# LaTeX2HTML 2008 (1.71) # Associate images original text with physical files. $key = q/partial_xzeta=-10^{-5};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_x\zeta=-10^{-5}$|; $key = q/16.6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 16.6$|; $key = q/c_{b5}=tilde{c}_{b5};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b5} = \tilde{c}_{b5}$|; $key = q/{displaymath}{array}{rcl}n&=&-dfrac{1}{4(2+d)(kappa^2R-L^2)}Bigg(4dkappa^2R,m-(1dkappa^2Rm+(2+d)(1+4m)alphaL^2Big)^2}right)quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} n &=& - \dfrac{1}{4 (2+d) (\kappa^2 R - L^... ...+ (2+d) (1+4m) \alpha L^2 \Big)^2 } \right) \quad . \end{array}\end{displaymath}|; $key = q/displaystyle{calN}=dfrac{c_1}{2};,quad{calN}_b=c_{b1}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal N} = \dfrac{c_1}{2} \; , \quad {\cal N}_b = c_{b1} \quad .$|; $key = q/langleRrangle=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \langle R \rangle =0$|; $key = q/displaystyled=-frac{2n}{2m+n-2c_{psi2}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle d = - \frac{2n}{2m+n-2c_{\psi 2}} \; , \quad$|; $key = q/rho_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \rho_0$|; $key = q/c_6=tilde{c}_6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_6 = \tilde{c}_6$|; $key = q/z_{bl};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_{bl}$|; $key = q/0.78;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.78$|; $key = q/displaystyleepsilon=(c_mu^0)^3K^frac{3}{2}L^{-1}(z+z_0)^{frac{3}{2}alpha-1}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon= (c_\mu^0)^3 K^\frac{3}{2} L^{-1} (z+z_0)^{\frac{3}{2}\alpha - 1} \quad .$|; $key = q/V_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ V_t$|; $key = q/displaystylel=kappafrac{(d_s+z_0^s)(d_b^text{Xing}+z_0^b)}{d_s+d_b^text{Xing}+z_};,quadd_b^text{Xing}=d_bexp{left(-betafrac{d_b}{H}right)};,quad    ;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l = \kappa \frac{(d_s+z_0^s)(d_b^\text{Xing}+z_0^b)} {d_s+d_b^\te... ... d_b^\text{Xing} = d_b \exp{\left(-\beta \frac{d_b}{H} \right)} \; , \quad     $|; $key = q/sigma_k^psi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma_k^\psi$|; $key = q/displaystylek=dfrac{u_*^2}{(c_mu^0)^2};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k = \dfrac{u_*^2}{(c_\mu^0)^2} \; , \quad$|; $key = q/vec{c}^{n+1}_i>0foralli;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \vec{c}^{n+1}_i>0 \forall i$|; $key = q/alpha=78^{circ};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha=78^{\circ}$|; $key = q/displaystyleoverline{S}_{ij}=dfrac{k}{epsilon}S_{ij};,quadoverline{Sigma}_{ij}=dpsilon}Sigma_{ij};,quadoverline{Z}_{ij}=dfrac{k}{epsilon}Z_{ij}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{S}_{ij} = \dfrac{k}{\epsilon} S_{ij} \; , \quad \overli... ...} \Sigma_{ij} \; , \quad \overline{Z}_{ij} = \dfrac{k}{\epsilon} Z_{ij} \quad .$|; $key = q/P-E;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P-E$|; $key = q/a_bi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_bi$|; $key = q/{calD}_U;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_U$|; $key = q/{calD}_k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_k$|; $key = q/D_i(vec{c});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ D_i(\vec{c})$|; $key = q/I=7;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I=7$|; $key = q/Deltat=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta t=1$|; $key = q/dot{S};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{S}$|; $key = q/1.75;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.75$|; $key = q/q_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q_s$|; $key = q/r;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ r$|; $key = q/displaystylepinleft(0,minleft({1,mathop{min}limits_{jinJ^n}left({-frac{c_j^n}{Deltat:f_j(t^n,vec{c}^n)}}right)}right)right).;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle p \in \left( 0, \min \left( {1,\mathop {\min }\limits_{j \in J^n}... ...( { - \frac{c_j^n }{\Delta t \: f_j (t^n,\vec{c}^n)}} \right)} \right) \right).$|; $key = q/z=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z=0$|; $key = q/0.0906;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.0906$|; $key = q/displaystylesigma_psi=frac{2kappa^2d}{(c_mu^0)^2(d+2)}nquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \sigma_\psi = \frac{2 \kappa^2 d}{(c_\mu^0)^2 (d+2)} n \quad .$|; $key = q/P_i(vec{c});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_i(\vec{c})$|; $key = q/c_3=2(tilde{c}_3+tilde{c}_4);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_3 = 2 ( \tilde{c}_3 + \tilde{c}_4 )$|; $key = q/nu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu$|; $key = q/displaystyled_{i,j}(vec{c})longrightarrow0;;;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle d_{i,j}(\vec{c}) \longrightarrow 0 \;\;$|; $key = q/11.9;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 11.9$|; $key = q/E_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ E_2$|; $key = q/tilde{Gamma}_B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_B$|; $key = q/displaystyleDelta_xTheta=Deltaxpartial_xTheta;,quadDelta_yTheta=Deltaypartial_yTheta;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Delta_x\Theta=\Delta x \partial_x\Theta \; , \quad \Delta_y\Theta=\Delta y \partial_y\Theta \; , \quad$|; $key = q/alpha=-2.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha=-2.5$|; $key = q/0.547;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.547$|; $key = q/C_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_1$|; $key = q/displaystyledot{rule{0mm}{4mm}langleu'_iu'_jrangle}-{calD}_{ij}=dfrac{{langleu'_(dfrac{dot{k}-{calD}_k}{k}+dfrac{dot{k}_b-{calD}_b}{k_b}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{ \rule{0mm}{4mm} \langle u'_i u'_j \rangle} - {\cal D}_{ij} ... ...{k} - {\cal D}_k }{k} + \dfrac{\dot{k}_b - {\cal D}_b }{k_b} \right) \; , \quad$|; $key = q/i=1,..,N_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ i=1,..,N_i$|; $key = q/d_l=d_u=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_l=d_u=0$|; $key = q/Q_{text{sour}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_{\text{sour}}$|; $key = q/l_epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_\epsilon$|; $key = q/displaystyle{vec{c}^{n+1}=vec{c}^n+Deltat:vec{f}(t^n,vec{c}^n)prodlimits_{jinJ^n}{frac{c_j^{n+1}}{c_j^n}}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\vec{c}^{n + 1} = \vec{c}^n + \Delta t \: \vec{f}(t^n,\vec{c}^n)\prod\limits_{j \in J^n} {\frac{c_j^{n + 1} }{c_j^n}}}$|; $key = q/H=50;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ H=50$|; $key = q/c_1=2slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_1 = 2/\lambda$|; $key = q/displaystylepartial_tX=left{{array}{ll}u&mbox{for}|X||; $key = q/epsilon_theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon_\theta$|; $key = q/displaystyleb_{ij}=-lambda_1overline{S}_{ij}-2lambda_2overline{Sigma}_{ij}-2lambda_3overline{Z}_{ij}-lambda_4overline{Gamma}_{ij}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle b_{ij} = - \lambda_1 \overline{S}_{ij} - 2 \lambda_2 \overline{\S... ...{ij} - 2 \lambda_3 \overline{Z}_{ij} - \lambda_4 \overline{\Gamma}_{ij} \quad .$|; $key = q/N_S^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_S^2$|; $key = q/T_a;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ T_a$|; $key = q/a_0=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_0=0$|; $key = q/tau_R=;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_R=$|; $key = q/nu'_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu'_t$|; $key = q/a_{b4}=2lambda_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b4} = 2 \lambda_0$|; $key = q/-1.74;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -1.74$|; $key = q/displaystylek_b={langleb'^2rangle}slash2=E_pN^2;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k_b = {\langle b'^2 \rangle} /2 = E_p N^2 \; , \quad$|; $key = q/displaystylek=K(z+z_0)^alpha;,quadl=L(z+z_0);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k = K (z+z_0)^\alpha \; , \quad l = L(z+z_0) \; , \quad$|; $key = q/d_l>0,d_u=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_l>0, d_u=0$|; $key = q/(tau_x^s,tau_y^s);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ (\tau_x^s,\tau_y^s)$|; $key = q/displaystylehat{c}_mu=dfrac{N_n}{D};,quadhat{c}_mu'=dfrac{N_b}{D};,quadGamma=dfrac{N_Gamma}{D};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \hat{c}_\mu = \dfrac{N_n}{D} \; , \quad \hat{c}_\mu' = \dfrac{N_b}{D} \; , \quad \Gamma = \dfrac{N_\Gamma}{D} \; , \quad$|; $key = q/0.58;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.58$|; $key = q/displaystyle{langlerhorangle}=hat{rho}(S,Theta,P)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\langle \rho \rangle} = \hat{\rho} (S,\Theta,P) \quad .$|; $key = q/displaystyle-frac{1}{rho_0}dfrac{partial{P}}{partial{y}}=-gdfrac{partial{zeta}}{partial{y}}+int_z^{zeta}dfrac{partial{B}}{partial{y}},dz';,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle - \frac{1}{\rho_0} \dfrac{\partial {P}}{\partial {y}}= -g \dfrac{... ...rtial {y}} +\int_z^{\zeta} \dfrac{\partial {B}}{\partial {y}} dz' \; , \quad$|; $key = q/{langlew'theta'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle w'\theta' \rangle} $|; $key = q/l_u;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_u$|; $key = q/pm0.1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \pm 0.1$|; $key = q/d_{i,j}geq0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_{i,j}\geq 0$|; $key = q/F_c;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F_c$|; $key = q/c_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_1$|; $key = q/e=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ e=1$|; $key = q/displaystylel=kappa,min(d_s+z_0^s,d_b+z_0^b);,quad    ;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l = \kappa \min(d_s+z_0^s,d_b+z_0^b) \; , \quad     $|; $key = q/c_{mu}^0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\mu}^0$|; $key = q/{langlew's'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle w's' \rangle} $|; $key = q/0.101;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.101$|; $key = q/kappaapprox0.4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \kappa \approx 0.4$|; $key = q/K_V;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ K_V$|; $key = q/tilde{Gamma}_Theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_\Theta$|; $key = q/-1.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -1.5$|; $key = q/R=c_mu^0slashc_mu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ R=c_\mu^0/c_\mu$|; $key = q/B_{is};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_{is}$|; $key = q/N^2=1cdot10^{-4};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N^2=1 \cdot 10^{-4}$|; $key = q/c_5=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_5 = 0$|; $key = q/tau^S_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau^S_R$|; $key = q/displaystyleGamma_{ij}=-left(G_{ij}-frac{2}{3}Gdelta_{ij}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Gamma_{ij} = - \left( G_{ij} - \frac{2}{3} G \delta_{ij} \right) \; , \quad$|; $key = q/tilde{c}_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_3$|; $key = q/a_3=2lambda_3slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_3 = 2 \lambda_3/\lambda$|; $key = q/displaystyleF_k=-dfrac{nu_t}{sigma_k}dfrac{partial{k}}{partial{z}}=-dfrac{c_mu}{sigma_k}K^frac{3}{2}Lalpha(z+z_0)^{frac{3}{2}alpha};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_k = - \dfrac{\nu_t}{\sigma_k} \dfrac{\partial {k}}{\partial {z}... ...c_\mu}{\sigma_k} K^\frac{3}{2} L \alpha (z+z_0)^{\frac{3}{2} \alpha} \; , \quad$|; $key = q/d=-1.2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d = -1.2$|; $key = q/P=epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P=\epsilon$|; $key = q/-2.0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -2.0$|; $key = q/Ri=N^2slashM^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri=N^2/M^2$|; $key = q/beta=;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \beta =$|; $key = q/displaystyle{calD}_Theta=dfrac{partial}{partial{z}}left(left(nu^Theta_t+nu^Thetaight)dfrac{partial{Theta}}{partial{z}}-tilde{Gamma}_Thetaright)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_\Theta = \dfrac{\partial}{\partial {z}} \left( \left( \n... ...\dfrac{\partial {\Theta}}{\partial {z}} - \tilde{\Gamma}_\Theta \right) \quad .$|; $key = q/{displaymath}{array}{rcl}dfrac{P}{epsilon}&=&-2b_{13}overline{S}_U-2b_{23}overli}{epsilon}dfrac{overline{N}^2}{overline{T}}{array}quad.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \dfrac{P}{\epsilon} &=& - 2 b_{13} \overli... ...silon} \dfrac{\overline{N}^2}{\overline{T}} \end{array} \quad .\end{displaymath}|; $key = q/zeta=d_;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \zeta=d_$|; $key = q/displaystyleP_b=-GN^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P_b = -G N^2 \quad .$|; $key = q/displaystyledot{overline{q^2l}}={calD}_l+l(E_1P+E_3G-E_2Fepsilon);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\overline{q^2 l}} = {\cal D}_l + l ( E_1 P + E_3 G - E_2 F \epsilon ) \; , \quad$|; $key = q/c_{psi3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\psi 3}$|; $key = q/=2tau;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ = 2 \tau$|; $key = q/d_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_b$|; $key = q/displaystyleoverline{Gamma}_{ij}=Gamma_{ij}slashepsilon={pmatrix}-frac{2}{3}gamm}gamma_3{pmatrix};,quadgamma_i=-dfrac{{langleu'_ib'rangle}}{epsilon};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{\Gamma}_{ij} = \Gamma_{ij} / \epsilon = \begin{pmatrix}... ...{pmatrix} \; , \quad \gamma_i = - \dfrac{ {\langle u'_i b' \rangle} }{\epsilon}$|; $key = q/displaystyleL_{ij}={pmatrix}0&0&S_U0&0&S_V0&0&0{pmatrix};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle L_{ij} = \begin{pmatrix}0 & 0 & S_U \ 0 & 0 & S_V \ 0 & 0 & 0 \ \end{pmatrix} \; , \quad$|; $key = q/{calN}_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N}_b$|; $key = q/{displaymath}{array}{rcl}{calN}b_{11}&=&-left(dfrac{a_2}{3}+a_3right)b_{13}overlS}_U-dfrac{1}{2}a_1overline{S}_V-a_5gamma_2quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} {\cal N} b_{11} &=& - \left( \dfrac{a_2}{3... ...rac{1}{2} a_1 \overline{S}_V - a_5 \gamma_2 \quad . \end{array}\end{displaymath}|; $key = q/X_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ X_0$|; $key = q/l=kappa(z+z_0);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l=\kappa(z+z_0)$|; $key = q/displaystyleleft(dfrac{partial{U}}{partial{z}}right)^2approxfrac{(barU_{j+1}-barU_j)(tildeU_{j+1}-tildeU_j)}{(z_{j+1}-z_j)^2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left( \dfrac{\partial {U}}{\partial {z}} \right)^2 \approx \frac{(\bar U_{j+1}-\bar U_j) (\tilde U_{j+1}-\tilde U_j)}{(z_{j+1}-z_j)^2}$|; $key = q/-1.05;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -1.05$|; $key = q/P+G=epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P+G=\epsilon$|; $key = q/c_{b4}=tilde{c}_{b4};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b4} = \tilde{c}_{b4}$|; $key = q/zeta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \zeta$|; $key = q/displaystylekappa=sqrt{dfrac{E_2-E_1+1}{S_lB_1}}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \kappa = \sqrt{\dfrac{E_2 - E_1 + 1}{S_l B_1}} \quad .$|; $key = q/partial_sT=-0.0575;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_s T=-0.0575$|; $key = q/{displaymath}{array}{l}int_{z_0}^{z_0+l_u(z_0)}(b(z_0)-b(z))dz=k(z_0);,quad[4mm]int_{z_0-l_d(z_0)}^{z_0}(b(z)-b(z_0))dz=k(z_0){array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{l} \int_{z_0}^{z_0+l_u(z_0)} (b(z_0)-b(z)) dz =... ... \int_{z_0-l_d(z_0)}^{z_0} (b(z)-b(z_0)) dz =k(z_0) \end{array}\end{displaymath}|; $key = q/zeta_i={langleu'_ib'rangle}slashsqrt{(kk_b)};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \zeta_i = {\langle u'_i b' \rangle} / \sqrt{(k k_b)}$|; $key = q/tilde{c}_{b1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_{b1}$|; $key = q/tilde{c}_{b2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_{b2}$|; $key = q/Gamma;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Gamma$|; $key = q/{langleu'_ib'rangle}=alphagh_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle u'_i b' \rangle} = \alpha g h_i$|; $key = q/zeta_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \zeta_i$|; $key = q/{displaymath}{array}{rcl}c_i^{n+1}&=&displaystylec_i^n+Deltatleft{P_ileft(underle{c}^nright)-D_ileft(underline{c}^nright)right}.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} c_i^{n+1} &=& \displaystyle c_i^n + \Delta... ...\right) - D_i\left(\underline{c}^n\right) \right\}. \end{array}\end{displaymath}|; $key = q/displaystyledfrac{partial{w}}{partial{sigma}}Big|_{sigma=1}=-5hw_phi(1)dfrac{B_f}{u_*^4};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {w}}{\partial {\sigma}} \Big\vert _{\sigma=1} = - 5 h w_\phi(1) \dfrac{B_f}{u_*^4} \; , \quad$|; $key = q/a_2=2lambda_2slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_2 = 2 \lambda_2/\lambda$|; $key = q/1.92;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.92$|; $key = q/p_{i,i}=d_{i,i}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,i}=d_{i,i}=0$|; $key = q/Q_e;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_e$|; $key = q/displaystyleleft{{i:f_i(t^n,vec{c}^n)+f_i(t^{n+1},vec{c}^{(1)})<0,iin{1,...,I}}right}.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left\{ {i:f_i (t^n ,\vec{c}^n ) + f_i (t^{n+1} ,\vec{c}^{(1)} ) < 0, i \in \{ 1,...,I\} } \right\}.$|; $key = q/C_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_3$|; $key = q/1.3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.3$|; $key = q/tildez;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde z$|; $key = q/displaystylenu_t^{SI}=nu'^{SI}_t=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t^{SI} = \nu'^{SI}_t = 0 \; , \quad$|; $key = q/lambda_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_0$|; $key = q/N^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N^2$|; $key = q/tau=kslashepsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau=k/\epsilon$|; $key = q/c_{mu}';MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\mu}'$|; $key = q/displaystyledot{V}={calD}_V-gdfrac{partial{zeta}}{partial{y}}+int_z^{zeta}dfrac{{partial{y}},dz'-frac{1}{tau^V_R}(V-V_{obs})-C_fVsqrt{U^2+V^2};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{V} = {\cal D}_V - g \dfrac{\partial {\zeta}}{\partial {y}} +... ...tial {y}} dz' - \frac{1}{\tau^V_R}(V-V_{obs})-C_f V \sqrt{U^2+V^2} \; , \quad$|; $key = q/displaystyleb_{13}=-dfrac{1}{2}hat{c}_muoverline{S}_U;,quadb_{23}=-dfrac{1}{2}ha{c}_muoverline{S}_V;,quadgamma_3=hat{c}_mu'overline{N}^2-Gamma;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle b_{13} = - \dfrac{1}{2} \hat{c}_\mu \overline{S}_U \; , \quad b_{... ...line{S}_V \; , \quad \gamma_3 = \hat{c}_\mu' \overline{N}^2 - \Gamma \; , \quad$|; $key = q/overline{T};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{T}$|; $key = q/displaystyleK_Vpartial_zc_ibig|_{z=eta}=F^s_i,qquadK_Vpartial_zc_ibig|_{z=-H}=-F^b_i,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle K_V\partial_z c_i\big\vert _{z=\eta} = F^s_i,\qquad K_V\partial_z c_i\big\vert _{z=-H} = -F^b_i,$|; $key = q/etaapprox100;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \eta \approx 100$|; $key = q/gamma_{i-1}|; $key = q/displaystyle=;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle =$|; $key = q/Phi_phi(zeta);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi_\phi(\zeta)$|; $key = q/d_u;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_u$|; $key = q/_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ _1$|; $key = q/S;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S$|; $key = q/u'_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ u'_i$|; $key = q/tilde{c}_6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_6$|; $key = q/lambda_0r=lambda_8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_0 r = \lambda_8$|; $key = q/2times2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2\times 2$|; $key = q/0.85;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.85$|; $key = q/omega;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \omega$|; $key = q/displaystylec_{mu}=c_{mu}^0,qquadc'_{mu}=frac{c_{mu}^0}{Pr_t};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle c_{\mu}=c_{\mu}^0,\qquad c'_{\mu}=\frac{c_{\mu}^0}{Pr_t}$|; $key = q/{displaymath}{array}{l}l_k(z_0)=text{Min}(l_d(z_0),l_u(z_0));,quad[4mm]l_{epsiloz_0)=left(l_d(z_0)l_u(z_0)right)^frac{1}{2}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{l} l_k(z_0)= \text{Min} ( l_d(z_0),l_u(z_0)) \;... ...=\left( l_d(z_0)l_u(z_0)\right)^\frac{1}{2} \quad . \end{array}\end{displaymath}|; $key = q/c_mu=c_mu^0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_\mu=c_\mu^0$|; $key = q/c_{epsilon3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\epsilon 3}$|; $key = q/Deltat;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta t$|; $key = q/H=d_b+d_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ H=d_b+d_s$|; $key = q/d_l;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_l$|; $key = q/displaystyleb_{ij}=dfrac{{langleu'_iu'_jrangle}}{2k}-dfrac{1}{3}delta_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle b_{ij} = \dfrac{ {\langle u'_i u'_j \rangle} }{2k} - \dfrac{1}{3} \delta_{ij} \; , \quad$|; $key = q/H=15;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ H=15$|; $key = q/q^2l;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q^2 l$|; $key = q/c_{psi1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\psi 1}$|; $key = q/B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B$|; $key = q/displaystyleN_Theta^2=-dfrac{g}{rho_0}dfrac{partial{rho}}{partial{z}}Big|_{S}=galpha(Theta,S,P_R)dfrac{partial{Theta}}{partial{z}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle N_\Theta^2 = - \dfrac{g}{\rho_0} \dfrac{\partial {\rho}}{\partial... ...{S} = g \alpha(\Theta,S,P_R) \dfrac{\partial {\Theta}}{\partial {z}} \; , \quad$|; $key = q/displaystyleN^2=-dfrac{g}{rho_0}dfrac{partial{rho}}{partial{z}}=dfrac{partial{B}}{partial{z}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle N^2 = - \dfrac{g}{\rho_0} \dfrac{\partial {\rho}}{\partial {z}} = \dfrac{\partial {B}}{\partial {z}} \; , \quad$|; $key = q/displaystyledfrac{partial{(b_{13}^2+b_{23}^2)^frac{1}{2}}}{partial{overline{S}}}frac{partial{tilde{c}_muoverline{S}}}{partial{overline{S}}}ge0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {(b_{13}^2 + b_{23}^2)^\frac{1}{2}}}{\partial {\o... ...\partial {\tilde{c}_\mu \overline{S}}}{\partial {\overline{S}}}\ge 0 \; , \quad$|; $key = q/displaystyleG'(1)=dfrac{B_f}{u_*^4}nu_{ti}^phiBig|_{z=z_{bl}}-dfrac{1}{w(1)}dfrac{partial{nu_{ti}^phi}}{partial{z}}Big|_{z=z_{bl}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G'(1) = \dfrac{B_f}{u_*^4} \nu_{ti}^\phi \Big\vert _{z=z_{bl}} - ... ...\dfrac{\partial {\nu_{ti}^\phi}}{\partial {z}} \Big\vert _{z=z_{bl}} \; , \quad$|; $key = q/Deltax;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta x$|; $key = q/dot{k_b};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{k_b}$|; $key = q/-lambda_0rgamma_3overline{N}^2delta_{i3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -\lambda_0 r \gamma_3 \overline{N}^2 \delta_{i3}$|; $key = q/A;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ A$|; $key = q/lambda_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_1$|; $key = q/displaystylePhi^b_i=-dfrac{lambda_5}{2}dfrac{epsilon}{k}{langleu'_ib'rangle};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi^b_i = - \dfrac{\lambda_5}{2} \dfrac{\epsilon}{k} {\langle u'_i b' \rangle} \; , \quad$|; $key = q/0.643;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.643$|; $key = q/displaystyle{calD}_psi=dfrac{partial}{partial{z}}left(dfrac{nu_t}{sigma_psi}dfrac{partial{psi}}{partial{z}}right)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_\psi = \dfrac{\partial}{\partial {z}} \left( \dfrac{\nu_t}{\sigma_\psi} \dfrac{\partial {\psi}}{\partial {z}} \right) \quad .$|; $key = q/N^2=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N^2=0$|; $key = q/L_{text{sour}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ L_{\text{sour}}$|; $key = q/c_{i}=2w_{i}Deltatslash(h_i+h_{i+1});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{i} = 2 w_{i} \Delta t / (h_i+h_{i+1})$|; $key = q/F^b_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F^b_i$|; $key = q/c_{b5}=C_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b5} = C_3$|; $key = q/displaystyledfrac{partial{B}}{partial{y}}approxfrac{B(S+Delta_yS,Theta+Delta_yTheta,P)-B(S,theta,P)}{Deltay};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {B}}{\partial {y}} \approx \frac{B(S+\Delta_yS,\Theta+\Delta_y\Theta,P)-B(S,\theta,P)}{\Delta y} \; , \quad$|; $key = q/{calD}_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_{ij}$|; $key = q/0.2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.2$|; $key = q/displaystyle=dfrac{a_1}{{calN}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle = \dfrac{a_1}{{\cal N}} \; , \quad$|; $key = q/eta_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \eta_2$|; $key = q/{calN}=(c_1+c_1^*)slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N} = (c_1 + c_1^*)/2$|; $key = q/l=c_2h_m;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l=c_2 h_m$|; $key = q/displaystyle{calN}_bgamma_3=-dfrac{a_{b1}-a_{b2}}{2}gamma_1overline{S}_U-dfrac{ae{N}^2+dfrac{a_{b3}}{3}overline{N}^2-a_{b5}gamma_3overline{N}^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal N}_b \gamma_3 = - \dfrac{a_{b1} - a_{b2}}{2} \gamma_1 \over... ...}^2 + \dfrac{a_{b3}}{3} \overline{N}^2 - a_{b5} \gamma_3 \overline{N}^2 \quad .$|; $key = q/Ri^{infty}=0.25;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri^{\infty}=0.25$|; $key = q/0.00336;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.00336$|; $key = q/overline{Gamma}_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{\Gamma}_{ij}$|; $key = q/sigma_psi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma_\psi$|; $key = q/displaystyleleft.{array}{rcl}c_i^{(1)}&=&displaystylec_i^n+Deltatleft{P_ileft(unline{c}^nright)-D_ileft(underline{c}^{(1)}right)right}.{array}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left. \begin{array}{rcl} c_i^{(1)} &=& \displaystyle c_i^n + \De... ...}^n\right) - D_i\left(\underline{c}^{(1)}\right) \right\}. \end{array} \right\}$|; $key = q/0.0032;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.0032$|; $key = q/dot{Theta};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{\Theta}$|; $key = q/displaystyleN^2=N_Theta^2+N_S^2;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle N^2 = N_\Theta^2 + N_S^2 \; , \quad$|; $key = q/displaystylenudfrac{partial{X}}{partial{z}}=F_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu \dfrac{\partial {X}}{\partial {z}} = F_b$|; $key = q/displaystyledfrac{partial{X}}{partial{t}}=P-QX,quadP,Q>0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {X}}{\partial {t}} =P-QX, \quad P,Q > 0$|; $key = q/displaystyleepsilon=dfrac{(c_mu^0)^3k^frac{3}{2}}{kappa(z+z_0)};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon= \dfrac{(c_\mu^0)^3 k^\frac{3}{2}}{\kappa (z+z_0)} \; , \quad$|; $key = q/{langlerule{3mm}{0mm}rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle \rule{3mm}{0mm} \rangle} $|; $key = q/S_U=partialUslashpartialz;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S_U=\partial U / \partial z$|; $key = q/c_1=2tilde{c}_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_1 = 2 \tilde{c}_1$|; $key = q/slashL;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ /L$|; $key = q/L;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ L$|; $key = q/0.74;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.74$|; $key = q/displaystyledfrac{lambda_5}{2}gamma_{i}=-lambda_6overline{S}_{ij}gamma_j-lambda_N}_{j}+dfrac{2}{3}overline{N}_{i}-lambda_0overline{T}delta_{i3}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\lambda_5}{2}\gamma_{i} = - \lambda_6 \overline{S}_{ij} \g... ...j} + \dfrac{2}{3} \overline{N}_{i} - \lambda_0 \overline{T} \delta_{i3} \quad .$|; $key = q/theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \theta$|; $key = q/displaystyleF_epsilon=-dfrac{nu_t}{sigma_epsilon}dfrac{partial{epsilon}}{partialgma_epsilon}K^2left(dfrac{3}{2}alpha-1right)(z+z_0)^{2alpha-1};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_\epsilon = - \dfrac{\nu_t}{\sigma_\epsilon} \dfrac{\partial {\e... ...silon} K^2 \left( \dfrac{3}{2} \alpha -1 \right) (z+z_0)^{2\alpha-1} \; , \quad$|; $key = q/c_{psi2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\psi 2}$|; $key = q/20pm10;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 20 \pm 10$|; $key = q/displaystyleB=-gfrac{{langlerhorangle}-rho_0}{rho_0};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle B=-g\frac{ {\langle \rho \rangle} -\rho_0}{\rho_0} \; , \quad$|; $key = q/-3|; $key = q/displaystylenu_t^{SI}=nu'^{SI}_t=5cdot10^{-3}left(1-left(frac{R_i}{0.7}right)^2right)^3;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t^{SI} = \nu'^{SI}_t = 5 \cdot 10^{-3} \left( 1-\left(\frac {R_i} {0.7}\right)^2\right)^3 \; , \quad$|; $key = q/Y;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Y$|; $key = q/0.318;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.318$|; $key = q/24;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 24$|; $key = q/sigma=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma=0$|; $key = q/dot{overline{q^2l}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{\overline{q^2 l}}$|; $key = q/Delta^{bio}t=7200;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta^{bio} t = 7200$|; $key = q/displaystyleh_k=Dfrac{mbox{tanh}left((d_l+d_u)frac{k}{M}-d_lright)+mbox{tanh}(d_l)}{mbox{tanh}(d_l)+mbox{tanh}(d_u)}-1quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle h_k = D\frac{\mbox{tanh}\left( (d_l+d_u)\frac{k}{M}-d_l\right) +\mbox{tanh}(d_l)}{\mbox{tanh}(d_l)+\mbox{tanh}(d_u)}-1 \quad .$|; $key = q/0.786;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.786$|; $key = q/11.6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 11.6$|; $key = q/{calN}=(c_1+c^*_1)slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N} = (c_1 + c^*_1)/2$|; $key = q/1.46;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.46$|; $key = q/D;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ D$|; $key = q/displaystyle;;c_ilongrightarrow0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \;\; c_i \longrightarrow 0$|; $key = q/displaystylel=frac{kappatildez}{1+frac{kappatildez}{c_2cdoth_m}}(1-R_f)^e;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l = \frac {\kappa \tilde z} {1+\frac {\kappa \tilde z} {c_2 \cdot h_m}} (1-R_f)^e$|; $key = q/overline{N};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{N}$|; $key = q/Q;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q$|; $key = q/S_l;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S_l$|; $key = q/displaystyleP_i(vec{c})=sum^I_{j=1}p_{i,j}(vec{c}),;;;;D_i(vec{c})=sum^I_{j=1}d_{i,j}(vec{c}),;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P_i(\vec{c}) = \sum^I_{j=1} p_{i,j}(\vec{c}), \;\;\;\; D_i(\vec{c}) = \sum^I_{j=1} d_{i,j}(\vec{c}),$|; $key = q/R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ R$|; $key = q/displaystyledisplaystylefrac{C}{C_0}=left(frac{-z}{D+z_0+z}right)^{-w_cslash(kappau_*)},;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \displaystyle \frac{C}{C_0} = \left(\frac{-z}{D+z_0+z} \right)^{-w_c/(\kappa u_*)},$|; $key = q/(P+G)slashepsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ (P+G)/\epsilon$|; $key = q/displaystyleoverline{N}_i=dfrac{k^2}{epsilon^2}dfrac{partial{B}}{partial{x_i}};,quadoverline{T}=dfrac{kk_b}{epsilon^2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{N}_i = \dfrac{k^2}{\epsilon^2} \dfrac{\partial {B}}{\partial {x_i}} \; , \quad \overline{T} = \dfrac{k k_b}{\epsilon^2} \quad .$|; $key = q/E_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ E_1$|; $key = q/^{-2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ ^{-2}$|; $key = q/B_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_2$|; $key = q/z_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_0$|; $key = q/displaystyledfrac{P+G}{epsilon}=hat{c}_mu(alpha_M,alpha_N)alpha_M-hat{c}'_mu(alpha_M,alpha_N)alpha_N=1;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{P+G}{\epsilon} = \hat{c}_\mu(\alpha_M,\alpha_N) \alpha_M - \hat{c}'_\mu(\alpha_M,\alpha_N) \alpha_N = 1 \; , \quad$|; $key = q/displaystylel=kappafrac{(d_s+z_0^s)(d_b+z_0^b)}{d_s+d_b+z_0^b+z_0^s};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l=\kappa \frac{(d_s+z_0^s) (d_b+z_0^b)} {d_s+d_b+z_0^b+z_0^s} \; , \quad$|; $key = q/0.107;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.107$|; $key = q/{displaymath}{array}{rcl}{calN}&=&dfrac{P+G}{epsilon}+dfrac{c_1}{2}-1[5mm]{calN}ac{1}{2r}left(dfrac{P_b}{epsilon_b}-1right)quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} {\cal N} &=& \dfrac{P+G}{\epsilon} + \dfra... ...} \left( \dfrac{P_b}{\epsilon_b} -1 \right) \quad . \end{array}\end{displaymath}|; $key = q/displaystylequadp_{i,i}(vec{c})=d_{i,i}(vec{c})=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \quad p_{i,i}(\vec{c}) = d_{i,i}(\vec{c})=0,$|; $key = q/fDeltat;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ f\Delta t$|; $key = q/c_1=B_1slash(3A_1);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_1 = B_1 / (3 A_1)$|; $key = q/displaystylePhi^b_i=-dfrac{B_1}{6A_2}dfrac{epsilon}{k}{langleu'_ib'rangle}+C_2(S_{ij}+W_{ij}){langleu'_jb'rangle}-2C_3k_bdelta_{i3};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi^b_i = - \dfrac{B_1}{6 A_2} \dfrac{\epsilon}{k} {\langle u'_i... ... S_{ij} + W_{ij} ) {\langle u'_j b' \rangle} - 2 C_3 k_b \delta_{i3} \; , \quad$|; $key = q/X^{n+1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ X^{n+1}$|; $key = q/hat{c}_mu^0=(c_mu^0)^4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \hat{c}_\mu^0 = (c_\mu^0)^4$|; $key = q/Q_E;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_E$|; $key = q/sigma_k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma_k$|; $key = q/displaystyle{langleu'w'rangle}=-nu_tdfrac{partial{u}}{partial{z}}+tilde{Gamma}_U'b'rangle}=-nu^B_tdfrac{partial{B}}{partial{z}}+tilde{Gamma}_B;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\langle u' w' \rangle} = - \nu_t \dfrac{\partial {u}}{\partial {... ...e} = - \nu^B_t \dfrac{\partial {B}}{\partial {z}} + \tilde{\Gamma}_B \; , \quad$|; $key = q/displaystylepsi=(c_mu^0)^pK^mL^n(z+z_0)^{malpha+n}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \psi= (c_\mu^0)^p K^m L^n (z+z_0)^{m \alpha + n} \quad .$|; $key = q/displaystylez_0^b=0.1frac{nu}{u_*^b}+0.03h_0^b+z_aquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle z_0^b = 0.1 \frac{\nu}{u_*^b} + 0.03 h_0^b + z_a \quad .$|; $key = q/alpha_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_b$|; $key = q/displaystyledisplaystylefrac{X^{n+1}_1-X^n_1}{Deltat}-frac{nu^n_1frac{X^{n+sigmaX^{n+sigma}_{1}}{0.5(h^{n+1}_{2}+h^{n+1}_1)}-F_b}{h^{n+1}_1}=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \displaystyle \frac{X^{n+1}_1-X^n_1}{\Delta t} -\frac{\nu^n_1\fra... ...2}-X^{n+\sigma}_{1}}{0.5(h^{n+1}_{2}+h^{n+1}_1)} -F_b}{h^{n+1}_1} =0 \; , \quad$|; $key = q/displaystyleP_b=epsilon_bquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P_b = \epsilon_b \quad .$|; $key = q/c_{b2}=C_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b2} = C_2$|; $key = q/{displaymath}{array}{rcl}D&=&d_0+d_1overline{N}^2+d_2overline{S}^2+d_3overline{Nb0}+n_{b1}overline{N}^2+n_{b2}overline{S}^2quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} D &=& d_0 + d_1 \overline{N}^2 + d_2 \over... ...{b1} \overline{N}^2 + n_{b2} \overline{S}^2 \quad . \end{array}\end{displaymath}|; $key = q/{displaymath}{array}{rcl}dfrac{text{d}}{text{d}{z}}left(dfrac{c_mu}{sigma_k^{psi{psi2}(c_mu^0)^{p+3}k^{m+frac{1}{2}}l^{n-1}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \dfrac{\text{d} }{\text{d} {z}} \left( \df... ...} (c_\mu^0)^{p+3} k^{m+\frac{1}{2}} l^{n-1} \quad . \end{array}\end{displaymath}|; $key = q/tilde{Gamma}_S;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_S$|; $key = q/displaystyleN_S^2=-dfrac{g}{rho_0}dfrac{partial{rho}}{partial{z}}Big|_{Theta}=-gbeta(Theta,S,P_R)dfrac{partial{S}}{partial{z}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle N_S^2 = - \dfrac{g}{\rho_0} \dfrac{\partial {\rho}}{\partial {z}}... ...\Theta} = - g \beta(\Theta,S,P_R) \dfrac{\partial {S}}{\partial {z}} \; , \quad$|; $key = q/k=q^2slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k= q^2 /2$|; $key = q/r=c_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ r=c_b$|; $key = q/tilde{c}_{b3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_{b3}$|; $key = q/lambda_7;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_7$|; $key = q/A_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ A_2$|; $key = q/nu^B_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^B_t$|; $key = q/d_;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_$|; $key = q/c_j;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_j$|; $key = q/displaystyleRi=dfrac{alpha_N}{alpha_M}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Ri = \dfrac{\alpha_N}{\alpha_M} \quad .$|; $key = q/tau_x=0.1027;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_x= 0.1027$|; $key = q/displaystyleK^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle K^n$|; $key = q/Delta^{bio}t=200;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta^{bio} t = 200$|; $key = q/displaystylepsi=(c_mu^0)^pk^ml^n;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \psi = (c_\mu^0)^p k^m l^n \; , \quad$|; $key = q/displaystylevec{c}^{n+1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \vec{c}^{n + 1}$|; $key = q/Q_{out};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_{out}$|; $key = q/z_0^bapprox0.0025;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_0^b\approx 0.0025$|; $key = q/c_mu^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_\mu^$|; $key = q/gamma_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_0$|; $key = q/scalebox{0.5}{includegraphics{figuresslashgridvert.eps}};FSF=1.6;AAT/; $cached_env_img{$key} = q|\scalebox{0.5}{\includegraphics{figures/gridvert.eps}}|; $key = q/0.833;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.833$|; $key = q/r=c_b=B_1slashB_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ r=c_b=B_1/B_2$|; $key = q/displaystyler^+_{i}=dfrac{Y_i-Y_{i-1}}{Y_{i+1}-Y_{i}};,quadr^-_{i}=dfrac{Y_{i+2}-Y_{i+1}}{Y_{i+1}-Y_{i}}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle r^+_{i} = \dfrac{Y_i - Y_{i-1}}{Y_{i+1}-Y_{i}} \; , \quad r^-_{i} = \dfrac{Y_{i+2} - Y_{i+1}}{Y_{i+1}-Y_{i}} \quad .$|; $key = q/displaystylez=-H.qquad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle z=-H.\qquad$|; $key = q/displaystyleS_{ij}=frac{1}{2}left(L_{ij}+L_{ji}right);,quadW_{ij}=frac{1}{2}left(L_{ij}-L_{ji}right);MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle S_{ij} = \frac{1}{2} \left( L_{ij} + L_{ji} \right) \; , \quad W_{ij} = \frac{1}{2} \left( L_{ij} - L_{ji} \right)$|; $key = q/D_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ D_{ij}$|; $key = q/d;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d$|; $key = q/displaystyle{calD}_S=dfrac{partial}{partial{z}}left(left(nu^S_t+nu^Sright)dfrac{partial{S}}{partial{z}}-tilde{Gamma}_Sright)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_S = \dfrac{\partial}{\partial {z}} \left( \left( \nu^S_t... ...S \right) \dfrac{\partial {S}}{\partial {z}} - \tilde{\Gamma}_S \right) \quad .$|; $key = q/displaystyle(S-S_0)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (S - S_0) \quad .$|; $key = q/displaystylefrac{X^{n+1}-X^n}{Deltat}=P^n-Q^nX^{n+1};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \frac{X^{n+1}-X^n}{\Delta t}=P^n-Q^nX^{n+1} \; , \quad$|; $key = q/beta=2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \beta = 2$|; $key = q/10;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 10$|; $key = q/displaystyleR_f=frac{c'_{mu}}{c_{mu}}R_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle R_f=\frac{c'_{\mu}}{c_{\mu}}R_i$|; $key = q/C_f;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_f$|; $key = q/B(z);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B(z)$|; $key = q/U;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ U$|; $key = q/displaystylePr_t=Pr_t^0expleft(-frac{Ri}{Pr_t^0Ri^{infty}}right)-frac{Ri}{Ri^{infty}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Pr_t = Pr_t^0 \exp\left(-\frac{Ri}{Pr_t^0 Ri^{\infty}}\right) -\frac{Ri}{Ri^{\infty}} \; , \quad$|; $key = q/displaystyleR_i<0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle R_i < 0 \quad .$|; $key = q/q^2slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q^2/2$|; $key = q/{calN}_b=c_{b1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N}_b=c_{b1}$|; $key = q/c_{epsilon1}=c_{epsilon3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\epsilon 1}=c_{\epsilon 3}$|; $key = q/I;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I$|; $key = q/{displaymath}{array}{rcl}d_0&=&36{calN}^3{calN}_b^2;,quad[3mm]d_1&=&84a_5a_{b3}{&=&3(3a_3^2-a_2^2)(a_{b2}^2-a_{b1}^2){calN}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} d_0 &=& 36 {\cal N}^3 {\cal N}_b^2 \; , \q... ..._3^2 - a_2^2)(a_{b2}^2 - a_{b1}^2) {\cal N} \quad . \end{array}\end{displaymath}|; $key = q/displaystylequadL_{ij}=dfrac{partial{U_i}}{partial{x_j}}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \quad L_{ij} = \dfrac{\partial {U_i}}{\partial {x_j}} \quad .$|; $key = q/0.0864;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.0864$|; $key = q/displaystylec_{mu}=;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle c_{\mu}=$|; $key = q/alpha_N;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_N$|; $key = q/z_0^b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_0^b$|; $key = q/0.5577;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.5577$|; $key = q/Theta_{obs};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Theta_{obs}$|; $key = q/displaystyletau=dfrac{k}{epsilon};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \tau = \dfrac{k}{\epsilon} \; , \quad$|; $key = q/displaystylealpha_Mledfrac{d_0n_0+(d_0n_1+d_1n_0)alpha_N+(d_1n_1+d_4n_0)alpha_N^d_4n_1alpha_N^3}{d_2n_0+(d_2n_1+d_3n_0)alpha_N+d_3n_1alpha_N^2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_M \le \dfrac{d_0 n_0 + (d_0 n_1 + d_1 n_0) \alpha_N + (d_1... ...pha_N^3}{ d_2 n_0 + (d_2 n_1 + d_3 n_0 ) \alpha_N + d_3 n_1 \alpha_N^2} \quad .$|; $key = q/2.58;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2.58$|; $key = q/K;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ K$|; $key = q/-0.67;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -0.67$|; $key = q/p_{i,j};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,j}$|; $key = q/displaystyle-d_5+n_2-left(d_3-n_1+n_{b2}right)Ri_c-left(d_4+n_{b1}right)Ri_c^2=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle -d_5 + n_2 - \left( d_3 - n_1 + n_{b2}\right) Ri_c - \left( d_4 + n_{b1} \right) Ri_c^2 = 0 \quad .$|; $key = q/Deltay;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta y$|; $key = q/displaystyleP+G-epsilon=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P+G-\epsilon=0 \quad .$|; $key = q/d_{i,j}(vec{c});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_{i,j}(\vec{c})$|; $key = q/i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ i$|; $key = q/^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ ^{-1}$|; $key = q/I_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I_0$|; $key = q/tau_y^s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_y^s$|; $key = q/displaystyledot{U}={calD}_U-gdfrac{partial{zeta}}{partial{x}}+int_z^{zeta}dfrac{{partial{x}},dz'-frac{1}{tau^U_R}(U-U_{obs})-C_fUsqrt{U^2+V^2};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{U} = {\cal D}_U - g \dfrac{\partial {\zeta}}{\partial {x}} +... ...tial {x}} dz' - \frac{1}{\tau^U_R}(U-U_{obs})-C_f U \sqrt{U^2+V^2} \; , \quad$|; $key = q/0.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.5$|; $key = q/tilde{Gamma}_{(U,V,B)};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_{(U,V,B)}$|; $key = q/QX;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ QX$|; $key = q/a_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_i$|; $key = q/sigma=0.6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma =0.6$|; $key = q/{displaymath}{array}{rcl}c_i^{n+1}&=&displaystylec_i^n+Deltatleft{P_ileft(underlnderline{c}^nright)frac{c_i^{n+1}}{c_i^n}right}.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} c_i^{n+1} &=& \displaystyle c_i^n + \Delta... ...rline{c}^n\right) \frac{c_i^{n+1}}{c_i^n} \right\}. \end{array}\end{displaymath}|; $key = q/x;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ x$|; $key = q/{displaymath}{array}{rcl}D&=&d_0+d_1overline{N}^2+d_2overline{S}^2+d_3overline{N1overline{N}^2+g_2overline{S}^2)overline{T}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} D &=& d_0 + d_1 \overline{N}^2 + d_2 \over... ...ne{N}^2 + g_2 \overline{S}^2 ) \overline{T} \quad . \end{array}\end{displaymath}|; $key = q/d_{i,i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_{i,i}$|; $key = q/{displaymath}{array}{rcl}alpha&=&dfrac{5kappaB_1^{frac{1}{2}}S_l+left(12E_2left(S_q(E_2-B_1kappa^2S_l)^2}right)^frac{1}{2};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \alpha &=& \dfrac{5 \kappa B_1^{\frac{1}{2... ..._1 \kappa^2 S_l )^2} \right)^\frac{1}{2} \; , \quad \end{array}\end{displaymath}|; $key = q/nu'_t=nu^B_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu'_t=\nu^B_t$|; $key = q/N_Theta^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_\Theta^2$|; $key = q/displaystyledot{B}={calD}_B;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{B} = {\cal D}_B \; , \quad$|; $key = q/dot{B};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{B}$|; $key = q/b_{ij}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b_{ij} = 0$|; $key = q/E_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ E_3$|; $key = q/0.5477;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.5477$|; $key = q/displaystylec'_{mu}=c_{mu}f(R_f)=c_{mu}frac{1}{P_r^0}(1-R_f)^{1slash2}.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle c'_{\mu}=c_{\mu} f(R_f)=c_{\mu} \frac{1}{P_r^0}(1-R_f)^{1/2}.$|; $key = q/a_2=1-c_3slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_2=1 - c_3/2$|; $key = q/displaystyleX_P=alpha_{sg}C_f|u|^3;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle X_P = \alpha_{sg} C_f \vert u\vert^3 \; , \quad$|; $key = q/gamma_{N_i}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_{N_i}=0$|; $key = q/displaystyleP_b=-{langlew'b'rangle}dfrac{partial{B}}{partial{z}}=-{langlew'b'rangle}N^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P_b = - {\langle w'b' \rangle} \dfrac{\partial {B}}{\partial {z}} = - {\langle w'b' \rangle} N^2 \quad .$|; $key = q/Y_{text{obs}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Y_{\text{obs}}$|; $key = q/S_q;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S_q$|; $key = q/^{-3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ ^{-3}$|; $key = q/z=-(D+z_0)slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z=-(D+z_0)/2$|; $key = q/{calD}_S;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_S$|; $key = q/p_{i,i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,i}$|; $key = q/displaystylenu_t=kappau_*(-z)frac{D+z_0+z}{D+z_0};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t = \kappa u_* (-z)\frac{D+z_0+z}{D+z_0}$|; $key = q/F_k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F_k$|; $key = q/a_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_2$|; $key = q/2.36;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2.36$|; $key = q/c_{b1}=tilde{c}_{b1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b1} = \tilde{c}_{b1}$|; $key = q/0.48;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.48$|; $key = q/displaystyleRi_b=dfrac{(B_r-B(z_{bl}))d}{|{boldsymbolU}_r-{boldsymbolU}(z_{bl})|^2+V_t^2(z_{bl})};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Ri_b = \dfrac{(B_r - B(z_{bl})) d} {\vert {\boldsymbol U}_r - {\boldsymbol U}(z_{bl}) \vert^2 + V_t^2(z_{bl})} \; , \quad$|; $key = q/displaystyledot{S}={calD}_S-frac{1}{tau^S_R}(S-S_{obs});,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{S} = {\cal D}_S - \frac{1}{\tau^S_R}(S-S_{obs}) \; , \quad$|; $key = q/k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k$|; $key = q/{calN}_b=c_{b1}=lambda_5slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N}_b = c_{b1} = \lambda_5/2$|; $key = q/overline{N}_{i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{N}_{i}$|; $key = q/partial_xTheta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_x\Theta$|; $key = q/c^*_1=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c^*_1=0$|; $key = q/c_{dd};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{dd}$|; $key = q/displaystyleN_noverline{S}^2-N_boverline{N}^2-D=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle N_n \overline{S}^2 - N_b \overline{N}^2 - D = 0 \; , \quad$|; $key = q/displaystylepartial_tc_i=P_i(vec{c})-D_i(vec{c}),;;i=1,ldots,I,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \partial_t c_i = P_i(\vec{c}) -D_i(\vec{c}), \;\; i = 1,\ldots,I,$|; $key = q/displaystylevec{c}^n+Deltat:vec{f}(t^n,vec{c}^n)prodlimits_{jinJ^n}{frac{{c_j^{(1)}}}{{c_j^n}}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \vec{c}^n + \Delta t \: \vec{f}(t^n ,\vec{c}^n )\prod\limits_{j \in J^n } {\frac{{c_j^{(1)} }}{{c_j^n }}}$|; $key = q/displaystylehat{c}_mu^0=dfrac{a_2^2-3a_3^2+3a_1{calN}}{3{calN}^2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \hat{c}_\mu^0 = \dfrac{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}}{3 {\cal N}^2} \quad .$|; $key = q/{boldsymbolcdot};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\boldsymbol \cdot}$|; $key = q/displaystyleP=nu_t(M^2+alpha_wN^2)+X_P;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P = \nu_t (M^2 + \alpha_w N^2) + X_P \; , \quad$|; $key = q/{calD}_l;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_l$|; $key = q/displaystyleI(z)=I_0bigg(Ae^{zslasheta_1}+(1-A)e^{zslasheta_2}bigg)B(z).;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle I(z) = I_0 \bigg(Ae^{z/\eta_1}+(1-A)e^{z/\eta_2}\bigg)B(z).$|; $key = q/qslashl;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q/l$|; $key = q/displaystyleF_k=etau_*^3quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_k = \eta u_*^3 \quad .$|; $key = q/nu_t=N(z+z_0)^gamma;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_t = N (z+z_0)^\gamma$|; $key = q/displaystylevec{h}(t^n,t^{n+1},vec{c}^n,vec{c}^{(1)})=frac{1}{2}left({vec{f}(t^n+1},vec{c}^{(1)})}right)prodlimits_{kinK^n}{frac{c_k^n}{c_k^{(1)}}}.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \vec{h}(t^n,t^{n + 1},\vec{c}^n,\vec{c}^{(1)}) = \frac{1}{2}\left... ...1},\vec{c}^{(1)})} \right)\prod\limits_{k \in K^n} {\frac{c_k^n }{c_k^{(1)} }}.$|; $key = q/0.2545;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.2545$|; $key = q/_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ _{ij}$|; $key = q/nu^S_t=nu^Theta_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^S_t = \nu^\Theta_t$|; $key = q/displaystylePhi_{ij}=-c_1tau^{-1}_uk;b_{ij}+c_2kS_{ij}+c_3kSigma_{ij}+c_4kZ_{ij}+c_5kN_{ij}+c_6Gamma_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_{ij} = - c_1 \tau^{-1}_u k \; b_{ij} + c_2 k S_{ij} + c_3 k \Sigma_{ij} + c_4 k Z_{ij} + c_5 k N_{ij} + c_6 \Gamma_{ij} \; , \quad$|; $key = q/16.55;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 16.55$|; $key = q/S_V=partialVslashpartialz;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S_V=\partial V / \partial z$|; $key = q/displaystylel=kappa(d_b+z_0^b)sqrt{1-frac{d_b-z_0^s}{H}};,quad   ;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l = \kappa (d_b+z_0^b) \sqrt{1-\frac{d_b-z_0^s}{H}} \; , \quad    $|; $key = q/alpha_N=overline{N}^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_N = \overline{N}^2 $|; $key = q/displaystyle(alphaL)^2=frac{2}{3}(c_mu^0)^2R,sigma_k^psi;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (\alpha L)^2 = \frac{2}{3} (c_\mu^0)^2 R \sigma_k^\psi \; , \quad$|; $key = q/W_y;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ W_y$|; $key = q/nu^Theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^\Theta$|; $key = q/displaystyleF_c(z_{bl})=Ri_;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_c(z_{bl}) = Ri_$|; $key = q/^{circ};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ ^{\circ}$|; $key = q/k<;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k<$|; $key = q/displaystylenu_t=hat{c}_mudfrac{k^2}{epsilon};,quadnu'_t=hat{c}_mu'dfrac{k^2}{epsilon};,quadtilde{Gamma}=epsilonGamma;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t = \hat{c}_\mu \dfrac{k^2}{\epsilon} \; , \quad \nu'_t = \ha... ...u' \dfrac{k^2}{\epsilon} \; , \quad \tilde{\Gamma} = \epsilon \Gamma \; , \quad$|; $key = q/P_b=epsilon_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_b=\epsilon_b$|; $key = q/alpha_M(alpha_N);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M(\alpha_N)$|; $key = q/a_1=lambda_1slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_1 = \lambda_1/\lambda$|; $key = q/{displaymath}{array}{rcl}{calN}&=&6E_2left(2S_l-S_qright)+B_1kappa^2S_lleft(13S_)+B_1kappa^2S_l(S_l+12S_q)right)^frac{1}{2}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} {\cal N} &=& 6 E_2 \left( 2 S_l - S_q \rig... ...pa^2 S_l (S_l + 12 S_q )\right)^\frac{1}{2} \quad . \end{array}\end{displaymath}|; $key = q/tilde{c}_{b5};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_{b5}$|; $key = q/N;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N$|; $key = q/lambda_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_3$|; $key = q/{calL}_z;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal L}_z$|; $key = q/displaystylePhi_{ij}=-2tilde{c}_1epsilonb_{ij}+left(dfrac{4}{3}(tilde{c}_3+tilde}+2left(tilde{c}_3-tilde{c}_4right)kZ_{ij}+tilde{c}_6Gamma_{ij}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_{ij} = - 2 \tilde{c}_1 \epsilon b_{ij} + \left( \dfrac{4}{3}... ...t( \tilde{c}_3 - \tilde{c}_4 \right) k Z_{ij} + \tilde{c}_6 \Gamma_{ij} \quad .$|; $key = q/i=0,..,N_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ i=0,..,N_i$|; $key = q/M^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ M^2$|; $key = q/0.5544;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.5544$|; $key = q/displaystylel_a=gamma_0frac{int_{-H}^etak^frac{1}{2}zdz}{int_{-H}^etak^frac{1}{2}dz};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l_a = \gamma_0 \frac{\int_{-H}^\eta k^\frac{1}{2} z dz} {\int_{-H}^\eta k^\frac{1}{2} dz}$|; $key = q/q;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q$|; $key = q/z_i^{n+1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_i^{n+1}$|; $key = q/Q_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_s$|; $key = q/sigma=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma=1$|; $key = q/_{ij}=2kb_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ _{ij} = 2 k b_{ij}$|; $key = q/displaystyleG(sigma)=sigmaBig[1+sigmaBig(left(sigma-2right)+left(3-2sigmaright)G(1)+left(sigma-1right)G'(1)Big)Big];MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G(\sigma) = \sigma \Big[ 1 + \sigma \Big( \left( \sigma - 2 \righ... ... \left( 3 - 2 \sigma \right) G(1) + \left( \sigma - 1 \right) G'(1) \Big) \Big]$|; $key = q/d=epsilonh;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d=\epsilon h$|; $key = q/displaystylez=zeta,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle z=\zeta,$|; $key = q/0.7;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.7$|; $key = q/displaystyleP=dfrac{1}{2}P_{ii};,quadG=dfrac{1}{2}G_{ii}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P = \dfrac{1}{2} P_{ii} \; , \quad G = \dfrac{1}{2} G_{ii} \quad .$|; $key = q/w=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ w=0$|; $key = q/F_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F_{ij}$|; $key = q/c_2=4slash3(tilde{c}_3+tilde{c}_4)-tilde{c}_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_2 = 4/3 ( \tilde{c}_3 + \tilde{c}_4 )- \tilde{c}_2$|; $key = q/gamma_0=-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_0=-1$|; $key = q/Ri=Ri_c;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri=Ri_c$|; $key = q/displaystyle(1-R_f)=[(tildeR_i^2+1)^{1slash2}-tildeR_i]^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (1-R_f)=[(\tilde R_i^2+1)^{1/2}-\tilde R_i]^2$|; $key = q/G;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ G$|; $key = q/displaystylei=j.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle i = j.$|; $key = q/displaystyledot{rule{0mm}{4mm}langleu'_iu'_jrangle}-{calD}_{ij}=P_{ij}+G_{ij}+F_{ij}+Phi_{ij}-epsilon_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{ \rule{0mm}{4mm} \langle u'_i u'_j \rangle} - {\cal D}_{ij} = P_{ij} + G_{ij} + F_{ij} + \Phi_{ij} - \epsilon_{ij} \; , \quad$|; $key = q/Phi^+_{i}=Phi^-_{i}=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi^+_{i}=\Phi^-_{i}=1$|; $key = q/0.127;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.127$|; $key = q/0.173;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.173$|; $key = q/eta_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \eta_1$|; $key = q/{displaymath}{array}{rcl}P_{ij}=-2kSigma_{ij}-2kZ_{ij}+dfrac{2}{3}Pdelta_{ij}-df+dfrac{2}{3}Pdelta_{ij}-dfrac{4}{3}kS_{ij};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} P_{ij} = - 2 k \Sigma_{ij} - 2 k Z_{ij} + ... ...3} P \delta_{ij} - \dfrac{4}{3} k S_{ij} \; , \quad \end{array}\end{displaymath}|; $key = q/P_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_R$|; $key = q/{calD}_Theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_\Theta$|; $key = q/P_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_b$|; $key = q/ktau;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k \tau$|; $key = q/displaystylealpha=-dfrac{1}{rho_0}left(dfrac{partial{rho_{is}}}{partial{T}}right)_S=dfrac{1}{g}left(dfrac{partial{B_{is}}}{partial{T}}right)_S;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha = - \dfrac{1}{\rho_0} \left( \dfrac{\partial {\rho_{is}}}{... ...dfrac{1}{g} \left( \dfrac{\partial {B_{is}}}{\partial {T}} \right)_S \; , \quad$|; $key = q/displaystyledfrac{partial{Y}}{partial{t}}=-wdfrac{partial{Y}}{partial{z}}=-left(dfrac{partial{F}}{partial{z}}-Ydfrac{partial{w}}{partial{z}}right).;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {Y}}{\partial {t}} = - w\dfrac{\partial {Y}}{\par... ...frac{\partial {F}}{\partial {z}} - Y\dfrac{\partial {w}}{\partial {z}} \right).$|; $key = q/d_l=0,d_u>0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_l=0, d_u>0$|; $key = q/l_a;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_a$|; $key = q/partial_yzeta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_y\zeta$|; $key = q/C;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C$|; $key = q/tau_y;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_y$|; $key = q/{displaymath}{array}{rcl}Phi^b_i&=&-tilde{c}_{b1}dfrac{epsilon}{k};{langleu'_ib'}{partial{x_j}}-2tilde{c}_{b5}k_bdelta_{i3}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \Phi^b_i &=& - \tilde{c}_{b 1} \dfrac{\eps... ... {x_j}} - 2 \tilde{c}_{b 5} k_b \delta_{i3} \quad . \end{array}\end{displaymath}|; $key = q/displaystylePhi_{ij}=-dfrac{2}{lambda}epsilonb_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_{ij} = - \dfrac{2}{\lambda} \epsilon b_{ij} \; , \quad$|; $key = q/P_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_{ij}$|; $key = q/kappaapprox0.433;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \kappa \approx 0.433$|; $key = q/C_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_0$|; $key = q/Phi^b_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi^b_i$|; $key = q/displaystyleF_psi=-dfrac{nu_t}{sigma_psi}dfrac{partial{psi}}{partial{z}}=-dfrac{n(c_mu^0)^{p+1}kappa^{n+1}}{sigma_psi}k^{m+frac{1}{2}}(z+z_0)^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_\psi = - \dfrac{\nu_t}{\sigma_\psi} \dfrac{\partial {\psi}}{\pa... ...dfrac{ n (c_\mu^0)^{p+1} \kappa^{n+1}}{\sigma_\psi} k^{m+\frac{1}{2}} (z+z_0)^n$|; $key = q/c_{epsilon1};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\epsilon 1}$|; $key = q/e;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ e$|; $key = q/displaystylepsi=(c_mu^0)^pkappa^nk^mleft(z+z_0right)^n;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \psi = (c_\mu^0)^p \kappa^n k^m \left( z+z_0 \right)^n \; , \quad$|; $key = q/displaystyle-frac{1}{rho_0}dfrac{partial{P}}{partial{x}}=-gdfrac{partial{zeta}}{partial{x}}+int_z^{zeta}dfrac{partial{B}}{partial{x}},dz';MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle - \frac{1}{\rho_0} \dfrac{\partial {P}}{\partial {x}}= -g \dfrac{... ...{\zeta}}{\partial {x}} +\int_z^{\zeta}\dfrac{\partial {B}}{\partial {x}} dz'$|; $key = q/11.04;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 11.04$|; $key = q/displaystyleY_i^{n+1}=Y_i^n-dfrac{Deltat}{h_i}left(F^n_{i}-F^n_{i-1}-Y_i^nleft(w_k-w_{k-1}right)right).;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Y_i^{n+1} = Y_i^n - \dfrac{\Delta t}{h_i} \left( F^n_{i} - F^n_{i-1} -Y_i^n \left(w_k-w_{k-1} \right)\right).$|; $key = q/p_{i,j}geq0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,j}\geq 0$|; $key = q/10.8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 10.8$|; $key = q/Y_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Y_i$|; $key = q/c_i^{min};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_i^{\min}$|; $key = q/b_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b_{ij}$|; $key = q/{calD}_epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_\epsilon$|; $key = q/Delta^{phy}t=7200;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta^{phy} t = 7200$|; $key = q/tau_x;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_x$|; $key = q/h_i={langleu'_itheta'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ h_i= {\langle u'_i \theta' \rangle} $|; $key = q/{displaymath}{array}{rcl}dfrac{1}{l}dfrac{text{d}{l}}{text{d}{t}}&=&-left(dfrac{rac{1}{n}c_{psi_3}-dfrac{m}{n}right)Gright)quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \dfrac{1}{l} \dfrac{\text{d} {l}}{\text{d}... ...c_{\psi_3} - \dfrac{m}{n} \right) G \right) \quad . \end{array}\end{displaymath}|; $key = q/tau_u;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_u$|; $key = q/F;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F$|; $key = q/q^2slash2=k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q^2/2=k$|; $key = q/{displaymath}{array}{rcl}d_0&=&36{calN}^3{calN}_b^2;,quad[3mm]d_1&=&84a_5a_{b3}{=&3(3a_3^2-a_2^2)(a_{b2}^2-a_{b1}^2){calN};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} d_0 &=& 36 {\cal N}^3 {\cal N}_b^2 \; , \q... ...2 - a_2^2)(a_{b2}^2 - a_{b1}^2) {\cal N} \; , \quad \end{array}\end{displaymath}|; $key = q/partial_xzeta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_x\zeta$|; $key = q/{calN}=1slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N} = 1/\lambda$|; $key = q/(1-R_f);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ (1-R_f)$|; $key = q/{ttKPP_IP_FC};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\tt KPP\_IP\_FC}$|; $key = q/c_6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_6$|; $key = q/a_3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_3$|; $key = q/1.33;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.33$|; $key = q/{displaymath}{array}{rcl}g_0&=&36a_{b4}{calN}^3{calN}_b;,quad[3mm]g_1&=&36a_5a_{]g_2&=&12a_{b4}(3a_3^2-a_2^2){calN}{calN}_bquad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} g_0 &=& 36 a_{b4} {\cal N}^3 {\cal N}_b \;... ...{b4} ( 3 a_3^2 - a_2^2) {\cal N} {\cal N}_b \quad . \end{array}\end{displaymath}|; $key = q/displaystyle{calD}_l=dfrac{partial}{partial{z}}left(qlS_ldfrac{partial{q^2l}}{partial{z}}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_l = \dfrac{\partial}{\partial {z}} \left( q l S_l \dfrac{\partial {q^2 l}}{\partial {z}} \right) \; , \quad$|; $key = q/{displaymath}{array}{rcl}{calN}_bgamma_1&=&-dfrac{a_{b1}+a_{b2}}{2}gamma_3overlic{a_{b3}}{3}overline{N}^2-a_{b4}overline{T}quad.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} {\cal N}_b \gamma_1 &=& - \dfrac{a_{b1} + ... ...3}}{3} \overline{N}^2 - a_{b4} \overline{T} \quad . \end{array}\end{displaymath}|; $key = q/c_mu^0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_\mu^0$|; $key = q/displaystyle(z_{bl})-Ri_cRi_;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (z_{bl}) - Ri_c Ri_$|; $key = q/displaystyleq^2l=2kappak(z+z_0);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle q^2 l = 2 \kappa k (z+z_0) \; , \quad$|; $key = q/displaystyledfrac{partial{X}}{partial{t}}-dfrac{partial{}}{partial{z}}left(nudfrac{partial{X}}{partial{z}}right)=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {X}}{\partial {t}} - \dfrac{\partial {}}{\partial {z}} \left( \nu \dfrac{\partial {X}}{\partial {z}} \right) = 0 \; , \quad$|; $key = q/a_3=1-c_4slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_3=1 - c_4/2$|; $key = q/z_i^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_i^n$|; $key = q/displaystylealpha_M=dfrac{3{calN}^2}{a_2^2-3a_3^2+3a_1{calN}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_M = \dfrac{3 {\cal N}^2}{a_2^2 - 3 a_3^2 + 3 a_1 {\cal N}} \; , \quad$|; $key = q/1.07;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.07$|; $key = q/{langleu'w'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle u' w' \rangle} $|; $key = q/a_5=1slash2-c_6slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_5=1/2 - c_6/2$|; $key = q/c'_mu=c'^0_mu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c'_\mu=c'^0_\mu$|; $key = q/omegaproptoepsilonslashk;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \omega \propto \epsilon / k$|; $key = q/l=q^3slash(B_1epsilon);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l=q^3/(B_1 \epsilon)$|; $key = q/displaystylenu_t^phi=hw_phi(sigma)G(sigma);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t^\phi = h w_\phi(\sigma) G(\sigma) \; , \quad$|; $key = q/gamma_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_i$|; $key = q/Q_B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_B$|; $key = q/lambda_5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_5$|; $key = q/p_{i,i}not=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,i}\not= 0$|; $key = q/slashrho_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ /\rho_0$|; $key = q/{calL}_z=overline{z};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal L}_z= \overline{z}$|; $key = q/displaystylefrac{k}{k_0}=Aleft(frac{t}{tau_0}right)^d;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \frac{k}{k_0} = A \left( \frac{t}{\tau_0} \right)^d \; , \quad$|; $key = q/displaystyleepsilon=frac{q^3}{B_1l};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon = \frac{q^3}{B_1 l} \; , \quad$|; $key = q/2slash3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2/3$|; $key = q/{calD}^b_{i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}^b_{i}$|; $key = q/G';MSF=1.6;AAT/; $cached_env_img{$key} = q|$ G'$|; $key = q/frac{3}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \frac{3}{2}$|; $key = q/displaystyleR=frac{-w_c}{u_*}.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle R =\frac{-w_c}{u_*}.$|; $key = q/partial_yS;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_yS$|; $key = q/displaystylepartial_tc_i+partial_zleft(m_ic_i-K_Vpartial_zc_iright)=P_i(vec{c})-D_i(vec{c}),;;i=1,ldots,I,quadj,k=1,ldots,3,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \partial_t c_i + \partial_z \left(m_i c_i - K_V \partial_z c_i\right) = P_i(\vec{c}) -D_i(\vec{c}), \;\; i = 1,\ldots,I, \quad j,k = 1,\ldots,3,$|; $key = q/kproptou^2_*;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k\propto u^2_*$|; $key = q/c'_mu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c'_\mu$|; $key = q/displaystyle(T-T_0)+;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (T - T_0) +$|; $key = q/epsilon_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon_b$|; $key = q/I=10;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I=10$|; $key = q/2.00;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2.00$|; $key = q/-1.3|; $key = q/displaystyleP_{ij}=-{langleu'_iu'_mrangle}dfrac{partial{U_j}}{partial{x_m}}-{lan=delta_{i3}{langleu'_jb'rangle}+delta_{j3}{langleu'_ib'rangle};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle P_{ij} = - {\langle u'_i u'_m \rangle} \dfrac{\partial {U_j}}{\pa... ...3} {\langle u'_j b' \rangle} + \delta_{j3} {\langle u'_i b' \rangle} \; , \quad$|; $key = q/displaystyle{calD}_U=dfrac{partial}{partial{z}}left(left(nu_t+nuright)dfrac{partial{U}}{partial{z}}-tilde{Gamma}_Uright)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_U = \dfrac{\partial}{\partial {z}} \left( \left( \nu_t + \nu \right) \dfrac{\partial {U}}{\partial {z}} - \tilde{\Gamma}_U \right) \quad .$|; $key = q/{displaymath}{array}{rcl}n_{b0}&=&12a_{b3}{calN}^3{calN}_b;,quad[3mm]n_{b1}&=&121(a_2-3a_3)-4(a_2^2-3a_3^2)){calN}{calN}_b;,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} n_{b0} &=& 12 a_{b3} {\cal N}^3 {\cal N}_b... ... 4 (a_2^2-3 a_3^2) ) {\cal N} {\cal N}_b \; , \quad \end{array}\end{displaymath}|; $key = q/C_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_2$|; $key = q/displaystylesigma=dfrac{d}{h};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \sigma = \dfrac{d}{h} \; , \quad$|; $key = q/c_{b3}=C_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b3} = C_2$|; $key = q/0.8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.8$|; $key = q/1.0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.0$|; $key = q/i,j;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ i,j$|; $key = q/-1.09;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -1.09$|; $key = q/alpha=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha=1$|; $key = q/n+1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ n+1$|; $key = q/{displaymath}{array}{rcll}F_{i}&=&w_{i}left(Y_i+dfrac{1}{2}Phi^+_{i}left(1-|c_{i{i+1}right)right)quad&text{for}quadw_{i}<0;,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcll} F_{i} &=& w_{i} \left(Y_i + \dfrac{1}{2} ... ...ight) \quad & \text{for} \quad w_{i} < 0 \; , \quad \end{array}\end{displaymath}|; $key = q/displaystyleh_i=(gamma_i-gamma_{i-1})D,qquadi=1,dots,N_i;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle h_i=(\gamma_i-\gamma_{i-1})D, \qquad i=1,\dots,N_i \; , \quad$|; $key = q/nu_Y;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_Y$|; $key = q/displaystyleJ^n=left{{i:f_i(t^n,vec{c}^n)<0,iin{1,...,I}}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle J^n = \left\{ {i:f_i (t^n,\vec{c}^n) < 0,i \in \{1,...,I\}} \right\}$|; $key = q/displaystyleF_k=-dfrac{nu_t}{sigma_k}dfrac{partial{k}}{partial{z}}=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_k = - \dfrac{\nu_t}{\sigma_k} \dfrac{\partial {k}}{\partial {z}} = 0 \quad .$|; $key = q/tilde{c}_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_1$|; $key = q/y;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ y$|; $key = q/Phi_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi_{ij}$|; $key = q/m_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ m_i$|; $key = q/k-epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k-\epsilon$|; $key = q/displaystyledfrac{partial{Y}}{partial{t}}=dfrac{partial{}}{partial{z}}left(nu_Ydight)-frac{1}{tau_R}(Y-Y_{obs})+YL_{text{sour}}+Q_{text{sour}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {Y}}{\partial {t}} = \dfrac{\partial {}}{\partial... ... - \frac{1}{\tau_R}(Y-Y_{obs}) + Y L_{\text{sour}} + Q_{\text{sour}} \; , \quad$|; $key = q/0.3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.3$|; $key = q/N_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_{ij}$|; $key = q/Theta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Theta$|; $key = q/1leqileqI;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1\leq i\leq I$|; $key = q/displaystylehat{c}_mu=(c^0_mu)^3c_mu;,quadhat{c}_mu'=(c^0_mu)^3c'_mu;,quadquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \hat{c}_\mu = (c^0_\mu)^3 c_\mu \; , \quad \hat{c}_\mu' = (c^0_\mu)^3 c'_\mu \; , \quad \quad .$|; $key = q/displaystylealpha_M=frac{k^2}{epsilon^2}M^2;,quadalpha_N=frac{k^2}{epsilon^2}N^2;,quadalpha_b=frac{kk_b}{epsilon^2};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_M = \frac{k^2}{\epsilon^2} M^2 \; , \quad \alpha_N = \frac{k^2}{\epsilon^2} N^2 \; , \quad \alpha_b = \frac{kk_b}{\epsilon^2} \; , \quad$|; $key = q/nu_{ti}^phi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_{ti}^\phi$|; $key = q/displaystylerho=rho_0+;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \rho = \rho_0 +$|; $key = q/1.8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.8$|; $key = q/alpha_M;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M$|; $key = q/{calD}_psi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_\psi$|; $key = q/displaystylew_phi=dfrac{kappau_*}{Phi_phi(zeta)}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle w_\phi = \dfrac{\kappa u_*}{\Phi_\phi (\zeta)} \quad .$|; $key = q/displaystyleY_i^{n+1}=Y_i^n-dfrac{Deltat}{h_i}left(F^n_{i}-F^n_{i-1}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Y_i^{n+1} = Y_i^n - \dfrac{\Delta t}{h_i} \left( F^n_{i} - F^n_{i-1} \right) \; , \quad$|; $key = q/{langleu'u'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle u'u' \rangle} $|; $key = q/alpha;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha$|; $key = q/displaystyleq^2l=2KL(z+z_0)^{alpha+1}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle q^2 l = 2 K L (z+z_0)^{\alpha + 1} \quad .$|; $key = q/Q_h;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_h$|; $key = q/0.20;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.20$|; $key = q/displaystyle{calD}_V=dfrac{partial}{partial{z}}left(left(nu_t+nuright)dfrac{partial{V}}{partial{z}}-tilde{Gamma}_Vright)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_V = \dfrac{\partial}{\partial {z}} \left( \left( \nu_t + \nu \right) \dfrac{\partial {V}}{\partial {z}} - \tilde{\Gamma}_V \right) \quad .$|; $key = q/displaystylePhi_{ij}=-dfrac{B_1}{3A_1}epsilonb_{ij}+4C_1kS_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_{ij} = - \dfrac{B_1}{3 A_1} \epsilon b_{ij} + 4 C_1 k S_{ij} \; , \quad$|; $key = q/frac{1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \frac{1}{2}$|; $key = q/Q_{tot};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_{tot}$|; $key = q/F=wY;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F=wY$|; $key = q/displaystylenu_t^{IW}=10^{-4};,quadnu'^{IW}_t=5cdot10^{-5}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t^{IW} = 10^{-4} \; , \quad \nu'^{IW}_t = 5 \cdot 10^{-5} \quad .$|; $key = q/Phi_phi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi_\phi$|; $key = q/d_{i,j,k};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_{i,j,k}$|; $key = q/r_1^{max}=1.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ r_1^{\max}=1.5$|; $key = q/displaystyle=0.5,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle =0.5,$|; $key = q/C_f^{max};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_f^{\max}$|; $key = q/n_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ n_2$|; $key = q/-3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -3$|; $key = q/1.44;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.44$|; $key = q/displaystylenudfrac{partial{X}}{partial{z}}=F_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu \dfrac{\partial {X}}{\partial {z}} = F_s$|; $key = q/c_i(t)geq0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_i(t) \geq 0$|; $key = q/(U,V);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ (U,V)$|; $key = q/l_{epsilon};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_{\epsilon}$|; $key = q/tilde{c}_{b4};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_{b4}$|; $key = q/displaystyleF_l=-S_lqldfrac{partial{q^2l}}{partial{z}}=-2sqrt{2}S_l(alpha+1)K^frac{3}{2}L^2(z+z_0)^{frac{3}{2}alpha+1};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_l = - S_l q l \dfrac{\partial {q^2 l}}{\partial {z}} = - 2 \sqrt{2} S_l (\alpha+1) K^\frac{3}{2} L^2 (z+ z_0)^{\frac{3}{2} \alpha + 1} \; , \quad$|; $key = q/displaystyle{calD}_epsilon=dfrac{partial}{partial{z}}left(dfrac{nu_t}{sigma_epsilon}dfrac{partial{epsilon}}{partial{z}}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_\epsilon = \dfrac{\partial}{\partial {z}} \left( \dfrac{... ...}{\sigma_\epsilon} \dfrac{\partial {\epsilon}}{\partial {z}} \right) \; , \quad$|; $key = q/0.225;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.225$|; $key = q/E_p;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ E_p$|; $key = q/k_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k_0$|; $key = q/t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ t$|; $key = q/sigma_epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma_\epsilon$|; $key = q/{displaymath}{array}{rcl}dfrac{langleu'^2rangle}{k}&=&dfrac23+dfrac{1}{mathcal{Ndfrac{a_2}{3}-a_3right)P+dfrac83a_5Gright);,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \dfrac{\langle u'^2 \rangle}{k} &=& \dfrac... ...{3}-a_3\right) P +\dfrac83 a_5 G \right) \; , \quad \end{array}\end{displaymath}|; $key = q/displaystyledfrac{partial{B}}{partial{x}}approxfrac{B(S+Delta_xS,Theta+Delta_xTheta,P)-B(S,Theta,P)}{Deltax};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {B}}{\partial {x}} \approx \frac{B(S+\Delta_xS,\Theta+\Delta_x\Theta,P)-B(S,\Theta,P)}{\Delta x} \; , \quad$|; $key = q/displaystyle{calD}_q=dfrac{partial}{partial{z}}left(qlS_qdfrac{partial{q^2slash2}}{partial{z}}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_q = \dfrac{\partial}{\partial {z}} \left( q l S_q \dfrac{\partial {q^2/2}}{\partial {z}} \right) \; , \quad$|; $key = q/displaystyleL=dfrac{u_*^3}{kappaB_f}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle L = \dfrac{u_*^3}{\kappa B_f} \quad .$|; $key = q/z;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z$|; $key = q/displaystylenu_t^{SI}=nu'^{SI}_t=5cdot10^{-3};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t^{SI} = \nu'^{SI}_t = 5 \cdot 10^{-3} \; , \quad$|; $key = q/nu_{ti}^{phi};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_{ti}^{\phi}$|; $key = q/displaystylez_0^s=alphafrac{(u_*^s)^2}{g}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle z_0^s=\alpha \frac{(u_*^s)^2}{g} \quad .$|; $key = q/displaystyle(c_mu^0)^{-2}=frac{1}{2}B_1^frac{2}{3}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (c_\mu^0)^{-2} = \frac{1}{2} B_1^\frac{2}{3} \quad .$|; $key = q/0.08;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.08$|; $key = q/a_5=lambda_4slashlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_5 = \lambda_4/\lambda$|; $key = q/epsilon^b_{i}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon^b_{i}=0$|; $key = q/displaystylePhi_{ij}=-2tilde{c}_1epsilonb_{ij}-tilde{c}_2kS_{ij}-tilde{c}_3left(4left(D_{ij}-dfrac{2}{3}Pdelta_{ij}right)+tilde{c}_6Gamma_{ij};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_{ij} = - 2 \tilde{c}_1 \epsilon b_{ij} - \tilde{c}_2 k S_{ij... ..._{ij} - \dfrac{2}{3} P \delta_{ij} \right) + \tilde{c}_6 \Gamma_{ij} \; , \quad$|; $key = q/B_f;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_f$|; $key = q/displaystylealpha_M=overline{S}^2;,quadalpha_N=overline{N}^2;,quadalpha_b=overline{T}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_M = \overline{S}^2 \; , \quad \alpha_N = \overline{N}^2 \; , \quad \alpha_b = \overline{T} \quad .$|; $key = q/w=partial_tzeta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ w=\partial_t\zeta$|; $key = q/dot{V};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{V}$|; $key = q/displaystyle{calD}_S=S(P-E),;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_S = S (P-E),$|; $key = q/W_x;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ W_x$|; $key = q/0.47;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.47$|; $key = q/rho_a;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \rho_a$|; $key = q/N_n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_n$|; $key = q/tau_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_R$|; $key = q/displaystyledfrac{partial{}}{partial{z}}=-frac{1}{h}dfrac{partial{}}{partial{sigma}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {}}{\partial {z}} = - \frac{1}{h} \dfrac{\partial {}}{\partial {\sigma}} \; , \quad$|; $key = q/B_f<0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_f < 0$|; $key = q/displaystylez=zeta,qquad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle z=\zeta,\qquad$|; $key = q/S_{obs};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ S_{obs}$|; $key = q/s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ s$|; $key = q/alpha_M=overline{S}^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M = \overline{S}^2$|; $key = q/varepsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \varepsilon $|; $key = q/1.22;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.22$|; $key = q/dot{{langlecdotsrangle}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{ {\langle \cdots \rangle} }$|; $key = q/Pslashepsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P/\epsilon$|; $key = q/10.1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 10.1$|; $key = q/{calN}=c_1slash2-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N} = c_1/2-1$|; $key = q/l_k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_k$|; $key = q/N_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_b$|; $key = q/displaystyleJ^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle J^n$|; $key = q/1.35;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.35$|; $key = q/displaystyleG'(1)=-dfrac{G(1)}{w(1)}dfrac{partial{w}}{partial{sigma}}Big|_{sigma1}{w(1)}dfrac{partial{nu_{ti}^phi}}{partial{z}}Big|_{z=z_{bl}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G'(1) = - \dfrac{G(1)}{w(1)} \dfrac{\partial {w}}{\partial {\sigm... ...\dfrac{\partial {\nu_{ti}^\phi}}{\partial {z}} \Big\vert _{z=z_{bl}} \; , \quad$|; $key = q/lambda_6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_6$|; $key = q/{langlerhorangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle \rho \rangle} $|; $key = q/displaystylepartial_tC+partial_zleft(w_cC-nu_tpartial_zCright)=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \partial_t C + \partial_z\left(w_c C - \nu_t \partial_z C\right) = 0,$|; $key = q/{calN};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal N}$|; $key = q/tilde{c}_4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_4$|; $key = q/displaystyler=frac{kappa}{lnleft(frac{0.5h_1+z_0^b}{z^b_0}right)};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle r=\frac{\kappa}{\ln \left( \frac{0.5h_1+z_0^b}{z^b_0} \right)} \; , \quad$|; $key = q/{displaymath}{array}{rcl}n_0&=&36a_1{calN}^2{calN}_b^2;,quad[3mm]n_1&=&-12a_5a_{alN}^2+24a_5a_{b4}(a_2+3a_3){calN}{calN}_b;,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} n_0 &=& 36 a_1 {\cal N}^2 {\cal N}_b^2 \; ... ...5 a_{b4} (a_2+3 a_3) {\cal N} {\cal N}_b \; , \quad \end{array}\end{displaymath}|; $key = q/Ri_c;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri_c$|; $key = q/displaystyleF=1+E_2left(dfrac{l}{kappa{calL}_z}right)^2;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F = 1 + E_2 \left( \dfrac{l}{\kappa {\cal L}_z} \right)^2 \; , \quad$|; $key = q/displaystyleq^2slash2=dfrac{u_*^2B_1^frac{2}{3}}{2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle q^2/2= \dfrac{u_*^2 B_1^\frac{2}{3}}{2} \quad .$|; $key = q/0.6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.6$|; $key = q/gamma_0=0.2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_0 = 0.2$|; $key = q/alpha=-2.0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha=-2.0$|; $key = q/nu_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_t$|; $key = q/partial_yB;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_yB$|; $key = q/R=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ R=1$|; $key = q/displaystyleepsilon=C_{epsilon}k^{3slash2}l_{epsilon}^{-1};,quadC_{epsilon}=0.7quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon=C_{\epsilon} k^{3/2} l_{\epsilon}^{-1} \; , \quad C_{\epsilon}=0.7 \quad .$|; $key = q/dot{k};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{k}$|; $key = q/Ri_{st};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri_{st}$|; $key = q/displaystyleF_psi=-dfrac{nu_t}{sigma_psi}dfrac{partial{psi}}{partial{z}}=-dfrac{+nright)K^{m+frac{1}{2}}L^{n+1}(z+z_0)^{(m+frac{1}{2})alpha+n};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_\psi = - \dfrac{\nu_t}{\sigma_\psi} \dfrac{\partial {\psi}}{\pa... ... \right) K^{m+\frac{1}{2}} L^{n+1} (z+z_0)^{(m+\frac{1}{2})\alpha+n} \; , \quad$|; $key = q/V_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ V_1$|; $key = q/W;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ W$|; $key = q/overline{N}^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{N}^2$|; $key = q/m;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ m$|; $key = q/displaystylelim_{alpha_Mrightarrow0}hat{c}_mu=hat{c}_mu^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \lim_{\alpha_M \rightarrow 0} \hat{c}_\mu = \hat{c}_\mu^$|; $key = q/displaystyle{vec{c}^{n+1}=vec{c}^n+Deltat:vec{h}(t^n,t^{n+1},vec{c}^n,vec{c}^{(1)})prodlimits_{kinK^n}{frac{c_k^{n+1}}{c_k^n}}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\vec{c}^{n + 1} = \vec{c}^n + \Delta t \: \vec{h}(t^n,t^{n + 1},\vec{c}^n,\vec{c}^{(1)})\prod\limits_{k \in K^n} {\frac{c_k^{n + 1} }{c_k^n }}}$|; $key = q/overline{N}_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{N}_i$|; $key = q/displaystyleinot=j;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle i \not= j$|; $key = q/c_{i,k};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{i,k}$|; $key = q/displaystyleoverline{S}_U=dfrac{k}{epsilon}S_U;,quadoverline{S}_V=dfrac{k}{epsilon}S_V;,quadoverline{N}^2=overline{N_3};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{S}_U = \dfrac{k}{\epsilon} S_U \; , \quad \overline{S}_V = \dfrac{k}{\epsilon} S_V \; , \quad \overline{N}^2 = \overline{N_3} \; , \quad$|; $key = q/{calD}_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_b$|; $key = q/B_fge0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_f \ge 0$|; $key = q/Phi^+_{i}=Phi^-_{i}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi^+_{i}=\Phi^-_{i}=0$|; $key = q/displaystyledfrac{P}{epsilon}=hat{c}_mualpha_M=1quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{P}{\epsilon} = \hat{c}_\mu \alpha_M = 1 \quad .$|; $key = q/epsilon;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon$|; $key = q/c_{ed};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{ed}$|; $key = q/displaystyleQ_{tot}=Q_E+Q_H+Q_B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Q_{tot}=Q_E+Q_H+Q_B$|; $key = q/P;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P$|; $key = q/c_{b1}=B_1slash(6A_2);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b1} = B_1 / (6 A_2)$|; $key = q/sigma;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma$|; $key = q/displaystyle{langlerhorangle}=hat{rho}(Theta,S,P_R);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\langle \rho \rangle} = \hat{\rho} (\Theta,S,P_R) \; , \quad$|; $key = q/0.80;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.80$|; $key = q/displaystyleG=-nu^B_tN^2+tilde{Gamma}_B;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G=-\nu^B_t N^2 + \tilde{\Gamma}_B \; , \quad$|; $key = q/{displaymath}{array}{rcl}alpha&=&-dfrac{4n{(sigma_k^psi)}^frac{1}{2}}{(1+4m){(sisi2}))^frac{1}{2}}{12n^2}right)^frac{1}{2};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \alpha &=& - \dfrac{4 n {(\sigma_k^\psi)}^... ...frac{1}{2} }{12 n^2} \right)^\frac{1}{2} \; , \quad \end{array}\end{displaymath}|; $key = q/tau^Theta_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau^\Theta_R$|; $key = q/Phi^-_{i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi^-_{i}$|; $key = q/alpha_M=overline{S}^2=overline{S}^2_U+overline{S}^2_V;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M = \overline{S}^2 = \overline{S}^2_U +\overline{S}^2_V$|; $key = q/1.00;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.00$|; $key = q/0.555;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.555$|; $key = q/c_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_i$|; $key = q/l_d;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ l_d$|; $key = q/q_a;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q_a$|; $key = q/nu^Theta_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^\Theta_t$|; $key = q/displaystylesigma_psi=dfrac{n^2kappa^2}{(c_mu^0)^2(c_{psi2}-c_{psi1})};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \sigma_\psi = \dfrac{n^2 \kappa^2}{(c_\mu^0)^2 (c_{\psi 2}-c_{\psi 1})}$|; $key = q/overline{S}^2Ri;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{S}^2 Ri$|; $key = q/displaystyler=dfrac{k_b}{epsilon_b}dfrac{epsilon}{k};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle r = \dfrac{k_b}{\epsilon_b} \dfrac{\epsilon}{k}$|; $key = q/displaystyleK=left(-dfrac{sigma_k}{c_mualphaL}F_kright)^frac{2}{3}dfrac{1}{z_0^alpha};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle K = \left( -\dfrac{\sigma_k}{c_\mu \alpha L} F_k \right)^\frac{2}{3} \dfrac{1}{z_0^\alpha} \; , \quad$|; $key = q/V;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ V$|; $key = q/displaystyleC_f=left{{array}{ll}C_f^{max}&mbox{for}|X|=X_{max};,quad0&mbox{else}quad.{array}right.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle C_f = \left\{ \begin{array}{ll} C_f^{\max} & \mbox{ for } \vert X\vert=X_{\max} \; , \quad \ 0 & \mbox{ else} \quad . \end{array} \right.$|; $key = q/displaystyledot{k_b}={calD}_b+P_b-epsilon_b;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{k_b} = {\cal D}_b + P_b - \epsilon_b \; , \quad$|; $key = q/kappa(z+z_0);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \kappa(z+z_0)$|; $key = q/P_r^0=0.7143;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ P_r^0=0.7143$|; $key = q/d_5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_5$|; $key = q/G_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ G_{ij}$|; $key = q/overline{S}_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{S}_{ij}$|; $key = q/displaystylek_b=dfrac{k_bepsilon}{kepsilon_b}dfrac{k}{epsilon}P_b=rdfrac{k}{epsilon}P_b=c_bdfrac{k}{epsilon}P_b;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k_b = \dfrac{k_b \epsilon}{k \epsilon_b} \dfrac{k}{\epsilon} P_b = r \dfrac{k}{\epsilon} P_b = c_b \dfrac{k}{\epsilon} P_b \; , \quad$|; $key = q/alpha_B=overline{T};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_B = \overline{T}$|; $key = q/Gamma_B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Gamma_B$|; $key = q/displaystyledot{omega}={calD}_omega+frac{omega}{k}(c_{omega_1}P+c_{omega_3}G-c_{omega2}epsilon);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\omega} = {\cal D}_\omega + \frac{\omega}{k} ( c_{\omega_1} P + c_{\omega_3} G - c_{\omega 2} \epsilon ) \; , \quad$|; $key = q/epsilon=E(z+z_0)^beta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon = E(z+z_0)^\beta$|; $key = q/n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ n$|; $key = q/beta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \beta$|; $key = q/a_{b2}=lambda_7;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b2} = \lambda_7$|; $key = q/a_1=1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_1=1$|; $key = q/tilde{Gamma}_B=epsilonGamma;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_B = \epsilon \Gamma$|; $key = q/displaystylegamma_3=gamma=-dfrac{{langlew'b'rangle}}{epsilon}=-dfrac{G}{epsilon}=dfrac{R_f}{1-R_f};,quadR_f=-dfrac{G}{P};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \gamma_3 = \gamma = - \dfrac{ {\langle w'b' \rangle} }{\epsilon} ... ...ac{G}{\epsilon} = \dfrac{R_f}{1-R_f} \; , \quad R_f = - \dfrac{G}{P} \; , \quad$|; $key = q/1leqkleqN;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1\leq k\leq N$|; $key = q/a_1=2slash3-c_2slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_1 = 2/3 - c_2/2$|; $key = q/T_w;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ T_w$|; $key = q/displaystyledot{Theta}={calD}_Theta-frac{1}{tau^Theta_R}(Theta-Theta_{obs})+frac{1}{C_prho_0}dfrac{partial{I}}{partial{z}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\Theta} = {\cal D}_\Theta - \frac{1}{\tau^\Theta_R}(\Theta-\Theta_{obs}) + \frac{1}{C_p \rho_0} \dfrac{\partial {I}}{\partial {z}} \; , \quad$|; $key = q/lambda_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_2$|; $key = q/tau_x^s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_x^s$|; $key = q/p_e;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_e$|; $key = q/dot{overline{q^2slash2}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{\overline{q^2/2}}$|; $key = q/0.9;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.9$|; $key = q/hat{c}_mu^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \hat{c}_\mu^$|; $key = q/{displaymath}{array}{rcl}P&=&-{langleu'w'rangle}dfrac{partial{U}}{partial{z}}-{lrtial{z}};,quad[3mm]G&=&{langlew'b'rangle};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} P &=& - {\langle u'w' \rangle} \dfrac{\par... ...ad \ [3mm] G &=& {\langle w'b' \rangle} \; , \quad \end{array}\end{displaymath}|; $key = q/p;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p$|; $key = q/tau_b=tau;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_b=\tau$|; $key = q/displaystylep_{i,j}(vec{c})=d_{j,i}(vec{c}),;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle p_{i,j}(\vec{c}) = d_{j,i}(\vec{c}),$|; $key = q/displaystyleleft.{array}{rcl}c_i^{(1)}&=&displaystylec_i^n+Deltatleft{sumlimits_{c}^{(1)}right)right)dfrac{c_i^{n+1}}{c_i^{(1)}}right}.{array}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left. \begin{array}{rcl} c_i^{(1)} &=& \displaystyle c_i^n + \De... ...(1)}\right) \right) \dfrac{c_i^{n+1}}{c_i^{(1)}} \right\}. \end{array} \right\}$|; $key = q/displaystyleG(sigma)=a_0+a_1sigma+a_2sigma^2+a_3sigma^3quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G(\sigma) = a_0 + a_1 \sigma + a_2 \sigma^2 + a_3 \sigma^3 \quad .$|; $key = q/N_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ N_i$|; $key = q/displaystylevec{c}^{(1)};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \vec{c}^{(1)}$|; $key = q/3.28;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 3.28$|; $key = q/displaystyledot{overline{q^2slash2}}={calD}_q+P+G-epsilon;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\overline{q^2/2}} = {\cal D}_q + P + G - \epsilon \; , \quad$|; $key = q/X;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ X$|; $key = q/alpha_M^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M^2$|; $key = q/R_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ R_i$|; $key = q/c_{epsilon2};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\epsilon 2}$|; $key = q/displaystyleK=left(-dfrac{F_q}{sqrt{2}S_qalphaL}right)^frac{2}{3}dfrac{1}{z_0^alpha};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle K = \left( - \dfrac{F_q}{\sqrt{2} S_q \alpha L} \right)^\frac{2}{3} \dfrac{1}{z_0^\alpha} \; , \quad$|; $key = q/F^s_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F^s_i$|; $key = q/X_P;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ X_P$|; $key = q/displaystylePhi_phi=1+zeta;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi_\phi = 1 + \zeta \; , \quad$|; $key = q/displaystyleRi_b=dfrac{(B(z_{bl})-B_r)d}{|{boldsymbolU}(z_{bl})-{boldsymbolU}_r|^2+V_t^2(z_{bl})};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Ri_b = \dfrac{(B(z_{bl})-B_r) d} {\vert {\boldsymbol U}(z_{bl})-{\boldsymbol U}_r \vert^2 + V_t^2(z_{bl})} \; , \quad$|; $key = q/partial_xB;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_xB$|; $key = q/Lneqkappa;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ L \neq \kappa$|; $key = q/displaystylek=dfrac{u_*^2}{(c_mu^0)^2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k= \dfrac{u_*^2}{(c_\mu^0)^2} \quad .$|; $key = q/57.3,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 57.3 $|; $key = q/{displaymath}{array}{ll}c_{mu}=c_mu^0;,quad[3mm]c_{mu}'=dfrac{c_{mu}}{Pr_t^0},dfh2}};,quad&Rigeq0c_{mu}'=c_{mu};,quad&Ri<0;,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{ll} c_{\mu} = c_\mu^0 \; , \quad \ [3mm] c_{\m... ...\ c_{\mu}'= c_{\mu} \; , \quad & Ri < 0 \; , \quad \end{array}\end{displaymath}|; $key = q/c_{psi_1}=m;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\psi _1}=m$|; $key = q/displaystyle{calN}_bgamma_{i}=-a_{b1}overline{S}_{ij}gamma_j-a_{b2}overline{W}_{j}+dfrac{1}{3}a_{b3}overline{N}_{i}-a_{b4}overline{T}delta_{i3}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal N}_b \gamma_{i} = - a_{b 1} \overline{S}_{ij} \gamma_j - a_... ...dfrac{1}{3} a_{b 3} \overline{N}_{i} - a_{b 4} \overline{T} \delta_{i3} \quad .$|; $key = q/displaystylenu_t=nu_t^{IW}+nu_t^{SI};,quadnu'_t=nu'^{IW}_t+nu'^{SI}_t;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t = \nu_t^{IW} + \nu_t^{SI} \; , \quad \nu'_t= \nu'^{IW}_t + \nu'^{SI}_t \; , \quad$|; $key = q/>;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ >$|; $key = q/h_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ h_i$|; $key = q/Rirightarrow0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri \rightarrow 0$|; $key = q/nu_t^phi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu_t^\phi$|; $key = q/0.33;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.33$|; $key = q/scalebox{0.5}{includegraphics{figuresslashgridtime.eps}};FSF=1.6;AAT/; $cached_env_img{$key} = q|\scalebox{0.5}{\includegraphics{figures/gridtime.eps}}|; $key = q/^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ ^2$|; $key = q/C_{pa};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ C_{pa}$|; $key = q/displaystyleX^{n+1}_i=X^n_i(1-DeltatQ^n_i)+DeltatP_i^n;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle X^{n+1}_i= X^n_i(1-\Delta tQ^n_i)+\Delta t P_i^n \; , \quad$|; $key = q/{calD}_V;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\cal D}_V$|; $key = q/alpha_Mrightarrow0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M \rightarrow 0$|; $key = q/alpha_M=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_M=0$|; $key = q/1.68;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.68$|; $key = q/Delta^{bio}t=1800;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Delta^{bio} t = 1800$|; $key = q/B_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ B_1$|; $key = q/tildeU_j=frac12(hatU_j+U_j);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde U_j=\frac12(\hat U_j+U_j)$|; $key = q/scalebox{0.5}{includegraphics{figuresslashstructure_gotmbio.eps}};FSF=1.6;AAT/; $cached_env_img{$key} = q|\scalebox{0.5}{\includegraphics{figures/structure_gotmbio.eps}}|; $key = q/0.96;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.96$|; $key = q/vec{h}(t^n,t^{n+1},vec{c}^n,vec{c}^{(1)});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \vec{h}(t^n,t^{n + 1},\vec{c}^n,\vec{c}^{(1)})$|; $key = q/c_mu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_\mu$|; $key = q/displaystyledisplaystylefrac{X^{n+1}_i-X^n_i}{Deltat}-frac{nu^n_ifrac{X^{n+sigmaX^{n+sigma}_{i-1}}{0.5(h^{n+1}_i+h^{n+1}_{i-1})}}{h^{n+1}_i}=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \displaystyle \frac{X^{n+1}_i-X^n_i}{\Delta t} -\frac{\nu^n_i\fra... ...i}-X^{n+\sigma}_{i-1}}{0.5(h^{n+1}_i+h^{n+1}_{i-1})}}{h^{n +1}_i} =0 \; , \quad$|; $key = q/sigma=0.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \sigma=0.5$|; $key = q/Q_H;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_H$|; $key = q/0.62;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.62$|; $key = q/partial_sS=0.0425;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_s S=0.0425$|; $key = q/{langlew'b'rangle};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\langle w'b' \rangle} $|; $key = q/p_{i,j}(vec{c});MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,j}(\vec{c})$|; $key = q/displaystylefrac{1}{l}dfrac{text{d}{l}}{text{d}{t}}=frac{1}{n}frac{1}{psi}dfrac{{psi}}{text{d}{t}}-frac{m}{n}frac{1}{k}dfrac{text{d}{k}}{text{d}{t}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \frac{1}{l} \dfrac{\text{d} {l}}{\text{d} {t}} = \frac{1}{n} \fra... ...si}}{\text{d} {t}} - \frac{m}{n} \frac{1}{k} \dfrac{\text{d} {k}}{\text{d} {t}}$|; $key = q/kappa=0.4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \kappa = 0.4$|; $key = q/displaystylealpha_M^2left(-d_5+n_2-left(d_3-n_1+n_{b2}right)Ri-left(d_4+n_{b1}riright)+alpha_Mleft(-d_2+n_0-left(d_1+n_{b0}right)Riright)-d_0=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_M^2 \left( -d_5 + n_2 - \left( d_3 - n_1 + n_{b2}\right) R... ..._M \left( -d_2 + n_0 - \left( d_1 + n_{b0} \right) Ri \right) - d_0 = 0 \quad .$|; $key = q/overline{W}_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{W}_{ij}$|; $key = q/f;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ f$|; $key = q/displaystyleleft{{i:f_i(t^n,vec{c}^n)<0,iin{1,...,I}}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left\{ {i:f_i (t^n ,\vec{c}^n ) < 0, i \in \{ 1,...,I\} } \right\}$|; $key = q/k_b={langleb'^2rangle}slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k_b= {\langle b'^2 \rangle} /2$|; $key = q/a_{b1}=lambda_6;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b1} = \lambda_6$|; $key = q/-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -1$|; $key = q/2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2$|; $key = q/Ri;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri$|; $key = q/p_{i,j,k};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p_{i,j,k}$|; $key = q/-2.5;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ -2.5$|; $key = q/displaystylec_{psi1}=m;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle c_{\psi 1} = m \; , \quad$|; $key = q/displaystyleF_epsilon=-dfrac{nu_t}{sigma_epsilon}dfrac{partial{epsilon}}{partial{z}}=dfrac{(c_mu^0)^4}{sigma_epsilon}dfrac{k^2}{z+z_0};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_\epsilon = - \dfrac{\nu_t}{\sigma_\epsilon} \dfrac{\partial {\e... ...rtial {z}} = \dfrac{(c_\mu^0)^4}{\sigma_\epsilon} \dfrac{k^2}{z+z_0} \; , \quad$|; $key = q/{displaymath}{array}{rcl}tau_x^s&=&c_{dd}rho_aW_xWtau_y^s&=&c_{dd}rho_aW_yWQ_e&=rho_aW(q_s-q_a)Q_h&=&c_{hd}C_{pa}rho_aW(T_w-T_a){array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \tau_x^s &=& c_{dd} \rho_a W_x W \ \ \ta... ...q_a) \ \ Q_h &=& c_{hd} C_{pa} \rho_a W (T_w-T_a) \end{array}\end{displaymath}|; $key = q/tilde{Gamma}_V;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_V$|; $key = q/displaystyle{calD}_B=dfrac{partial}{partial{z}}left((nu^B_t+nu^B)dfrac{partial{B}}{partial{z}}-tilde{Gamma}_Bright)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_B = \dfrac{\partial}{\partial {z}} \left( (\nu^B_t+\nu^B) \dfrac{\partial {B}}{\partial {z}} - \tilde{\Gamma}_B \right) \quad .$|; $key = q/displaystyledot{psi}={calD}_psi+frac{psi}{k}(c_{psi_1}P+c_{psi_3}G-c_{psi2}epsilon);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\psi} = {\cal D}_\psi + \frac{\psi}{k} ( c_{\psi_1} P + c_{\psi_3} G - c_{\psi 2} \epsilon ) \; , \quad$|; $key = q/0.12;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.12$|; $key = q/0.4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.4$|; $key = q/c_{b2}=tilde{c}_{b2}+tilde{c}_{b3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b2} = \tilde{c}_{b2} + \tilde{c}_{b3}$|; $key = q/inot=j;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ i\not= j$|; $key = q/displaystyleleft(alpham+nright)left(left(dfrac{1}{2}+mright)alpha+nright)L^2=left(c_mu^0right)^2R,sigma_psic_{psi2}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left( \alpha m + n \right) \left( \left( \dfrac{1}{2} + m \right... ...a + n \right) L^2 = \left(c_\mu^0 \right)^2 R \sigma_\psi c_{\psi 2} \quad .$|; $key = q/w_phi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ w_\phi$|; $key = q/{displaymath}{array}{rcl}Phi^b_i&=&-c_{b1}tau^{-1}_b;{langleu'_ib'rangle}+c_{b2}ial{B}}{partial{x_j}}-2c_{b5}k_bdelta_{i3};,quad{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} \Phi^b_i &=& - c_{b 1} \tau^{-1}_b \; {\la... ...rtial {x_j}} - 2 c_{b 5} k_b \delta_{i3} \; , \quad \end{array}\end{displaymath}|; $key = q/{displaymath}{array}{rcl}U&=&{langleurangle}quadtext{forthex-velocity}V&=&{langlancy}S&=&{langlesrangle}quadtext{forthesalinity}{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} U &=& {\langle u \rangle} \quad \text{for ... ...& {\langle s \rangle} \quad \text{for the salinity} \end{array}\end{displaymath}|; $key = q/{c_mu}';MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {c_\mu}'$|; $key = q/displaystyledot{rule{0mm}{4mm}langleu'_ib'rangle}-{calD}^b_{i}=-{langleu'_iu'_mrU_i}}{partial{x_m}}+F^b_i+2delta_{i3}k_b+Phi^b_i-epsilon^b_{i};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{ \rule{0mm}{4mm} \langle u'_i b' \rangle} - {\cal D}^b_{i} =... ...rtial {x_m}} + F^b_i + 2 \delta_{i3} k_b + \Phi^b_i - \epsilon^b_{i} \; , \quad$|; $key = q/nu^S;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^S$|; $key = q/displaystyleR_i>0.7;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle R_i>0.7 \; , \quad$|; $key = q/M;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ M$|; $key = q/_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ _2$|; $key = q/psi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \psi$|; $key = q/displaystyleD_{ij}=-{langleu'_iu'_mrangle}dfrac{partial{U_m}}{partial{x_j}}-{langleu'_ju'_mrangle}dfrac{partial{U_m}}{partial{x_i}}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle D_{ij} = - {\langle u'_i u'_m \rangle} \dfrac{\partial {U_m}}{\pa... ...}} - {\langle u'_j u'_m \rangle} \dfrac{\partial {U_m}}{\partial {x_i}} \quad .$|; $key = q/nu^B;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^B$|; $key = q/b_{23};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b_{23}$|; $key = q/displaystyleF_l=-S_lqldfrac{partial{q^2l}}{partial{z}}=-2sqrt{2}S_lkappa^2k^frac{3}{2}(z+z_0);MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_l = - S_l q l \dfrac{\partial {q^2 l}}{\partial {z}} = - 2 \sqrt{2} S_l \kappa^2 k^\frac{3}{2} (z+z_0)$|; $key = q/alpha=-;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha = -$|; $key = q/displaystylebeta=dfrac{1}{rho_0}left(dfrac{partial{rho_{is}}}{partial{S}}right)_T=-dfrac{1}{g}left(dfrac{partial{B_{is}}}{partial{S}}right)_T;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \beta = \dfrac{1}{\rho_0} \left( \dfrac{\partial {\rho_{is}}}{\pa... ...dfrac{1}{g} \left( \dfrac{\partial {B_{is}}}{\partial {S}} \right)_T \; , \quad$|; $key = q/displaystyledisplaystylefrac{X^{n+1}_{N_i}-X^n_{N_i}}{Deltat}-frac{F_s-nu^n_{N_i_{N_i-1}}{0.5(h^{n+1}_{N_i}+h^{n+1}_{N_i-1})}}{h^{n+1}_{N_i}}=;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \displaystyle \frac{X^{n+1}_{N_i}-X^n_{N_i}}{\Delta t} -\frac{F_s... ...gma}_{N_i-1}}{0.5(h^{n+1}_{N_i}+h^{n+1}_{N_i-1})}}{h^{n +1}_{N_i}} = \; , \quad$|; $key = q/q^2=2k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ q^2=2k$|; $key = q/displaystylephi={langlephirangle}+phi';,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \phi = {\langle \phi \rangle} + \phi' \; , \quad$|; $key = q/displaystylek=(c_mu^0)^{-3},l^2(c_muM^2-c'_muN^2);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k= (c_\mu^0)^{-3} l^2 ( c_\mu M^2 - c'_\mu N^2 ) \; , \quad$|; $key = q/X_{max};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ X_{\max}$|; $key = q/d_{i,j}(vec{c})=f(vec{c})c_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_{i,j}(\vec{c}) = f(\vec{c}) c_i$|; $key = q/tau_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_0$|; $key = q/displaystyleRi_{st}=dfrac{c_mu}{{c_mu}'}dfrac{c_{psi2}-c_{psi1}}{c_{psi2}-c_{psi3}}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Ri_{st}=\dfrac{c_\mu}{{c_\mu}'} \dfrac{c_{\psi 2} - c_{\psi 1}}{c_{\psi 2} - c_{\psi 3}} \quad .$|; $key = q/kappa;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \kappa$|; $key = q/Ri=overline{N}^2slashoverline{S}^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri = \overline{N}^2 / \overline{S}^2$|; $key = q/2.78;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 2.78$|; $key = q/{displaymath}{array}{l}displaystyled_tleft(sum_{i=1}^Ic_iright)=sum_{i=1}^Ileft(}^Ileft(p_{i,i}(vec{c})-d_{i,i}(vec{c})right)=0.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{l} \displaystyle d_t\left(\sum_{i=1}^I c_i \rig... ...\left(p_{i,i}(\vec{c}) - d_{i,i}(\vec{c})\right)=0. \end{array}\end{displaymath}|; $key = q/alpha_{sg};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_{sg}$|; $key = q/nu^B_t=nu^Theta_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^B_t = \nu^\Theta_t$|; $key = q/{displaymath}{array}{rcl}n_0&=&36a_1{calN}^2{calN}_b^2;,quad[3mm]n_1&=&-12a_5a_{;,quad[3mm]n_2&=&9a_1(a_{b2}^2-a_{b1}^2){calN}^2{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} n_0 &=& 36 a_1 {\cal N}^2 {\cal N}_b^2 \; ... ...[3mm] n_2 &=& 9 a_1 (a_{b2}^2 - a_{b1}^2){\cal N}^2 \end{array}\end{displaymath}|; $key = q/b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b$|; $key = q/partial_yTheta;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_y\Theta$|; $key = q/h;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ h$|; $key = q/tilde{Gamma}_U;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{\Gamma}_U$|; $key = q/11.2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 11.2$|; $key = q/tau^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau^$|; $key = q/displaystyleDeltat>frac{X^n}{X^nQ^n-P^n}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Delta t > \frac{X^n}{X^nQ^n-P^n} \quad .$|; $key = q/3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 3$|; $key = q/c_2=4C_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_2 = 4 C_1$|; $key = q/dot{psi};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{\psi}$|; $key = q/0.24;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.24$|; $key = q/tau_x=1.027;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau_x= 1.027$|; $key = q/20,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 20 $|; $key = q/g;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ g$|; $key = q/displaystylel=left(frac{1}{kappa(d_s+z_0^s)}+frac{1}{kappa(d_b+z_0^b)}+dfrac{1}{l_a}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle l = \left( \frac{1}{\kappa (d_s+z_0^s)} +\frac{1}{\kappa (d_b+z_0^b)} +\dfrac{1}{l_a} \right) \; , \quad$|; $key = q/h_k;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ h_k$|; $key = q/displaystylepartial_tu-partial_z(nu_tpartial_zu)=-gpartial_xzeta-C_fu|u|;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \partial_t u - \partial_z(\nu_t \partial_z u) = -g\partial_x\zeta-C_fu\vert u\vert \; , \quad$|; $key = q/kl;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ kl$|; $key = q/displaystylePhi^+_{i}=alpha_{i}+beta_{i}r^+_{i};,quadPhi^-_{i}=alpha_{i}+beta_{i}r^-_{i};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Phi^+_{i} = \alpha_{i} + \beta_{i} r^+_{i} \; , \quad \Phi^-_{i} = \alpha_{i} + \beta_{i} r^-_{i} \; , \quad$|; $key = q/displaystyle{calN}b_{ij}=-a_1overline{S}_{ij}-a_2overline{Sigma}_{ij}-a_3overline{Z}_{ij}-a_4overline{N}_{ij}-a_5overline{Gamma}_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal N} b_{ij} = - a_1 \overline{S}_{ij} - a_2 \overline{\Sigma}_{ij} - a_3 \overline{Z}_{ij} - a_4 \overline{N}_{ij} - a_5 \overline{\Gamma}_{ij}$|; $key = q/c_i+c_i^{min};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_i+c_i^{\min}$|; $key = q/alpha=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha=0$|; $key = q/Ri_{st}approx0.25;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Ri_{st}\approx0.25$|; $key = q/b';MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b'$|; $key = q/lambda;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda$|; $key = q/R_f=-GslashP;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ R_f=-G/P$|; $key = q/rho;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \rho$|; $key = q/tilde{c}_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tilde{c}_2$|; $key = q/1.96;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.96$|; $key = q/partial_xS;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \partial_xS$|; $key = q/a_{b5}=ra_{b4};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b5} = r a_{b4}$|; $key = q/dot{U};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{U}$|; $key = q/nu^S_t;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \nu^S_t$|; $key = q/displaystylek=K(z+z_0)^alphaquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle k= K (z+z_0)^\alpha \quad .$|; $key = q/scalebox{0.6}{includegraphics{figuresslashmixerA.eps}};FSF=1.6;AAT/; $cached_env_img{$key} = q|\scalebox{0.6}{\includegraphics{figures/mixerA.eps}}|; $key = q/displaystylefrac{q^2}{2}=k=K(z+z_0)^alphaquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \frac{q^2}{2} = k = K (z+z_0)^\alpha \quad .$|; $key = q/tau;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau$|; $key = q/hat{c}_mu;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \hat{c}_\mu$|; $key = q/overline{S};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{S}$|; $key = q/g=9.81;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ g=9.81$|; $key = q/displaystyleleft.{array}{rcl}c_i^{(1)}&=&displaystylec_i^n+Deltatleft{P_ileft(un{c}^{(1)}right)right)dfrac{c_i^{n+1}}{c_i^{(1)}}right}.{array}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left. \begin{array}{rcl} c_i^{(1)} &=& \displaystyle c_i^n + \De... ...{(1)}\right)\right) \dfrac{c_i^{n+1}}{c_i^{(1)}} \right\}. \end{array} \right\}$|; $key = q/Phi^+_{i};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \Phi^+_{i}$|; $key = q/z_0=lslashL;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_0 = l / L$|; $key = q/a_{b3}=2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b3} = 2$|; $key = q/U_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ U_1$|; $key = q/displaystyledfrac{partial{P}}{partial{z}}+g{langlerhorangle}=0;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {P}}{\partial {z}} + g {\langle \rho \rangle} = 0 \; , \quad$|; $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|$ l$|; $key = q/F_q;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ F_q$|; $key = q/displaystyle(z_{bl})=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle (z_{bl}) = 0 \quad .$|; $key = q/displaystylealpha_{i}=dfrac{1}{2}+dfrac{1}{6}left(1-2|c_{i}|right);,quadbeta_{i}=dfrac{1}{2}-dfrac{1}{6}left(1-2|c_{i}|right)quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \alpha_{i} = \dfrac{1}{2} + \dfrac{1}{6} \left( 1- 2 \vert c_{i} ... ...{i} = \dfrac{1}{2} - \dfrac{1}{6} \left( 1- 2 \vert c_{i} \vert \right) \quad .$|; $key = q/displaystyleSigma_{ij}=S_{im}b_{mj}+S_{jm}b_{mi}-frac{2}{3}S_{mn}b_{mn}delta_{ij};,quadZ_{ij}=W_{im}b_{mj}+W_{jm}b_{mi};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Sigma_{ij} = S_{im} b_{mj} + S_{jm} b_{mi} - \frac{2}{3} S_{mn} b_{mn} \delta_{ij} \; , \quad Z_{ij} = W_{im} b_{mj} + W_{jm} b_{mi} \; , \quad$|; $key = q/a_{b5}=2lambda_8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_{b5} = 2 \lambda_8$|; $key = q/displaystylenu_t=c_muk^frac{1}{2}l;,quadnu^B_t=c_mu'k^frac{1}{2}l;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \nu_t = c_\mu k^\frac{1}{2} l \; , \quad \nu^B_t = c_\mu' k^\frac{1}{2} l \; , \quad$|; $key = q/displaystyleF_q=-S_qqldfrac{partial{k}}{partial{z}}=0quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_q = - S_q q l \dfrac{\partial {k}}{\partial {z}} = 0 \quad .$|; $key = q/L|; $key = q/displaystyle{langlew'phi'rangle}=-nu_t^phidfrac{partial{{langlephirangle}}}{partial{z}}+tilde{Gamma}_phi;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\langle w'\phi' \rangle} = - \nu_t^\phi \dfrac{\partial { {\langle \phi \rangle} }}{\partial {z}} + \tilde{\Gamma}_\phi \; , \quad$|; $key = q/displaystyledfrac{partial{Y}}{partial{t}}=-dfrac{partial{F}}{partial{z}};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dfrac{\partial {Y}}{\partial {t}} = - \dfrac{\partial {F}}{\partial {z}} \; , \quad$|; $key = q/overline{z};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{z}$|; $key = q/d_s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_s$|; $key = q/I_{PAR};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I_{PAR}$|; $key = q/Pr_t^0=0.74;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Pr_t^0=0.74$|; $key = q/displaystyleRi_b(z_{bl})=dfrac{Ri_text{top}(z_{bl})}{Ri_text{bot}(z_{bl})}=Ri_c;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Ri_b(z_{bl}) = \dfrac{Ri_\text{top}(z_{bl})}{Ri_\text{bot}(z_{bl})} = Ri_c \; , \quad$|; $key = q/overline{N}_1=overline{N}_2=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \overline{N}_1=\overline{N}_2=0$|; $key = q/1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1$|; $key = q/u_*;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ u_*$|; $key = q/displaystyleX^{n+1}=frac{X^n+DeltatP^n}{1+DeltatQ^n}quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle X^{n+1}= \frac{X^n+\Delta t P^n}{1+\Delta t Q^n} \quad .$|; $key = q/I(z);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ I(z)$|; $key = q/1.02;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.02$|; $key = q/alpha_w;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \alpha_w$|; $key = q/displaystyleepsilon_b=dfrac{1}{c_b}dfrac{epsilon}{k}k_bquad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon_b = \dfrac{1}{c_b} \dfrac{\epsilon}{k} k_b \quad .$|; $key = q/epsilon_{ij}=2slash3epsilondelta_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon_{ij} = 2/3 \epsilon \delta_{ij}$|; $key = q/c_2=0.065;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_2=0.065$|; $key = q/displaystyletilde{Gamma}_phi=nu_t^phigamma_phi;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \tilde{\Gamma}_\phi = \nu_t^\phi \gamma_\phi \; , \quad$|; $key = q/w;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ w$|; $key = q/d_l>0,d_u>0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ d_l>0, d_u>0$|; $key = q/displaystyledot{epsilon}={calD}_epsilon+frac{epsilon}{k}(c_{epsilon1}P+c_{epsilon3}G-c_{epsilon2}epsilon);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{\epsilon} = {\cal D}_\epsilon + \frac{\epsilon}{k} ( c_{\epsilon 1} P + c_{\epsilon 3} G - c_{\epsilon 2} \epsilon ) \; , \quad$|; $key = q/c_3=2(tilde{c}_3-tilde{c}_4);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_3 = 2 ( \tilde{c}_3 - \tilde{c}_4 )$|; $key = q/0.92;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.92$|; $key = q/displaystyleleft.{array}{rcl}c_i^{(1)}&=&displaystylec_i^n+Deltatleft{P_ileft(un16left{c_i^{(1)}+2c_i^{(2)}+2c_i^{(3)}+c_i^{(4)}right}.{array}right};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \left. \begin{array}{rcl} c_i^{(1)} &=& \displaystyle c_i^n+\Delt... ...\left\{c_i^{(1)}+2c_i^{(2)}+2c_i^{(3)}+c_i^{(4)} \right\}. \end{array} \right\}$|; $key = q/f(R_f);MSF=1.6;AAT/; $cached_env_img{$key} = q|$ f(R_f)$|; $key = q/displaystyleQ_s=Q_{tot}(1-0.62C+0.0019beta)(1-alpha),;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle Q_s=Q_{tot} (1-0.62 C + 0.0019 \beta) (1-\alpha),$|; $key = q/=(c_mu^0)^3c_mu^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ = (c_\mu^0)^3 c_\mu^$|; $key = q/displaystyle{calD}_k=dfrac{partial}{partial{z}}left(dfrac{nu_t}{sigma_k}dfrac{partial{k}}{partial{z}}right);,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle {\cal D}_k = \dfrac{\partial}{\partial {z}} \left( \dfrac{\nu_t}{\sigma_k} \dfrac{\partial {k}}{\partial {z}} \right) \; , \quad$|; $key = q/{displaymath}{array}{rcl}z_i^{n+1}&=&z^n_i+partial_znu_t(z^n_i)Deltat&+&Rleft{2rrtial_znu_t(z^n_i)Deltat)Deltatright}^{1slash2},{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} z_i^{n+1} &=& z^n_i + \partial_z \nu_t (z^... ...al_z \nu_t (z^n_i)\Delta t) \Delta t\right\}^{1/2}, \end{array}\end{displaymath}|; $key = q/r_1^{max}=2.0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ r_1^{\max}=2.0$|; $key = q/displaystyleF_q=-S_qqldfrac{partial{k}}{partial{z}}=-sqrt{2}S_qK^frac{3}{2}alphaL(z+z_0)^{frac{3}{2}alpha};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F_q = - S_q q l \dfrac{\partial {k}}{\partial {z}} = - \sqrt{2} S_q K^\frac{3}{2} \alpha L (z+ z_0)^{\frac{3}{2} \alpha} \; , \quad$|; $key = q/gamma_phi;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \gamma_\phi$|; $key = q/a_4=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_4=0$|; $key = q/Q_{in};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Q_{in}$|; $key = q/{displaymath}{array}{rcl}c_i^{n+1}&=&displaystylec_i^n+Deltatleft{sumlimits_{staderline{c}^nright)dfrac{c_i^{n+1}}{c_i^n}right}.{array}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|\begin{displaymath}\begin{array}{rcl} c_i^{n+1} &=& \displaystyle c_i^n + \Delta... ...line{c}^n\right) \dfrac{c_i^{n+1}}{c_i^n} \right\}. \end{array}\end{displaymath}|; $key = q/displaystyleu_*^b=rsqrt{U_1^2+V_1^2};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle u_*^b = r \sqrt{U_1^2 + V_1^2} \; , \quad$|; $key = q/epsilonslashkš;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \epsilon /kš$|; $key = q/c_{mu};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{\mu}$|; $key = q/c_{hd};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{hd}$|; $key = q/tau^U_R;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \tau^U_R$|; $key = q/(c_mu^0)^2=0.3;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ (c_\mu ^0)^2=0.3$|; $key = q/w_c;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ w_c$|; $key = q/k_b;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ k_b$|; $key = q/displaystylevec{c}^n+frac{{Deltat}}{2}left({vec{f}(t^n,vec{c}^n)+vec{f}(t^{n+1},{c}^{(1)})}right)prodlimits_{kinK^n}{frac{{c_k^{n+1}}}{{c_k^{(1)}}}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \vec{c}^n + \frac{{\Delta t}}{2}\left( {\vec{f}(t^n ,\vec{c}^n ) ... ...{(1)} )} \right)\prod\limits_{k \in K^n } {\frac{{c_k^{n + 1} }}{{c_k^{(1)} }}}$|; $key = q/A_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ A_1$|; $key = q/displaystyleX^{n+sigma}=sigmaX^{n+1}+(1-sigma)X^n.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle X^{n+\sigma}=\sigma X^{n+1}+(1-\sigma)X^n.$|; $key = q/displaystyleDelta_xS=Deltaxpartial_xS;,quadDelta_yS=Deltaypartial_yS;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \Delta_xS=\Delta x \partial_xS \; , \quad \Delta_yS=\Delta y \partial_yS \; , \quad$|; $key = q/250;;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 250\;$|; $key = q/lambda_4;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_4$|; $key = q/displaystyleM^2=left(dfrac{partial{U}}{partial{z}}right)^2+left(dfrac{partial{V}}{partial{z}}right)^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle M^2 = \left( \dfrac{\partial {U}}{\partial {z}} \right)^2 + \left( \dfrac{\partial {V}}{\partial {z}} \right)^2 \quad .$|; $key = q/z_0^s;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_0^s$|; $key = q/langleR^2rangle=r;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \langle R^2 \rangle=r$|; $key = q/displaystyleG(1)=dfrac{1}{hw_phi(1)}nu_{ti}^phi(z_{bl});,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle G(1) = \dfrac{1}{h w_\phi(1)} \nu_{ti}^\phi(z_{bl}) \; , \quad$|; $key = q/1.25;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 1.25$|; $key = q/b^;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b^$|; $key = q/Pr_t^0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ Pr_t^0$|; $key = q/displaystylefrac{X^{n+1}-X^n}{Deltat}=P^n-Q^nX^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \frac{X^{n+1}-X^n}{\Delta t}=P^n-Q^nX^n$|; $key = q/0.1;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.1$|; $key = q/displaystyleoverline{T}=rgamma_3overline{N}^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{T} = r \gamma_3 \overline{N}^2 \quad .$|; $key = q/a_4=c_5slash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ a_4=c_5/2$|; $key = q/z_a;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ z_a$|; $key = q/{ttKPP_TWOPOINT_REF};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ {\tt KPP\_TWOPOINT\_REF}$|; $key = q/c_i(0)geq0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_i(0) \geq 0$|; $key = q/psi_0;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \psi_0$|; $key = q/displaystyledot{k}={calD}_k+P+G-epsilon;,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \dot{k} = {\cal D}_k + P + G - \epsilon \; , \quad$|; $key = q/displaystyle0|; $key = q/0.038;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ 0.038$|; $key = q/dot{epsilon};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \dot{\epsilon}$|; $key = q/p:=prodlimits_{jinJ^n}{{c_j^{n+1}}slash{c_j^n}};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ p: = \prod\limits_{j \in J^n}{{c_j^{n + 1}}/{c_j^n}}$|; $key = q/b_{13};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ b_{13}$|; $key = q/c_{b3}=tilde{c}_{b2}-tilde{c}_{b3};MSF=1.6;AAT/; $cached_env_img{$key} = q|$ c_{b3} = \tilde{c}_{b2} - \tilde{c}_{b3}$|; $key = q/displaystyleF^n_{i}=dfrac{1}{Deltat}int_{z^text{Face}_{i}-wDeltat}^{z^text{Face}_{i}}Y(z')dz'quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle F^n_{i} = \dfrac{1}{\Delta t} \int_{z^\text{Face}_{i} - w \Delta t}^{z^\text{Face}_{i}} Y(z') dz' \quad .$|; $key = q/1|; $key = q/displaystyleepsilon=(c_mu^0)^3frac{k^{frac{3}{2}}}{l};,quad;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \epsilon = (c_\mu^0)^3 \frac{k^{\frac{3}{2}}}{l} \; , \quad$|; $key = q/displaystyleJ^n=left{{i:f_i(t^n,vec{c}^n)<0,iin{1,...,I}}right},;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle J^n = \left\{ {i:f_i (t^n,\vec{c}^n) < 0,i \in \{1,...,I\}} \right\},$|; $key = q/lambda_8;MSF=1.6;AAT/; $cached_env_img{$key} = q|$ \lambda_8$|; $key = q/displaystyletildeR_i=frac{0.5}{P_r^0}R_i;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \tilde R_i=\frac{0.5}{P_r^0} R_i$|; $key = q/displaystyleoverline{N}_3=dfrac{k^2}{epsilon^2}dfrac{partial{B}}{partial{z}}=dfrac{k^2}{epsilon^2}N^2quad.;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle \overline{N}_3 = \dfrac{k^2}{\epsilon^2} \dfrac{\partial {B}}{\partial {z}} = \dfrac{k^2}{\epsilon^2} N^2 \quad .$|; $key = q/displaystyleg(p)=prodlimits_{jinJ^n}{left({1+frac{Deltat:f_j(t^n,vec{c}^n)}{c_j^n}p}right)}-p=0,;MSF=1.6;AAT/; $cached_env_img{$key} = q|$\displaystyle g(p) = \prod\limits_{j \in J^n} {\left( {1 + \frac{\Delta t \: f_j(t^n,\vec{c}^n)}{c_j^n }p} \right)} - p = 0,$|; 1;