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).

$$\varepsilon _{i,j}^{n+1} = \varepsilon _{i,j}^{N} + dt \left\{ [\mbox{DKADV}]_{i,j}^{n} + [\mbox{DKDIF}]_{i,j}^{N} + [\mbox{DKNLD}]_{i,j}^{N} \right.$$

$$+ \left. [\mbox{DKBP}]_{i,j}^{n} + [\mbox{DKSP}]_{i,j}^{n} - \frac{C_{\epsilon}}{l}(\varepsilon _{i,j}^{N})^{\frac{3}{2}} \right\}$$ (41)

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

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

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

$$\mbox{DKBP}_{i,j}^{n} = -\frac{g}{\Theta _{0,j}}K_{i,j}^{n} \frac{1}{\Delta z_{j}}\left[ ... ...+\frac{1}{2}}) - (\theta _{i,j-\frac{1}{2}}+\Theta _{0,j-\frac{1}{2}}) \right],$$ (43)

$$\mbox{DKSP}_{i,j}^{n} = 2K_{i,j}^{n}\left[ \left( \frac{u_{i+\frac{1}{2},j}^{n}-u_{i-\fra... ...i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}}{\Delta z_{j}} \right)^{2} \right]$$

$$+ \frac{2}{3}\varepsilon _{i,j}^{n} \left[ \frac{u_{i,j+\frac{1}{... ..._{j}} + \frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]$$

$$+ K_{i,j}^{n} \left[ \frac{u_{i,j+\frac{1}{2}}^{n}-u_{i,j-\frac{1... ...} + \frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]^{2}$$ (44)

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

3.2 Surface flux parameterization

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

$$F_{u,i} = - \rho _{0}C_{D,i}\vert u_{i,\frac{1}{2}}\vert u_{i,\frac{1}{2}},$$ (45)

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

where

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

The bulk Richardson number is calculated as follows.

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