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.

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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 $S(t)$ and $r(t)$ affect the droplet growth rate $dr/dt$, which is encapsulated in $G(S, r)$. Then this growth determines how much supersaturation drops, which gives you the new $dS/dt$**.

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!

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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

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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”

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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

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5. 🔍 Visual Summary:

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

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💡 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:

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

$$\frac{dS}{dt} = \alpha w - \text{condensation sink}$$

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

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

$$\frac{dr}{dt} = \frac{G \cdot S}{r}$$

• Small $r$: fast growth

• Large $S$: fast growth

• So growth rate depends directly on both $S$ and $r$

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

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3. Droplet growth reduces supersaturation:

$$\frac{dS}{dt}=\alpha w-\beta \sum (r_i\cdot \frac{d{r}_i}{dt})$$

This means:

$$\frac{dS}{dt} = \alpha w - \beta \sum \left( r_i \cdot \frac{G \cdot S}{r_i} \right) = \alpha w - \beta G S N \quad \text{(if all } r_i \text{ equal)}$$

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

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

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🧠 So the Full Feedback Loop Is:

1. Updraft (w) → ↑ Cooling → ↑ Supersaturation $S$
   

2. ↑ $S$ → ↑ Droplet growth rate $dr/dt \propto S/r$
   

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

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

This self-limiting loop results in:

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

• A set of droplets activated at their respective critical $S_c$

• Evolution of droplet size distribution