Cloud Parcel Modelling – Part 2: Water Vapor Budget and Droplet Growth

 

🌧️ Part 2 – Water Vapor Budget and Droplet Growth

Title: How Supersaturation and Aerosols Control Cloud Condensation Rate

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📉 Equation of Condensation

The rate at which water vapor condenses into droplets is given by:

$$\frac{dw_c}{dt} = \frac{4\pi \rho_w}{\rho_a} \sum_{i=1}^{n} N_i r_i^2 \frac{dr_i}{dt}$$

Where:

• $\rho_w$: density of liquid water

• $\rho_a$: air density

• $N_i$: number concentration of droplets of radius $r_i$

• $\frac{dr_i}{dt}$: growth rate of each droplet

This expression links individual droplet growth to the total condensation rate in the parcel.

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☁️ Droplet Growth from Supersaturation

Each droplet grows as water vapor diffuses onto its surface:

$$\frac{dr_i}{dt} = \frac{G}{r_i}(S - S_{eq})$$

Where:

• $S$: ambient supersaturation

• $S_{eq}$: equilibrium saturation (depends on droplet curvature and solute)

• $G$: combined thermal-diffusional resistance factor

The coefficient $G$ is:

$$G = \left( \frac{1}{\frac{\rho_w R T}{p^* D_v M_w} + \frac{L \rho_w}{k_a T} \left(\frac{L M_w}{R T} - 1 \right)} \right)$$

This term incorporates heat and mass transfer limitations, governing how fast droplets can grow under given atmospheric conditions.

Symbol	Description
ρw	Density of liquid water (≈ 1000 kg/m³)
R	Universal gas constant (≈ 8.314 J/mol·K)
T	Temperature (K)
p⁎	Saturation vapor pressure (Pa)
Dv	Diffusivity of water vapor in air (m²/s)
Mw	Molar mass of water (≈ 0.018 kg/mol)
Lw	Latent heat of vaporization (≈ 2.5 × 10⁶ J/kg)
ka	Thermal conductivity of air (W/m·K)

This equation balances mass diffusion resistance (left term) and thermal resistance (right term), both of which slow down droplet growth. A higher G means faster droplet growth due to easier vapor transport and heat removal.

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📈 Supersaturation Budget Equation

To track how supersaturation evolves in the cloud parcel, the following equation is used:

$$\frac{dS}{dt} = \frac{1}{w_v^*} \left[ \frac{dw_v}{dt} - (S + 1) \left( \frac{\partial w_v^*}{\partial T} \frac{dT}{dt} + \frac{\partial w_v^*}{\partial p_a} \rho_a g V \right) \right]$$

This balances:

• Supply of water vapor: $\frac{dw_v}{dt}$

• Demand for condensation: due to cooling (via $\frac{dT}{dt}$) and pressure drop (via ascent)

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🔗 Link to Aerosol Activation

• If many droplets activate, total surface area increases → $\frac{dw_c}{dt}$ increases

• This leads to faster latent heat release, influencing parcel temperature and supersaturation evolution

• Aerosol type matters:

  • Salt → easy activation

  • Black carbon → poor activation

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✅ Key Insight

Cloud condensation is not automatic. It’s controlled by a dynamic interplay between:

• Supersaturation

• Droplet number and size

• Aerosol chemical properties

This interaction ultimately governs how fast clouds form, grow, and reflect sunlight.

Referrence(s) or reading(s):

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

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