Changes between Version 7 and Version 8 of doc/tec/lsm

Timestamp:

Apr 4, 2016 3:31:17 PM (9 years ago)

Author:

maronga

Comment:

--

doc/tec/lsm

@@ -45 +45 @@
 == Soil model ==
 
+== Technical details ==
+The discretized and linearized energy budget equation in PALM reads
+{{{
+#!Latex
+\begin{equation*}
+T_\mathrm{0,p} = \dfrac{A \Delta t + C_\mathrm{sk} T_0}{C_\mathrm{sk} + B \Delta t} 
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+\begin{equation*}
+A = R_\mathrm{n} + 3 \sigma T_0^4 + \dfrac{f_H}{\Pi} T_1 + f_{LE} \left( q_1 - q_s + \dfrac{d q_s}{d T} T_0 \right) + \Lambda T_\mathrm{soil}
+\end{equation*}
+}}}
+and
+{{{
+#!Latex
+\begin{equation*}
+B = \Lambda + f_{LE} \dfrac{d q_s}{d T} + \dfrac{f_H}{\Pi} + 4 \sigma T_0^3
+\end{equation*}
+}}}
+with (in order of occurence):
+{{{
+#!Latex
+$T_\mathrm{0,p}$:  prognostic skin temperature\\
+$\Delta t$: time step\\
+$C_\mathrm{sk}$: skin heat capacity\\
+$T_\mathrm{0}$: skin temperature before time step\\
+$R_\mathrm{n}$: net radiation at the surface\\
+$\sigma$: Stefan-Boltzmann constant\\
+$f_{H} = \dfrac{\rho c_\mathrm{p}}{r_\mathrm{a}}$\\
+$\Pi$: conversion factor from actual temperature to potential temperature\\
+$T_1$: temperature at first grid level\\
+$f_{LE} = \dfrac{\rho l_\mathrm{v}}{r_\mathrm{a} + r_\mathrm{s}}$\\
+$q_1$: specific humidity at first grid level\\
+$q_s$: saturation specific humidity at the surface\\
+$\Lambda$: heat conductivity of the skin layer\\
+$T_\mathrm{soil}$: temperature of the uppermost soil layer
+}}}
+
+
+
+
+
 == Usage ==