### Equilibrium states

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

 (81)

which is polynomial equation in and . This expression can be used to replace one of the latter two variables by the other. An interesting consequence is the fact that all non-dimensional turbulent quantities can be expressed in terms of the Richardson number only. Replacing by in (81), a quadratic equation for in terms for can be established (see e.g. Cheng et al. (2002). Using the definitions given in section 4.26, this equation can be written as

 (82)

The solution for can, via (81), be used to expressed also in terms of . This implies that also the stability functions and hence the complete solution of the problem only depends on .

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

 (83)

Solutions of this equation for some popular models are given in table 1. For , equilibrium models predict complete extinction of turbulence. For non-equilibrium models solving dynamical equations like (152), however, has no direct signifcance, because turbulence may be sustainned by turbulent transport and/or the rate term.

Table 1: Critical Richardson number for some models
 GL78 KC94 CHCD01A CHCD01B CCH02

Karsten Bolding 2012-12-28