Editing Openai/680a25f8-9f18-8004-a42d-cd36374ba000 (section)

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* Particle: 20 µm diameter spherical ice crystal (radius r=10−5 mr = 10^{-5} \, \text{m}r=10−5m)
* Altitude: 10 km (~250 hPa)
* Ambient Air Temperature: –50 °C (223 K)
* Ice Crystal Density: 917 kg/m³
* Cross-Sectional Area: A=πr2=3.14×10−10 m2A = \pi r^2 = 3.14 \times 10^{-10} \, \text{m}^2A=πr2=3.14×10−10m2
* Surface Area: 4πr2=1.26×10−9 m24\pi r^2 = 1.26 \times 10^{-9} \, \text{m}^24πr2=1.26×10−9m2
* Volume: 43πr3=4.19×10−15 m3\frac{4}{3}\pi r^3 = 4.19 \times 10^{-15} \, \text{m}^334πr3=4.19×10−15m3
* Mass: m0=3.84×10−12 kgm_0 = 3.84 \times 10^{-12} \, \text{kg}m0=3.84×10−12kg

==== - Incident Solar Flux: ~1300 W/m² at 10 km (after atmospheric attenuation) ====
* Effective Solar Absorptivity: 0.25–0.35 for 20 µm ice particles
* Absorbed Solar Power: Psolar≈(400–450) W/m2×3.14×10−10 m2≈1.3×10−7 WP_{\text{solar}} \approx (400\text{–}450) \, \text{W/m}^2 \times 3.14\times10^{-10} \, \text{m}^2 \approx 1.3\times10^{-7} \, \text{W}Psolar≈(400–450)W/m2×3.14×10−10m2≈1.3×10−7W
* Atmospheric Longwave Absorption: ~544 W/m² incoming → absorbed ≈ 1.7×10⁻⁷ W.
* Total Absorbed Power: Pabs≈3×10−7 WP_{\text{abs}} \approx 3 \times 10^{-7} \, \text{W}Pabs≈3×10−7W

==== - Latent Heat of Sublimation: Ls=2.83×106 J/kgL_s = 2.83\times10^6 \, \text{J/kg}Ls=2.83×106J/kg ====
* Diffusion Coefficient at 250 hPa and 223 K: D≈2.5×10−4 m2/sD \approx 2.5\times10^{-4} \, \text{m}^2/\text{s}D≈2.5×10−4m2/s
* Effective Sublimation Cooling Power: Sublimation consumes ~2.56×10⁻⁷ W (~86% of absorbed energy).
* Net Effect: - Particle stabilizes at a lower temperature (~230–233 K) instead of rising to ~243 K. - Cooling is a dominant, first-order process, not a small correction.

==== - Vapor pressure over ice (Murphy & Koop, 2005 / Clausius-Clapeyron): - At 223 K (ambient): 4.02 Pa - At 243 K: ~11.6–12 Pa - At 230.7 K (after cooling): ~7.1 Pa ====
* Required RHi to prevent sublimation: - At 243 K: ~373% - At 230.7 K: ~177%
* Typical Ambient RHi: 140–160% in contrail regions.

✅ Thus, even after sublimation cooling, ambient humidity remains far too low to prevent net sublimation.

==== Using the mass transfer limited rate: ====

dmdt=−4πrDMRT(ei(Tc)−e∞)\frac{dm}{dt} = -4\pi r D \frac{M}{RT} (e_i(T_c) - e_{\infty})dtdm=−4πrDRTM(ei(Tc)−e∞)
where:
* ei(Tc)e_i(T_c)ei(Tc) = vapor pressure at particle temperature
* e∞e_{\infty}e∞ = ambient vapor pressure (RHi × saturation pressure at ambient T)

Results:
* For RHi = 100%: - Sublimation time ≈ 4.25 seconds (static) - ≈ 2.98 seconds (dynamic adjustment of RHi from 177% down to 100% after nucleation)
* For RHi = 160%: - Sublimation time ≈ ~33 seconds (static), but dynamic processes rapidly lower RHi inside contrail plumes.

✅ Typical particle survival time under sunlight: 3–5 seconds, sometimes extending toward ~30 s under high RHi, but only temporarily.

==== - Local vapor depletion inside contrails accelerates sublimation: - Newly nucleated ice crystals pull down RHi toward 100% within ~0.05 seconds inside dense wakes (N ≈ 10⁵/cm³). ====
* Thus, even in humid wakes, sublimation rapidly reasserts itself.

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