Based on limited studies (Sternberg, 1972), constant value of C^j on the 

 order of 0.004 have been used in the past. Recent studies of tidal currents 

 in a shallow estuary found C^j to be on the order of 0.035 (Brown and 

 Trask, 1980). A more rigorous way of determining C^ is to utilize the 

 Monin-Obukhov similarity relationships valid within the constant flux layer. 

 If z is within the constant flux layer above the bottom, C^j can be shown to 

 be (Sheng, 1980): 



-2 



0. = k^ 



in(zi/Zo) + ♦s(z^/L) 



(2.20) 



where k is the von Karman constant, Zq is the roughness height, and ^^ is a 

 stability function characterizing the stability of the bottom boundary layer 

 (Businger et al . 1971; Lewellen, 1977), and L=u /kga^w'T' is the 

 Monin-Obukhov similarity length, where Oy is the coefficient of volumetric 



expansion and w'T' is the vertical heat flux. 



It can be shown that the stability may increase (unstable case) or 

 decrease (stable case) the drag coefficient by as much as 40%. Indeed, this 

 definition of C^ should also be used within the constant flux layers below the 

 free surface as well as above the air-sea interface. Huang and Sloss (1981) 

 used a Richardson-number-dependent C^jg suggested by Deardorff (1968) in a 

 study of the monthly mean circulation in Lake Ontario. 



Others (Forristal et al . , 1977; Blumberg and Mellor, 1981) used linear 

 bottom stress laws instead of the quadratic stress law (2.18). It should be 

 pointed out that the bottom stress is relatively unimportant in deep waters. 

 In relatively shallow coastal waters, however, the bottom stress can be of the 

 same order of magnitude as the wind stress. In such cases, numerically 

 modeled currents are quite sensitive to the bottom stress formula used in the 

 model. The no-slip condition, (u,v,w) = 0, has been used by many modelers, 

 but is valid only if the laminar sublayer is adequately resolved by an 

 extremely fine vertical (< 1 cm) grid in the numerical model. This will be 

 discussed further. 



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