1.2. Shortwave Radiation¶

Incoming solar radiation \(R_{\mathrm{sw}}^\downarrow =\) swdown_Wm2 [W m-2; Driver: downward shortwave radiation at the surface] is partitioned among vegetation, ground, and the atmosphere. ADELM separates shortwave radiation into visible / photosynthetically active radiation (PAR) and near-infrared (NIR), and solves a two-stream radiative-transfer problem that accounts for direct beam transmission, diffuse scattering, and canopy-ground multiple interactions (Fig. 1.2.1). The main objective is to diagnose the net absorbed shortwave radiation by vegetation and by ground. These net fluxes satisfy the shortwave conservation relation:

(1.2.1)¶\[\sum_{\Lambda} \left( R_{\mathrm{sw},\Lambda}^{\mu} + R_{\mathrm{sw},\Lambda}^{\mathrm{d}} \right) = \left(R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v}} + R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g}}\right) + \sum_{\Lambda} \left( R_{\mathrm{sw},\Lambda}^{\mu}\,\alpha^{\mu}_{\Lambda} + R_{\mathrm{sw},\Lambda}^{\mathrm{d}}\,\alpha^{\mathrm{d}}_{\Lambda} \right)\]

Here, \(R_{\mathrm{sw},\Lambda}^{\mu}\) and \(R_{\mathrm{sw},\Lambda}^{\mathrm{d}}\) denote the incident direct-beam and diffuse shortwave radiation in spectral band \(\Lambda\), respectively. \(R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v}}\) denotes net shortwave radiation absorbed by vegetation, \(R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g}}\) denotes net shortwave radiation absorbed by ground, and \(\alpha^{\mu}_{\Lambda}\) and \(\alpha^{\mathrm{d}}_{\Lambda}\) denote the effective direct-beam and diffuse albedo of the vegetation-ground system for spectral band \(\Lambda\).

../../_images/radiation.svg — Fig. 1.2.1 Schematic diagram of shortwave (red; direct beam shown as solid lines and diffuse radiation as dashed lines) and longwave (blue) radiation absorbed, transmitted, and reflected by vegetation and the ground.¶

1. Partition of incoming solar radiation¶

The incident shortwave flux is split into visible and near-infrared bands:

(1.2.2)¶\[R_{\mathrm{sw},\mathrm{vis}}^\downarrow = f_{\mathrm{vis}}\,R_{\mathrm{sw}}^\downarrow, \qquad R_{\mathrm{sw},\mathrm{nir}}^\downarrow = (1-f_{\mathrm{vis}})\,R_{\mathrm{sw}}^\downarrow\]

where \(f_{\mathrm{vis}}\) is vis_fraction_of_shortwave [default: 0.5; Fixed parameter: fraction of incoming shortwave radiation in the visible (400-700 nm) band]. \(R_{\mathrm{sw},\mathrm{vis}}^\downarrow\) is the incident PAR at the canopy top and is diagnosed as incoming_par_Wm2 [W m-2; Flux: incoming photosynthetically active radiation at the canopy top].

Within each band \(\Lambda\), the flux is split into direct-beam and diffuse components:

(1.2.3)¶\[R_{\mathrm{sw},\Lambda}^{\mu} = f_{\mathrm{dir}}\,R_{\mathrm{sw},\Lambda}^\downarrow, \qquad R_{\mathrm{sw},\Lambda}^{\mathrm{d}} = (1-f_{\mathrm{dir}})\,R_{\mathrm{sw},\Lambda}^\downarrow\]

where \(f_{\mathrm{dir}}\) is direct_beam_fraction [default: 0.5; Fixed parameter: fraction of incoming shortwave radiation that is direct beam].

Throughout this page, \(\mathrm{LAI}\) denotes the one-sided leaf area index (lai [m2 m-2; Driver: leaf area index]), and

(1.2.4)¶\[\mathrm{VAI} = \mathrm{LAI} + \mathrm{SAI}\]

denotes the total vegetation area index (vegetation_area_index [m2 m-2; Diagnostic: total vegetation area index: one-sided leaf area index plus stem area index]), where \(\mathrm{SAI}\) is the one-sided stem area index (stem_area_index [m2 m-2; PFT-based parameter: one-sided stem area index]).

2. Two-stream radiative transfer¶

Governing equations¶

The canopy is treated as a plane-parallel medium with cumulative vegetation area index \(\tau\) measured downward from the canopy top (\(\tau = 0\)) to the ground (\(\tau = \mathrm{VAI}\)). The two-stream approximation gives coupled ODEs for the upward and downward diffuse fluxes \(I^\uparrow\) and \(I^\downarrow\) (normalized to unit incoming radiation):

(1.2.5)¶\[\begin{split}\frac{dI^\uparrow}{d\tau} &= b\,I^\uparrow - c_1\,I^\downarrow - S^\uparrow \\ \frac{dI^\downarrow}{d\tau} &= -b\,I^\downarrow + c_1\,I^\uparrow + S^\downarrow\end{split}\]

The ODE coefficients are:

(1.2.6)¶\[\begin{split}b &= 1 - \omega_\Lambda\,(1 - \beta_\Lambda^{\mathrm{d}}) \\ c_1 &= \omega_\Lambda\,\beta_\Lambda^{\mathrm{d}} \\ S^\uparrow &= \bar{\mu}\,K_{\mathrm{dir}}\,\omega_\Lambda\,\beta_\Lambda^\mu \\ S^\downarrow &= \bar{\mu}\,K_{\mathrm{dir}}\,\omega_\Lambda\,(1 - \beta_\Lambda^\mu)\end{split}\]

\(b\) is the loss rate of each stream (absorption plus backscatter into the opposite direction). \(c_1\) is the forward-scatter coupling that transfers energy from one stream to the other. \(S^\uparrow\) and \(S^\downarrow\) are the direct-beam source terms: intercepted direct radiation rescattered upward and downward, respectively, with \(\bar{\mu}\,K_{\mathrm{dir}}\) being the direct-beam optical depth per unit \(\tau\) scaled by the mean diffuse path length.

The canopy optical coefficients and geometry used here are defined in Optical Properties.

Symbol	ADELM variables
\(\omega_{\mathrm{vis}}\)	`canopy_single_scattering_albedo_vis` [Diagnostic: single-scattering albedo of the canopy in the visible band]
\(\omega_{\mathrm{nir}}\)	`canopy_single_scattering_albedo_nir` [Diagnostic: single-scattering albedo of the canopy in the near-infrared band]
\(\beta_{\mathrm{vis}}^\mu\)	`canopy_direct_beam_upscatter_parameter_vis` [Diagnostic: upscatter parameter for direct beam radiation in the visible band]
\(\beta_{\mathrm{nir}}^\mu\)	`canopy_direct_beam_upscatter_parameter_nir` [Diagnostic: upscatter parameter for direct beam radiation in the near-infrared band]
\(\beta_{\mathrm{vis}}^{\mathrm{d}}\)	`canopy_diffuse_upscatter_parameter_vis` [Diagnostic: upscatter parameter for diffuse radiation in the visible band]
\(\beta_{\mathrm{nir}}^{\mathrm{d}}\)	`canopy_diffuse_upscatter_parameter_nir` [Diagnostic: upscatter parameter for diffuse radiation in the near-infrared band]
\(K_{\mathrm{dir}}\)	`canopy_direct_beam_extinction_coefficient` [Diagnostic: optical depth of the direct beam per unit leaf and stem area index]
\(\bar{\mu}\)	`canopy_diffuse_optical_depth_scale` [Diagnostic: average inverse optical depth for diffuse radiation]

Analytical solution¶

The general solution is the sum of the homogeneous solution and a particular solution driven by the direct-beam sources. The homogeneous system (\(S^\uparrow = S^\downarrow = 0\)) has exponential solutions \(e^{\pm h\tau}\) with eigenvalue:

(1.2.7)¶\[h = \frac{\sqrt{b^2 - c_1^2}}{\bar{\mu}}\]

\(h > 0\) always holds since \(b > c_1\) by construction. The two homogeneous modes represent diffuse radiation propagating downward (\(e^{+h\tau}\)) and upward (\(e^{-h\tau}\)) through the canopy. Define also \(p_1 = b + \bar{\mu}h\) and \(p_2 = b - \bar{\mu}h\), which appear in the boundary-value solution below.

The particular solution for the direct-beam forcing decays as \(e^{-K_{\mathrm{dir}}\tau}\). Its amplitude involves the denominator:

(1.2.8)¶\[\sigma = (\bar{\mu}\,K_{\mathrm{dir}})^2 - (b^2 - c_1^2)\]

which measures the separation between the direct-beam decay rate and the diffuse eigenvalue. \(\sigma\) vanishes in the degenerate case where the two rates coincide; the solver avoids this numerically.

Two boundary conditions close the system:

Top (\(\tau = 0\)): \(I^\uparrow(0) = 0\), no upward diffuse flux enters from above.
Bottom (\(\tau = \mathrm{VAI}\)): \(I^\uparrow(\mathrm{VAI}) = \alpha_{\mathrm{g},\Lambda}\,I^\downarrow(\mathrm{VAI})\), the ground reflects the downward diffuse flux upward with albedo \(\alpha_{\mathrm{g},\Lambda}\).

Symbol	ADELM variables
\(\alpha_{\mathrm{g},\mathrm{vis}}^\mu\)	`ground_direct_beam_albedo_vis` [Diagnostic: effective ground albedo for direct beam radiation in the visible band, accounting for snow cover]
\(\alpha_{\mathrm{g},\mathrm{nir}}^\mu\)	`ground_direct_beam_albedo_nir` [Diagnostic: effective ground albedo for direct beam radiation in the near-infrared band, accounting for snow cover]
\(\alpha_{\mathrm{g},\mathrm{vis}}^{\mathrm{d}}\)	`ground_diffuse_albedo_vis` [Diagnostic: effective ground albedo for diffuse radiation in the visible band, accounting for snow cover]
\(\alpha_{\mathrm{g},\mathrm{nir}}^{\mathrm{d}}\)	`ground_diffuse_albedo_nir` [Diagnostic: effective ground albedo for diffuse radiation in the near-infrared band, accounting for snow cover]

Enforcing these two conditions on the general solution gives a 2×2 linear system for the homogeneous-mode amplitudes, which is solved analytically. Two attenuation factors at the full canopy depth \(\mathrm{VAI}\) appear in the solution:

(1.2.9)¶\[s_1 = e^{-h\,\mathrm{VAI}}, \qquad s_2 = e^{-K_{\mathrm{dir}}\,\mathrm{VAI}}\]

\(s_2\) (s2) is the Beer-Lambert direct-beam transmittance through the full canopy. \(s_1\) (s1) is the attenuation of the diffuse eigenmode over the same depth.

Direct-beam solution¶

The direct-beam forcing adds a particular solution that decays as \(e^{-K_{\mathrm{dir}}\tau}\). The particular-solution amplitudes at \(\tau = 0\) are:

(1.2.10)¶\[\begin{split}h_1 &= -S^\uparrow\,(b - \bar{\mu}\,K_{\mathrm{dir}}) - c_1\,S^\downarrow \\ h_4 &= -S^\downarrow\,(b + \bar{\mu}\,K_{\mathrm{dir}}) - c_1\,S^\uparrow\end{split}\]

\(h_1\) contributes to canopy reflectance; \(h_4\) contributes to ground irradiance.

Enforcing the boundary conditions on the full solution (homogeneous plus particular) yields a 2×2 linear system. Define the ground boundary terms:

(1.2.11)¶\[u_1 = b - \frac{c_1}{\alpha_{\mathrm{g},\Lambda}}, \qquad u_2 = b - c_1\,\alpha_{\mathrm{g},\Lambda}, \qquad u_3 = S^\downarrow + c_1\,\alpha_{\mathrm{g},\Lambda}\]

The system determinants are:

(1.2.12)¶\[\begin{split}d_1 &= \frac{p_1\,(u_1 - \bar{\mu}h)}{s_1} - p_2\,(u_1 + \bar{\mu}h)\,s_1 \\ d_2 &= \frac{u_2 + \bar{\mu}h}{s_1} - (u_2 - \bar{\mu}h)\,s_1\end{split}\]

The homogeneous amplitudes for \(I^\uparrow\) (\(h_2, h_3\)) and \(I^\downarrow\) (\(h_5, h_6\)) are:

(1.2.13)¶\[\begin{split}h_2 &= \frac{1}{d_1}\left[ \frac{u_1 - \bar{\mu}h}{s_1} \left(S^\uparrow - \frac{h_1(b + \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right) - p_2\left(S^\uparrow - c_1 - \frac{h_1(u_1 + \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right)s_2 \right] \\ h_3 &= \frac{-1}{d_1}\left[ (u_1 + \bar{\mu}h)\,s_1 \left(S^\uparrow - \frac{h_1(b + \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right) - p_1\left(S^\uparrow - c_1 - \frac{h_1(u_1 + \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right)s_2 \right] \\ h_5 &= \frac{-1}{d_2}\left[ \frac{h_4(u_2 + \bar{\mu}h)}{\sigma\,s_1} + \left(u_3 - \frac{h_4(u_2 - \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right)s_2 \right] \\ h_6 &= \frac{1}{d_2}\left[ \frac{h_4(u_2 - \bar{\mu}h)\,s_1}{\sigma} + \left(u_3 - \frac{h_4(u_2 - \bar{\mu}K_{\mathrm{dir}})}{\sigma}\right)s_2 \right]\end{split}\]

The output fractions per unit incident direct radiation are:

(1.2.14)¶\[\begin{split}f_{tdd} &= s_2 \\ f_{tid} &= \frac{h_4\,s_2}{\sigma} + h_5\,s_1 + \frac{h_6}{s_1} \\ \alpha^{\mu}_{\Lambda} &= \frac{h_1}{\sigma} + h_2 + h_3\end{split}\]

\(f_{tdd}\) (ftdd) is the unscattered direct-beam transmittance through the full canopy. \(f_{tid}\) (ftid) is the downward diffuse flux at the ground generated by scattering of the direct beam within the canopy. \(\alpha^{\mu}_{\Lambda}\) (albd) is the combined canopy-and-ground albedo for direct radiation.

Vegetation absorption \(\bar{I}_\Lambda^\mu\) (fabd) follows from energy conservation:

(1.2.15)¶\[\bar{I}_\Lambda^\mu = 1 - \alpha^{\mu}_{\Lambda} - (1-\alpha_{\mathrm{g},\Lambda})\,f_{tdd} - (1-\alpha_{\mathrm{g},\Lambda})\,f_{tid}\]

Diffuse solution¶

For pure diffuse forcing there is no particular solution; the general solution consists of the two homogeneous modes only. The same boundary-value structure applies with the direct-beam sources set to zero. Using the same \(u_1, u_2, d_1, d_2\) (now with \(\alpha_{\mathrm{g},\Lambda}\) for the diffuse ground albedo), the integration constants are:

(1.2.16)¶\[\begin{split}h_7 &= \frac{c_1\,(u_1 - \bar{\mu}h)}{d_1\,s_1}, \qquad h_8 = \frac{-c_1\,(u_1 + \bar{\mu}h)\,s_1}{d_1} \\ h_9 &= \frac{u_2 + \bar{\mu}h}{d_2\,s_1}, \qquad h_{10} = \frac{-(u_2 - \bar{\mu}h)\,s_1}{d_2}\end{split}\]

\(h_7, h_8\) are the homogeneous amplitudes for \(I^\uparrow\); \(h_9, h_{10}\) for \(I^\downarrow\). The output fractions per unit incident diffuse radiation are:

(1.2.17)¶\[\begin{split}f_{tii} &= h_9\,s_1 + \frac{h_{10}}{s_1} \\ \alpha^{\mathrm{d}}_{\Lambda} &= h_7 + h_8\end{split}\]

\(f_{tii}\) (ftii) is the diffuse transmittance to the ground surface. \(\alpha^{\mathrm{d}}_{\Lambda}\) (albi) is the canopy-and-ground albedo for diffuse radiation.

Vegetation absorption \(\bar{I}_\Lambda^{\mathrm{d}}\) (fabi) follows from energy conservation:

(1.2.18)¶\[\bar{I}_\Lambda^{\mathrm{d}} = 1 - \alpha^{\mathrm{d}}_{\Lambda} - (1-\alpha_{\mathrm{g},\Lambda})\,f_{tii}\]

3. Absorbed shortwave fluxes¶

For each band \(\Lambda\), vegetation absorbs contributions from both the direct beam and the diffuse stream:

(1.2.19)¶\[R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v},\Lambda} = R_{\mathrm{sw},\Lambda}^{\mu}\,\bar{I}_{\Lambda}^{\mu} + R_{\mathrm{sw},\Lambda}^{\mathrm{d}}\,\bar{I}_{\Lambda}^{\mathrm{d}}\]

Summing over the two spectral bands gives canopy_net_swrad_Wm2 [W m-2; Flux: shortwave radiative flux absorbed by the canopy]:

(1.2.20)¶\[R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v}} = R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v},\mathrm{vis}} + R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v},\mathrm{nir}}\]

The leaf APAR \(\mathrm{APAR}_{\mathrm{leaf}}\) (leaf_apar_Wm2 [W m-2; Flux: photosynthetically active radiation absorbed by leaf surfaces]) scales the canopy APAR by the leaf fraction of VAI:

(1.2.21)¶\[\mathrm{APAR}_{\mathrm{leaf}} = R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v},\mathrm{vis}} \times \frac{\mathrm{LAI}}{\mathrm{VAI}}\]

The ground receives three downward flux components at \(\tau = \mathrm{VAI}\): the unscattered direct beam (\(f_{tdd}\)), diffuse generated by scattering of the direct beam within the canopy (\(f_{tid}\)), and transmitted incident diffuse (\(f_{tii}\)). Each is attenuated by the ground albedo:

(1.2.22)¶\[R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g},\Lambda} = \left( R_{\mathrm{sw},\Lambda}^{\mu}\,f_{tdd} + R_{\mathrm{sw},\Lambda}^{\mu}\,f_{tid} + R_{\mathrm{sw},\Lambda}^{\mathrm{d}}\,f_{tii} \right) (1-\alpha_{\mathrm{g},\Lambda})\]

Summing over the two spectral bands gives soil_net_swrad_Wm2 [W m-2; Flux: shortwave radiative flux absorbed by the soil surface (visible plus near-infrared)]:

(1.2.23)¶\[R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g}} = R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g},\mathrm{vis}} + R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g},\mathrm{nir}}\]

The total shortwave absorbed by the land surface, net_swrad_Wm2 [W m-2; Flux: net shortwave radiative flux absorbed by the land surface], is:

(1.2.24)¶\[R^{\mathrm{n}}_{\mathrm{sw}} = R^{\mathrm{n}}_{\mathrm{sw},\mathrm{v}} + R^{\mathrm{n}}_{\mathrm{sw},\mathrm{g}}\]