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appropriate integration limits forµ. From equation 2.12 upwellingE↑ and downwellingE↓

flux-densities can be derived accordingly. These are sometimes called irradiances or slightly incorrect just fluxes. They are important for defining transmission T and reflection R:

T(x, y) = E↓(x, y, z = 0)

E↓(x, y, z =T OA)

;R(x, y) = E↑(x, y, z =T OA)−E↑(x, y, z = 0)

E↓(x, y, z =T OA)

(2.13)

T OAstands for Top Of Atmosphere and represents the upper boundary of the considered domain. Similar to the net flux-density is the definition of the actinic flux:

Fact(r, t) = 2π Z 0 1 Z −1 I(r, µ, ϕ, t)dµdϕ (2.14)

The connection of the net-flux with heating-rates is given by the first law of thermody- namics, see Zdunkowski and Bott (2004):

cp

dT(r, t)

dt =−α(r, t)∇·(J

q_{(r, t) +E}

N(r, t)) +αJ(r, t)· ·∇v(r, t) (2.15) All terms except −α∇·EN refer to non-radiative heat sources and can therefore be ne- glected in this context. One derives:

∂T(r, t) ∂t rad =− 1 ρ(r, t)cp ∇·EN(r, t) (2.16)

The different components of the net-flux can be interpreted as fluxes across the coordinate layers. For example Ez is the flux density balance with respect to the light crossing the surface with z =const. in vertical direction. In contrast to that, the actinic flux describes the omnidirectional power across the unit sphere towards the sphere’s center, which can be a molecule for example. Therefore, it is of importance when one is interested in the ra- diative budget with respect to a point. This is the case when calculating photodissociation coefficients in atmospheric chemistry.

2.2

Definition of Optical Properties

The effects of radiation interacting with matter can be divided in three major phenomena: 1.) absorption, 2.) scattering, 3.) emission.

Absorption

The absorption of a photon, meaning the reception of the photon’s energy by a molecule or a particle, is described by the absorption coefficientσa and the mass absorption coefficient

χa. Both are connected by the absorber density ρa via:

A measure for the probability of absorption within the path incrementds is the differential absorption optical depth dτa:

dτa(r, t) = σa(r, t)ds (2.18)

Scattering

The process of scattering changes the direction of propagation and, in case of inelastic scattering, the energy and thus the frequency of the photon. Analogously to the absorption, the differential scattering optical depth is defined as:

dτs(r, t) = σs(r, t)ds (2.19) withσs being the scattering coefficient. Apart from that, the differential scattering optical depth can be defined as:

dτs(r, t) = σs,d(r, ν0 →ν,Ω0 →Ω, t)dsdνdΩ (2.20) where σs,d is the differential scattering coefficient. It describes the change of the direction of propagation from Ω0 to Ω and the shift in frequency from ν0 to ν. Both scattering coefficients are connected:

σs(r, t) = ∞ Z 0 Z 4π σs,d(r, ν0 →ν,Ω0 →Ω, t)dΩdν (2.21)

The differential scattering coefficient can be split up itself:

σs,d(r, ν0 →ν,Ω0 →Ω, t) = 1

4πσs(r, t)p(r, ν

0 _→

ν,Ω0 →Ω, t) (2.22)

The phase function p describes the dependence between the sets of variables (ν0,Ω0) and (ν,Ω). It can be regarded as the probability density distribution for the scattering from the primed to the unprimed state. To this end it is normalized:

1 4π ∞ Z 0 Z 4π p(r, ν0 →ν,Ω0 →Ω, t)dΩdν = 1 (2.23)

In the atmosphere usually the reciprocity of the way the light travels is assumed. That means that the phase function does not depend on the values of Ω0 and Ω in an absolute manner but rather on the angle between those two which is called scattering angle Θ. It is:

2.2 Definition of Optical Properties 15

Extinction

The addition of absorption and scattering coefficient leads to the extinction coefficient:

σt(r, t) = σa(r, t) +σs(r, t) (2.25) It describes the combined attenuation of light propagating in a certain direction Ω and frequency ν by both processes. Therefore, the extinction optical depth becomes:

dτt(r, t) =σt(r, t)ds=dτa(r, t) +dτs(r, t) (2.26) This property is usually referred to as differential optical depth. The relation between scattering and extinction is called single scattering albedo ωo:

ωo(r, t) = σs(r, t)

σt(r, t)

= 1−σa(r, t)

σt(r, t)

(2.27) It gives the fraction of scattering processes in relation to all interactions. A medium with

ωo = 1 is referred to as conservative medium.

Emission

Emission is the release of photons from a source within the medium. The source can be either of thermal or artificial nature. The source function J is given as:

J(r,Ω, t) = hνj (2.28)

where j is the emission coefficient, describing the number of emitted photons per volume, time, solid angle interval, and frequency interval. The source function gives the released power per volume, solid angle interval, and frequency interval.

Integrated optical depth

The optical depth not only serves as optical property but also as coordinate, see Figure 2.4. The above introduced extinction differential optical depth can be written as:

dτ(r, t) =σt(r, t)ds

The component ofdτ(r, t) along the z-axis is referred to as normal optical depthdτ⊥(r, t):

dτ⊥(r, t) =σt(r, t)ds⊥

If we assume solar illumination and the atmosphere is considered to be plane-parallel, the optical properties are then independent of the lateral coordinates, one can connect the optical depth with the angleθ∗_o:

dτ(z, t) = 1

µ∗

o

If the normal component ds⊥ is chosen to be oriented in the opposite direction than z,

which is natural as θo[π₂, π],ds⊥=−dz, it follows that:

dτ(z, t) =−σt(z, t)

µ∗

o

dz

As cosθ∗_o =−cosθo, one derives for the integrated optical depth from TOA to position p:

τ(z, t) = p Z 0 σt(s, t)ds= 1 µ∗ o pz Z 0 σt(s, t)ds⊥= 1 µo z Z T OA σt(z0, t)dz0 (2.29) = 1 |µo| T OA Z z σt(z0, t)dz0 (2.30)

It was assumed that the origin of the coordinate s (s = 0) is located at TOA. For the last equality in equation 2.29 to hold it was used that |µo| = µ∗o. The optical depth which is finally deduced is the slant optical depth along the solar beam in an horizontally homogeneous atmosphere. However, optical depth is supposed to refer to the normal optical depth from hereafter:

τ(z, t) = z Z

T OA

σt(z0, t)dz0 (2.31)

The total optical depth is then derived by integrating from TOA to the lower boundary

z = 0.

Asymmetry factor

The last property, the asymmetry factor g is the first moment of the phase function. It is defined as: g(r, t) = 1 4π 2π Z 0 1 Z −1

cos Θp(r, ν0 →ν,cos Θ, t)d(cos Θ)dφ (2.32)

= 1 2

1

Z

−1

cos Θp(r, ν0 →ν,cos Θ, t)d(cos Θ) (2.33)

where Θ is the scattering angle as in equation 2.24, and the phase function was assumed to be independent of the azimuthal direction φ.

The asymmetry factor is a measure of the shape of the phase function. If g > 0 most of the light is scattered into the forward hemisphere, if g < 0 backscattering dominates. In the atmosphere the former is usually the case. Isotropic and pure molecular (Rayleigh) scattering are characterized by g = 0, see below.

2.3 Optical Properties of the Atmosphere 17