SECT. 5] SURGES 625 



may now find new, and we hope better, approximations ^<2), Ux^^K Uy'''^^ ', and 

 so on. If the non-Hnear residues are small enough, as compared with the linear 

 terms in the left-hand members of the equations, we may expect the iterative 

 procedure to converge rather rapidly. (This method has been applied by 

 Proudman, 1955, and Schonfeld, 1955.) 



The above procedure makes it also possible to investigate the effects of 

 coupling between astronomical and atmospheric effects. Since the first approxi- 

 mation is a solution of a set of linear differential equations, we can write it as 

 the sum of an astronomical component and an atmospheric component : 



U(l) = Uas<l)+Uat<l). 



Substituting these expressions in X'l) and YC^^i we obtain expressions which 

 may be written as 



yd) = ras<l)+Fat(l)+ras,at<l), 



where Xas.at^^^ and Fas.at^^^ stand for the sums of all terms that are products 

 of both astronomical and atmospheric factors, thus causing coupling effects. 

 Consequently, when writing Xd) and F^^) in this way it is possible to write 

 the second approximation of t, and U as the sum of a purely astronomical com- 

 ponent, a purely atmospheric component and a mixed component containing 

 the coupling effects : 



^(2) = ^as(2) + U(2) + Uat<2), 

 U(2) = Uas(2)+Uat(2)+Uas,at(2). 



In the following treatment we shall mainly concern ourselves with the 

 linearized equations, which are in many cases sufficiently adequate for studying 

 atmospherically induced surges, and shall ignore the coupling effects mentioned 

 above. 



In shallow-sea areas, where strong tidal currents prevail, these may even be 

 of advantage for the linearization of the bottom stress terms in the dynamical 

 equations. It has been shown (Staatscommissie Zuiderzee, 1926; Bowden, 

 1953) that in this case the non-periodic component of the bottom stress, 

 caused by a drift current superimposed on the tidal currents, is roughly pro- 

 portional to the strength of the drift current, the factor of proportionality 

 being itself proportional to the amplitude of the tidal current (which is supposed 

 to be sufficiently large relative to the drift current). In treating the effects of 

 atmospheric origin separately we may, therefore, in such cases, for the bottom 

 stress, use the following linearized formula 



lb = -mia + rpVlh, (10) 



where r is a friction parameter which may depend on x and y and has to be 

 determined empirically (cf. Bowden, 1956; Reid, 1956a; Weenink, 1958). 



21— s. I 



