Cloud Parcel Modelling – Part 1: Temperature Change and Equation (a. basic)

Equation:

$$\frac{dT}{dt}=-\frac{gV}{c_p}-\frac{L}{c_p}\frac{d{w}_v}{d\mathrm{t}}$$

does describe the temperature change of the rising air parcel as it rises adiabatically (i.e., without external heat exchange, but with condensation possible), and it is a core part of the cloud parcel model. 

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✅ What the Equation Represents:

This is the thermodynamic energy equation for a rising parcel of moist air. It quantifies how the temperature $T$ of the parcel evolves with time $t$ due to two main physical processes:

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🔹 1. Adiabatic Expansion Cooling:

$$-\frac{gV}{c_{\mathrm{p}}}$$

• As the parcel rises with velocity $V$, it moves into lower pressure.

• It expands and does work on its surroundings.

• This expansion uses internal energy, lowering the parcel's temperature.

• This term is always negative, causing cooling.

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🔹 2. Latent Heating (or Cooling):

$$-\frac{L}{c_p}\frac{d{w}_v}{d\mathrm{t<span\; class="vlist-s"></span>}}$$

• If water vapor condenses ($dw_v/dt < 0$), latent heat is released: warming the parcel.

• If evaporation occurs ($dw_v/dt > 0$), latent heat is absorbed: cooling the parcel.

• The sign of $\frac{dw_v}{dt}$ determines if this term is positive (heating) or negative (cooling).

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🧠 Summary:

The Equation (1) is the temperature evolution equation in the cloud parcel model.
It expresses how temperature changes over time due to:

1. Adiabatic expansion (cooling),

2. Latent heat effects (heating or cooling depending on condensation/evaporation).

This equation is coupled with other equations in the model, such as:

• Vertical motion,

• Water vapor budget,

• Droplet activation and growth,

• Supersaturation evolution.

So it’s not the entire cloud parcel model, but it is a central thermodynamic component of it.

ADDITIONAL or CLARIFYING THINKING......

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🔍 1. Why Not Use an Absolute Value (+ (L/cp) * |dwv/dt|) in the Latent Heating Term?

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You’re exactly right in terms of the physical meaning — but the reason it’s not written with an absolute value is because in physics, we usually let the sign of the variables themselves carry the meaning, especially when the sign reflects a physical process (like condensation or evaporation).

✅ Your Interpretation (Correct in Meaning)

Yes, the second term in the temperature equation:

dT/dt = -gV/cp - (L/cp) * (dwv/dt)

...is a positive contribution to dT/dt when water vapor is condensing, i.e., when:

dwv/dt < 0

This represents latent heating from condensation, and it corresponds to:

+ (L/cp) * |dwv/dt|

🤔 So Why Not Write It That Way?

Because:

• Using signs preserves generality:
  If dwv/dt > 0 (evaporation), the parcel is absorbing water vapor, and the process cools the parcel. So, the same equation can model both condensation (heating) and evaporation (cooling) — just by letting dwv/dt take on its proper sign.
• Thermodynamic convention:
  In physics and fluid dynamics, we let variable signs reflect the physical process. If we wrote:

      dT/dt = -gV/cp + (L/cp) * |dwv/dt|
      

  ...we’d lose the ability to model cooling from evaporation without changing the equation structure.
• Physical consistency:
  The minus sign means energy is leaving the vapor phase:
  • dwv/dt < 0 → condensation → releases heat → parcel heats up
  • dwv/dt > 0 → evaporation → absorbs heat → parcel cools down

🧠 Summary Table

Process	dwv/dt	Latent Heat Term	Effect on dT/dt
Condensation	< 0	Positive	Heating
Evaporation	> 0	Negative	Cooling

✅ Conclusion

Your absolute-value version does reflect the idea of latent heating correctly for condensation. But the standard form of the equation is more general and compact. It naturally handles both heating and cooling effects, just by keeping track of the sign of dwv/dt.

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🔍 2. Why Are Both Terms Negative?

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🔹 First Term: $-\frac{gV}{c_p}$

• $g$: acceleration due to gravity (> 0)

• $V$: updraft velocity (> 0 if the parcel is rising)

• $c_p$: specific heat at constant pressure (> 0)

So:

• This whole term is negative.

• It represents adiabatic cooling due to expansion as the parcel rises into lower pressure.

Why negative?
Because when a parcel rises (positive $V$), it expands and cools, so $\frac{dT}{dt} < 0$. This is dry adiabatic cooling.

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🔹 Second Term: $-\frac{L}{c_p} \frac{dw_v}{dt}$

• $L > 0$: latent heat of vaporization/condensation

• $c_p > 0$ again

• $\frac{dw_v}{dt}$: change in water vapor mixing ratio

This term accounts for latent heating or cooling, depending on whether condensation or evaporation occurs.

Why negative?
Because:

• If $\frac{dw_v}{dt} < 0$, meaning condensation (vapor is decreasing), then:

  $$-\frac{L}{c_p}\frac{d{w}_v}{dt}>0$$

  So this term becomes positive, i.e., heating (condensation releases heat).

• If $\frac{dw_v}{dt} > 0$, meaning evaporation, then:

  $$-\frac{L}{c_p}\frac{d{w}_v}{dt}<0$$

  So this term becomes negative, i.e., cooling (evaporation absorbs heat).

So, again — the minus sign is included to let the sign of $\frac{dw_v}{dt}$ determine whether the effect is heating or cooling.

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🧠 Big Picture: The Two Effects

Term	Sign in Formula	Physical Meaning	Sign of Contribution
$-\frac{gV}{c_p}$	Negative	Expansion cooling (always)	Negative
$-\frac{L}{c_p} \frac{dw_v}{dt}$	Negative	Condensation: heating Evaporation: cooling	Can be positive or negative depending on $\frac{dw_v}{dt}$

✅ Final Summary:

• The negative sign in both terms is intentional:

  • The first term is always cooling (expansion during ascent).

  • The second term's sign is determined by the physics:

    • Condensation (vapor decreases) → heating

    • Evaporation (vapor increases) → cooling

Keeping both negative signs in the equation allows the formula to:

1. Reflect that ascent leads to cooling,

2. Let latent heating/cooling be flexible, depending on whether condensation or evaporation is occurring.

This is a general and elegant way to write the temperature evolution.

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🔍 3. What does it mean by higher cp?

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🔍 Equation (1) again:

$$\frac{dT}{dt} = -\frac{gV}{c_p} - \frac{L}{c_p} \frac{dw_v}{dt}$$

Both terms have $\frac{1}{c_p}$, so the magnitude of temperature change depends inversely on $c_p$.

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🔥 What happens when $c_p$ is higher?

• A higher $c_p$ means it takes more heat to change the air temperature by 1 K.

• Therefore, for the same amount of energy (e.g., from adiabatic expansion or condensation):

  • The change in temperature ($dT/dt$) is smaller in magnitude.

  • The parcel cools or heats more slowly.

So:

        A higher $c_p$ makes the air temperature change less for the same energy gain or loss.

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❄️ What happens when $c_p$ is lower?

• The same amount of energy leads to a larger temperature change.

• So the parcel will cool or heat more quickly.

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✅ “A higher $c_p$ means the air temperature doesn’t increase (or decrease) easily for a given energy change. Therefore, the rate of temperature decrease during adiabatic ascent is smaller, not larger. So the air cools more slowly, not faster.”

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📘 Summary:

cp	Effect on dT/dt	Cooling Rate
Higher	Smaller magnitude	Slower
Lower	Larger magnitude	Faster

So, higher c_p stabilizes temperature against rapid changes.

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🔍 4. Water Vapor and Liquid Water Balance

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🔄 In Equation (1), the second term represents latent heating from condensation. Since any loss of water vapor corresponds directly to gain in liquid water content, we use the conservation relation:

$$\frac{d{w}_v}{dt}=-\frac{d{w}_c}{dt}$$

This means that the rate at which vapor decreases equals the rate at which liquid water forms. As condensation proceeds, latent heat is released, warming the parcel and partially offsetting the cooling due to ascent.

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