2. Storm Surge Equations . 



The vertically integrated form of the quasi-linear long wave 

 equations in a Cartesian system are well known (e.g., Welander, 

 1961) . The appropriate forms for storm surge computation in the 

 stretched shelf coordinate system are as follows: 



3Q C 



at 



f Qn 



Fp 3S* 



(H - H_) 



(40) 



3Q n 



3t 



fQ«^ 



gJJ 3_ 



Fv 3T* 



(H - HJ 



lr r* 



(41) 



and 



3H 

 3t 



-? L,. cic* (PQc*J 



■u 3S 



1_ 3 



v 3T* 



(FQ T J] 



(42) 



where Q is the volume transport per unit width, t is the wind 

 stress divided by the water density, a is the bottom resistance 

 stress divided by water density, f is the Coriolis parameter, D 

 is depth of water, H is the sea surface elevation relative to mean 

 sea level, and Hg is the hydrostatic elevation of the sea surface 

 corresponding to the atmospheric pressure anomaly. The term F 'is 

 a variable scale factor associated with the conformal part of the 

 orthogonal curvilinear coordinate system. Specifically, 



F = [(If) * 



l 3? ; J 



(43) 



where x and y are given by the transformation equations. The 

 terms p and v , are additional scale factors that relate to the 

 transformation of g to S* and n to T* , respectively, and 

 are given by: 



3S 



(44) 



and 



= ? n dS n 3T 



3S 3T 3T* 

 n 



(45) 



Furthermore, F 2 yv corresponds to the Jacobian of the transforma- 

 tion in the sense that 



A = f f F 2 yv dS* dT J 

 R* 



(46) 



65 



