# A Novel GPU-Based Acceleration Algorithm for a Longwave Radiative Transfer Model
## 3 Model Description and GPU Overview  
### RRTMG_LW Model  
As a critical process affecting our planet's climate, radiative transfer is the transport of energy by electromagnetic waves through a gas. Atmospheric and Environmental Research (AER) developed the RRTM and RRTMG. Their calculations exhibit an effective accuracy equivalent to that provided by the LBLRTM, but their computational cost is lower. In view of these advantages, the RRTM or RRTMG was adopted as the radiative transfer schemes of many climate models, such as the WRF and GRAPES [32].  
The RRTM uses the correlated k-distribution method to calculate the broad-band radiative fluxes. In this method, the radiative spectrum is first divided into bands. Because of the rapid variation of absorption lines within the bands of gas molecules, the values of the absorption intensities within each band are further binned into a cumulative distribution function of the intensities. This distribution function is then discretized by using \(g\) intervals for integration within each band to obtain the band radiative fluxes, which are further integrated across the bands to obtain the total radiative flux to calculate atmospheric radiative heating or cooling. The \(g\) points are the discretized absorption intensities within each band.  
The RRTM_LW is the RRTM for infrared radiation (longwave, LW); RRTM_SW is the RRTM for solar radiation (shortwave, SW). RRTM_LW has 16 bands and 256 \(g\) points. RRTM_SW has 16 bands and 224 \(g\) points. To speed up the calculations for climate and weather models, the spectral resolutions of RRTM_LW and RRTM_SW are further coarsened for applications in GCMs as RRTMG_LW and RRTMG_SW. RRTMG_LW has 16 bands and 140 \(g\) points. RRTMG_SW has 16 bands and 112 \(g\) points [33].  
Below, we briefly describe the radiation flux and heating/cooling rate for calculating radiative transfer through a planetary atmosphere. The spectrally averaged outgoing radiance from an atmospheric layer is expressed using the following formula:  
\[I_{v}(\mu)=\frac{1}{v_{2}-v_{1}}\int_{v_{1}}^{v_{2}}dv\left\{I_{0}(v)+\int_{T_ {v}}^{1}[B(v,\theta(T_{v}^{\prime}))-I_{0}(v)]dT^{\prime}\right\}. \tag{1}\]  
In this expression, \(\mu\) is the zenith direction cosine; \(v\) is the wavenumber; \(\theta\) is temperature; \(v_{1}\) and \(v_{2}\) are the beginning and ending wavenumbers of the spectral interval, respectively; \(B(v,\theta)\) is the Planck function at \(v\) and \(\theta\); \(I_{0}\) is the radiance incoming to the layer; \(T_{v}\) is the transmittance for the layer optical path; \(T^{\prime}_{v}\) is the transmittance at a point along the optical path in the layer.  
Equation (1) can derive the following Equation (2) under the necessary assumptions. In Equation (2), \(g\) is the fraction of the absorption coefficient; \(Beff(g,T_{g})\) is an effective Planck function for the layer; \(\rho\) is the absorber density in the layer; \(P\) is layer pressure; \(\Delta z\) is the vertical thickness of the layer; \(\varphi\) is the angle of the optical path in the azimuthal direction; \(k(g,P,\theta)\) is the absorption coefficient at \(P\) and \(\theta\).  
\[I_{g}(\mu,\varphi)=\int_{0}^{1}dg\left\{Beff(g,T_{g})+[I_{0}(g)-Beff(g,T_{g})] exp[-k(g,P,\theta)\frac{\rho\Delta z}{cos\varphi}]\right\}. \tag{2}\]  
The monochromatic net flux is  
\[F_{v}=F_{v}^{+}-F_{v}^{-}, \tag{3}\]  
where \(F_{v}^{+}=2\pi\int_{0}^{1}I_{v}(\mu)\mu d\mu\) and \(F_{v}^{-}=2\pi\int_{0}^{-1}I_{v}(\mu)d\mu\).  
The total net flux is obtained by integrating over \(v\)  
\[F_{net}=F_{net}^{+}-F_{net}^{-}. \tag{4}\]  
The radiative heating (or cooling) rate is expressed as  
\[\frac{d\theta}{dt}=-\frac{1}{c_{p}\rho}\frac{dF_{net}}{dz}=\frac{g}{c_{p}} \frac{dF_{net}}{dP}, \tag{5}\]  
where \(c_{p}\) is the specific heat at a constant pressure; \(P\) is pressure; \(g\) is the gravitational acceleration; \(\rho\) is the air density in a given layer [30].