fix(P3): use per-regime material density in the ice aspect-ratio closure

[haakon-e]

Summary

The oblate aspect ratio ϕ = c/a = 3√π·m/(4·ρ·A^{3/2}) was evaluated with the size-dependent effective density ρ = m/V_sphere(D) instead of the particle's material density. With the effective density the particle mass cancels, ϕ comes out prolate (>1) for every nonspherical size, and the min(1, ϕ) clamp then pinned ϕ ≡ 1 everywhere — disabling the cbrt(ϕ) aspect-ratio fall-speed correction entirely. Use the per-regime material density and drop the clamp.

What changed (src/P3_particle_properties.jl, docs, tests)

• New ϕ_material_density(state, D): returns ρ_i in the small-spherical, dense-nonspherical, and partially-rimed regimes, and ρ_g in the graupel regime (D_gr ≤ D < D_cr).
• ϕᵢ now divides by ϕ_material_density(state, D) and no longer clamps; the docstring keeps the one-line formula and links to the docs.
• ice_density docstring: removed the stale "Needed for aspect ratio calculation, so we assume zero liquid fraction" clause (ϕᵢ no longer calls it).

Mechanism / why

ϕ = 3√π·m/(4·ρ·A^{3/2}) compares the particle's mass-to-area shape factor against the density of the solid material it is made of. Substituting the effective density m/V_sphere(D) replaces ρ with a quantity that itself carries the mass, cancelling m and collapsing the closure to a near-constant ≈ 1; the min(1, ⋅) clamp then masked the residual >1 excursions, so ϕ ≡ 1 and cbrt(ϕ) ≡ 1. Within each P3 mass regime the material density is constant, so ϕ tracks m/A^{3/2}: spherical in the spherical regimes (ϕ = 1), oblate (ϕ < 1) in the dense-nonspherical regime.

Residual ϕ > 1 band just above D_th (why no clamp)

ϕ slightly exceeds 1 (peak ≈ 1.2) in a narrow band immediately above D_th. The projected area is discontinuous at D_th (it drops from the spherical law (π/4)D² to the nonspherical law γ·D^σ, ≈ 0.88× smaller) while the mass stays continuous, so m/A^{3/2} is inflated just past the threshold. cbrt(ϕ) stays finite and bounded there (cbrt(1.2) ≈ 1.06), so a clamp is unnecessary and would re-introduce the ϕ ≡ 1 pathology in that band. Reconciling the mass and area laws at D_th (so the area is continuous too) is a separate P3 area power-law item, deliberately left as a follow-on.

Validation

• Effect: large unrimed ice fall speeds drop by up to ~4× — the intended oblate correction. Mass-weighted velocities drop most (mass weighting emphasises the large nonspherical sizes where ϕ is smallest).

Caveats / follow-on

• The residual ϕ > 1 band above D_th remains (follow-on: reconcile the area law).

[haakon-e]

This change is part of the following stack:

• Make the pointwise 2M+P3 tendencies ForwardDiff-differentiable #725
  • feat(Rosenbrock): unified RosenbrockAverage substepping framework + 2M+P3 exact-AD mode #726
    • feat(Rosenbrock): 1M donor / coupled / exact modes on the unified framework #731
      • feat(Rosenbrock): post-solve per-process verbose diagnostics #732
        • fix(Rosenbrock): saturation adjustment on max(S_ice, S_liq) #737
          • fix(P3): use per-regime material density in the ice aspect-ratio closure #740 ◀
            • feat(P3): Gauss-Legendre quadrature, integrand breakpoints -> GL(6) default #741
              • perf(BMT): recover the primal tendency from the Rosenbrock Jacobian pass #742
                • feat(BMT): analytic ManualJacobian for the 2M+P3 Rosenbrock substep #743
                  • feat(BMT): coupled Tier-3 donor linearization for the 2M+P3 ManualJacobian #744

_{Change managed by git-spice.}

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 93.27%. Comparing base (090fd49) to head (c8b0dba).

Additional details and impacted files

@@                   Coverage Diff                   @@
##           he/rosenbrock-modes     #740      +/-   ##
=======================================================
+ Coverage                93.21%   93.27%   +0.05%     
=======================================================
  Files                       57       57              
  Lines                     3185     3197      +12     
=======================================================
+ Hits                      2969     2982      +13     
+ Misses                     216      215       -1     

Components	Coverage Δ
src	94.00% <100.00%> (+0.05%)	⬆️
ext	69.47% <ø> (ø)

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[haakon-e]

Folded into #741