Module 11
Radiative Heating in the Atmosphere
11.1
Introduction
The atmosphere is heated by the absorption of solar radiation during daytime and it
depends on the solar zenith angle. The calculation of zenith angle – a function of
latitude, hour angle and declination – is very basic to solar radiation; and its computation
can be seen in standard textbooks on solar radiation mentioned earlier. The atmosphere is
also heated (cooled) due to absorption (emission) of infrared radiation by water vapour
and the well-mixed carbon dioxide. The daytime solar heating and the nighttime infrared
heating/cooling give rise to diurnal cycle of heating. However, the heat deposited in
different layers is horizontally transported by the winds and also it is mixed vertically due
to turbulence. But the vertical mixing is strong in the troposphere, which results from
spatially and temporally evolving turbulent zones of boundary layer and those
characterized by dry and moist convection. Ozone is mainly concentrated in the
stratosphere and the mesosphere (i.e. middle and upper atmosphere) and absorbs solar
radiation in the ultraviolet (UV) region, visible, near infrared and Chappuis bands.
Oxygen in the middle and upper atmosphere is also an important absorber of radiation in
UV and visible bands thereby affecting the solar flux reaching the troposphere. Besides,
there are other gases that too absorb solar radiation and heat the atmosphere. Water
vapour (WV) is highly variable but mainly confined to the lower troposphere, because
water vapour mixing ratios of 1.0 g kg −1 are found close to 600 hPa level or at about
4.0 km height in the atmosphere. Thus there are three dominant absorber gases present in
three different layers (Fig. 11.1) of the atmosphere, which absorb radiation strongly in
three distinct wavelength bands. Ozone absorbs radiation in the part of spectrum
0.20 − 0.85 µ m ; while water vapour ( 0.82 − 3.20 µ m ) and CO2 ( 2.0 − 3.50 µ m ) absorb
in near-infrared region (1.0 − 4.0 µ m) . The emission of longwave radiation will however
result in cooling of an atmospheric layer. For an elaborate discussion on the
solar/terrestrial absorption bands of atmospheric gases, one may refer to any of the
standard textbooks on this topic (e.g. Liou: An Introduction to Atmospheric Radiation;
Houghton: Physics of Atmospheres; Andrews: An Introduction to Atmospheric Physics).
The recently proposed value of the solar constant ( S0 = 1366 Wm −2 ) for radiative
transfer calculations is obtained as the sum of solar spectral fluxes; their spectral
distribution is shown in Fig. 11.2 as a function of wavelength. Except in the ultraviolet
region ( < 0.4 µ m ) where one notices important variability, it is of great significance to
observe from Fig. 11.2 that the Planck function curve computed by taking the emitting
temperature of 5800 K of the Sun’s photosphere fits accurately well with the solar
spectrum irradiance ( Wm −2 µ m −1 ) curve. It thus enables us to create a look-up table for
solar spectral irradiances ( Sλ ) from the Planck function to perform accurate and rapid
solar radiative transfer calculations in the atmosphere if the absorber gas profiles are
already prescribed or predetermined from atmospheric chemistry models. The integral of
Sλ over the wavelength intervals will sum up to S0 ; that is,
1
∞
5 µm
0
0.2 µ m
−2
∫ Sλ d λ = S0 = 1366 Wm ; and
∫
Sλ d λ =1359.58 Wm −2
(11.1)
More complete and comprehensive models of climate and weather forecasting not only
include dynamics and physics (surface processes, clouds and radiation) of atmosphere but
also compute interactively aerosols and the trace gas constituents that evolve with the
simulated meteorology which in turn affects the radiative heating.
Layer 1
Ozone
0.20 − 0.85 µ m
1.06 − 1.58 µ m
Middle and Upper
Atmosphere
Layer 2
Carbon dioxide
2.0 − 3.5 µ m
Water vapour
0.82 − 3.20 µ m
Lower Stratosphere
Upper Troposphere
Fig. 11.1 A schematic showing the
distribution of dominant absorbing
gases in three different layers of the
atmosphere and their effective
absorption wavebands. Note that in
Layer-1, O2 is also an important
absorbing gas of UV radiation. It is
to be noted that CO2 is well-mixed in
the atmosphere, but its absorption
exceeds that by other gases in the
Layer-2.
Layer 3
Lower
Troposphere
In effect, nearly 70% of the incoming solar radiation is absorbed by earthatmosphere system (albedo = 0.3) and in order to maintain a constant global equilibrium
temperature, all that energy received from the sun has to be emitted back to space as
thermal radiation. The thermal radiation from the earth-atmosphere system is emitted in
the infrared part of the electromagnetic spectrum. Hence, while computing the radiative
heating of an atmospheric layer, the downward flux of radiation from the sun and the
upward flux of the terrestrial radiation are necessary. These fluxes can be computed from
the upward and downward propagating intensities of radiation in the atmosphere. No
doubt, we deal separately the heating of cloudy and cloud-free (clear) regions both for
solar and infrared thermal radiation bands while performing radiative transfer
computations.
11.2
Solar heating of the atmosphere
The incident solar radiation of spectral flux density Fsν (∞) of wavenumber ν falls
on earth at an inclination with angle θ to the local zenith. For radiative transfer
calculations, a plane parallel atmosphere is assumed so the normal spectral flux density,
as shown in Fig. 11.3, is given by cosθ Fsν (∞) . The net flux density centred at any
wavenumber ν is the difference of downward and upward flux densities at that
wavenumber. Thus, the net flux density at any level z is given by,
Net spectral flux density:
Fν (z) = Fν↓ (z) − Fν↑ (z) with ν = ν / c
(11.2)
2
∂Fν (z)
F (z) − Fν (z + Δz)
= lim ν
Δz →0
∂z
Δz
−
Divergence of flux:
(11.3)
∞
Net flux: F(z) = cosθ ∫ Fν (z)dν (summation over all wavenumbers)
0
(11.4)
−1
−2
Solar Irradiance Sλ (W m µ m )
Visible(40%)
UV
Observed
0.4 − 0.7 µ m
Planck Function
2000
Tsun = 5800 K
S0 = 1366 Wm −2
5
∑ S Δλ = 1359.58 Wm
1500
0.2 µ m
λ
100
∑ S Δλ = 6.42 Wm
5 µm
1000
λ
−2
−2
(50% Solar flux in λ > 0.7µ m)
500
Near IR
0
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
5.0
Wavelength λ ( µ m)
Fig. 11.2 Solar irradiance at the top of the atmosphere (light brown) and the Planck curve
computed with Tsun = 5800 K. Note the variability in the ultraviolet region ( λ < 0.4 µ m )
The radiative heating rate is equal to divergence of the net flux in an atmospheric
layer; that is, the rate of heating of an atmospheric layer can be computed from the ratio
of net heat flux and the thickness of the atmospheric layer under consideration. Hence,
− ρaC p
∂T ∂F(z)
∂F (z)
=
= cosθ ∫ ν dν
∂t
∂z
∂z
ν
(11.5)
In the above expression, ρa is the density of air and C p is its specific heat at constant
pressure. Taking the discretization of the atmosphere as shown in Fig. 11.3, the discrete
form of the eq. (11.5) is given as,
cosθ ∑ Δ Fν (z) Δ ν
∂T ΔF(z)
ν
− ρaC p
=
=
(11.6)
∂t
Δz
Δz
Since the mass of atmospheric constituents is involved in calculating the optical depth
and transmission through atmospheric layers, an appropriate form of eq. (11.6) could also
be derived. However, evaluation of integrals for optical depth is straightforward in the
pressure coordinate, so (11.6) may be written in pressure coordinate as
3
∂T
ΔF(z)
=−
∂t
C p ( ρa Δ z)
∂T
g ΔF( p)
=
∂t C p Δ p
⇒
(11.7)
Local Zenith
Fsν (∞)
z=∞
θ
cosθ Fsν (∞)
To
F ↓ + dF ↓
F ↑ + dF ↑
F(z + δ z)
z
p=0
p −δ p
dz
p p
Fig. 11.3 Schematic of calculating heating
rates in the atmosphere. z = ∞ refers to the
top of the atmosphere and z = 0 to the earth
surface. Radiative balance for a layer of
thickness dz in the earth’s atmosphere is
also shown; and from the divergence of net
flux, radiative heating of the layer can then
be calculated.
Note that,
F(z) = cosθ
F(z)
F↓
z=0
F
↑
ps
∫ Fν (z)dν
ν
Bottom
Let du be the absorber mass in a layer of thickness dz given by du = ρ dz, where ρ is
ρ
the density of the absorber gas with mixing ratio q =
. It is then possible to express
ρa
(11.7) in a form with u as an independent coordinate and we may write,
∂T
ΔF(z)
ρΔF(z)
q ΔF(z)
=−
=−
=−
;
∂t
C p ( ρa Δ z)
C p ρa ( ρΔ z)
C p ( ρΔ z)
Since Δu = ρΔz and q is the mixing ratio, the above equation can be written as
∂T
q ΔF(u)
=−
∂t
Cp Δ u
(11.8)
It must be noted that the calculations of discrete derivatives are accomplished in the
direction of increasing z with origin at the earth surface.
As an example let us compute heating as a result of absorption of solar radiation by
ozone in the Hartley band (200 − 300 nm) . All the solar energy absorbed will go into
increasing the kinetic energy of the molecule; that is, the temperature of the atmospheric
layer. To a beginner it may appear that the temperature (heating) profile would attain a
4
maximum where ozone concentration peaks. But in reality, a maximum in the heating
profile occurs where optical depth is equal to unity ( χ = 1 ), which is well above the level
of peak absorber concentration. In such a situation heating would exhibit a Chapman
layer. The maximum in ozone concentration is found at about 22 km in the stratosphere,
but heating peaks at 50 km i.e. at the stratopause level (Fig. 1.1). Solar energy deposition
above and below the Chapman layer reduces drastically and therefore a well-defined
layered structure of solar energy deposition develops in the atmosphere. The discussion
of the Chapman layer is given only for the ozone, but combined effect of absorbing gases
is necessary to be taken into account in its discussion. Heating in middle atmosphere is
mainly due to ozone absorption, therefore, it is an important observation in the
atmosphere. Also note that, ozone mainly forms both in stratosphere (10-50 km) and
mesosphere (50-80 km) by photochemical reactions, which deplete the incident beam of
its energy. Surely, the complication arising from scattering of solar radiation is
completely ignored in our calculations, and the absorption coefficient is taken
independent of pressure as well. The heating due to absorption of ozone can be calculated
from eq. (11.5) or its discrete analogue (11.6) but we require two important data, viz., the
absorption coefficient kν at frequency ν and the solar flux density at this frequency at a
height z . In case of solar radiation, the upward irradiance Fν↑ (z) = 0 ; hence eq. (11.5)
takes the form given below by eq. (11.9); that is, solar heating is determined by the
divergence of the downward flux from the Sun:
∂T
∂Fν↓ (z)
ρaC p
= − cosθ ∫
dν
∂t
∂z
ν
(11.9)
To obtain the spectral flux density Fν↓ (z) , the optical depth χν (z) is computed from
solar absorption cross-section of ozone and its number concentration as follows:
∞
χν (z) = ∫ k ν# ( z′ )n( z′ )secθ dz′
z
(11.10)
kν# (z) is absorption cross-section and n(z) is the number density of O3 . The downward
solar irradiance at a height z is then obtained as
∂Fs↓ν (z)
dχ
= Fs↓ν (∞){− ν }exp{− χν (z)} .
(11.11)
∂z
dz
d χν
On substituting the expression for
from (11.10), the eq. (11.9) can be written as
dz
∂T
(11.12)
ρaC p
= ∫ Fsν↓ (∞) k ν# (z)n(z)exp{− χν (z)}dν .
∂t ν
Fs↓ν (z) = Fs↓ν (∞)exp{− χν (z)} ⇒
On assuming kν# independent of z , calculations considerably simplify in (11.12). In the
general case, χν (z) is obtained by numerical evaluation of the integral on the right hand
side using the profiles of absorption cross-section k # and the number density n(z) of O3 .
The integral over wavenumbers on the right in eq. (11.12) is also evaluated numerically.
For this purpose, one may apply the trapezoidal rule that uses layer averages of a variable
quantity, which is calculated from its values at the end points of intervals of subdivision.
5
The complete integral is represented as a summation, which gives the final evaluation of
the quantity.
For calculating the heating rate due to ozone absorption of solar radiation in the
0.2 − 0.3 µ m interval, let us take typical values as follows:
O3 number density n = 1.0 × 1017 molec /m 3 ; pressure = 1.0 hPa; T = 270.7 K
ρa = 0.0013 kg m −3 at 1.0 hPa; C p = 1005 J kg −1K −1
Solar irradiance (average over 0.2 − 0.3 µ m ) = 70.0Wm −2 µ m −1 ; Average O3 absorption
cross-section k ν# = 4.614 × 10 −22 m 2 /molec ; Δν = 0.1 µ m ; Optical depth χν = 1 ;
These quantities may be substituted in the discrete form of the eq. (11.12) to get the
maximum heating rate due to ozone absorption at 50 km in the atmosphere where ozone
concentration is lower than its peak value at about 22 km in the stratosphere. The
radiative heating from the discrete form of (11.12) is obtained by a straightforward
evaluation of the integral by taking out the terms that are independent of wavenumber
and we have
∂T
ρaC p
= Fsν↓ (∞)k ν# n(z) exp{− χν (z)} Δ ν
∂t
Substitute in the above equation the values of the variables appearing on the right hand
side and we get,
ρaC p
∂T
W
m2
molec −1
= (70.0 2
)(4.614 × 10 −22
)(1.0 × 1017
) (e )(0.1µ m)
∂t
m µm
molec
m3
W
W
= (70)(4.614 × 10 −22 )(1.0 × 1017 ) (0.3679)(0.1) 3 = 1.1882 × 10 −4 3
m
m
ρa = 0.0013 kg m −3 at 1.0 hPa; C p = 1005 J kg −1K −1 ; hence ρaC p = 1.3065 J K −1m −3
∂T 1.1882 × 10 −4 K
=
= 9.0949 × 10 −5 K s −1
∂t
1.3065
s
That is, the heating rate due to ozone absorption of ultraviolet radiation in the Hartley
band (200 − 300 nm) is about 8.0 K day −1 in the stratosphere at 50 km above the earth
surface. This shows that stratospheric heating is mainly due to absorption of UV by O3 .
Therefore, we have
Table 11.1 Absorption cross-section
kν# by Ozone
(Data: J. Malicet et al., J. Atmospgeric Chem. 1995; J. Brion et al. 1993, Chem. Phys. Lett.)
Wavelength (nm)
200
210
220
230
240
250
260
270
280
290
300
kν#
3.1540e-19
5.7160e-19
1.7850e-18
4.4760e-18
8.3119e-18
1.1146e-17
1.0708e-17
7.8556e-18
3.8834e-18
1.3369e-18
3.6380e-19
Wavelength (nm)
310
320
330
340
450
500
550
600
650
700
750
kν#
8.7162e-20
2.8538e-20
3.2276e-21
1.6102e-21
1.7010e-21
1.1760e-21
3.3140e-21
5.0694e-21
2.5000e-21
8.7454e-22
4.2745e-22
6
The absorption cross-sections kν# of ozone are tabulated for wavelengths 200 – 750 nm
below in Table 11.1 for the sake of completeness. Solar heating of the atmosphere from
the absorption of incident solar radiation by other absorbing gases is also computed in the
similar manner. However, for direct solar beam, the integration over the solid angle (or
the diffuse approximation) is not required.
11.3
Infrared radiative cooling/heating of the atmosphere
The expressions (10.48) and (10.49) derived in Module 10 are directly amenable to
numerical computations; these expressions read as
∞
∞ z
∂τ ( z′, z)
(11.13)
F ↑ (z) = ∫ π Bν (0)τ ν (0, z)dν + ∫ ∫π Bν ( z′ ) ν
dz′dν
∂ z′
0
0 0
∞∞
∂τ ν ( z′, z)
dz′dν .
∂
z
′
0 z
The infrared radiative cooling/heating is then calculated as,
F ↓ (z) = − ∫ ∫ π Bν ( z′ )
− ρaC p
∂T ∂F(z) ∂[F ↑ (z) − F ↓ (z)]
=
=
∂t
∂z
∂z
(11.14)
(11.15)
For calculating the fluxes it is important to have the transmission function τ ν through the
layer, which is calculated by two methods
(i)
Narrow-band method
(ii)
k-distribution method
(i) In the narrow-band method, the transmission function τ ν is defined as
⎧ −Su ⎛
Su ⎞ ⎫
1+
⎜
⎟⎬.
⎩ δ ⎝ πα ⎠ ⎭
τ ν (u) = exp ⎨
(11.16)
The S represent a mean line strength over the interval Δν ; δ is a mean line spacing, α
is a mean line half-width, and u is the weighted absorber amount by diffusivity factor
(= 1.66) . R.M. Goody (1952) first derived this expression using a random model. The
πα
S
advantage of this statistical model is that we require the two parameters
and
to
δ
δ
calculate the transmission function from (11.16) for different absorbents in the
atmosphere: two H2O rotational bands, the 15 µ m CO2 band, the 9.6 µ m O3 band and the
6.3µ m H2O vibration band. The key requirement of the band models is that the
transmission function must be treated constant over the entire band interval Δν . That is
why the current weather prediction models use large number of narrow bands to calculate
the infrared cooling or heating for producing short-, medium- or long-range weather
forecasts. These two parameters have been tabulated in Table 11.2 given below.
(ii) The k-distribution method is based on the grouping of the absorption spectral
transmittance according to kν , the absorption coefficients. Radiative transfer calculations
with this method are faster than the band models because in a homogeneous atmosphere,
the spectral transmittance is independent of ordering of k for a given spectral interval.
7
Thus frequency intervals can be grouped according to line strengths. The expression for
the spectral transmittance is as follows
τ ν (u) =
∞
1
e−kν u dν = ∫0 f (k) e−ku dk
∫
Δν Δν
kmin → 0 , k max → ∞
and
∫
∞
0
(11.17)
f (k)dk = 1.
The computations performed with k-distribution or correlated-k method produce results,
which differ only by 1% from the line-by-line integration of the radiative transfer model.
The radiative transfer calculations essentially can be learnt with the Microsoft Excel.
Some problems on infrared radiative transfer have been given as exercises, and one can
use the Excel to obtain their answers. The discrete form of (11.12) and (11.13) may be
written in terms of the broadband emissivity and the data given in Table 11.2 may used
for computing the transmissivity function for the principal absorbers.
Table 11.2 Parameters in the transmission function
τ ν (u)
of given path length
Spectral interval (cm )
H2O rotational band
S / δ (cm 2/ g)
πα / δ
40 - 160
160 - 280
280 - 380
380 - 500
500 - 600
600 - 720
720 - 800
800 - 900
7210.30
6024.80
1614.10
139.03
21.64
2.919
0.386
0.0715
0.182
0.094
0.081
0.080
0.068
0.060
0.059
0.067
718.7
0.448
6.99 e+2
1.40 e+2
2.79 e+2
4.66 e+2
5.11 e+2
3.72 e+2
2.57 e+2
6.05 e+2
7.69 e+2
2.79 e+2
5.0
5.0
5.0
5.5
5.8
8.0
6.1
8.4
8.3
6.7
25.65
134.4
632.9
331.2
434.1
136.0
35.65
9.015
1.529
0.089
0.230
0.320
0.296
0.452
0.359
0.165
0.104
0.116
−1
CO2 15 µ m band
585 - 752
u
O3 9.6 µ m band
1000.0 - 1006.5
1006.5 - 1013.0
1013.0 - 1019.5
1019.5 - 1026.0
1026.0 - 1032.5
1032.5 - 1390.0
1390.0 - 1045.5
1045.5 - 1052.0
1052.0 - 1058.5
1058.5 – 1065.0
H2O 6.3 µ m
1200 - 1350
1350 -1450
1450 - 1550
1550 - 1650
1650 - 1750
1750 - 1850
1850 - 1950
1950 - 2050
2050 - 2200
8
11.4
Cloud-radiation interaction in the atmosphere
Clouds form in the atmosphere as a consequence of complex interactions between
thermodynamical and dynamical processes. The cloud parameters are computed from
microphysical, radiative and thermodynamic state variables, and the liquid water content
(LWC) of the cloud. Since some of the cloud parameters can now be derived from the
well-documented satellite datasets, one can also use them directly in calculating the cloud
radiative properties such as cloud optical thickness and emissivity for determining
heating rates. The liquid water path (W) of a cloud layer bounded by z = z1 and z = z2
in the vertical is defined as
z2
W = ∫ ρa lcld dz (gm −2 );
z1
lcld = q cld − qsat ,
(11.18)
where lcld is the LWC which is the excess of cloud specific humidity over saturated air
in the cloud layer, and ρa is the air density. For longwave radiative transfer calculations,
cloud emissivity ( ε ) is determined as
ε =1− e−kW
(11.19)
where the absorption coefficient k is set equal to 0.13 m2g-1 for all cloud types. The
shortwave cloud radiative properties are characterized by an asymmetry factor, a single
scattering albedo, and a cloud optical thickness. The asymmetry factor is set equal to 0.86
and 0.91 for the two 0.2 – 0.68 µ m and 0.68 – 4.0 µ m bands, respectively. The cloud
optical thickness ( χ ) is defined in the model according to the Mie theory as
3W
(11.20)
χ=
2re
where re is the effective radius of cloud droplets which is taken as 20 µ m for water
clouds and 40 µ m for the ice clouds. However, the present day numerical models
interactively compute the cloud droplet sizes. Assuming random or maximum cloud
overlaps in a cloudy column, atmospheric radiative transfer calculations are performed in
a plane parallel atmosphere. A complete theory of radiative transfer in a cloudy
atmosphere is beyond the scope of these lecture notes, but the interested reader could
refer to the standard textbooks for further reading.
The atmosphere can be heated in a variety of ways involving several complicated
processes. When one considers only the dry atmosphere, most of the solar radiation will
be absorbed at the earth surface, and the air parcels after receiving this energy in the form
of sensible heat will become unstable and move vertically up (updrafts) in order to cool
down. Thus, adiabatic ascent during daytime, when ground is hot, will act as a heat sink
(cooling), while nighttime descent will act as a heat source for maintaining the
temperature of lower layers as the ground cools. However, the situation is different when
one considers the moist atmosphere. The descending air columns are cloud free; but in
rising atmospheric columns, clouds form and act as reflecting surfaces for the incoming
solar radiation but they also act as surfaces that uniformly emit infrared radiation in all
directions. Thus, the atmospheric column below the cloud will be heated as a result of
infrared radiation trapping of surface emissions but the column above the cloud top will
9
be cooled due to excess infrared emission over what it absorbs radiation in the shortwave.
One may infer that cloud tops become cooler and cloud base become warmer as a result
of cloud radiation interaction. This situation is manifested in the vertical growth of cloud
as the vertical gradient between the base and top of the cloud amplifies.
Two things happen when clouds are present in an atmospheric column. First, if
the clouds are precipitating, the ground will cools down and there will be reduction of the
thermal emission from the ground. Second, the incoming solar radiation will be reflected
back to space at the cloud tops, diminishing the supply of energy into the surface–
atmospheric (or the ocean–atmosphere) column. This may lead to an overall cooling
(clouds heat the atmosphere and cool the surface) of the column producing stable lapse
rate conditions and thereby in due course, the atmospheric column would turn again
cloud free. In fact, with such complex cloud radiation interactions, clouds are endowed
with a strong autoregulatory mechanism.
There are broadly three types of clouds, which are classified according to their level
in the atmosphere as low, middle and high clouds. The International Satellite Cloud
Climatology Project (ISCCP) has given a finer classification of clouds according to their
radiative properties. In the ISCCP classification, low clouds (cumulus, strato cumulus and
stratus) are found from 1000 hPa to 680 hPa; middle clouds (alto cumulus, alto stratus
and nimbostratus) extend from 680 hPa to 440 hPa; and the high clouds (cirrus, cirrocumulus, cirrostratus and deep convection) are found above 440 hPa up to the
tropopause. The low clouds are close to the surface or just above the well-mixed
atmospheric boundary layer. That is why low clouds emit radiation at temperatures close
to surface temperatures with large infrared radiation flux. Also the incoming solar
radiation at the cloud tops is re-radiated back to space. Hence, low clouds will produce
cooling of the atmospheric column above. In case of middle clouds, solar radiation will
be reflected back to space at the cloud tops diminishing energy input to the column
below. The clouds also radiate uniformly in all direction in the infrared. As a result the
middle clouds cool the column above and heat the atmospheric column below. Now, it
can be easily argued that the high clouds will heat the atmospheric column below, though
they (being ice clouds) are 50% permeable to the upward longwave radiation.
10