| 47 | == Technical details == |
| 48 | The discretized and linearized energy budget equation in PALM reads |
| 49 | {{{ |
| 50 | #!Latex |
| 51 | \begin{equation*} |
| 52 | T_\mathrm{0,p} = \dfrac{A \Delta t + C_\mathrm{sk} T_0}{C_\mathrm{sk} + B \Delta t} |
| 53 | \end{equation*} |
| 54 | }}} |
| 55 | with |
| 56 | {{{ |
| 57 | #!Latex |
| 58 | \begin{equation*} |
| 59 | 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} |
| 60 | \end{equation*} |
| 61 | }}} |
| 62 | and |
| 63 | {{{ |
| 64 | #!Latex |
| 65 | \begin{equation*} |
| 66 | B = \Lambda + f_{LE} \dfrac{d q_s}{d T} + \dfrac{f_H}{\Pi} + 4 \sigma T_0^3 |
| 67 | \end{equation*} |
| 68 | }}} |
| 69 | with (in order of occurence): |
| 70 | {{{ |
| 71 | #!Latex |
| 72 | $T_\mathrm{0,p}$: prognostic skin temperature\\ |
| 73 | $\Delta t$: time step\\ |
| 74 | $C_\mathrm{sk}$: skin heat capacity\\ |
| 75 | $T_\mathrm{0}$: skin temperature before time step\\ |
| 76 | $R_\mathrm{n}$: net radiation at the surface\\ |
| 77 | $\sigma$: Stefan-Boltzmann constant\\ |
| 78 | $f_{H} = \dfrac{\rho c_\mathrm{p}}{r_\mathrm{a}}$\\ |
| 79 | $\Pi$: conversion factor from actual temperature to potential temperature\\ |
| 80 | $T_1$: temperature at first grid level\\ |
| 81 | $f_{LE} = \dfrac{\rho l_\mathrm{v}}{r_\mathrm{a} + r_\mathrm{s}}$\\ |
| 82 | $q_1$: specific humidity at first grid level\\ |
| 83 | $q_s$: saturation specific humidity at the surface\\ |
| 84 | $\Lambda$: heat conductivity of the skin layer\\ |
| 85 | $T_\mathrm{soil}$: temperature of the uppermost soil layer |
| 86 | }}} |
| 87 | |
| 88 | |
| 89 | |
| 90 | |
| 91 | |