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$\displaystyle \frac{ \phi_{i, k(w)} V_{i, k(w)} - \phi_{i, k-1(w)} V_{i, k-1(w)} }
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$\displaystyle \frac{9}{8} \frac{\phi_{i+1,k} - \phi_{i,k}}{\Delta x}
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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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$\displaystyle \frac{9}{8} \frac{H_{i(u), k+1} - H_{i(u), k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{V_{i+1, k(w)} - V_{i, k(w)}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{V_{i, k(w)} - V_{i, k-1(w)}}{\Delta x} - \Dinv{24}
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{\newpage\clearpage
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i-1(u), k(w)} }{\Delta
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i(u), k-1(w)} }{\Delta
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4495}%
$\displaystyle \frac{9}{8} \frac{\phi_{i, k(w)} - \phi_{i, k-1(w)}}{\Delta x} - \Dinv{24}
\frac{\phi_{i, k+1(w)} - \phi_{i, k-2(w)}}{\Delta z},
\Deqlab{RisanZ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4501}%
$\displaystyle \frac{9}{8}
\frac{ \phi_{i(u), k} H_{i(u), k}
- \phi_{i-1(u), k} H_{i-1(u), k}}{\Delta x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4505}%
$\displaystyle - \Dinv{24}
\frac{\phi_{i+1(u), k} H_{i+1(u), k}
- \phi_{i-2(u), k} H_{i-2(u), k} }{\Delta z},
\Deqlab{RisanZ-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4511}%
$\displaystyle \frac{9}{8}
\frac{\phi_{i, k(w)} V_{i, k(w)}
- \phi_{i, k-1(w)} V_{i, k-1(w)} }{\Delta x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4515}%
$\displaystyle - \Dinv{24}
\frac{\phi_{i, k+1(w)} V_{i, k+1(w)}
- \phi_{i, k-2(w)} V_{i, k-2(w)}}{\Delta z}.
\Deqlab{RisanZ-3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4522}%
$\displaystyle \DP{u_{i(u), k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4526}%
$\displaystyle - u_{i(u), k} \left( \DP{u}{x} \right)_{i(u), k}
- w_{i(u), k} \left( \DP{u}{z} \right)_{i(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4530}%
$\displaystyle - c_{pv} \overline{\theta}_{i(u), k}
\left( \DP{\Pi}{x} \right)_{i(u), k}
+ \left[ D_u \right]_{i(u), k}
+ \left[ D_u^{sfc} \right]_{i(u), k}.
\Deqlab{RisanAA}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4533}%
$\displaystyle \left[ D_u \right]_{i(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4537}%
$\displaystyle 2 \left[
\DP{}{x}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{u}{x} \right)_{i,k}
\right\}
\right]_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4539}%
$\displaystyle + \left[ \DP{}{z} \left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \DP{w}{x} \right)_{i(u),k(w)}
+ \left( K_{m} \right)_{i(u),k(w)}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4543}%
$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{x} \right)_{i(u),k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4547}%
$\displaystyle \left[ D_u^{sfc} \right]_{i(u), k}
=
\delta_{1k}
\left[ \DP{}{z} (C_D V_{sfc} u_{z_1}) \right]_{i(u),k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4550}%
$\displaystyle \DP{w_{i, k(w)}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4554}%
$\displaystyle - u_{i, k(w)} \left( \DP{w}{x} \right)_{i, k(w)}
- w_{i, k(w)} \left( \DP{w}{z} \right)_{i, k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4558}%
$\displaystyle - c_{pv} \overline{\theta}_{i, k(w)}
\left( \DP{\Pi}{z} \right)_{i, k(w)}
+ \left[ D_w \right]_{i, k(w)}
+ g
\left(
\frac{\theta^{\prime}}{\overline{\theta}}
- \frac{R}{p_0}
\frac{\overline{\theta}}{\overline{\Pi}^{c_{vv} / R}}
\rho_s
\right)_{i, k(w)}.
\Deqlab{RisanAB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4561}%
$\displaystyle \left[ D_w \right]_{i, k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4565}%
$\displaystyle 2 \left[
\DP{}{z}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{w}{z} \right)_{i,k}
\right\}
\right]_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4567}%
$\displaystyle +\left[ \DP{}{x}\left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \DP{w}{x} \right)_{i(u),k(w)}
+ \left( K_{m} \right)_{i(u),k(w)}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4571}%
$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{z} \right)_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4574}%
$\displaystyle \DP{\Pi_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4578}%
$\displaystyle - \frac{\overline{c^2}_{i,k}}{c_{pv} \overline{\rho}_{i,k}
\overline{\theta}^2_{i,k}}
\left[ \left( \DP{}{x} (\overline{\rho} \overline{\theta} u)
\right)_{i,k} +
\left( \DP{}{z} (\overline{\rho} \overline{\theta} w)
\right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4580}%
$\displaystyle + \frac{\overline{c^2}_{i,k}}{c_{pv} \overline{\rho}_{i,k}
\overline{\theta}_{i,k}}
\left( \frac{L}{c_{pv} \overline{T}_{i,k} } -1  \right)
\left[ M_{cond} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4584}%
$\displaystyle + \frac{\overline{c^2}_{i,k}}{c_{pv} \overline{\theta}_{i,k}^2
\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right).
\Deqlab{RisanAD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4587}%
$\displaystyle \DP{{\rho_s}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4591}%
$\displaystyle - \left[ \DP{}{x} \left( \rho_s u \right) \right]_{i,k}
- \left[ \DP{}{z} \left( \rho_s w \right) \right]_{i,k}
+ \left[ M_{cond} \right]_{i,k}
+ \left[ M_{fall} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4595}%
$\displaystyle + \left[ D_{\rho_s} \right]_{i,k}.
\Deqlab{RisanAE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4600}%
$\displaystyle \left[ M_{cond} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4604}%
$\displaystyle \frac{ 4 \pi {r_d}_{i,k} \overline{\rho}_{i,k} N^{*} }
{{R_h}_{i,k}} (S_{i,k} - 1),
\Deqlab{TrisanS2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4608}%
$\displaystyle {R_h}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4612}%
$\displaystyle \frac{L^2}{k R [\theta^2]_{i,k} [\Pi^2]_{i,k}},
\Deqlab{TrisanS3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4616}%
$\displaystyle S_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4620}%
$\displaystyle \frac{{p_0}^{\frac{c_{pv}}{R}} \Pi_{i,k}}
{\left[ p_{\ast} \right]_{i,k}},
\Deqlab{TrisanS4}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4624}%
$\displaystyle \left[ M_{fall} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4628}%
$\displaystyle \left[
\DP{}{z} \left( \rho_s V_{term} \right)
\right]_{i,k}
\Deqlab{TrisanV2},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4632}%
$\displaystyle {V_{term}}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4636}%
$\displaystyle \left[ C_{sc} \right]_{i,k}
\frac{2 \left[ r_d^2 \right]_{i,k} g \rho_I}{9 \eta_{i,k} }
\Deqlab{TrisanV3},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4640}%
$\displaystyle {r_d}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4644}%
$\displaystyle \left(
r_{aero}^3 + \frac{3 {\rho_s}_{i,k}}
{4 \rho_I \pi \overline{\rho}_{i,k} N^{*}}
\right)^{1/3}
\Deqlab{TrisanV3-2},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4648}%
$\displaystyle \left[ C_{sc} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4652}%
$\displaystyle 1 + \frac{4}{3} \left[ K_n \right]_{i,k},
\Deqlab{TrisanV4}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4656}%
$\displaystyle \left[ K_n \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4660}%
$\displaystyle \frac{\lambda_{i,k}}{{r_d}_{i,k}},
\Deqlab{TrisanV4-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4664}%
$\displaystyle \lambda_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4668}%
$\displaystyle \frac{k_B \overline{\theta}_{i,k}}
{\sqrt{2} \pi \sigma^2 p_0 \overline{\Pi}^{c_{vv}/R}_{i,k}},
\Deqlab{TrisanV5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4672}%
$\displaystyle \eta_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4676}%
$\displaystyle \eta_{ref}
\left(
\frac{T_{ref} + C_{CO_2}}
{\overline{\theta}_{i,k} \overline{\Pi}_{i,k} + C_{CO_2}}
\right)
\left(
\frac{\overline{\theta}_{i,k} \overline{\Pi}_{i,k} }{T_{ref}}
\right)^{3/2},
\Deqlab{TrisanV6}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4680}%
$\displaystyle \left[ D_{\rho_s} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4684}%
$\displaystyle \left[
u_{i(u),k}  \left( \DP{\rho_s}{x} \right)_{i(u),k}
\right]_{i,k}
+
\left[
w_{i,k(w)} \left( \DP{\rho_s}{z} \right)_{i,k(w)}
\right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4687}%
$\displaystyle \DP{\theta_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4691}%
$\displaystyle - u_{i,k} \left( \DP{\theta}{x} \right)_{i,k}
- w_{i,k} \left( \DP{\theta}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4695}%
$\displaystyle + \Dinv{\overline{\Pi}_{i,k}} \frac{L
[M_{cond}]_{i,k}}{\overline{\rho}_{i,k} c_{pv}}
+ \Dinv{\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right)
+ \left[ D_{\theta} \right]_{i,k}
+ \left[ D_{\theta}^{sfc} \right]_{i,k}.
\Deqlab{RisanAC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4700}%
$\displaystyle \left[ D_{\theta} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4704}%
$\displaystyle \left[
u_{i(u),k}  \left( \DP{\theta}{x} \right)_{i(u),k}
\right]_{i,k}
+
\left[
w_{i,k(w)} \left( \DP{\theta}{z} \right)_{i,k(w)}
\right]_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4708}%
$\displaystyle \left[ D_{\theta}^{sfc} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4712}%
$\displaystyle \delta_{1k} \left\{
\DP{}{z} [C_D V_{sfc} \rho (T_{z_1} - T_{sfc})]
\right\}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4715}%
$\displaystyle \DP{{K_m}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4719}%
$\displaystyle - u_{i,k} \left( \DP{{K_m}}{x} \right)_{i,k}
- w_{i,k} \left( \DP{{K_m}}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4721}%
$\displaystyle - \frac{3 g {C_m}^2 l^2}{2}
\left[
\Dinv{\overline{\theta}_{i,k}}
\left( \DP{\theta}{z} \right)_{i,k}
- \frac{R}{p_0}
\left(
\frac{\overline{\theta}}{\overline{\Pi}^{c_v/R}}
\right)_{i,k}
\left( \DP{\rho_s}{z} \right)_{i,k}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4723}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2_{i,k}
+ \left( \DP{w}{z} \right)^2_{i,k} \right]
+ \frac{{C_m}^2 l^2}{2} \left( \DP{u}{z} + \DP{w}{x} \right)^2_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4725}%
$\displaystyle - \frac{{K_m}_{i,k}}{3} \left( \DP{u}{x} + \DP{w}{z} \right)_{i,k}
+ \Dinv{2} \left[
\left( \DP[2]{{K_m}^2}{x} \right)_{i,k}
+ \left( \DP[2]{{K_m}^2}{z} \right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4729}%
$\displaystyle + \left( \DP{{K_m}}{x} \right)^2_{i,k}
+ \left( \DP{{K_m}}{z} \right)^2_{i,k}
- \frac{C_{\varepsilon}}{2 C_m l^2} {K_m}^2_{i,k}.
\Deqlab{RisanAF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4740}%
$\displaystyle \DP{\phi}{t} + U \DP{\phi}{x} = 0 \Deqlab{TrisanA}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4744}%
$\displaystyle \frac{U \Delta t}{\Delta x} \leq 1 \Deqlab{TrisanB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4746}%
$ U$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure770}%
\begin{figure} \begin{center}
\Depsf[78mm]{ps/timesplit.ps}
  
 \end{center}
  \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4761}%
$ \Pi^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4765}%
$\displaystyle \Pi^{\prime} - \alpha {\rm Div}  \Deqlab{SoundDecay1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4767}%
$ {\rm Div} \equiv (\partial u / \partial x) + (\partial w / \partial z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4769}%
$ \alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4773}%
$\displaystyle \alpha = 5.0 \times 10^{-7} \times \frac{{\rm min} [(\Delta x)^2, (\Delta  z)^2]}{\Delta \tau} \Deqlab{SoundDecay2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4786}%
$\displaystyle \nu_H \DP[2]{\phi}{x} + \nu_V \DP[2]{\phi}{z} \Deqlab{NumDisp1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4788}%
$ \nu_H$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4790}%
$ \nu_V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4794}%
$\displaystyle \nu_H = \nu_V = 1.0 \times 10^{-4} \times \frac{{\rm min} [(\Delta x)^2,  (\Delta z)^2]}{\Delta t} \Deqlab{NumDisp2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4804}%
$ u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4806}%
$ w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4810}%
$ \phi^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4812}%
$ \phi^{\tau +
\Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4814}%
$ \phi^{t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4816}%
$ \phi^{\tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4818}%
$ \phi^{t -
\Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4820}%
$ \phi^{\tau - \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4824}%
$\displaystyle u_{i(u),k}^{\tau + \Delta \tau} = u_{i(u),k}^{\tau}    - \left( c_{pv} \overline{\theta} \DP{[\Pi^{\prime}]^{\tau}}{x}     \right)_{i,u(k)} \Delta \tau   + \left( c_{pv} \overline{\theta} \DP{[{\rm Div}]^{\tau}}{x}     \right)_{i,u(k)} \Delta \tau   + {F_u}_{i(u),k}^{t} \Delta \tau    \Deqlab{TrisanC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4826}%
$ F_u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4830}%
$\displaystyle {F_u}_{i(u),k}^{t} = - \left( u^t \DP{u^t}{x} \right)_{i(u), k}   - \left( w^t \DP{u^t}{z} \right)_{i(u), k}   + \left[ D_u \right]_{i(u),k}^{t - \Delta t}    + \left[ D_u^{sfc} \right]_{i(u),k}^{t}   \Deqlab{TrisanD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4832}%
$ t - \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4835}%
$\displaystyle w_{i,k(w)}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4839}%
$\displaystyle w_{i,k(w)}^{\tau}
- c_{pv} \overline{\theta}_{i,k(w)}
\left[ \beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} + (1 -
\beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]_{i,k(w)} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4841}%
$\displaystyle + c_{pv} \overline{\theta}_{i,k(w)}
\left[ \DP{(\alpha {\rm Div})}{z} \right]_{i,k(w)} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4845}%
$\displaystyle + {F_w}_{i,k(w)}^{t} \Delta \tau, \Deqlab{TrisanE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4848}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4852}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{x} (\overline{\rho} \overline{\theta}
u^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4854}%
$\displaystyle - \beta \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau
- (1 - \beta) \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4856}%
$\displaystyle + \left[ \frac{\overline{c^2}}{c_{pv} \overline{\theta}}
\left( \frac{L}{c_{pv} \overline{T}} - 1 \right)
\frac{M_{cond}^{\tau}}{\overline{\rho}} \right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4860}%
$\displaystyle + \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} )
\right]_{i,k} \Delta \tau
\Deqlab{TrisanF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4862}%
$ \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4864}%
$ \beta = 0.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4866}%
$ F_w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4869}%
$\displaystyle {F_w}_{i,k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4873}%
$\displaystyle - \left( u^t \DP{w^t}{x} \right)_{i, k(w)}
- \left( w^t \DP{w^t}{z} \right)_{i, k(w)}
+ g
\frac{ [\theta^{\prime}]_{i,k(w)}^{t}}{\overline{\theta}_{i,k(w)}}
- g
\frac{R}{p_0}
\frac{\overline{\theta}_{i, k(w)}}
{ \overline{\Pi}^{c_{vv} / R}_{i, k(w)} }
[\rho_s]_{i, k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4877}%
$\displaystyle + \left[ D_w \right]_{i,k(w)}^{t - \Delta t}
\Deqlab{TrisanG}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4879}%
$ u^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4881}%
$ w^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4885}%
$\displaystyle \begin{pmatrix}
A_1 & B_2 & \cdots & 0 & 0 \\
C_1 & A_2 & \cdots & 0 & 0 \\
\colon & \colon &  & \colon & \colon \\
0 & 0 & \cdots & A_{k_m - 1} & B_{k_m} \\
0 & 0 & \cdots & C_{k_m - 1} & A_{k_m}
\end{pmatrix}
\begin{pmatrix}
\Pi^{\prime}_{1,1} & \Pi^{\prime}_{2,1} & \cdots & \Pi^{\prime}_{k_m-1,1} &
\Pi^{\prime}_{k_m,1} \\
\Pi^{\prime}_{1,2} & \Pi^{\prime}_{2,2} & \cdots & \Pi^{\prime}_{k_m-1,2} &
\Pi^{\prime}_{k_m,2} \\
\colon & \colon & & \colon & \colon \\
\Pi^{\prime}_{1,k_m-1} & \Pi^{\prime}_{2,k_m-1} & \cdots &
\Pi^{\prime}_{k_m-1,k_m-1} & \Pi^{\prime}_{k_m,k_m-1} \\
\Pi^{\prime}_{1,k_m} & \Pi^{\prime}_{2,k_m} & \cdots &
\Pi^{\prime}_{k_m-1,k_m} & \Pi^{\prime}_{k_m,k_m}
\end{pmatrix} ^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4890}%
$\displaystyle =
\begin{pmatrix}
D_{1,1} & D_{2,1} & \cdots & D_{k_m-1,1} & D_{k_m,1} \\
D_{1,2} & D_{2,2} & \cdots & D_{k_m-1,2} & D_{k_m,2} \\
\colon & \colon & & \colon & \colon \\
D_{1,k_m-1} & D_{2,k_m-1} & \cdots & D_{k_m-1,k_m-1} & D_{k_m,k_m-1} \\
D_{1,k_m} & D_{2,k_m} & \cdots & D_{k_m-1,k_m} & D_{k_m,k_m}
\end{pmatrix} ^{\tau}
\Deqlab{TrisanH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4892}%
$ k_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4897}%
$\displaystyle A_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4901}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_1
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,1(w)},
\Deqlab{TrisanI}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4903}%
$\displaystyle A_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4907}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_k
\Dinv{(\Delta z)^2}
[ ( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k(w)}
+ ( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k-1(w)}
],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4911}%
$\displaystyle (k = 2, 3, \dots , k_m -1 ) \Deqlab{TrisanJ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4915}%
$\displaystyle A_{k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4919}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k_m}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k_m-1(w)},
\Deqlab{TrisanK}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4923}%
$\displaystyle B_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4927}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k-1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k-1(w)},
\Deqlab{TrisanL}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4931}%
$\displaystyle C_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4935}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k+1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k(w)},
\Deqlab{TrisanM}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4937}%
$\displaystyle D_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4941}%
$\displaystyle [ \Pi^{\prime} ]_{i,1}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{1}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,1}
- \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{1}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,1}
+ F_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4945}%
$\displaystyle - \beta \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{1}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,0(w)}
E_{i,0(w)}, \Deqlab{TrisanN}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4948}%
$\displaystyle D_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4952}%
$\displaystyle [ \Pi^{\prime} ]_{i,k}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,k}
- \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{k}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,k}
+ F_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4956}%
$\displaystyle (k = 2, 3, \dots , k_m -1 )
\Deqlab{TrisanO}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4958}%
$\displaystyle D_{i,k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4962}%
$\displaystyle [ \Pi^{\prime} ]_{i,k_m}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k_m}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,k_m}
- \left[ \frac{\overline{c^2} \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{k_m}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4966}%
$\displaystyle + F_{i,k_m}
+ \beta \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k_m}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k_m(w)}
E_{i,k_m(w)}. \Deqlab{TrisanP}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4971}%
$\displaystyle E_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4975}%
$\displaystyle \left[ \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right]_{i,k(w)}
- (1 - \beta )
\left[ \DP{[\Pi^{\prime}]^{\tau}}{z} \right]_{i,k(w)}
+ \left[ \frac{F^{t}_{w}}{c_{pv} \overline{\theta}}
\right]_{i,k(w)},
\Deqlab{TrisanQ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4977}%
$\displaystyle F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4981}%
$\displaystyle - \beta
\left(
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right)_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4983}%
$\displaystyle \Big[
\DP{}{z}
\Big\{ \overline{\rho} \overline{\theta}
\Big( w^{\tau} - c_{pv} \overline{\theta} \Delta \tau
\left\{
(1 - \beta) \DP{[\Pi^{\prime}]^{\tau}}{z}
- \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau \Big) \Big\} \Big]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4985}%
$\displaystyle + \left[
\frac{\overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho} \overline{\theta}}
\left( \frac{L}{c_{pv} \overline{T}} -1 \right)
\right]_{k}
[M_{cond}^{\tau}]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4989}%
$\displaystyle + \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta}^2
\overline{\Pi} } \right]_{k}
\left[ ( Q_{rad}^{\tau} + Q_{dis}^{\tau} ) \right]_{i,k}
\Deqlab{TrisanR}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4991}%
$ \tau + \Delta \tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4999}%
$\displaystyle {\rho_s}_{i,k}^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5003}%
$\displaystyle {\rho_s}_{i,k}^{\tau}
+ \left[ M_{fall} \right]_{i,k}^{\tau} \Delta \tau
+ \left[ F_{\rho_s} \right]_{i,k}^{t} \Delta \tau,
\Deqlab{TrisanU-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5007}%
$\displaystyle {\rho_s}_{i,k}^{**}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5011}%
$\displaystyle {\rm FillNeg.} [{\rho_s}_{i,k}^{*}],
\Deqlab{TrisanU-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5015}%
$\displaystyle {\rho_s}_{i,k}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5019}%
$\displaystyle {\rho_s}_{i,k}^{**}
+ \left[ M_{cond} \right]_{i,k}^{**} \Delta \tau.
\Deqlab{TrisanU-3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5021}%
$ {\rm FillNeg.} [{\rho_s}_{i,k}^{*}]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5023}%
$ {\rho_s}_{i,k}^{**}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5025}%
$ F_{\rho_s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5030}%
$\displaystyle \left[ F_{\rho_s} \right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5034}%
$\displaystyle - \left[ \DP{}{x} ({\rho_s}^t u^t) \right]_{i,k}
- \left[ \DP{}{z} ({\rho_s}^t w^t) \right]_{i,k}
+ {D_{\rho_s}}_{i,k}^{t}.
\Deqlab{TrisanV}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5038}%
$\displaystyle {\theta^{\prime}}_{i,k}^{\tau + \Delta \tau}  = {\theta^{\prime}}_{i,k}^{\tau} + {F_{\theta}}_{i,k}^{t} \Delta \tau +     \Dinv{\overline{\Pi}_{i,k}}     \frac{L \left[ M_{cond} \right]^{**}_{i,k}}    {c_{pv} \overline{\rho}_{i,k}} \Delta \tau   \Deqlab{TrisanS}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5040}%
$ F_{\theta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5043}%
$\displaystyle {F_{\theta}}_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5047}%
$\displaystyle - u_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{x} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{z} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{\overline{\theta}^t}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5051}%
$\displaystyle + \Dinv{\overline{\Pi}_{i,k}}
( Q_{rad} + Q_{dis})_{i,k}^{t}
+ [{D_{\theta}}]_{i,k}^{t - \Delta t}
+ [{D_{\theta}^{sfc}}]_{i,k}^{t}
\Deqlab{TrisanT}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5055}%
$\displaystyle {K_m}_{i,k}^{t+ \Delta t} = {K_m}_{i,k}^{t- \Delta t} + 2 \Delta t [F_{K_m}]_{i,k}^{t} \Deqlab{TrisanW}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5058}%
$\displaystyle [F_{K_m}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5062}%
$\displaystyle - \left[ u_{i(u),k} \left( \DP{K_m}{x} \right)_{i(u),k} \right]_{i,k}^t
- \left[ w_{i,k(w)} \left( \DP{K_m}{z} \right)_{i,k(w)} \right]_{i,k}^t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5064}%
$\displaystyle - \frac{3 g {C_m}^2 l^2}{2}
\left[
\Dinv{\overline{\theta}}
\left( \DP{\theta}{z} \right)_{i,k(w)}
- \frac{R}{p_0}
\left(
\frac{\overline{\theta}}{\overline{\Pi}^{c_v/R}}
\right)
\left( \DP{\rho_s}{z} \right)_{i,k(w)}
\right]^{t - \Delta t}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5066}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2 \right]_{i,k}^{t -
\Delta t}
+ {C_m}^2 l^2 \left[ \left( \DP{w}{z} \right)^2 \right]_{i,k}^{t -
\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5068}%
$\displaystyle + \frac{{C_m}^2 l^2}{2} \left[
\left\{ \left( \DP{u}{z} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \left( \DP{w}{x} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
\right]^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5070}%
$\displaystyle - \Dinv{3} [K_m]_{i,k}^{t - \Delta t}
\left[ \left( \DP{u}{x} \right)_{i,k}^{t - \Delta t}
+ \left( \DP{w}{z} \right)_{i,k}^{t - \Delta t}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5072}%
$\displaystyle + \Dinv{2} \left[
\left\{ \DP{}{x} \left( \DP{K_m^2}{x} \right)_{i(u),k}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \DP{}{z} \left( \DP{K_m^2}{z} \right)_{i,k(w)}
\right\}_{i,k}^{t - \Delta t} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5074}%
$\displaystyle + \left[ \left\{ \left( \DP{K_m}{x} \right)^2 \right\}_{i(u),k}
\right]_{i,k}^{t - \Delta t}
+ \left[ \left\{ \left( \DP{K_m}{z} \right)^2 \right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5078}%
$\displaystyle - \frac{C_{\varepsilon}}{2 C_m l^2} [K_m^2]_{i,k}^{t - \Delta t}
\Deqlab{TrisanX}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5080}%
$ t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5090}%
$ t + \Delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5096}%
$ \theta^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5100}%
$ \rho_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5102}%
$ K_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5106}%
$ \overline{\phi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5110}%
$\displaystyle \overline{\phi^t} = \phi^t + \mu_a (\overline{\phi^{t - \Delta t}} - 2  \phi^t + \phi^{t + \Delta t})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5112}%
$ \mu_a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5114}%
$ \mu_a = 0.1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5118}%
$ \bf {17}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5120}%
$ \bf {100}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5122}%
$ \bf {29}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5124}%
$ \bf {35}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5126}%
$ \bf {12}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5128}%
$ \bf {50}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5130}%
$ \bf {44}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5132}%
$ \bf {120}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\setcounter{equation}{0}
\setcounter{chapter}{1}

\renewcommand{\thechapter}{\Alph{chapter}}

\renewcommand{\theequation}{\thechapter .\arabic{equation}}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5142}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\theta} }
\DP{}{x} ( u^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5144}%
$\displaystyle - \beta \Bigg[
\frac{ \overline{c^2} }{c_{pv} \overline{\rho}
\overline{\theta}^2 }
\DP{}{z}
\Bigg(
\overline{\rho} \overline{\theta}
\Bigg\{
w^{\tau}
- c_{pv} \overline{\theta}
\left[
\beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
+ (1 - \beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]
\Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5146}%
$\displaystyle + c_{pv} \overline{\theta}
\left[ \DP{(\alpha {\rm Div})}{z} \right] \Delta \tau
+ {F_w}^{t} \Delta \tau \Bigg\}
\Bigg)
\Bigg]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5148}%
$\displaystyle - (1 - \beta) \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5154}%
$\displaystyle + \left[ \frac{ \overline{c^2} }{c_{pv} \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} )
\right]_{i,k} \Delta \tau
\Deqlab{TrisanG-Deriv1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5156}%
$ \left[ \Pi^{\prime}
\right]^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5159}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau + \Delta \tau}
- \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\left[
\DP{}{z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right\}
\right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5163}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- \left[
\frac{ \overline{c^2} }{c_{pv} \overline{\theta} }
\right]_{k}
\left[
\DP{u^{\tau + \Delta \tau}}{x}
\right]_{i,k}
\Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5165}%
$\displaystyle - \beta
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5173}%
$\displaystyle + \left[
\frac{\overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho} \overline{\theta}}
\left( \frac{L}{c_{pv} \overline{T}} -1 \right)
\right]_{k}
[M_{cond}^{\tau}]_{i,k}
+ \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta}^2
\overline{\Pi} } \right]_{k}
\left[ ( Q_{rad}^{\tau} + Q_{dis}^{\tau} ) \right]_{i,k}
\Deqlab{TrisanG-Deriv2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5186}%
$\displaystyle + \left[
\frac{\overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho} \overline{\theta}}
\left( \frac{L}{c_{pv} \overline{T}} -1 \right)
\right]_{k}
[M_{cond}^{\tau}]_{i,k}
+ \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta}^2
\overline{\Pi} } \right]_{k}
\left[ ( Q_{rad}^{\tau} + Q_{dis}^{\tau} ) \right]_{i,k}
\Deqlab{TrisanG-Deriv3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5193}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- (1 - \beta)
\left[ \frac{ \overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho}
\overline{\theta}^2 } \right]_k
\left[ \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau}) \right]_{i,k}
- \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta} }
\right]_{k}
\left[
\DP{u^{\tau + \Delta \tau}}{x}
\right]_{i,k}
+ F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5197}%
$\displaystyle \Deqlab{TrisanG-Deriv4}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5199}%
$ k=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5201}%
$ k=k_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5205}%
$ w_{i,1(w)} = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5211}%
$\displaystyle - c_{pv} \overline{\theta}_{i,0(w)}
\left[
\beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
+ (1 - \beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5213}%
$\displaystyle + c_{pv} \overline{\theta}_{i,0(w)}
\left[ \DP{(\alpha {\rm Div})}{z} \right]_{i,0(w)}
+ {F_w}_{i,0(w)}^{t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5217}%
$\displaystyle \beta
\left[ \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} \right]_{i,0(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5221}%
$\displaystyle \left[ \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right]_{i,0(w)}
- (1 - \beta )
\left[ \DP{[\Pi^{\prime}]^{\tau}}{z} \right]_{i,0(w)}
+ \left[ \frac{F^{t}_{w}}{c_{pv} \overline{\theta}}
\right]_{i,0(w)}
\Deqlab{TrisanG-Deriv5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5230}%
$\displaystyle \left[ \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right]_{i,k(w)}
- (1 - \beta )
\left[ \DP{[\Pi^{\prime}]^{\tau}}{z} \right]_{i,k(w)}
+ \left[ \frac{F^{t}_{w}}{c_{pv} \overline{\theta}}
\right]_{i,k(w)}
\Deqlab{TrisanG-Deriv6}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5235}%
$\displaystyle \beta
\left[ \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} \right]_{i,0(w)}
= E_{i,0(w)}.
\Deqlab{TrisanG-Deriv7}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5240}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,1}^{\tau + \Delta \tau}
- \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}
\left[
\DP{}{z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right\}
\right]_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5244}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,1}^{\tau + \Delta \tau}
- \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5246}%
$\displaystyle \Dinv{\Delta z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)_{1(w)}
\left[ \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} \right]_{i,1(w)}
- (c_{pv} \overline{\rho} \overline{\theta}^2)_{0(w)}
\left[ \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} \right]_{i,0(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5252}%
$\displaystyle \Dinv{\Delta z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)_{1(w)}
\Dinv{\Delta z}
\left(
\left[ \Pi^{\prime} \right]_{i,2}^{\tau + \Delta \tau}
- \left[ \Pi^{\prime} \right]_{i,1}^{\tau + \Delta \tau}
\right)
- (c_{pv} \overline{\rho} \overline{\theta}^2)_{0(w)}
\Dinv{\beta} E_{i,0(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5256}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{1(w)}
\Dinv{(\Delta z)^2}
\left[ \Pi^{\prime} \right]_{i,2}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5258}%
$\displaystyle + \left\{
1 +
\beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{1(w)}
\right\}
\left[ \Pi^{\prime} \right]_{i,1}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5262}%
$\displaystyle + \beta
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}
\Dinv{\Delta z}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{0(w)}
E_{i,0(w)}
\Deqlab{TrisanG-Deriv8}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5271}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,1}^{\tau}
- (1 - \beta)
\left[ \frac{ \overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho}
\overline{\theta}^2 } \right]_1
\left[ \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau}) \right]_{i,1}
- \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta} }
\right]_{1}
\left[
\DP{u^{\tau + \Delta \tau}}{x}
\right]_{i,1}
+ F_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5275}%
$\displaystyle + \beta
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{1}
\Dinv{\Delta z}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{0(w)}
E_{i,0(w)}
\Deqlab{TrisanG-Deriv9}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5279}%
$ w_{i,k_m(w)} = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5285}%
$\displaystyle - c_{pv} \overline{\theta}_{i,k_m(w)}
\left[ \beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} + (1 -
\beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]_{i,k_m(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5287}%
$\displaystyle + c_{pv} \overline{\theta}_{i,k_m(w)}
\left[ \DP{(\alpha {\rm Div})}{z} \right]_{i,k_m(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5289}%
$\displaystyle + {F_w}_{i,k_m(w)}^{t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5291}%
$\displaystyle E_{i,k_m(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5295}%
$\displaystyle \beta
\left[ \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} \right]_{i,k_m(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5301}%
$\displaystyle \left[ \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right]_{i,k_m(w)}
- (1 - \beta )
\left[ \DP{[\Pi^{\prime}]^{\tau}}{z} \right]_{i,k_m(w)}
+ \left[ \frac{F^{t}_{w}}{c_{pv} \overline{\theta}}
\right]_{i,k_m(w)}
\Deqlab{TrisanG-Deriv10}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5306}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k_m}^{\tau + \Delta \tau}
- \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}
\left[
\DP{}{z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right\}
\right]_{i,k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5310}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k_m}^{\tau + \Delta \tau}
- \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5312}%
$\displaystyle \Dinv{\Delta z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m(w)}
\left[
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right]_{i,k_m(w)}
- (c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m - 1(w)}
\left[
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right]_{i,k_m - 1(w)}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5318}%
$\displaystyle \Dinv{\Delta z}
\left\{
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m(w)}
\Dinv{\beta} F_{i,k_m(w)}
- (c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m - 1(w)}
\Dinv{\Delta z}
\left[
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k_m}
- [\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k_m - 1}
\right]
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5322}%
$\displaystyle \left\{
1 +
\beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m - 1(w)}
\right\}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5324}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m - 1(w)}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k_m - 1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5328}%
$\displaystyle - \beta
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m(w)}
F_{i,k_m(w)}
\Deqlab{TrisanG-Deriv11}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5337}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k_m}^{\tau}
- (1 - \beta)
\left[ \frac{ \overline{c^2} \Delta \tau }
{c_{pv} \overline{\rho}
\overline{\theta}^2 } \right]_{k_m}
\left[ \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau}) \right]_{i,k_m}
- \left[
\frac{ \overline{c^2} \Delta \tau }{c_{pv} \overline{\theta} }
\right]_{k_m}
\left[
\DP{u^{\tau + \Delta \tau}}{x}
\right]_{i,k_m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5341}%
$\displaystyle + F_{i,k}
+ \beta
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k_m}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k_m(w)}
F_{i,k_m(w)}
\Deqlab{TrisanG-Deriv12}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5343}%
$ k = 1, k_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5352}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{\Delta z}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k(w)}
\left[
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right]_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5354}%
$\displaystyle + \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{\Delta z}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k - 1(w)}
\left[
\DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z}
\right]_{i,k - 1(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5360}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k(w)}
\left[
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k+1}
- [\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5362}%
$\displaystyle + \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k - 1(w)}
\left[
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k}
- [\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k-1}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5366}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k(w)}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5368}%
$\displaystyle + \left\{
1
+ \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
\left[
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k(w)}
+ (c_{pv} \overline{\rho} \overline{\theta}^2)_{k - 1(w)}
\right]
\right\}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5372}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k - 1(w)}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k-1}
\Deqlab{TrisanG-Deriv13}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5379}%
$\displaystyle - \beta^2
\left[
\frac{\overline{c^2} \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}^2}
\right]_{k}
\Dinv{(\Delta z)^2}
(c_{pv} \overline{\rho} \overline{\theta}^2)_{k - 1(w)}
[\Pi^{\prime}]^{\tau + \Delta \tau}_{i,k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5387}%
$\displaystyle \Deqlab{TrisanG-Deriv14}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5395}%
$\displaystyle =
\begin{pmatrix}
D_{1,1} & D_{2,1} & \cdots & D_{k_m-1,1} & D_{k_m,1} \\
D_{1,2} & D_{2,2} & \cdots & D_{k_m-1,2} & D_{k_m,2} \\
\colon & \colon & & \colon & \colon \\
D_{1,k_m-1} & D_{2,k_m-1} & \cdots & D_{k_m-1,k_m-1} & D_{k_m,k_m-1} \\
D_{1,k_m} & D_{2,k_m} & \cdots & D_{k_m-1,k_m} & D_{k_m,k_m}
\end{pmatrix} ^{\tau},
\Deqlab{TrisanG-Deriv15}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5404}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_1
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,1(w)},
\Deqlab{TrisanG-Deriv16}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5414}%
$\displaystyle (k = 2, 3, \dots , k_m -1 )
\Deqlab{TrisanG-Deriv17}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5422}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k_m}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k_m-1(w)},
\Deqlab{TrisanG-Deriv18}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5430}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k-1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k-1(w)},
\Deqlab{TrisanG-Deriv19}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5438}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k+1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k(w)},
\Deqlab{TrisanG-Deriv20}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5448}%
$\displaystyle - \beta \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{1}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,0(w)}
E_{i,0(w)},
\Deqlab{TrisanG-Deriv21}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5459}%
$\displaystyle (k = 2, 3, \dots , k_m -1 )
\Deqlab{TrisanG-Deriv22}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5469}%
$\displaystyle + F_{i,k_m}
+ \beta \left[ \frac{\overline{c^2} (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k_m}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k_m(w)}
E_{i,k_m(w)}.
\Deqlab{TrisanG-Deriv23}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\setcounter{equation}{0}
\setcounter{chapter}{2}

\renewcommand{\thechapter}{\Alph{chapter}}

\renewcommand{\theequation}{\thechapter .\arabic{equation}}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5473}%
$\displaystyle QSUMPN(i,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5477}%
$\displaystyle 1.0 /$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5479}%
$\displaystyle \{
0.75 \ast [
{\rm max}( 0.0, \rho_s(i-1,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5481}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i+1,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5483}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k-1) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5485}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k+1) )
]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5487}%
$\displaystyle + 0.25 \ast [
{\rm max}( 0.0, \rho_s(i-2,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5489}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i+2,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5491}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k-2) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5495}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k+2) )
]
\}
\Deqlab{A}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5497}%
$\displaystyle DQFILL(i,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5501}%
$\displaystyle - {\rm min}( 0.0, \rho_s(i,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5503}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5505}%
$\displaystyle * \{
0.75 * [
{\rm min}( 0.0, \rho_s(i-1,k) ) * QSUMPN(i-1,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5507}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i+1,k) ) * QSUMPN(i+1,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5509}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k-1) ) * QSUMPN(i,k-1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5511}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k+1) ) * QSUMPN(i,k+1)
]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5513}%
$\displaystyle + 0.25 * [
{\rm min}( 0.0, \rho_s(i-2,k) ) * QSUMPN(i-2,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5515}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i+2,k) ) * QSUMPN(i+2,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5517}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k-2) ) * QSUMPN(i,k-2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5519}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k+2) ) * QSUMPN(i,k+2)
]
\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5523}%
$\displaystyle \Deqlab{B}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5527}%
$\displaystyle \hat{\rho_s} (i,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5531}%
$\displaystyle \rho_s (i,k) + DQFILL (i,k)
\Deqlab{C}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5535}%
$ \hat{\rho_s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5537}%
$ DQFILL(i,k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5539}%
$ 3/16$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5541}%
$ 1/16$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5543}%
$ (i,k) = (M,N)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5547}%
$ (i,k) = (M-2,N), (M-1,N), (M,N), (M+1,N), (M+2,N), (M,N-2), (M,N-1),
(M,N+1), (M,N+2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5549}%
$ DQFILL(i,k) = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5554}%
$\displaystyle DQFILL (M-2,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5558}%
$\displaystyle 0.25 * \rho_s(M-2,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5562}%
$\displaystyle \Deqlab{D}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5564}%
$\displaystyle DQFILL (M-1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5568}%
$\displaystyle 0.75 * \rho_s(M-1,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5572}%
$\displaystyle \Deqlab{E}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5574}%
$\displaystyle DQFILL (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5578}%
$\displaystyle - \rho_s (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5582}%
$\displaystyle \Deqlab{F}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5584}%
$\displaystyle DQFILL (M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5588}%
$\displaystyle 0.75 * \rho_s(M+1,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5592}%
$\displaystyle \Deqlab{G}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5594}%
$\displaystyle DQFILL (M+2,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5598}%
$\displaystyle 0.25 * \rho_s(M+2,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5602}%
$\displaystyle \Deqlab{H}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5604}%
$\displaystyle DQFILL (M,N-2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5608}%
$\displaystyle 0.25 * \rho_s(M,N-2) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5612}%
$\displaystyle \Deqlab{I}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5614}%
$\displaystyle DQFILL (M,N-1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5618}%
$\displaystyle 0.75 * \rho_s(M,N-1) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5622}%
$\displaystyle \Deqlab{J}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5624}%
$\displaystyle DQFILL (M,N+1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5628}%
$\displaystyle 0.75 * \rho_s(M,N+1) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5632}%
$\displaystyle \Deqlab{K}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5634}%
$\displaystyle DQFILL (M,N+2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5638}%
$\displaystyle 0.25 * \rho_s(M,N+2) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5642}%
$\displaystyle \Deqlab{L}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5645}%
$\displaystyle QSUMPN (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5649}%
$\displaystyle \{
0.75 \ast [
\rho_s(M-1,N)
+ \rho_s(M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5651}%
$\displaystyle + \rho_s(M,N-1)
+ \rho_s(M,N+1)
]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5653}%
$\displaystyle + 0.25 \ast [
\rho_s(M-2,N)
+ \rho_s(M+2,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5657}%
$\displaystyle + \rho_s(M,N-2)
+ \rho_s(M,N+2)
]
\}^{-1}
\Deqlab{M}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5660}%
$\displaystyle \sum_{i} \sum_{k} \hat{\rho_s} (i, k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5664}%
$\displaystyle \sum_{i} \sum_{k}
\left[ \rho_s (i,k) + DQFILL (i,k) \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5668}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5670}%
$\displaystyle + DQFILL (M-2,N)
+ DQFILL (M-1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5672}%
$\displaystyle + DQFILL (M,N)
+ DQFILL (M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5674}%
$\displaystyle + DQFILL (M+2,N)
+ DQFILL (M,N-2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5676}%
$\displaystyle + DQFILL (M,N-1)
+ DQFILL (M,N+1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5678}%
$\displaystyle + DQFILL (M,N+2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5682}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)
- \rho_s (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5684}%
$\displaystyle + \rho_s (M,N) * QSUMPN(M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5686}%
$\displaystyle \{
0.75
* [ \rho_s(M-1,N) + \rho_s(M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5688}%
$\displaystyle + \rho_s(M,N+1) + \rho_s(M,N+1) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5690}%
$\displaystyle + 0.25
* [ \rho_s(M-2,N) + \rho_s(M+2,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5692}%
$\displaystyle + \rho_s(M,N+2) + \rho_s(M,N+2) ]
\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5698}%
$\displaystyle + \rho_s (M,N) * QSUMPN(M,N) * QSUMPN(M,N)^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay5704}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)
\Deqlab{N}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
