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\Dtitle{2 次元非静力学モデルの離散化}
\Dauthor{杉山耕一朗, 小高正嗣, 北守太一}
\Ddate{2006 年 8 月 18 日}




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$\displaystyle + g \frac{\sum [q_{v}]_{i,k(w)}/M_{v}}{1/M_{d}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6540}%
$\displaystyle - g \frac{\sum [q_{v}]_{i,k(w)} + \sum [q_{c}]_{i,k(w)} + \sum [q_{r}]_{i,k(w)}}
{1 + \sum [\bar{q_{v}}]_{i,k(w)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6543}%
$\displaystyle \DP{\pi_{i,k}}{t}
+ \frac{\overline{c}_{i,k}^{2}}{{c_{p}}_{d}
\overline{\rho}_{i,k} [\overline{\theta_{v}}^{2}]_{i,k}}
\left[
\DP{\overline{\rho} \overline{\theta_{v}} u}{x} +
\DP{\overline{\rho} \overline{\theta_{v}} u}{z}
\right]_{i,k}
= 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline703}%
$\overline{c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath532}%
\begin{displaymath}
  \overline{c}_{i,k}^{2} = \frac{{c_{p}}_{d} R_{d}}{c_{v}}
                  \overline{\pi}_{i,k} [\overline{\theta_{v}}]_{i,k}.
 \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6548}%
$\displaystyle \DP{\theta_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6550}%
$\displaystyle - u_{i,k}\left[\DP{\theta}{x}\right]_{i,k}
- w_{i,k}\left[\DP{\theta}{z}\right]_{i,k}
- w_{i,k}\left[\DP{\overline{\theta}}{z}\right]_{i,k}
+ \left[{\rm Turb}.{\theta}\right]_{i,k}
+ \left[{\rm Turb}.{\overline{\theta}}\right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6551}%
$\displaystyle + \Dinv{\overline{\pi}_{i,k}}
\left([Q_{cnd}]_{i,k} + [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6554}%
$\displaystyle \DP{[q_{v}]_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6556}%
$\displaystyle - u_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
- w_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
- w_{i,k} \left[\DP{\bar{q_{v}}}{x} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6557}%
$\displaystyle + [{\rm Src}.q_{v}]_{i,k}
+ [{\rm Turb}.{\overline{q_{v}}}]_{i,k} + [{\rm Turb}.{q_{v}}]_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6558}%
$\displaystyle \DP{[q_{c}]_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6560}%
$\displaystyle - u_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
- w_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
+ [{\rm Src}.q_{c}]_{i,k} + [{\rm Turb}.{q_{c}}]_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6561}%
$\displaystyle \DP{[q_{r}]_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6563}%
$\displaystyle - u_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
- w_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
+ [{\rm Src}.q_{r}]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6564}%
$\displaystyle + [{\rm Fall}.q_{r}]_{i,k}
+ [{\rm Turb}.{q_{r}}]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline805}%
$1(u)\sim im(u)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline807}%
$2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6573}%
$\displaystyle u_{0(u), k} = u_{im(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6574}%
$\displaystyle u_{-1(u), k} = u_{im-1(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6575}%
$\displaystyle u_{im+1(u), k} = u_{1(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6576}%
$\displaystyle u_{im+2(u), k} = u_{2(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline809}%
$k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline811}%
$-1 \leq k \leq km + 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6584}%
$\displaystyle u_{0(u),k} = u_{im(u),k} = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6585}%
$\displaystyle u_{-1(u),k} = - u_{1(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6586}%
$\displaystyle u_{im+1(u),k} = - u_{im-1(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6587}%
$\displaystyle u_{im+2(u),k} = - u_{im-2(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6589}%
$\displaystyle \pi_{0,k} = - \pi_{1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6590}%
$\displaystyle \pi_{-1,k} = - \pi_{1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6591}%
$\displaystyle \pi_{im+1,k} = - \pi_{im,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6592}%
$\displaystyle \pi_{im+2,k} = - \pi_{im-1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6602}%
$\displaystyle \pi_{0,k} =  \pi_{1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6603}%
$\displaystyle \pi_{-1,k} =  \pi_{1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6604}%
$\displaystyle \pi_{im+1,k} =  \pi_{im,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6605}%
$\displaystyle \pi_{im+2,k} =  \pi_{im-1,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2249}%
$\tau +\Delta \tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2253}%
$\pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2255}%
$u, \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2257}%
$w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2259}%
$F_u, F_w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6620}%
$\displaystyle \DP{u_{i(u),k}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v}
\DP{(\pi - \alpha Div )}{x}\right]_{i(u),k}
+ [F_{u}]_{i(u),k}^{t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6621}%
$\displaystyle \DP{w_{i,k(w)}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v}
\DP{(\pi - \alpha Div )}{z}\right]_{i,k(w)}
+ [F_{w}]_{i,k(w)}^{t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6622}%
$\displaystyle \DP{\pi_{i,k}}{t}
+ \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\DP{(\bar{\rho} \bar{\theta}_{v} w)}{z}\right]_{i,k}
=
-  \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u}{x}\right]_{i,k}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2263}%
$\alpha Div$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2265}%
$F_{u}, F_{w}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6627}%
$\displaystyle [F_{u}]_{i(u),k}^{t} =
- \left[{\rm Adv}.{u}\right]_{i(u),k}^{t}
+ \left[{\rm Turb}.{u}\right]_{i(u),k}^{t - \Delta t}
+ \left[{\rm Diff}.u\right]_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6628}%
$\displaystyle \left[F_{w}\right]_{i,k(w)}^{t} =
- \left[{\rm Adv}.{w}\right]_{i,k(w)}^{t}
+ [{\rm Buoy}]_{i,k(w)}^{t}
+ \left[{\rm Turb}.{w}\right]_{i,k(w)}^{t - \Delta t}.
+ \left[{\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2267}%
$t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6633}%
$\displaystyle u^{\tau + \Delta \tau}_{i(u),k}
=  u^{\tau}_{i(u), k}
-   \left[
\bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
\DP{\pi^{\tau}}{x}
- \DP{(\alpha Div)^{\tau}}{x}
\right\}
\right]_{i(u),k}
+
F_{u,i(u),k}^{t} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2277}%
$u^{\tau +\Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6641}%
$\displaystyle w^{\tau + \Delta \tau}
=  w^{\tau}
- \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
\beta \DP{\pi^{\tau + \Delta \tau}}{z}
+ (1 - \beta) \DP{\pi^{\tau}}{z}
- \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6643}%
$\displaystyle \pi^{\tau + \Delta \tau}
+ \beta \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau + \Delta \tau})}{z}
=
\pi^{\tau}
- (1 - \beta) \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
-  \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u^{\tau + \Delta \tau}}{x}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2279}%
$w^{\tau +
\Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6646}%
$\displaystyle \pi^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6647}%
$\textstyle -$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6648}%
$\displaystyle \beta^{2}
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\DP{}{z}
\left\{
\left(\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}\right)
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6650}%
$\displaystyle \pi^{\tau}
-(1 - \beta)
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
-  \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\DP{u^{\tau + \Delta \tau}}{x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6651}%
$\displaystyle - \beta  \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\left[
\DP{}{z}
\bar{\rho} \bar{\theta}_{v}
\left\{
w^{\tau}
- \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
(1 - \beta) \DP{\pi^{\tau}}{z}
- \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6653}%
$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6654}%
$\displaystyle \hspace{10mm}+ \left[
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
+
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\right]
\pi^{\tau + \Delta \tau}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6655}%
$\displaystyle \hspace{10mm}+ \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6656}%
$\displaystyle =
\pi^{\tau}_{i,k}
- (1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,k}
-\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{k}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6657}%
$\displaystyle \hspace{2mm} - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\left[
\DP{}{z}
\left( \bar{\rho} \bar{\theta}_{v} \right)_{i,k(w)}
\left\{
w^{\tau}_{i,k(w)}
\right. \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6658}%
$\displaystyle \hspace{10mm}
\left. \left.
- \left( \bar{c_{p}} \bar{\theta}_{v} \right)_{i,k(w)} \Delta \tau
\left\{
(1 - \beta)  \DP{\pi^{\tau}}{z}
-  \DP{(\alpha Div)^{\tau}}{z}
\right\}_{i,k(w)}
+ \left( F_{w}^{t} \right)_{i,k(w)} \Delta \tau
\right\}
\right]_{i,k}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6662}%
$\displaystyle w(i,0(w)) = 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6663}%
$\displaystyle w(i,km(w)) = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2283}%
$k(w) = 0(w)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6666}%
$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6668}%
$\displaystyle \left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,0(w)}
- (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)_{i,0(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6669}%
$\textstyle \equiv$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6670}%
$\displaystyle E_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6672}%
$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,2}
+
\left\{
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\} \pi^{\tau + \Delta \tau}_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6674}%
$\displaystyle \pi^{\tau}_{i,1}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,1}
-
\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{1}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6675}%
$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\left[
\DP{}{z}
\bar{\rho} \bar{\theta}_{v}
\left\{
w^{\tau}
- \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
(1 - \beta) \DP{\pi^{\tau}}{z}
- \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right]_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6676}%
$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{i,0(w)}
E_{i,0(w)}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2285}%
$k(w) = km(w)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6679}%
$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6681}%
$\displaystyle \left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,km(w)}
- (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)_{i,km(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,km(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6683}%
$\displaystyle E_{i,km(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6685}%
$\displaystyle \left\{
1 +
\beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6686}%
$\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6688}%
$\displaystyle \pi^{\tau}_{i,km}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,km}
-
\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{km}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6689}%
$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\left[
\DP{}{z}
\bar{\rho} \bar{\theta}_{v}
\left\{
w^{\tau}
- \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
(1 - \beta) \DP{\pi^{\tau}}{z}
- \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right]_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6690}%
$\displaystyle + \frac{\beta}{\Delta z}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6693}%
$\displaystyle \left(\begin{array}{cccc}
A_{1} & B_{2}  &        &   0      \\
C_{1} & \ddots & \ddots &          \\
& \ddots & \ddots & B_{km}   \\
0   &        & C_{km-1} & A_{km} \\
\end{array}
\right)
\left(\begin{array}{cccc}
\pi_{1,1} & \pi_{2,1}  & \cdots & \pi_{im,1} \\
\pi_{1,2} & \ddots    & \ddots & \vdots        \\
\vdots   &           & \ddots & \vdots        \\
\pi_{1, km} & \cdots & \cdots & \pi_{im, km} \\
\end{array}
\right)^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6694}%
$\displaystyle =
\left(\begin{array}{cccc}
D_{1,1} & D_{2,1}  & \cdots & D_{im,1} \\
D_{1,2} & \ddots    & \ddots & \vdots        \\
\vdots   &           & \ddots & \vdots        \\
D_{1,km} & \cdots & \cdots & D_{im,km} \\
\end{array}
\right)^{\tau}
.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6697}%
$\displaystyle A_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6699}%
$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
+
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6700}%
$\displaystyle (k = 2, 3, \cdots km-1),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6701}%
$\displaystyle A_{1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6703}%
$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6704}%
$\displaystyle A_{km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6706}%
$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6707}%
$\displaystyle B_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6709}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k-1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6710}%
$\displaystyle (k = 2, 3, \cdots km),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6711}%
$\displaystyle C_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6713}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k+1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6714}%
$\displaystyle (k = 1, 2, \cdots km-1),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6715}%
$\displaystyle D_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6717}%
$\displaystyle \pi^{\tau}_{i,k}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,k}
-\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{k}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,k}
+ F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6719}%
$\displaystyle D_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6721}%
$\displaystyle \pi^{\tau}_{i,1}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,1}
-
\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{1}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,1}
+ F_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6722}%
$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{i,0(w)}
E_{i,0(w)},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6723}%
$\displaystyle D_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6725}%
$\displaystyle \pi^{\tau}_{i,km}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\left\{
\DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
\right\}_{i,km}
-
\left(
\frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\right)_{km}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,km}
+ F_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6726}%
$\displaystyle + \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6728}%
$\displaystyle E_{i,k(w)} \equiv
\left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,k(w)}
- (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)_{i,k(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6730}%
$\displaystyle F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6736}%
$\displaystyle F_{u,i(u),k}^{t} =
- \left[ {\rm Adv}.u \right]_{i(u),k}^{t}
+ \left[{\rm Turb}.{u} \right]_{i(u),k}^{t-\Delta t}
+ \left[{\rm Diff}.u \right]_{i(u),k}^{t-\Delta t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6737}%
$\displaystyle F_{w,i,k(w)} =
- \left[ {\rm Adv}.w \right]_{i(u),k}^{t}
+ \left[ {\rm Buoy} \right]_{i, k(w)}^{t}
+ \left[ {\rm Turb}.{w} \right]_{i,k(w)}^{t - \Delta t}
+ \left[ {\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6739}%
$\displaystyle \left[ {\rm Adv}.{u} \right]_{i(u),k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6741}%
$\displaystyle u_{i(u),k}^{t} \left[\DP{u}{x}\right]_{i(u),k}^{t}
+ w_{i(u),k}^{t} \left[\DP{u}{z}\right]_{i(u),k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6742}%
$\displaystyle \left[ {\rm Adv}.{w} \right]_{i,k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6744}%
$\displaystyle u_{i,k(w)}\left[\DP{w}{x}\right]_{i,k(w)}^{t}
+ w_{i,k(w)}\left[\DP{w}{z}\right]_{i,k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6746}%
$\displaystyle [{\rm Buoy}]^{t}_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6748}%
$\displaystyle g \frac{\theta_{i,k(w)}^{t}}{\overline{\theta}_{i,k(w)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6749}%
$\displaystyle + g \frac{\sum [q_{v}]_{i,k(w)}^{t}/M_{v}}{1/M_{d}
+ \sum  [\bar{q_{v}}]_{i,k(w)}/M_{v}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6750}%
$\displaystyle - g \frac{\sum [q_{v}]_{i,k(w)}^{t}
+ \sum [q_{c}]_{i,k(w)}^{t} + \sum [q_{r}]_{i,k(w)}^{t}}
{1 + \sum [\bar{q_{v}}]_{i,k(w)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6752}%
$\displaystyle \left[ {\rm Turb}.{u} \right]_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6754}%
$\displaystyle 2 \left[
\DP{}{x}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{u}{x} \right)_{i,k}
\right\}
\right]_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6755}%
$\displaystyle +\left[ \DP{}{z}\left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \DP{w}{x} \right)_{i(u),k(w)}
+ \left( K_{m} \right)_{i(u),k(w)}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6756}%
$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{x} \right)_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6757}%
$\displaystyle \left[ {\rm Turb}.{w} \right]_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6759}%
$\displaystyle 2 \left[
\DP{}{z}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{w}{z} \right)_{i,k}
\right\}
\right]_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6760}%
$\displaystyle +\left[ \DP{}{x}\left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \DP{w}{x} \right)_{i(u),k(w)}
+ \left( K_{m} \right)_{i(u),k(w)}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6761}%
$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{z} \right)_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6763}%
$\displaystyle \left[ {\rm Diff}.u \right]_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6765}%
$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{u}{x}\right)_{i,k} \right\}_{i(u),k}^{t - \Delta t}
+ \nu_{v} \left\{ \DP{}{z} \left(\DP{u}{z}\right)_{i(u),k(w)} \right\}_{i(u),k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6766}%
$\displaystyle \left[ {\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6768}%
$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{w}{x}\right)_{i(u),k(w)} \right\}_{i,k(w)}^{t - \Delta t}
+ \nu_{v} \left\{ \DP{}{z} \left(\DP{w}{z}\right)_{i,k} \right\}_{i,k(w)}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2289}%
$K_{m}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2291}%
$\nu_{h}, \nu_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6772}%
$\displaystyle \nu_{h} = \frac{\alpha_{h} \Delta x^{2}}{\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6773}%
$\displaystyle \nu_{v} = \frac{\alpha_{v} \Delta z^{2}}{\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2293}%
$\Delta x, \Delta z$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2295}%
$\alpha_{h}, \alpha_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6777}%
$\displaystyle \alpha_{h}  \le \Dinv{8}, \hspace{3em}
\alpha_{v}  \le \Dinv{8}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3036}%
$F$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6781}%
$\displaystyle \theta_{i,k}^{t + \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6783}%
$\displaystyle \theta_{i,k}^{t - \Delta t} + 2 \Delta t  [F_{\theta}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6784}%
$\displaystyle \left[ q_{v} \right]_{i,k}^{t+\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6786}%
$\displaystyle \left[ q_{v} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{v}}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6787}%
$\displaystyle \left[ q_{c} \right]_{i,k}^{t+\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6789}%
$\displaystyle \left[ q_{c} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{c}}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6790}%
$\displaystyle \left[ q_{r} \right]_{i,k}^{t+\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6792}%
$\displaystyle \left[ q_{r} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{r}}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6794}%
$\displaystyle [F_{\theta}]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6796}%
$\displaystyle - \left[{\rm Adv}.{\theta}\right]_{i,k}^{t}
- \left[{\rm Adv}.{\bar{\theta}}\right]_{i,k}^{t}
+ \left[{\rm Turb}.{\theta} \right]_{i,k}^{t - \Delta t}
+ \left[{\rm Turb}.{\bar{\theta}} \right]_{i,k}^{t - \Delta t}
+ \left[{\rm Diff}.{\theta} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6797}%
$\displaystyle + [Q_{cnd}]_{i,k}^{t}
+ [Q_{rad}]_{i,k}^{t-\Delta t}
+ [Q_{dis}]_{i,k}^{t-\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6798}%
$\displaystyle \left[F_{q_{v}}\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6800}%
$\displaystyle - \left[ {\rm Adv}.q_{v} \right]_{i,k}^{t}
- \left[ {\rm Adv}.\bar{q_{v}} \right]_{i,k}^{t}
+ \left[ {\rm Turb}.q_{v} \right]_{i,k}^{t - \Delta t}
+ \left[ {\rm Turb}.\bar{q_{v}} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6801}%
$\displaystyle + \left[ {\rm Diff}.q_{v} \right]_{i,k}^{t - \Delta t}
+ \left[ EV_{rv} \right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6802}%
$\displaystyle \left[F_{q_{c}}\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6804}%
$\displaystyle - \left[ {\rm Adv}.q_{c} \right]_{i,k}^{t}
+ \left[ {\rm Turb}.q_{c} \right]^{t-\Delta t}
+ \left[ {\rm Diff}.q_{c} \right]^{t-\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6805}%
$\displaystyle - \left[ CN_{cr} + CL_{cr} \right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6806}%
$\displaystyle \left[F_{q_{r}}\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6808}%
$\displaystyle - \left[ {\rm Adv}.q_{r} \right]_{i,k}^{t}
+ \left[ {\rm Turb}.q_{r} \right]_{i,k}^{t-\Delta t}
+ \left[ {\rm Diff}.q_{c} \right]^{t-\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6809}%
$\displaystyle + \left[CN_{cr} + CL_{cr} - EV_{rv} \right]_{i,k}^{t}
+ \left[ PR_{r} \right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3038}%
$CN_{vc}, EV_{cv}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3040}%
$\theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3042}%
$q_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3044}%
$q_{c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3046}%
$q_{r}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6817}%
$\displaystyle \left[{\rm Adv}.{\phi}\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6819}%
$\displaystyle \left[
u_{i(u),k}  \left[ \DP{\phi}{x} \right]_{i(u),k}
\right]_{i,k}^{t}
+
\left[
w_{i,k(w)} \left[ \DP{\phi}{z} \right]_{i,k(w)}
\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6821}%
$\displaystyle \left[{\rm Adv}.{\bar{\phi}}\right]_{i,k}^{t} =
\left[
w_{i,k(w)} \left[ \DP{\overline{\phi}}{z} \right]_{i,k(w)}
\right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6823}%
$\displaystyle \left[{\rm Turb}.{\phi} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6825}%
$\displaystyle \left[ \DP{}{x}
\left\{
\left( K_{h} \right)_{i(u),k}
\left( \DP{\phi}{x} \right)_{i(u),k}
\right\}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6826}%
$\displaystyle + \left[ \DP{}{z}\left\{
\left( K_{h} \right)_{i,k(w)}
\left( \DP{\phi }{z} \right)_{i,k(w)}
\right\} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6828}%
$\displaystyle \left[{\rm Turb}.{\bar{\phi}} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6830}%
$\displaystyle \left[ \DP{}{z}\left\{
\left( K_{h} \right)_{i,k(w)}
\left( \DP{\overline{\phi}}{z} \right)_{i,k(w)}
\right\} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6832}%
$\displaystyle \left[ {\rm Diff}_{\phi} \right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6834}%
$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{\phi}{x}\right)_{i(u),k} \right\}_{i,k}^{t - \Delta t}
+ \nu_{v} \left\{ \DP{}{z} \left(\DP{\phi}{z}\right)_{i,k(w)} \right\}_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3050}%
$K_{h}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3054}%
$Q_{cnd}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2625}%
\begin{displaymath}
 [Q_{cnd}]_{i,k}^{t} 
  = - \left[ \frac{L}{{c_{p} \bar{\pi}}_{d} \bar{\pi}} EV_{rv} \right]_{i,k}^{t} 
  = - \frac{L}{{c_{p}}_{d} \bar{\pi}} 
  \left\{
   4.85 \times 10^{-2} ([q_{vsw}]_{i,k}^{t} - [q_{v}]_{i,k}^{t}) 
    (\bar{\rho}_{i,k} [q_{r}]_{i,k})^{0.65}
  \right\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3056}%
$Q_{dis}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2655}%
\begin{displaymath}
  [Q_{dis}]_{i,k}^{t-\Delta t} 
   = \frac{1}{{c_{p}}_{d} \bar{\pi}}
   \frac{C_{\varepsilon}}{l}
            \frac{(K_{m,i,k}^{t-\Delta t})^{3}}{(C_{m}l)^{3}}
   = \frac{1}{{c_{p}}_{d} \bar{\pi}}
   \frac{(K_{m,i,k}^{t-\Delta t})^{3}}{{C_{m}}^{2} l^{4}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3058}%
$l=(\Delta x\Delta z)^{1/2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3060}%
$[Q_{rad}]_{i,k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3062}%
$CN_{cr}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3064}%
$CL_{cr}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6846}%
$\displaystyle CN_{cr} = \frac{10^{6} [\bar{\rho}]_{i,k} \left( [q_{c}]_{i,k}^{t}\right)^{3}}
{60 (2 [q_{c}]_{i,k}^{t} + 2.66 \times 10^{-8} N_{0}/ [\bar{\rho}]_{i,k} D_{0})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6847}%
$\displaystyle \hspace{2em}
N_{0} = 5.0 \times 10^{7},
\hspace{1em}
D_{0} = 0.366$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6848}%
$\displaystyle \left[ CL_{cr} \right]_{i,k}^{t}
= 2.2   [q_{c}]_{i,k}^{t}
\left(
[\bar{\rho}]_{i,k}
\left[ q_{r} \right]_{i,k}^{t}
\right)^{0.875}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3066}%
$EV_{rv}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6851}%
$\displaystyle \left[ EV_{rv} \right]_{i,k}^{t} =
4.85 \times 10^{-2} ([q_{vsw}]_{i,k}^{t} - [q_{v}]_{i,k}^{t})
([\bar{\rho}]_{i,k} [q_{r}]_{i,k}^{t})^{0.65}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3068}%
$PR_{r}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6854}%
$\displaystyle \left[
PR_{r}
\right]_{i,k}^{t}
= \Dinv{[\bar{\rho}]_{i,k}} \DP{}{z}([\bar{\rho}]_{i,k}
[U_{r}]_{i,k}^{t}
[q_{r}]_{i,k}^{t}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6855}%
$\displaystyle [U_{r}]_{i,k}^{t} = 12.2 ([q_{r}]_{i,k}^{t})^{0.125}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3070}%
$-CN_{vc} + EV_{cv}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3072}%
$dS=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3074}%
$\mu_{vapar} = \mu_{condensed phase}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3076}%
$\mu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3078}%
$*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3080}%
$[\theta]^{*}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3082}%
$[q_{v}]^{*}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3084}%
$[q_{c}]^{*}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3086}%
$[q_{r}]^{*}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6868}%
$\displaystyle \Delta q_{c} = MAX\{0, [q_{v}]^{*} - q_{vsw}([\theta]^{*})\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3088}%
$\Delta q_{c} > 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3090}%
$q_{c}^{*} > 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6875}%
$\displaystyle \left[ \theta \right]^{t + \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6877}%
$\displaystyle \theta^{*} +
\frac
{ \gamma ( [q_{v}]^{*} -  q_{vsw}([\theta]^{*})) }
{ 1 + \gamma \DP{q_{vsw}([\theta]^{*})}{\theta} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6878}%
$\displaystyle \left[ q_{v} \right]^{t + \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6880}%
$\displaystyle [q_{v}]^{*} + \frac{[\theta]^{*} - [\theta]^{t + \Delta t}}{\gamma},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6881}%
$\displaystyle \left[ q_{c} \right]^{t + \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6883}%
$\displaystyle [q_{v}]^{*} + [q_{c}]^{*} - [q_{v}]^{t + \Delta t} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3098}%
$\gamma = L_{v}/(c_{p} \Pi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3100}%
$[q_c]^{t + \Delta t} > 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3104}%
$q_{c}^{t + \Delta t} < 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6889}%
$\displaystyle \left[ \theta \right]^{t + \Delta t} =
[\theta]^{*}  - \gamma [q_{c}]^{*},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6890}%
$\displaystyle \left[ q_{v} \right]^{t + \Delta t},
= [q_{v}]^{*} + [q_{c}]^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6891}%
$\displaystyle \left[ q_{c} \right]^{t + \Delta t}
= 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6894}%
$\displaystyle {\rm NH_{3}} + {\rm H_{2}S} \rightarrow {\rm NH_{4}SH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6896}%
$\displaystyle K_{p}
\equiv  \ln(p_{\rm NH_{3}} \cdot p_{\rm H_{2}S})
= 61.781 - \frac{10834}{T}  - \ln{10^{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3106}%
$T$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3108}%
$_{4}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3110}%
$X$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6901}%
$\displaystyle (p_{\rm NH_{3}} - X) ( p_{\rm H_{2}S} - X )
= e^{k_{p}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6902}%
$\displaystyle X^{2} - (p_{\rm NH_{3}} + p_{\rm H_{2}S}) X
+  p_{\rm NH_{3}} \cdot p_{\rm H_{2}S}
- e^{k_{p}} = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6904}%
$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
\pm \sqrt{   (p_{\rm NH_{3}} + p_{\rm H_{2}S})^{2}
- 4 (p_{\rm NH_{3}} \cdot p_{\rm H_{2}S} - e^{K_{p}}) }
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6905}%
$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
\pm \sqrt{   (p_{\rm NH_{3}} - p_{\rm H_{2}S})^{2}
+ 4 e^{K_{p}} }
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3112}%
$\exp{(K_{p})} \approx 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6908}%
$\displaystyle X = {\rm min}(P_{\rm NH_3}, P_{\rm H_{2}S} )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6911}%
$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
- \sqrt{   (p_{\rm NH_{3}} - p_{\rm H_{2}S})^{2}
+ 4 e^{K_{p}} }.
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6914}%
$\displaystyle {\rm min}(P_{\rm NH_{3}}, P_{\rm H_{2}S}) > X, \hspace{1em}
X < - \frac{M_{\rm NH_4SH} \cdot {\rm min}(P_{\rm NH_{3}}, P_{\rm
H_{2}S})}{M_{d} P_{all} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3118}%
$X = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6921}%
$\displaystyle \left[ q_{\rm NH_3} \right]^{t + \Delta t}
= [q_{\rm NH_3}]^{*} + \Delta q_{\rm NH_3},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6922}%
$\displaystyle \left[ q_{\rm H_2S} \right]^{t + \Delta t}
= [q_{\rm H_2S}]^{*} + \Delta q_{\rm H_2S},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6923}%
$\displaystyle \left[ q_{\rm NH_4SH} \right]^{t + \Delta t}
= [q_{\rm NH_3}]^{*} + [q_{\rm H_2S}]^{*}
- [q_{\rm NH_3}]^{t + \Delta t} - [q_{\rm H_2S}]^{t + \Delta t} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6924}%
$\displaystyle \left[ \theta \right]^{t + \Delta t}
=
\theta^{*} + \gamma \left([q_{\rm NH_3}]^{*} + [q_{\rm H_2S}]^{*}
- [q_{\rm NH_3}]^{t + \Delta t} - [q_{\rm H_2S}]^{t + \Delta t}
\right).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3128}%
$\gamma = L_{\rm NH_4SH}/({c_{p}}_{d} \Pi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3130}%
$\Delta q_{\rm NH_3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3132}%
$\Delta q_{\rm H_2S}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3136}%
$_3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3138}%
$_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6933}%
$\displaystyle [K_{m}]_{i,k}^{t + \Delta t} = [K_{m}]_{i,k}^{t - \Delta t}
+ 2 \Delta t [F_{K_m}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6935}%
$\displaystyle [F_{K_m}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6937}%
$\displaystyle - [{\rm Adv}.K_m]_{i,k}^{t}
+ [{\rm Buoy}.K_m]_{i,k}^{t - \Delta t}
+ [{\rm Shear}.K_m]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6938}%
$\displaystyle + [{\rm Turb}.K_m]_{i,k}^{t - \Delta t}
+ [{\rm Disp}.K_m]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3542}%
$t - \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3544}%
$F_{K_m}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6943}%
$\displaystyle [{\rm Adv}.K_m]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6945}%
$\displaystyle \left\{
u_{i(u),k} \left( \DP{K_{m}}{x} \right)_{i(u), k}
\right\}_{i,k}^{t}
+
\left\{
w_{i,k(w)} \left( \DP{K_{m}}{z} \right)_{i, k(w)}
\right\}_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6946}%
$\displaystyle \left[{\rm Buoy}.K_m\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6948}%
$\displaystyle - \left\{
\frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}}
\left(\DP{\theta_{el}}{z} \right)_{i,k(w)}
\right\}_{i,k}^{t-\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6949}%
$\displaystyle \left[{\rm Shear}.K_m\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6951}%
$\displaystyle \left( C_{m}^{2} l^{2} \right)_{i,k}
\left[
\left\{
\left( \DP{u}{x} \right)^{2}
\right\}_{i(u),k}
\right]_{i,k}^{t - \Delta t}
+ \left( C_{m}^{2} l^{2} \right)_{i,k}
\left[
\left\{
\left( \DP{w}{z} \right)^{2}
\right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6952}%
$\displaystyle + \left(  \frac{ C_{m}^{2} l^{2} }{2} \right)_{i,k}
\left[
\left\{
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\}_{i,k}^{t-\Delta t}
+
\left\{
\left( \DP{w}{x} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
\right]^{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6953}%
$\displaystyle - \left( \frac{K_{m}}{3} \right)_{i,k}^{t - \Delta t}
\left\{
\left( \DP{u}{x} \right)_{i,k}^{t-\Delta t}
+
\left( \DP{w}{z} \right)_{i,k}^{t-\Delta t}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6954}%
$\displaystyle \left[{\rm Turb}.K_m\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6956}%
$\displaystyle \Dinv{2}
\left[
\left\{
\DP{}{x}
\left(
\DP{K_{m}^{2}}{x}
\right)_{i(u),k}
\right\}_{i,k}^{t-\Delta t}
+
\left\{
\DP{}{z}
\left(
\DP{K_{m}^{2}}{z}
\right)_{i,k(w)}
\right\}_{i,k}^{t-\Delta t}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6957}%
$\displaystyle +
\left[
\left\{
\left( \DP{K_{m}}{x}\right)^{2}
\right\}_{i(u),k}
\right]_{i,k}^{t - \Delta t}
+
\left[
\left\{
\left(\DP{K_{m}}{z}\right)^{2}
\right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6958}%
$\displaystyle \left[{\rm Disp}.K_m\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6960}%
$\displaystyle - \Dinv{2 l^{2}} \left( K_{m}^{2} \right)_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3546}%
$C_{\varepsilon} = C_{m} = 0.2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3548}%
$l = \left(\Delta x \Delta z \right)^{1/2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3550}%
$\theta_{el}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6965}%
$\displaystyle \theta_{el}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6967}%
$\displaystyle \overline{ \theta_{v}} + \theta_{v}^{'} \;\;\; (for \;\; q_{c} = 0)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6970}%
$\displaystyle \overline{\theta_{v}} + \theta_{v}^{'} + \frac{ \sum L
q_{v}}{{c_p}_{d} \bar{\pi}}
\;\;\; (for \;\; q_{c} > 0)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6972}%
$\displaystyle \overline{\theta_{v}} + \theta_{v}^{'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6974}%
$\displaystyle \bar{\theta_{v}}
\left\{
1 + \frac{\theta}{\bar{\theta}}
+ \frac{\sum q_{v}/M_{v}}{1/M_{d} + \sum  \bar{q_{v}}/M_{v}}
- \frac{\sum q_{v} + \sum q_{c} + \sum q_{r}}
{1 + \sum \bar{q_{v}}}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3616}%
$N_{\tau}\equiv 2\Delta t/\Delta
\tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3618}%
$ u^{t + \Delta t}_{i(u),k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6979}%
$\displaystyle u^{*}_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6981}%
$\displaystyle u^{\tau + (N_{\tau}-1)\Delta \tau}_{i(u), k}
-   \left[
\bar{c_{p}} \bar{\theta}_{v} \Delta \tau
\left\{
\DP{\pi^{\tau + (N_{\tau}-1)\Delta \tau}}{x}
- \DP{(\alpha Div)^{\tau + (N_{\tau}-1)\Delta \tau}}{x}
\right\}
\right]_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6982}%
$\displaystyle +
F_{u,i(u),k}^{t} \Delta \tau,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6983}%
$\displaystyle u^{t+\Delta t}_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6985}%
$\displaystyle (1-2 \gamma)u^{t}_{i(u),k} +
\gamma (u^{*}_{i(u),k} + u^{t -\Delta t}_{i(u),k})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3620}%
$\gamma$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6989}%
$\displaystyle \DP{\phi}{t} = -{\rm Adv}.\phi + \cdots + \gamma_{h}(x) (\phi - \phi_{e})
+ \gamma_{v}(z) (\phi - \phi_{e})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3721}%
$\phi_{e}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3725}%
$2 \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6995}%
$\displaystyle [\pi]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.\pi]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x)  + \gamma_{v}(z)
\right\} (\pi - \bar{\pi})^{t - \Delta t}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6996}%
$\displaystyle [u]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.u]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x)  + \gamma_{v}(z)
\right\} [u]^{t - \Delta t}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6997}%
$\displaystyle [w]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.w]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x)  + \gamma_{v}(z)
\right\} [w]^{t - \Delta t}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6998}%
$\displaystyle [\theta]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.\theta]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x)  + \gamma_{v}(z)
\right\} [\theta]^{t - \Delta t}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3727}%
$\bar{\pi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3729}%
$\gamma_{h}, \gamma_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3731}%
$d_{h}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3735}%
$0 \leq x \leq
x_{max}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7005}%
$\displaystyle \gamma_{h} = \alpha_{h} \left( 1 - \frac{x}{d_{h}}\right)^{3}
\hspace{5em} (x <  d_{h}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7006}%
$\displaystyle \gamma_{h} = 0  \hspace{10em} ( d_{h} \leq x \leq  x_{max} - d_{h}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7007}%
$\displaystyle \gamma_{h} = \alpha_{h} \left( 1 - \frac{(x -
x_{max})}{d_{h}}\right)^{3}
\;\; (x > x_{max} - d_{h}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3737}%
$d_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3741}%
$0 \leq z \leq z_{max}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7012}%
$\displaystyle \gamma_{v} = 0  \hspace{12em} ( z \leq  z_{max} - d_{v}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7013}%
$\displaystyle \gamma_{v} = \alpha_{v} \left
( 1 - \cos{\frac{\pi (z - z_{max})}{d_{v}}}
\right)^{3}
\;\; (z > z_{max} - d_{v}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3743}%
$\alpha_h, \alpha_v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3747}%
$1/\alpha_{h}, 1/\alpha_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3749}%
$d_{h}, d_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\appendix
\stepcounter{chapter}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7021}%
$\displaystyle \Deqref{uwpi:sabun}\mbox{left side}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7023}%
$\displaystyle \pi^{\tau + \Delta \tau}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7024}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7025}%
$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k-1(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7028}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
\left(
\frac{\pi^{\tau + \Delta \tau}_{i,k+1}
- \pi^{\tau + \Delta \tau}_{i,k}}{\Delta z}
\right)
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7029}%
$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\left(
\frac{\pi^{\tau + \Delta \tau}_{i,k}
- \pi^{\tau + \Delta \tau}_{i,k-1}}{\Delta z}
\right)
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7032}%
$\displaystyle + \left[
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)}
+
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\right]
\pi^{\tau + \Delta \tau}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7033}%
$\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{k}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k-1}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7039}%
$\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4429}%
$k = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7046}%
$\displaystyle \pi^{\tau + \Delta \tau}_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7047}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,1(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7048}%
$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{0(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7051}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\left(
\frac{\pi^{\tau + \Delta \tau}_{i,2}
- \pi^{\tau + \Delta \tau}_{i,1}}{\Delta z}
\right)
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7054}%
$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7055}%
$\displaystyle +
\left\{
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\} \pi^{\tau + \Delta \tau}_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7056}%
$\displaystyle + \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
\bar{\theta}_{v}^{2}}
\right)_{1}
\Dinv{\Delta z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{0(w)}
E_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7062}%
$\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,km(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7069}%
$\displaystyle \pi^{\tau + \Delta \tau}_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7070}%
$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7071}%
$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km-1(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7074}%
$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7075}%
$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}
{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
\right)_{km}
\Dinv{\Delta z}
\left\{
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\left(
\frac{\pi^{\tau + \Delta \tau}_{i,km}
- \pi^{\tau + \Delta \tau}_{i,km-1}}{\Delta z}
\right)
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4616}%
$2\Delta t/\Delta \tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7083}%
$\displaystyle \DP{u}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7085}%
$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}
\DP{\pi}{x} + \alpha\DP{D}{x} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7086}%
$\displaystyle \DP{w}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7088}%
$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}
\DP{\pi}{z} + \alpha\DP{D}{z}
+ g\frac{\theta}{\overline{\theta}_{v}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7089}%
$\displaystyle \DP{\pi}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7091}%
$\displaystyle - \frac{\overline{c}^{2}}{\overline{c}_{p}
\overline{\rho}\overline{\theta}_{v}^{2}}
\left[\DP{\overline{\rho}\overline{\theta}_{v}u}{x} +
\DP{\overline{\rho}\overline{\theta}_{v}w}{z}
\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7092}%
$\displaystyle \DP{\theta}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7094}%
$\displaystyle -w\DP{\overline{\theta}_{v}}{z},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7095}%
$\displaystyle D$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7097}%
$\displaystyle \DP{u}{x}+\DP{w}{z}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7099}%
$\displaystyle \DP{D}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7101}%
$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}\Dlapla \pi
+ g\DP{}{z}\left(\frac{\theta}{\overline{\theta}_{v}}\right)
+ \alpha \Dlapla D,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7102}%
$\displaystyle \Dlapla$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7104}%
$\displaystyle \DP[2]{}{x} + \DP[2]{}{z}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4618}%
$u=\hat{u}e^{i(kx+lz -\omega t)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4521}%
\begin{displaymath}
  \omega ^{4} + i\alpha (k^{2}+l^{2})\omega ^{3}  - 
  [\overline{c}^{2}(k^{2}+l^{2}) + N^{2}] \omega ^{2}
  - i\alpha k^{2}N^{2}\omega + \overline{c}^{2}k^{2}N^{2}=0, 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4620}%
$N^{2} = \frac{g}{\overline{\theta}_{v}}\DP{
\overline{\theta}_{v}}{z}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4614}%
\begin{displaymath}
 \omega ^{2} = \frac{N^{2}k^{2}}{k^{2}+l^{2}},
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4622}%
$\alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4626}%
$\epsilon \equiv \alpha \sqrt{k^{2}+l^{2}}/\overline{c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4628}%
$\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4630}%
$\epsilon$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4557}%
\begin{displaymath}
 \frac{\alpha \Delta \tau}{\mbox{Min}(\Delta x^{2}, \Delta z^{2})} \leq \frac{1}{2}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4567}%
\begin{eqnarray*}
  \mbox{Max}(\epsilon) 
    &=& \frac{2\alpha}{\mbox{Min}(\Delta x, \Delta z)\overline{c}} \nonumber \\
    &=& \frac{2\alpha \Delta \tau}{\mbox{Min}(\Delta x^{2}, \Delta z^{2})}
        \left/
        \frac{\overline{c}\Delta \tau}{\mbox{Min}(\Delta x, \Delta z)}
        \right.
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4638}%
$\mbox{Max}(\epsilon) \leq 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4580}%
\begin{displaymath}
    \frac{\alpha \Delta \tau}{\mbox{Min}(\Delta x^{2}, \Delta z^{2})}
    \leq \frac{1}{2}
        \frac{\overline{c}\Delta \tau}{\mbox{Min}(\Delta x, \Delta z)}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5021}%
$(i,j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5017}%
\begin{displaymath}
  \left[\DP{u}{x}\right]_{i,k}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5025}%
$(i(u),j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5029}%
$(u_{i(u),j})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5033}%
$-\Delta x/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5039}%
$(u_{i,j})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7139}%
$\displaystyle u _{i(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7141}%
$\displaystyle u_{i,j}
+ \left[\DP{u}{x} \right]_{i,k}\frac{\Delta x}{2}
+ \frac{1}{2!}\left[\DP[2]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{2}
+ \frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7142}%
$\displaystyle + \frac{1}{4!}\left[\DP[4]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{4}
+ \frac{1}{5!}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}
+ \cdots .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5041}%
$(i-1(u),j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5045}%
$(u_{i-1(u),j})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5047}%
$u_{i,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7148}%
$\displaystyle u_{i-1(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7150}%
$\displaystyle u_{i,j}
- \left[\DP{u}{x} \right]_{i,k}\frac{\Delta x}{2}
+ \frac{1}{2!}\left[\DP[2]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{2}
- \frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5049}%
$-$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7154}%
$\displaystyle u _{i(u),j} - u_{i-1(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7156}%
$\displaystyle \left[\DP{u}{x} \right]_{i,k}\Delta x
+ \frac{1}{3}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{3}
+ \frac{1}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7157}%
$\displaystyle + O[(\Delta x)^{7}]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7162}%
$\displaystyle \left[\DP{\pi}{x} \right]_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7164}%
$\displaystyle \frac{\pi _{i+1,j}  - \pi _{i,j}}{\Delta x}
- \frac{1}{24}\left[\DP[3]{\pi}{x} \right]_{i(u),k}
\left(\Delta x\right)^{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7165}%
$\displaystyle - \frac{1}{1920}\left[\DP[5]{\pi}{x} \right]_{i(u),k}
\left(\Delta x\right)^{4}
+ O[(\Delta x)^{6}]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5057}%
$\left(\Delta x\right)^{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4797}%
\begin{displaymath}
\left[\DP{u}{x} \right]_{i(u),k} = 
  \frac{u_{i+1,j} - u_{i,j}}{\Delta x}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4805}%
\begin{displaymath}
   \left|\frac{1}{24}\left[\DP[3]{u}{x} \right]_{i(u),k}
       \left(\Delta x\right)^{2}\right|
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5063}%
$\pm
\Delta 3x/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5067}%
$(u _{i+1(u),j}, u
_{i-2(u),j})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7177}%
$\displaystyle u _{i+1(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7179}%
$\displaystyle u_{i,j}
+ \left[\DP{u}{x} \right]_{i,k}\frac{3\Delta x}{2}
+ \frac{1}{2!}\left[\DP[2]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{2}
+ \frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7180}%
$\displaystyle + \frac{1}{4!}\left[\DP[4]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{4}
+ \frac{1}{5!}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{5}
+ \cdots .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7181}%
$\displaystyle u _{i-2(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7183}%
$\displaystyle u_{i,j}
- \left[\DP{u}{x} \right]_{i,k}\frac{3\Delta x}{2}
+ \frac{1}{2!}\left[\DP[2]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{2}
- \frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7187}%
$\displaystyle u _{i+1(u),j} - u_{i-2(u),j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7189}%
$\displaystyle \left[\DP{u}{x} \right]_{i,k}3\Delta x
+ \frac{1}{3}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{3}
+ \frac{1}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5073}%
$\times 27 -$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5075}%
$(\Delta x)^{3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7194}%
$\displaystyle 27(u _{i(u),j} - u_{i-1(u),j}) - (u _{i+1(u),j} - u_{i-2(u),j})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7196}%
$\displaystyle \left[\DP{u}{x} \right]_{i,k}24\Delta x
- \frac{216}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7202}%
$\displaystyle \left[\DP{u}{x}\right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7204}%
$\displaystyle \frac{9}{8}\left(\frac{u_{i,j}  - u _{i-1,j}}{\Delta x}\right)
- \frac{1}{24}\left(\frac{u_{i+1,j}  - u _{i-2,j}}{\Delta x}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7205}%
$\displaystyle + \frac{3}{640}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\Delta x\right)^{4}
+ O[(\Delta x)^{6}]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5083}%
$\left(\Delta x\right)^{4}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4973}%
\begin{displaymath}
\left[\DP{u}{x}\right]_{i,k} = 
  \frac{9}{8}\left(\frac{u_{i,j}  - u _{i-1,j}}{\Delta x}\right) 
  - \frac{1}{24}\left(\frac{u_{i+1,j}  - u _{i-2,j}}{\Delta x}\right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4988}%
\begin{displaymath}
   \left|\frac{3}{640}\left[\DP[5]{u}{x} \right]_{i(u),k}
       \left(\Delta x\right)^{4}\right|
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7211}%
$\displaystyle d\theta + \gamma dq_{v} = 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5259}%
$[\theta]^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5261}%
$[q_{v}]^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5263}%
$[q_{c}]^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5265}%
$[q_{r}]^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7222}%
$\displaystyle ([\theta]^{t + \Delta t} - [\theta]^{*}) =
\gamma ([q_{v}]^{*} - [q_{v}]^{t + \Delta t})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7224}%
$\displaystyle [q_{v}]^{t + \Delta t} = q_{sw}([\theta]^{t + \Delta t})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5267}%
$q_{vsw}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5269}%
$q_{vsw}([\theta]^{t + \Delta t})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7229}%
$\displaystyle q_{sw}([\theta]^{t + \Delta t})
= q_{sw}([\theta]^{*})
+ \DP{q_{vsw}([\theta]^{*})}{\theta} ([\theta]^{t + \Delta t} - [\theta]^{*})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7231}%
$\displaystyle ([\theta]^{t + \Delta t} - [\theta]^{*}) =
\gamma \left(
[q_{v}]^{*} - q_{sw}([\theta]^{*})
- \DP{q_{vsw}([\theta]^{*})}{\theta}
([\theta]^{t + \Delta t} - [\theta]^{*})
\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7232}%
$\displaystyle \left(
1 + \gamma \DP{q_{vsw}([\theta]^{*})}{\theta}
\right)  ([\theta]^{t + \Delta t} - [\theta]^{*}) =
\gamma \left(
[q_{v}]^{*} - q_{sw}([\theta]^{*})
\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7233}%
$\displaystyle [\theta]^{t + \Delta t} =  [\theta]^{*} +
\gamma  \frac{[q_{v}]^{*} - q_{vsw}([\theta]^{*})}
{   1 + \gamma \DP{q_{vsw}([\theta]^{*})}{\theta}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7235}%
$\displaystyle ([\theta]^{t + \Delta t} - [\theta]^{*}) =
\gamma ([q_{v}]^{*} - [q_{v}]^{t + \Delta t}) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7236}%
$\displaystyle [q_{v}]^{t + \Delta t}  =
[q_{v}]^{*} - ([\theta]^{t + \Delta t} - [\theta]^{*}) / \gamma.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7238}%
$\displaystyle ([q_{v}]^{*} + [q_{c}]^{*})
= ([q_{v}]^{t + \Delta t} + [q_{c}]^{t + \Delta t}) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7239}%
$\displaystyle [q_{c}]^{t + \Delta t}  =
[q_{v}]^{*} + [q_{c}]^{*} - [q_{v}]^{t + \Delta t}  .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7242}%
$\displaystyle \DP{\exp{1/T}}{T} = \Dinv{T} e^{1/T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7247}%
$\displaystyle \ln p^{*}= (A - B / (C + T - T_{0})) * \log(10.0) + \ln(133.322)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5454}%
$p^{*}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5458}%
$T_{0} = 273.15$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5460}%
$A, B, C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5462}%
$^{\circ}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5464}%
$_{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5468}%
$_{3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7279}%
$\displaystyle K_{p}
= \ln(p_{\rm NH_{3}} \cdot p_{\rm H_{2}S})
= 61.781 - \frac{10834}{T}  - \ln{10^{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7283}%
$\displaystyle \DD{p_{v}}{T} =  \frac{p_{v} L_{v}}{R T^{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5470}%
$L_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7286}%
$\displaystyle L_{v} = \DD{\ln p_{v}}{T} {R_{v} T^{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5472}%
$R_{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7289}%
$\displaystyle L_{v} = \left\{
\frac{B \ln(10.0)}{ (C + T - T_{0})^{2} }
\right\} R_{v} T^{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5474}%
$_4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7298}%
$\displaystyle s_{\rm NH_4SH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7300}%
$\displaystyle - \DP{\mu_{\rm NH_4SH}}{T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7302}%
$\displaystyle - \DP{}{T}
\left(
\mu_{\rm NH_3} + \mu_{\rm H_2S} + RT K_{p}
- RT \ln {p_{0}}^{2}
\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7304}%
$\displaystyle s_{\rm NH_3} + s_{\rm H_2S}  - RT \DP{K_{p}}{T} - R K_{p}
- RT \ln {p_{0}}^{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5482}%
$K_{p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7309}%
$\displaystyle s_{\rm NH_4SH}  + s_{\rm NH_4SH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7311}%
$\displaystyle - \DP{\mu_{\rm NH_3} + \mu_{\rm H_2S}}{T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7313}%
$\displaystyle - \DP{}{T}
\left(
\mu^{\circ}_{\rm NH_3} + \mu^{\circ}_{\rm H_2S}
+ RT \ln( p_{\rm NH_3} p_{\rm H_2S}) - RT \ln {p_{0}}^{2}
\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7315}%
$\displaystyle - \DP{}{T}
\left(
\mu^{\circ}_{\rm NH_3} + \mu^{\circ}_{\rm H_2S}
+ RT K_{p} - RT \ln {p_{0}}^{2}
\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7317}%
$\displaystyle s_{\rm NH_3} + s_{\rm H_2S}   - R K_{p}
- RT \ln {p_{0}}^{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7319}%
$\displaystyle \Delta s
= RT \DP{K_{p}}{T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7321}%
$\displaystyle L_{\rm NH_4SH} = T \Delta s = RT^{2} \DP{K_{p}}{T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7324}%
$\displaystyle L_{\rm NH_4SH} =  \frac{10834}{T^2} {R T^{2}} =  10834 R$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5731}%
$d$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5733}%
$v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5737}%
$q_{x}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5739}%
$p_{x}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7333}%
$\displaystyle q_{x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7335}%
$\displaystyle \frac{\rho_{x}}{\rho_{d}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7337}%
$\displaystyle \left(
\frac{R_{d} T}{p} \frac{1/M_{d} + \sum_{v} q_{v}/M_{v}}{1/M_{d}}
\right)
\frac{p_{x}}{R_{x}T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7339}%
$\displaystyle \frac{p_{x}}{p} M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7340}%
$\displaystyle q_{x} \left(1 - \frac{p_{x}}{p} \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7342}%
$\displaystyle \frac{p_{x}}{p} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7345}%
$\displaystyle \frac{p_{x}}{p}\frac{ M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v})  }
{\left(1 - \frac{p_{x}}{p} \right) }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7348}%
$\displaystyle \frac{ p_{x} M_{x}  (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v}) }{p - p_{x}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7351}%
$\displaystyle \frac{ p_{x} M_{x}  (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v}) }
{p_{0} \pi^{{c_{p}}_{d}/R_{d}} - p_{x}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5741}%
$x_{x}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7354}%
$\displaystyle q_{x} =
\frac{ x_{x} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v})}{ 1 - x_{x} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7358}%
$\displaystyle p_{x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7360}%
$\displaystyle \frac{q_{x} p}{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7363}%
$\displaystyle \frac{q_{x}p_{0} \pi^{{c_{p}}_{d}/R_{d}}  }{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7366}%
$\displaystyle x_{x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7368}%
$\displaystyle \frac{q_{x}}{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5729}%
\begin{displaymath}
  \mbox{dcstaff@gfd-dennou.org}
 \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}


\end{document}