(1) Hydrodynamic Equations . Equations that describe the storm surge 

 generation processes are the continuity equation expressing conservation of 

 mass and the equations of motion expressing Newton's second law. The deriva- 

 tions are not presented here; references are cited below. The equations of 

 motion and continuity given here represent a simplification of the more 

 complete equations. A more simplified form is obtained by vertically integra- 

 ting all governing equations and then expressing everything in terms of either 

 the mean horizontal current velocity or volume transport. Vertically 

 integrated equations are generally preferred in storm surge calculations since 

 interest is centered in the free-surface motion and mean horizontal flow. 

 Integration of the equations for the storm surge problem are given by Haurwitz 

 (1951), Welander (1961), Fortak (1962), Platzman (1963), Reid (1964), and 

 Harris (1967). 



The equations given here are obtained by assuming 



(a) Vertical accelerations are negligible 



(b) Curvature of the earth and effects of surface 

 waves can be ignored 



(c) The fluid is inviscid 



(d) The bottom is fixed and impermeable 



The notation and the coordinate scheme employed are shown schematically in 

 Figure 3-56. D is the total water depth at time t , and is given by D = d + 

 S , where d is the undisturbed water depth and S the height of the free 

 surface above or below the undisturbed depth resulting from the surge. The 

 Cartesian coordinate axes, x and y , are in the horizontal plane at the 

 undisturbed water level, and the z axis is directed positively upward. 

 The X axis is taken normal to the shoreline (positive in the shoreward 

 direction), and the y axis is taken alongshore (positive to the left when 

 facing the shoreline from the sea) ; is the angle of wind measured 

 counterclockwise from the x axis; W is windspeed. 



The differential equations appropriate for tropical or extratropical storm 

 surge problems on the open coast and in enclosed and semienclosed basins are 

 as follows: 



If + -^"+ ^= fv - gD -p + gD 1^ + gD ^^ + ^ - ^ + W P (3-77) 

 dt 3x dy dx 3x 3x p p x 



0) CO <u 



U-l Oi (U CO CO JJ 



O coo ViT3rHC0<Un) 



B -H 1-1 OldJ-HoJ 0) u oi 



CO >H CO MjJH-H U 4-1 



04-I O U 01 U U W r-{ 



•HC -H <U CUe-CtO rH 



u <u M cj >ooa) em 



Og O cfl CUS-i*-! 13 o^w 



ojo o 14-1 Mn!4-io c; 4-1 c 



>S! U CQ«)PL,-H4-I-H 



13 o <; 13 o m 



9V ^^ VV ^^ o:y ^ 3S dE, 3c '^ sy "^ by 



-~ + -^^+ -5—^= - fU - gD -^ + gD ^ + gD -7^ + + W, P (3-78) 



9t 3y 3x '' 3y " 3y " 3y p p y ^ ■' 



3-119 



