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 Incoming links: Dew-point depression

Let

e_{v}

e_{v} / p

is constant. Its dew point

T_{D}

is defined by

E (T_{D}) = e_{v}

, where

E

\frac{L _{v}}{R _{v} T _{D}^{2}} \frac{d T _{D}}{d z} = \frac{d l n e _{v}}{d z} = \frac{d l n p}{d z} ≃ - \frac{g}{R _{d} T} .

The last approximation uses hydrostatic balance, the ideal gas law, and an environmental temperature close to the parcel's. Rearrangement yields

\frac{d l n T _{D}}{d z} = - \frac{g R _{v}}{L _{v} R _{d}} \frac{T _{D}}{T} .

For a modest initial dew-point depression,

T_{D} / T ≃ 1

in absolute temperature, so

T_{D} (z) ≃ T_{D, s} e^{- z / H}, H = \frac{L _{v} R _{d}}{g R _{v}} .

With

L_{v} ≃ 2.5 \times 1 0^{6} J k g^{- 1}

R_{d} ≃ 287 J k g^{- 1} K^{- 1}

, and

R_{v} ≃ 461.5 J k g^{- 1} K^{- 1}

H ≃ 1.58 \times 1 0^{5} m

. This dew point scale is much longer than the atmospheric pressure scale height.

The exponential function is an approximation, not the exact consequence of a dry adiabatic process. Retaining

T = T_{s} - Γ_{d} z

gives

p / p_{s} = (T / T_{s})^{c_{p} / R_{d}}

and the useful comparison

\frac{1}{T _{D} ( z )} = \frac{1}{T _{D, s}} - \frac{R _{v}}{L _{v}} \frac{c _{p}}{R _{d}} ln \frac{T ( z )}{T _{s}} .

It also shows explicitly that expansion lowers the dew point even before any water vapor condenses.

6 K

, so

z_{LCL} ≃ 125 \times 6 m = 750 m .

Keeping

T_{D, s} = 276.15 K

H ≃ 158.5 km

and

c_{p} = 1004 J k g^{- 1} K^{- 1}

gives approximately

747 m

, consistent with the rounded estimate.

The real cloud base can differ because entrainment changes the parcel's atmospheric mixing ratio and temperature, surface humidity varies, and the environmental pressure profile need not match the assumed parcel profile. Finite cloud-droplet activation requires some atmospheric supersaturation, so the first visible cloud droplets need not occur exactly at the thermodynamic lifting condensation level. A parcel must also actually be lifted that far; environmental static stability can prevent this. The constant-latent heat and first-order Taylor expansion approximations produce smaller numerical differences.