Explicit models for vertical shear and stratification

In the following, we restrict ourselves to flows with vertical shear and stratification, and assume that mean quantities are horizontally homogeneous. Under these conditions, (64) yiels $\overline{N}_1=\overline{N}_2=0$ and

$$\overline{N}_3 = \dfrac{k^2}{\epsilon^2} \dfrac{\partial {B}}{\partial {z}} = \dfrac{k^2}{\epsilon^2} N^2 \quad .$$ (67)

The velocity gradient simplifies to

$$L_{ij} = \begin{pmatrix}0 & 0 & S_U \\ 0 & 0 & S_V \\ 0 & 0 & 0 \\ \end{pmatrix} \; , \quad$$ (68)

where $S_U=\partial U / \partial z$ and $S_V=\partial V / \partial z$ are the vertical shear in $U$ and $V$, respectively.

Under these conditions, and using the conventions

$$\overline{S}_U = \dfrac{k}{\epsilon} S_U \; , \quad \overline{S}_V = \dfrac{k}{\epsilon} S_V \; , \quad \overline{N}^2 = \overline{N_3} \; , \quad$$ (69)

(59) reduces to

$$\begin{array}{rcl} {\cal N} b_{11} &=& - \left( \dfrac{a_2}{3... ...rac{1}{2} a_1 \overline{S}_V - a_5 \gamma_2 \quad . \end{array}$$ (70)

Similarly, for the mixing efficiencies, (63) yields

$$\begin{array}{rcl} {\cal N}_b \gamma_1 &=& - \dfrac{a_{b1} + ... ...3}}{3} \overline{N}^2 - a_{b4} \overline{T} \quad . \end{array}$$ (71)

In geophysical applications, a reasonable assumption is $P_b=\epsilon_b$ to elimmate the dependence of (71) on $\overline{T}$. From (158), using (62) and (66), it follows that $\overline{T}$ can be expressed in the form

$$\overline{T} = r \gamma_3 \overline{N}^2 \quad .$$ (72)

With the help of (72), the last of (71) can be re-written as

$${\cal N}_b \gamma_3 = - \dfrac{a_{b1} - a_{b2}}{2} \gamma_1 \over... ...}^2 + \dfrac{a_{b3}}{3} \overline{N}^2 - a_{b5} \gamma_3 \overline{N}^2 \quad .$$ (73)

Note that the new parameter $a_{b5} = r a_{b4}$ depends on the time scale ratio, $r$, and is, in general, not constant. Nevertheless, constant $r=c_b$ is frequently assumed (see below).

In the general case, (70) and (71) can be inverted directly to yield a solution of the form

$$b_{13} = - \dfrac{1}{2} \hat{c}_\mu \overline{S}_U \; , \quad b_{... ...line{S}_V \; , \quad \gamma_3 = \hat{c}_\mu' \overline{N}^2 - \Gamma \; , \quad$$ (74)

from which, by insertion into (70) and (71), all other quantities can be determined. Since ${\cal N}$ and ${\cal N}_b$ defined in (65) have been treated as known, the solution is not yet completely explicit. In the numerical scheme of GOTM, they are updated from their values at past time steps. By identifying

$$\nu_t = \hat{c}_\mu \dfrac{k^2}{\epsilon} \; , \quad \nu'_t = \ha... ...u' \dfrac{k^2}{\epsilon} \; , \quad \tilde{\Gamma} = \epsilon \Gamma \; , \quad$$ (75)

(74) corresponds in form exactly to (43). Note that, adopting the equilibrium assumption (72), the dependence on $\Gamma$ drops in (74). From (44) and (75), and using the definition of the dissipation rate (153), it is clear that

$$\hat{c}_\mu = (c^0_\mu)^3 c_\mu \; , \quad \hat{c}_\mu' = (c^0_\mu)^3 c'_\mu \; , \quad \quad .$$ (76)

The structure of the dimensionless parameter functions apearing in (74) is given by

$$\hat{c}_\mu = \dfrac{N_n}{D} \; , \quad \hat{c}_\mu' = \dfrac{N_b}{D} \; , \quad \Gamma = \dfrac{N_\Gamma}{D} \; , \quad$$ (77)

where the numerators and the denominator are polynomials of the square of the shear number, $\alpha_M = \overline{S}^2 = \overline{S}^2_U +\overline{S}^2_V$, the square of the buoyancy number, $\alpha_N = \overline{N}^2$, the mixed scalar, $\alpha_B = \overline{T}$, and the functions ${\cal N}$ and ${\cal N}_b$. The latter two functions depend on the production-to-dissipation ratios for $k$ and $k_b$, which for vertical shear and stratification can be written as

$$\begin{array}{rcl} \dfrac{P}{\epsilon} &=& - 2 b_{13} \overli... ...silon} \dfrac{\overline{N}^2}{\overline{T}} \end{array} \quad .$$ (78)

Once $k$ and $k_b$ (and their dissipation ratios, $\epsilon$ and $\epsilon_b$) are known, also the time scale ratio $r$ defined in (66) can be computed, and the problem can be solved. Different possibilities to derive these quantities are discussed in the following.