Cloud Parcel Modeling – Part 3: Supersaturation Evolution, Droplet Size Distribution, and and the Core Understanding in Cloud Parcel Modeling

☁️ Cloud Parcel Modeling – Part 3: Droplet Size Distribution and Supersaturation Evolution

📌 Core Idea:

Once cloud droplets activate, the system evolves in a coupled way — supersaturation changes, droplets grow, and their size distribution shapes the cloud’s microphysical and radiative properties.

1. 🌡️ Supersaturation (S) is Dynamic, Not Static

As an air parcel ascends:

  • It cools → water vapor condenses → latent heat is released
  • Droplets grow by condensation → reduces ambient water vapor
  • This causes a feedback: Supersaturation increases, peaks, and then declines

Equation:

 dS/dt = α ⋅ w - β ⋅ G(S, r)

Where:

  • α ~ adiabatic cooling rate (from updraft w)
  • β ⋅ G ~ condensation sink (depends on total surface area of droplets)

✅Thinking: What does dS/dt and G(S,r) relationship infer?

The relationship infer that current values of S(t)S(t) and r(t)r(t) affect the droplet growth rate dr/dtdr/dt, which is encapsulated in G(S,r)G(S, r). Then this growth determines how much supersaturation drops, which gives you the new dS/dtdS/dt**.

This is the core of cloud parcel modeling. More about the S, r, and dr/dt loop, look the last or bottom part of this page.

From Nenes et al. (2001):
Supersaturation rarely stays constant. There is a competition between generation (cooling) and depletion (condensation).

Also, note: the above dS/dt equation is a simplified form of the original dS/dt equation. Check here!

2. ☔️ Droplet Size Distribution (DSD)

After activation, each droplet grows based on available vapor. But not all droplets grow equally:

  • Early-activated CCN grow larger
  • Late-activated CCN grow less (due to vapor depletion)

Shape of DSD:

  • Initially narrow
  • Broadens over time due to CCN variety and activation time differences

3. ⛘ Kinetic Limitations on Activation

Nenes et al. (2001) introduced the concept of competition effect:

  • In polluted clouds, many CCN → strong condensation sink
  • → Supersaturation peak suppressed
  • → Not all CCN activate (even if their Sc < peak S)

This effect depends on:

  • CCN number
  • Updraft speed
  • CCN properties (size, κ, solubility)

Key term: “Kappa-Kinetic Effect”

4. 🧠 Implications for Cloud Albedo

More droplets → higher albedo. But due to kinetic limitations:

  • More CCN ≠ more activated droplets
  • Cloud parcel models help predict actual Nd, not just potential CCN

5. 🔍 Visual Summary:

Updraft ↑ → Cooling ↑ → S ↑ → CCN activate → Droplet growth →
Condensation → S drops → Feedback stabilizes → DSD established

💡 Takeaway Summary Table:

Aspect Physical Driver Modeled in Parcel?
Supersaturation Peak Adiabatic cooling & CCN growth ✅
Activation CCN properties + S(t) ✅
Droplet Growth Vapor availability ✅
Kinetic Limitation Vapor competition ✅ (if included)
Hygroscopic Swelling RH < 100% ❌ (not standard)

ADDITIONAL NOTES - THE CORE OF CLOUD PARCEL MODELING......

🔁 Feedback Loop Between S, r, and dr/dt

Let’s follow the chain:

1. Supersaturation S(t)S(t) grows due to updraft cooling:

dSdt=αw−condensation sink\frac{dS}{dt} = \alpha w - \text{condensation sink}

Initially, the condensation sink is small (no droplets yet), so S increases.

2. Once S>ScS > S_c (critical supersaturation), droplets start to activate and grow:

drdt=G⋅Sr\frac{dr}{dt} = \frac{G \cdot S}{r}

  • Small rr: fast growth

  • Large SS: fast growth

  • So growth rate depends directly on both SS and rr

Thus, the growth term drdt\frac{dr}{dt} is nonlinear.

3. Droplet growth reduces supersaturation:

dSdt=αw−β∑(ri⋅dridt)

This means:

dSdt=αw−β∑(ri⋅G⋅Sri)=αw−βGSN(if all ri equal)\frac{dS}{dt} = \alpha w - \beta \sum \left( r_i \cdot \frac{G \cdot S}{r_i} \right) = \alpha w - \beta G S N \quad \text{(if all } r_i \text{ equal)}

N= Number concentration of cloud droplets (e.g., droplets per cm³ or m³)

Now SS starts to decline because droplets are growing and consuming vapor.

🧠 So the Full Feedback Loop Is:

  1. Updraft (w) → ↑ Cooling → ↑ Supersaturation SS

  2. ↑ SS → ↑ Droplet growth rate dr/dt∝S/rdr/dt \propto S/r

  3. ↑ Growth → ↑ Condensation of vapor → ↓ Water vapor in air

  4. ↓ Water vapor → ↓ SS (via the sink term in dS/dtdS/dt)

This self-limiting loop results in:

  • A peak in supersaturation (maximum SmaxS_{\text{max}})

  • A set of droplets activated at their respective critical ScS_c

  • Evolution of droplet size distribution


Referrence(s) or reading(s):

1. Nenes, A., Ghan, S., Abdul-Razzak, H., Chuang, P. Y., & Seinfeld, J. H. (2001). Kinetic limitations on cloud droplet formation and impact on cloud albedo. Journal of Geophysical Research: Atmospheres, 106(D6), 7629–7639. https://doi.org/10.1029/2000JD900091

2. McFiggans, G., Artaxo, P., Baltensperger, U., Coe, H., Facchini, M. C., Feingold, G., Fuzzi, S., Gysel, M., Laaksonen, A., Lohmann, U., Mentel, T. F., Murphy, D. M., O’Dowd, C. D., Snider, J. R., & Weingartner, E. (2006). The effect of physical and chemical aerosol properties on warm cloud droplet activation. Atmospheric Chemistry and Physics, 6, 2593–2649. https://doi.org/10.5194/acp-6-2593-2006

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