Radiative transfer

The RTE solvers integrate the radiative transfer equation through a plane-parallel, horizontally homogeneous column and return the upward and downward radiative fluxes and the heating rate. This page states the equations they solve and how they solve them; the optical properties that enter as coefficients are described under Optics.

The radiative transfer equation

Radiation is described by the spectral radiance $I_\lambda$, the power per unit area, solid angle, and wavelength flowing in a given direction $\Omega$. Along a ray, the medium changes the radiance by three processes: extinction (absorption plus scattering out of the beam), thermal emission, and scattering into the beam from all other directions. In terms of the extinction optical depth $\tau_\lambda$ measured along the ray, the radiative transfer equation reads

\[\frac{dI_\lambda(\Omega)}{d\tau_\lambda} = (1 - \omega_{0,\lambda})\, B_\lambda(T) - I_\lambda(\Omega) + \omega_{0,\lambda} \int_{4\pi} P_\lambda(\Omega', \Omega)\, I_\lambda(\Omega')\, d\Omega',\]

where $B_\lambda(T)$ is the Planck function at the local temperature, $\omega_{0,\lambda}$ is the single-scattering albedo — the fraction of the extinction due to scattering — and $P_\lambda(\Omega', \Omega)$ is the phase function, the probability of scattering from direction $\Omega'$ into $\Omega$. The emission term $B_\lambda$ follows from Kirchhoff's law under local thermodynamic equilibrium (absorptivity = emissivity at every wavelength), which is assumed to hold throughout the atmosphere RRTMGP models. The scattering integral couples all directions, which is what makes the general equation expensive; in the atmosphere, it matters for clouds and aerosols in both the longwave and shortwave and for Rayleigh scattering in the shortwave. The two-stream approximation below reduces the angular coupling to two hemispheric fluxes.

For longwave transfer in clear skies, scattering is negligible ($\omega_{0,\lambda} \to 0$), and the equation reduces to the Schwarzschild equation,

\[\frac{dI_\lambda}{d\tau_\lambda} = B_\lambda(T) - I_\lambda,\]

with $\tau_\lambda$ now the absorption optical depth.

Two-stream approximation

Resolving the full angular dependence of $I_\lambda$ is too costly for a climate model, so the flux calculation collapses the radiation field into an upward flux $F_\lambda^\uparrow$ and a downward flux $F_\lambda^\downarrow$. In the plane-parallel column, the vertical optical depth $\widehat{\tau}_\lambda$ increases downward from zero at the top of the atmosphere. Neglecting scattering, the two streams obey

\[\frac{dF_\lambda^\uparrow}{d\widehat{\tau}_\lambda} = D\,(F_\lambda^\uparrow - \pi B_\lambda), \qquad \frac{dF_\lambda^\downarrow}{d\widehat{\tau}_\lambda} = D\,(\pi B_\lambda - F_\lambda^\downarrow),\]

where the diffusivity factor $D$ accounts for the slantwise paths of a diffuse field: replacing the range of zenith angles by a single effective secant $D$ closes the hemispheric integral that relates flux to radiance. With scattering, the streams couple through a layer reflectance, and RRTMGP forms the two-stream transmittance and reflectance from the optical thickness, single scattering albedo, and asymmetry parameter following Meador and Weaver [12].

Angular discretization

The longwave no-scattering solver does better than a single diffusivity angle: it integrates the Schwarzschild equation along a small set of discrete zenith angles and sums the results with Gauss quadrature weights, so the hemispheric flux is $F_\lambda = \sum_i w_i\, I_\lambda(\mu_i)$. RRTMGP uses the Gauss-Jacobi-5 nodes of Hogan [13] with one to four angles (AngularDiscretization). The default single angle has secant $D \approx 1.64$, close to Elsasser's classic diffusivity factor of $1.66$, which the two-stream longwave solver adopts following Fu et al. [14]; adding angles improves the accuracy of the angular integral.

Radiative heating rate

The fluxes feed back on temperature through the divergence of the net upward flux $F^{\mathrm{net}} = F^\uparrow - F^\downarrow$. A layer warms where the net flux converges and cools where it diverges:

\[\rho\, c_p \frac{\partial T}{\partial t} = -\frac{dF^{\mathrm{net}}}{dz}.\]

Because the temperature tendency scales as $1/\rho$, a given flux divergence warms thin, high-altitude air far more than dense air near the surface — one reason ozone's shortwave absorption heats the stratosphere so effectively. heating_rate returns $\partial T/\partial t$ in K/s.

How the equations are solved

The continuous equations are solved by discretizing the column into layers of uniform optical properties, within which they admit exact solutions. The resulting sweeps run independently for every spectral quadrature point and column, which is what makes the solvers parallelize well.

The no-scattering longwave solver integrates the Schwarzschild equation along each quadrature angle exactly: crossing a layer of vertical optical depth $\widehat{\tau}$ multiplies the radiance by the slant-path transmittance $t = e^{-D\widehat{\tau}}$ and adds the layer's own emission,

\[I_{\mathrm{out}} = t\, I_{\mathrm{in}} + S,\]

where the source $S$ follows from a Planck function that varies linearly in optical depth across the layer (Clough et al. [15]). One downward and one upward sweep of this update per quadrature angle yields the fluxes.

In the two-stream solver, the coupled equations for $F^\uparrow$ and $F^\downarrow$ within a layer have exponential solutions $e^{\pm k\tau}$, which reduce each layer $i$ to a diffuse reflectance $R_i$, a transmittance $T_i$, and upward and downward source terms $S_i^{\pm}$ (Meador and Weaver [12]; Toon et al. [16] for the thermal source). The layers are then coupled by the adding method of Shonk and Hogan [17]. With $\alpha_i$ the albedo of the atmosphere–surface system below level $i$ and $G_i$ its upward emission, a first pass climbs from the surface,

\[\begin{aligned} \alpha_{i+1} &= R_i + T_i^2 \beta_i \alpha_i,\\ G_{i+1} &= S_i^{+} + T_i \beta_i \left(G_i + \alpha_i S_i^{-}\right), \end{aligned}\]

where $\beta_i = (1 - \alpha_i R_i)^{-1}$ sums the infinite series of reflections between layer $i$ and the medium below it. A second pass descends from the top-of-atmosphere boundary condition, computing the downwelling flux

\[F_i^{\downarrow} = \beta_i \left(T_i F_{i+1}^{\downarrow} + R_i G_i + S_i^{-}\right) + F_{i,\mathrm{dir}}^{\downarrow},\]

where the last term is the direct solar beam, attenuated by Beer's law, and from it the upwelling flux $F_i^{\uparrow} = \alpha_i F_i^{\downarrow} + G_i$. The result is the discrete counterpart of the integral solution of the transfer equation: the flux at each level accumulates every layer's emission, transmitted and reflected through the layers between the emitting layer and that level.

No-scattering and two-stream solvers

RRTMGP provides two solver families, distinguished by whether they represent scattering:

• The no-scattering solver carries the optical thickness $\tau$ alone (OneScalar optics). It integrates absorption and emission over the Gauss angles above and is used for clear-sky longwave transfer, where scattering is negligible; in the shortwave, it reduces to Beer's-law extinction of the direct solar beam.
• The two-stream solver carries the optical thickness, single scattering albedo, and asymmetry parameter (TwoStream optics) and represents multiple scattering. It is required wherever scattering is not negligible: clouds and aerosols in both bands, and Rayleigh scattering in the shortwave.

Both families exist for the longwave and the shortwave (NoScatLWRTE / TwoStreamLWRTE and NoScatSWRTE / TwoStreamSWRTE), and solve_lw! / solve_sw! dispatch on the workspace type.

Boundary conditions

The sweep needs conditions at both ends of the column. At the surface, the longwave upwelling radiance is the surface Planck emission scaled by the surface emissivity, and the shortwave reflection is set by the direct and diffuse albedos; the incident solar flux and the cosine of the solar zenith angle enter at the top. Any prescribed incident diffuse flux at the top defaults to zero, as does the downwelling longwave flux at the top.

RRTMGP can also insert an isothermal boundary layer: an extra layer between the host model's top and the minimum pressure of the gas-optics tables, held at the model-top temperature, that represents the radiative effect of the atmosphere above the model lid. It is enabled through RRTMGPGridParams, and every getter masks it off so the host reads and writes arrays sized to its own grid (see The getter contract).