Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology

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6.4 Analytical solution of TSMs for a homogeneous layer 175

where the following constants have been introduced

= A

− λ

exp(−λτ

i−1

), β

=−A exp(−λτ

)

=−A

+ λ

exp(λτ

i−1

), β

= A exp(λτ

)

=−β

exp



−

τ



− β

, β

=−β

exp



−

τ



− β

A =



− λ

exp(λτ

) −

+ λ

exp(−λτ

)



−1

(6.82)

According to (6.78) the radiative ﬂux densities leaving the layer τ

are given

(τ

i−1

) = C

+ C

+ α

S(τ

i−1

)

−

(τ

) = C

+ C

+ α

S(τ

i−1

)exp



−

τ



(6.83)

with

= exp(λτ

i−1

), γ

= exp(−λτ

i−1

)

(α

+ λ)

exp(λτ

), γ

(α

− λ)

exp(−λτ

)

(6.84)

From (6.65) and (6.80) we obtain for the solar radiation ﬂux density at τ

S(τ

) = S(τ

i−1

)exp



−

τ



(6.85)

Substituting (6.81) into (6.83) and combining the result with (6.85) we obtain the

compact matrix notation







(τ

i−1

)

−

(τ

)

S(τ

)

























(τ

)

−

(τ

i−1

)

S(τ

i−1

)







(6.86)

with

= β

+ β

, a

= β

+ β

, a

= β

+ β

+ α

= β

+ β

, a

= β

+ β

, a

= β

+ β

+ α

= 0, a

= exp



−

τ



(6.87)

176 Two-stream methods for the solution of the RTE

From (6.82), (6.84), and (6.87) it may be easily seen that

= a

= A

2λ

, a

= a

= A[exp(λτ

) − exp(−λτ

)] (6.88)

Thus we ﬁnally obtain the analytical solution to the TSM as







(τ

i−1

)

−

(τ

)

S(τ

)













00a













(τ

)

−

(τ

i−1

)

S(τ

i−1

)







(6.89)

The a

occurring in this equation are functions of the optical properties of the layer

τ

. They have the following physical meaning:

: transmission coefﬁcient for diffuse radiation,

: reﬂection coefﬁcient for diffuse radiation,

: reﬂection coefﬁcient for the primary scattered parallel solar radiation,

: transmission coefﬁcient for the primary scattered parallel solar radiation,

: transmission coefﬁcient for the direct parallel solar radiation.

6.5 Approximate treatment of scattering in the infrared spectral region

It is well-known that scattering of solar radiation by cloud droplets and aerosol

particles is a dominant factor affecting the Earth’s planetary albedo and thus the

global climate. In the previous chapters we have discussed various calculation

methods to determine solar radiances. To obtain the ﬂux densities from the radiances

is a relatively simple calculational procedure. A review of the literature shows that

the scattering of thermal radiation by clouds and aerosol layers is often neglected

in weather and climate models for two main reasons: (i) with the exception of the

atmospheric window region ranging from about 8–12.5

µm, long-wave radiative

transfer is dominated by absorption and emission processes due to atmospheric

water vapor and some other trace gases and by water droplets and ice particles;

and (ii) multiple scattering calculations require a large amount of computer time

in comparison to situations where the scattering part of the source function in the

RTE can be neglected.

In the following we will show how to include multiple scattering in the infrared

window region in addition to absorption and emission by modifying the two stream

equation (6.63). The same method, if desired, can also be applied to any other

part of the long-wave spectrum. Since the solar and the infrared spectrum can be

separated, say at 4

µm, all we need to do is to replace the solar scattering term in

equation (6.63) by the Planckian emission B(τ ). Analogously to (6.63) we obtain

6.5 Approximate treatment of scattering in the IR region 177

for the infrared spectral region with multiple scattering







dτ

−

dτ











−α









−





+ α





−B(τ )

B(τ )





(6.90)

with

= α

, α

= α

, α

π(1 −ω

)

¯µ

(6.91)

The terms α

,α

are given by (6.37) where again b

= b

−

and µ

= µ

−

= ¯µ.As

before, we suppress any reference to wave number or wavelength. In order to handle

the emission term in an effective way we make an Eddington type approximation

for the thermal radiation, cf. (6.42)

B(τ ) = B

+ B

τ (6.92)

where B

and B

are constants.

For a homogeneous layer the complete mathematical solution to the system

(6.90) is found in the same way as in the short-wave radiation case. The solution of

the homogeneous part of this differential equation is given by (6.74). To determine

a particular solution of the inhomogeneous system, instead of (6.75), we now sub-

stitute a trial solution of the type of the inhomogeneous term (6.92) into (6.90), i.e.

+, p

(τ ) = α

+ α

τ , E

−, p

(τ ) = α

+ α

τ (6.93)

By comparing in the resulting equations coefﬁcients of zero-th and ﬁrst order in τ

we obtain four linear equations for the unknown constants α

, i = 4,...,7 which

may be easily solved yielding

= γ

+ γ

, α

= γ

− γ

, α

= α

= γ

with γ

− α

, γ

− α

(6.94)

Hence, the general solution of (6.90) may be written as

(τ )=C

exp(λτ) +C

exp(−λτ) + γ

+ B

τ ) + γ

−

(τ )=C

(α

+ λ)

exp(λτ) +C

(α

− λ)

exp(−λτ) + γ

+ B

τ ) − γ

(6.95)

The determination of the integration constants C

and C

is performed in the

same way as in the short-wave situation. First we use (6.95) to calculate the radia-

tive ﬂux densities entering the homogeneous layer τ

= τ

− τ

i−1

. Introducing

178 Two-stream methods for the solution of the RTE

the abbreviations

(τ ) = E

(τ ) − γ

+ B

τ ) − γ

−

(τ ) = E

−

(τ ) − γ

+ B

τ ) + γ

(6.96)

we obtain

(τ

) = C

exp(λτ

) + C

exp(−λτ

)

−

(τ

i−1

) = C

(α

+ λ)

exp(λτ

i−1

) + C

(α

− λ)

exp(−λτ

i−1

)

(6.97)

By comparing (6.97) with (6.79) we see that formally both systems differ only in

the solar radiation term which is missing in (6.97). Hence, it is an easy task to ﬁnd

the integration constants by dropping the solar radiation term in (6.81)

= β

(τ

) + β

−

(τ

i−1

), C

= β

(τ

) + β

−

(τ

i−1

)

(6.98)

The constants β

follow from (6.82).

Utilizing (6.84) in (6.97) the radiative ﬂuxes leaving the layer τ

may be

formulated as

(τ

i−1

) = C

+ C

−

(τ

)= C

+ C

(6.99)

Substitution of (6.98) into these equations results in





(τ

i−1

)

−

(τ

)

















(τ

)

−

(τ

i−1

)





(6.100)

where the a

are given by (6.87).

Finally, we need to set up proper boundary conditions at the top of the atmosphere

(τ = 0) and at the Earth’s surface (τ = τ

). Denoting the albedo of the ground by

we obtain from (6.96)

(τ

) = A

−

(τ

) + π B(T

)(1 − A

) − γ

+ B

) − γ

−

(0) = E

−

(0) − γ

+ γ

(6.101)

Usually we set E

−

(0) = 0.

As in previous cases, the atmosphere will be subdivided into numerous

homogeneous layers. The vertical variation of temperature is accounted for by

permitting the Planck function to vary according to the linearization (6.92). By

using a sufﬁciently large number N of homogeneous sublayers, we can represent

with high accuracy the non-isothermal structure of the atmosphere. The upward and

downward directed ﬂux densities of each layer are obtained by constructing from

(6.100) a (2N × 2N )-tridiagonal equation system. This system may be solved by

means of standard numerical procedures. Finally, the results for

(τ

) are used in

(6.96) yielding E

(τ

6.6 Approximations for partial cloud cover 179

6.6 Approximations for partial cloud cover

The radiative transfer models described in the previous sections are only suitable for

horizontally homogeneous layers. In numerical weather prediction models dealing

with partial cloudiness within a numerical grid box, particular problems arise since

in the layers which are partially ﬁlled with clouds the assumption of horizontal

homogeneity is void. In addition to the horizontal inhomogeneity within a grid

box, the radiation scheme has to account for the vertical distribution of fractional

cloud cover within a numerical grid column.

In order to treat the situation with partial cloudiness more realistically, assump-

tions about the vertical distribution of the partial cloud cover have to be introduced.

In combination with two-stream radiative transfer two particular approximations

are widely employed to treat the overlap of contiguous cloud layers. These are the

concepts of random overlap and maximum overlap. For random overlap the com-

bined partial cloud cover of both layers is obtained by multiplying the cloud covers

of each individual layer. Apparently, this concept was ﬁrst used in the radiative

transfer model by Manabe and Strickler (1964). The maximum overlap assumption

means that the combined partial cloud cover of two vertically adjacent cloud layers

is arranged in such a way that the cloudy portions of both layers overlap maximally.

This scheme has ﬁrst been employed in the two-stream ﬂux transfer by Geleyn and

Hollingsworth (1979). In the following we will illustrate both cases in detail.

6.6.1 Partial cloud cover with random overlap

In the random overlap concept it is assumed that the clouds of contiguous lay-

ers overlap in a random way. If in a real situation cloudy layers are separated by

cloud-free regions then it seems physical to postulate that these layers are statisti-

cally independent. For simplicity let us ﬁrst consider only the transmission of the

downward directed diffuse radiation.

Figure 6.3 illustrates the random overlap assumption. At the bottom of the

partially cloudy layer i − 1 the two radiative ﬂux densities E

−

(τ

i−1

) and E

−

(τ

i−1

)

emanate, whereby the superscripts c and f denote the cloudy and cloud-free regions,

respectively. These two ﬂuxes are added yielding the single ﬂux E

−

(τ

i−1

) =

−

(τ

i−1

) + E

−

(τ

i−1

). If C

is the cloud cover of layer i then it is assumed that

the fraction C

−

(τ

i−1

) enters the cloudy portion of this layer, whereas the remain-

ing fraction (1 − C

−

(τ

i−1

) propagates through the clear sky portion. For the

cloudy and cloud-free parts of the downward radiation at τ

we obtain

−

(τ

)

= a

(1 − C

−

(τ

i−1

), E

−

(τ

)

= a

−

(τ

i−1

) (6.102)

Here, a

and a

are, respectively, the transmission coefﬁcients for diffuse radiation

of the cloud-free and the cloudy part of layer i . The two relations (6.102) may be

180 Two-stream methods for the solution of the RTE

i−1

1 C

1 − C

i−1

E (

)

E (

i−1

)

(

i−1

)

(

)

(1 C

)E (

) C

E ()

−

i−2

i−1

−

Fig. 6.3 Transfer of the downwelling diffuse radiation between two individual

sublayers assuming random overlap of the partial cloud fractions C

i−1

and C

combined to a single equation for the total downward radiative ﬂux density

−

(τ

) = E

−

(τ

) + E

−

(τ

) =



(1 − C

) + a



−

(τ

i−1

) (6.103)

This means that in the random overlap approximation we have to replace a

(6.89) by

= (1 − C

+ C

(6.104)

The transfer of the diffuse upward ﬂux densities and the solar radiation can be

treated in exactly the same manner so that similar relations apply to the other a

coefﬁcients of (6.89). In summary, we obtain

= (1 − C

+ C

, j, k = 1, 2, 3 (6.105)

Consider, for example, the direct solar radiation through two adjacent cloud

layers of cloudiness C

i−1

and C

. The emerging directly transmitted solar radiation

at the base of layer i is proportional to the product C

i−1

according to the rule that

statistically independent probabilities multiply. This explains the name ‘random’.

6.6.2 Partial cloud cover with maximum overlap

The assumption of maximum overlap of vertically adjacent clouds is reasonably

well justiﬁed for a situation for which the clouds in the various layers are formed

6.6 Approximations for partial cloud cover 181

i−1

−

−−

−

1 C

i−1

−1

−

1 C

i−1

−

i−1

1 C

i+1

1 C

i+1

1 C

i+1

1 C

i+1

1 b

(

i−1

i−2

)

−

(

i−1

)

()

−

i−1

()

(a)

(b)

Fig. 6.4 Downward (a) and upward (b) transmission of diffuse radiation through

two adjacent layers with different partial cloud cover for the maximum overlap

concept.

by the same physical process. Again we ﬁrst consider only the transmission of

downward directed diffuse radiation in two contiguous sublayers i −1 and i with

cloud covers C

i−1

and C

. The situation is illustrated in Figure 6.4(a). Two different

cases must be treated as shown in the left and right panel of this ﬁgure. In one case

(left panel) the lower layer has the larger cloud cover, that is C

> C

i−1

, whereas

in the other case (right panel) the situation is reversed.

From Figure 6.4(a) it is seen that the transfer of the downward radiation can be

formulated as follows. The cloudy part of layer i − 1 transmits E

−

(τ

i−1

) whereas

the cloudless part transmits E

−

(τ

i−1

). Both parts of the transmitted radiation must

now be distributed between the cloudy and cloudless parts of the lower layer i. Let

the parameter b

refer to that fraction of E

−

(τ

i−1

) leaving the base of layer i − 1

and entering the cloudy part of layer i. This fraction is equal to 1 if C

≥ C

i−1

(left panel of Figure 6.4(a)) and equal to C

i−1

if C

< C

i−1

(right panel of the

ﬁgure). Both conditions can be combined yielding

min

(

, C

i−1

)

i−1

(6.106)

182 Two-stream methods for the solution of the RTE

Inspection of the right panel of Figure 6.4(a) shows that in this particular situation

the fraction (1 − b

−

(τ

i−1

) enters the cloud free part of layer i.

Let us now consider the clear sky part in the left panel of Figure 6.4(a). The

diffuse radiation E

−

(τ

i−1

) emanating from layer i − 1 will be subdivided between

the cloudy part C

− C

i−1

and the clear sky region 1 − C

. This is done by means

of the parameter b

which in the left panel is given by b

= (1 − C

)/(1 − C

i−1

In the right panel of Figure 6.4(a) where C

i−1

> C

we have b

= 1. Both cases

can again be combined as

1 − max

(

, C

i−1

)

1 − C

i−1

(6.107)

The diffuse upwelling radiation E

(τ

) and the solar radiation are treated in

the same way as E

−

(τ

i−1

). While for the solar radiation also the parameters b

and b

are applied, for E

(τ

) the coefﬁcients b

and b

have to be utilized, see

Figure 6.4(b). Analogously to (6.106) and (6.107) they are given as

1 − max

(

, C

i+1

)

1 − C

i+1

, b

min

(

, C

i+1

)

i+1

(6.108)

The parameters b

, j = 1,...,4 are now introduced on the right-hand side of

(6.89) to obtain the cloud free and cloudy parts of the radiation ﬁeld.

For the cloud

free part we have







(τ

i−1

)

−

(τ

)

(τ

)













00a













(τ

) + (1 − b

(τ

)

−

(τ

i−1

) + (1 − b

−

(τ

i−1

)

(τ

i−1

) + (1 − b

(τ

i−1

)







(6.109)

while the cloudy parts of the radiative ﬂuxes are written as







(τ

i−1

)

−

(τ

)

(τ

)













00a













(1 − b

(τ

) + b

(τ

)

(1 − b

−

(τ

i−1

) + b

−

(τ

i−1

)

(1 − b

(τ

i−1

) + b

(τ

i−1

)







(6.110)

It is noteworthy that a particular coefﬁcient b

is set equal to 1 if an undetermined expression 0/0 occurs. This

follows from physical reasoning or from applying l’Hˆopital’s rule.

6.7 The classical emissivity approximation 183

These two systems of linear equations determine the ﬂux densities E

(τ

i−1

(τ

i−1

), E

−

(τ

), E

−

(τ

), S

(τ

), and S

(τ

) that emerge from the homogeneous

layer τ

as a function of the corresponding incoming ﬂux densities E

(τ

), E

−

(τ

i−1

), E

−

(τ

i−1

), S

(τ

i−1

), and S

(τ

i−1

). At an arbitrary level i the

physically relevant ﬂux densities are the sum of the individual cloudy and cloud free

contributions

(τ

) = E

(τ

) + E

(τ

)

−

(τ

) = E

−

(τ

) + E

−

(τ

)

S(τ

) = S

(τ

) + S

(τ

)

(6.111)

A comment is due regarding the ground reﬂection which, for simplicity, is

assumed to be isotropic, i.e. the ground is a Lambertian reﬂector. In this case

the diffuse upwelling ﬂux densities are given by

(τ

) = A



−

(τ

) + S

(τ

)



, E

(τ

) = A

−

(τ

) + S

(τ

)]

(6.112)

Such a situation may arise if ground fog occurs. Note that the downward ﬂux

densities leaving the lowest layer from which they emerged, after reﬂection enter

the same section of cloudy and cloud-free air.

Due to the fact that the systems (6.109) and (6.110) are coupled, for an

inhomogeneous atmosphere with N homogeneous sublayers we have to solve a

6N -dimensional linear matrix system. This means that in case of partially cloudy

layers the maximum overlap treatment costs about twice the computational time

as the original TSM which assumes 100% cloud cover in each cloudy layer. In

contrast to this, the numerical effort of the random overlap approach is almost the

same as in the original TSM since in both cases the same number of 3N coupled

equations has to be solved.

6.7 The classical emissivity approximation

In Section 2.6.2 we have shown that in case of a purely absorbing atmosphere it is

possible to obtain an analytical solution to the RTE, see (2.123). We have already

stated that in the infrared spectral region, apart from the atmospheric window,

scattering processes may savely be neglected in the calculation of atmospheric

radiative transfer in many cases. However, the analytic solution is still relatively

elaborate since it consists of integrals of the transmission function over the absorbing

mass. Furthermore, it is important to note that (2.123) is a spectral equation.

This

Recall that in all equations of Chapter 2 for simplicity the index ν has been omitted.

184 Two-stream methods for the solution of the RTE

means that we need to perform another integration over the entire infrared spectral

region in order to obtain the total infrared radiative heating rate.

In this section we will present the classical emissivity method which is one of the

fastest ways to ﬁnd simple approximate values of the infrared radiative ﬂux densities

in a nonscattering atmosphere. The major simpliﬁcation of the emissivity method

consists in the decoupling of the frequency integration from the integration over

the absorbing mass. To derive the classical ﬂux-emissivity equations, we assume

that the temperature and pressure dependency of the mass absorption coefﬁcient

abs,ν

(p, T ) can be expressed as

abs,ν

(p, T ) = κ

abs,ν

, T

)

(6.113)

The sufﬁx 0 refers to standard conditions. In the following chapter we will present

a detailed discussion of different forms of the absorption coefﬁcient. There it will

be shown that the so-called Lorentz absorption coefﬁcient closely resembles the

form (6.113).

Utilizing (6.113) the argument of the ﬂux-transmission function T

f,ν

, as given by

(2.142), may be written as





abs,ν

(t)dt = κ

abs,ν

, T

)





dt = κ

abs,ν

, T

)(w



− w) (6.114)

The integral





p/p

√

/Tdt which is symbolically written as w



− w is known

as the reduced absorber mass. This is equivalent to the deﬁnition

dw =

du =

abs

ds =−

abs

(6.115)

see also (2.118). With the help of (6.114) the ﬂux-transmission function can be

approximated as

f,ν

(u, u



) = 2E







abs,ν

(t)dt



≈ 2E

[κ

abs,ν

, T

)(w



− w)] = T

f,ν

(w, w



)

(6.116)

where w



≥w. Since E

(0) = 1/2 we obtain the physically correct transmission

function T

(w, w



) = 1 if no absorption takes place.

Integration over ν yields for the upward and downward directed ﬂux densities

as given by (2.143)