A. 圧力方程式 (3.9) の左辺の空間微分の書き下し

appendix-a

(3.9) 左辺の変形を行う.

$$\Deqref{uwpi:sabun}\mbox{left side} = \pi^{\tau + \Delta \tau}_{i,k}$$

$$- \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...right)_{k(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,k(w)} \right\}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...t)_{k-1(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,k-1(w)} \right\}$$

$$= \pi^{\tau + \Delta \tau}_{i,k}$$

$$- \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...elta \tau}_{i,k+1} - \pi^{\tau + \Delta \tau}_{i,k}}{\Delta z} \right) \right\}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...elta \tau}_{i,k} - \pi^{\tau + \Delta \tau}_{i,k-1}}{\Delta z} \right) \right\}$$

$$= \left\{ - \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b... ...} \bar{\theta}_{v}^{2} \right)_{k(w)} \right\} \pi^{\tau + \Delta \tau}_{i,k+1}$$

$$+ \left[ 1 + \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}... ...theta}_{v}^{2} \right)_{k-1(w)} \right\} \right] \pi^{\tau + \Delta \tau}_{i,k}$$

$$+ \left\{ - \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{... ...bar{\theta}_{v}^{2} \right)_{k-1(w)} \right\} \pi^{\tau + \Delta \tau}_{i,k-1}.$$ (A.1)

A.1 下部境界

下部境界($k(w) = 0(w)$)について考える. この時 (3.7) 式は,

$$\beta \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,0(w)} = \left[ \left( \DP{(\alpha Div)^{\tau}}{z} \right) - (1 - \beta) \... ...) + \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right) \right]_{i,0(w)}$$

$$\equiv E_{i,0(w)}$$ (A.2)

となるので, (3.9) 式の左辺は, $k = 1$ の場合には,

$$\Deqref{uwpi:sabun}\mbox{left side} = \pi^{\tau + \Delta \tau}_{i,1}$$

$$- \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...right)_{1(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,1(w)} \right\}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...right)_{0(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,0(w)} \right\}$$

$$= \pi^{\tau + \Delta \tau}_{i,1}$$

$$- \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...\Delta \tau}_{i,2} - \pi^{\tau + \Delta \tau}_{i,1}}{\Delta z} \right) \right\}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...right)_{0(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,0(w)} \right\}$$

$$= \left\{ - \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b... ...ho} \bar{\theta}_{v}^{2} \right)_{1(w)} \right\} \pi^{\tau + \Delta \tau}_{i,2}$$

$$+ \left\{ 1 + \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}... ...ho} \bar{\theta}_{v}^{2} \right)_{1(w)} \right\} \pi^{\tau + \Delta \tau}_{i,1}$$

$$+ \beta \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b... ...z} \left( \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} \right)_{0(w)} E_{i,0(w)}$$

A.2 上部境界

上部境界($k(w) = km(w)$)について考える. (3.9) 式の左辺は,

$$\beta \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,km(w)} = \left[ \left( \DP{(\alpha Div)^{\tau}}{z} \right) - (1 - \beta) \... ... + \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right) \right]_{i,km(w)}$$

$$\equiv E_{i,km(w)}$$ (A.3)

となるので, (3.9) 式の左辺は, $k(w) = km(w)$ の場合には,

$$\Deqref{uwpi:sabun}\mbox{left side} = \pi^{\tau + \Delta \tau}_{i,km}$$

$$- \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ...ght)_{km(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,km(w)} \right\}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}... ..._{km-1(w)} \left( \DP{\pi^{\tau + \Delta \tau}}{z} \right)_{i,km-1(w)} \right\}$$

$$= \pi^{\tau + \Delta \tau}_{i,km}$$

$$- \beta \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b... ... \left( \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} \right)_{km(w)} E_{i,km(w)}$$

$$+ \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}} {\bar{c_{p... ...ta \tau}_{i,km} - \pi^{\tau + \Delta \tau}_{i,km-1}}{\Delta z} \right) \right\}$$

$$= \left\{ 1 + \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{... ...\bar{\theta}_{v}^{2} \right)_{km-1(w)} \right\} \pi^{\tau + \Delta \tau}_{i,km}$$

$$+ \left\{ - \beta^{2} \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{... ...ar{\theta}_{v}^{2} \right)_{km-1(w)} \right\} \pi^{\tau + \Delta \tau}_{i,km-1}$$

$$- \beta \left( \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b... ... \left( \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} \right)_{km(w)} E_{i,km(w)}$$