622 GROEN AND GROVES [CHAP. 17 



and Tzy will not be taken into consideration in the present theory. Therefore, 

 Tzx and Tzy will simply be written as tx and Ty, together forming the horizontal 

 vector T. Assuming static equilibrium in the vertical, we have 



Jz 

 ^x,yP = ^x,yP0 + gp0^x,y^+ gV x, y p{z') dz' , 



where po = atmospheric pressure at the sea surface, t, = height of the sea surface 

 above zero level, and V^;, 2/ = horizontal component of the gradient operator. 



The above equations, together with the equation of continuity and the 

 boundary conditions, also form the basis of the theory of wind-driven sea 

 currents, which is dealt with extensively elsewhere in this book (Section 3). For 

 the present theory of surges we shall concentrate our attention mainly on the 

 phenomena in more or less shallow sea areas and shall neglect differences of 

 density (/> = const.), thus confining ourselves to quasi-barotropic systems of 

 motion. Furthermore, we shall make the problem a two-dimensional one by 

 integrating the equations of motion vertically. For this purpose, the acceleration 

 terms may now with a sufficient degree of accuracy be approximated by 



8ux — dUx — dUx dux 

 and 



dUy dUy dUy dUy 



where the bar over a symbol means the vertical average of the quantity con- 

 cerned. Integrating them vertically from the bottom, z= —h, to the surface, 

 z = ^, and dividing the result by the depth, H = h + t„ we obtain : 



d/ Ux\ ^ Ux d / U„ , 



et, 1 dpo dq Tax - Tbx .ON 

 ^ 8x p 8x 8x p{h + C) 



8t\h + CJ h + C 8x\h + C 



N Uy\^'Ux8IU_^ , 



8t\h + ^J h + C 8x\h + t, 



81 1 8pQ 8q Tgy-Tby 



h + C ^ 8y p 8y 8y^ p{h + Q' ^ ^ 



where U = udz={h + Qu = volume transport, 



J-h 



Xa = surface stress vector ( = the stress exerted by the atmosphere on 

 the sea), 



Tb — bottom stress vector ( = the stress exerted by the sea on the 

 bottom). 



