| | 11 | === Radiation budget === |
| | 12 | The radiation budget at the Earth's surface reads |
| | 13 | {{{ |
| | 14 | #!Latex |
| | 15 | \begin{equation*} |
| | 16 | R_\mathrm{n} = SW_\mathrm{in} - SW_\mathrm{out} + LW_\mathrm{in} - LW_\mathrm{out} |
| | 17 | \end{equation*} |
| | 18 | }}} |
| | 19 | with ''R'',,n,,, ''SW'',,in,,, ''SW'',,out,,, ''LW'',,in,,, ''LW'',,out,, being the net radiation, shortwave incoming (downward), shortwave outgoing (upward), longwave incoming (downward), and longwave outgoing (upward) flux, respectively. |
| | 20 | |
| | 21 | === Parameterization of radiative fluxes === |
| | 22 | |
| | 23 | The shortwave outgoing radiative flux depends on the surface albedo ''α'': |
| | 24 | {{{ |
| | 25 | #!Latex |
| | 26 | \begin{equation*} |
| | 27 | SW_\mathrm{out} = \alpha\ SW_\mathrm{in} |
| | 28 | \end{equation*} |
| | 29 | }}} |
| | 30 | The longwave outgoing radiative flux can be parameterized as |
| | 31 | {{{ |
| | 32 | #!Latex |
| | 33 | \begin{equation*} |
| | 34 | LW_\mathrm{out} = \epsilon\ \sigma\ T_0^4 |
| | 35 | \end{equation*} |
| | 36 | }}} |
| | 37 | where ''ε'' is the surface emissivity and ''σ'' = 5.67 * 10^-8^ W m^-2^ K^-4^ is the Stefan-Boltzmann constant. ''T'',,0,, is the skin temperature provided by the solver for the surface energy budget. |
| | 38 | |
| | 39 | The longwave incoming radiative flux follows a very simple parameterization: |
| | 40 | {{{ |
| | 41 | #!Latex |
| | 42 | \begin{equation*} |
| | 43 | LW_\mathrm{in} = \epsilon_\mathrm{atm}\ \sigma\ T_1^4 |
| | 44 | \end{equation*} |
| | 45 | }}} |
| | 46 | with |
| | 47 | {{{ |
| | 48 | #!Latex |
| | 49 | $\epsilon_\mathrm{atm} = 0.8$: Emissivity of the atmosphere\\ |
| | 50 | $T_1$: Temperature at first grid level |
| | 51 | }}} |
| | 52 | |
| | 53 | The shortwave incoming radiative flux is calculated from |
| | 54 | {{{ |
| | 55 | #!Latex |
| | 56 | \begin{equation*} |
| | 57 | SW_\mathrm{in} = S_0\ \tau\ \sin(\varPsi) |
| | 58 | \end{equation*} |
| | 59 | }}} |
| | 60 | with |
| | 61 | {{{ |
| | 62 | #!Latex |
| | 63 | $S_0 = 1368 W m^{-2}$: Solar constant\\ |
| | 64 | $\varPsi$: Zenith angle |
| | 65 | }}} |
| | 66 | and the transmissivity of the atmosphere ''τ'': |
| | 67 | {{{ |
| | 68 | #!Latex |
| | 69 | \begin{equation*} |
| | 70 | \tau = 0.6 + 0.2\ \sin(\varPsi) |
| | 71 | \end{equation*} |
| | 72 | }}} |
| | 73 | ''ψ'' is the cosine of the zenith angle, depending on time, date and location. Input parameters are |
| | 74 | * day of year (1-365) |
| | 75 | * UTC time (0:00 - 23:59) |
| | 76 | * Longitude (-180° - 180°) |
| | 77 | * Latitude (-90° - 90°) |
| | 78 | |
| | 79 | ''ψ'' is calculated from the following system of equations: |
| | 80 | {{{ |
| | 81 | #!Latex |
| | 82 | \begin{equation*} |
| | 83 | D = \arcsin \left[ d1 * \sin\left(d2 * day - d3\right) \right] |
| | 84 | \end{equation*} |
| | 85 | }}} |
| | 86 | where D is the declination of the sun with |
| | 87 | {{{ |
| | 88 | #!Latex |
| | 89 | $d1 = \sin \left(\dfrac{23.45 * \pi}{180}\right)$\\ |
| | 90 | $d2 = \dfrac{2 \pi}{365}$\\ |
| | 91 | $d3 = 81\ d2$ |
| | 92 | }}} |
| | 93 | The hour angle is then given by |
| | 94 | {{{ |
| | 95 | #!Latex |
| | 96 | \begin{equation*} |
| | 97 | H = 2 \pi \left(\dfrac{t_\mathrm{UTC}}{86400}\right) + lon - \pi |
| | 98 | \end{equation*} |
| | 99 | }}} |
| | 100 | where ''t'',,UTC,, is the current UTC time and ''lon'' is longitude. |
| | 101 | |
| | 102 | The cosine of the zenith angle ''Z'' is computed as |
| | 103 | {{{ |
| | 104 | #!Latex |
| | 105 | \begin{equation*} |
| | 106 | Z = \sin(lat) \sin(D) + \cos(lat) \cos(D) \cos(H) |
| | 107 | \end{equation*} |
| | 108 | }}} |
| | 109 | where ''lat'' is latitude. |
| | 110 | |
