Calculation of Cumulus Base Height

Calculation of Cumulus Base Height

This answer to an inquiry is a simple illustration of efficiency using MKS units for calculation. Conversion from metric to English units was unNecessary until finishing the calculation. It also illustrates customization of a general analysis for specific situations. 
> Can anyone confirm this for me:
> 
> Cloud height (ft.) = 227 x N
> 
> N = ground temp (degrees F.) - dew point(degrees F.)
> --------------
> Also any other misc. about dew point is welcome.

The derivation in Atmospheric Thermodynamics shows that for dry adiabatic ascent :

dTd = (Cp Td2)/(.62197 Lv) dT/T

T : temp
Td : dew point
Cp : specific heat @ constant pressure
Lv : vaporization latent heat

Thus, the equation you cite depends on initial dew point. The authors choose T ~ Td ~ 273 K, but cumulus convection is most relevant for T ~ 300 K, Td ~ 289 K, so their approximation can be slightly improved. Using finite differences (MKS units) :

dTd = ((1006.3)(289)2)/((.62197)(2466400)(300)) dT

dTd = .18263 dT

® (Tds - Tdo) = .18263 (Ts - To)

'o' referRing to values at ground, and 's' to values at saturation point (LCL).

For these conditions, the dry adiabatic lapse rate is ~ 9.66 °K/km = .00966 °K/m , so

dT = - .00966 dZ

Z : height above ground

® (Ts - To) = -.00966 (Zs - Zo)

from ground to the saturation point (LCL).

Because Ts = Tds, you can calculate :

Zs - Zo = (To - Tdo)/((.00966)(1 - .18263)) = 126.6 (To - Tdo).

Using English units,

Zs - Zo = (126.6)(3.2808 ft/m)(Td - To)(5/9 °F/°C) = 230.8 (Td - To)

Confirmed , so long as using it you are aware of factors such as mixing ascent beginning from various altitudes, etc., which cause real cloud base to slightly differ from this (probably slightly higher, perhaps

Zs - Zo = 250 (Td - To)

Note that because of Td2 in the original equation, increasing Td increases cloud base estimation for a specific dew point depression.


Text is copyright of Joseph Bartlo, though may be used with proper crediting.

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