\documentclass{article}
\begin{document}


{ \bf enerbil.f90 equations }

\vspace {10mm}
{ \bf enerbil\_begin }

\vspace {10mm}
enerbilbegin1.tex

\begin{equation}
    ps^{t} = T_S C_p
\end{equation}

\vspace {10mm}
enerbilbegin2.tex

\begin{equation}
    \frac{\delta q_s^{t}}{\delta t} = \delta q_s \frac{(p_l)^\kappa}{C_p^{air}}
\end{equation}

\vspace {10mm}
enerbilbegin3.tex

\begin{equation}
    R^{LW}_{abs} = \epsilon R^{LW}_{\downarrow}
\end{equation}

\vspace {10mm}
enerbilbegin4.tex

\begin{equation}
    R_{net} = R^{LW}_{\downarrow} + S^{SW}_{net} - \epsilon \sigma T^4 (1-\epsilon) R^{LW}_{\downarrow}
\end{equation}

\vspace {10mm}
{ \bf enerbil\_surftemp }

\vspace {10mm}
enerbilsurftemp1.tex

\begin{equation}
    U = max \{ U_{min}, \sqrt{u^2 + v^2} \}
\end{equation}

\vspace {10mm}
enerbilsurftemp2.tex

\begin{equation}
    z_{ikt}=\frac{1}{\rho_{air} U q_{c}}; \quad z_{ikq}=\frac{1}{\rho_{air} U q_{c}}
\end{equation}

\vspace {10mm}
enerbilsurftemp3.tex

\begin{equation}
    H^{t} = \frac{B_T^{orc} - ps^{t}}{z_{ikt} - A_T^{orc}}
\end{equation}

\vspace {10mm}
enerbilsurftemp4.tex

\begin{equation}
    \lambda E _{sub}^{t} = \lambda E_{sub}^0 \frac{B_q^{orc} - q_{s, sat}}{z_{ikq}-A_q^{orc}}
\end{equation}

\vspace {10mm}
enerbilsurftemp5.tex

\begin{equation}
    \lambda E _{evap}^{t} = \lambda E_{evap}^0  \left( 1-\beta_{v1} \beta_v \left( \frac {  B_q^{orc} - \alpha_v q_{s, sat}  }{  z_{ikq} - A_q^{orc}  }  \right) \right)
\end{equation}

\vspace {10mm}
enerbilsurftemp6.tex

\begin{equation}
    R^{net}_{sns} = \left( \frac{1}{C_p} \right) 4 \epsilon \sigma \left( \left( \frac{1}{C_p} \right) ps_{t} ^3 \right)
\end{equation}

\vspace {10mm}
enerbilsurftemp7.tex

\begin{equation}
    H_{sns} = \frac{1} { \left( \frac{1}{\rho_{air} S q_{c}} - A_T^{orc} \right)}
\end{equation}

\vspace {10mm}
enerbilsurftemp8.tex

\begin{equation}
    \lambda E_{sns}^{sub} = \lambda E_{sub}^0 \beta_{v1} \frac{1}{C_p} \left( \frac{ ({\delta q_s^{t}}/{\delta t}) } {z_{ikq} - A_q^{orc} } \right)
\end{equation}

\vspace {10mm}
enerbilsurftemp9.tex

\begin{equation}
    \lambda E_{sns}^{evap} = \lambda E_{evap}^0 (1-\beta_{v1}) \beta_v \alpha_v \frac{1}{C_p} \left( \frac{ ({\delta q_s^{t}}/{\delta t}) } {z_{ikq} - A_q^{orc} } \right)
\end{equation}

\vspace {10mm}
enerbilsurftemp10.tex

\begin{equation}
    \Sigma E^{t} = R_{net} + H^{t} + {\lambda E}_{sub}^{t} + {\lambda E}_{evap}^{t} + G
\end{equation}

\vspace {10mm}
enerbilsurftemp11.tex

\begin{equation}
    \Sigma E^{sns} = R^{sns} + H^{sns} + {\lambda E}_{sub}^{sns} + {\lambda E}_{evap}^{sns} + G
\end{equation}

\vspace {10mm}
enerbilsurftemp12.tex

\begin{equation}
    \Delta \theta = \frac {\Delta t (\Sigma E^{t})} {\left( \frac {1}{C_p^{air}} (C_p^{soil} + \Delta T) (\Sigma E^{sns}) \right)}
\end{equation}

\vspace {10mm}
enerbilsurftemp13.tex

\begin{equation}
    ps_{t+\Delta t} = ps_{t} + \Delta \theta
\end{equation}

\vspace {10mm}
enerbilsurftemp14.tex

\begin{equation}
    q_{s,sat}^{t+\Delta t} = q_{s,sat} + \left( \left( \frac{1}{C_p^{air}} \right) \left( \frac{\delta q_s^{t}}{\delta t} \right) \delta\theta \right)
\end{equation}

\vspace {10mm}
enerbilsurftemp15.tex

\begin{equation}
    T_{s}^{t+\Delta t} = \frac {ps^{t+\Delta t}}{C_p^{air}}
\end{equation}

\vspace {10mm}
enerbilsurftemp16.tex

\begin{equation}
    E_{pot}^{air,t+\Delta t} = z_{ikt} (H^{t} - H^{sns} \Delta \theta) + ps^{t+\Delta t}
\end{equation}

\vspace {10mm}
enerbilsurftemp17.tex

\begin{equation}
    {\lambda E} ^{evap} = ({\lambda E}_{evap}^{t} - {\lambda E} _{evap} ^{sns} \Delta \theta) + ({\lambda E}_{sub}^{t} - {\lambda E}_{sub}^{sns} \Delta \theta)
\end{equation}

\vspace {10mm}
enerbilsurftemp18.tex

\begin{equation}
    q_{air}^{t+\Delta t}(ji) = q_{air}(ji)
\end{equation}

\vspace {10mm}
enerbilsurftemp19.tex

\begin{equation}
   q_{air}^{t+\Delta t} = z_{ikq} \frac {1} {(\lambda E^{evap}_0  (1 - \beta_{v1}) \beta_1 \alpha_v + \lambda E^{sub}_0 \beta_{v1})} {\lambda E} ^{evap} + q_{s,sat}^{t+\Delta t} 
\end{equation}

\vspace {10mm}
{ \bf surf\_land\_orchidee (links from LMDZ to ORCHIDEE)} 

\vspace {10mm}
surflandLMDZ1.tex

\begin{equation}
    A_T^{orc} = B_T^{lmdz} \Delta t
\end{equation}

\vspace {10mm}
surflandLMDZ2.tex

\begin{equation}
    B_T^{orc} = A_T^{lmdz}
\end{equation}

\vspace {10mm}
surflandLMDZ3.tex

\begin{equation}
    A_q^{orc} = B_q^{lmdz} \Delta t
\end{equation}

\vspace {10mm}
surflandLMDZ4.tex

\begin{equation}
    B_q^{orc} = A_q^{lmdz}
\end{equation}

\vspace {10mm}
{ \bf enerbil\_flux} 

\vspace {10mm}
enerbilflux1.tex

\begin{equation}
    U=max\{U_{min}, \sqrt{u^2+v^2} \}
\end{equation}

\vspace {10mm}
enerbilflux2.tex

\begin{equation}
    q_c=|v|q_{c, drag}
\end{equation}

\vspace {10mm}
enerbilflux3.tex

\begin{equation}
    R^{LW}_{\uparrow}=\epsilon \sigma T_{sol}^4+\epsilon 4 \sigma T_{sol}^3(T_{s}^{t+\Delta t}-T_{s})
\end{equation}

\vspace {10mm}
enerbilflux4.tex

\begin{equation}
    R^{LW}_{\uparrow}=R^{LW}_{\uparrow}+(1-\epsilon)R^{LW}_{\downarrow}
\end{equation}

\vspace {10mm}
enerbilflux5.tex

\begin{equation}
    T_{s}^{rad}=\epsilon \sigma T_{s}^4 + R^{LW}_{\uparrow}
\end{equation}

\vspace {10mm}
enerbilflux6.tex

\begin{equation}
    q_{surf}=\beta_{v,1} (q_{s,sat}^{t+\Delta t})+(1-\beta_{v,1})\beta_v\alpha_v(q_{s,sat}^{t+\Delta t})
\end{equation}

\vspace {10mm}
enerbilflux7.tex

\begin{equation}
    q_s = max \{q_s, q_{air} \}
\end{equation}

\vspace {10mm}
enerbilflux8.tex

\begin{equation}
    R_{net}=R^{LW}_{\downarrow}+R^{SW}_{net}-R^{LW}_{\uparrow}
\end{equation}

\vspace {10mm}
enerbilflux9.tex

\begin{equation}
    vev_{app} = dt(\rho)q_c \beta_{v,1}(q_{s,sat}^{t+\Delta t}-q_{air})+\Delta t(\rho)q_c(1-\beta_{1,v}\beta_v(\alpha_v)q_{s,sat}^{t+\Delta t}-q_{air})
\end{equation}

\vspace {10mm}
enerbilflux10.tex

\begin{equation}
   H=\lambda E_0^{sub}(\rho)q_c(\beta_{v,1})(q_{s,sat}^{t+\Delta t}-q_{air}+\lambda E_0^{evap}(\rho)q_c(1-\beta_{v,1})\beta_v(\alpha_v)q_{s,sat}^{t+\Delta t}-q_{air})
\end{equation}

\vspace {10mm}
enerbilflux11.tex

\begin{equation}
   \lambda E^{sub}=\lambda E_0^{sub} (\rho)q_c\beta_{v,1}(q_{s,sat}^{t+\Delta t}-q_{air})
\end{equation}

\vspace {10mm}
enerbilflux12.tex

\begin{equation}
    H = \rho q_c (ps^{t+\Delta t}- E^{pot}_{air})
\end{equation}

\vspace {10mm}
enerbilflux13.tex

\begin{equation}
   R^{LW}_{net} = R^{LW}_{\downarrow}-R^{LW}_{\uparrow}
\end{equation}

\vspace {10mm}
enerbilflux14.tex

\begin{equation}
   E^{pot}_{air} = max \{0, \Delta t \rho q_c (q_{s,sat}^{t+\Delta t} -q_{air}) \}
\end{equation}

\vspace {10mm}
enerbilflux15.tex

\begin{equation}
   T_{air}= \frac{E^{pot}_{air}}{c_{p, air}}
\end{equation}









{\bf enerbil\_evapveg.f90}

\vspace {10mm}
enerbilevapveg1.tex

\begin{equation}
    U = max\{U_{min}, \sqrt{u^2 + v^2} \}
\end{equation}

\vspace {10mm}
enerbilevapveg2.tex

\begin{equation}
   E^{snow} = \beta_{v1} \Delta t \rho U q_c^{drag} (q_{sol,s}^{t+\Delta t}- q_{air})
\end{equation}

\vspace {10mm}
enerbilevapveg3.tex

\begin{equation}
   E^{soil} = (1-\beta_{v1})\beta_{v4} \Delta t \rho U q_c^{drag} (q_{sol,s}^{t+\Delta t} - q_{air})
\end{equation}

\vspace {10mm}
enerbilevapveg4.tex

\begin{equation}
    U = max\{U_{min}, \sqrt{u^2 + v^2} \}
\end{equation}

\vspace {10mm}
enerbilevapveg5.tex

\begin{equation}
   xx = \Delta t (1-\beta_{v1}) (q_{s,sat}^{t+\Delta t} - q_{a}) \rho S q_c^{drag}
\end{equation}

\vspace {10mm}
enerbilevapveg6.tex

\begin{equation}
   (Interception) = xx \beta_{v2}
\end{equation}

\vspace {10mm}
enerbilevapveg7.tex

\begin{equation}
   (Transpiration) = xx \beta_{v3}
\end{equation}


\vspace {10mm}
enerbilevapveg8.tex

\begin{equation}
   U = max \{U_{min}, \sqrt{u^2 + v^2} \}
\end{equation}


\vspace {10mm}
enerbilevapveg9.tex

\begin{equation}
   (Assimilation) = \beta_{v,CO_2} \Delta t \rho S q_c^{drag} (\chi_{CO_2}^{canopy} - \bar{Ci})
\end{equation}





\end{document}