Atmospheric Radiation
Properties of Radiation
Wavelength | \[ \lambda = \frac{c}{\nu} \] | ||||
Maxwell's Equations | \[ \vec{\nabla} \cdot \vec{E} = 0\ \ \ (Gauss's\ Law) \] \[ \vec{\nabla} \cdot \vec{H} = 0\ \ \ (Gauss's\ Law) \] \[ \vec{\nabla} \times \vec{E} = -\frac{\mu}{c} \frac{\partial \vec{H}}{\partial t}\ \ \ (Faraday's\ Law) \] \[ \vec{\nabla} \times \vec{H} = \frac{\varepsilon}{c} \frac{\partial \vec{E}}{\partial t}\ \ \ (Amp\grave{e}re's\ Law) \] | ||||
Speed of Light | \[ c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \] | ||||
Complex Index of Refraction | \[ N = \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}} = \frac{c}{c'} \] \[ N = n_r + i\, n_i \] | ||||
Volume Absorption Coefficient | \[ \beta_a = \frac{4 \pi\, \nu\, n_i}{c} = \frac{4 \pi\, n_i}{\lambda} \] | ||||
Photon Energy | \[ E = h \nu \] | ||||
Broadband Flux | \[ F(\lambda_1 , \lambda_2) = \int_{\lambda_1}^{\lambda_2} F_{\lambda}\, d\lambda \] | ||||
Solid Angle | \[ \omega = \iint \limits_{\phi\ \theta} \sin{\theta}\, d\theta\, d\phi \] | ||||
Intensity | \[ I = \frac{\delta F}{\delta \omega} \] | ||||
Stokes Vector | \[ \vec{I} = \begin{pmatrix} I\\ Q\\ U\\ V \end{pmatrix} \] | ||||
Upward Flux | \[ F^{\uparrow} = \int_{0}^{2\pi} \int_{0}^{\pi/2} I^{\uparrow}(\theta,\phi) \cos{\theta} \sin{\theta}\, d\theta\, d\phi \] | ||||
Downward Flux | \[ F^{\downarrow} = - \int_{0}^{2\pi} \int_{\pi/2}^{\pi} I^{\downarrow}(\theta,\phi) \cos{\theta} \sin{\theta}\, d\theta\, d\phi \] | ||||
Net Flux | \[ F^{net} = F^{\uparrow} - F^{\downarrow} \] | ||||
Insolation | \begin{eqnarray} W &=& S_0 \left( \frac{a}{r} \right)^2 \int_{t_{sunrise}}^{t_{sunset}} \cos{\theta_0 (t)}\, dt \\ &=& S_0 \left( \frac{a}{r} \right)^2 \int_{-H}^{H} \left( \sin{\phi}\sin{\delta} + \cos{\phi}\cos{\delta}\sin{h} \right)\, \frac{dh}{\omega} \\ &\simeq& \frac{S_0}{\pi} \left( \frac{a}{r} \right)^2 \left( H\sin{\phi}\sin{\delta} + \cos{\phi}\cos{\delta}\sin{H} \right) \end{eqnarray} | ||||
Solar Constant | \[ S_0 = 1370 \, Wm^{-2} \] | ||||
Sun-Earth Factor |
| ||||
Solar Declination |
| ||||
Half-Day |
| ||||
Electromagnetic Spectrum
Electromagnetic Waves |
| ||||||||||||||||||||||||
Ultraviolet Bands |
| ||||||||||||||||||||||||
Visible Bands |
| ||||||||||||||||||||||||
Infrared Bands |
| ||||||||||||||||||||||||
Microwave Bands |
| ||||||||||||||||||||||||
Wavenumber | \[ \tilde{\nu} = \frac{1}{\lambda} = \frac{\nu}{c} \] | ||||||||||||||||||||||||
Reflection and Refraction
Transmittance | \[ t(x) = \frac{I(x)}{I_0} = e^{-\beta_a x} \] |
Absorption Depth | \[ D = \frac{1}{\beta_a} = \frac{\lambda}{4 \pi\, n_i} \] |
Dielectric Constant | \[ \epsilon = \frac{\varepsilon}{\varepsilon_0} = N^2 \] |
Snel's Law | \[ \frac{\sin{\Theta_t}}{N_1} = \frac{\sin{\Theta_i}}{N_2} \] |
Fresnel Relations | \[ R_{\parallel} = \left| \frac{\cos{\Theta_t} - m\cos{\Theta_i}}{\cos{\Theta_t} + m\cos{\Theta_i}} \right|^2 \] \[ R_{\perp} = \left| \frac{\cos{\Theta_i} - m\cos{\Theta_t}}{\cos{\Theta_i} + m\cos{\Theta_t}} \right|^2 \] |
Angle of Transmittance | \[ \cos{\Theta_t} = \sqrt{1 - \left( \frac{\sin{\Theta_i}}{m} \right)^2 } \] |
Complex Index of Refraction | \[ m = \frac{N_2}{N_1} \] |
Total Reflectivity | \[ R = R_{\parallel} + R_{\perp} \] |
Brewster Angle | \[ \Theta_B = \sin^{-1}{ \sqrt{ \frac{m^2}{m^2+1} }} \] |
Natural Surfaces
Reflectivity and Absorptivity | \[ a_{\lambda} + r_{\lambda} = 1 \] |
Reflected Flux | \[ F_{\lambda,r} = r_{\lambda}F_{\lambda,0} \] |
Absorbed Flux | \[ F_{\lambda,abs} = (1-r_{\lambda}) F_{\lambda,0} = a_{\lambda} F_{\lambda,0} \] |
Graybody Reflectivity | \[ r = \frac{ F_{r} }{ F_{i} } \] |
Graybody Absorptivity | \[ a = 1 - r \] |
Shortwave Albedo | \[ r_{sw} = 1 - a_{sw} \] |
Reflected Isotropic Intensity | \[ I^{\uparrow} = \frac{F_r}{\pi} = \frac{rF_i}{\pi} = \frac{rS_0 \cos{\theta_i}}{\pi} \] |
Bidirectional Reflection Function (BDRF) | \[ \rho (\theta_i, \phi_i, \theta_r, \phi_r) = \frac{I^{\uparrow}(\theta_r, \phi_r)}{S_0 \cos{\theta_i}} \] |
Thermal Emission
Planck's Function | \[ B_{\lambda}(T) = \frac{2hc^2}{\lambda^5 \left( e^{hc/k_B \lambda T}-1 \right) } \] |
Wien's Displacement Law | \[ \lambda_{max} = \frac{2897}{T} \] |
Stefan-Boltzmann Law | \[ F_{BB}(T) = \sigma T^4 \] |
Rayleigh-Jeans Approximation | \[ B_{\lambda}(T) \simeq \frac{2ck_B}{\lambda^4}T \ \ \ \ \ (\lambda > 1mm) \] |
Monochromatic Emissivity | \[ \varepsilon_{\lambda} = \frac{I_{\lambda}}{B_{\lambda}(T)} \] |
Graybody Emissivity | \[ \varepsilon = \frac{F}{\sigma T^4} \] |
Kirchoff's Law | \[ \varepsilon_{\lambda}(\theta,\phi) = a_{\lambda}(\theta,\phi) \] |
Brightness Temperature | \[ T_B = B_{\lambda}^{-1} \left[ I_{\lambda} \right] = B_{\lambda}^{-1} \left[ \varepsilon B_{\lambda}(T) \right] \] \[ T_B = \varepsilon T \ \ \ \ \ (\lambda > 1mm) \] |
Radiative Cooling | \[ \frac{dT}{dt}_{rad} \simeq \frac{F_{net,\ base} - F_{net,\ top}}{c_p \rho \Delta z} \] |
Atmospheric Transmission
Volume Extinction Coefficient | \[ \beta_e = \beta_a + \beta_s \] | ||||||||||||||||||||||||||||||||||||
Single Scatter Albedo | \[ \tilde{\omega} = \frac{\beta_s}{\beta_e} = \frac{\beta_s}{\beta_s + \beta_a} \] | ||||||||||||||||||||||||||||||||||||
Beer's Law | \[ I_{\lambda}(s_2) = I_{\lambda}(s_1) \exp \left[ - \int_{s_1}^{s_2} \beta_e (s)\, ds \right] = I_{\lambda}(s_1) e^{-\tau\, (s_1,s_2)} = I_{\lambda}(s_1) t\, (s_1,s_2) \] | ||||||||||||||||||||||||||||||||||||
Optical Thickness | \[ \tau\, (s_1,s_2) = \int_{s_1}^{s_2} \beta_e (s)\, ds \] \[ \tau\, (s_1,s_N) = \tau\, (s_1,s_2) + \tau\, (s_2,s_3) + \cdots + \tau\, (s_{N-1},s_N) \] | ||||||||||||||||||||||||||||||||||||
Transmittance | \[ t\, (s_1,s_2) = e^{-\tau\, (s_1,s_2)} \] \[ t\, (s_1,s_N) = t\, (s_1,s_2) \cdot t\, (s_2,s_3) \cdot \cdots \cdot t\, (s_{N-1},s_N) \] | ||||||||||||||||||||||||||||||||||||
Absorptance | \[ a = 1-t \ \ \ \ \ (\tilde{\omega} = 0) \] | ||||||||||||||||||||||||||||||||||||
Volume Coefficients | \[ \beta_e = \rho k_e = N \sigma_e \] \[ \beta_a = \rho k_a = N \sigma_a \] \[ \beta_s = \rho k_s = N \sigma_s \] \[ \left[ \frac{m^2}{m^3} \right] = \left[ \frac{m^2}{kg} \right] \left[ \frac{kg}{m^3} \right] = \left[ \frac{1}{m^3} \right] \left[ \frac{m^2}{1} \right] \] | ||||||||||||||||||||||||||||||||||||
Extinction Cross-Section | \[ \sigma_e = Q_e\, A \] | ||||||||||||||||||||||||||||||||||||
Path Distance | \[ s = \frac{z}{\mu} \] | ||||||||||||||||||||||||||||||||||||
Propagation Direction | \[ \mu = \left| \cos{\theta} \right| \] | ||||||||||||||||||||||||||||||||||||
Optical Thickness | \[ \tau\, (z_1,z_2) = \int_{z_1}^{z_2} \beta_e (z)\, dz \] \[ \tau\, (z_1,z_N) = \tau\, (z_1,z_2) + \tau\, (z_2,z_3) + \cdots + \tau\, (z_{N-1},z_N) \] | ||||||||||||||||||||||||||||||||||||
Transmittance | \[ t\, (z_1,z_2) = e^{-\tau\, (z_1,z_2)/\mu} \] \[ t\, (z_1,z_N) = t\, (z_1,z_2) \cdot t\, (z_2,z_3) \cdot \cdots \cdot t\, (z_{N-1},z_N) \] | ||||||||||||||||||||||||||||||||||||
Absorption Bands |
| ||||||||||||||||||||||||||||||||||||
Mass Path | \[ u(z) = \int_{z}^{\infty} \rho (z')\, dz' \] | ||||||||||||||||||||||||||||||||||||
Weighting Function | \[ W(z) = \frac{dt(z)}{dz} \] | ||||||||||||||||||||||||||||||||||||
Total Transmittance | \[ t = t_{dir} + t_{diff} \] | ||||||||||||||||||||||||||||||||||||
Radiative Conservation | \[ t_{dir} + t_{diff} + r + a = 1 \] | ||||||||||||||||||||||||||||||||||||
Direct Transmittance | \[ t_{dir} = e^{-\tau^* / \mu} \] | ||||||||||||||||||||||||||||||||||||
Liquid Water Path | \[ L = \int_{z_{bot}}^{z_{top}} \rho_w (z)\, dz = \int_{z_{bot}}^{z_{top}} \int_{0}^{\infty} n(r) \left[ \rho_l \frac{4 \pi}{3}r^3 \right] \, dr\, dz \] | ||||||||||||||||||||||||||||||||||||
Total Droplets | \[ N = \int_{0}^{\infty} n(r) \, dr \] | ||||||||||||||||||||||||||||||||||||
Surface Area of Droplets | \[ A_{sfc} = \int_{0}^{\infty} n(r) \left[ 4 \pi r^2 \right] \, dr \] | ||||||||||||||||||||||||||||||||||||
Liquid Water Content | \[ \rho_w = \int_{0}^{\infty} n(r) \left[ \rho_l \frac{4 \pi}{3}r^3 \right] \, dr \] | ||||||||||||||||||||||||||||||||||||
Volume Extinction Coefficient | \[ \beta_e = \int_{0}^{\infty} n(r) \left[ Q_e(r) \pi r^2 \right] \, dr \] | ||||||||||||||||||||||||||||||||||||
Mass Extinction Coefficient | \[ k_e = \frac{\beta_e}{\rho_w} \simeq \frac{3}{2 \rho_l r_{eff}} \] | ||||||||||||||||||||||||||||||||||||
Cloud Optical Thickness | \[ \tau^* \simeq \frac{3L}{2 \rho_l r_{eff}} \] | ||||||||||||||||||||||||||||||||||||
Atmospheric Emission
Schwarzchild's Equation | \[ \frac{dI}{ds} = \beta_a (B-I) \] \[ I(0) = I(\tau')e^{-\tau'} + \int_{0}^{\tau'} Be^{-\tau}\, d\tau \] |
Non-Scattering RTE (Down) | \[ I^{\downarrow }(0) = I^{\downarrow}(\infty) t^* + \int_{0}^{\infty} B(z) W^{\downarrow}(z) \, dz \] \[ W^{\downarrow }(z) = -\frac{dt(0,z)}{dz} = \frac{\beta_a(z)}{\mu}t(0,z) \] |
Non-Scattering RTE (Up) | \[ I^{\uparrow }(\infty) = I^{\uparrow}(0) t^* + \int_{0}^{\infty} B(z) W^{\uparrow}(z) \, dz \] \[ W^{\uparrow }(z) = \frac{dt(z,\infty)}{dz} = \frac{\beta_a(z)}{\mu}t(z,\infty) \] |
Molecular Absorption
Dominant Transitions |
| ||||||||||||
Linear and Rotational Motion |
| ||||||||||||
Rotational Frequency | \[ \nu = \frac{\Delta E}{h} = \frac{h}{4\pi^2 I}(J+1) \] | ||||||||||||
Broadband Fluxes
Equivalent Width | \[ W = \int_{\Delta \tilde{\nu}_i} 1-e^{-\tau_{\tilde{\nu}}} \, d\tilde{\nu} = \mathcal{A} \Delta \tilde{\nu} = (1-\mathcal{T}) \Delta \tilde{\nu} \] | ||
Equivalent Width Limits |
| ||
Radiative Heating Rate | \[ \mathcal{H} = -\frac{1}{\rho(z) c_p} \frac{\partial F^{net}}{\partial z}(z) \] | ||
RTE with Scattering
Scattering Phase Function | \[ \frac{1}{4\pi} \int_{4\pi} p(\hat{\Omega}',\hat{\Omega}) \, d\omega' = 1 \] \[ \frac{1}{2} \int_{-1}^{1} p(\cos{\Theta}) \, d\cos{\Theta} = 1 \] | ||||||
Isotropic Scattering | \[ p(\cos{\Theta}) = 1 \] | ||||||
Scattering RTE | \[ dI = dI_{ext} + dI_{emit} + dI_{scat} \] \[ dI = -\beta_e Ids + \beta_a Bds + \frac{\beta_s}{4\pi} \int_{4\pi} p(\hat{\Omega}',\hat{\Omega}) \, d\omega' \] \[ \frac{dI(\hat{\Omega})}{d\tau} = I(\hat{\Omega}) - \left( 1-\tilde{\omega} \right)B - \frac{\tilde{\omega}}{4\pi} \int_{4\pi} p(\hat{\Omega}',\hat{\Omega}')\, I(\hat{\Omega}) \, d\omega' \] | ||||||
Asymmetry Parameter | \[ g = \frac{1}{4\pi} \int_{4\pi} p(\cos{\Theta}) \cos{\Theta} \, d\omega \] | ||||||
Henyey-Greenstein Phase Function | \[ p_{HG}(\cos{\Theta}) = \frac{1-g^2}{\left( 1+g^2-2g\cos{\Theta} \right)^{3/2}} \] | ||||||
Single-Scattering RTE | \[ \mu \frac{dI(\mu,\phi)}{d\tau} = I(\mu,\phi) - \frac{\tilde{\omega}}{4\pi} \int_{0}^{2\pi} \int_{-1}^{1} p(\mu,\phi;\mu',\phi')\, I(\mu',\phi') \, d\mu' d\phi' \] | ||||||
Full RTE | \begin{eqnarray} \mu \frac{dI(\mu,\phi)}{d\tau} &=& I(\mu,\phi) - (1-\tilde{\omega})B \\ && - \frac{\tilde{\omega}}{4\pi} \int_{0}^{2\pi} \int_{-1}^{1} p(\mu,\phi;\mu',\phi')\, I(\mu',\phi') \, d\mu' d\phi' \\ && -\frac{\tilde{\omega}}{4\pi} \pi F_0 p(\mu,\phi;\mu',\phi')\, I(\mu',\phi') \, d\mu' d\phi' \end{eqnarray} | ||||||
RTE Solution Methods |
| ||||||
Size Parameter | \[ x = \frac{2 \pi r}{\lambda} \] | ||||||
Scattering Types |
|
||||||
Rayleigh Phase Function | \[ p(\Theta) = \frac{3}{4} \left( 1 + \cos^2{\Theta} \right) \] | ||||||
Scattering Cross-Section | \[ \sigma_s \propto \frac{r^6}{\lambda^4}\ \ \ \ (Rayleigh\ Scattering) \] \[ \frac{\sigma_{s,\ blue}}{\sigma_{s,\ red}} \simeq \frac{\left( 650 \right)^4}{\left( 450 \right)^4} \simeq 4 \] | ||||||
Reflectivity Factor | \[ Z = \int_{0}^{\infty} n(D) D^6 \, dD \] \[ Z\ \left[ dBZ \right] = 10\log_{10}(Z) \] | ||||||
Equivalent Reflectivity Factor | \[ Z \simeq 0.20Z\ \ \ \ (Ice\ Particles) \] | ||||||
Marshall-Palmer Size Distribution | \[ n(D_0) = n_0 \exp{(-\Lambda D_0)} \] \[ n_0 = 8000 \] \[ \Lambda = 4.1R^{-0.21} \] | ||||||
Z-R Relationship | \[ Z = 200R^{1.6} \] | ||||||
Depolarization Ratio | \[ \delta = \frac{I_{\perp}}{I_{\parallel}} \] | ||||||
Microwave Radiometer | \[ T_B = \varepsilon T = \left[ 1-e^{-\tau} \right]T \] | ||||||
Total Optical Thickness | \[ \tau \simeq \tau_O + k_LL + k_VV \] \[ \tau \simeq Oxygen + Liquid + Vapor \] | ||||||
Multiple Scattering
Azimuthally Averaged RTE | \[ \mu \frac{dI(\mu,\phi)}{d\tau} = I(\mu,\phi) - \frac{\tilde{\omega}}{2} \int_{-1}^{1} p(\mu,\mu')\, I(\mu') \, d\mu' \] |
Two-Stream Solutions | \[ I^{\uparrow}(\tau) = \frac{r_{\infty}I_0}{e^{\Gamma \tau^*}-r^2_{\infty} e^{-\Gamma \tau^*}} \left[ e^{\Gamma(\tau^*-\tau)} - e^{-\Gamma(\tau^*-\tau)} \right] \] \[ I^{\downarrow}(\tau) = \frac{I_0}{e^{\Gamma \tau^*}-r^2_{\infty} e^{-\Gamma \tau^*}} \left[ e^{\Gamma(\tau^*-\tau)} - r^2_{\infty} e^{-\Gamma(\tau^*-\tau)} \right] \] |
Two-Stream Parameters | \[ \Gamma = 2\sqrt{1-\tilde{\omega}} \sqrt{1-\tilde{\omega}g} \] \[ r_{\infty} = \frac{ \sqrt{1-\tilde{\omega}g} - \sqrt{1-\tilde{\omega}} }{ \sqrt{1-\tilde{\omega}g} + \sqrt{1-\tilde{\omega}} } \] |
Two-Stream Solutions | \[ I^{\uparrow}(\tau) = \frac{I_0(1-g)(\tau^*-\tau)}{1+(1-g)\tau^*} \] \[ I^{\downarrow}(\tau) = \frac{I_0 \left[ 1 + (1-g)(\tau^*-\tau) \right] }{1+(1-g)\tau^*} \] |
Cloud-Top Albedo | \[ r = \frac{I^{\uparrow}(0)}{I^{\downarrow}(0)} \] |
Adding-Doubling Solutions | \[ \tilde{r} = r + \frac{r_{sfc}t^2}{1-r_{sfc}r} \] \[ \tilde{t} = \frac{t}{1-r_{sfc}r} \] |
Adding-Doubling Solutions | \[ \tilde{r} = r_1 + \frac{r_{2}t_1^2}{1-r_{1}r_2} \] \[ \tilde{t} = \frac{t_1 t_2}{1-r_{1}r_2} \] |