Inverse-Square Singularity: 1/r^2
Three hot sources orbit over an absorbing plane. Intensity spikes as 1/r^2 near each one; the r^2-weighted view of the same field stays bounded.
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The inverse-square law governs radiation intensity, gravitational force, and electrostatic fields. As distance to a point source approaches zero, these quantities diverge as 1/r^2, creating numerical challenges for simulations.
The Physics
Inverse-Square Law
Radiation intensity from a point source follows:
I(r) = P / (4 * pi * r^2)This applies to: light from stars, gravitational fields, electric fields, sound intensity, and any phenomenon spreading spherically.
Why It Matters
Climate models must compute radiative transfer through atmospheres. Near point sources (stars, heating elements), the 1/r^2 singularity requires extremely fine resolution or special handling.
Intensity vs Weighted View
Intensity (k=0)
I = P / (4*pi*r^2) * exp(-kappa*r)- - Diverges as 1/r^2 toward a point source
- - Held at its surface value inside each finite source
- - Heats the temperature field
Weighted view (k=-2)
sum_s I_s * r_s^2 = sum_s P_s exp(-kappa*r_s) / (4*pi)- - Each source term weighted by r_s^2
- - Bounded by sum_s P_s / (4 pi)
- - A readout of the same field; nothing else changes
Note: an earlier version multiplied the intensity itself by r (k=-1) and reported an "improvement". That changes the field (it is no longer inverse-square), so it is not a numerical method for the same physics. Weighting by r^2 makes the readout bounded; the intensity still diverges.
Benchmark Context
The climt_radiation row of the local CASCADE suite is a synthetic toy, not a CliMT run: one 2-D test function with a singular point, relabelled per domain, scored by how close the optimizer got to that point. Every 1/r^2 row in that file shares the same value. It is a local benchmark, not citable, and this browser demo does not reproduce it.
Mathematical Note
The project's diagnostic weight is r^(-k). For a 1/r^2 term, k=-2 gives (1/r^2) * r^2 = 1, so the weighted readout is bounded; k=-1 leaves 1/r, still singular.
On r > 0 the same fact reads D_BG[r^-2] = e^-2, a constant log-slope. Neither statement removes the singularity of the physical intensity.
Math checked against local simulation utilities. View more demos | All tools