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$\displaystyle \frac{9}{8} \frac{\phi_{i,k+1} - \phi_{i,k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{H_{i(u), k} - H_{i-1(u), k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{H_{i(u), k+1} - H_{i(u), k}}{\Delta x} - \Dinv{24}
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{\newpage\clearpage
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$\displaystyle \frac{9}{8} \frac{V_{i+1, k(w)} - V_{i, k(w)}}{\Delta x} - \Dinv{24}
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{\newpage\clearpage
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$\displaystyle \frac{9}{8} \frac{V_{i, k(w)} - V_{i, k-1(w)}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i-1(u), k(w)} }{\Delta
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{\newpage\clearpage
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i(u), k-1(w)} }{\Delta
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$\displaystyle \DP{u_{i(u), k}}{t}$%
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$\displaystyle - u_{i(u), k} \left( \DP{u}{x} \right)_{i(u), k}
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$\displaystyle - c_p \overline{\theta}_{i(u), k} \left( \DP{\Pi}{x} \right)_{i(u),
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{\newpage\clearpage
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$\displaystyle \DP{w_{i, k(w)}}{t}$%
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$\displaystyle - u_{i, k(w)} \left( \DP{w}{x} \right)_{i, k(w)}
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{\newpage\clearpage
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$\displaystyle - c_p \overline{\theta}_{i, k(w)} \left( \DP{\Pi}{z} \right)_{i,
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+ \left[ D_w \right]_{i, k(w)}, \Deqlab{RisanAB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2108}%
$\displaystyle \DP{\theta_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2112}%
$\displaystyle - u_{i,k} \left( \DP{\theta}{x} \right)_{i,k}
- w_{i,k} \left( \DP{\theta}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2116}%
$\displaystyle + \Dinv{\overline{\Pi}_{i,k}} \frac{L
[M_{cond}]_{i,k}}{\overline{\rho}_{i,k} c_p}
+ \Dinv{\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right)
+ \left[ D_{\theta} \right]_{i,k}, \Deqlab{RisanAC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2118}%
$\displaystyle \DP{\Pi_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2122}%
$\displaystyle - \frac{\overline{c}^2_{i,k}}{c_p \overline{\rho}_{i,k}
\overline{\theta}^2_{i,k}}
\left[ \left( \DP{}{x} (\overline{\rho} \overline{\theta} u)
\right)_{i,k} +
\left( \DP{}{z} (\overline{\rho} \overline{\theta} w)
\right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2124}%
$\displaystyle + \frac{\overline{c}^2_{i,k}}{c_p \overline{\rho}_{i,k}
\overline{\theta}_{i,k}}
\left( \frac{L}{c_p \overline{T}_{i,k} } -1  \right)
\left[ M_{cond} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2128}%
$\displaystyle + \frac{\overline{c}^2_{i,k}}{c_p \overline{\theta}_{i,k}^2
\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right),
\Deqlab{RisanAD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2132}%
$\displaystyle \DP{{\rho_s}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2136}%
$\displaystyle - \left( \rho_s u \right)_{i,k}
- \left( \rho_s w \right)_{i,k}
+ \left[ M_{cond} \right]_{i,k}
+ \left[ M_{fall} \right]_{i,k}
+ \left[ D_{\rho_s} \right]_{i,k},
\Deqlab{RisanAE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2138}%
$\displaystyle \DP{{K_m}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2142}%
$\displaystyle - u_{i,k} \left( \DP{{K_m}}{x} \right)_{i,k}
- w_{i,k} \left( \DP{{K_m}}{z} \right)_{i,k}
- \frac{3 g {C_m}^2 l^2}{2 \overline{\theta}}_{i,k}
\left( \DP{\theta^{\prime}}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2144}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2_{i,k}
+ \left( \DP{w}{z} \right)^2_{i,k} \right]
+ \frac{{C_m}^2 l^2}{2} \left( \DP{u}{z} + \DP{w}{x} \right)^2_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2146}%
$\displaystyle - \frac{{K_m}_{i,k}}{3} \left( \DP{u}{x} + \DP{w}{z} \right)_{i,k}
+ \Dinv{2} \left[
\left( \DP[2]{{K_m}^2}{x} \right)_{i,k}
+ \left( \DP[2]{{K_m}^2}{z} \right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2150}%
$\displaystyle + \left( \DP{{K_m}}{x} \right)^2_{i,k}
+ \left( \DP{{K_m}}{z} \right)^2_{i,k}
- \frac{C_{\varepsilon}}{2 C_m l^2} {K_m}^2_{i,k}.
\Deqlab{RisanAF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2161}%
$\displaystyle \DP{\phi}{t} + U \DP{\phi}{x} = 0 \Deqlab{TrisanA}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2165}%
$\displaystyle \frac{U \Delta t}{\Delta x} \leq 1 \Deqlab{TrisanB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2167}%
$ U$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure454}%
\begin{figure} \begin{center}
\Depsf[78mm]{ps/timesplit.ps}
  
 \end{center}
  \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2182}%
$ \Pi^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2186}%
$\displaystyle \Pi^{\prime} - \alpha \left( \DP{u}{x} + \DP{w}{z} \right)  \Deqlab{SoundDecay1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2188}%
$ \alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2192}%
$\displaystyle \alpha = 5.0 \times 10^{-7} \times \frac{{\rm min} [(\Delta x)^2, (\Delta  z)^2]}{\Delta \tau} \Deqlab{SoundDecay2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2202}%
$\displaystyle \nu_H \DP[2]{\phi}{x} + \nu_V \DP[2]{\phi}{z} \Deqlab{NumDisp1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2204}%
$ \nu_H$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2206}%
$ \nu_V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2210}%
$\displaystyle \nu_H = \nu_V = 5.0 \times 10^{-4} \times \frac{{\rm min} [(\Delta x)^2,  (\Delta z)^2]}{\Delta t} \Deqlab{NumDisp2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2213}%
$ \nu_H = \nu_V = 1.0 \times 10^{-4} \times {\rm min} [(\Delta x)^2,
 (\Delta z)^2] / \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2223}%
$ u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2225}%
$ w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2229}%
$ \phi^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2231}%
$ \phi^{\tau +
\Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2233}%
$ \phi^{t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2235}%
$ \phi^{\tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2237}%
$ \phi^{t -
\Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2239}%
$ \phi^{\tau - \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2243}%
$\displaystyle u_{i(u),k}^{\tau + \Delta \tau} = u_{i(u),k}^{\tau}    - \left( c_p \overline{\theta} \DP{[\Pi^{\prime}]^{\tau}}{x}     \right)_{i,u(k)} \Delta \tau   + {F_u}_{i(u),k}^{t} \Delta \tau \Deqlab{TrisanC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2245}%
$ F_u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2249}%
$\displaystyle {F_u}_{i(u),k}^{t} = - \left( u^t \DP{u^t}{x} \right)_{i(u), k}   - \left( w^t \DP{u^t}{z} \right)_{i(u), k}   + \left[ D_u \right]_{i(u),k}^{t - \Delta t} \Deqlab{TrisanD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2251}%
$ t - \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2254}%
$\displaystyle w_{i,k(w)}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2258}%
$\displaystyle w_{i,k(w)}^{\tau} - c_p \overline{\theta}_{i,k(w)}
\left[ \beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} + (1 -
\beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]_{i,k(w)} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2262}%
$\displaystyle + {F_w}_{i,k(w)}^{t} \Delta \tau, \Deqlab{TrisanE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2265}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2269}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- \left[ \frac{ \overline{c}^2 }{c_p \overline{\rho}
\overline{\theta}^2 } \DP{}{x} (\overline{\rho} \overline{\theta}
u^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2271}%
$\displaystyle - \beta \left[ \frac{ \overline{c}^2 }{c_p \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau
- (1 - \beta) \left[ \frac{ \overline{c}^2 }{c_p \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2273}%
$\displaystyle + \left[ \frac{\overline{c}^2}{c_p \overline{\theta}}
\left( \frac{L}{c_p \overline{T}} - 1 \right)
\frac{M_{cond}^{\tau}}{\overline{\rho}} \right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2277}%
$\displaystyle + \left[ \frac{ \overline{c}^2 }{c_p \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} )
\right]_{i,k} \Delta \tau
\Deqlab{TrisanF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2279}%
$ \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2281}%
$ \beta = 0.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2283}%
$ F_w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2287}%
$\displaystyle {F_w}_{i,k(w)}^{t} = - \left( u^t \DP{w^t}{x} \right)_{i, k(w)}   - \left( w^t \DP{w^t}{z} \right)_{i, k(w)}   + g \frac{ [\theta^{\prime}]_{i,k(w)}^{t}   }{\overline{\theta}_{i,k(w)}}    + \left[ D_w \right]_{i,k(w)}^{t - \Delta t} \Deqlab{TrisanG}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2289}%
$ u^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2291}%
$ w^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2295}%
$\displaystyle \begin{pmatrix}
A_1 & B_2 & \cdots & 0 & 0 \\
C_1 & A_2 & \cdots & 0 & 0 \\
\colon & \colon &  & \colon & \colon \\
0 & 0 & \cdots & A_{km - 1} & B_{km} \\
0 & 0 & \cdots & C_{km - 1} & A_{km}
\end{pmatrix}
\begin{pmatrix}
\Pi^{\prime}_{1,1} & \Pi^{\prime}_{2,1} & \cdots & \Pi^{\prime}_{km-1,1} &
\Pi^{\prime}_{km,1} \\
\Pi^{\prime}_{1,2} & \Pi^{\prime}_{2,2} & \cdots & \Pi^{\prime}_{km-1,2} &
\Pi^{\prime}_{km,2} \\
\colon & \colon & & \colon & \colon \\
\Pi^{\prime}_{1,km-1} & \Pi^{\prime}_{2,km-1} & \cdots &
\Pi^{\prime}_{km-1,km-1} & \Pi^{\prime}_{km,km-1} \\
\Pi^{\prime}_{1,km} & \Pi^{\prime}_{2,km} & \cdots &
\Pi^{\prime}_{km-1,km} & \Pi^{\prime}_{km,km}
\end{pmatrix} ^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2300}%
$\displaystyle =
\begin{pmatrix}
D_{1,1} & D_{2,1} & \cdots & D_{km-1,1} & D_{km,1} \\
D_{1,2} & D_{2,2} & \cdots & D_{km-1,2} & D_{km,2} \\
\colon & \colon & & \colon & \colon \\
D_{1,km-1} & D_{2,km-1} & \cdots & D_{km-1,km-1} & D_{km,km-1} \\
D_{1,km} & D_{2,km} & \cdots & D_{km-1,km} & D_{km,km}
\end{pmatrix}
\Deqlab{TrisanH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2302}%
$ km$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2307}%
$\displaystyle A_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2311}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_1
\Dinv{(\Delta z)^2}
( c_p \overline{\rho} \overline{\theta}^2)_{i,1(w)},
\Deqlab{TrisanI}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2313}%
$\displaystyle A_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2317}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_k
\Dinv{(\Delta z)^2}
[ ( c_p \overline{\rho} \overline{\theta}^2)_{i,k(w)}
+ ( c_p \overline{\rho} \overline{\theta}^2)_{i,k-1(w)}
],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2321}%
$\displaystyle (k = 2, 3, \dots , km -1 ) \Deqlab{TrisanJ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2325}%
$\displaystyle A_{km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2329}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{km}
\Dinv{(\Delta z)^2}
( c_p \overline{\rho} \overline{\theta}^2)_{i,km-1(w)},
\Deqlab{TrisanK}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2333}%
$\displaystyle B_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2337}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{k-1}
\Dinv{(\Delta z)^2}
( c_p \overline{\rho} \overline{\theta}^2)_{i,k-1(w)},
\Deqlab{TrisanL}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2341}%
$\displaystyle C_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2345}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{k+1}
\Dinv{(\Delta z)^2}
( c_p \overline{\rho} \overline{\theta}^2)_{i,k(w)},
\Deqlab{TrisanM}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2347}%
$\displaystyle D_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2351}%
$\displaystyle [ \Pi^{\prime} ]_{i,1}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{1}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,1}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} } \right]_{1}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,1}
+ F_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2355}%
$\displaystyle - \beta \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{1}
\Dinv{\Delta z} (c_p \overline{\rho} \overline{\theta}^2)_{i,0(w)}
E_{i,0(w)}, \Deqlab{TrisanN}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2358}%
$\displaystyle D_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2362}%
$\displaystyle [ \Pi^{\prime} ]_{i,k}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{k}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,k}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} } \right]_{k}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,k}
+ F_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2366}%
$\displaystyle (k = 2, 3, \dots , km -1 )
\Deqlab{TrisanO}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2368}%
$\displaystyle D_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2372}%
$\displaystyle [ \Pi^{\prime} ]_{i,km}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{km}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,km}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_p
\overline{\rho} } \right]_{km}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2376}%
$\displaystyle + F_{i,km}
+ \beta \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_p
\overline{\rho} \overline{\theta}^2} \right]_{km}
\Dinv{\Delta z} (c_p \overline{\rho} \overline{\theta}^2)_{i,km(w)}
E_{i,km(w)}. \Deqlab{TrisanP}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2381}%
$\displaystyle E_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2385}%
$\displaystyle - (1 - \beta )\left[ \DP{[\Pi^{\prime}]^{\tau}}{z}
\right]_{i,k(w)}
+ \left[ \frac{F^{t}_{w}}{c_p \overline{\theta}}
\right]_{i,k(w)}, \Deqlab{TrisanQ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2387}%
$\displaystyle F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2391}%
$\displaystyle \Big[ - \Delta \tau \beta \frac{\overline{c}^2}{c_p
\overline{\rho} \overline{\theta}}
\DP{}{z} \Big\{ \overline{\rho} \overline{\theta}
\Big( w^{\tau} - \Delta \tau (1 - \beta) c_p \overline{\theta}
\DP{[\Pi^{\prime}]^{\tau}}{z} + F_{w}^{t} \Delta \tau \Big) \Big\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2395}%
$\displaystyle + \frac{\overline{c}^2}{c_p \overline{\theta}} \left(\frac{L}{c_p
\overline{T}} -1 \right) \frac{M_{cond}^{\tau}}{\overline{\rho}}
\Delta \tau
+ \frac{ \overline{c}^2 }{c_p \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} ) \Delta \tau
\Big] \Deqlab{TrisanR}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2397}%
$ \tau + \Delta \tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2403}%
$\displaystyle {\theta^{\prime}}_{i,k}^{\tau + \Delta \tau}  = {\theta^{\prime}}_{i,k}^{\tau} + {F_{\theta}}_{i,k}^{t} \Delta \tau +     \Dinv{\overline{\Pi}_{i,k}}     \frac{L {M_{cond}^{\tau}}_{i,k}}{c_p \overline{\rho}_{i,k}} \Delta    \tau   \Deqlab{TrisanS}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2405}%
$ F_{\theta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2408}%
$\displaystyle {F_{\theta}}_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2412}%
$\displaystyle - u_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{x} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{z} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{\overline{\theta}^t}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2416}%
$\displaystyle + \frac{\overline{{\theta^{\prime}}}_{i,k}}{\overline{T}_{i,k}}
( Q_{rad} + Q_{dis})_{i,k}^{t} + {D_{\theta}}_{i,k}^{t - \Delta t}
\Deqlab{TrisanT}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2420}%
$\displaystyle {\rho_s}_{i,k}^{\tau + \Delta \tau} = {\rho_s}_{i,k}^{\tau} + {M_{cond}}_{i,k}^{\tau} \Delta \tau   + {F_{\rho_s}}_{i,k}^{t} \Delta \tau \Deqlab{TrisanU}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2422}%
$ F_{\rho_s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2426}%
$\displaystyle {F_{\rho_s}}_{i,k}^{t} = - \left[ \DP{}{x} ({\rho_s}^t u^t) \right]_{i,k}   - \left[ \DP{}{z} ({\rho_s}^t w^t) \right]_{i,k}   + {M_{fall}}_{i,k}^{t}   + {D_{\rho_s}}_{i,k}^{t} \Deqlab{TrisanV}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2430}%
$\displaystyle {K_m}_{i,k}^{t+ \Delta t} = {K_m}_{i,k}^{t+ \Delta t} + 2 \Delta [F_{K_m}]_{i,k}^{t} \Deqlab{TrisanW}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2433}%
$\displaystyle [F_{K_m}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2437}%
$\displaystyle - \left[ u_{i(u),k} \left( \DP{K_m}{x} \right)_{i(u),k} \right]_{i,k}^t
- \left[ w_{i,k(w)} \left( \DP{K_m}{z} \right)_{i,k(w)} \right]_{i,k}^t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2439}%
$\displaystyle - \left[
\frac{3 g {C_m}^2 l^2}{2 \overline{\theta}}
\left( \DP{\theta^{\prime}}{z} \right)_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2441}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2 \right]_{i,k}^{t -
\Delta t}
+ {C_m}^2 l^2 \left[ \left( \DP{w}{z} \right)^2 \right]_{i,k}^{t -
\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2443}%
$\displaystyle + \frac{{C_m}^2 l^2}{2} \left[
\left\{ \left( \DP{u}{z} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \left( \DP{w}{x} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
\right]^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2445}%
$\displaystyle - \Dinv{3} [K_m]_{i,k}^{t - \Delta t}
\left[ \left( \DP{u}{x} \right)_{i,k}^{t - \Delta t}
+ \left( \DP{w}{z} \right)_{i,k}^{t - \Delta t}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2447}%
$\displaystyle + \Dinv{2} \left[
\left\{ \DP{}{x} \left( \DP{K_m^2}{x} \right)_{i(u),k}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \DP{}{z} \left( \DP{K_m^2}{z} \right)_{i,k(w)}
\right\}_{i,k}^{t - \Delta t} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2449}%
$\displaystyle + \left[ \left\{ \left( \DP{K_m}{x} \right)^2 \right\}_{i(u),k}
\right]_{i,k}^{t - \Delta t}
+ \left[ \left\{ \left( \DP{K_m}{z} \right)^2 \right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2453}%
$\displaystyle - \frac{C_{\varepsilon}}{2 C_m l^2} [K_m^2]_{i,k}^{t - \Delta t}
\Deqlab{TrisanX}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2455}%
$ t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2465}%
$ t + \Delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2471}%
$ \theta^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2475}%
$ K_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2479}%
$ \overline{\phi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2483}%
$\displaystyle \overline{\phi^t} = \phi^t + \mu_a (\overline{\phi^{t - \Delta t}} - 2  \phi^t + \phi^{t + \Delta t})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2485}%
$ \mu_a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2487}%
$ \mu_a = 0.1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
