Using delta-M scaling

Delta-M scaling (Wiscombe 1977) makes strongly forward-peaked phase functions tractable with a modest number of streams. It is enabled with the use_delta_m flag and requires one extra Legendre moment per layer. The theory is in Delta-M scaling.

adrt::ADConfig cfg(20, 8);
cfg.use_delta_m = true;        // BEFORE allocate(): provisions 2*M + 1 moments
cfg.allocate();

cfg.setHenyeyGreenstein(0.85); // fills all 2*M + 1 moments, incl. chi_{2M}
for (int l = 0; l < 20; ++l) {
  cfg.delta_tau[l]          = 0.5;
  cfg.single_scat_albedo[l] = 0.99;
}

adrt::RTOutput r = adrt::solve(cfg);

When to use it

• Do use delta-M for aerosol/cloud layers with \(g \gtrsim 0.7\); it can reduce the streams needed for a converged flux by a large factor.

• It has no effect on Rayleigh scattering (\(f = 0\) for \(M \geq 2\)).

• The truncation fraction is \(f = \chi_{2M}\), so a custom phase function must fill the moment at index 2*num_quadrature for scaling to activate.

Consistency of derived quantities

The direct beam and all cumulative optical depths use the scaled optical depth \(\tau^*\). The heating rate / flux divergence, however, uses the unscaled single-scattering albedo, because the absorption optical depth is invariant under the scaling. You therefore get physically correct heating rates with or without delta-M.