N. V. M. Odd, Wallingford, UK and J. G. Rodger, Reading, UK
The contributors carried out research and devised a method of predicting vertical mixing in stratified flows in the 1970s,25 which was based in part on the work of Kent and Pritchard.26 Our mixing length method25 requires a minimum amount of calibration provided the numerical scheme resolves the necessary details of the flow and does not cause an unacceptable degree of numerical smoothing.
The author's equation (9) that was used to evaluate the coefficient of eddy viscosity, Az, appears to be unrealistically insensitive to the Richardson number, Ri. Kent and Prichard26 would have used (1 + βRi)−2 rather than (1 + βRi)−1/2. In that case, the best empirical value for b is about 10, not 0·75. The parameter a is equivalent to the square of the Von Karman constant (κ = 0·4) and is expected to have a value of 0·16 and not 0·015.
It would seem that Hsu et al.17,19 calibrated the model using a rather basic turbulence closure model. If one wishes to use Az as a time-varying variable, we would recommend
where the local gradient Richardson number, Ri, has an upper limit of 1 (Az is insensitive to Ri values > 1). This equation gives Az values similar to those quoted by the author if one makes assumptions about the depth of flow.
However, it is much better to use a mixing length representation as follows
where
Because the mixing length, lm, is a measure of the vertical eddy size, it is filtered and controlled in size by the effect of the halocline (maximum density gradient) at which Ri has its maximum value, Rimax. This means that the momentum mixing length, lm, in stratified tidal flows in estuaries is often almost constant in size over the whole depth of flow (except for the bed boundary layer) and equal to the size predicted at the depth where, and if, Ri has a maximum value. If Ri does not have a maximum within the water depth, lm varies with the local Richardson number.
One other unexpected aspect of the author's paper is the assumption that Kz is nearly equal to Az in a stratified tidal flow. The paper assumes that the turbulent Schmidt number (Kz/Az or lc/lm) is constant at 5/6 = 0·83.
The contributors' research25 showed lc/lm falling from a value of 1 for a local Richardson number Ri=0 to a value of about 0·1 for Ri=1. The data fitted an earlier theory of Ellison for conditions in the atmosphere27 which, when modified, gave
The discussers found the best-fit value for the critical flux Richardson number, Rfc,to be 0·08. The turbulent solute flux is then given by
where c is the concentration for the solute (or suspended matter).
