V. NEARSHORE CIRCULATION SYSTEMS 



No theoretical methods existed before 1969 to predict mean water sur- 

 face and current variations in the nearshore region when circulation cells 

 and rip currents were present. Only a relatively crude model of rip currents 

 was available as devised by Arthur (1962). For steady flow, neglecting bot- 

 tom frictional resistance, lateral mixing, and the driving force terms (ra- 

 diation stress gradients), Arthur (1962) showed that the nonlinear, convection 

 acceleration terms in the vorticity transport equation increase with increase 

 in water depth. This translated into a narrowing and strengthening of a sea- 

 wind current, i.e., a rip current model near the root. Bowen (1967, 1969b) 

 followed many of the general assumptions suggested by Arthur (1962) , except 

 he included the resistance and driving force terms. The analytic model of 

 nearshore cell circulation and rip currents incorporating radiation stress 

 principles as devised by Bowen (1967, 1969b) became the forerunner for all 

 subsequent efforts. Central in the theory are the equations employed. 



1. Fundamental Equations of Motion in Two Dimensions . 



The conservation of mass, momentxmi, and energy equations for waves super- 

 imposed on a current was first given by Whitham (1962) 3'*. The complete 

 derivations can be found in Liu and Mei (1975) or Phillips (1977)^^. The 

 water is assumed homogeneous and incompressible of constant density. Coriolis 

 accelerations are neglected. The velocity field is assumed independent of 

 the water depth so that only two-dimensional (horizontal) motion is allowed. 

 Most importantly, the velocity field is considered as the sum of the mean and 

 wave-induced variation. Turbulence due to random velocity fluctuations at 

 scales far below the wave motions is neglected. 



The applicable equations are derived by averaging the Navier- Stokes dif- 

 ferential motion equations and the mass equation over depth and over time, in 

 that order. Battjes (1974a) has a detailed discussion of the averaging pro- 

 cess. A key ingredient is the Liebnitz rule for differentiation of a definite 

 integral. The resulting equations have appeared in two different forms in the 

 literature. 



a. Eulerian Form . As used first by Bowen (1967, 1969b), the equations 

 are written in terms of the depth-averaged Eulerian mean velocities u, v, and 

 mean water surface deviation, n. The accelerations form to the total or sub- 

 stantive derivative, meaning the equations are in Eulerian form. 



Mass 



8n , , 8u , - 8h 8v . - 9h „ , ,- .qq\ 



-r— + h-r-+u^- + h-r-+v^— = h=d+Ti (98) 



dt dX dX dy dy 



^'^WHITl'IAN, G.B., "Mass, Momentum and Energy Flux in Water Waves," Journal 

 of Fluid MeohanicSt Vol. 12, 1962, pp. 135-147 (not in bibliography) . 

 PHILLIPS, O.M. , The Dynamics of the Upper Ocean^ Cambridge University 

 Press, Cambridge, Mass., 2d ed. , 19 77 (not in bibliography). 



114 



