The definition of equivalent potential temperature assumes that all condensation

Question

The definition of equivalent potential temperature assumes that all condensation results in the appearance of liquid water, regardless of the temperature. If freezing occurs, additional latent heat is released, which further raises the temperature of the parcel. Assume that a parcel is at saturation at a temperature of freezing and a pressure of 900 hPa.

a) What is the equivalent potential temperature e according to the traditional definition: e = exp(Lw/(cpTLcL)) where cp = heat capacity of dry air at a constant pressure (1005 J/kgK)

b) By how much would e increase if all condensate were assumed to freeze (answer 1.5 K)

Explanation / Answer

A)

e = exp(Lw/(cpTLcL))

Now, = T(Po/P)^(R/Cp)

T = Freezing temperature = 0 C = 273 K

Po = 1000 hpa

P = 900 hpa

R/Cp = 0.286 for air

therefore = 273(1000/900)^(0.286) = 281.35 K

now, it is at saturation level therefore, Lw = 0

therefore, e = exp(0) = = 281.35 K

B)

The value of rvs at 0 C and 900 mb is about 4.3 g kg-1 = 0.0043 kg kg-1

L = latent heat of freezing = 3.34*10^5 J/kg

dq = -Ld(rvs) = cp dT

integrating we get,

change in T = -L * rvs / cp

substituting value in above equation we get,

change in T = 3.34*10^5*0.0043 / 1005 = 1.429 K

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