SECT. 5] SURGES 623 



Besides the equations of motion we have the equation of continuity, whicli 

 after vertical integration yields 



8lh dUy ^ _dC 



dx^ dy ~ dt' ^ ' 



Finally we have, at boundaries of the sea area considered, boundary condi- 

 tions of two sorts : 



(i) along a closed boundary (coast) the normal component Un of the volume 

 transport vector U vanishes, U n — ; 



(ii) along an open boundary ^ or C/^ or a quantitative relation between 

 them is given. 



As for the zero level of z, it is supposed that 2 = is the sea-surface level in 

 the case of no astronomical (tidal) and atmospheric effects being present, so 

 that l,{x, y, t) may be considered as a disturbance of sea-surface height. 



Of the surface and bottom stress vectors the first one is mainly determined 

 by the wind velocity W (defined at a certain height above the sea surface). 

 Strictly speaking one should use the relative velocity W — uo, where uo denotes 

 the surface current velocity, which might be expressed by means of U and 

 h-{-t,; we shall, however, discard this detail here and assume Xa to be roughly 

 proportional to W^ x air density, the proportionality factor being dependent 

 on the vertical stability of the air mass and also, to some degree, on W itself. 



The bottom stress t^ must be expressed by means of the dependent variables 

 U and I, and of Xa in order to make the two-dimensional equations accessible to 

 mathematical treatment. The physically most simple current model for this 

 purpose is one which uses a vertically constant eddy viscosity (see, e.g., 

 Welander, 1957). That this assumption fails, however, is most easily seen in the 

 case of zero volume transport (U = 0) and negligible transverse currents (%;^0, 

 Ta2/ = 0). As is well known, the stress exerted by the bottom would in this case, 

 under the above assumption, become 0.5 x the surface stress, whereas in reality, 

 in turbulent flow, it is mostly less than 0.1 x the surface stress (see e.g. Schal- 

 kwijk, 1947; Hunt, 1956; Weenink, 1958). We shall denote the bottom stress 

 present in this case (U = 0) by t^'^^ and assume (see Bowden, 1953) 



Tft(O) = — mTa(w<|l). 



If a volume transport is present, tj, may in fair approximation be written as the 

 sum of Tb^°^ and a stress that is roughly proportional to pu'^, or : 



T. = -mT, + ^^^^, (6) 



where the proportionality factor s may still depend somewhat on the depth 

 (e.g., a formula with s proportional to (A-f^)-'/^ has been used by Freeman, 

 Baer and Jung, 1957) and on the degree of turbulence (influenced by wind and 

 tidal currents). This formula presupposes a vertical current profile which, 



