2 Atmospheric model

2.1 Equation of motion

Before making finite difference equations, equation (1)$\sim$(3) show in Part I are transformed as follows.

$$\begin{eqnarray*} \DP{u}{t} &=& -\DP{\hat{P}}{x} + \alpha , \\ \DP{v}{t} &=& \beta , \\ \DP{w}{t} &=& -\DP{\hat{P}}{z} \end{eqnarray*}$$

where,

$$\begin{eqnarray*} \hat{P} &\equiv& c_{p}\Theta _{0}\pi - \gamma , \\ \alpha &... ... - w\DP{w}{z} + g\frac{\theta}{\Theta _{0}} + D(w)\right) \Dd z \end{eqnarray*}$$

In this formulation, the time dependence of upper and lower boundary conditions are disappeared.

The advection terms $\mbox{D[UVW]ADV}$ are evaluated by the combination scheme of flux and advection forms. The time integration is performed by the forward scheme for the friction term $[\mbox{D[UVW]VIS}]_{i+\frac{1}{2},j}^{N}$, $[\mbox{D[UVW]NLV}]_{i+\frac{1}{2},j}^{N}$, and combination of the leap-frog and forward scheme for the other terms. The calculation method of pressure term $\hat{P}$ are shown in 第2.3節.

$$u_{i+\frac{1}{2},j}^{n+1} = u_{i+\frac{1}{2},j}^{N} + dt \left\{ \frac{\hat{P}_{i+1,j}-\hat{P}_{i,j}}{\Delta x} + \alpha _{i+\frac{1}{2},j} \right\},$$ (1)

$$v_{i+\frac{1}{2},j}^{n+1} = v_{i+\frac{1}{2},j}^{N} + dt \beta _{i+\frac{1}{2},j}^{n},$$ (2)

$$w_{i,j+\frac{1}{2}}^{n+1} = w_{i,j+\frac{1}{2}}^{N} + dt \frac{\hat{P}_{i,j+1}-\hat{P}_{i,j}}{\Delta z_{j+\frac{1}{2}}}.$$ (3)

$$N = \left\{ \begin{array}{lcl} n-1 & \mbox{for} & \mbox{l... ... \Delta t & \mbox{for} & \mbox{forward}. \end{array} \right.$$ (4)

$$\alpha _{i+\frac{1}{2},j} = - ( \gamma _{i+1,j} - \gamma _{i,j}) + [\mbox{DUADV}]_{i+\frac{1}... ... + [\mbox{DUPRS}]_{i+\frac{1}{2},j}^{n} + [\mbox{DUCOLI}]_{i+\frac{1}{2},j}^{n}$$

$$+ [\mbox{DUVIS}]_{i+\frac{1}{2},j}^{N} + [\mbox{DUNLV}]_{i+\frac{1}{2},j}^{N} ,$$ (5)

$$\beta _{i+\frac{1}{2},j}^{n} = [\mbox{DVADV}]_{i+\frac{1}{2},j}^{n} + [\mbox{DVCOLI}]_{i+\frac{1... ... + [\mbox{DVVIS}]_{i+\frac{1}{2},j}^{N} + [\mbox{DVNLV}]_{i+\frac{1}{2},j}^{N},$$ (6)

$$\gamma _{i,j} = \sum _{j'=0}^{j} \left( [\mbox{DWADV}]_{i,j'-\frac{1}{2}}^{n} + [... ...{DWVIS}]_{i,j'-\frac{1}{2}}^{N} + [\mbox{DWNLV}]_{i,j'-\frac{1}{2}}^{N} \right.$$

$$\left. + [\mbox{BUOY}]_{i,j'-\frac{1}{2}}^{n} \right)\Delta z_{j'-\frac{1}{2}}$$ (7)

$$\mbox{DUADV}_{i+\frac{1}{2},j}^{n} = - \frac{1}{\Delta x}\left\{ \left(u_{i+1,j}^{n}u_{i+1,j}^{n} - u_{i-1,j}^{n}u_{i-1,j}^{n}\right) \right.$$

$$+ \left. \left( \rho _{0,j+\frac{1}{2}} u_{i+\frac{1}{2},j+\frac{... ...\frac{1}{2},j-\frac{1}{2}}^{n} \right) \Delta x\right/(\rho _{0,j}\Delta z_{j})$$

$$\left. - u_{i+\frac{1}{2},j}^{n} \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0}\right) _{i+\frac{1}{2},j}^{n} \right\},$$ (8)

$$\mbox{DVADV}_{i+\frac{1}{2},j}^{n} = - \left\{ \left(v_{i+1,j}^{n}u_{i+1,j}^{n} - v_{i-1,j}^{n}u_{i-1,j}^{n}\right) \right.$$

$$+ \left. \left( \rho _{0,j+\frac{1}{2}} v_{i+\frac{1}{2},j+\frac{... ...\frac{1}{2},j-\frac{1}{2}}^{n} \right) \Delta x\right/(\rho _{0,j}\Delta z_{j})$$

$$\left. \left. - v_{i+\frac{1}{2},j}^{n} \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0}\right)_{i+\frac{1}{2},j}^{n} \right\} \right/ \Delta x,$$ (9)

$$\mbox{DWADV}_{i,j+\frac{1}{2}}^{n} = - \frac{1}{\Delta x}\left\{ \left(w_{i+\frac{1}{2},j+\frac{1}{2}}... ...ac{1}{2},j+\frac{1}{2}}^{n} u_{i-\frac{1}{2},j+\frac{1}{2}}^{n} \right) \right.$$

$$+ \left. \left( \rho _{0,j+1}w_{i,j+1}^{n}w_{i,j+1}^{n} - \rho _{... ...^{n} \right) \Delta x \right/ (\rho _{0,j+\frac{1}{2}}\Delta z_{j+\frac{1}{2}})$$

$$\left. - w_{i,j+\frac{1}{2}}^{n} \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0,j}\right) _{i,j+\frac{1}{2}}^{n} \right\},$$ (10)

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

$$+ \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}} \left[ \left. \r... ...1}^{n}-u_{i+\frac{1}{2},j}^{n}\right) \right/\Delta z_{j+\frac{1}{2}} - \right.$$

$$\left. \left. \left. \rho _{0,j-\frac{1}{2}} K_{i+\frac{1}{2},j-\... ...{i+\frac{1}{2},j-1}^{n}\right) \right/\Delta z_{j-\frac{1}{2}} \right]\right\},$$ (11)

$$\mbox{DVVIS}_{i+\frac{1}{2},j}^{n} = \frac{1}{(\Delta x)^{2}} \left\{ \left[ K_{i+1,j}^{n} \left(v_{i+... ...n} \left(v_{i+\frac{1}{2},j}^{n}-v_{i-\frac{1}{2},j}^{n}\right) \right] \right.$$

$$+ \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}} \left[ \left. \r... ...1}^{n}-u_{i+\frac{1}{2},j}^{n}\right) \right/\Delta z_{j+\frac{1}{2}} - \right.$$

$$\left. \left. \left. \rho _{0,j-\frac{1}{2}} K_{i+\frac{1}{2},j-\... ...{i+\frac{1}{2},j-1}^{n}\right) \right/\Delta z_{j-\frac{1}{2}} \right]\right\},$$ (12)

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

$$+ \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}} \left[ \left. \r... ...j+1}^{n} \left(w_{i,j+1}^{n}-w_{i,j}^{n}\right) \right/\Delta z_{j+1} - \right.$$

$$\left. \left. \left. \rho _{0,j}K_{i,j}^{n} \left(w_{i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}\right) \right/\Delta z_{j-1} \right] \right\}.$$ (13)

$$\mbox{DUNLV}_{i+\frac{1}{2},j}^{n} = \left\{ \left[ \left(u_{i+\frac{3}{2},j}^{n}-u_{i+\frac{1}{2},j}^... ...t(u_{i+\frac{1}{2},j+1}^{n}-u_{i+\frac{1}{2},j}^{n} \right)^{3} \right. \right.$$

$$\left. \left. \left. - \left(u_{i+\frac{1}{2},j}^{n}-u_{i+\frac{1... ...] \right\} \right/ (16.0 \cdot 10^{3} \cdot \rho _{0,j}\Delta z_{j}/\Delta x ),$$ (14)

$$\mbox{DVNLV}_{i+\frac{1}{2},j}^{n} = \left\{ \left[ \left(v_{i+\frac{3}{2},j}^{n}-v_{i+\frac{1}{2},j}^... ...t(v_{i+\frac{1}{2},j+1}^{n}-v_{i+\frac{1}{2},j}^{n} \right)^{3} \right. \right.$$

$$\left. \left. \left. - \left(v_{i+\frac{1}{2},j}^{n}-v_{i+\frac{1... ...] \right\} \right/ (16.0 \cdot 10^{3} \cdot \rho _{0,j}\Delta z_{j}/\Delta x ),$$ (15)

$$\mbox{DWNLV}_{i,j+\frac{1}{2}}^{n} = \left\{ \left[ \left(w_{i+1,j+\frac{1}{2}}^{n}-w_{i,j+\frac{1}{2}... ...ft(w_{i,j+\frac{1}{2}}^{n}-w_{i-1,j+\frac{1}{2}}^{n}\right)^{3} \right] \right.$$

$$\left. \left. + 0.1 \left[ \left(w_{i,j++\frac{3}{2}}^{n}-w_{i,j+... ...,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}\right)^{3} \right] \right\} \right/$$

$$(16.0 \cdot 10^{3} \cdot \rho _{0,j+\frac{1}{2}} \Delta z_{j+\frac{1}{2}}/\Delta x ),$$ (16)

$$\mbox{BUOY}_{i,j+\frac{1}{2}}^{n} = \frac{g}{\Theta _{0,j+\frac{1}{2}}} \theta _{i,j+\frac{1}{2}}^{n},$$ (17)

$$\mbox{DUCOLI}_{i+\frac{1}{2},j}^{n} = - f v_{i+\frac{1}{2},j}^{n},$$ (18)

$$\mbox{DVCOLI}_{i+\frac{1}{2},j}^{n} = + f u_{i+\frac{1}{2},j}^{n}.$$ (19)

$$\begin{eqnarray*} && u_{i+\frac{1}{2},j+\frac{1}{2}}^{n} = 0.5\left(u_{i+\frac... ...w_{i,j}^{n}} {\rho _{0,j+\frac{1}{2}}\Delta z_{j+\frac{1}{2}}}. \end{eqnarray*}$$

2.2 Thermodynamics equation

The advection terms $[\mbox{DTADV}]_{i,j}^{n}, [\mbox{DTAD0}]_{i,j}^{n}$ of equation (5) in Part I are evaluated by forth order centered scheme. In time integration, the forward scheme is adapted for the friction terms $[\mbox{DTDIF}]_{i,j}^{N}$, $[\mbox{DTDI0}]_{i,j}^{N}$, the radiative heating term $Q_{rad,i,j}^{N}$ and the dissipative heating term $Q_{dis,i,j}^{N}$. The calculation method of radiative heating term is shown in 第5節.

$$\theta _{i,j}^{n+1} = \theta _{i,j}^{N} + dt \left\{ \frac{\Theta _{0,j}}{T_{0,j}}(Q_{rad,i,j}^{N} + Q_{dis,i,j}^{N}) \right.$$

$$+ \left. [\mbox{DTADV}]_{i,j}^{n} + [\mbox{DTAD0}]_{i,j}^{n} + [\mbox{DTDIF}]_{i,j}^{N} + [\mbox{DTDI0}]_{i,j}^{N} \right\}$$ (20)

$$Q_{dis,i,j}^{N} = \frac{C_{\epsilon}}{lc_{p}} (\varepsilon _{i,j}^{N})^{\frac{3}{2}},$$ (21)

$$\mbox{DTADV}_{i,j}^{n} = - \left\{ \frac{1}{\rho _{0,j}\Delta x}\... ...c{1}{2},j)}^{n} + \frac{1}{24}F\theta _{x(i-\frac{3}{2},j)}^{n} \right] \right.$$

$$+ \left. \frac{1}{\rho _{0,j}\Delta z_{j}}\left[ - \frac{1}{24}F\... ...c{1}{2})}^{n} + \frac{1}{24}F\theta _{z(i,j-\frac{3}{2})}^{n} \right] \right\},$$ (22)

$$F\theta _{x(i+\frac{1}{2},j)}^{n} = \rho _{0,j}u_{i+\frac{1}{2},j... ...^{n} + \frac{9}{16}\theta _{i,j}^{n} - \frac{1}{16}\theta _{i-1,j}^{n} \right),$$

$$F\theta _{z(i,j+\frac{1}{2})}^{n} = \rho _{0,j+\frac{1}{2}}w_{i,j... ...^{n} + \frac{9}{16}\theta _{i,j}^{n} - \frac{1}{16}\theta _{i,j-1}^{n} \right).$$

$$\mbox{DTAD0}_{i,j}^{n} = - \left\{ \frac{1}{\rho _{0,j}\Delta x}\... ...(i-\frac{1}{2},j)}^{n} + \frac{1}{24}F_{x(i-\frac{3}{2},j)}^{n} \right] \right.$$

$$+ \left. \frac{1}{\rho _{0,j}\Delta z_{j}}\left[ - \frac{1}{24}F\... ...c{1}{2})}^{n} + \frac{1}{24}F\Theta _{0,z(j-\frac{3}{2})}^{n} \right] \right\},$$ (23)

$$F_{x(i+\frac{1}{2},j)}^{n} = \rho _{0,j}u_{i+\frac{1}{2},j}^{n},$$

$$F\Theta _{0,z(j+\frac{1}{2})}^{n} = \rho _{0,j+\frac{1}{2}}w_{i,j... ...eta _{0,j+1} + \frac{9}{16}\Theta _{0,j} - \frac{1}{16}\Theta _{0,j-1} \right).$$

$$\mbox{DTDIF}_{i,j}^{n} = \frac{1}{(\Delta x)^{2}} \left[ \tilde{K}_{i+\frac{1}{2},j}^{n}(\... ...ilde{K}_{i-\frac{1}{2},j}^{n}(\theta _{i,j}^{n} -\theta _{i-1,j}^{n}) \right] +$$

$$\frac{1}{\rho _{0,j}\Delta z_{j}} \left[ \rho _{0,j+\frac{1}{2}} ... ...c{ (\theta _{i,j}^{n} -\theta _{i,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}} \right],$$ (24)

$$\mbox{DTDI0}_{i,j}^{n} = \frac{1}{\rho _{0,j}\Delta z_{j}... ...{n} -\Theta _{0,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}} \right],$$ (25)

$$\tilde{K}_{i+\frac{1}{2},j}^{n} = 0.5(\tilde{K}_{i+1,j}^{... ...{2}}^{n} = 0.5(\tilde{K}_{i,j+1}^{n} +\tilde{K}_{i,j}^{n} ).$$

2.3 Diagonostic equation of pressure function

The diagnostic equation of nodimensional pressure function is solved by using the dimension reduction method. Before making the finite difference equation, equation (9) show in Part I is transformed as follows.

$$\left(\DP[2]{}{x} +\frac{1}{\rho _{0}}\DP{}{z}\rho _{0}\DP{... ...\rho _{0}}\Ddiv \rho _{0}\Dvect{v}\right) + \DP{}{x}\alpha ,$$

The finite difference form of the pressure equation can be written in matrix form as follows.

$$\Dvect{D_{x}P} + \Dvect{D_{z}P} = \Dvect{S}$$ (26)

where $\Dvect{P}, \Dvect{D_{x}}, \Dvect{D_{z}}, \Dvect{S}$ are matrixes whose elements are finite difference form of following terms.

$$\hat{P}, \quad \DP[2]{}{x}, \quad \frac{1}{\rho _{0}}\DP{... ...frac{1}{\rho _{0}}\Ddiv \Dvect{v}\right) + \DP{}{x}\alpha ,$$

(26) can be rewritten by using the eigenvalue matrix $\Dvect{\Lambda }$ and the eigenvector matrix $\Dvect{V}$ of $\Dvect{D_{z}}$.

$$\Dvect{V}\Dvect{D_{x}H} + \Dvect{V\Lambda H} = \Dvect{S}.$$

where $\Dvect{D_{z}V} = \Dvect{V\Lambda }$ and $\Dvect{P} = \Dvect{V}\cdot \Dvect{H}$. The final form of matrix equation is as follows.

$$\left(\Dvect{D_{x}} + \Dvect{\Lambda }\right) \Dvect{H} = \Dvect{V}^{-1}\Dvect{S},$$ (27)

In calculating elements of matrix $\Dvect{D}_{z}$, the vertical derivative in

$$\frac{1}{\rho _{0}}\DP{}{z}\rho _{0}\DP{}{z} \pi$$

are evaluated by the second and forth order centered schemes because the space differencing in the continuity equation is evaluated by the forth order centered scheme while that in the pressure gradient term is evaluated by the second order centered scheme. Therefore $\Dvect{D}_{z}$ is represented as a band matrix whose elements $A_{i,j}$ are given as follows.

$$A_{i,i+2} = -\frac{1}{\rho _{0}\Delta z_{i}}\left( \frac{1}{24\Delta z_{i+\frac{3}{2}}}\frac{(\rho _ {0,i+2}+\rho _{0,i+1})}{2}\right),$$ (28)

$$A_{i,i+1} = \frac{1}{\rho _{0}\Delta z_{i}}\left( \frac{1}{24\Delta z_{i+\fra... ...rac{9}{8\Delta m_{i+\frac{1}{2}}}\frac{(\rho _ {0,i+1}+\rho _{0,i})}{2}\right),$$ (29)

$$A_{i,i} = - \frac{1}{\rho _{0}\Delta z_{i}}\left( \frac{9}{8\Delta zm_{i+\f... ...rac{9}{8\Delta z_{i-\frac{1}{2}}}\frac{(\rho _ {0,i}+\rho _{0,i-1})}{2}\right),$$ (30)

$$A_{i+1,i} = \frac{1}{\rho _{0}\Delta z_{i}}\left( \frac{1}{24\Delta z_{i-\fra... ...rac{9}{8\Delta z_{i-\frac{1}{2}}}\frac{(\rho _ {0,i}+\rho _{0,i-1})}{2}\right),$$ (31)

$$A_{i+2,i} = -\frac{1}{\rho _{0}\Delta z_{i}}\left( \frac{1}{24\Delta z_{i-\frac{3}{2}}}\frac{(\rho _ {0,i-1}+\rho _{0,i-2})}{2} \right).$$ (32)

The boundary conditions are $\DP{}{z}=0$ at the lower and upper boundary.

The horizontal differencing is evaluated by using the Fourier expansion.

$$\Dvect{H} = \sum_{k=1}^{NX/2-1} \left[\Dvect{H}\right]_{k_{x}},$$ (33)

$$\Dvect{V}^{-1}\Dvect{S} = \sum_{k=1}^{NX/2-1} \left[\Dvect{V}^{-1}\Dvect{S}\right]_{k_{x}},$$ (34)

$$\left(\Dvect{D_{x}} + \Dvect{\Lambda }\right) = \sum_{k_{x}=1}^{NX/2-1} \left[\Dvect{D_{x}} + \Dvect{\Lambda }\right]_{k_{x}}$$ (35)

$$\left[\Dvect{H}\right]_{k_{x}} = \left[\Dvect{D_{x}} + \D... ...ht]_{k_{x}}^{-1} \left[\Dvect{V}^{-1}\Dvect{S}\right]_{k_{x}}$$ (36)

2.4 Basic state equations

The basic state pressure ($P_{0,j}$) and density ($\rho _{0,j}$) are calculated by the hydrostatic equation and the equation of state given the basic state temperature $T_{0,j}$.

$$\ln P_{0,j} = \ln P_{00} - \sum _{j=1}^{j}\frac{g}{RT_{0,j}}\Delta z_{j},$$ (37)

$$\rho _{0,j} = \frac{P_{0,j}}{RT_{0,j}}.$$ (38)

$\Pi_{0,j}$, $\Theta _{0,j}$ are calculated by using $P_{0,j}$, $\rho _{0,j}$ as follows.

$$\Pi _{0,j} = \left(\frac{P_{0,j}}{P_00}\right)^{\kappa },$$ (39)

$$\Theta _{0,j} = \frac{T_{0,j}}{\Pi _{0,j}}.$$ (40)