SECT. 5] SURGES 631 



If, finally, the field of x is irrotational, we are left with 



V2</. = 0, 



which, together with (30), implies that ^ is constant all over the area, so that 

 U = in this special case and C = ^t, according to (2.")). 



In the case of a sea area having one or more open boundaries, things become 

 much more complicated. Since we are dealing with the case of quasi-equilibrium, 

 we need not here consider time-dependent boundary conditions. 



If the normal component Un of U is given along the boundary, this means 

 that the tangential derivative ('(f)/ rs is given: in other words, that the variation 

 of (f> along that boundary is known. 



If C is given along the open boundary, either as a known function <^b{s) of 

 place (s) along the boundary or as a constant, as in equation (13), this means 

 that the tangential derivative (i/cs along the boundary is given, or: 



k — + i— = — - — = r]{s), 

 ex cy (is 



where ]c = dxfds, l = dylds and ry(s) is a known function of 5. By (22) and (23) this 

 condition becomes a condition for </> of the following general form : 



A-|.x| = m ,32) 



where A', L and F are known functions of s (the place coordinate along the 

 boundary), K and L being constants if h is constant along the boundary. 



If the open boundary is a narrow opening which acts only as a "leak", we 

 have a boundary condition of the type of equation (14) where the transport 

 Tab through the opening may now be written : 



Tab 

 so that we have 



= f Unds = (f>i,-(f)A, (33) 



Ja 



<^B = (^A + A(^*AB-^**), (34) 



where ^a may be put equal to zero, if we like. 



As an example we take a model sea as shown in Fig. 9. Along the line CD, 

 where the sea communicates freely with the deep open ocean, it is supposed 

 that ^ = (see section 3-B-a), which means that condition (32) becomes: 



f^ + rh-'^^ + p-Wx ^ alongCD, (35) 



ex <^y 



where the a:-axis is assumed to have the direction of CD. 



At the opening AB condition (34) holds, ^*ab and ^** having been defined 

 at the end of section 3-B-a. 



The current field may be seen as consisting of three components : (a) a current 

 "caused" by curl t, according to the first term of the right-hand member of 



