Cloud Parcel Modelling - additional notes (1a): full form equation of dS/dt

 Yes — the simplified form of 

$\frac{dS}{dt} = \alpha w - \beta G(S, r)$  is indeed a reduction or approximation of the more complete form derived by Nenes et al. (2001). Let's clarify how they relate historically and mathematically:

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🧪 1. The Full Form Comes First

Nenes et al. (2001) derived the full, thermodynamically consistent form of the supersaturation evolution equation:

$$\frac{dS}{dt} = \frac{1}{w^*_v} \cdot \frac{dw_v}{dt} - (S+1) \left( \frac{\partial w^*_v}{\partial T} \cdot \frac{dT}{dt} + \frac{\partial w^*_v}{\partial p_a} \cdot \frac{dp_a}{dt} \right)$$

This equation:

• Comes from differentiating the definition of $S = \frac{w_v}{w^*_v} - 1$
  

• Tracks both physical vapor content and the thermodynamic state

• Is exact under the assumptions of parcel theory, ideal gas law, and known thermodynamics

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🔄 2. The Simplified Form Is a Parameterization

The simplified equation:

S​=αw−βG(S,r)

is a recast version built to:

• Represent the same physics in a numerically simple way

• Separate the source (cooling by ascent) and the sink (droplet growth), where:

• $\alpha w$ represents supersaturation production by updraft velocity $w$

• $\beta G(S,r)$represents supersaturation depletion due to droplet growth (condensation sink)

• Be more usable in numerical parcel models, where the main variables are $S(t)$, $r(t)$, and $w(t)$

This compact expression  is derived by assuming:

1. Supersaturation production is driven mainly by updraft-induced cooling, i.e., temperature decrease due to vertical motion.
2. Vapor depletion is dominated by condensation onto droplets, neglecting other sinks.
3. Thermodynamic influences such as temperature and pressure changes are encapsulated in the constants α and β, assuming:
   • a dry adiabatic ascent before condensation becomes significant, and
   • a quasi-constant lapse rate.
4. Droplet growth is coupled to supersaturation via the microphysical law dr/dt = G(S, r)/r, forming a feedback loop between vapor and droplet size.

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📘 Summary of Relation:

Full Equation (Nenes 2001)	Simplified Version
Thermodynamically exact	Reduced, model-friendly
Derived from $S=w_v\mathrm{/}{w}_v^{\ast }-1$	Encapsulates key terms into $\alpha$, $\beta$, and $G(S,r)$
Contains temperature & pressure derivatives	Assumes standard lapse rate and constant parameters
Suitable for rigorous theory	Suitable for fast cloud parcel simulations

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✅ So Yes:

The simplified form was developed after and based on the full Nenes (2001) formulation, to make cloud parcel models computationally efficient without losing the essence of supersaturation dynamics.

📚 Interpretation & Relationship

• The full Nenes form is more physically accurate and accounts for vertical motion, cooling, and pressure effects.

• The simplified form is widely used for parcel models (e.g., cloud microphysics parameterizations) because it’s easier to couple to droplet growth.

G(S, r) in this case is shorthand for the integrated growth rate of all droplets, which in physical terms is derived from the term:

$$\sum \frac{d{r}_i}{dt}{r}_{\mathrm{i}}$$

in the microphysical sense, which comes from water mass conservation across all droplets.