Algebraic Models

The key assumptions in deriving algebraic models have been formulated by Rodi (1976) and Gibson and Launder (1976). These authors suggested to simplify the right hand sides of (46) and (49) according to

$\displaystyle \dot{ \rule{0mm}{4mm} \langle u'_i u'_j \rangle} - {\cal D}_{ij} ...
...{k} - {\cal D}_k }{k} + \dfrac{\dot{k}_b - {\cal D}_b }{k_b} \right) \; , \quad$ (58)

which are reasonable approximations in many situations. Moreover, (58) can be shown to hold exactly in stably stratified, homogeneous shear flows, when the flow approaches the so-called weak-equilibrium limit, see Shih et al. (2000). Using (58) and the pressure-strain model (51), it can be shown after some algebra that the transport equations for the momentum flux (46) reduces to

$\displaystyle {\cal N} b_{ij} = - a_1 \overline{S}_{ij} - a_2 \overline{\Sigma}_{ij} - a_3 \overline{Z}_{ij} - a_4 \overline{N}_{ij} - a_5 \overline{\Gamma}_{ij}$ (59)

in dimensionless form. The $ a_i$ relate to the coefficients used in (51) according to $ a_1 = 2/3 - c_2/2$, $ a_2=1 - c_3/2$, $ a_3=1 - c_4/2$, $ a_4=c_5/2$, and $ a_5=1/2 - c_6/2$. The dimensionless, traceless and symmetric tensors appearing on the right hand side of (59) are defined as

$\displaystyle \overline{S}_{ij} = \dfrac{k}{\epsilon} S_{ij} \; , \quad \overli...
...} \Sigma_{ij} \; , \quad \overline{Z}_{ij} = \dfrac{k}{\epsilon} Z_{ij} \quad .$ (60)

Additionally,

$\displaystyle \overline{\Gamma}_{ij} = \Gamma_{ij} / \epsilon = \begin{pmatrix}...
...{pmatrix} \; , \quad \gamma_i = - \dfrac{ {\langle u'_i b' \rangle} }{\epsilon}$ (61)

has been introduced in (59) for convenience. Here, the $ \gamma_i$ correspond to the mixing efficiencies in each coordinate direction, respectively. Note, that the vertical component,

$\displaystyle \gamma_3 = \gamma = - \dfrac{ {\langle w'b' \rangle} }{\epsilon} ...
...ac{G}{\epsilon} = \dfrac{R_f}{1-R_f} \; , \quad R_f = - \dfrac{G}{P} \; , \quad$ (62)

can be identified with the classical mixing efficiency used in many studies of stratified fluids.

Most authors proceed know in deriving, with the help of (58), a dimensionless equation for the normalised turbulent buoyancy flux, $ \zeta_i = {\langle u'_i b' \rangle} / \sqrt{(k k_b)}$, see So et al. (2002), Jin et al. (2003). It can be shown, however, that the resulting algebraic equations alternatively can be expressed, without further assumptions, in the form of equations for the mixing efficiencies,

$\displaystyle {\cal N}_b \gamma_{i} = - a_{b 1} \overline{S}_{ij} \gamma_j - a_...
...dfrac{1}{3} a_{b 3} \overline{N}_{i} - a_{b 4} \overline{T} \delta_{i3} \quad .$ (63)

Since efficiencies $ \gamma_i$ are the primary variables appearing on the right hand side of (59) through the presence of the tensor $ \overline{\Gamma}_{ij}$ defined in (61), and since they are variables with a clear physical interpretation, we prefer (63) to a mathematicall equivalent equation for the normalised buoyancy flux, $ \zeta_i$.

The new dimensionless quantities entering the problem via (63) are

$\displaystyle \overline{N}_i = \dfrac{k^2}{\epsilon^2} \dfrac{\partial {B}}{\partial {x_i}} \; , \quad \overline{T} = \dfrac{k k_b}{\epsilon^2} \quad .$ (64)

Note that the vertical component of $ \overline{N}_i$ can be identified with the square of the buoyancy frequency, $ N^2$, made dimensionless with the dynamic dissipation time scale $ \tau=k/\epsilon$.

(59) and (63) are linear in $ b_{ij}$ and $ \gamma_i$, with a non-linear coupling introduced by the terms

\begin{displaymath}\begin{array}{rcl} {\cal N} &=& \dfrac{P+G}{\epsilon} + \dfra...
...} \left( \dfrac{P_b}{\epsilon_b} -1 \right) \quad . \end{array}\end{displaymath} (65)

The production-to-dissipation ratios appearing in these expression are exclusively related to known quantities and thus introduce no new independent variables. However, the time scale ratio,

$\displaystyle r = \dfrac{k_b}{\epsilon_b} \dfrac{\epsilon}{k}$ (66)

needs to be described.

(59) and (63) are a system of 9 coupled algebraic equations for the anisotropies $ b_{ij}$ and the mixing efficiencies $ \gamma_i$, depending solely on the non-dimensional tensors $ \overline{S}_{ij}$, $ \overline{W}_{ij}$, the vector $ \overline{N}_{i}$, and the scalar $ \overline{T}$. This system is linear, if $ {\cal N}$ and $ {\cal N}_b$ are treated as knowns and if the nonlinear term $ N_{ij}$ in (59) is neglected, $ a_4=0$. No closed solution of the complete system in three dimensions has been reported so far in the literature. Nevertheless, separate solutions in three dimensions for (59) and (63), respectively, have been reported (see Jin et al. (2003) and the references therein).

In geophysical applications, the system (59) and (63) can be considerably simplified by assuming that the fluid is horizontally homogeneous (boundary layer approximation), and closed solutions can be obtained (see Cheng et al. (2002)). The procedure to obtain such solutions is discussed in the following subsection.

Karsten Bolding 2012-01-24