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When do contrails cool the atmosphere?
Judith Rosenow and Hartmut Fricke
Institute of Logistics and Aviation
Technische Universit¨
at Dresden
Dresden, Germany
Abstract—One of the main causes of aviation-induced climate
change is the formation of contrails. To quantify how much of
an impact contrails in general have on global warming, lots
of thorough research has been done. Individual contrails might
even decide on a cooling and a warming effect. This effect can
be in the same order of magnitude as the climate impact of
aviation-emitted carbon dioxides and nitrogen oxides. Therefore,
three-dimensional optical and meteorological investigations of
individual contrails have been performed. This paper focuses on
conditions of contrails with a cooling effect on global warming.
We found, that a flat sun position during sunrise and sunset
increases the possibility of a solar cooling effect dominating
the terrestrial heating effect. Individual contrails additionally
benefit from flight paths between East and West due to an
increased travelling distance of photons through the contrail. On
a small scale, the outcomes can be applied to arbitrary trajectory
optimization tools with a focus on environmental optimization.
On a large scale, the results can be used in environmentally
optimized Air Traffic Flow Management.
Keywords—Contrails, Aviation, Environmental Impact, Radia-
tive Forcing
I. INTRODUCTION
When water vapour emissions and ambient humidity con-
dense around exhausted soot particles and atmospheric con-
densation nuclei in a cold ambient atmosphere, a type of
human-induced cloud known as a condensation trail (contrail)
is formed, satisfying the Schmidt-Appleman-criterion [1, 2].
According to the World Meteorological Organization [3], these
man-made ice clouds transform into long-lasting cirrus clouds
known as ”Cirrus homogenitus” in an ice-supersaturated envi-
ronment. Contrails function as a barrier to the energy budget of
the Earth’s atmosphere [4–6]. They scatter incoming shortwave
solar radiation back to the sky (resulting in a cooling effect)
and absorb and emit outgoing longwave terrestrial radiation
back to the Earth’s surface (yielding a warming effect in the
lower layer of the atmosphere) [5, 7–9].
The dominant effect can be described by the radiative
forcing RF , as an imbalance in the net energy exchange
between Earth and the atmosphere in the tropopause (taking
into account the instantaneous reaction of the stratosphere) [5].
The precise amount of contrails’ RF is yet unknown and is
dependent on flight efficiency, the environment, and the time
of day. Recent combinations of several modelling techniques
to simulate the effects of contrails on global warming result in
a warming net effect for 2010 of RFContrail = 0.05 W m−2
with accuracy between −0.02 and +0.15 W m−2[5]. Single
studies provide a precise assessment of the environmental
consequences of contrails, including their cooling-related neg-
ative net effects (e.g., RFContrail =−0.007 to +0.02 W m−2
for 2005 [8]. Using global climate models and historical
air traffic data, accurate projections of the global contrail
radiative forcing for the year 2000 of RFContrail ≈0.03
(−0.01 to +0.08) W m−2[8] have been improved for 2010
to RFContrail ≈0.02 W m−2(−0.01 to +0.03) W m−2[5]
considering an increased air traffic volume by 22 % between
2005 and 2010. In a global climate model for the year 2002,
Burkhardt and K¨
archer [9] estimated that the contribution of
contrails and contrail cirrus to aviation’s radiative forcing was
RFContrail = 0.03 W m−2.
However, the uncertainties in determining the net radiative
forcing of contrails arise, among other things, from the fact
that contrails can cool the troposphere under certain condi-
tions. This happens as soon as the solar cooling effect dom-
inates the warming terrestrial effect. Therefore, the radiative
effect of single contrails has to be investigated. This paper
investigates in formation conditions of cooling contrails by
applying a three-dimensional optical model to single contrails.
The question arises, when do contrails cool the atmosphere?
Can we use this knowledge to minimize the climate impact of
aviation?
Even hydrogen-powered aircraft can induce contrails
by emitting water and fulfilling the Schmidt-Appleman-
criterion [10].
A. State of the Art
There are two main approaches for investigating the effect
of contrails on global warming. First, local investigations in
single contrails as done by Gounou et al. [11] and Forster
et al. [12], examined the radiative effect of single contrails,
concentrating on the significance of large solar zenith angles
at sunset and sunrise by applying a Monte Carlo code for
photon transport in a coarse spatial grid and by ignoring the
effect of flight performance on the optical characteristics of
the contrail. At least, phenomena like multiple scattering are
into account. Schumann et al. [13, 14] developed the Con-
trail Cirrus prediction tool CoCiP, an empiric and parametric
radiative forcing model calculating the radiative extinction of
single contrails with a low dependency on solar zenith angle
and particle radius. The time of day is only reflected in the
weak dependence on the solar zenith angle. Optical properties
are only parameterized for radiant fluxes which have already
been integrated over a hemisphere. This means that no angular
dependence due to the time of day or the spatial orientation
of the contrail can be taken into account.
Assuming a constant optical depth, Avila and Sherry [15]
applied a model created by Schumann et al. [14] to assess
the radiative forcing of individual contrails. Here, the optical
depth, width, and particle diameter of contrails are generally
parameterized. According to Schumann, an effective particle
radius is roughly calculated for each contrail class, and each
effective radius relates to particular optical characteristics. The
fact that the solar zenith angle was taken into account in the
study by Avila and Sherry is a significant advantage.
Rosenow [16, 17] has developed thorough investigations
with a granular spatial resolution and considering all possible
solar zenith angles in order to allow investigations of all day
times.
Second, climate models are used for global investigations by
treating contrails as an endless homogeneous artificial cloud
layer. For example, the Adjusted Forcing AF as an imbalance
of the Earth-atmosphere energy system has been calculated
using satellite data, taking into account a completed transition
of contrails into cirrus after stratospheric temperatures and
adjusted to regain a radiative equilibrium in the stratosphere
(assuming zero further radiative heating rates) [18, 19]. Here,
it is possible to distinguish between regions with low and high
demand for air traffic as well as the daily cycle of contrails
and cirrus [19]. AFContrail = 0.045 to 0.075 W m−2has
been quantified for contrails and contrail-induced cirrus using
satellite data from 2006.
A contrail and contrail-induced cirrus AFContrail for the
year 2010 of AFContrail = 0.05 (0.02 to 0.15) W m−2is
widely accepted [5] for the year 2010 based on a combination
of modelled and satellite data-based estimates with tolerances
in spreading rate, contrails optical depth, ice-particle shape,
and radiative transfer, as well as accounting for the ongoing
increase in air traffic [20].
Assuming RFContrail = 0.049 W m−2in 2006 and taking
into account a change in cruising altitudes, an increase in air
traffic distance until 2050 by a factor of 4compared to 2006,
an increase in alternative fuels (with reduced soot emission),
and anticipated changes in propulsion efficiency in 2050,
Bock and Burkhardt [21] predicted a global future contrail
radiative forcing of RFContrail = 0.159 W m−2. The global
distribution of the anticipated air traffic distance was taken
into consideration when determining the contrail formation’s
distribution. However, global averages were once more used
to calculate the effect of those contrails on global warming.
Chen and Gettelman [22] investigated modelling the effect
of current contrails on the size and structure of cirrus contrail
ice crystals. They projected a 7-fold increase in contrail cirrus
radiative forcing for 2050 (i.e. RFContrail = 0.087 W m−2)
compared to 2006, assuming an average increase in air traffic
movements by a factor of 4till 2050.
II. RA DI ATIV E EXTINCTION OF THE CONTRAIL
Before the optical properties of a contrail can be examined,
the micro-physical properties have to be calculated along its
whole life cycle. In order to focus on the radiative impact
of cooling contrails, we refer to external publications, which
describe the modelling of the life cycle [16, 17, 23].
The radiation, extinguished by the contrail, comes from
all directions in space. The direction Ω(θ, ϕ)is described
by the zenith angle θ= [0,π]and by the azimuthal angle
ϕ= [0,2π]. Radiative extinction takes place by scattering
(re-directing), absorption (conversion of photons into intrinsic
energy) and re-emission (conversion of intrinsic energy into
photons with a wavelength according to the higher temperature
after absorption of photons) of radiation after interaction with
atmospheric molecules, such as contrail ice crystals.
A cooling effect occurs when radiation coming from above
is scattered into the upper hemisphere by the contrail (back-
ward scattered). Heating effects occur when either radiation
coming from below is scattered into the lower hemisphere,
or when radiation is absorbed by the contrail (independent of
the direction of incidence). The further the path of a photon
through the contrail is (the flatter the angle of incidence), the
higher the probability that several scattering events (so-called
multi-scattering events) occur to a photon before it is either
absorbed or leaves the contrail again.
Radiative extinction due to scattering, absorption, and emis-
sion within the contrail is calculated using a Monte Carlo
Simulation to take into account multiple scattering events that
are likely, especially for large solar zenith angles θ[rad] [16].
This calculation depends on the non-constant geometrical and
micro-physical characteristics of the contrail. In the simula-
tion, 107individual photons of a specific wavelength λand
coming from a specific direction (Ω) are tracked on their
way through the contrail between location s1and s2(see
Figure 1). For each direction of incoming photons, the number
of backward scattered photons (i.e., scattered into the initial
hemisphere) is compared with the number of absorbed photons
(see Figure 2). Therefore, Beer’s law
Iλ(s2)
Iλ(s1)= exp −Zs2
s1
−QeAp(s)np(s) ds=Nout
Nin
(1)
is utilized, where Iλ(s1)stands for radiation coming from
direction Ω(θ, ϕ)and Iλ(s2)describes the radiation transmit-
ted in the same direction without extinction. The term radiation
can be specified by solar intensities [W m−2sr−1] and by
terrestrial irradiances [W m−2]. The extinction efficiency [-]
Qe(s) = Qs(s) + Qa(s)(2)
as sum of scattering efficiency Qs(s)and absorption ef-
ficiency Qa(s)depends on location s, wavelength, particle
size and particle shape [16, 24]. Qs(s)and Qa(s)determine
the probability that a scattering or an absorption event takes
place at location s.Ap(s)stands for the projected particle area
[m2] and the number of ice-particles npis hereafter called
ice-particle number density [m−3]. Equation 1 is interpreted
as number ratio Nout
Nin of extinguished photons of a specific
wavelength λ[µm] coming from a direction Ωor from a
specific solid angle dω
dω= sin θdθdϕ(3)
and is independent of the solar and terrestrial radiation
components reaching the contrail [16, 25].
In the simulation, a random number determines the location
s, where the next extinguishing event takes place. Qs(s)and
Qa(s)decide which event takes place and the asymmetry
parameter gHG(s)decides on the scattering angle ϑof a single
scattering event by assuming a distribution of scattering angles
following the Henyey-Greenstein phase function PHG(cos ϑ)
depending on the asymmetry parameter gHG(s)
PHG(cos ϑ) = 1
4π
1−g2
HG
(1 + g2
HG −2gHG cos ϑ)3/2(4)
The Henyey-Greenstein phase function satisfies isotropic
scattering for gHG = 0, as well as forward and backward
scattering for gHG = 1 and gHG =−1, respectively. For solar
wavelengths, ice crystals are usually characterized by a strong
forward scattering gHG ≈0.8[16].
For ice crystals in high altitude cirrus clouds, Qs(s),Qa(s)
and gHG(s)are parameterized by Wyser et al. [24] and Yang
et al. [26] as functions of wavelength, ice-particle size, shape
and density. For a mixture of typical ice crystal shapes with an
ice particle radius of rp= 10−5m, scattering is most likely
in the solar spectrum and absorption is typical for terrestrial
bands (see Table I with example efficiencies). Hence, for a
cooling contrail, the backward scattering of solar photons must
compensate the absorption of terrestrial photons.
TABLE I
SOLAR AND TERRESTRIAL SCATTERING Qs,ABSORPTION Qa
EFFI CIE NC IES A ND AS YM MET RY PARA ME TER S gHG(s)FO R IC E PARTI CLE
RADIUS OF rp= 10−5M. VALU ES AR E PARA ME TER IZ ED BY [24, 26]
Wavelength [µm] QaQsgHG
λ= 0.55 Qa= 0.009 Qs= 1.96 gHG = 0.74
λ= 10.471 Qa≈0.24 Qs≈0.13 gHG = 0.83
For each direction of incoming photons, the number ratio
Nout
Nin (Equation 1) is used to determine a weighted number
ratio Si(λ, t, dω)[m] by
Si(λ, t, dω) = Nout
Nin
win sin α, (5)
where win = 6ˆσh(t)denotes the irradiated width of the
contrail as a function of the horizontal standard deviation of
the contrail width ˆσh(t)[16], see Figure 1. Since the intensity
of the radiation depends on the angle between incoming radi-
ation and the irradiated surface, αdefines the angle between
the length axis of the contrail and the incoming photons
cos α= sin θcos ϕ. (6)
The simulation is repeated for all spatial directions, de-
scribed by θ= [0, π]and ϕ= [0,2π]with dθ= dϕ= 2°.
Si(λ, t, dω)is accumulated to number ratios of backward
scattered Sbphotons with a cooling (blue) and heating (red)
σh
Figure 1. Geometry of the radiative extinction simulation. Photons (black
arrows) irradiate the contrail (grey) along the width win at the angle αto
the longitudinal axis of the contrail. The photon’s interactions with the ice
particles in the contrail are simulated within the observation circle with radius
win between s1and s2.
effect, forward scattered Sf(black) and absorbed Sabs (red)
photons (see Figure 2).
Figure 2. Extinction of radiation (solar: straight arrows, terrestrial: wavy
arrows) when photons pass through the contrail. Cooling effects are marked
in blue, and warming effects are marked in red.
The weighted number ratios in Figure 3 clearly indicate
a strong dependence of the solar cooling (back-scattering)
potential on the incoming angles θand ϕ. Hence, only large
solar zenith angles (i.e. horizontal photon transport) hold a
cooling potential of contrails.
The longer the travel distance of photons through the
contrail (i.e. the larger αand ϕin Figure 3), the higher
the probability that an extinguishing event takes place. With
increasing travel distance and with increasing α, the number
of out-scattered photons increases, although a dominating
forward scattering is expected [24, 26, 27]. This means that
compared to vertical photon transport at noon, more photons
will be scattered during horizontal photon transport during
sunrise and dusk. The power with which the contrail is
irradiated, which will be largest at midday and minimum at
night, also affects the contrail’s radiative extinction.
The terrestrial radiative extinction is dominated by absorp-
tion, i.e. heating (Figure 4). The weighted number ratios of
absorbed terrestrial photons S↓↑a(heating effect in Figure 4)
is in the order of magnitude as the weighted number ratios
0 10 20 30 40 50 60 70 80 90
Azimuthal anlge
φ
[
°
]
0
100
200
300
400
500
600
700
800
Sf [m]
θ
=1
°
θ
=23
°
θ
=45
°
θ
=67
°
θ
=89
°
0 10 20 30 40 50 60 70 80 90
Azimuthal anlge
φ
[
°
]
0
20
40
60
80
100
120
140
Sb [m]
Figure 3. Number ratios of forward Sf(left) and backward Sb(right)
scattered photons to the total number of photons (Neval.= 107), weighted
by the sine of αand by the width win. The simulation is done for a solar
wavelength λ= 0.55 µm. Only large zenith angles θenable a cooling contrail
effect. The cooling effect is supported for large azimuthal angles ϕbetween
the contrail axis and incoming photons.
5 10 15 20
Wavelength [
μ
m]
100
200
300
400
500
600
S
f
[m]
5 10 15 20
Wavelength [
μ
m]
50
100
150
200
250
300
350
S
a
[m]
Figure 4. Wavelength-specific simulated number ratios of forward scattered
Sf(left) and absorbed Sa(right) photons to the total number of photons
(Neval.= 107), weighted by the sine of αand by the width win. The
simulation is done for vertical photon transport.
of backward scattered solar photons S↓b(cooling effect in
Figure 3).
In order to answer the question under which condi-
tions which effect dominates, the wavelength- and direction-
dependent solar intensities or terrestrial irradiances that hit the
contrail must first be calculated and then weighted with the
extinguished number ratios. For example, for radiation coming
from above and getting scattered into the upper hemisphere the
extinguished power P↓b[W m−1nm−1] is calculated by
P↓b=I S↓bΩ,(7)
where I[mW sr−1m−2nm−1] denotes the irradiance coming
from the particular solid angle dω.
The advantage of the developed method is that the extin-
guished number ratios (Beer’s law, Equation 1) are calculated
independently of the unaffected, direction- and wavelength-
specific radiation reaching the contrail and can be subse-
quently combined with any radiation values in Equation 7.
For applying, Equation 7 the unaffected solar intensities and
terrestrial irradiances are approximated following the results
of a radiative transfer model.
III. ATMOSPHERIC RADIATIVE TRANSFER
The contrail extinguishes radiation from three different
sources. First, direct solar intensity coming from a single
direction Ω(θ, ϕ)is primarily scattered by the contrail. On the
way from the sun to the contrail, this direct solar intensity
might be scattered by molecules of the atmosphere in all
spatial directions. The result is diffuse solar irradiance from
all spatial solid angles dω. The solar irradiance and intensity
are also absorbed by the Earth’s surface and by preferably
triatomic molecules of the atmosphere and re-emitted at a
longer wavelength. These are terrestrial irradiances. The longer
the wavelength of the irradiances, the higher the probability
of being absorbed by the atmosphere and Earth’s surface.
Approximating the sun and the Earth as a black body, the
expected radiative irradiances can be calculated with Planck’s
function assuming mean temperatures of 5750 K of the sun
and 288 K of the Earth’s surface. In this case, solar irradiances
are expected between 0.2<λ<1µmwith a maximum
at around λsol = 0.55 µm. Terrestrial irradiances should be
considered between 3<λ<100 µmwith a maximum at
around λterr = 10.471 µm[16]. For comprehensibility, the
investigations in this study are focused on the two maximum
wavelengths λsol and λterr representing the solar and terres-
trial spectrum. Wavelength, longitude, latitude, altitude, the
presence of clouds, time of day, and season all affect solar
intensities and terrestrial irradiances before reaching the con-
trail. The radiative transfer software package libRadtran [28] is
utilized to calculate this atmospheric radiative transfer. Details
on the calculations of solar intensity and terrestrial irradiances
are provided in [17].
Due to the high computational effort for the solution of the
radiative transfer equation, it makes sense to approximate the
radiation values beforehand and then fall back on tabulated
values.
A. Terrestrial Radiative Transfer
The terrestrial wavelength spectrum (3≤λ≤100 µm)
is modelled with the Two Stream Approximation (TSA) [29],
because of a weak angular dependency of terrestrial radiation
is anticipated since a contribution of direct irradiance is
missing [30]. With TSA all fractions of radiation from a single
hemisphere are azimuthally averaged over the half-space (with
the solid angle dω= 2π) and considered as a single irradiance
F[W m−2] (Fdown and Fup). Figure 5 right, indicates max-
imum terrestrial irradiances at λterr = 10.471 µm of Fup =
17.3507 mW/(m2nm) and Fdown = 0.0199 mW/(m2nm).
The TSA, however, does not allow a distinction between
different surfaces (and temperatures) of the Earth (e.g., snow,
ocean, desert,...). The impact of the Earth’s surface on terres-
trial radiation at flight altitude is under current investigation
by the authors.
B. Solar Radiative Transfer
Simplification to lambda max In the solar wavelength spec-
trum (0≤λ≤4µm) a TSA is not applicable, because of
its large dependency on the angle of direct solar intensity.
The radiative transfer solver DISORT (DIScrete Ordinate
Radiative Transfer solver) is used for the angular-dependent
calculation of direct solar intensities Idir(λ, t, lon,lat,Ω)
[mW sr−1m−2nm−1]. The direct beam Ω(θ, ϕ)is described
by an infinitesimal solid angle Ω(θ, ϕ)
Diffuse solar irradiances Idiff (λ, t, lon,lat,dω)
[mW sr−1m−2nm−1] depending on longitude, latitude,
altitude, time of the day year, and solid angle dω[31] are
pre-calculated with DISORT with an angular discretization of
dθ= dϕ= 2°.
In this study, the angular dependence of Idiff with high irra-
diances coming from θ= 90°, as discussed by Rosenow [16],
will be neglected for the investigation of location, time and
season of flights with cooling contrails, because, the impact
of the direct beam is orders of magnitude greater. For this
reason, we approximate hemispherically averaged solar diffuse
irradiances Idiff (θ, λ)as a function of solar zenith angle and
wavelength.
C. Approximation of Solar Radiative Transfer
From Figures 3 and 5 left, follows a strong dependence
of solar radiative extinction on the one hand on the position
of the sun and on the other hand on the direction from which
the photons irradiate the contrail. Since direct solar radiation is
coming only from a single direction, Idir(θ, λ)is approximated
as a function of solar zenith angle and wavelength.
From applying Lambert’s cosine law, where the radiant
intensity is directly proportional to the cosine of the angle
between the direction of the incident photons and the surface
normal to the zenith angle θ[32], a dependence of the
direct solar intensity from the sinus of zenith angle θcan
be expected [30], besides the dependence on wavelength.
Therewith, Idir(θ, λ)is approximated by
Idir(θ) = asin(b θ +c),(8)
where a(λ= 550 nm) = 1763,b(λ= 550 nm) = 0.01718
and b(λ= 550 nm) = 1.62 are wavelength-specific parame-
ters. The dependence of θcombines the dependence of Idir on
latitude, time of day and season and clearly shows the decrease
in intensity with increasing zenith angle. It follows that at
zenith angles with a high probability of backward scattering
(i.e.θ≈90°), lower intensities radiate onto the contrail.
Hemispherically averaged diffuse irradiances Idiff(θ, λ)are
parameterised depending on the position of the sun and the
reflectivity of the Earth’s surface. The reflectivity for diffuse
upward irradiances is parameterized for 18 surface types
defined by the surface library of the International Geosphere
Biosphere (IGBP) from the NASA CERES/SARB Surface
Properties Project [33]. Figure 6 shows the global distribution
of the surface types. Most important for this study are forest
classes 1-5, as well as classes 10-13 (see also Table II for class
description).
Diffuse downward irradiances above snow (IGBP 15) are
twice as high as the irradiances of the other surface types,
which hardly differ from each other at a sensor height of
0 25 50 75
solar zenith angle
θ
[°]
0
500
1000
1500
I
dir
[mW/(m
2
nm)]
0 50000 100000
Wavelength [nm]
0
5
10
15
Terrestrial irradiance [mW/(m2nm)]
Fdown
Fup
Figure 5. Left: Solar direct intensities as a function of θmodelled with
DISORT. Right: terrestrial irradiances at 10 km altitude coming from the
upper (Fdown) and lower (Fup) hemisphere modelled with TSA.
Figure 6. Global distribution of surface types defined by the surface library
of the International Geosphere Biosphere Project (IGBP) [33].
TABLE II
IGBP LAND COVER CLASSIFICATION SYSTEM [33].
Nr. Class name Nr. Class name
1 Evergreen needleleaf forests 10 Grasslands
2 Evergreen broadleaf forests 11 Permanent wetlands
3 Deciduous needleleaf forests 12 Croplands
4 Deciduous broadleaf forests 13 Urban and built-up
5 Mixed forests 14 Cropland/natural
6 Closed shrublands 15 Snow and ice
7 Open shrublands 16 Barren
8 Woody savannas 17 Water bodies
9 Savannas 18 Tundra
10 km. Due to the small deviation, the mean value of all
the other diffuse downward irradiances is approximated by
a single exponential function with two terms of the form
Idiff (θ) = o1exp(p1θ) + o2exp(p2θ),(9)
where o1(λ= 550 nm) = −6.821 10−5,p1(λ=
550 nm) = 0.1464,o2(λ= 550 nm) = 40,63 and p2(λ=
550 nm) = −1.3 10−3are wavelength-specific parameters.
Diffuse upward irradiances of individual surface types deviate
more strongly from one another and are parameterised indi-
vidually for applying Equation 9.
Due to the strongly deviating values and the deviating
0 25 50 75
solar zenith angle
θ
[°]
0
250
500
750
1000
1250
1500
I
diff, up
[mW/(m
2
nm)]
0 25 50 75
solar zenith angle
θ
[°]
20
40
60
I
diff, down
[mW/(m
2
nm)]
Figure 7. Solar diffuse radiances as function of θmodelled with DISORT in
10 km altitude coming from the lower (Iup) (left) and upper (Idown) (right)
hemisphere. The colour map follows Figure 6 with dashed: snow (IGBP 15),
dash-dotted: Urban (IGBP 13), dotted: Water (IGBP 17).
course, diffuse downward irradiances above snow (IGBP 15)
are considered separately. Here, a third-degree polynomial
function in the form
Idiff,snow (θ) = q1θ3+q2θ2+q3θ+q4(10)
is used with q1(λ= 550 nm) = −3.863 10−5,q2(λ=
550 nm) = −2.375 10−3,q3(λ= 550 nm) = −0.03289 and
q4(λ= 550 nm) = 70.02 for downward irradiances above
snow and q1(λ= 550 nm) = 8.893 10−4,q2(λ= 550 nm) =
−0.296,q3(λ= 550 nm) = 2.372 and q4(λ= 550 nm) =
1.520 for upward irradiances above snow.
IV. CONDITIONS OF COOLING CONTRAILS
Finally, the parameterized radiative quantities Idiff ,Idir
and Fas a function of wavelength and surface type can
be combined with the number ratios Si(Equation 5) of
extinguished photons in order to quantify the balance between
cooling and heating effects.
From Section III-A follow terrestrial irradiances at λterr =
10.471 µm of Fup = 17.3507 mW/(m2nm) and Fdown =
0.0199 mW/(m2nm). The combination with Sifrom Figure 4
yield net terrestrial heating effect of
Pterr,net = 32684.6 mW/(m nm).(11)
Obviously and supported by Equation 8 follows no de-
pendence of Idir on the land surface class. Hence, the most
important cooling contribution is constant for all surface
classes but strongly depends on θ.
On the one hand, according to Lambert’s cosine law, Equa-
tions 8 to 10, as well as Figures 5 and 7 show a decrease in
radiation for large θ. On the other hand, the number ratios of
upward scattered photons increase with θ. Hence, only small
amounts of intensities coming from large θcan contribute to a
net cooling effect. Figures 8 to 10 identify a net cooling effect
above nearly all surfaces for 70°≤θ≤80°.
This effect has to be compensated mainly by the solar direct
cooling effect. Although the cooling effects (Figure 8) are
inferior to the warming effects (Figure 9, for large solar zenith
angles between 70°≤θ≤80° the cooling effect dominates
above most of the surfaces.
0 20 40 60 80
solar zenith angle
θ
[°]
2.5
5.0
7.5
10.0
12.5
15.0
17.5
Land surface class
−38000.0 <
x
≤ − 36000.0
−36000.0 <
x
≤ − 34000.0
−34000.0 <
x
≤ − 32000.0
−32000.0 <
x
≤ − 30000.0
−30000.0 <
x
≤ − 28000.0
−28000.0 <
x
≤ − 26000.0
−26000.0 <
x
≤ − 24000.0
−24000.0 <
x
≤ − 22000.0
−22000.0 <
x
≤ − 20000.0
−20000.0 <
x
≤ − 18000.0
−18000.0 <
x
≤ − 16000.0
Figure 8. Cooling impact [mW/(m nm)] of extinguished energy of the sun and
terrestrial irradiances and sun direct intensities as characteristic of solar zenith
attitude and surface type. For all surface types, maximum cooling effects occur
for large solar zenith angles between 70°≤θ≤80°.
0 20 40 60 80
solar zenith angle
θ
[°]
2.5
5.0
7.5
10.0
12.5
15.0
17.5
Land surface class
28000.0 <
x
≤ 32000.0
32000.0 <
x
≤ 36000.0
36000.0 <
x
≤ 40000.0
40000.0 <
x
≤ 44000.0
44000.0 <
x
≤ 48000.0
48000.0 <
x
≤ 52000.0
52000.0 <
x
≤ 56000.0
56000.0 <
x
≤ 60000.0
60000.0 <
x
≤ 64000.0
64000.0 <
x
≤ 68000.0
68000.0 <
x
≤ 72000.0
72000.0 <
x
≤ 76000.0
76000.0 <
x
≤ 80000.0
80000.0 <
x
≤ 84000.0
84000.0 <
x
≤ 88000.0
88000.0 <
x
≤ 92000.0
Figure 9. Heating effects [mW/(mnm)] of extinguished solar and terrestrial
photons as a function of solar zenith angle and surface type. Red contours
indicate maximum warming effects above the snow. Dark blue negative values
indicate cooling effects for most surface types at large solar zenith angles
between 70°≤θ≤80°.
V. LOCATIONS AND TIME OF COOLING CONTRAILS
From a scientific point of view, the information on solar
zenith angles and surface types with a high potential for
cooling contrails might be interesting. However, from an oper-
ational point of view, some information on times and locations,
characterized by large θmight be more applicable. For this
reason, highly frequented air spaces are investigated regarding
0 20 40 60 80
solar zenith angle
θ
[°]
2.5
5.0
7.5
10.0
12.5
15.0
17.5
Land surface class
−4000.0 <
x
≤ 0.0
0.0 <
x
≤ 4000.0
4000.0 <
x
≤ 8000.0
8000.0 <
x
≤ 12000.0
12000.0 <
x
≤ 16000.0
16000.0 <
x
≤ 20000.0
20000.0 <
x
≤ 24000.0
24000.0 <
x
≤ 28000.0
28000.0 <
x
≤ 32000.0
32000.0 <
x
≤ 36000.0
36000.0 <
x
≤ 40000.0
40000.0 <
x
≤ 44000.0
44000.0 <
x
≤ 48000.0
48000.0 <
x
≤ 52000.0
52000.0 <
x
≤ 56000.0
56000.0 <
x
≤ 60000.0
60000.0 <
x
≤ 64000.0
Figure 10. Net effect [mW/(mnm)] of extinguished power of solar and
terrestrial irradiances and solar direct intensities as a function of solar zenith
angle and surface type. The net warming effect (red contours) is still maximum
above the snow. Light blue negative values indicate net cooling effects for
most surface types at large solar zenith angles between 70°≤θ≤80°.
Earth surface types and θ. First, θalong the North Atlantic
Track System (NATS) at latitudes ≈55° N is investigated in
a sun chart (Figure 11). It becomes clear that along the major
latitudes in the northern hemisphere, solar zenith angles of
70°≤θ≤80° are common. From September to March the
sun does not reach smaller angles θ, and during the summer
months, the sun only shines horizontally on the contrail during
sunrise and sunset. Note, the solar elevation in Figure 11 is
defined as 90°−θ.
Figure 11. Sun chart diagram for latitude φ= 55° N. The axis label of solar
elevation is defined as 90°−θ. Hence, the solar elevation of 20corresponds
to θ= 80°. Along the North Atlantic tracks, the sun is irradiating contrails
with 70°≤θ≤80° between September and March the whole day and for
the rest year during sunrise and sunset. The diagram is generated with a tool
provided by www.sunearthtools.com.
Figure 12 approximates situations in the Northern hemi-
sphere with 70°≤θ≤80°, where contrails could cool the
atmosphere. Those conditions do not occur near the equator.
However, condensation trails are less likely to form at these
latitudes due to higher air temperatures. In mid-latitudes, those
conditions are fulfilled in the winter time, in the vicinity of the
North pole, 70°≤θ≤80° occurs mainly in the summer time.
From this follows, that air spaces with highly dense air traffic
(i.e. the NATS mid-latitudes in the Northern hemisphere) hold
the potential of contrails with a cooling effect. The probability
of the cooling effect can be increased when flying during
sunrise and sunset.
VI. CONCLUSION
In this study, conditions of cooling contrails have been
investigated. Those conditions can be described at the best by
the solar zenith angle θ. We found, the optimal conditions of
cooling contrails at 70°≤θ≤80°, because horizontal photon
transport increases the probability of direct solar radiation
scattered into the upper hemisphere. However, the larger the
Figure 12. Situations with solar zenith angles between 70°≤θ≤80°
described by latitude, month and time of day. For example ”whole” stands
for all day, and ”0.5 hours” means half an hour before and after sunrise
and sunset. Conditions with 70°≤θ≤80° have been identified as most
beneficial for a dominating cooling effect of contrails
solar zenith angle, the lower the solar intensity irradiated
on a horizontal contrail surface. For this reason, θ= 90°
does not yield a maximum cooling effect. The cooling effect
of contrails is described by the energy budget at contrail
altitude, which is disturbed due to radiative extinction within
the contrail. Absorption of photons and backward scattering
of photons coming from the lower hemisphere lead to a
warming effect which must be compensated by a cooling
effect described by backward scattered photons coming from
the upper hemisphere. While terrestrial long-wave radiation
coming mainly from the lower hemisphere is mainly absorbed,
solar direct short-wave radiation coming from the sun is
scattered with a higher probability. In our approximation,
the terrestrial warming effect is independent of the time and
the location of the contrail. A solar warming effect due to
scattering and absorption of diffuse solar radiation coming
from all directions in space strongly depends on the Earth’s
surface type beyond the contrail. The solar cooling effect as a
result of scattered direct solar radiation significantly depends
on latitude, time of the day and year and is summarized in
Figure 12.
Finally, we found, that especially mid and high latitudes
hold the potential to induce cooling contrails, due to frequently
occurring large solar zenith angles. This statement is supported
by a higher possibility of contrail formation in those regions.
Fortunately, a significant amount of air traffic is taking place in
mid and high latitudes. Furthermore, we found that the impact
of the land surfaces is not significant, except on snow surfaces,
because the contribution of diffuse solar radiation to the energy
budget is of minor importance, compared to the impact of the
direct beam.
However, in this study, a few simplifications have been
carried out, which impact the results and could not be com-
pletely quantified. For example, the reduction of the investi-
gation to two single wavelengths has been barely broached by
Rosenow [16] but requires additional investigations. Second,
the simplification to hemispherically averaged solar diffuse
and terrestrial irradiances, although supported by [30] is cur-
rently under investigation by the authors. Finally, an additional
natural cloud cover would weaken both the cooling and the
warming effect. This was not considered in this study.
However, this paper clearly emphasises that the radiation
effect of contrails is strongly direction and sun position
dependent and can hardly be quantified in two-dimensional
studies with infinitely extended contrails.
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