Priciples of Active and Passive Remote Measurements ...
109
The scattering cross section of a molecule is
sc 83m2-1
(f>. = 7r 3N2>..4
(5.44)
where m is the refractive index of air and N the density of molecules. Typically, the resulting
optical depth is J>. = JA ~ 0.01>..-4 for P = Po = 1013 hPa, else J>. = E..J~.
Po
5.4.2 Scattering by spherical particles
Whereas gas molecules have very few interactions between each others, interactions of radiation
with water and aerosols particles can not be described at the molecular scale, since multiple
interactions between molecules can not be described simply. These particles are thus considered
as a continuous medium characterized by its complex refractive index, and the Maxwell theory
which describes the transfer of an electromagnetic wave through a surface is used. The Mie
oscat 5
4
3
2
o
m = 1,33 - O,Oi
m= 1,50 - O,Oi
r---~----'-----'----.----, 5
o
10
20
30
40
Figure 5.7: Mie scattering efficiency.
4
3
2
o
50
theory develops the electromagnetic wave in spherical harmonics to solve exactly Maxwell
equations in the case of spherical surfaces. This is approximately the case for water droplets
and most aerosols, where observations agree with Mie theory. The Mie scattering efficiency for
one particle is the non dimensional quantity QSC( m, r / >..) such as
(f~C(r) = 7rr2Qsc(m,r/>")
(5.45)
Similarly, we define the absorption and extinction efficiencies. In these expressions, r is the
particle radius and m its complex refractive index. Figure 5.7 shows the variation of QSc as a
function of the so-called Mie parameter x = 47r(m - 1) r / >.. for two values of m. Note that QSc
essentially depends on (m - l)r/>.. and tends to 2 for large particles, according to Babinet's
principle (Goody and Young, 1989, p301).
Aerosol and cloud particles occur randomly with different sizes, and consequently, the scattering
events are incoherent. If n( r) is the particle size distribution, the mean optical coefficients for
many particles are
sc fooo(f~C(r)n(r)dr
(f>. = fo oo n(r)dr
( 5.46)