2n-Stream Thermal Emissions from Clouds

We analyze thermal emission of radiation in homogeneous clouds. The clouds have negligible horizontal variation, but the temperature can vary vertically. The radiation at optical depth $\tau$ above the cloud bottom is characterized by intensity values $I(\mu_i,\tau)$ at $2n$ Gauss-Legendre direction cosines $\mu_i$, the roots of the $2n$th Legendre polynomial, $P_{2n}(\mu_i)=0$. Scattering matrices describe the intensity fraction with incoming direction cosine $\mu_{i'}$ that is scattered to intensity with outgoing direction cosine $\mu_{i}$. Green's-function matrices describe radiation generated by infinitesimally thin layers of cloud particulates, thermally emitting at the source optical depth $\tau'$. For thin isothermal clouds of optical thickness $\tau_c\le1$, thermal emission is mainly determined by the single scattering albedo $\tilde \omega$. The emitted intensity is limb brightened and depends little on the scattering phase function. For optically thick isothermal clouds, with $\tau_c \gg 1$, purely absorbing clouds with $\tilde\omega = 0$ become perfect blackbodies and emit isotropic Planck intensity. Thick isothermal clouds with moderate single scattering albedos emit substantially less thermal radiation than black bodies and the emission depends strongly on the anisotropy of the scattering phase function. Clouds with strong forward scattering are ``blackest" and emit the most thermal radiation. There is limb darkening which is most pronounced for strongly forward-scattering phase functions. The limb darkening in thick, isothermal, scattering clouds is not due to temperature gradients, as it is for the Sun. With scattering, nearly vertical radiation produced by thermal emission escapes through the surface more efficiently than nearly horizontal radiation. There is limb darkening even for purely absorbing particulates for clouds with hot interiors.