Equilibrium states

Some authors use simplifying assumptions to derive more compact forms of the expressions for the solution in (74). 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 (74), Canuto et al. (2001) e.g. assumed $P_b=\epsilon_b$ and constant $r$. Under these conditions, because of (72), the dependence on $\overline{T}$ dissapears, and the counter-gradient term $\Gamma_B$ in (74) drops. It was further assumed that $P+G=\epsilon$ in (65) only, leading to ${\cal N} = (c_1 + c_1^*)/2$ and ${\cal N}_b=c_{b1}$. These particularly simple expressions linearize the system, and a fully explicit solution can be obtained, provided $k$ and $\epsilon$ are known. Burchard and Bolding (2001) adopted the solution of Canuto et al. (2001) and complemented it by $k$ and $\epsilon$ computed from dynamical equations (`$k$-$\epsilon$ model').

In contrast, Canuto et al. (2001) and Cheng et al. (2002) decided for a further simplification. They solved (74) with $k$ and $\epsilon$ from algebraic expressions. In their case, $k$ followed from the approximation $P+G=\epsilon$ of (150) (see section 4.7.30), and $\epsilon$ from a prescribed length-scale.

Using (74), (77), and (78), it is easy to show that the assumption $(P+G)/\epsilon$ leads to

$$N_n \overline{S}^2 - N_b \overline{N}^2 - D = 0 \; , \quad$$ (79)

which is polynomial equation in $\overline{S}$ and $\overline{N}$. 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 $Ri = \overline{N}^2 / \overline{S}^2$ only. Replacing $\overline{N}^2$ by $\overline{S}^2 Ri$ in (79), a quadratic equation for $\alpha_M = \overline{S}^2$ in terms for $Ri$ can be established (see e.g. Cheng et al. (2002). Using the definitions given in section 4.7.39, this equation can be written as

$$\alpha_M^2 \left( -d_5 + n_2 - \left( d_3 - n_1 + n_{b2}\right) R... ..._M \left( -d_2 + n_0 - \left( d_1 + n_{b0} \right) Ri \right) - d_0 = 0 \quad .$$ (80)

The solution for $\alpha_M$ can, via (79), be used to expressed also $\overline{N}^2$ in terms of $Ri$. This implies that also the stability functions and hence the complete solution of the problem only depends on $Ri$.

Investigating the solution of the quadratic equation (80), it can be seen that $\alpha_M$ becomes infinite if the factor in front of $\alpha_M^2$ vanishes. This is the case for a certain value of the Richardson number, $Ri=Ri_c$, following from

$$-d_5 + n_2 - \left( d_3 - n_1 + n_{b2}\right) Ri_c - \left( d_4 + n_{b1} \right) Ri_c^2 = 0 \quad .$$ (81)

Solutions of this equation for some popular models are given in table 1. For $Ri=Ri_c$, equilibrium models predict complete extinction of turbulence. For non-equilibrium models solving dynamical equations like (150), however, $Ri_c$ 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
$0.47$	$0.24$	$0.85$	$1.02$	$0.96$