Changes between Version 11 and Version 12 of doc/tec/lsm

Timestamp:

Apr 4, 2016 4:28:37 PM (9 years ago)

Author:

maronga

Comment:

--

doc/tec/lsm

@@ -105 +105 @@
 
 == Soil model ==
-The soil model consists of prognostic equations for the soil temperature and the volumetric soil moisture which are solved for multiple layers. By default, the soil model consists of four layers, in which the vertical heat and water transport is modelled using the Fourier law of diffusion and Richards' equation, respectively. Also, root fractions can be assigned to each soil layer to to account for the explicit water withdrawal of plants used for transpiration from the respective soil layer. 
+The soil model consists of prognostic equations for the soil temperature and the volumetric soil moisture which are solved for multiple layers. The soil model only takes into account vertical transport within the soil and no ice phase is considered. By default, the soil model consists of four layers, in which the vertical heat and water transport is modelled using the Fourier law of diffusion and Richards' equation, respectively. Also, root fractions can be assigned to each soil layer to to account for the explicit water withdrawal of plants used for transpiration from the respective soil layer. 
 
 === Soil heat transport ===
+The prognostic equation for soil temperature reads
+{{{
+#!Latex
+\begin{equation*}
+(\rho C)_\mathrm{soil} \dfrac{\partial T_\mathrm{soil}}{\partial t} = \dfrac{\partial}{\partial z} \left( \lambda_T \dfrac{\partial T_\mathrm{soil}}{\partial z}\right)
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+$(\rho C)_\mathrm{soil}$: Volumetric heat capacity of the soil layer\\
+$\lambda_T$: Thermal heat conductivity of the soil layer
+}}}
+The volumetric heat capacity is calculated as
+{{{
+#!Latex
+\begin{equation*}
+   \lambda_T = Ke\ (\lambda_{T,\mathrm{sat}} - \lambda_{T,\mathrm{dry}}) + \lambda_{T,\mathrm{dry}}
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+$\lambda_{T,\mathrm{sat}}$: Heat conductivity of saturated soil\\
+$\lambda_{T,\mathrm{dry}}$: Heat conductivity of dry soil
+$Ke$: Kersten number
+}}}
+The heat conductivity of saturated soil is given by
+{{{
+#!Latex
+\begin{equation*}
+\lambda_{T,\mathrm{sat}} = \lambda_{T,\mathrm{sm}}^{1-m_\mathrm{soil,sat}}\ \lambda_\mathrm{m}
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+$\lambda_{T,\mathrm{sm}}$: Heat conductivity of the soil matrix\\
+$\lambda_\mathrm{m}$: Heat conductivity of water\\
+}}}
+The Kersten number is calculated as
+{{{
+#!Latex
+\begin{equation*}
+Ke = \log_{10} \left[max\left(0.1, \dfrac{m_\mathrm{soil}}{m_\mathrm{sat}}\right) \right] + 1
+\end{equation*}
+}}}
+
 
 === Soil moisture transport ===
-
-
-
-For more details, see \citet{viterbo1995} and \citet{balsamo2009}.
-
-
+The prognostic equation for volumetric soil moisture reads
+{{{
+#!Latex
+\begin{equation*}
+\dfrac{\partial m_\mathrm{soil}}{\partial t} = \dfrac{\partial}{\partial z} \left( \lambda_m \dfrac{\partial m_\mathrm{soil}}{\partial z} - \gamma\right) + S_m
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+$\lambda_m$: Hydraulic diffusion coefficient\\
+$\gamma$: Hydraulic conductivity\\
+$S_m$: Sink term due to root extraction 
+}}}
+
+The hydraulic diffusion coefficient is calculated after Clapp and Hornberger (1978) as
+{{{
+#!Latex
+\begin{equation*}
+\lambda_m = \dfrac{b \gamma_\mathrm{sat}(-\Psi_\mathrm{sat})}{m_\mathrm{sat}}\left(\dfrac{m_\mathrm{soil}}{m_\mathrm{sat}}\right)^{b+2}
+\end{equation*}
+}}}
+with
+{{{
+#!Latex
+$b = 6.04$: exponent\\
+$\gamma_\mathrm{sat}$: Hydraulic conductivity at saturation\\
+$\Psi_\mathrm{sat} = -338 m$: Matrix potential at saturation
+}}}
+
+The hydraulic conductivity is calculated either after Van Genuchten (1980) (as in H-TESSEL):
+{{{
+#!Latex
+\begin{equation*}
+ \gamma = \gamma_\mathrm{sat} \dfrac{\left[(1 + (\alpha h)^n)^{1 - 1/n} - (\alpha h)^{n-1}\right]^2}{(1+ (\alpha h)^n)^{(1 - 1/n)(l + 2)}}
+\end{equation*}
+}}}
+where
+{{{
+#!Latex
+$h$: Pressure head\\
+$n$: Van Genuchten coefficient\\
+$l$: Van Genuchten coefficient
+}}}
+and
+{{{
+#!Latex
+\begin{equation}
+m_\mathrm{soil}(h) = m_\mathrm{res} + \dfrac{m_\mathrm{sat} - m_\mathrm{res}}{(1 + (\alpha h)^n)^{1 - 1/n}} 
+\end{equation}
+}}}
+or after Clapp anf Hornberger (1978):
+{{{
+#!Latex
+\begin{equation}
+\gamma = \gamma_\mathrm{sat} \left(\dfrac{m_\mathrm{soil}}{m_\mathrm{sat}}\right)^{2b + 3}
+\end{equation}
+}}}
+
+For more details, see also Viterbo et al. (1995) and Balsamo et al. (2009).
 
 == Technical details ==