Radiatve Transfer Equations

Radiation heating rate

Radiative fluxes enter the thermal energy equation as follows:

\[\begin{align} \partial_t (\rho e) = - \nabla \cdot F_{net} \end{align}\]

where $F_{net} = \sum_j (F^{+}_j - F^{-}_j)$, and $F^{+}_j$ and $F^{-}_j$ denote the sum of shortwave and longwave upward and downward fluxes for spectral interval j, respectively.

RTE solver

solve_lw! and solve_sw! solve the radiative transfor equation for longwave and shortwave radiation.

Computing fluxes through a vertically layered atmosphere

rte_lw_2stream! and rte_sw_2stream! compute longwave and shortwave radiative fluxes through a vertically layered atmosphere. The equations for upward and downward fluxes are calculated following [1]:

\[\begin{align} F^{+}_i = \alpha_i F^{-}_i + G_i \\ F^{-}_i = \beta_i (T_i F^{-}_{i+1} + R_i G_i + S^{-}_i) + F^{-}_{i, dir} \\ \alpha_{i+1} = R_i + T_i^2 \beta_i \alpha_i \\ G_{i+1} = S^{+}_i + T_i \beta_i (G_i + \alpha_i S^{-}_i) \\ \end{align}\]

where $\beta_k = 1 / (1 - \alpha_k R_k)$. $T$ and $R$ are transmittance and reflectance, and $S^{+}$ and $S^{-}$ represent upward and downward source functions. $F^-_{dir}$ is the downward direct shortwave flux. The subscript $i$ represents the level.

Computing transmittance, reflectance, and total source functions

Longwave

lw_2stream_coeffs calculates transmittance ($T$), reflectance ($R$), and source functions ($S^+$ and $S^-$) from optical properties for longwave radiation. Transmittance and reflectance are calculated following [2]:

\[\begin{align} T = \frac{2 k e^{-k \tau}}{k + \gamma_1 + (k - \gamma_1) e^{-2 k \tau}} \\ R = \frac{\gamma_2 (1 - e^{-2 k \tau})}{k + \gamma_1 + (k - \gamma_1) e^{-2 k \tau}} \end{align}\]

where $\tau$ is the optical depth, and $k = \sqrt{\gamma_1^2 - \gamma_2^2}$. $\gamma_1$ and $\gamma_2$ are coupling coefficients in the two-stream approximation that describe how isotropic the phase function is, and are determined by the optical properties (single scattering albedo and asymmetry parameter).

$S^+$ and $S^-$ are calculated from optical properties and Planck source functions following [3].

Shortwave

sw_2stream_coeffs calculates diffuse transmittance ($T$), diffuse reflectance ($R$), direct transmittance ($T_{dir}$), and direct reflectance ($R_{dir}$) from optical properties for shortwave radiation. $T$ and $R$ are calculated following the same equations as for the longwave radiation. $T_{dir}$ and $R_{dir}$ is calculated following [2]:

\[\begin{align} T_{dir} = - A [(1 + k \mu) (\alpha_1 + k \gamma_4) e^{-\tau/\mu} - (1 - k \mu) (\alpha_1 - k \gamma_4) e^{-2k\tau} e^{-\tau/\mu} - 2k (\gamma_4 + \alpha_1 \mu) e^{-\tau/\mu}] \\ R_{dir} = A [(1 - k \mu) (\alpha_2 + k \gamma_3) - (1 + k \mu) (\alpha_2 - k \gamma_3) e^{-2 k \tau} - 2 (k \gamma_3 - \alpha_2 k \mu) e^{-k \tau} e^{-\tau/\mu}] \end{align}\]

where $A = \omega_0 / (1 - k^2 \mu^2) / [k (1 + e^{-2 k \tau}) + \gamma_1 (1 - e^{-2 k \tau})]$. $\omega_0$ is the single scattering albedo. $\mu$ is the cosine of solar zenith angle. $\gamma_3$, and $\gamma_4$ are coefficients in the two-stream approximation and determined by the optical properties. They are constrained to $\gamma_3 + \gamma_4 = 1$ by energy conservation.

The direct downward flux ($F^{-}_{dir}$) and source functions ($S^+$ and $S^-$) for each level are calculated from $T_{dir}$ and $R_{dir}$:

\[\begin{align} F^-_{dir, i} = e^{-\tau/\mu} F^-_{dir, i+1} \\ S^+_i = R_{dir, i} F^-_{dir, i+1} \\ S^-_i = T_{dir, i} F^-_{dir, i+1} \end{align}\]