2 大気モデル

2.1 運動方程式

第 1 部に示した式(1)$\sim$(3)は以下のように変形した後に離散化する.

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

ただし,

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

である. このように変形するのはモデル上下端において $\pi$ の境界条件を時 間変化させないようにするためである.

これらを以下のように離散化する. 移流項 $\mbox{D[UVW]ADV}$ は保存型と移流 型の混合型で計算する. 摩擦項 $[\mbox{D[UVW]VIS}]_{i+\frac{1}{2},j}^{N}$, $[\mbox{D[UVW]NLV}]_{i+\frac{1}{2},j}^{N}$ の時間積分は前進差分, その他の 項の積分は leap frog スキームと前進差分の組み合わせを用いて行う. 圧力項 $\hat{P}$ の導出について第2.3節を参照.

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

移流項 $[\mbox{DTADV}]_{i,j}^{n}, [\mbox{DTAD0}]_{i,j}^{n}$ は 4 次中央 差分で空間離散化する. 摩擦項 $[\mbox{DTDIF}]_{i,j}^{N}$, $[\mbox{DTDI0}]_{i,j}^{N}$ と放射加熱項 $Q_{rad,i,j}^{N}$ と散逸加熱項 $Q_{dis,i,j}^{N}$ の時間積分は常に前進差分を用いて行う. 放射加熱項の計算方法は第5節を参照.

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

$$+ \left. [\mbox{DTADV}]_{i,j}^{n} + [\mbox{DTAD0}]_{i,j}^{n} + [\mbox{DTDIF}]_{i,j}^{N} + [\mbox{DTDI0}]_{i,j}^{N} \right\}$$ (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 圧力診断式

圧力の診断式()を第2.1節で行った変形 にあわせて以下のように変形する.

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

これを dimension-reduction 法を用いて解く. 上の式を適当に離散化すると,

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

と書くことができる. ただし $\Dvect{P}, \Dvect{D_{x}}, \Dvect{D_{z}}, \Dvect{S}$ はそれぞれ,

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

を離散化した行列である. 行列 $\Dvect{D_{z}}$ の固有値を $\lambda_{i}$, 固有ベクトルを $V_{i}(z)$ とし, 固有列ベクトルを並べた行列を $\Dvect{V}$, 対角成分が $\lambda_{i}$ である行列を $\Dvect{\Lambda }$ とすると, $\Dvect{D_{z}V} = \Dvect{V\Lambda }$ となる. $\Dvect{P} = \Dvect{V}\cdot \Dvect{H}$ と展開すると,

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

したがって,

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

となる.

固有値を $\lambda_{i}$ と固有ベクトルを $V_{i}(z)$ を求めるために必要な $z$ 方向の係数行列 $\Dvect{D}_{z}$ は

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

を差分化して与える. 最初の微分は 2 次中央差分, 次の微分は 4 次中央差分で 解く. これは連続の式は 4 次中央差分, 圧力の微分は 2 次中央差分であること による. よって係数行列は 5重対角行列になる. 係数行列の成分を $A_{ij}$ と すると以下のようになる.

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

境界条件は上下壁で $\DP{}{z}=0$ となるようにする.

$x$ 方向についても適当な固有関数に展開して各モードに対する展開係数 を求める. ここでは三角関数で展開する.

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

温度分布 $T_{j}$ を与え, 静水圧平衡の式と状態方程式を用いて $P_{0,j}$ と $\rho _{0,j}$ を計算する.

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

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

$P_{0,j}, \rho _{0,j}$ を求めた後 $\Pi _{0,j}, \Theta _{0,j}$ を計算する.

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

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