SECT. 5] SURGES 629 



If then only closed boundaries are present, it is clear that equations (15), (16), 

 (17) and the boundary condition are satisfied by 



U^ = Uy = 0, 



9P 

 where pi is determined by the condition that the integral of C over the whole 

 area vanishes, so that pi here is the average oi po over the area. 



If the area has one or more open boundaries, the solution may be different 

 from (18) because of currents being forced across an open boundary and 

 through the sea area considered. Also, if the sea-level height is prescribed along 

 an open boundary (see section 3-B-a), this will only be compatible with the 

 dynamical equations if currents are present there to compensate the atmos- 

 pheric pressure gradient force at the boundary. The solution of the problem 

 may then be constructed as the superposition of a pure pressure effect, described 

 by (18), and a pure current effect, determined by equations (15) and (16) with 

 zeros on their right-hand sides. The value of the integration constant ^i in 

 formula (18) will now, in general, not be the average of ^o over the sea area 

 considered but will depend on the open-boundary conditions. The current field, 

 determining the current effect on ^, is itself determined by the boundary 

 conditions and equations (15), (16) and (17). The last equation is equivalent 

 with the statement that U can be expressed by means of a stream function (f), 

 as follows : 



U. = -I U, = |. ,19, 



From equations (15) and (16), t, may be eliminated by dividing by h and 

 cross-differentiating. We shall not here enter further into the problem of 

 finding the current field, since this problem will come back in a more general 

 form in connection with the wind effect. 



(u ) Wind effect 



Suppose now a wind-stress field Ta{x, y) to be present. Leaving the pressure 

 effect, which may be superposed on the wind effect, out of consideration, we 

 have the equations 



^¥x - f^~^Uy-sh-WU:c + p-Hl+m)h-^rax, (20) 



g^^ -fh-Wx-sh-^UUy + p-\l+m)h-^ray, (21) 



which follow from (15) and (16) with (7), or 



g^ = fh-Wy-rh-^U^ + p~Hl+m)h-Wa., (22) 



g^= -fh-W:c-rh-Wy + p-Hl+m)h-Way, (23) 



if the linear bottom stress formula (10) is used. 



