Some authors use simplifying assumptions to derive more compact forms of the expressions for the solution in (76). In the following, a few examples, which are special cases of the general solution discussed here, are reviewed.
In deriving their version of the general solution (76), Canuto et al. (2001) e.g. assumed and constant . Under these conditions, because of (74), the dependence on dissapears, and the counter-gradient term in (76) drops. It was further assumed that in (67) only, leading to and . These particularly simple expressions linearize the system, and a fully explicit solution can be obtained, provided and are known. Burchard and Bolding (2001) adopted the solution of Canuto et al. (2001) and complemented it by and computed from dynamical equations (`- model').
In contrast, Canuto et al. (2001) and Cheng et al. (2002) decided for a further simplification. They solved (76) with and from algebraic expressions. In their case, followed from the approximation of (152) (see section 4.17), and from a prescribed length-scale.
Using (76), (79), and (80), it is easy to show that the assumption leads to
Investigating the solution of the quadratic equation (82), it can be seen that becomes infinite if the factor in front of vanishes. This is the case for a certain value of the Richardson number, , following from
Karsten Bolding 2012-12-28