Discrete angle radiative transfer: 2. Renormalization approach for homogeneous and fractal clouds

The discrete angle radiative transfer systems discussed in part 1 readily lend themselves to approximation schemes in which simple scaling systems with known radiative transfer properties can be doubled in size yielding analytic expressions relating the transfer coefficients corresponding to the initial and doubled scale. This “real space renormalization” method can be viewed as a generalization of conventional invariant imbedding techniques to scaling systems. Analytic nonlinear doubling mappings are obtained for homogeneous square, cubic and triangular systems, as well as for a simple fractal system with both open and cyclic horizontal boundary conditions. The doubling mappings have both thick and thin cloud fixed points; to which the transmission and albedoes are respectively algebraically attracted and repelled, with universal (phase function independent) exponents we estimate analytically. The method is approximate since it systematically neglects small‐scale intensity gradients; however, the results are qualitatively correct, arid it therefore establishes the connection between the scaling of the cloud optical density field and the scaling of the corresponding transfer coefficients. We also discuss the limitations of the method; in part 3 we compare it with a numerical approach.