Cloud Physics 2020-2021: tutorial 4 condensation
CLOUD PHYSICS - tutorial 4
Condensational growth of a cloud droplet
The condensational growth of a cloud droplet by water vapor diffusion is described by:
rdr
dt = S − Seq Fd+ Fk
where
Fd= ρlRvT
Des(T ), Fk=
L RvT − 1
ρlL KT
D is the diffusion coefficient of water vapor in the air. It depends on temperature and pressure: D = 2.11 · 10−5TT
0
1.94p
0
p
m2s−1, where p0 = 101 325 Pa.
K is the thermal conductivity coefficient: K = 4.1868·10−3[5.69+0.017(T −T0)] W m−1K−1.
Seq = 1 OR Seq = r3− r3d
r3 − (1 − κ)r3dexp(A
T) ≈ 1 + A r − B
r3 where A(T ) = 2σ ρlRvT Assume that σ = 75.64 · 10−3N m−1.
1. Show how ξ = 1/(Fd+ Fk) depends on temperature, T , and pressure, p.
2. Calculate the droplet radius growth rate dr/dt as a function of r for different values of saturation S (0.1, 0.3, 0.5%) and Seq expressed either by the κ-K¨ohler relation (assume κ = 0.67, rd = 0.01µm), or assume that Seq = 1.
How does it depend on temperature and pressure?
3. Calculate r(t). How does it depend on saturation, temperature, pressure?
4. Corrections (see A Short Course in Cloud Physics by R.R. Rogers and M.K. Yau).
In condensational growth equation the diffusion and thermal conductivity coefficient have to be replaced by D0 and K0 as shown below.
D0 = D r
r + lβ, lβ = D β
2π RvT
1/2
, β = 0.04
K0 = K r
r + lα, lα = K αp
!(2πRdT )1/2
cv + Rd/2 , α = 1 where lα and lβ are the length scales.
Show how these length scales depend on temperature and pressure.
Show how these corrections influence the condensational growth of cloud droplets.
For that purpose show how ξ/ξcorr (ξ = 1/(Fd+ Fk)) depends on temperature and pressure.
1
