# LaTeX2HTML 2002-2-1 (1.70) # Associate images original text with physical files. $key = q/displaystyleT;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T$ |; $key = q/c_pmathcal{D}(Dvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| $c_p \mathcal{D}(\Dvect{v})$ |; $key = q/sumh=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum h =0$ |; $key = q/e^*(T=273mbox{K})=611;MSF=1.6;AAT/; $cached_env_img{$key} = q| $e^*(T=273\mbox{K}) = 611$ |; $key = q/tau_{S,i}(z);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tau_{S,i}(z)$ |; $key = q/displaystyle+wbiggl{-frac{1}{rho}DP{p}{r}-g+underbrace{2Omegaucosvarphi}_{(2)}+uerbrace{frac{u^2}{r}}_{(4)}+underbrace{frac{v^2}{r}}_{(5)}+F_rbiggl};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + w \biggl\{ - \frac{1}{\rho} \DP{p}{r} -g + \underbrace{ 2 \Omeg... ...race{ \frac{u^2}{r} }_{(4)} + \underbrace{ \frac{v^2}{r} }_{(5)} + F_r \biggl\}$ |; $key = q/bullet^v;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bullet^v$ |; $key = q/{displaymath}DdivDvect{A}=frac{1}{r^2cosvarphi}left[rDP{A_{lambda}}{lambda}+rDP{i}(cosvarphiA_{varphi})+cosvarphiDP{}{r}(r^2A_r)right],{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Ddiv \Dvect{A} = \frac{1}{r^2 \cos \varphi} \left[ r \... ...hi A_{\varphi}) + \cos \varphi \DP{}{r} ( r^2 A_r ) \right], \end{displaymath}$ |; $key = q/{displaymath}Dgradbullet=Dvect{e}_{lambda}frac{1}{rcosvarphi}DP{bullet}{lambda}+i}frac{1}{r}DP{bullet}{varphi}+Dvect{e}_rDP{bullet}{r},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Dgrad \bullet = \Dvect{e}_{\lambda} \frac{1}{r \cos \varp... ...c{1}{r} \DP{\bullet}{\varphi} + \Dvect{e}_r \DP{\bullet}{r}, \end{displaymath}$ |; $key = q/displaystylefrac{1}{a(1-mu^2)}DP{V}{lambda}-frac{1}{a}DP{U}{mu},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a ( 1- \mu^2 )} \DP{V}{\lambda} - \frac{1}{a} \DP{U}{\mu},$ |; $key = q/{displaymath}DdivDvect{A}=frac{1}{h_1h_2h_3}left[DP{}{xi_1}(h_2h_3A_1)+DP{}{xi_2}(h_1h_3A_2)+DP{}{xi_3}(h_1h_2A_3)right],{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Ddiv \Dvect{A} = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\x... ...\xi_2} ( h_1 h_3 A_2) + \DP{}{\xi_3} ( h_1 h_2 A_3) \right], \end{displaymath}$ |; $key = q/displaystylenabla^{2}_{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla^{2}_{\sigma}$ |; $key = q/displaystylerho^{d}R^dT,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \rho^{d} R^d T,$ |; $key = q/displaystyleDP{zeta}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\zeta}{t}$ |; $key = q/displaystyle+frac{1}{2}left{TMcdotMM-TPcdotMM-frac{1}{2}TPcdotTMcdotD_{k}right}DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ TM \cdot MM - TP \cdot MM - \frac{1}{2} TP \cdot TM \cdot D_{k} \right\} \Delta T_{k}$ |; $key = q/displaystylefrac{1}{acos^2varphi}DP{vcosphi}{lambda}-frac{1}{acosvarphi}DP{}{varphi}(ucosvarphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos^2 \varphi} \DP{v \cos \phi}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi)$ |; $key = q/T_vequivT(1+epsilon_vq);MSF=1.6;AAT/; $cached_env_img{$key} = q| $T_v \equiv T ( 1 + \epsilon_v q )$ |; $key = q/r;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/tau_i(z);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tau_i(z)$ |; $key = q/p^{*}_{0}=1.4times10^{11};MSF=1.6;AAT/; $cached_env_img{$key} = q| $p^{*}_{0} = 1.4 \times 10^{11}$ |; $key = q/displaystyleDP{D}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{D}{t}$ |; $key = q/displaystylefrac{p_{k-1}}{p_{k-1slash2}},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{p_{k-1}}{p_{k-1/2}},$ |; $key = q/displaystylehat{T}_{k-1}+hat{T}_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{T}_{k-1} + \hat{T}_{k},$ |; $key = q/displaystylesinvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sin \varphi$ |; $key = q/displaystylefrac{dsigma}{dt};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{d \sigma}{d t}$ |; $key = q/displaystyle+frac{E}{2}QPcdotMMcdotTM-frac{E}{2}TPcdotQM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{E}{2} QP \cdot MM \cdot TM - \frac{E}{2} TP \cdot QM$ |; $key = q/displaystylex_2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle x_2$ |; $key = q/q=rho^vslashrho;MSF=1.6;AAT/; $cached_env_img{$key} = q| $q=\rho^v/\rho$ |; $key = q/displaystyle-dot{sigma}frac{partialq}{partialsigma}+S_{q}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \dot{\sigma} \frac{\partial q }{\partial \sigma} + S_{q}.$ |; $key = q/{displaymath}q^*(T,p)=epsilone^*(T=273mbox{K})expleft[frac{L}{R_v}left(frac{1}{273}-frac{1}{T}right)right]frac{1}{p}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} q^*(T,p) = \epsilon e^*(T=273\mbox{K}) \exp \left[ \fra... ...eft( \frac{1}{273} - \frac{1}{T} \right) \right] \frac{1}{p} \end{displaymath}$ |; $key = q/Dvect{A};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{A}$ |; $key = q/displaystyleDD{}{t}left(frac{1}{2}Dvect{v}^2right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{}{t} \left( \frac{1}{2} \Dvect{v}^2 \right)$ |; $key = q/mathcal{D}(D);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mathcal{D}(D)$ |; $key = q/displaystyle-frac{F}{4}TP^2cdotMMleft(P_{k-1}M_{k-1}-P_{k}M_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)$ |; $key = q/displaystyleleft{1+frac{L}{c_p}left.DP{q^{*}}{T}right|_{k-1}right}DeltaT_{k-1}De+frac{L}{c_p}left{q^{*}(hat{T}_{k-1})-hat{q}_{k-1}right}Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left\{ 1+ \frac{L}{c_p} \left. \DP{q^{*}}{T} \right\vert _{k-1} ... ...{L}{c_p} \left\{ q^{*} (\hat{T}_{k-1} ) - \hat{q}_{k-1} \right\} \Delta p_{k-1}$ |; $key = q/{displaymath}frac{partialpi}{partialt}=-int_{0}^{1}Dvect{v}_{H}cdotnabla_{sigma}pidsigma-int_{0}^{1}Ddsigma,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{\partial \pi}{\partial t} = - \int_{0}^{1} \Dvect{v}... ...cdot \nabla_{\sigma} \pi d \sigma - \int_{0}^{1} D d \sigma , \end{displaymath}$ |; $key = q/displaystyleU;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle U$ |; $key = q/displaystylercosvarphisinlambda,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle r \cos \varphi \sin \lambda,$ |; $key = q/(r,lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $(r,\lambda,\phi)$ |; $key = q/p_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle(c_pT_{k}+Lq_{k})Deltap_{k}+(c_pT_{k-1}+Lq_{k-1})Deltap_{k-1}=(c_pha}+Lhat{q}_{k})Deltap_{k}+(c_phat{T}_{k-1}+Lhat{q}_{k-1})Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (c_p T_{k} + L q_{k}) \Delta p_{k} + (c_p T_{k-1} + L q_{k-1}) \D... ...hat{q}_{k}) \Delta p_{k} + (c_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}$ |; $key = q/displaystyleleft.DP{q^{*}}{T}right|_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left. \DP{q^{*}}{T} \right\vert _{k},$ |; $key = q/displaystyleqquad=-St,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad = - St,$ |; $key = q/displaystylex_1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle x_1$ |; $key = q/displaystylebar{T}(sigma)+T^{prime};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \bar{T}(\sigma) + T^{\prime}$ |; $key = q/displaystylefrac{1}{acosvarphi}DP{v}{lambda}-frac{1}{acosvarphi}DP{}{varphi}(ucosvarphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos \varphi} \DP{v}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi)$ |; $key = q/displaystyleq_{k-1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q_{k-1/2}$ |; $key = q/displaystyleleft{DeltaT_{k-1}+frac{L}{c_p}(q_{k-1}-hat{q}_{k-1})right}Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left\{ \Delta T_{k-1} + \frac{L}{c_p} (q_{k-1} - \hat{q}_{k-1} ) \right\} \Delta p_{k-1}$ |; $key = q/{displaymath}F_R(z)=left(piB(T_g)-piB(T_s)right){calT}^f(z,0)+piB(T(z_T)){calT}^f(z,z_T)-int_0^{z_T}DD{piB}{xi}{calT}^f(z,xi)dxi{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} F_R(z) = \left( \pi B(T_g) - \pi B(T_s) \right) {\cal T}^f(z... ...,z_T) - \int_0^{z_T} \DD{\pi B}{\xi} {\cal T}^f(z,\xi) d \xi \end{displaymath}$ |; $key = q/%latex2htmlidmarker7335setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7335 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystyleqquadleft.-frac{1}{2}left(P_{k-1}M_{k-1}-P_{k}M_{k}right)TP^2D_{k-1}c{1}{2}left(P_{k-1}M_{k-1}-P_{k}M_{k}right)TP^2D_{k}DeltaT_{k}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. - \frac{1}{2} \left( P_{k-1} M_{k-1} - P_{k} M_{k} ... ...2} \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) TP^2 D_{k} \Delta T_{k} \right.$ |; $key = q/displaystylequad+frac{E}{2}left(TP+DeltaT_{k-1}+DeltaT_{k}right)left(QM+D_{k-1}DeltaT_{k-1}-D_{k}DeltaT_{k}right)=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \frac{E}{2} \left( TP + \Delta T_{k-1} + \Delta T_{k} \right) \left( QM + D_{k-1} \Delta T_{k-1} - D_{k} \Delta T_{k} \right) = 0$ |; $key = q/u,v,w;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{E}{2}left(-MMcdotQP-frac{1}{2}TMcdotQPcdotD_{k}+MMcdotTMcdotD_{k}right)+frac{E}{2}(-TPcdotD_{k}+QM);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left( - MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot D_{k} + MM \cdot TM \cdot D_{k} \right) + \frac{E}{2} ( - TP \cdot D_{k} + QM )$ |; $key = q/displaystylefrac{U^{2}+V^{2}}{2(1-mu^{2})};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{U^{2}+V^{2}}{2(1-\mu^{2}) }$ |; $key = q/textstyleDvect{A}=;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle \Dvect{A} =$ |; $key = q/zeta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\zeta$ |; $key = q/Dvect{g};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{g}$ |; $key = q/displaystyleB;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle B$ |; $key = q/displaystyleP_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle P_{k-1}$ |; $key = q/q=rho_vslashrho;MSF=1.6;AAT/; $cached_env_img{$key} = q| $q=\rho_v/\rho$ |; $key = q/displaystyleDD{v}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{v}{t}$ |; $key = q/{displaymath}F_S^I=-S_0r_S^{-2}coszeta.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} F_S^I = - S_0 r_S^{-2} \cos \zeta . \end{displaymath}$ |; $key = q/zeta_0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\zeta_0$ |; $key = q/{displaymath}rhoDD{varepsilon}{t}=frac{p}{rho}left(DD{rho}{t}right)+rhoQ.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \rho \DD{\varepsilon}{t} = \frac{p}{\rho} \left( \DD{\rho}{t} \right) + \rho Q. \end{displaymath}$ |; $key = q/Dvect{v}slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v}/2$ |; $key = q/displaystyleqquadleft.+MMcdotTM(D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k})right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. + MM \cdot TM (D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} ) \right\}$ |; $key = q/{displaymath}DD{u}{t}=frac{uvtanvarphi}{a}+fv-frac{1}{rhoacosvarphi}DP{p}{lambda}+F_{lambda},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{u}{t} = \frac{uv \tan \varphi}{a} + fv - \frac{1}{\rho a \cos \varphi} \DP{p}{\lambda} + F_{\lambda}, \end{displaymath}$ |; $key = q/displaystyle=-St-left(1+gamma_{k-1}-kappafrac{Deltap_{k-1slash2}}{2p_{k-1slash2}t)frac{1}{1+gamma_{k-1}}frac{L}{c_p}Deltahat{Q}frac{1}{Deltap_{k-1}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = - St - \left( 1 + \gamma_{k-1} - \kappa \frac{\Delta p_{k-1/2}}... ...rac{1}{ 1 + \gamma_{k-1}} \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}$ |; $key = q/{displaymath}DD{bullet}{t}=DP{bullet}{t}+frac{v_1}{h_1}DP{bullet}{xi_1}+frac{v_2}{h_2}DP{bullet}{xi_2}+frac{v_3}{h_3}DP{bullet}{xi_3},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\bullet}{t} = \DP{\bullet}{t} + \frac{v_1}{h_1} \DP{... ...2} \DP{\bullet}{\xi_2} + \frac{v_3}{h_3} \DP{\bullet}{\xi_3}, \end{displaymath}$ |; $key = q/S;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleqquad+kappaTleft(DP{pi}{t}+Dvect{v}_{H}cdotnabla_{sigma}pi+frac{dot{sigma}}{sigma}right)+frac{Q^*}{C_p^d}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + \kappa T \left( \DP{\pi}{t} + \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi + \frac{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^*}{C_p^d}.$ |; $key = q/nabla^{N_D};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\nabla^{N_D}$ |; $key = q/%latex2htmlidmarker6973setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6973 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystyle+frac{E}{2}left{TPcdotQM+(TPcdotD_{k-1}+QM)DeltaT_{k-1}+(-TPcdotD_{k}+QM)DeltaT_{k}right}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{E}{2} \left\{ TP \cdot QM + ( TP \cdot D_{k-1} + QM ) \Delta T_{k-1} + ( - TP \cdot D_{k} + QM ) \Delta T_{k} \right\} = 0$ |; $key = q/Delta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Delta$ |; $key = q/displaystyleDeltahat{Q}equivleft{hat{q}_{k-1}-q^{*}(hat{T}_{k-1})right}Deltap_{k-1}+left{hat{q}_{k}-q^{*}(hat{T}_{k})right}Deltap_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta \hat{Q} \equiv \left\{ \hat{q}_{k-1} - q^{*} (\hat{T}_{k-1... ...Delta p_{k-1} + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\} \Delta p_{k}$ |; $key = q/e^*;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle=frac{-St-left(1+gamma_{k-1}-kappafrac{Deltap_{k-1slash2}}{2p_{k-1slrac{1+gamma_{k}}{1+gamma_{k-1}}frac{Deltap_{k}}{Deltap_{k-1}}right)};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = \frac{ - St - \left( 1 + \gamma_{k-1} - \kappa \frac{\Delta p_{... ...+ \gamma_{k} }{ 1 + \gamma_{k-1}} \frac{\Delta p_{k}}{\Delta p_{k-1}} \right) }$ |; $key = q/r_S;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylequadleft.+MMcdotTM(D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k})right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad \left. + MM \cdot TM (D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} ) \right\}$ |; $key = q/displaystylefrac{1}{a(1-mu^{2})}frac{partialV}{partiallambda}-frac{1}{a}frac{partialU}{partialmu};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a ( 1-\mu^{2} ) } \frac{\partial V}{\partial \lambda} - \frac{1}{a} \frac{\partial U}{\partial \mu}$ |; $key = q/{calD}'(Dvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal D}' (\Dvect{v})$ |; $key = q/displaystyleq^{*}(hat{T}_{k},p_{k})+left.DP{q^{*}}{T}right|_{hat{T}_{k}}cdotDeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (\hat{T}_{k},p_{k}) + \left. \DP{q^{*}}{T} \right\vert _{\hat{T}_{k}} \cdot \Delta T_{k}$ |; $key = q/displaystyleDP{T}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{T}{t}$ |; $key = q/displaystyle-frac{1}{acosvarphi}DP{}{varphi}(zetavcosvarphi)-frac{1}{acosvarphi}DP{}{lambda}(zetau);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( \zeta v \cos \varphi ) - \frac{1}{a \cos \varphi} \DP{}{\lambda} ( \zeta u )$ |; $key = q/displaystyleD;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle D$ |; $key = q/{displaymath}overline{F_S^I}simeq-S_0(0.127+0.183cos^2varphi),{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \overline{F_S^I} \simeq - S_0 ( 0.127 + 0.183 \cos^2 \varphi ) , \end{displaymath}$ |; $key = q/displaystyle-frac{E}{2}QPcdotMMcdotTM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} QP \cdot MM \cdot TM$ |; $key = q/displaystylembox{{sl{VA};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \mbox{\sl VA}$ |; $key = q/displaystylep^v;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle p^v$ |; $key = q/displaystyle+frac{1}{2}left{TMcdotMM-TPcdotMM-frac{1}{2}TPcdotTMcdotD_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ TM \cdot MM - TP \cdot MM - \frac{1}{2} TP \cdot TM \cdot D_{k} \right\}$ |; $key = q/displaystyleubiggl{-frac{1}{rhorcosvarphi}DP{p}{lambda}+underbrace{2vOmegasinvar{uv}{r}tanvarphi}_{(3)}-underbrace{frac{uw}{r}}_{(4)}+F_lambdabiggl};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle u \biggl\{ - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambda} + \un... ... \tan \varphi }_{(3)} - \underbrace{ \frac{u w}{r} }_{(4)} + F_\lambda \biggl\}$ |; $key = q/displaystyle+frac{E}{2}TPcdotQM+frac{E}{2}(TPcdotD_{k-1}+QM)DeltaT_{k-1}+frac{E}{2}(-TPcdotD_{k}+QM)DeltaT_{k}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{E}{2} TP \cdot QM + \frac{E}{2} ( TP \cdot D_{k-1} + QM ) \Delta T_{k-1} + \frac{E}{2} ( - TP \cdot D_{k} + QM ) \Delta T_{k} = 0$ |; $key = q/Q^*;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/St;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}Dequivfrac{1}{acosvarphi}DP{u}{lambda}+frac{1}{acosvarphi}DP{}{varphi}(vcosvarphi).{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} D \equiv \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi). \end{displaymath}$ |; $key = q/displaystyleDvect{i}A_x+Dvect{j}A_y+Dvect{k}A_z;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{i} A_x + \Dvect{j} A_y + \Dvect{k} A_z$ |; $key = q/displaystyleM_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle M_{k}$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}DP{UT'}{lambda}-frac{1}{a}DP{VT'}{mu}+T'D.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \DP{UT'}{\lambda} - \frac{1}{a} \DP{VT'}{\mu} + T' D.$ |; $key = q/displaystyleqquadtimes(hat{T}_{k-1}+DeltaT_{k-1}-hat{T}_{k}-DeltaT_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times (\hat{T}_{k-1} + \Delta T_{k-1} - \hat{T}_{k} - \Delta T_{k})$ |; $key = q/displaystyleqquadtimesleft{1-frac{1}{2}left(q^{*}(hat{T}_{k-1})+left.DP{q^{*}}{T+q^{*}(hat{T}_{k})+left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left\{ 1 - \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}... ...t{T}_{k}) + \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right) \right\}$ |; $key = q/displaystyleleft[left(-1-gamma_{k}right)-kappafrac{Deltap_{k-1slash2}}{2p_{k-1sl}{1+gamma_{k-1}}frac{Deltap_{k}}{Deltap_{k-1}}right)right]DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left[ \left( - 1 - \gamma_{k} \right) - \kappa \frac{\Delta p_{k... ... \gamma_{k-1}} \frac{\Delta p_{k}}{\Delta p_{k-1}} \right) \right] \Delta T_{k}$ |; $key = q/displaystyle1-q^{*}(hat{T}_{k-1}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 1 - q^{*} (\hat{T}_{k-1}),$ |; $key = q/displaystyle-frac{1}{rhorcosvarphi}DP{p}{lambda}+2Omegavsinvarphi-2Omegawcosvarphi+frac{uv}{r}tanvarphi-frac{uw}{r}+F_lambda,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambda} + 2 \Omega v \si... ...\Omega w \cos \varphi + \frac{u v}{r} \tan \varphi - \frac{u w}{r} + F_\lambda,$ |; $key = q/displaystyle+frac{1}{2}v_i^2DP{rho}{t}+frac{1}{2}v_i^2DP{}{x_j}(rhov_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} v_i^2 \DP{\rho}{t} + \frac{1}{2} v_i^2 \DP{}{x_j} ( \rho v_j )$ |; $key = q/{displaymath}frac{partialPhi}{partialsigma}=-frac{RT_v}{sigma}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{\partial \Phi}{\partial \sigma} = - \frac{RT_v}{\sigma} \end{displaymath}$ |; $key = q/Dvect{r};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{r}$ |; $key = q/displaystyleD_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle D_{k-1}$ |; $key = q/displaystyleTM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle TM$ |; $key = q/{displaymath}dot{sigma}=0atsigma=0,1.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \dot{\sigma} = 0 at \sigma = 0 , 1 . \end{displaymath}$ |; $key = q/{displaymath}DD{}{t}=DP{}{t}+frac{u}{acosvarphi}DP{}{lambda}+frac{v}{a}DP{}{varphi}+wDP{}{z},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{}{t} = \DP{}{t} + \frac{u}{a \cos \varphi} \DP{}{\lambda} + \frac{v}{a} \DP{}{\varphi} + w \DP{}{z}, \end{displaymath}$ |; $key = q/L;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}DP{zeta}{t}=-frac{1}{a(1-mu^2)}DP{VA}{lambda}-frac{1}{a}DP{UA}{mu},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{\zeta}{t} = -\frac{1}{a (1-\mu^2)} \DP{VA}{\lambda} -\frac{1}{a} \DP{UA}{\mu}, \end{displaymath}$ |; $key = q/{displaymath}DD{rho}{t}+frac{1}{rcosvarphi}DP{}{lambda}(u)+frac{1}{rcosvarphi}DP{}{varphi}(cosvarphiv)+frac{1}{r^2}DP{}{r}(r^2w)=0.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\rho}{t} + \frac{1}{r \cos \varphi} \DP{}{\lambda} ( ... ...i} ( \cos \varphi v) + \frac{1}{r^2} \DP{}{r} ( r^2 w ) = 0. \end{displaymath}$ |; $key = q/{displaymath}coszeta=cosvarphicosdelta_ScosH+sinvarphisindelta_S{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \cos \zeta = \cos \varphi \cos \delta_S \cos H + \sin \varphi \sin \delta_S \end{displaymath}$ |; $key = q/displaystyleqquadleft.+left(P_{k-1}M_{k-1}-P_{k}M_{k}right)left{-frac{1}{2}TP^2lD_{k}DeltaT_{k}right)+2TPcdotMM(DeltaT_{k-1}+DeltaT_{k})right}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) \lef... ... T_{k} \right) + 2 TP \cdot MM (\Delta T_{k-1} + \Delta T_{k}) \right\} \right]$ |; $key = q/displaystylequad+frac{1}{2}left{MM-frac{1}{2}left(D_{k-1}DeltaT_{k-1}+D_{k}Deltaht}left(TP+DeltaT_{k-1}+DeltaT_{k}right)(TM+DeltaT_{k-1}-DeltaT_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \frac{1}{2} \left\{ MM - \frac{1}{2} \left( D_{k-1} \Delt... ...TP + \Delta T_{k-1} + \Delta T_{k} \right) (TM + \Delta T_{k-1} - \Delta T_{k})$ |; $key = q/displaystyle=frac{(1+gamma_{k-1})Deltap_{k-1}St+left(1+gamma_{k-1}-kappafrac{Delsh2}}left{(1+gamma_{k-1})Deltap_{k-1}-(1+gamma_{k})Deltap_{k}right}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = \frac{ (1 + \gamma_{k-1}) \Delta p_{k-1} St + \left( 1 + \gamma... ...\{ (1 + \gamma_{k-1}) \Delta p_{k-1} - (1 + \gamma_{k}) \Delta p_{k} \right\} }$ |; $key = q/alpha_A;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\alpha_A$ |; $key = q/u,v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/q(T_{k-1},p_{k-1});MSF=1.6;AAT/; $cached_env_img{$key} = q| $q (T_{k-1}, p_{k-1})$ |; $key = q/D;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}rhoDD{Dvect{v}}{t}+Dgradp+rhoDgradPhi^*=Dvect{F}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \rho \DD{\Dvect{v}}{t} + \Dgrad p + \rho \Dgrad \Phi^* = \Dvect{F}. \end{displaymath}$ |; $key = q/{calT}^f(z_1,z_2);MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal T}^f(z_1,z_2)$ |; $key = q/Q;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/%latex2htmlidmarker7326setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7326 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/R;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/N_D;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/k_{S,i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $k_{S,i}$ |; $key = q/displaystyleDvect{i}left(DD{A_x}{t}right)_{{rm{a}+Dvect{j}left(DD{A_y}{t}right)_{{rm{a}+Dvect{k}left(DD{A_z}{t}right)_{{rm{a};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{i} \left( \DD{A_x}{t} \right)_{\rm a} + \Dvect{j} \left( \DD{A_y}{t} \right)_{\rm a} + \Dvect{k} \left( \DD{A_z}{t} \right)_{\rm a}$ |; $key = q/S_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}tau_{R,i}(z)=int_z^inftyk_{R,i}rhoqdz+int_z^inftybar{k}_{R,i}rhodz{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \tau_{R,i}(z) = \int_z^\infty k_{R,i} \rho q dz + \int_z^\infty \bar{k}_{R,i} \rho dz \end{displaymath}$ |; $key = q/displaystyleBleft(alphaDeltaT_{k}+betaright)+CT_{k}=St;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle B \left( \alpha \Delta T_{k} + \beta \right) + C T_{k} = St$ |; $key = q/{displaymath}DP{q^*}{T}(T,p)=frac{L}{R_vT^2}epsilone^*(T=273mbox{K})expleft[fracR_v}left(frac{1}{273}-frac{1}{T}right)right]frac{1}{p}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{q^*}{T} (T,p) = \frac{L}{R_v T^2} \epsilon e^*(T=2... ...ft( \frac{1}{273} - \frac{1}{T} \right) \right] \frac{1}{p}. \end{displaymath}$ |; $key = q/^{-2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $^{-2}$ |; $key = q/displaystyleleft.-alpha_g(1-alpha_A)F_S^Iexpleft(-tau_{S,i}(0)seczetaright)expleft(-(tau_{S,i}(0)-tau_i(z))seczeta_0right)right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left. - \alpha_g (1-\alpha_A) F_S^I \exp \left( - \tau_{S,i}(0) ... ... \right) \exp \left( - ( \tau_{S,i}(0)-\tau_i(z) ) \sec \zeta_0 \right) \right]$ |; $key = q/(u,v,w);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/frac{1}{acosvarphi}DP{}{lambda};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{1}{a \cos \varphi} \DP{}{\lambda}$ |; $key = q/displaystyle-frac{1}{acosvarphi}DP{}{lambda}left[dot{sigma}DP{u}{sigma}+frac{R^dT_v}{acosvarphip_s}DP{p_s}{lambda}-F_{lambda}-fvright];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigma} \DP... ...frac{R^d T_v}{a \cos \varphi p_s} \DP{p_s}{\lambda} - F_{\lambda} - f v \right]$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[+frac{R^dbar{T}_v}{a}(1-mu^2)DP{a(1-mu^2)}DP{}{lambda}left[+frac{R^dT'_v}{a}(1-mu^2)DP{pi}{mu}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d \bar{T}... ...^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a} (1-\mu^2) \DP{\pi}{\mu} \right]$ |; $key = q/displaystyle=-left[1+frac{L}{c_p}left.DP{q^{*}}{T}right|_{k}right]DeltaT_{k}Deltp_{k}-frac{L}{c_p}left{q^{*}(hat{T}_{k})-hat{q}_{k}right}Deltap_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = - \left[ 1 + \frac{L}{c_p} \left. \DP{q^{*}}{T} \right\vert _{k... ... \frac{L}{c_p} \left\{ q^{*} (\hat{T}_{k}) - \hat{q}_{k} \right\} \Delta p_{k},$ |; $key = q/displaystyleTleft(pslashp_{0}right)^{-kappa};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T \left( p/p_{0} \right)^{- \kappa}$ |; $key = q/displaystyle1-q^{*}(hat{T}_{k}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 1 - q^{*} (\hat{T}_{k}),$ |; $key = q/{displaymath}DD{q}{t}=S_q.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{q}{t} = S_q. \end{displaymath}$ |; $key = q/z=z_1,z_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylefrac{1}{a(1-mu^2)}DP{}{lambda}(Ubullet)-frac{bullet}{a(1-mu^2)}DP{U}{lambda}+frac{1}{amu}DP{}{phi}(Vbullet)-frac{bullet}{a}DP{V}{mu};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a (1-\mu^2)} \DP{}{\lambda} (U\bullet) -\frac{\bullet}{a... ...\lambda} +\frac{1}{a \mu} \DP{}{\phi} (V\bullet) -\frac{\bullet}{a} \DP{V}{\mu}$ |; $key = q/{displaymath}overline{coszeta}simeq0.410+0.590cos^2varphi.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \overline{\cos \zeta} \simeq 0.410 + 0.590 \cos^2 \varphi . \end{displaymath}$ |; $key = q/{displaymath}Ddiv{Dvect{v}}equivfrac{1}{acosvarphi}DP{u}{lambda}+frac{1}{acosvarphi}DP{v}{varphi}(vcosvarphi)+DP{w}{z}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Ddiv{\Dvect{v}} \equiv \frac{1}{a \cos \varphi} \DP{u}{\l... ...cos \varphi} \DP{v}{\varphi} ( v \cos \varphi ) + \DP{w}{z}. \end{displaymath}$ |; $key = q/{displaymath}nabla^2bullet=frac{1}{r^2cosvarphi}left[DP{}{lambda}left(frac{1}{coght)+DP{}{r}left(r^2cosvarphiDP{bullet}{r}right)right],{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \nabla^2 \bullet = \frac{1}{r^2 \cos \varphi} \left[ \D... ...}{r} \left( r^2 \cos \varphi \DP{\bullet}{r} \right) \right], \end{displaymath}$ |; $key = q/DeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Delta T_{k-1}$ |; $key = q/displaystyleq^{*}(T_{k},p_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (T_{k}, p_{k})$ |; $key = q/{displaymath}DP{pi}{t}+Dvect{v}_HcdotDgrad_{sigma}pi=-Dgrad_{sigma}cdotDvect{v}_H-DP{dot{sigma}}{sigma}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{\pi}{t} + \Dvect{v}_H \cdot \Dgrad_{\sigma} \pi = -\Dgrad_{\sigma} \cdot \Dvect{v}_H - \DP{\dot{\sigma}}{\sigma}. \end{displaymath}$ |; $key = q/displaystyleDvect{v}_HcdotDgrad_{sigma}bullet.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{v}_H \cdot \Dgrad_{\sigma} \bullet.$ |; $key = q/displaystylelnp_{S};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \ln p_{S}$ |; $key = q/mathcal{D}(T);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mathcal{D}(T)$ |; $key = q/displaystyleDeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta T_{k}$ |; $key = q/(xi_1,xi_2,xi_3)=(lambda,varphi,r);MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\xi_1, \xi_2, \xi_3) = (\lambda, \varphi, r)$ |; $key = q/displaystylepi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \pi$ |; $key = q/displaystylerho^{v}R^vT,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \rho^{v} R^v T,$ |; $key = q/displaystylequad+left{1-frac{1}{2}left(q^{*}(hat{T}_{k-1})+left.DP{q^{*}}{T}righ+q^{*}(hat{T}_{k})+left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \left\{ 1 - \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}) + \l... ...t{T}_{k}) + \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right) \right\}$ |; $key = q/F_S^I;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylefrac{1}{acos^2phi}left{DP{}{lambda}(ucosphibullet)-bulletDP{}{lambdaacosphi}left{DP{}{phi}(vcosphibullet)-bulletDP{}{phi}(vcosphi)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos^2 \phi} \left\{ \DP{}{\lambda} (u \cos \phi \bull... ...DP{}{\phi} (v \cos \phi \bullet) - \bullet \DP{}{\phi} ( v \cos \phi ) \right\}$ |; $key = q/displaystylequadleft.-TPcdotTMleft(frac{1}{2}D_{k-1}DeltaT_{k-1}+frac{1}{2}D_{k}DeltaT_{k}right)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad \left. - TP \cdot TM \left( \frac{1}{2} D_{k-1} \Delta T_{k-1} + \frac{1}{2} D_{k} \Delta T_{k} \right) \right\}$ |; $key = q/d_{H_2O};MSF=1.6;AAT/; $cached_env_img{$key} = q| $d_{H_2O}$ |; $key = q/displaystyleVA;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle VA$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}DP{Ubar{T}}{lambda}-frac{1}{a(1-mu^{2})}DP{UT'}{lambda}-frac{1}{a}DP{Vbar{T}}{mu}-frac{1}{a}DP{VT'}{mu}+bar{T}D+T'D;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \DP{U \bar{T}}{\lambda} - \frac{1}{a(1-\... ... \frac{1}{a} \DP{V \bar{T}}{\mu} - \frac{1}{a} \DP{VT'}{\mu} + \bar{T} D + T' D$ |; $key = q/{displaymath}DD{}{t}=DP{}{t}+frac{u}{rcosphi}DP{}{lambda}+frac{v}{r}DP{}{phi}+wDP{}{r},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{}{t} = \DP{}{t} + \frac{u}{r \cos \phi} \DP{}{\lambda} + \frac{v}{r} \DP{}{\phi} + w \DP{}{r}, \end{displaymath}$ |; $key = q/{displaymath}DD{Dvect{v}_a}{t}=DD{Dvect{v}}{t}+2Dvect{Omega}timesDvect{v}+Dvect{Omega}times(Dvect{Omega}timesDvect{r}),{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\Dvect{v}_a}{t} = \DD{\Dvect{v}}{t} + 2 \Dvect{\Omega}... ... + \Dvect{\Omega} \times ( \Dvect{\Omega} \times \Dvect{r} ), \end{displaymath}$ |; $key = q/%latex2htmlidmarker6951setcounter{footnote}{2}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6951 \setcounter{footnote}{2}\fnsymbol{footnote}$ |; $key = q/displaystyle-Dgrad^2_{sigma}(R^dbar{T}_vpi)-frac{1}{a(1-mu^2)}DP{}{lambda}left[fht]-frac{1}{a}DP{}{mu}left[frac{R^dT'_v}{a}(1-mu^2)DP{pi}{mu}right].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \Dgrad^2_{\sigma} ( R^d \bar{T}_v \pi ) - \frac{1}{a (1-\mu^2)}... ...c{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right].$ |; $key = q/displaystyle(zeta+f)V-dot{sigma}DP{U}{sigma}-frac{RT'}{a}DP{pi}{lambda}+F_{lambda}cosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle ( \zeta + f ) V - \dot{\sigma} \DP{U}{\sigma} - \frac{RT'}{a} \DP{\pi}{\lambda} + F_{\lambda} \cos \varphi$ |; $key = q/displaystyleuDD{u}{t}+vDD{v}{t}+wDD{w}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle u \DD{u}{t} + v \DD{v}{t} + w \DD{w}{t}$ |; $key = q/displaystyleqquad-lq_{k-1slash2}(1-q_{k-1slash2})(T_{k-1}-T_{k})+lT_{k-1slash2}(q_{k-1}-q_{k})=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad - l q_{k-1/2} (1-q_{k-1/2}) (T_{k-1} - T_{k}) + l T_{k-1/2} (q_{k-1} - q_{k}) = 0,$ |; $key = q/{displaymath}{calD}(D)=-K_{HD}left[(-1)^{N_Dslash2}nabla^{N_D}-left(frac{2}{a^2}right)^{N_Dslash2}right]D,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} {\cal D}(D) = - K_{HD} \left[ (-1)^{N_D/2} \nabla^{N_D} - \left( \frac{2}{a^2} \right)^{N_D/2} \right] D , \end{displaymath}$ |; $key = q/{displaymath}DD{Dvect{v}}{t}=-frac{1}{rho}Dgradp-2Dvect{Omega}timesDvect{v}-Dveca}times(Dvect{Omega}timesDvect{r})+DgradPhi^*+Dvect{F}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect... ...vect{\Omega} \times \Dvect{r} ) + \Dgrad \Phi^* + \Dvect{F}. \end{displaymath}$ |; $key = q/displaystyle-frac{F}{4}left[TP^2cdotMMleft(P_{k-1}M_{k-1}-P_{k}M_{k}right)+TP^2ctMMleft{-P_{k-1}D_{k-1}DeltaT_{k-1}+P_{k}D_{k}DeltaT_{k}right}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left[ TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k}... ...\{ - P_{k-1} D_{k-1} \Delta T_{k-1} + P_{k} D_{k} \Delta T_{k} \right\} \right.$ |; $key = q/displaystylefrac{1}{a^{2}(1-mu^{2})}frac{partial^{2}}{partiallambda^{2}}+frac{1}ac{partial}{partialmu}left[(1-mu^{2})frac{partial}{partialmu}right].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a^{2}(1-\mu^{2})} \frac{\partial^{2} }{\partial \lambda^... ...ial }{\partial \mu} \left[ (1-\mu^{2}) \frac{\partial }{\partial \mu} \right] .$ |; $key = q/H;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/dz;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}DD{}{t}T=frac{1}{c_p^drho}DD{p}{t}+frac{Q^*}{c_p^d}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{}{t} T = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d}. \end{displaymath}$ |; $key = q/displaystyleC;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle C$ |; $key = q/C_v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/%latex2htmlidmarker6861setcounter{footnote}{1}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6861 \setcounter{footnote}{1}\fnsymbol{footnote}$ |; $key = q/displaystyleT_{k}=frac{St-betaB}{C+alphaB};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_{k} = \frac{St - \beta B }{C + \alpha B}$ |; $key = q/displaystyle-frac{1}{a^2}DP{}{mu}left[(1-mu^2)DP{}{mu}(R^dbar{T}_vpi)right]-frac{1}{a}DP{}{mu}left[frac{R^dT'_v}{a}(1-mu^2)DP{pi}{mu}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a^2} \DP{}{\mu} \left[ (1-\mu^2) \DP{}{\mu} ( R^d \bar... ...ac{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$ |; $key = q/{displaymath}DD{q}{t}=S_q,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{q}{t} = S_q, \end{displaymath}$ |; $key = q/%latex2htmlidmarker6865setcounter{footnote}{2}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6865 \setcounter{footnote}{2}\fnsymbol{footnote}$ |; $key = q/displaystyleDvect{v}_{H}cdotnabla;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{v}_{H} \cdot \nabla$ |; $key = q/displaystyle-frac{R_d}{c_p}left(frac{T_{k-1}+T_{k}}{2}right)^2left(1-frac{q_{k-1{p_{k-1slash2}}(1-q_{k-1})-frac{p_{k}}{p_{k-1slash2}}(1-q_{k})right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{c_p} \left( \frac{T_{k-1} + T_{k}}{2} \right)^2 \lef... ...rac{p_{k-1}}{p_{k-1/2}} (1-q_{k-1}) - \frac{p_{k}}{p_{k-1/2}} (1-q_{k}) \right]$ |; $key = q/displaystyleqquadtimesleft(P_{k-1}M_{k-1}-P_{k-1}D_{k-1}DeltaT_{k-1}-P_{k}M_{k}+P_{k}D_{k}DeltaT_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left( P_{k-1} M_{k-1} - P_{k-1} D_{k-1} \Delta T_{k-1} - P_{k} M_{k} + P_{k} D_{k} \Delta T_{k} \right)$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[+frac{R^dT'_v}{a}(1-mu^2)DP{pi}{right]-frac{1}{a}DP{}{mu}left[-frac{R^dT'_v}{a}DP{pi}{lambda}right].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a... ...- \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right].$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[dot{sigma}DP{U}{sigma}+frac{R^dT_v}{a}DP{pi}{lambda}-F_{lambda}cosvarphi-fVright];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \dot{\sigma} \DP{U}... ... + \frac{R^d T_v}{a} \DP{\pi}{\lambda} - F_{\lambda} \cos \varphi - f V \right]$ |; $key = q/displaystyle(c_pT_{k-1}+Lq_{k-1})Deltap_{k-1}-(c_phat{T}_{k-1}+Lhat{q}_{k-1})Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (c_p T_{k-1} + L q_{k-1}) \Delta p_{k-1} - (c_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}$ |; $key = q/Dvect{A}=(A_1,A_2,A_3);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{A}=(A_1, A_2, A_3)$ |; $key = q/displaystyle-frac{F}{4}left{-TP^2cdotMMcdotP_{k-1}cdotD_{k-1}+left(P_{k-1}M_{k-1_{k}M_{k}right)left(-frac{1}{2}TP^2cdotD_{k-1}+2TPcdotMMright)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ - TP^2 \cdot MM \cdot P_{k-1} \cdot D_{k-1}... ...\right) \left(- \frac{1}{2} TP^2 \cdot D_{k-1} + 2 TP \cdot MM \right) \right\}$ |; $key = q/bar{T}_v(sigma);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bar{T}_v(\sigma)$ |; $key = q/displaystyle+left(1+gamma_{k-1}-frac{R}{c_p}frac{Deltap_{k-1slash2}}{2p_{k-1slas{1}{1+gamma_{k-1}}frac{L}{c_p}Deltahat{Q}frac{1}{Deltap_{k-1}}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \left( 1 + \gamma_{k-1} - \frac{R}{c_p} \frac{\Delta p_{k-1/2}}... ...1 + \gamma_{k-1}} \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}} \right)$ |; $key = q/k_{R,i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $k_{R,i}$ |; $key = q/displaystylegz;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle gz$ |; $key = q/displaystyleDD{s}{z}<0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{s}{z} < 0$ |; $key = q/%latex2htmlidmarker7325setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7325 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystyle-frac{1+gamma_{k}}{1+gamma_{k-1}}frac{Deltap_{k}}{Deltap_{k-1}}Delta}+frac{1}{1+gamma_{k-1}}frac{L}{c_p}Deltahat{Q}frac{1}{Deltap_{k-1}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1 + \gamma_{k}}{1+\gamma_{k-1}} \frac{\Delta p_{k}}{\Delt... ... \frac{1}{1+\gamma_{k-1}} \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}$ |; $key = q/displaystyleqquadtimesleft(q^{*}(hat{T}_{k-1})+left.DP{q^{*}}{T}right|_{k-1}Deltk-1}-q^{*}(hat{T}_{k})-left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right)=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left( q^{*} (\hat{T}_{k-1}) + \left. \DP{q^{*}}{T}... ... (\hat{T}_{k}) - \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right) = 0$ |; $key = q/{displaymath}zetaequivfrac{1}{acosvarphi}DP{v}{lambda}-frac{1}{acosvarphi}DP{}{varphi}(ucosvarphi).{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \zeta \equiv \frac{1}{a \cos \varphi} \DP{v}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi). \end{displaymath}$ |; $key = q/displaystylehat{T}_{k-1}-hat{T}_{k}+frac{L}{c_p}left{q^{*}(hat{T}_{k-1})-q^{*}(hc{Deltap_{k-1slash2}}{p_{k-1slash2}}frac{hat{T}_{k-1}+hat{T}_{k}}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{T}_{k-1} - \hat{T}_{k} + \frac{L}{c_p} \left\{ q^{*} (\hat{T... ...{c_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}$ |; $key = q/displaystylefrac{T_{k-1}+T_{k}}{2},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{T_{k-1} + T_{k}}{2},$ |; $key = q/displaystyleqquad+c_pleft(1-frac{q_{k-1}+q_{k}}{2}right)frac{T_{k-1}+T_{k}}{2}(T_{k-1}-T_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + c_p \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) \frac{T_{k-1} + T_{k}}{2} (T_{k-1} - T_{k})$ |; $key = q/displaystyleqquadleft.+2left(P_{k-1}M_{k-1}-P_{k}M_{k}right)TPcdotMMDeltaT_{k-1}+2left(P_{k-1}M_{k-1}-P_{k}M_{k}right)TPcdotMMDeltaT_{k}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. + 2 \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) TP... ...2 \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) TP \cdot MM \Delta T_{k} \right]$ |; $key = q/F_S;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{F}{4}left(TP+DeltaT_{k-1}+DeltaT_{k}right)^2left{MM-frac{1}{2}D_{k-1}DeltaT_{k-1}-frac{1}{2}D_{k}DeltaT_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left( TP + \Delta T_{k-1} + \Delta T_{k} \right)^2... ... - \frac{1}{2} D_{k-1} \Delta T_{k-1} - \frac{1}{2} D_{k} \Delta T_{k} \right\}$ |; $key = q/displaystyleDvect{i}'left(DD{A'_x}{t}right)_{{rm{r}+Dvect{j}'left(DD{A'_y}{t}rig}'A'_x+Dvect{Omega}timesDvect{j}'A'_y+Dvect{Omega}timesDvect{k}'A'_z;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{i}' \left( \DD{A'_x}{t} \right)_{\rm r} + \Dvect{j}' \left... ...+ \Dvect{\Omega} \times \Dvect{j}' A'_y + \Dvect{\Omega} \times \Dvect{k}' A'_z$ |; $key = q/displaystyle-frac{u^2tanvarphi}{a}-fu-frac{1}{rhor}DP{p}{varphi}+F_{varphi}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho r } \DP{p}{\varphi} + F_{\varphi}.$ |; $key = q/i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/times10^{-8};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\times 10^{-8}$ |; $key = q/^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $^{-1}$ |; $key = q/displaystylefrac{1}{acosvarphi}DP{}{lambda}(zetav)-frac{1}{acosvarphi}DP{}{varphi}(zetaucosvarphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos \varphi} \DP{}{\lambda} ( \zeta v ) - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( \zeta u \cos \varphi)$ |; $key = q/c_p^d;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{R_d}{p(1-q)}DD{}{z}left{p(1-q)right}+frac{1}{1-q}frac{c_p}{T}D{T^2}frac{q}{1-q}DD{T}{z}+frac{l}{T}DD{}{z}left(frac{q}{1-q}right)=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\} + \frac{1... ...\frac{q}{1-q} \DD{T}{z} + \frac{l}{T} \DD{}{z} \left( \frac{q}{1-q} \right) = 0$ |; $key = q/{displaymath}DP{}{t}left(frac{1}{2}rhov_i^2+rhoPhi^*right)+DP{}{x_j}left(frac{1}v_iright)=pDP{v_j}{x_j}-sigma_{ij}DP{v_i}{x_j}+F'_iv_i,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{}{t} \left( \frac{1}{2} \rho v_i^2 + \rho \Phi^* \right... ...ht) = p \DP{v_j}{x_j} - \sigma_{ij} \DP{v_i}{x_j} + F'_i v_i, \end{displaymath}$ |; $key = q/a_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylefrac{F}{4}TP^2cdotMMleft(P_{k-1}M_{k-1}-P_{k}M_{k}right)-frac{1}{2}MMcdotTPcdotTM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{F}{4} TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) - \frac{1}{2} MM \cdot TP \cdot TM$ |; $key = q/%latex2htmlidmarker7331setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7331 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystyle+left(1+gamma_{k-1}-kappafrac{Deltap_{k-1slash2}}{2p_{k-1slash2}}rig{1}{1+gamma_{k-1}}frac{L}{c_p}Deltahat{Q}frac{1}{Deltap_{k-1}}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \left( 1 + \gamma_{k-1} - \kappa \frac{\Delta p_{k-1/2}}{2 p_{k... ...1 + \gamma_{k-1}} \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}} \right)$ |; $key = q/displaystyleleft(DD{Dvect{A}}{t}right)_{{rm{a};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( \DD{\Dvect{A}}{t} \right)_{\rm a}$ |; $key = q/S_q;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle=-St;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = - St$ |; $key = q/T_v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/qll1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $q \ll 1$ |; $key = q/displaystyle-frac{E}{2}left(-MMcdotQP-frac{1}{2}TMcdotQPcdotD_{k}+MMcdotTMcdotD_{k}right)DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left( - MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot D_{k} + MM \cdot TM \cdot D_{k} \right) \Delta T_{k}$ |; $key = q/displaystyleDvect{i}'left(DD{A'_x}{t}right)_{{rm{a}+Dvect{j}'left(DD{A'_y}{t}rigct{j}'}{t}right)_{{rm{a}A'_y+left(DD{Dvect{k}'}{t}right)_{{rm{a}A'_z;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{i}' \left( \DD{A'_x}{t} \right)_{\rm a} + \Dvect{j}' \left... ...t{j}'}{t} \right)_{\rm a} A'_y + \left( \DD{\Dvect{k}'}{t} \right)_{\rm a} A'_z$ |; $key = q/{displaymath}rhoDD{v_i}{t}+DP{p}{x_i}-DP{tau_{ij}}{x_j}+rhoDP{Phi^*}{x_i}=F'_i,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \rho \DD{v_i}{t} + \DP{p}{x_i} - \DP{\tau_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = F'_i, \end{displaymath}$ |; $key = q/displaystylekappa;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \kappa$ |; $key = q/k;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleDP{T}{z}-frac{RT}{c_pp}DP{p}{z}+frac{L}{c_p}DP{q^*}{z}<0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{T}{z} - \frac{RT}{c_p p} \DP{p}{z} + \frac{L}{c_p} \DP{q^*}{z} < 0$ |; $key = q/displaystyle-frac{1}{a}DP{}{mu}left[-dot{sigma}DP{U}{sigma}-frac{R^dT_v}{a}DP{pi}{lambda}+F_{lambda}cosvarphi+fVright].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a} \DP{}{\mu} \left[ - \dot{\sigma} \DP{U}{\sigma} - \frac{R^d T_v}{a} \DP{\pi}{\lambda} + F_{\lambda} \cos \varphi + fV \right].$ |; $key = q/displaystyleq^{*}(hat{T}_{k-1},p_{k-1})+left.DP{q^{*}}{T}right|_{hat{T}_{k-1}}cdotDeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (\hat{T}_{k-1},p_{k-1}) + \left. \DP{q^{*}}{T} \right\vert _{\hat{T}_{k-1}} \cdot \Delta T_{k-1}$ |; $key = q/displaystyleRslashC_{p};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle R/C_{p}$ |; $key = q/N_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[dot{sigma}DP{V}{sigma}+frac{R^dT_v}{a}(1-mu^2)DP{pi}{mu}-F_{varphi}cosvarphi+fUright];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \dot{\sigma} \DP{V}... ...ac{R^d T_v}{a} (1-\mu^2) \DP{\pi}{\mu} - F_{\varphi} \cos \varphi + f U \right]$ |; $key = q/displaystyle-(zeta+f)U-dot{sigma}DP{V}{sigma}-frac{RT'}{a}(1-mu^{2})DP{pi}{mu}+F_{varphi}cosvarphi.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - ( \zeta + f ) U - \dot{\sigma} \DP{V}{\sigma} - \frac{RT'}{a} ( 1 - \mu^{2} ) \DP{\pi}{\mu} + F_{\varphi} \cos \varphi.$ |; $key = q/q^*(T_{k-1},p_{k-1});MSF=1.6;AAT/; $cached_env_img{$key} = q| $q^* (T_{k-1}, p_{k-1})$ |; $key = q/F_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}left(DD{psi}{t}right)_{{rm{a}=left(DD{psi}{t}right)_{{rm{r},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \left( \DD{\psi}{t} \right)_{\rm a} = \left( \DD{\psi}{t} \right)_{\rm r}, \end{displaymath}$ |; $key = q/displaystylefrac{p_{k}}{p_{k-1slash2}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{p_{k}}{p_{k-1/2}}$ |; $key = q/displaystyleSt;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle St$ |; $key = q/{displaymath}DD{Dvect{v}}{t}=-frac{1}{rho}Dgradp-2Dvect{Omega}timesDvect{v}+Dvect{g}+Dvect{F},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect{\Omega} \times \Dvect{v} + \Dvect{g} + \Dvect{F}, \end{displaymath}$ |; $key = q/displaystylefrac{1}{a(1-mu^2)}DP{}{lambda}(Ubullet)+frac{1}{amu}DP{}{phi}(Vbullet)+bulletD.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a (1-\mu^2)} \DP{}{\lambda} (U\bullet) +\frac{1}{a \mu} \DP{}{\phi} (V\bullet) +\bullet D.$ |; $key = q/R_v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylePhi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi$ |; $key = q/displaystyle-frac{1}{rhor}DP{p}{varphi}-2Omegausinvarphi-frac{u^2}{r}tanvarphi-frac{vw}{r}+F_varphi,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{\rho r} \DP{p}{\varphi} - 2 \Omega u \sin \varphi - \frac{u^2}{r} \tan \varphi - \frac{v w}{r} + F_\varphi,$ |; $key = q/{displaymath}DD{bullet}{t}=DP{bullet}{t}+frac{u}{rcosvarphi}DP{bullet}{lambda}+frac{v}{r}DP{bullet}{varphi}+wDP{bullet}{r},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\bullet}{t} = \DP{\bullet}{t} + \frac{u}{r \cos \varp... ...bda} + \frac{v}{r} \DP{\bullet}{\varphi} + w \DP{\bullet}{r}, \end{displaymath}$ |; $key = q/^{circ};MSF=1.6;AAT/; $cached_env_img{$key} = q| $^{\circ}$ |; $key = q/displaystyleDeltaT_{k-1}-DeltaT_{k}+gamma_{k-1}DeltaT_{k-1}-gamma_{k}DeltaT_{k}-{k-1}-frac{R}{c_p}frac{Deltap_{k-1slash2}}{2p_{k-1slash2}}DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta T_{k-1} - \Delta T_{k} + \gamma_{k-1} \Delta T_{k-1} - \ga... ...Delta T_{k-1} - \frac{R}{c_p} \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}} \Delta T_{k}$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}frac{partialUq}{partiallambda}-frac{1}{a}frac{partialVq}{partialmu}+qD;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \frac{\partial Uq}{\partial \lambda} - \frac{1}{a} \frac{\partial Vq}{\partial \mu} + q D$ |; $key = q/displaystyle+frac{1}{a}frac{R^dbar{T}_v}{a}frac{partial^2pi}{partialmupartiallambda}-frac{1}{a}DP{}{mu}left[-frac{R^dT'_v}{a}DP{pi}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{a} \frac{R^d \bar{T}_v}{a} \frac{\partial^2 \pi}{\part... ... - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$ |; $key = q/displaystylep;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle p$ |; $key = q/displaystyleDP{}{t}(rhov_i^2)+DP{}{x_j}(rhov_jv_i^2)-rhoDP{}{t}left(frac{1}{2}v_i^2right)-rhov_jDP{}{x_j}left(frac{1}{2}v_i^2right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{}{t} ( \rho v_i^2 ) + \DP{}{x_j} ( \rho v_j v_i^2 ) - \rho \D... ...frac{1}{2} v_i^2 \right) - \rho v_j \DP{}{x_j} \left( \frac{1}{2} v_i^2 \right)$ |; $key = q/displaystyle-frac{R_d}{p_{k-1slash2}}left(frac{T_{k-1}+T_{k}}{2}right)^2left(1-f{q_{k-1}+q_{k}}{2}right)left[p_{k-1}(1-q_{k-1})-p_{k}(1-q_{k})right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p_{k-1/2}} \left( \frac{T_{k-1} + T_{k}}{2} \right)^... ..._{k-1} + q_{k}}{2} \right) \left[ p_{k-1} (1-q_{k-1}) - p_{k} (1-q_{k}) \right]$ |; $key = q/{displaymath}tau_{S,i}(z)=int_z^inftyk_{S,i}rhoqdz+int_z^inftybar{k}_{S,i}rhodz{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \tau_{S,i}(z) = \int_z^\infty k_{S,i} \rho q dz + \int_z^\infty \bar{k}_{S,i} \rho dz \end{displaymath}$ |; $key = q/displaystylehat{T}_{k-1}+DeltaT_{k-1}-left(hat{T}_{k}+DeltaT_{k}right)+frac{L}{c_{k-1}-q^{*}(hat{T}_{k})-left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{T}_{k-1} + \Delta T_{k-1} - \left( \hat{T}_{k} + \Delta T_{k... ...{*} (\hat{T}_{k}) - \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right\}$ |; $key = q/{displaymath}F_S(0)=sum_i^{N_S}a_i(1-alpha_g)(1-alpha_A)F_S^Iexp(-tau_{S,i}(0)seczeta),{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} F_S(0) = \sum_i^{N_S} a_i (1-\alpha_g) (1-\alpha_A) F_S^I \exp ( - \tau_{S,i}(0)\sec \zeta ) , \end{displaymath}$ |; $key = q/q;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/bar{k}_{S,i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bar{k}_{S,i}$ |; $key = q/displaystylembox{{sl{UA};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \mbox{\sl UA}$ |; $key = q/Dvect{A}=(A_{lambda},A_{varphi},A_r);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{A}=(A_{\lambda}, A_{\varphi}, A_r)$ |; $key = q/displaystyleM_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle M_{k-1}$ |; $key = q/kappa=frac{R}{c_p};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\kappa = \frac{R}{c_p}$ |; $key = q/displaystylefrac{1}{acosvarphi}DP{u}{lambda}+frac{1}{acosvarphi}DP{}{varphi}(vcosvarphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi)$ |; $key = q/dS=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-(c_pT_{k}+Lq_{k})Deltap_{k}+(c_phat{T}_{k}+Lhat{q}_{k})Deltap_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - (c_p T_{k} + L q_{k}) \Delta p_{k} + (c_p \hat{T}_{k} + L \hat{q}_{k}) \Delta p_{k},$ |; $key = q/epsilonequivR^dslashR^v(=0.622);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\epsilon \equiv R^d/R^v(=0.622)$ |; $key = q/displaystylev_irhoDP{Phi^*}{x_i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle v_i \rho \DP{\Phi^*}{x_i}$ |; $key = q/textstyleequiv;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle \equiv$ |; $key = q/displaystyleqquad+DeltaT_{k-1}-DeltaT_{k}+frac{L}{c_p}left.DP{q^{*}}{T}right|_{keltap_{k-1slash2}}{p_{k-1slash2}}frac{DeltaT_{k-1}+DeltaT_{k}}{2}=0.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + \Delta T_{k-1} - \Delta T_{k} + \frac{L}{c_p} \left. \DP... ...\frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{\Delta T_{k-1} + \Delta T_{k}}{2} = 0.$ |; $key = q/displaystyleUA;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle UA$ |; $key = q/displaystyleQP;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle QP$ |; $key = q/displaystylefrac{partialT}{partialt};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{\partial T}{\partial t}$ |; $key = q/{displaymath}q^*(T,p)=frac{epsilone^*(T)}{p}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} q^*(T,p) = \frac{\epsilon e^*(T)}{p} . \end{displaymath}$ |; $key = q/displaystyledot{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \dot{\sigma}$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}frac{partialUT^{prime}}{partiallambda}-frac{1}{a}frac{partialVT^{prime}}{partialmu}+T^{prime}D;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \frac{\partial UT^{\prime}}{\partial \lambda} - \frac{1}{a} \frac{\partial VT^{\prime}}{\partial \mu} + T^{\prime} D$ |; $key = q/displaystyleF_S(z);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle F_S(z)$ |; $key = q/displaystylealphaDeltaT_{k}+beta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \alpha \Delta T_{k} + \beta$ |; $key = q/times10^6;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\times 10^6$ |; $key = q/DD{S}{z}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\DD{S}{z} =0$ |; $key = q/{displaymath}DP{}{t}left[rholeft(frac{1}{2}Dvect{v}^2+varepsilon+Phi^*right)righn+Phi^*right)v_j+pv_j-sigma_{ij}v_iright]=rhoQ+F'_iv_i,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{}{t} \left[ \rho \left( \frac{1}{2} \Dvect{v}^2 +... ...v_j + p v_j - \sigma_{ij}v_i \right] = \rho Q + F'_i v_i, \end{displaymath}$ |; $key = q/displaystyleq^{*}(hat{T}_{k-1})+q^{*}(hat{T}_{k}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (\hat{T}_{k-1}) + q^{*} (\hat{T}_{k}),$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}DP{UT'}{lambda}-frac{1}{a}DP{VT'}{mu}+T'D-dot{sigma}DP{T}{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \DP{UT'}{\lambda} - \frac{1}{a} \DP{VT'}{\mu} + T' D - \dot{\sigma} \DP{T}{\sigma}$ |; $key = q/displaystyletheta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \theta$ |; $key = q/{displaymath}DD{T}{t}=frac{1}{c_p^drho}DD{p}{t}+frac{Q^*}{c_p^d},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{T}{t} = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d}, \end{displaymath}$ |; $key = q/displaystyleleft(-1-gamma_{k}-frac{R}{c_p}frac{Deltap_{k-1slash2}}{2p_{k-1slash2}}right)DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( - 1 - \gamma_{k} - \frac{R}{c_p} \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}} \right) \Delta T_{k}$ |; $key = q/{displaymath}DD{c_pT}{t}=frac{1}{rho}DD{p}{t}+Q^*,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{c_p T}{t} = \frac{1}{\rho} \DD{p}{t} + Q^*, \end{displaymath}$ |; $key = q/{displaymath}{calD}(zeta)=-K_{HD}left[(-1)^{N_Dslash2}nabla^{N_D}-left(frac{2}{a^2}right)^{N_Dslash2}right]zeta,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} {\cal D}(\zeta) = - K_{HD} \left[ (-1)^{N_D/2} \nabla^{N_D} - \left( \frac{2}{a^2} \right)^{N_D/2} \right] \zeta , \end{displaymath}$ |; $key = q/{displaymath}p=rhoR^dT_v.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} p = \rho R^d T_v. \end{displaymath}$ |; $key = q/mathcal{D}(q);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mathcal{D}(q)$ |; $key = q/{displaymath}DP{rho^v}{t}+DP{}{x_j}(rho^vv_j)=S.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{\rho^v}{t} + \DP{}{x_j} ( \rho^v v_j ) = S. \end{displaymath}$ |; $key = q/frac{1}{acosvarphi}DP{}{varphi}cosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{1}{a \cos \varphi} \DP{}{\varphi} \cos \varphi$ |; $key = q/displaystyle-frac{F}{4}left{TP^2cdotMM-frac{1}{2}TP^2left(D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k}right)+2TPcdotMM(DeltaT_{k-1}+DeltaT_{k})right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ TP^2 \cdot MM - \frac{1}{2} TP^2 \left( D_{... ...} \Delta T_{k} \right) + 2 TP \cdot MM (\Delta T_{k-1} + \Delta T_{k}) \right\}$ |; $key = q/displaystyleleft{1+gamma_{k-1}right}DeltaT_{k-1}Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left\{ 1 + \gamma_{k-1} \right\} \Delta T_{k-1} \Delta p_{k-1}$ |; $key = q/DP{Phi^*}{t}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\DP{\Phi^*}{t}=0$ |; $key = q/displaystylefrac{u^2+v^2}{2}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{u^2 + v^2}{2}.$ |; $key = q/q^*(T,p);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{1}{a}frac{R^dbar{T}_v}{a}frac{partial^2pi}{partiallambdapartiaa(1-mu^2)}DP{}{lambda}left[+frac{R^dT'_v}{a}(1-mu^2)DP{pi}{mu}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a} \frac{R^d \bar{T}_v}{a} \frac{\partial^2 \pi}{\part... ...^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a} (1-\mu^2) \DP{\pi}{\mu} \right]$ |; $key = q/{displaymath}DD{rho}{t}=-rhoDdivDvect{v},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\rho}{t} = - \rho \Ddiv \Dvect{v}, \end{displaymath}$ |; $key = q/displaystylercosvarphicoslambda,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle r \cos \varphi \cos \lambda,$ |; $key = q/displaystyle+frac{1}{2}left{TMcdotMM+TPcdotMM-frac{1}{2}TPcdotTMcdotD_{k-1}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ TM \cdot MM + TP \cdot MM - \frac{1}{2} TP \cdot TM \cdot D_{k-1} \right\}$ |; $key = q/Dvect{v};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v}$ |; $key = q/displaystylefrac{u}{acosvarphi}left(frac{partial}{partiallambda}right)_{sigma}+frac{v}{a}left(frac{partial}{partialvarphi}right)_{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{u}{a \cos \varphi} \left( \frac{\partial }{\partial \lambda... ...sigma} + \frac{v}{a} \left( \frac{\partial }{\partial \varphi} \right)_{\sigma}$ |; $key = q/displaystyleDvect{e}_{lambda}left[DP{A_{lambda}}{t}+frac{u}{rcosvarphi}DP{A_{lamwDP{A_{lambda}}{r}+frac{u}{r}A_r-frac{utanvarphi}{r}A_{varphi}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{e}_{\lambda} \left[ \DP{A_{\lambda}}{t} + \frac{u}{r \cos ... ..._{\lambda}}{r} + \frac{u}{r} A_r - \frac{u \tan \varphi}{r} A_{\varphi} \right]$ |; $key = q/(Dvect{e}_{lambda},Dvect{e}_{varphi},Dvect{e}_{r});MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\Dvect{e}_{\lambda}, \Dvect{e}_{\varphi}, \Dvect{e}_{r})$ |; $key = q/displaystyle-frac{R_d}{p}T^2(1-q)DD{}{z}left{p(1-q)right}+c_pT(1-q)DD{T}{z}-lq(1-q)DD{T}{z}+lT(1-q)DD{q}{z}+lTqDD{q}{z}=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p} T^2 (1-q) \DD{}{z} \left\{ p (1-q) \right\} + c_p... ...q) \DD{T}{z} - l q (1-q) \DD{T}{z} + l T (1-q) \DD{q}{z} + l T q \DD{q}{z} = 0,$ |; $key = q/u;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleDD{u}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{u}{t}$ |; $key = q/displaystyle-frac{1}{a}DP{}{mu}left[frac{R^dbar{T}_v}{a}(1-mu^2)DP{pi}{mu}right]-frac{1}{a}DP{}{mu}left[frac{R^dT'_v}{a}(1-mu^2)DP{pi}{mu}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a} \DP{}{\mu} \left[ \frac{R^d \bar{T}_v}{a} ( 1-\mu^2... ...ac{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$ |; $key = q/displaystylefrac{L}{c_p},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{L}{c_p},$ |; $key = q/displaystyle-frac{1}{rho}Dvect{v}Dgrad{p}-gw-Dvect{v}cdotDvect{F}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{\rho} \Dvect{v} \Dgrad{p} - g w - \Dvect{v} \cdot \Dvect{F}.$ |; $key = q/Dvect{g}equivDgradPhi^*-Dvect{Omega}times(Dvect{Omega}timesDvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{g} \equiv \Dgrad \Phi^* - \Dvect{\Omega} \times ( \Dvect{\Omega} \times \Dvect{v})$ |; $key = q/(x_1,x_2,x_3);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylequad+frac{1}{2}left(MM-frac{1}{2}D_{k-1}DeltaT_{k-1}-frac{1}{2}D_{k}cdotTM+(DeltaT_{k-1}+DeltaT_{k})TM+(DeltaT_{k-1}-DeltaT_{k})TPright};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \frac{1}{2} \left( MM - \frac{1}{2} D_{k-1} \Delta T_{k-1... ...Delta T_{k-1} + \Delta T_{k}) TM + ( \Delta T_{k-1} - \Delta T_{k}) TP \right\}$ |; $key = q/displaystylequad-frac{E}{2}left(QP+D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k}right)left{MM-frac{1}{2}left(D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k}right)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad - \frac{E}{2} \left( QP + D_{k-1} \Delta T_{k-1} + D_{k} \D... ...\frac{1}{2} \left( D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} \right) \right\}$ |; $key = q/displaystyle-frac{1}{acosvarphi}DP{}{varphi}left[-cosvarphidot{sigma}DP{u}{sigma{R^dT_v}{ap_s}DP{p_s}{lambda}+F_{lambda}cosvarphi+fvcosvarphiright].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ - \cos \varphi \... ...a p_s} \DP{p_s}{\lambda} + F_{\lambda} \cos \varphi + f v \cos \varphi \right].$ |; $key = q/displaystyle(2Omegacosvarphiw-2Omegasinvarphiv)Dvect{e}_{lambda}+2OmegasinvarphiuDvect{e}_{varphi}-2OmegacosvarphiuDvect{e}_r.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle ( 2 \Omega \cos \varphi w - 2 \Omega \sin \varphi v) \Dvect{e}_{\... ...Omega \sin \varphi u \Dvect{e}_{\varphi} - 2 \Omega \cos \varphi u \Dvect{e}_r.$ |; $key = q/displaystyle-frac{R_d}{p(1-q)}DD{}{z}left{p(1-q)right}+frac{1}{1-q}frac{c_p}{T}D1-q}DD{T}{z}+frac{l}{T}frac{DD{q}{z}(1-q)-qDD{}{z}(1-q)}{(1-q)^2}=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\} + \frac{1... ...\DD{T}{z} + \frac{l}{T} \frac{\DD{q}{z} (1-q) - q \DD{}{z}(1-q)} {(1-q)^2} = 0,$ |; $key = q/displaystyleDrotDvect{A};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Drot \Dvect{A}$ |; $key = q/sim;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sim$ |; $key = q/{displaymath}Dgradbullet=left(frac{1}{h_1}DP{bullet}{xi_1},frac{1}{h_2}DP{bullet}{xi_2},frac{1}{h_3}DP{bullet}{xi_3}right),{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Dgrad \bullet = \left( \frac{1}{h_1} \DP{\bullet}{\xi_1},... ...{\bullet}{\xi_2}, \frac{1}{h_3} \DP{\bullet}{\xi_3} \right), \end{displaymath}$ |; $key = q/displaystylefrac{uvtanvarphi}{r}+fv-frac{1}{rhorcosvarphi}DP{p}{lambda}+F_{lambda};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{uv \tan \varphi}{r} + fv - \frac{1}{\rho r \cos \varphi} \DP{p}{\lambda} + F_{\lambda}$ |; $key = q/displaystylefrac{1}{a^2cos^2varphi}DP[2]{}{lambda}+frac{1}{a^2cosvarphi}DP{}{varphi}(cosvarphiDP{}{varphi}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a^2 \cos^2 \varphi} \DP[2]{}{\lambda} + \frac{1}{a^2 \cos \varphi} \DP{}{\varphi} ( \cos \varphi \DP{}{\varphi} ),$ |; $key = q/p_{k-1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $p_{k-1/2}$ |; $key = q/displaystylefrac{1}{acos^2varphi}DP{ucosphi}{lambda}+frac{1}{acosvarphi}DP{}{varphi}(vcosvarphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a \cos^2 \varphi} \DP{u \cos \phi}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi)$ |; $key = q/displaystyle-nabla^2_{sigma}(Phi+KE).;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \nabla^2_{\sigma} ( \Phi + KE).$ |; $key = q/displaystyleDP{S}{z}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{S}{z} = 0$ |; $key = q/displaystyleQM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle QM$ |; $key = q/displaystyleqquadleft.-TPcdotTMfrac{1}{2}D_{k-1}DeltaT_{k-1}-TPcdotTMfrac{1}{2}D_{k}DeltaT_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. - TP \cdot TM \frac{1}{2} D_{k-1} \Delta T_{k-1} - TP \cdot TM \frac{1}{2} D_{k} \Delta T_{k} \right\}$ |; $key = q/{displaymath}DP{Phi}{sigma}=-frac{R^dT_v}{sigma}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{\Phi}{\sigma} = - \frac{R^d T_v}{\sigma}. \end{displaymath}$ |; $key = q/displaystyleleft(DD{Dvect{A}}{t}right)_{{rm{r}+Dvect{Omega}timesDvect{A}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}.$ |; $key = q/displaystyleDeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta T_{k-1}$ |; $key = q/%latex2htmlidmarker7265setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7265 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/{displaymath}DP{rho}{t}+DP{}{x_j}(rhov_j)=0.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{\rho}{t} + \DP{}{x_j}( \rho v_j ) = 0. \end{displaymath}$ |; $key = q/displaystylembox{{sl{KE};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \mbox{\sl KE}$ |; $key = q/displaystyle-left{DeltaT_{k}+frac{L}{c_p}(q_{k}-hat{q}_{k})right}Deltap_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \left\{ \Delta T_{k} + \frac{L}{c_p} (q_{k} - \hat{q}_{k}) \right\} \Delta p_{k},$ |; $key = q/Dvect{v}_aequiv(dDvect{r}slashdt)_{{rm{a};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v}_a \equiv (d\Dvect{r}/dt)_{\rm a}$ |; $key = q/displaystyleDP{}{t}(rhoPhi^*)+DP{}{x_i}(rhoPhi^*v_i).;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{}{t} ( \rho \Phi^* ) + \DP{}{x_i} ( \rho \Phi^* v_i ).$ |; $key = q/displaystyle-frac{F}{4}left{-TP^2cdotMMcdotP_{k-1}cdotD_{k-1}+left(P_{k-1}M_{k-1ht)left(-frac{1}{2}TP^2cdotD_{k-1}+2TPcdotMMright)right}DeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ - TP^2 \cdot MM \cdot P_{k-1} \cdot D_{k-1}... ... \frac{1}{2} TP^2 \cdot D_{k-1} + 2 TP \cdot MM \right) \right\} \Delta T_{k-1}$ |; $key = q/epsilon=0.622;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\epsilon=0.622$ |; $key = q/displaystyleStequivhat{T}_{k-1}-hat{T}_{k}+frac{L}{c_p}left{q^{*}(hat{T}_{k-1})-c{Deltap_{k-1slash2}}{p_{k-1slash2}}frac{hat{T}_{k-1}+hat{T}_{k}}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle St \equiv \hat{T}_{k-1} - \hat{T}_{k} + \frac{L}{c_p} \left\{ q^{... ...{c_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}$ |; $key = q/displaystyle-frac{F}{4}left{TP^2cdotMMcdotP_{k}cdotD_{k}+left(P_{k-1}M_{k-1}-P_{k}M_{k}right)left(-frac{1}{2}TP^2cdotD_{k}+2TPcdotMMright)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ TP^2 \cdot MM \cdot P_{k} \cdot D_{k} + \le... ... \right) \left( - \frac{1}{2} TP^2 \cdot D_{k} + 2 TP \cdot MM \right) \right\}$ |; $key = q/c_p;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/Dvect{v}equiv(dDvect{r}slashdt)_{{rm{r};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v} \equiv (d\Dvect{r}/dt)_{\rm r}$ |; $key = q/{displaymath}frac{partialD}{partialt}=frac{1}{a(1-mu^{2})}frac{partialmbox{{sl{Unabla^{2}_{sigma}(Phi+Rbar{T}pi+mbox{{sl{KE})+{calD}(D){displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{\partial D}{\partial t} = \frac{1}{a(1-\mu^{2})} \... ...ma} ( \Phi + R \bar{T} \pi + \mbox{\sl KE} ) + {\cal D}(D) \end{displaymath}$ |; $key = q/displaystyleqquad+lfrac{T_{k-1}+T_{k}}{2}(q_{k-1}-q_{k})=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + l \frac{T_{k-1} + T_{k}}{2} (q_{k-1} - q_{k}) = 0,$ |; $key = q/displaystyleqquadleft.-TPcdotTMleft(frac{1}{2}D_{k-1}DeltaT_{k-1}+frac{1}{2}D_{k}DeltaT_{k}right)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. - TP \cdot TM \left( \frac{1}{2} D_{k-1} \Delta T_{k-1} + \frac{1}{2} D_{k} \Delta T_{k} \right) \right\}$ |; $key = q/displaystyleqquadtimesleft[frac{p_{k-1}}{p_{k-1slash2}}left(1-q^{*}(hat{T}_{k-1}-q^{*}(hat{T}_{k})-left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right)right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left[ \frac{p_{k-1}}{p_{k-1/2}} \left( 1 - q^{*} (... ...at{T}_{k}) - \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right) \right]$ |; $key = q/epsilon_vequiv1slashepsilon-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\epsilon_v \equiv 1/\epsilon -1$ |; $key = q/displaystyle+frac{1}{2}MMcdotTPcdotTM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} MM \cdot TP \cdot TM$ |; $key = q/displaystylev_iDP{}{t}(rhov_i)+v_iDP{}{x_j}(rhov_jv_i);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle v_i \DP{}{t} ( \rho v_i ) + v_i \DP{}{x_j} ( \rho v_j v_i )$ |; $key = q/{displaymath}DD{Dvect{v}}{t}=sum^3_{k=1}Dvect{e}_kleft[DP{v_k}{t}+sum^3_{j=1}frafrac{v_k}{h_k}frac{1}{h_j}DP{h_k}{xi_j}right)v_jright].{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\Dvect{v}}{t} = \sum^3_{k=1} \Dvect{e}_k \left[ \DP{v... ...{v_k}{h_k} \frac{1}{h_j} \DP{h_k}{\xi_j} \right) v_j \right]. \end{displaymath}$ |; $key = q/^{-4};MSF=1.6;AAT/; $cached_env_img{$key} = q| $^{-4}$ |; $key = q/RequivR^d(1+epsilon_vq);MSF=1.6;AAT/; $cached_env_img{$key} = q| $R \equiv R^d ( 1+\epsilon_v q )$ |; $key = q/displaystyle-frac{R_d}{p}T^2(1-q)DD{}{z}left{p(1-q)right}+c_pT(1-q)DD{T}{z}-lq(1-q)DD{T}{z}+lTDD{q}{z}=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p} T^2 (1-q) \DD{}{z} \left\{ p (1-q) \right\} + c_p T (1-q) \DD{T}{z} - l q (1-q) \DD{T}{z} + l T \DD{q}{z} = 0,$ |; $key = q/alpha_g;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\alpha_g$ |; $key = q/delta_v=epsilon_v^{-1}-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\delta_v = \epsilon_v^{-1} - 1$ |; $key = q/{displaystylesigma=frac{p}{p_s}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma= \frac{p}{p_s}$ |; $key = q/varphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\varphi$ |; $key = q/mathcal{D}(zeta);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mathcal{D}(\zeta)$ |; $key = q/p_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $p_{k-1}$ |; $key = q/displaystylefrac{1}{a(1-mu^2)}DP{U}{lambda}+frac{1}{a}DP{V}{mu}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a ( 1-\mu^2)} \DP{U}{\lambda} + \frac{1}{a} \DP{V}{\mu}.$ |; $key = q/displaystyleq*(T,p)=p^{*}_{0}expleft(-frac{L}{RT}right)frac{1}{p};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q*(T, p) = p^{*}_{0} \exp \left( - \frac{L}{RT} \right) \frac{1}{p}$ |; $key = q/q_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $q_{k}$ |; $key = q/displaystyle=frac{(1+gamma_{k-1})Deltap_{k-1}St+left(1+gamma_{k-1}-kappafrac{Delac{Deltap_{k-1slash2}}{2p_{k-1slash2}}right)(1+gamma_{k})Deltap_{k}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = \frac{ (1 + \gamma_{k-1}) \Delta p_{k-1} St + \left( 1 + \gamma... ...pa \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}} \right) (1 + \gamma_{k}) \Delta p_{k} }$ |; $key = q/v_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleq_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q_{k}$ |; $key = q/z;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylehat{T}_{k-1}-hat{T}_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{T}_{k-1} - \hat{T}_{k},$ |; $key = q/displaystyle-frac{bar{T}}{a(1-mu^{2})}DP{U}{lambda}-frac{1}{a(1-mu^{2})}DP{UT'}{frac{1}{a(1-mu^{2})}DP{U}{lambda}+frac{bar{1}}{a}DP{V}{mu}right]+T'D;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{\bar{T}}{a(1-\mu^{2})} \DP{U}{\lambda} - \frac{1}{a(1-\mu... ...1}{a(1-\mu^{2})} \DP{U}{\lambda} + \frac{\bar{1}}{a} \DP{V}{\mu} \right] + T' D$ |; $key = q/displaystyle-frac{1}{a}DP{}{mu}(zetaV)-frac{1}{a(1-mu^2)}DP{}{lambda}(zetaU);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle -\frac{1}{a}\DP{}{\mu} ( \zeta V ) -\frac{1}{a (1-\mu^2)} \DP{}{\lambda} ( \zeta U )$ |; $key = q/a;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}DrotDvect{A}=left(frac{1}{h_2h_3}left[DP{(h_3A_3)}{xi_2}-DP{(h_2A_2left[DP{(h_2A_2)}{xi_1}-DP{(h_1A_1)}{xi_2}right]right),{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Drot \Dvect{A} = \left( \frac{1}{h_2 h_3} \left[ \DP{(h_... ...DP{(h_2 A_2)}{\xi_1} - \DP{(h_1 A_1)}{\xi_2} \right] \right), \end{displaymath}$ |; $key = q/displaystyleqquadtimesfrac{1}{2}left(hat{T}_{k-1}+DeltaT_{k-1}+hat{T}_{k}+DeltaT_{k}right)(hat{T}_{k-1}+DeltaT_{k-1}-hat{T}_{k}-DeltaT_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \frac{1}{2} \left( \hat{T}_{k-1} + \Delta T_{k-1} +... ...lta T_{k} \right) (\hat{T}_{k-1} + \Delta T_{k-1} - \hat{T}_{k} - \Delta T_{k})$ |; $key = q/displaystylefrac{u}{acosphi}DP{bullet}{lambda}+frac{v}{a}DP{bullet}{phi};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{u}{a \cos\phi}\DP{\bullet}{\lambda} + \frac{v}{a} \DP{\bullet}{\phi}$ |; $key = q/displaystylee^{*}(T)=p^{*}_{0}expleft(-frac{L}{RT}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle e^{*}(T) = p^{*}_{0} \exp \left( - \frac{L}{RT} \right)$ |; $key = q/displaystyleDD{w}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{w}{t}$ |; $key = q/displaystylequad-frac{E}{2}left(QP+D_{k-1}DeltaT_{k-1}+D_{k}DeltaT_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad - \frac{E}{2} \left( QP + D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} \right)$ |; $key = q/displaystyleBDeltaT_{k-1}+CDeltaT_{k}=St;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle B \Delta T_{k-1} + C \Delta T_{k} = St$ |; $key = q/varepsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\varepsilon$ |; $key = q/displaystyleT_{k-1}-T_{k}+frac{L}{C_p}left{q^{*}(T_{k-1},p_{k-1})-q^{*}(T_{k},p_{C_p}frac{Deltap_{k-1slash2}}{p_{k-1slash2}}frac{T_{k-1}+T_{k}}{2}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_{k-1} - T_{k} + \frac{L}{C_p} \left\{ q^{*} (T_{k-1}, p_{k-1}) ... ...- \frac{R}{C_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{T_{k-1} + T_{k}}{2} =0$ |; $key = q/{displaymath}DP{q}{t}=-frac{1}{a(1-mu^{2})}DP{Uq}{lambda}-frac{1}{a}DP{Vq}{mu}+qD-dot{sigma}frac{partialq}{partialsigma}+S_{q}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{q}{t} = - \frac{1}{a(1-\mu^{2})} \DP{Uq}{\lambda} -... ... - \dot{\sigma} \frac{\partial q }{\partial \sigma} + S_{q}. \end{displaymath}$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[+frac{R^dT_v}{a}(1-mu^2)DP{pi}{mu}right]-frac{1}{a}DP{}{mu}left[-frac{R^dT_v}{a}DP{pi}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d T_v}{a}... ...] - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T_v}{a} \DP{\pi}{\lambda} \right]$ |; $key = q/T_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $T_{k}$ |; $key = q/%latex2htmlidmarker6968setcounter{footnote}{3}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6968 \setcounter{footnote}{3}\fnsymbol{footnote}$ |; $key = q/{displaymath}h_lambda=rcosvarphi,h_varphi=r,h_r=1.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} h_\lambda = r \cos \varphi, h_\varphi = r, h_r = 1. \end{displaymath}$ |; $key = q/displaystylefrac{U}{a(1-mu^{2})}left(frac{partial}{partiallambda}right)_{sigma}+frac{V}{a}left(frac{partial}{partialmu}right)_{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{U}{a ( 1 -\mu^{2} )} \left( \frac{\partial }{\partial \lamb... ...)_{\sigma} + \frac{V}{a} \left( \frac{\partial }{\partial \mu} \right)_{\sigma}$ |; $key = q/displaystyle+frac{ucosphi}{acos^2phi}DP{bullet}{lambda}+frac{vcosphi}{acosphi}DP{bullet}{phi};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{u \cos \phi }{a \cos^2 \phi}\DP{\bullet}{\lambda} + \frac{v \cos \phi }{a \cos \phi } \DP{\bullet}{\phi}$ |; $key = q/displaystyle-frac{R_d}{p}T^2(1-q)dleft{p(1-q)right}+c_pT(1-q)dT-lq(1-q)dT+lTdq=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p} T^2 (1-q) d \left\{ p (1-q) \right\} + c_p T (1-q) dT - l q (1-q) dT + l T dq = 0,$ |; $key = q/displaystyle+frac{1}{2}left{MMcdotTPcdotTM+left{(DeltaT_{k-1}+DeltaT_{k})TM+(DeltaT_{k-1}-DeltaT_{k})TPright}MMright.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ MM \cdot TP \cdot TM + \left\{ (\Delta T_{k-1} + \Delta T_{k}) TM + ( \Delta T_{k-1} - \Delta T_{k}) TP \right\} MM \right.$ |; $key = q/displaystyle-frac{E}{2}left(MMcdotQP-frac{1}{2}TMcdotQPcdotD_{k-1}+MMcdotTMcdotD_{k-1}right)DeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left( MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot D_{k-1} + MM \cdot TM \cdot D_{k-1} \right) \Delta T_{k-1}$ |; $key = q/{displaymath}e^*(T)=e^*(T=273mbox{K})expleft[frac{L}{R_v}left(frac{1}{273}-frac{1}{T}right)right],{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} e^*(T) = e^*(T=273\mbox{K}) \exp \left[ \frac{L}{R_v} \left( \frac{1}{273} - \frac{1}{T} \right) \right] , \end{displaymath}$ |; $key = q/displaystyleTP;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle TP$ |; $key = q/piBequivsigma_{SB}T^4;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\pi B \equiv \sigma_{SB} T^4$ |; $key = q/{displaymath}DP{}{t}(rhov_i)+DP{}{x_j}(rhov_iv_j)+DP{p}{x_i}-DP{sigma_{ij}}{x_j}+rhoDP{Phi^*}{x_i}=F'_i.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{}{t}(\rho v_i) + \DP{}{x_j}( \rho v_i v_j ) + \DP{p}{x_i} - \DP{\sigma_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = F'_i. \end{displaymath}$ |; $key = q/displaystyleMM;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle MM$ |; $key = q/Phi^*;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Phi^*$ |; $key = q/displaystyleT_v;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_v$ |; $key = q/muequivsinvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu \equiv \sin \varphi$ |; $key = q/displaystyle-frac{F}{4}left[TP^2cdotMMleft(P_{k-1}M_{k-1}-P_{k}M_{k}right)-TP^2cdotMMP_{k-1}D_{k-1}DeltaT_{k-1}+TP^2cdotMMP_{k}D_{k}DeltaT_{k}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left[ TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k}... ...P_{k-1} D_{k-1} \Delta T_{k-1} + TP^2 \cdot MM P_{k} D_{k} \Delta T_{k} \right.$ |; $key = q/{displaymath}frac{partialzeta}{partialt}=frac{1}{a(1-mu^{2})}frac{partialmbox{{s{1}{a}frac{partialmbox{{sl{UA}}{partialmu}+{calD}(zeta){displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{\partial \zeta}{\partial t} = \frac{1}{a(1-\mu^{2})... ...frac{\partial \mbox{\sl UA}}{\partial \mu} + {\cal D}(\zeta) \end{displaymath}$ |; $key = q/pi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\pi$ |; $key = q/Vequivcosphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $V \equiv \cos \phi$ |; $key = q/displaystyleDD{s}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{s}{t}$ |; $key = q/displaystyleq^{*}(hat{T}_{k-1})-q^{*}(hat{T}_{k}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (\hat{T}_{k-1}) - q^{*} (\hat{T}_{k}),$ |; $key = q/bullet^d;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bullet^d$ |; $key = q/{displaymath}p=rhoRT.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} p = \rho R T. \end{displaymath}$ |; $key = q/displaystylefrac{1}{a(1-mu^{2})}frac{partialU}{partiallambda}+frac{1}{a}frac{partialV}{partialmu};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a ( 1-\mu^{2} ) } \frac{\partial U}{\partial \lambda} + \frac{1}{a} \frac{\partial V}{\partial \mu}$ |; $key = q/displaystyleleft.DP{q^{*}}{T}right|_{k-1},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left. \DP{q^{*}}{T} \right\vert _{k-1},$ |; $key = q/displaystyle+Dvect{e}_{varphi}frac{1}{rcosvarphi}left[DP{}{r}(rcosvarphiA_{lambda})-DP{A_r}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \Dvect{e}_{\varphi} \frac{1}{r \cos \varphi} \left[ \DP{}{r} (r \cos \varphi A_{\lambda}) - \DP{A_r}{\lambda} \right]$ |; $key = q/Dvect{A}=Dvect{v}_{{rm{a};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{A}=\Dvect{v}_{\rm a}$ |; $key = q/displaystyleqquad+left(1-frac{q_{k-1}+q_{k}}{2}right)frac{T_{k-1}+T_{k}}{2}(T_{k-1}-T_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) \frac{T_{k-1} + T_{k}}{2} (T_{k-1} - T_{k})$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[frac{R^dT_v}{a}DP{pi}{lambda}right]-frac{1}{a}DP{}{mu}left[frac{R^dT_v}{a}(1-mu^2)DP{pi}{mu}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T_v}{a} \... ...rac{1}{a} \DP{}{\mu} \left[ \frac{R^d T_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$ |; $key = q/displaystyle-frac{F}{4}left{TP^2cdotMMcdotP_{k}cdotD_{k}+left(P_{k-1}M_{k-1}-P_{}right)left(-frac{1}{2}TP^2cdotD_{k}+2TPcdotMMright)right}DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ TP^2 \cdot MM \cdot P_{k} \cdot D_{k} + \le... ...t( - \frac{1}{2} TP^2 \cdot D_{k} + 2 TP \cdot MM \right) \right\} \Delta T_{k}$ |; $key = q/c_p=c_v+R;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/sigma;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}DP{Uq}{lambda}-frac{1}{a}DP{Vq}{mu}+qD;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \DP{Uq}{\lambda} - \frac{1}{a} \DP{Vq}{\mu} + q D$ |; $key = q/displaystyleDP{}{t}left(frac{1}{2}rhov_i^2right)+DP{}{x_j}(frac{1}{2}rhov_jv_i^2)+frac{1}{2}v_i^2left{DP{rho}{t}+DP{}{x_j}(rhov_j)right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{}{t} \left( \frac{1}{2} \rho v_i^2 \right) + \DP{}{x_j} ( \fr... ...2 ) + \frac{1}{2} v_i^2 \left\{ \DP{\rho}{t} + \DP{}{x_j} ( \rho v_j ) \right\}$ |; $key = q/F_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}DD{T}{t}=frac{1}{c_p^drho}DD{p}{t}+frac{Q^*}{c_p^d}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{T}{t} = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d}. \end{displaymath}$ |; $key = q/displaystylePhi^*left{DP{rho}{t}+DP{}{x_i}(rhov_i)right}+rhoDP{Phi^*}{t}+v_irhoDP{Phi^*}{x_i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi^* \left\{ \DP{\rho}{t} + \DP{}{x_i}(\rho v_i) \right\} + \rho \DP{\Phi^*}{t} + v_i \rho \DP{\Phi^*}{x_i}$ |; $key = q/displaystyleqquadtimes(TM+DeltaT_{k-1}-DeltaT_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times (TM + \Delta T_{k-1} - \Delta T_{k})$ |; $key = q/displaystyle-frac{F}{4}left{TP^2+2TPcdot(DeltaT_{k-1}+DeltaT_{k})right}left(MM-frac{1}{2}D_{k-1}DeltaT_{k-1}-frac{1}{2}D_{k}DeltaT_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{F}{4} \left\{ TP^2 + 2 TP \cdot (\Delta T_{k-1} + \Delta ... ...M - \frac{1}{2} D_{k-1} \Delta T_{k-1} - \frac{1}{2} D_{k} \Delta T_{k} \right)$ |; $key = q/displaystylequadtimesleft{left(P_{k-1}M_{k-1}-P_{k}M_{k}right)-P_{k-1}D_{k-1}DeltaT_{k-1}+P_{k}D_{k}DeltaT_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad \times \left\{ \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) - P_{k-1} D_{k-1} \Delta T_{k-1} + P_{k} D_{k} \Delta T_{k} \right\}$ |; $key = q/{displaymath}DP{}{t}(rhovarepsilon)+DP{}{x_j}(rhovarepsilonv_j)=-pDP{v_j}{x_j}+sigma_{ij}DP{v_i}{x_j}+rhoQ.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{}{t} ( \rho \varepsilon ) + \DP{}{x_j} ( \rho \varepsi... ... ) = - p \DP{v_j}{x_j} + \sigma_{ij} \DP{v_i}{x_j} + \rho Q. \end{displaymath}$ |; $key = q/displaystyle+vbiggl{-frac{1}{rhor}DP{p}{varphi}-underbrace{2Omegausinvarphi}_{(1u^2}{r}tanvarphi}_{(3)}-underbrace{frac{vw}{r}}_{(5)}+F_varphibiggl};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + v \biggl\{ - \frac{1}{\rho r} \DP{p}{\varphi} - \underbrace{ 2 ... ... \tan \varphi }_{(3)} - \underbrace{ \frac{v w}{r} }_{(5)} + F_\varphi \biggl\}$ |; $key = q/displaystyle-frac{1}{a(1-mu^2)}DP{}{lambda}left[frac{R^dbar{T}_v}{a}DP{pi}{lambdc{1}{a(1-mu^2)}DP{}{lambda}left[frac{R^dT'_v}{a}DP{pi}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d \bar{T}_v... ...a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$ |; $key = q/displaystyle+frac{U}{a(1-mu^2)}DP{bullet}{lambda}+frac{V}{a}DP{bullet}{mu};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{U}{a (1-\mu^2)} \DP{\bullet}{\lambda} + \frac{V}{a} \DP{\bullet}{\mu}$ |; $key = q/%latex2htmlidmarker7334setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7334 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystylemu;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \mu$ |; $key = q/displaystyle-frac{E}{2}left{QPcdotMMcdotTM+MMcdotQPDeltaT_{k-1}-MMcdotQPDeltaT_{rac{1}{2}D_{k-1}DeltaT_{k-1}-TMcdotQPfrac{1}{2}D_{k}DeltaT_{k}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left\{ QP \cdot MM \cdot TM + MM \cdot QP \Delta T... ...}{2} D_{k-1} \Delta T_{k-1} - TM \cdot QP\frac{1}{2} D_{k} \Delta T_{k} \right.$ |; $key = q/displaystyle-frac{1}{a}DP{}{mu}left[dot{sigma}DP{V}{sigma}+frac{R^dT_v}{a}(1-mu^2)DP{pi}{mu}-F_{varphi}cosvarphi+fUright];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a} \DP{}{\mu} \left[ \dot{\sigma} \DP{V}{\sigma} + \frac{R^d T_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} - F_{\varphi} \cos \varphi + f U \right]$ |; $key = q/displaystyleT_{k-1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_{k-1/2}$ |; $key = q/displaystyleleft{1+gamma_{k-1}right}DeltaT_{k-1}Deltap_{k-1}=-left(1+gamma_{k}right)DeltaT_{k}Deltap_{k}+frac{L}{c_p}Deltahat{Q},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left\{ 1 + \gamma_{k-1} \right\} \Delta T_{k-1} \Delta p_{k-1} =... ... + \gamma_{k} \right) \Delta T_{k} \Delta p_{k} + \frac{L}{c_p} \Delta \hat{Q},$ |; $key = q/{calF}_lambda,{calF}_varphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal F}_\lambda, {\cal F}_\varphi$ |; $key = q/displaystyle-frac{R_d}{p_{k-1slash2}}T_{k-1slash2}^2(1-q_{k-1slash2})left[p_{k-1_{k}(1-q_{k})right]+c_p(1-q_{k-1slash2})T_{k-1slash2}(T_{k-1}-T_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p_{k-1/2}} T_{k-1/2}^2 (1-q_{k-1/2}) \left[ p_{k-1} ... ...k-1}) - p_{k} (1-q_{k}) \right] + c_p (1-q_{k-1/2}) T_{k-1/2} (T_{k-1} - T_{k})$ |; $key = q/displaystyleF;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle F$ |; $key = q/sigma_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma_{ij}$ |; $key = q/displaystyleDP{}{t}left(frac{1}{2}rhov_i^2right)+DP{}{x_j}(frac{1}{2}rhov_jv_i^2).;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{}{t} \left( \frac{1}{2} \rho v_i^2 \right) + \DP{}{x_j} ( \frac{1}{2} \rho v_j v_i^2 ).$ |; $key = q/displaystylefrac{R_d}{c_p},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{R_d}{c_p},$ |; $key = q/displaystyle-(zeta+f)U-dot{sigma}frac{partialV}{partialsigma}-frac{RT^{prime}}{a}(1-mu^{2})frac{partialpi}{partialmu}+{calF}_{varphi}cosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - ( \zeta + f ) U - \dot{\sigma} \frac{\partial V}{\partial \sigm... ...- \mu^{2} ) \frac{\partial \pi}{\partial \mu} + {\cal F}_{\varphi} \cos \varphi$ |; $key = q/displaystyleq^{*}(hat{T}_{k},p_{k})+left.DP{q^{*}}{T}right|_{hat{T}_{k}}cdotDeltaT_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^{*} (\hat{T}_{k},p_{k}) + \left. \DP{q^{*}}{T} \right\vert _{\hat{T}_{k}} \cdot \Delta T_{k},$ |; $key = q/displaystyleDD{Dvect{A}}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DD{\Dvect{A}}{t}$ |; $key = q/p;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/textstyle=;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle =$ |; $key = q/b_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}frac{partialpi}{partialt}+Dvect{v}_{H}cdotnabla_{sigma}pi=-nabla_{s}cdotDvect{v}_{H}-frac{partialdot{sigma}}{partialsigma}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{\partial \pi}{\partial t} + \Dvect{v}_{H} \cdot \na... ...Dvect{v}_{H} - \frac{\partial \dot{\sigma}}{\partial \sigma} \end{displaymath}$ |; $key = q/displaystyle-dot{sigma}DP{T}{sigma}+kappaTleft(DP{pi}{t}+Dvect{v}_{H}cdotnabla_{sigma}pi+frac{dot{sigma}}{sigma}right)+frac{Q^*}{C_{p}}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \dot{\sigma} \DP{T}{\sigma} + \kappa T \left( \DP{\pi}{t} + \Dv... ...bla_{\sigma} \pi + \frac{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^*}{C_{p}}.$ |; $key = q/T_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $T_{k-1}$ |; $key = q/displaystyle=-left(1+gamma_{k}right)DeltaT_{k}Deltap_{k}+frac{L}{c_p}left[left{hDeltap_{k-1}+left{hat{q}_{k}-q^{*}(hat{T}_{k})right}Deltap_{k}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = - \left( 1 + \gamma_{k} \right) \Delta T_{k} \Delta p_{k} + \fr... ...{k-1} + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\} \Delta p_{k} \right]$ |; $key = q/h_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/N_S;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/p=p^d+p^v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle+Dvect{e}_rfrac{1}{rcosvarphi}left[DP{A_{varphi}}{lambda}-DP{}{varphi}(cosvarphiA_{lambda})right],;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \Dvect{e}_r \frac{1}{r \cos \varphi} \left[ \DP{A_{\varphi}}{\lambda} - \DP{}{\varphi} (\cos \varphi A_{\lambda}) \right],$ |; $key = q/delta_S;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\delta_S$ |; $key = q/displaystyle-R_dDD{}{t}left{ln(p(1-q))right}+frac{1}{1-q}c_pDD{}{t}(lnT)+lDD{}{t}left(frac{1}{T}frac{q}{1-q}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - R_d \DD{}{t} \left\{ \ln (p (1-q)) \right\} + \frac{1}{1-q} c_p \DD{}{t} (\ln T) + l \DD{}{t} \left( \frac{1}{T} \frac{q}{1-q} \right)$ |; $key = q/%latex2htmlidmarker7263setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7263 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystylerhoR^d(1+epsilon_vq)T,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \rho R^d ( 1 + \epsilon_v q ) T,$ |; $key = q/displaystyle(zeta+f)V-dot{sigma}frac{partialU}{partialsigma}-frac{RT^{prime}}{a}frac{partialpi}{partiallambda}+{calF}_{lambda}cosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle ( \zeta + f ) V - \dot{\sigma} \frac{\partial U}{\partial \sigma}... ...me}}{a} \frac{\partial \pi}{\partial \lambda} + {\cal F}_{\lambda} \cos \varphi$ |; $key = q/displaystyle-frac{E}{2}left(MMcdotQP-frac{1}{2}TMcdotQPcdotD_{k-1}+MMcdotTMcdotD_{k-1}right)+frac{E}{2}(TPcdotD_{k-1}+QM);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left( MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot ... ...-1} + MM \cdot TM \cdot D_{k-1} \right) + \frac{E}{2} ( TP \cdot D_{k-1} + QM )$ |; $key = q/displaystyleP_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle P_{k}$ |; $key = q/displaystylefrac{q_{k-1}+q_{k}}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{q_{k-1} + q_{k}}{2}$ |; $key = q/{displaymath}dot{sigma}=-sigmafrac{partialpi}{partialt}-int_{0}^{sigma}Ddsigma-int_{0}^{sigma}Dvect{v}_{H}cdotnabla_{sigma}pidsigma,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \dot{\sigma} = - \sigma \frac{\partial \pi}{\partial t}... ...^{\sigma} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma , \end{displaymath}$ |; $key = q/displaystyle-frac{1}{acosvarphi}DP{}{lambda}left[dot{sigma}DP{v}{sigma}+frac{R^dT_v}{ap_s}DP{p_s}{varphi}-F_{varphi}+furight];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigma} \DP{v}{\sigma} + \frac{R^d T_v}{a p_s} \DP{p_s}{\varphi} - F_{\varphi} + f u \right]$ |; $key = q/displaystyleqquad-lfrac{q_{k-1}+q_{k}}{2}left(1-frac{q_{k-1}+q_{k}}{2}right)(T_{k-1}-T_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad - l \frac{q_{k-1} + q_{k}}{2} \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) (T_{k-1} - T_{k})$ |; $key = q/displaystyle+frac{1}{2}left{TMcdotMM+TPcdotMM-frac{1}{2}TPcdotTMcdotD_{k-1}right}DeltaT_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ TM \cdot MM + TP \cdot MM - \frac{1}{2} TP \cdot TM \cdot D_{k-1} \right\} \Delta T_{k-1}$ |; $key = q/displaystylenabla^2_{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla^2_{\sigma}$ |; $key = q/displaystyleV;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle V$ |; $key = q/displaystyle2Dvect{Omega}timesDvect{v};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 2 \Dvect{\Omega} \times \Dvect{v}$ |; $key = q/{displaymath}{calD}(T)=-(-1)^{N_Dslash2}K_{HD}nabla^{N_D}T,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} {\cal D}(T) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} T , \end{displaymath}$ |; $key = q/{displaymath}DP{D}{t}=frac{1}{a(1-mu^2)}DP{UA}{lambda}-frac{1}{a}DP{VA}{mu}-Dgrad^2_{sigma}(Phi+Rbar{T}_vpi+KE).{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DP{D}{t} = \frac{1}{a (1-\mu^2)} \DP{UA}{\lambda} - \fra... ...A}{\mu} - \Dgrad^2_{\sigma} ( \Phi + R \bar{T}_v \pi + KE ). \end{displaymath}$ |; $key = q/displaystyleT_{k-1}-T_{k}+frac{L}{c_p}left{q(T_{k-1},p_{k-1})-q^{*}(T_{k},p_{k})c_p}frac{Deltap_{k-1slash2}}{p_{k-1slash2}}frac{T_{k-1}+T_{k}}{2}=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_{k-1} - T_{k} + \frac{L}{c_p} \left\{ q (T_{k-1}, p_{k-1}) - q^... ...\frac{R}{c_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{T_{k-1} + T_{k}}{2} = 0,$ |; $key = q/displaystyle=-left[DeltaT_{k}+frac{L}{c_p}left{q^{*}(T_{k})+left.DP{q^{*}}{T}right|_{k}DeltaT_{k}-hat{q}_{k}right}right]Deltap_{k},;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = - \left[ \Delta T_{k} + \frac{L}{c_p} \left\{ q^{*} (T_{k}) + \... ...}{T} \right\vert _{k} \Delta T_{k} - \hat{q}_{k} \right\} \right] \Delta p_{k},$ |; $key = q/displaystyle(rho^dR^d+rho^vR^v)T;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (\rho^d R^d + \rho^v R^v) T$ |; $key = q/{displaymath}nabla^2bullet=frac{1}{h_1h_2h_3}left[DP{}{xi_1}left(frac{h_2h_3}{h_i_3}left(frac{h_1h_2}{h_3}DP{bullet}{xi_3}right)right],{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \nabla^2 \bullet = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\... ...eft( \frac{h_1 h_2}{h_3} \DP{\bullet}{\xi_3} \right) \right], \end{displaymath}$ |; $key = q/f;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{1}{rho}DP{p}{r}-g+2Omegaucosvarphi+frac{u^2}{r}+frac{v^2}{r}+F_r.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{\rho} \DP{p}{r} -g + 2 \Omega u \cos \varphi + \frac{u^2}{r} + \frac{v^2}{r} + F_r.$ |; $key = q/{displaymath}frac{1}{e^*_v}DP{e^*_v}{T}=frac{L}{R_vT^2}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \frac{1}{e^*_v} \DP{e^*_v}{T} = \frac{L}{R_v T^2} \end{displaymath}$ |; $key = q/fequiv2Omegasinvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $f \equiv 2\Omega \sin \varphi$ |; $key = q/kappa=RslashC_p;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\kappa = R/C_p$ |; $key = q/displaystyleleft(-1-gamma_{k}-kappafrac{Deltap_{k-1slash2}}{2p_{k-1slash2}}right)DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( - 1 - \gamma_{k} - \kappa \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}} \right) \Delta T_{k}$ |; $key = q/(q,T);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{1}{a}DP{}{mu}left[-frac{R^dbar{T}_v}{a}DP{pi}{lambda}right]-frac{1}{a}DP{}{mu}left[-frac{R^dT'_v}{a}DP{pi}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d \bar{T}_v}{a} \DP{\pi... ... - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$ |; $key = q/{displaymath}Dvect{v}_a=Dvect{v}+Dvect{Omega}timesDvect{r}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \Dvect{v}_a = \Dvect{v} + \Dvect{\Omega} \times \Dvect{r}. \end{displaymath}$ |; $key = q/displaystyleDvect{i}'A'_x+Dvect{j}'A'_y+Dvect{k}'A'_z;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{i}' A'_x + \Dvect{j}' A'_y + \Dvect{k}' A'_z$ |; $key = q/DeltaT_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Delta T_{k}$ |; $key = q/{displaymath}{calT}^f(z_1,z_2)=sum_{i=1}^{N_R}b_iexpleft(-delta_R|tau_{R,i}(z_1)-tau_{R,i}(z_2)|right){displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} {\cal T}^f(z_1,z_2) = \sum_{i=1}^{N_R} b_i \exp \left( - \delta_R \vert \tau_{R,i}(z_1) - \tau_{R,i}(z_2) \vert \right) \end{displaymath}$ |; $key = q/displaystylep^d;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle p^d$ |; $key = q/displaystyle-dot{sigma}frac{partialq}{partialsigma}+S_{q}+{calD}(q);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \dot{\sigma} \frac{\partial q }{\partial \sigma} + S_{q} + {\cal D}(q)$ |; $key = q/F'_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleKE;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle KE$ |; $key = q/{displaymath}DD{rho}{t}+rhoDdivDvect{v}=0.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{\rho}{t} + \rho \Ddiv \Dvect{v} = 0. \end{displaymath}$ |; $key = q/displaystyle-frac{R_d}{p(1-q)}DD{}{z}left{p(1-q)right}+frac{1}{1-q}frac{c_p}{T}Dac{l}{T}frac{1}{(1-q)}DD{q}{z}+frac{l}{T}frac{q}{(1-q)^2}DD{q}{z}=0.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\} + \frac{1... ...{l}{T} \frac{1}{(1-q)} \DD{q}{z} + \frac{l}{T} \frac{q}{(1-q)^2} \DD{q}{z} = 0.$ |; $key = q/displaystyle2Omega(Dvect{e}_{varphi}cosvarphi+Dvect{e}_rsinvarphi)times(uDvect{e}_{lambda}+vDvect{e}_{varphi}+wDvect{e}_r);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 2 \Omega ( \Dvect{e}_{\varphi} \cos \varphi + \Dvect{e}_r \sin \varphi) \times ( u \Dvect{e}_{\lambda} + v \Dvect{e}_{\varphi} + w \Dvect{e}_r)$ |; $key = q/psi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\psi$ |; $key = q/sigma_{SB};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma_{SB}$ |; $key = q/displaystyle1-frac{1}{2}q^{*}(hat{T}_{k-1})-frac{1}{2}q^{*}(hat{T}_{k}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 1 - \frac{1}{2} q^{*} (\hat{T}_{k-1}) - \frac{1}{2} q^{*} (\hat{T}_{k}),$ |; $key = q/displaystyleucosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle u \cos \varphi$ |; $key = q/displaystyleT(1+epsilon_vq);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T ( 1+\epsilon_v q )$ |; $key = q/displaystyleleft[DeltaT_{k-1}+frac{L}{c_p}left{q^{*}(hat{T}_{k-1})+left.DP{q^{*}}{T}right|_{k-1}DeltaT_{k-1}-hat{q}_{k-1}right}right]Deltap_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left[ \Delta T_{k-1} + \frac{L}{c_p} \left\{ q^{*} (\hat{T}_{k-1... ...ight\vert _{k-1} \Delta T_{k-1} - \hat{q}_{k-1} \right\} \right] \Delta p_{k-1}$ |; $key = q/displaystyleqquadleft.+MMcdotTMD_{k-1}DeltaT_{k-1}+MMcdotTMD_{k}DeltaT_{k}right};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \left. + MM \cdot TM D_{k-1} \Delta T_{k-1} + MM \cdot TM D_{k} \Delta T_{k} \right\}$ |; $key = q/displaystylequad-frac{L}{c_p}frac{1}{2}left(q^{*}(hat{T}_{k-1})+left.DP{q^{*}}{T_{k-1}+q^{*}(hat{T}_{k})+left.DP{q^{*}}{T}right|_{k}DeltaT_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad - \frac{L}{c_p} \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}) + ... ...^{*} (\hat{T}_{k}) + \left. \DP{q^{*}}{T} \right\vert _{k} \Delta T_{k} \right)$ |; $key = q/displaystyle-R_dDD{}{t}(lnp_d)+left(1+frac{q}{1-q}right)c_pDD{}{t}(lnT)+lDD{}{t}frac{r}{T};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - R_d \DD{}{t} (\ln p_d) + \left(1 + \frac{q}{1-q} \right) c_p \DD{}{t} (\ln T) + l \DD{}{t} \frac{r}{T}$ |; $key = q/v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/%latex2htmlidmarker6864setcounter{footnote}{2}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6864 \setcounter{footnote}{2}\fnsymbol{footnote}$ |; $key = q/C_p;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/T'_v;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaymath}0=-frac{1}{rho}DP{p}{z}-g,{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} 0 = - \frac{1}{\rho} \DP{p}{z} - g, \end{displaymath}$ |; $key = q/displaystyleqquad-frac{R}{c_p}frac{Deltap_{k-1slash2}}{p_{k-1slash2}}frac{hat{T}_{k-1}+DeltaT_{k-1}+hat{T}_{k}+DeltaT_{k}}{2}=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad - \frac{R}{c_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}} \frac{\hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k}}{2} = 0,$ |; $key = q/{displaymath}{calD}(q)=-(-1)^{N_Dslash2}K_{HD}nabla^{N_D}q.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} {\cal D}(q) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} q . \end{displaymath}$ |; $key = q/bar{k}_{R,i};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bar{k}_{R,i}$ |; $key = q/displaystyleqquad+frac{L}{c_p}frac{T_{k-1}+T_{k}}{2}(q_{k-1}-q_{k})=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad + \frac{L}{c_p} \frac{T_{k-1} + T_{k}}{2} (q_{k-1} - q_{k}) = 0$ |; $key = q/{displaymath}0=-frac{1}{rho}DP{p}{z}-g.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} 0 = - \frac{1}{\rho} \DP{p}{z} - g. \end{displaymath}$ |; $key = q/displaystyleq_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q_{k-1}$ |; $key = q/delta_R=1.5;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\delta_R = 1.5$ |; $key = q/displaystylesum_i^{N_S}a_ileft[(1-alpha_A)F_S^Iexpleft(-tau_{S,i}(z)seczetaright)right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sum_i^{N_S} a_i \left[ (1-\alpha_A) F_S^I \exp \left( - \tau_{S,i}(z) \sec \zeta \right) \right.$ |; $key = q/{calD}(zeta),{calD}(D),{calD}(T),{calD}(q);MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal D}(\zeta), {\cal D}(D), {\cal D}(T), {\cal D}(q)$ |; $key = q/%latex2htmlidmarker7304setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7304 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/dot{sigma};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\sigma}$ |; $key = q/q_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $q_{k-1}$ |; $key = q/g;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/varepsilon=c_vT;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\varepsilon = c_v T$ |; $key = q/displaystyleDP{q}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{q}{t}$ |; $key = q/Uequivucosphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $U \equiv u \cos \phi$ |; $key = q/displaystyle-R_dfrac{1}{p(1-q)}DD{}{t}left{p(1-q)right}+frac{1}{1-q}frac{c_p}{T}}DD{T}{t}frac{q}{1-q}+frac{1}{T}DD{}{t}left(frac{q}{1-q}right)right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - R_d \frac{1}{p (1-q)} \DD{}{t} \left\{ p (1-q) \right\} + \frac... ...T}{t} \frac{q}{1-q} + \frac{1}{T} \DD{}{t} \left( \frac{q}{1-q} \right) \right]$ |; $key = q/lambda;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\lambda$ |; $key = q/displaystylefrac{partialq}{partialt};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{\partial q}{\partial t}$ |; $key = q/displaystyle+Dvect{e}_rleft[DP{A_r}{t}+frac{u}{rcosvarphi}DP{A_r}{lambda}+frac{vvarphi}+wDP{A_r}{r}-frac{v}{r}A_{varphi}-frac{u}{r}A_{lambda}right].;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \Dvect{e}_r \left[ \DP{A_r}{t} + \frac{u}{r \cos \varphi} \DP{A... ...i} + w \DP{A_r}{r} - \frac{v}{r} A_{\varphi} - \frac{u}{r} A_{\lambda} \right].$ |; $key = q/seczeta_0=1.66;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sec \zeta_0 = 1.66$ |; $key = q/Q^*=Q+c_pmathcal{D}(Dvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| $Q^*=Q+c_p \mathcal{D}(\Dvect{v})$ |; $key = q/T^2(1-q)^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylequad+frac{L}{c_p}frac{1}{2}left(hat{T}_{k-1}+DeltaT_{k-1}+hat{T}_{k}+DeltaT_{k}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \frac{L}{c_p} \frac{1}{2} \left( \hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k} \right)$ |; $key = q/displaystylersinvarphi,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle r \sin \varphi,$ |; $key = q/displaystylequad+frac{E}{2}left{TPcdotQM+left(D_{k-1}DeltaT_{k-1}-D_{k}DeltaT_{k}right)TP+(DeltaT_{k-1}+DeltaT_{k})QMright}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \quad + \frac{E}{2} \left\{ TP \cdot QM + \left( D_{k-1} \Delta T... ...D_{k} \Delta T_{k} \right) TP + (\Delta T_{k-1} + \Delta T_{k}) QM \right\} = 0$ |; $key = q/rho^v;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\rho^v$ |; $key = q/{displaymath}left(DD{Dvect{A}}{t}right)_{{rm{a}=left(DD{Dvect{A}}{t}right)_{{rm{r}+Dvect{Omega}timesDvect{A}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \left( \DD{\Dvect{A}}{t} \right)_{\rm a} = \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}. \end{displaymath}$ |; $key = q/displaystyleDP{}{t}(rhov_i^2)+DP{}{x_j}(rhov_jv_i^2)-DP{}{t}left(frac{1}{2}rhov_i^2right)-DP{}{x_j}left(frac{1}{2}v_i^2rhov_jright);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{}{t} ( \rho v_i^2 ) + \DP{}{x_j} ( \rho v_j v_i^2 ) - \DP{}{t... ...1}{2} \rho v_i^2 \right) - \DP{}{x_j} \left( \frac{1}{2} v_i^2 \rho v_j \right)$ |; $key = q/displaystyle+frac{1}{2}left{MMcdotTPcdotTM+TMcdotMMDeltaT_{k-1}+TMcdotMMDeltaT_{k}+TPcdotMMDeltaT_{k-1}-TPcdotMMDeltaT_{k}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{1}{2} \left\{ MM \cdot TP \cdot TM + TM \cdot MM \Delta T... ...MM \Delta T_{k} + TP \cdot MM \Delta T_{k-1} - TP \cdot MM \Delta T_{k} \right.$ |; $key = q/phi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\phi$ |; $key = q/l;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle=frac{St+left(1+gamma_{k-1}-kappafrac{Deltap_{k-1slash2}}{2p_{k-1slaight)frac{1+gamma_{k}}{1+gamma_{k-1}}frac{Deltap_{k}}{Deltap_{k-1}}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle = \frac{ St + \left( 1 + \gamma_{k-1} - \kappa \frac{\Delta p_{k-... ...frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}} \frac{\Delta p_{k}}{\Delta p_{k-1}} }$ |; $key = q/%latex2htmlidmarker6967setcounter{footnote}{3}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 6967 \setcounter{footnote}{3}\fnsymbol{footnote}$ |; $key = q/bullet;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bullet$ |; $key = q/displaystyle-frac{E}{2}left{QPcdotMMcdotTM+MMcdotQP(DeltaT_{k-1}-DeltaT_{k})+TMc-frac{1}{2}D_{k-1}DeltaT_{k-1}-frac{1}{2}D_{k}DeltaT_{k}right)right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{E}{2} \left\{ QP \cdot MM \cdot TM + MM \cdot QP ( \Delta... ...c{1}{2} D_{k-1} \Delta T_{k-1} - \frac{1}{2} D_{k} \Delta T_{k} \right) \right.$ |; $key = q/{displaymath}DD{v}{t}=-frac{u^2tanvarphi}{a}-fu-frac{1}{rhoa}DP{p}{varphi}+F_{varphi},{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \DD{v}{t} = - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho a } \DP{p}{\varphi} + F_{\varphi}, \end{displaymath}$ |; $key = q/(xi_1,xi_2,xi_3);MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\xi_1, \xi_2, \xi_3)$ |; $key = q/displaystyleq^*(T_{k-1},p_{k-1}),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle q^* (T_{k-1}, p_{k-1}),$ |; $key = q/displaystyleqquad-frac{L}{c_p}frac{q_{k-1}+q_{k}}{2}left(1-frac{q_{k-1}+q_{k}}{2}right)(T_{k-1}-T_{k});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad - \frac{L}{c_p} \frac{q_{k-1} + q_{k}}{2} \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) (T_{k-1} - T_{k})$ |; $key = q/Dvect{Omega};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{\Omega}$ |; $key = q/epsilon_v;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\epsilon_v$ |; $key = q/displaystyleleft(-1-gamma_{k}-frac{R}{c_p}frac{Deltap_{k-1slash2}}{2p_{k-1slash2}{c_p}frac{Deltap_{k-1slash2}}{2p_{k-1slash2}}right)DeltaT_{k-1}=-St;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( - 1 - \gamma_{k} - \frac{R}{c_p} \frac{\Delta p_{k-1/2}}{2... ...frac{R}{c_p} \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}} \right) \Delta T_{k-1} = - St$ |; $key = q/displaystyleE;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle E$ |; $key = q/displaystyle-frac{R_d}{c_p}frac{1}{4}left(hat{T}_{k-1}+DeltaT_{k-1}+hat{T}_{k}+DeltaT_{k}right)^2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{R_d}{c_p} \frac{1}{4} \left( \hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k} \right)^2$ |; $key = q/R^d;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylevcosvarphi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle v \cos \varphi$ |; $key = q/%latex2htmlidmarker7332setcounter{footnote}{6}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7332 \setcounter{footnote}{6}\fnsymbol{footnote}$ |; $key = q/displaystyle+Dvect{e}_{varphi}left[DP{A_{varphi}}{t}+frac{u}{rcosvarphi}DP{A_{vawDP{A_{varphi}}{r}+frac{v}{r}A_r+frac{utanvarphi}{r}A_{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \Dvect{e}_{\varphi} \left[ \DP{A_{\varphi}}{t} + \frac{u}{r \co... ..._{\varphi}}{r} + \frac{v}{r} A_r + \frac{u \tan \varphi}{r} A_{\lambda} \right]$ |; $key = q/displaystyleDeltaT_{k-1}=-frac{1+gamma_{k}}{1+gamma_{k-1}}frac{Deltap_{k}}{Delta}+frac{1}{1+gamma_{k-1}}frac{L}{c_p}Deltahat{Q}frac{1}{Deltap_{k-1}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta T_{k-1} = - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}} \fr... ...rac{1}{ 1 + \gamma_{k-1}} \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}$ |; $key = q/displaystyleqquadtimesleft[P_{k-1}left(M_{k-1}-D_{k-1}DeltaT_{k-1}right)-P_{k}left(M_{k}-D_{k}DeltaT_{k}right)right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left[ P_{k-1} \left( M_{k-1} - D_{k-1} \Delta T_{k-1} \right) - P_{k} \left( M_{k} - D_{k} \Delta T_{k} \right) \right]$ |; $key = q/displaystyleqquadtimesleft{MMcdotTM+(DeltaT_{k-1}-DeltaT_{k})MM+left(-frac{1}{2}D_{k-1}DeltaT_{k-1}-frac{1}{2}D_{k}DeltaT_{k}right)TMright};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \qquad \times \left\{ MM \cdot TM + ( \Delta T_{k-1} - \Delta T_{... ...1}{2} D_{k-1} \Delta T_{k-1} - \frac{1}{2} D_{k} \Delta T_{k}\right)TM \right\}$ |; $key = q/Dvect{e}_i;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{e}_i$ |; $key = q/T_{k-1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $T_{k-1/2}$ |; $key = q/displaystyle-frac{1}{a(1-mu^{2})}DP{UT}{lambda}-frac{1}{a}DP{VT}{mu}+TD;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a(1-\mu^{2})} \DP{UT}{\lambda} - \frac{1}{a} \DP{VT}{\mu} + T D$ |; $key = q/Dvect{A}=Dvect{r};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{A}=\Dvect{r}$ |; $key = q/displaystylefrac{1}{a(1-mu^2)}DP{}{lambda}(zetaV)-frac{1}{a}DP{}{mu}(zetaU);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{a (1-\mu^2)} \DP{}{\lambda} ( \zeta V ) - \frac{1}{a} \DP{}{\mu} ( \zeta U )$ |; $key = q/displaystylex_3;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle x_3$ |; $key = q/displaystyle-frac{1}{acosvarphi}DP{}{varphi}left[cosvarphidot{sigma}DP{v}{sigma}{ap_s}DP{p_s}{varphi}cosvarphi-F_{varphi}cosvarphi+fucosvarphiright];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ \cos \varphi \do... ..._s}{\varphi} \cos \varphi - F_{\varphi} \cos \varphi + f u \cos \varphi \right]$ |; $key = q/displaystylegamma_{k}equivfrac{L}{c_p}left.DP{q^{*}}{T}right|_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \gamma_{k} \equiv \frac{L}{c_p} \left. \DP{q^{*}}{T} \right\vert _{k}$ |; $key = q/displaystyleDvect{e}_{lambda}frac{1}{r}left[DP{A_r}{varphi}-DP{}{r}(rA_{varphi})right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Dvect{e}_{\lambda} \frac{1}{r} \left[ \DP{A_r}{\varphi} - \DP{}{r}(r A_{\varphi}) \right]$ |; $key = q/(hat{q},hat{T});MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\hat{q}, \hat{T})$ |; $key = q/displaystyle-frac{1}{a^2(1-mu^2)}DP[2]{}{lambda}(R^dbar{T}_vpi)-frac{1}{a(1-mu^2)}DP{}{lambda}left[frac{R^dT'_v}{a}DP{pi}{lambda}right];MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{1}{a^2 (1-\mu^2)} \DP[2]{}{\lambda} ( R^d \bar{T}_v \pi )... ...a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$ |; $key = q/e^*(T);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylezeta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \zeta$ |; $key = q/displaystyle-dot{sigma}frac{partialT}{partialsigma}+kappaTleft(frac{partialpi}{pgma}}{sigma}right)+frac{Q}{C_{p}}+{calD}(T)+{calD}^{prime}(Dvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \dot{\sigma} \frac{\partial T }{\partial \sigma} + \kappa T \le... ...\sigma } \right) + \frac{Q}{C_{p}} + {\cal D}(T) + {\cal D}^{\prime}(\Dvect{v})$ |; 1;