Cloud Parcel Modelling - Additional Note (1b): Two forms of simplified dS/dt equations

 Yes, you're absolutely right — the two forms:

$$\frac{dS}{dt} = \alpha w - \beta G \sum r_i \quad \text{and} \quad \frac{dS}{dt} = \alpha \cdot w - \beta \cdot G(S, r)$$

are conceptually the same, but they reflect different levels of explicitness and generality in how supersaturation sinks are treated.

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🔍 Breakdown of the Terms:

Term	Meaning
$\alpha w$	Supersaturation production from adiabatic cooling due to vertical velocity $w$
$\beta G \sum r_i$	Condensation sink, where each droplet of radius $r_i$ grows and removes water vapor
$G$	Growth factor from condensation theory (depends on T, p, S)
$\sum r_i$ or $G(S, r)$	Total growth contribution from droplets, which is a function of both supersaturation and droplet sizes

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✅ So, what’s the difference?

Form	Description	Use Case
$\frac{dS}{dt} = \alpha w - \beta G \sum r_i$	Simplified, where $G$ is treated as a constant or known value and the droplet sizes are known	Conceptual models, constant-parameter parcel models
$\frac{dS}{dt} = \alpha w - \beta G(S, r)$	More general, where G depends on the evolving supersaturation $S(t)$ and radii $r_i(t)$	Full cloud parcel models (like Nenes 2001), numerical integration

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🧠 Intuition:

• In both forms, supersaturation increases due to rising motion ($\alpha w$),

• and decreases due to droplet growth, which consumes vapor and releases heat (represented by $\beta G \sum r_i$ or $\beta G(S, r)$).

But when you write:

$$\frac{dS}{dt} = \alpha w - \beta \cdot G(S, r)$$

…you're acknowledging that this term dynamically depends on supersaturation and droplet evolution — the feedback loop central to Nenes et al. (2001).

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🧪 Bottom Line:

✅ Yes — they are equivalent in structure, but the latter form $\boxed{dS/dt = \alpha w - \beta G(S, r)}$ is more general and needed when doing full parcel simulations where $S(t)$ and $r(t)$ evolve simultaneously.