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3 Turbulent parameterization

3.1 Subgrid turbulent mixing parameterization

The space differencing in the turbulent kinetic energy equation (equation (10) in Part I) is evaluated by the forth order centered scheme for advection terms and the second order centered scheme for other terms. In time integration, the forward scheme is adapted for the friction terms. Representations of $[\mbox{DKADV}]_{i,j}^{n}$ and $[\mbox{DKDIF}]_{i,j}^{N}$ are same as those of (22) and (24).

$\displaystyle \varepsilon _{i,j}^{n+1}$ $\textstyle =$ $\displaystyle \varepsilon _{i,j}^{N} + dt \left\{
[\mbox{DKADV}]_{i,j}^{n} +
[\mbox{DKDIF}]_{i,j}^{N} +
[\mbox{DKNLD}]_{i,j}^{N} \right.$  
    $\displaystyle + \left.
[\mbox{DKBP}]_{i,j}^{n} +
[\mbox{DKSP}]_{i,j}^{n} -
\frac{C_{\epsilon}}{l}(\varepsilon _{i,j}^{N})^{\frac{3}{2}} \right\}$ (41)


$\displaystyle \mbox{DKNLD}_{i,j}^{n}$ $\textstyle =$ $\displaystyle \frac{1}{(\Delta x)^{2}}
\left[
K_{NLD,i+\frac{1}{2},j}^{n}(\vare...
...{i,j}^{n} -\varepsilon_{i-1,j}^{n})
\right] + \frac{1}{\rho _{0,j}\Delta z_{j}}$  
    $\displaystyle \left[ \rho _{0,j+\frac{1}{2}}
K_{NLD,i,j+\frac{1}{2}}^{n}
\frac{...
...epsilon_{i,j}^{n}-\varepsilon_{*i,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}}
\right],$ (42)

\begin{eqnarray*}
K_{NLD,i+\frac{1}{2},j}^{n}
&=&
\mbox{MIN}
\left[ K_{NLD...
... /2000, \\
K_{NLD,max} &=& 0.2\frac{(\Delta x)^{2}}{\Delta t}.
\end{eqnarray*}


$\displaystyle \mbox{DKBP}_{i,j}^{n}$ $\textstyle =$ $\displaystyle -\frac{g}{\Theta _{0,j}}K_{i,j}^{n}
\frac{1}{\Delta z_{j}}\left[
...
...+\frac{1}{2}}) -
(\theta _{i,j-\frac{1}{2}}+\Theta _{0,j-\frac{1}{2}}) \right],$ (43)
$\displaystyle \mbox{DKSP}_{i,j}^{n}$ $\textstyle =$ $\displaystyle 2K_{i,j}^{n}\left[
\left(
\frac{u_{i+\frac{1}{2},j}^{n}-u_{i-\fra...
...i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}}{\Delta z_{j}}
\right)^{2}
\right]$  
    $\displaystyle + \frac{2}{3}\varepsilon _{i,j}^{n} \left[
\frac{u_{i,j+\frac{1}{...
..._{j}} +
\frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]$  
    $\displaystyle + K_{i,j}^{n}
\left[
\frac{u_{i,j+\frac{1}{2}}^{n}-u_{i,j-\frac{1...
...} +
\frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]^{2}$ (44)


\begin{displaymath}
u_{i,j+\frac{1}{2}}^{n} =
0.5\left(u_{i+\frac{1}{2},j+\fra...
...rac{1}{2}}^{n} +
w_{i+\frac{1}{2},j-\frac{1}{2}}^{n}\right).
\end{displaymath}

3.2 Surface flux parameterization

The finite difference form of the surface flux are as follows.

$\displaystyle F_{u,i}$ $\textstyle =$ $\displaystyle - \rho _{0}C_{D,i}\vert u_{i,\frac{1}{2}}\vert u_{i,\frac{1}{2}},$ (45)
$\displaystyle F_{\theta,i}$ $\textstyle =$ $\displaystyle \rho _{0}C_{D,i}\vert u_{i,\frac{1}{2}}\vert(T_{sfc,i}-T_{i,1}).$ (46)

where


\begin{displaymath}
C_{D,i} = \left\{
\begin{array}{lcl}
C_{Dn}\left( 1 -
\f...
...B,i})^{2}}& for & \mbox{Ri}_{B,i} \geq 0,
\end{array}\right.
\end{displaymath} (47)

The bulk Richardson number is calculated as follows.

\begin{displaymath}
\mbox{Ri}_{B,i} \equiv \frac{gz_{1}(\Theta _{sfc,i}-\Theta _
{i,1})}{\overline{\Theta }_{0,1}u_{i,\frac{1}{2}}}.
\end{displaymath} (48)


next up previous
: 4 Dust transport : Two dimensional anelastic model : 2 Atmospheric model
Odaka Masatsugu 平成19年4月26日