Inverse-Square Singularity: 1/r^2
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.
The CliMT climate modeling framework encounters this in thermal radiation calculations.
How NNC Regularizes 1/r^2
Classical (k=0)
I = P / (4*pi*r^2)- - Diverges as r^-2 near sources
- - Requires mesh refinement near sources
- - Timestep constraints for stability
NNC (k=-1)
I_nnc = I * r = P / (4*pi*r)- - Regularization factor r^(-k) = r
- - Reduces to 1/r singularity (weaker)
- - 83.6% improvement in approach distance
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Verified Results
83.6%
Closer Approach
from CASCADE 21-sim suite
k=-1
NNC Transform
regularization factor = r
CliMT
Physics Domain
climate/radiation modeling
Real-World Applications
Climate Modeling
Radiative transfer through Earth's atmosphere involves computing absorption and emission at many wavelengths. Point sources and strong gradients benefit from NNC regularization.
Astrophysics
Stellar evolution, accretion disks, and galactic dynamics all involve 1/r^2 gravitational and radiative fields. NNC enables closer approach to dense objects.
Computer Graphics
Point light rendering, global illumination, and ray tracing all use inverse-square falloff. NNC can accelerate lighting calculations near light sources.
Where 1/r^2 Appears
Gravity
F = GMm/r^2Electrostatics
E = kQ/r^2Light Intensity
I = P/(4*pi*r^2)Sound
I = P/(4*pi*r^2)All of these phenomena share the same mathematical singularity structure, and all benefit from the same NNC regularization approach.
Mathematical Foundation
From Grossman & Katz (1972), the NNC regularization factor for a singularity of form r^n is:
regularization_factor = r^(-k)For a 1/r^2 singularity (n=-2), choosing k=-1 gives:
r^(-(-1)) = rMultiplying: (1/r^2) * r = 1/r. The singularity is reduced from r^-2 to r^-1, a significant improvement that allows 83.6% closer approach to sources.