed, 

 the lower velocity attained will correspond to a smaller Coriolis force and the current 

 will be deflected contra solem. However, if the depth decreases (i.e. the bottom rises) 

 the velocity must increase; this will give an increase in the Coriolis force and a deflec- 

 tion of the current cum sole. The equihbrium state of equation (XIII.4) will continue 

 for each stream line only when the current follows the depth lines of the bottom. 

 If the depth is variable, (XII. 16) will be replaced by the continuity equation 



di (dhu 8hv\ 



Under stationary conditions the equations of motion (XIII.4) will then give the con- 

 dition 



8h dC 8h dl 



This relation states that if the depth varies then steady frictionless currents are only 

 possible if the topography of the sea surface on a relative scale accords with that of the 

 sea bottom. The currents must thus run parallel to the bathymetric curves; the strength 

 of the current is, however, free and depends only on the absolute gradient of the 

 ^-values. If there are coastal limits, the boundary condition requires that the depth 

 should be constant along the outer boundary (the coast). 



Since the continuity equation for currents in an ocean partly or completely covering 

 the spherical Earth has a diff'erent form (equation (X.27), the conditions for steady 

 currents will also be different. The equations of motion for the meridional and zonal 

 velocity components will now be {R = Earth radius, & = 90° — ^ = zenith distance): 



g ^l g^l 



U = — 75-^ 5 ^Y ^"^ ^' = fD aQ - (XIII. 10) 



fR sm § 8A fR dd 



For a variable depth // and taking into account that h is always small compared with 

 7?, the continuity equation will have the form 



dl 1 Idh sin du dhv\ 



The condition for a frictionless steady current is then under these conditions 



8h 8i 8h 8t 8t 



The first two terms are identical with the condition for planar co-ordinates (equation 

 XIII.9); they thus include only the efl'ects of variable depth. The third term 

 h tan d{8t,j8X) takes into account the eff'ect of the spherical shape of the Earth; it is 

 largest in the equatorial regions (§ close to 90°) and vanishes at the poles {d = 0°). 

 Some special cases can be selected to illustrate the two efl"ects. 



(1) If the depth of the sea is constant, the conditional equation is satisfied only if 

 8l,j8X = 0, i.e., only if zo«a/ currents are possible (along latitude circles). 



