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geophysical institute
GEOPHYSICAL INSTITUTE
UNIVERSITY OF ALASKA, FAIRBANKS
Improved Contrail Forecasting Techniques for the
Subarctic Setting of Fairbanks, Alaska
Gerd Wendler and Martin Stuefer
UAG R-329
Aug 2002
1
Improved Contrail Forecasting Techniques for the subarctic setting of
Fairbanks, Alaska
by
G. Wendler and M. Stuefer,
with help from
B. Moore, J. Broussard, C. Cole, J. Curtis, S. Nakanishi, M. Robb, H. Stone
ABSTRACT .......................................................................................................................................................... 3
INTRODUCTION .................................................................................................................................................. 4
OBSERVATIONAL METHODS .............................................................................................................................. 5
Digital photography......................................................................................................................................... 5
FAA flight data ................................................................................................................................................ 7
Radiosonde data from Fairbanks ................................................................................................................... 8
CONTRAIL DATABASE AND OVER-FLIGHT CHARACTERISTICS .......................................................................... 9
ATMOSPHERIC CONDITIONS AND STATISTICS.................................................................................................. 14
VALIDATION OF EXISTING CONTRAIL FORECASTING ALGORITHMS WITH UAF DATASET ............................. 19
Appleman Algorithm ..................................................................................................................................... 19
Hanson Algorithm ......................................................................................................................................... 21
Schumann Algorithm .................................................................................................................................... 22
Model verification.......................................................................................................................................... 25
DISCUSSION ..................................................................................................................................................... 27
FUTURE WORK................................................................................................................................................. 32
REFERENCES .................................................................................................................................................... 33
APPENDIX A..................................................................................................................................................... 35
THIS RESEARCH WAS SUPPORTED BY THE UNIVERSITY PARTNERING FOR OPERATIONAL
SUPPORT (UPOS) IN COLLABORATION WITH THE APPLIED PHYSICS LABORATORY, JOHNS
HOPKINS UNIVERSITY, BY A GRANT FROM DOD.
2
Abstract
Jet contrails can be frequently observed in the subarctic setting of Fairbanks, Alaska,
much like in the contiguous United States. Since March of 2000, continuous digital
imagery of the sky was obtained, supported by FAA flight data and radiosonde ascents at
the Fairbanks International Airport. There were a total of 2504 over-flights (March 2000July 2002) at Fairbanks, but for a great number of these, contrail observations were not
possible due to clouds and/or darkness. For 590 cases, the formation of contrails could be
confirmed; their life span varied widely from a few seconds to several hours. In general,
cold temperatures and high relative humidity at flight level favored the formation of
contrails. These conditions are frequently found in the upper troposphere close to the
tropopause.
Using our substantial database, different existing algorithms were tested and, in part,
improved in order to predict contrail formation and lifetime. The best results were
obtained with an algorithm described by Schumann (1996) and an aircraft specific
contrail factor of 0.036 g/kgK. For contrails within 4 hours of the radiosonde ascents, a
combined hit rate for correctly forecasting the occurrence and non-occurrence of contrails
of 92% was obtained.
3
Introduction
Aircraft contrails have been investigated since the advancement of high-flying aircrafts.
Large differences in contrail occurrence can be observed; some of them disappear after a
few seconds, others last for longer time periods and might drift with the winds aloft,
while some might spread over parts of the sky or even the whole sky. Schmidt (1941)
and Appleman (1953) have performed pioneer work in describing the thermodynamics
involved in contrail formation. The water vapor saturation pressure of air is a logarithmic
function of the temperature; therefore cold air masses are especially susceptible for the
formation of contrails (e.g. Schrader 1997). At very low temperatures, contrails can form
in relatively dry air; however, they will only persist if the ambient air temperature is
moist and supersaturated with respect to ice. Further, the stability of the air and the wind
shear are also of importance. Aircraft characteristics such as aircraft speed, engine type,
fuel type and consumption along with the sulfur content of the fuel play a major role in
the characteristics of contrails (Busen and Schumann 1995). For significant horizontal
spreading across the sky, a strong wind shear aloft combined with super saturation with
respect to ice is essential.
Observations of contrails have been carried out by ground-based observers, all sky
cameras and lidar instruments (Sassen et al 2001), and also from satellites (Carleton and
Lamb 1986). Travis et al. (1997) have done modeling efforts, and a comprehensive study
(IPCC 1999) discusses possible influences of increasing air traffic on the structure of the
atmosphere and the climate of the earth.
Contrail formation facilitates the detection of an aircraft, which is important for military
purposes. Further, contrails have a strong influence on radiation budget and as such, are
of importance for climate change (Seinfeld 1998). In the short wave region of the solar
spectrum, cirrus contrails tend to scatter the incoming radiation (mostly forward) (e.g.
Gayet et al. 1996). In the infrared region they contribute to the back radiation of the
atmosphere. Kuhn (1970) carried out an early investigation on the infrared radiation,
while Smith et al. (1998) performed more recent studies. Combined, these processes can
contribute either to cooling or warming, depending on the time of the day, season and
latitude. However, integrated over the whole Earth and the annual cycle, they contribute
to warming (Seinfeld 1998).
In the present study, we examine the atmospheric conditions favorable to the
formation/non-formation of contrails for the subarctic setting of Fairbanks, which lies
below the much traveled flight corridor from Europe to Asia. We have collected the most
comprehensive data set on contrail formation for the subarctic, and one of the largest
worldwide, on which our analyses is based.
4
Observational Methods
Digital photography
Continuous all sky digital dome camera imagery has been taken since 1 March 2000.
This camera is directed at the zenith and equipped with a fish-eye lens. It is situated
below a transparent plexiglas dome on the roof of the Geophysical Institute of the
University of Alaska in Fairbanks. The temporal spacing between successive images is
2.5 minutes. The characteristics of the typical dome camera image and the camera
position are comprised in Table 1.
Table 1: All sky digital imagery characteristics
Size
Resolution
Colors
File Format
File Length
Exposure Time
Scanner
Latitude/ Longitude
Altitude
1024 x 1536
72 x 72 ppi
16.7 Mill. Colors (32 Bit)
JPEG
>150 KB (185 KB typical during day)
1/250 sec
KODAK DC265 ZOOM DIGITAL
64° 51' 22" N / 147° 50' 58" W
225 m a.s.l.
In Fig. 1 a typical example is presented for 13:21h LT on 31 July 2002. This contrail
lasted more than 10 minutes. In addition to single image analysis, we produced daily
digital movies from all the images, so that the occurrence, evolution, and duration of
contrails can be evaluated.
Direct, visual observations of aircraft over-flights occasionally reveal the formation of
contrails, which last only a few seconds. These cases will not necessarily be captured by
the operational all-sky camera as the time intervals of obtaining images is 2.5 minutes.
Hence, the flight pass in relation to the station is important. If the flight takes place at the
edge of our visibility range, it is unlikely that we will obtain an image. However, if the
flight is more toward zenith, the time the plane is visible will likely be longer than the
time interval between two subsequent images.
In addition to the all-sky camera, we installed an Axis Network 2110 camera on the 3rd
floor of the International Arctic Research Center building adjacent to the Geophysical
Institute. The camera faces south towards the Alaska Range, and supplies additional
information for this field of view.
5
Figure 1: All sky digital dome camera image on 31 July 2002 showing a long lasting
contrail.
6
FAA flight data
Using Flyte Trax 2001 Version 2.0 software we are able to track all commercial flights
overhead Fairbanks in real time. Besides aircraft position, the flight identification
number, aircraft type, departure and destination airport, flight altitude, ground-speed and
estimated time of arrival at the destination are obtained. For contrails analysis, all
daytime aircraft passages within 50 miles of Fairbanks are recorded. Figure 2 shows an
example print- screen of the flight tracking system.
Figure 2: Example of flights in the Fairbanks area at 23:54 h GMT, 27 March 2002. The
circles are centered at Fairbanks with a radius spacing of 10 miles (16 km). Flight KLM
9196 from Anchorage to Amsterdam, flying of an altitude of 9450 m produced a contrail
lasting only few seconds.
We have recently changed our FlyteCom service, which will summarize all daily flights
in the Fairbanks area (daily e-mailed), reducing the time needed for data processing, and
hence leaving more time for data analysis.
7
Radiosonde data from Fairbanks
Atmospheric soundings are carried out at Fairbanks Airport twice a day at 00:00 h and
12:00 h GMT nominal time.
Sounding data are available online via
http://weather.uwyo.edu/upperair/sounding.html. The main station characteristics are
given in Table 2.
Table 2: Radiosonde data, Fairbanks
Station identifier
Station number
Latitude
Longitude
Station elevation
PAFA
70261
64.81 N
-147.86 E
138 m a.s.l
Figure 3: Atmospheric sounding carried out from the Fairbanks International Airport at
00:00 h GMT, 27 March 2002.
8
Figure 3 shows the Fairbanks temperature, dew-point temperature and wind profiles up to
an altitude of 30,350 m for 27 March 2002. A number of atmospheric indices
characterizing the atmospheric stability are also presented.
Contrail database and over-flight characteristics
Systematic observations of contrails started on 1 March 2000; each observation includes
the following parameters:
1. Date and time of the position of closest approach of an aircraft to the University
of Alaska, Fairbanks. The GMT and local time are included.
2. Aircraft specific data at the time of passing Fairbanks:
• Aircraft identification (ID)
• Origin and destination of flight
• Route direction
• Aircraft type
• Altitude
• Speed
• Duration to destination airport
• Closest approach to the University of Alaska Fairbanks (estimated from
flight visualization program).
3. Contrail data
• Contrail formed (yes/no)
• Number of digital dome camera frames showing the contrail
• Duration of contrail visibility derived from dome camera pictures
• Duration derived from direct (by eye) observation (short/middle/long
lasting)
• Duration derived from direct observation in minutes
• Total duration representing the larger value of direct observation or domecamera duration in minutes
• The degree of determination in case of a 'no contrail' observation. If the
aircraft can be spotted directly and there exists no contrail (rare case), the
degree of determination is 100%. A sky already covered by clouds, 'nocontrail' observation based only on digital dome camera images (also no
aircraft visible) reduces the degree of determination.
• Comments about the dissolving process, the drifting of contrails, the
observing person and additional information such as the present Fairbanks
weather situation (visibility) or hand-held pictures taken of the aircraft.
From 1 March 2000 to 31 July 2002, 2504 over-flights have occurred. The most frequent
direction of the flights was towards the West (33.5 %) with destination airports in Japan.
A number of 29.9 % of the aircrafts originated in Anchorage going north to Europe via
the polar route; such a case was depicted in Figure 2. Nearly 1/5 of all flights came from
Europe going south to Anchorage, while all other directions were less frequent. See
Figure 4 for more details.
9
Figure 4: Directional frequency distribution of aircraft over-flights at Fairbanks, Alaska,
between March 2000 and June 2002.
The most frequently observed airplane types are Boeing 747-200 and Boeing 747-400,
together comprising 65 % of all observations. Typical cruising speeds are between 400
knots (741 km/h) and 550 knots (1019 km/h); 96% of all flights fall in this speed range.
In Figure 5 a frequency distribution of the airplane speeds is presented. The cruising
altitudes vary from flight-level 210 (6,400 m) to flight-level 450 (13,716m). Nineteen
percents of all flights of our database are between flight-levels 280 (8,534m) and 300
(9,144m), and 40.9 % of the flights are between flight-levels 340 (10,363m) and 360
(10,973m). The most frequent altitudes are flight levels 290 (8,840m) and 360 (10,973
m). The observed aircraft types overhead Fairbanks are shown in Table 3.
10
Table 3: Aircraft types and frequencies
Aircraft Type
Boeing 737-200
Boeing 737-Q
Boeing 747-200
Boeing 747-300
Boeing 747-400
Boeing 767-300
Boeing 777-200
MD11
Airbus 340
Airbus 343
others
unknown
#
119
97
848
119
643
14
230
30
7
11
61
281
Percentage
4.8%
3.9%
34.5%
4.8%
26.1%
0.6%
9.3%
1.2%
0.3%
0.4%
2.5%
11.4%
70%
60%
50%
40%
30%
20%
10%
0%
551600
501550
451500
401450
351400
301350
251300
201250
Groundspeed (knots)
Figure 5: Histogram presentation of aircraft cruising speeds at Fairbanks, Alaska, March
2000-July 2002
11
The closest horizontal distance between the aircrafts and our point of observation at UAF
is
estimated
from
the
'FlyteComm'
visualization
software
(http://www.flytecomm.com/products/flytetrax.html) in order to ensure correct
identification of contrail observations. The technical specifications of the digital dome
camera and the fish-eye lens allow a direct observation of contrails within a radial
spacing of about 16 km depending on the weather situation and the sun-angle. Sometimes
contrails drift with prevailing high wind speeds into the camera field-of-view, therefore
we record over-flight data within a range of 50 km to UAF.
Prior to 1 January 2002 the contrails analysis is mostly based on dome camera imagery.
As the camera picture frequency is 2.5 minutes, the separation of 'no contrail' cases from
short contrails, which lasted only a few seconds, is not always possible. Therefore in
2002 we have increased direct observations 'by eye'.
Figure 6: Histogram presentation of observed contrail/no-contrail cases with different
lifetimes, Fairbanks, Alaska, March 2000-July 2002. A non-linear scale is applied
to the x-axis.
For comparison with atmospheric sounding data, the contrail database has been restricted
to cases with accurate observations of the contrail duration (lifetime) and to observations
within 4 hours of the sounding ascent. According to the duration we classified 3 different
contrail types: the short duration contrails with a lifetime less or equal to 1 minute, the
12
medium duration contrails lasting between 1 and 10 minutes and the long duration
contrails lasting longer than 10 minutes. This reference database contains a number of 20
'no contrails', which refer to cases when an aircraft has been spotted while not forming a
contrail; these cases require 'by eye' observations and excellent visibility. A number of
223 contrails were confirmed, ranging from short tails, which dissolve within a few
seconds, to long lasting ones, which could be seen in some cases for several hours. Our
database includes 78 short-lasting contrails, 47 medium duration contrails (<10 minutes)
and 98 contrails lasting longer than 10 minutes (see Figure 6). The maximum observed
duration has been 6 hours, which occurred on 23 May 2002 when a Boeing 777-200 overflying Fairbanks from East to West formed a contrail at 10,360 m altitude.
Table 4: Altitude distribution of the different contrail classes relative to 10,000m level
and to the height of the tropopause.
at or below 10000 m
above 10000 m
at or below tropopause
above tropopause
no
contrail
20.0%
80.0%
25.0%
75.0%
short
duration
41.0%
59.0%
42.3%
57.7%
medium
duration
61.7%
38.3%
59.6%
40.4%
Altitude above/below Tropopause (m)
4000
long
duration
53.1%
46.9%
68.4%
31.6%
no contrail
short duration
medium duration
long duration
3000
2000
1000
0
0%
10%
20%
30%
40%
50%
60%
-1000
-2000
-3000
-4000
Figure 7: Altitude distribution of different contrail classes in reference to the tropopause.
Positive values refer to contrails (no-contrails) above the tropopause.
13
Table 5: Number of contrail cases in different altitude ranges above/at/below the
tropopause
Altitude Range
(m)
>3500
2501 - 3500
1501 - 2500
501 - 1500
-499 - 500
-1499 - -500
-2499 - -1500
-3499 - -2500
<= -3500
sum
no
contrail
2
3
5
5
0
2
2
1
0
20
short
medium
long
duration duration duration
0
0
0
0
0
0
4
2
0
29
7
5
25
19
59
15
15
29
5
4
4
0
0
0
0
0
1
78
47
98
The flight levels show statistically significant different frequencies for the 'no-contrails'
compared to contrail cases. At flight-level 290 (8,840m), the preferred altitude of the
lower traffic corridor overhead Fairbanks, a frequency peak is obtained for the contrail
cases. 'No contrails' have been observed frequently at altitudes above 10,000m (Table 4,
Table 5). Medium and long lasting contrails show an almost symmetric frequency
distribution around the tropopause (Figure 7). Short lasting contrails are most frequently
observed above the tropopause, whereas no-contrails showed 2 maxima; the dominant
frequency maximum is in the lower stratosphere, while a secondary maximum was found
in the upper troposphere. At the height of the tropopause, no 'no-contrails' were observed.
The tropopause altitudes were calculated for all atmospheric soundings available between
March 2000 and July 2002. Figure 8 indicates daily and seasonal variations; we obtained
average tropopause levels overhead Fairbanks for summer of 10,650m (June-August) and
for winter of 9,470m (December-February).
Atmospheric conditions and statistics
Atmospheric characteristics favorable for contrails formation are investigated for the
contrails reference database from June 2000 to July 2002.
Statistical analysis has been carried out for temperature, humidity, wind data and
different indices including stability and wind-shear data. Histograms of temperature,
relative humidity, mixing ratio and mixing ratio deficit were calculated (Figures 9-12).
The mixing ratio deficit was defined as the difference between saturation mixing ratio
and the measured mixing ratio.
The temperatures at aircraft altitudes derived from atmospheric soundings ranged from
-35ºC to -70ºC (Figure 9). Almost all contrail observations occurred at altitudes with
14
temperatures below -45ºC; one long lasting contrail was observed at a temperature of
-35ºC indicating an error in the temperature measurement of this specific radiosonde
ascent. Eighty-five percent of the no-contrails occurred at temperatures above -52.5ºC.
This temperature threshold of -52.5ºC for the separation between contrails and nocontrails yields to a “success rate” of 83%.
The humidity measurements at aircraft altitudes show significant differences especially
between no-contrails and long lasting contrails (see Figure 10). Eighty-five percent of ‘no
contrails’ were observed with relative humidity values of less than 25%. In contrast 97%
of the 'long-lasting contrails’ occurred during situations with relative humidity values
higher than 25%. Nevertheless, a success rate for the separation between no-contrails and
all contrails of only 75% was derived, as 64 % of the short lasting contrails were
observed at a relative humidity less than 25%.
Figure 11 shows mixing ratio values; the mixing ratio is in contrast to the relative
humidity an absolute measure of the water content in air (g water vapor/kg air). Values
are small for no-contrail cases, and increase with the increasing lifetime of the contrails.
For example, the most frequent value for long lasting contrails is about four times as high
(0.04g/kg) as for no contrail occurrences (about 0.01g/kg).
The mixing ratio deficit is statistically significant different between the contrail and nocontrail cases (Figure 12). Mixing ratio deficits of less than 0.15 g/kg were derived for
98.7% of all contrail cases, whereas 70% of the no-contrails showed deficits above 0.15
g/kg. A mixing ratio deficit of 0.10% as threshold between contrail and no-contrail cases
yields a hit rate of 88.5%. The figure indicates similar mixing ratio deficit frequencies for
the middle and long lasting contrails.
Atmospheric indices were calculated using the 250 hPa and 500 hPa sounding levels. The
indices contain temperature, dew point, wind direction and wind speed data. We
compared the indices with our contrail observations; the statistical analysis revealed no
significant differences between contrail classes. This indicates that for the formation of
contrails, stability criteria of the atmosphere are of secondary importance. However, we
are still investigating their effect on spreading of contrails.
15
Figure 8: Altitudes of the tropopause derived from atmospheric soundings at Fairbanks
between March 2000 and July 2002. The red line shows the 7-point running average.
16
70%
no contrail
60%
medium duration
short duration
long duration
Frequency
50%
40%
30%
20%
10%
0%
-70
-65
-60
-55
-50
-45
-40
-35
Temperature (ºC)
Figure 9: Temperatures at flight level for different contrail lifetimes and “no-contrail”
cases at Fairbanks, Alaska, March 2000 – July 2002.
60%
no contrail
short duration
50%
medium duration
Frequency
long duration
40%
30%
20%
10%
0%
0
10
20
30
40
50
60
70
80
Humidity (%)
Figure 10: Relative humidity at flight level for different contrail lifetimes and “nocontrail” cases at Fairbanks, Alaska, March 2000 – July 2002
17
50%
no contrail
45%
short duration
medium duration
40%
long duration
Frequency
35%
30%
25%
20%
15%
10%
5%
0%
0
0.02
0.04
0.06
0.08
0.1
Mixing Ratio (g/kg)
Figure 11: The mixing ratio at flight level for different contrail lifetimes and “nocontrail” cases at Fairbanks, Alaska, March 2000 – July 2002
long duration
medium duration
short duration
no contrail
100%
90%
80%
Frequency
70%
60%
50%
40%
30%
20%
10%
0%
0.24
0.20
0.16
0.12
0.08
0.04
0.00
Mixing ratio deficit (g/kg)
Figure 12: The saturation mixing ratio deficit for different contrail lifetimes and “nocontrail” cases at Fairbanks, Alaska, March 2000 – July 2002
18
Validation of existing contrail forecasting algorithms with UAF dataset
Schumann (1996) has given an overview of the history of aircraft condensation contrails
research and the different studies, which have been carried out in order to understand the
factors responsible for the contrail formation. The objectives of previous research efforts
have not been uniform. For example, the military is interested to avoid flight levels at
which contrails form, as it facilitates the detection of aircraft. Hence, scientists working
for the Air Force, have developed contrail forecast algorithms, which are in operational
use (Shull 1998). Furthermore, due to the steadily increasing aircraft traffic, contrails are
of interest for scientists investigating atmospheric radiation transfer processes, the
chemical state of the atmosphere, and their potential for climatic change (Grassl 1990).
Appleman Algorithm
Pioneer work to describe the thermodynamics of the contrails formation process has been
carried out by Schmidt (1941) and Appleman (1953). The Schmidt/Appleman algorithm
provides a threshold decision for contrails, and most of the succeeding algorithms are
based on their investigations. Theoretical calculations were used for describing the
thermodynamics and the possible phase changes, which might occur when mixing the
aircraft engine exhaust with the ambient air. The input parameters for the contrail forecast
model are the ambient air pressure, temperature and relative humidity. By considering the
entrainment of the heated and moist exhaust gas to the ambient air, a 'critical' temperature
is calculated as the threshold temperature to determine if saturation occurs. A contrail is
composed of ice crystals and to visibly detect a contrail, a minimum ice crystal content of
0.004 gm-3 is assumed. The contrail formation requires saturation with respect to water, a
phase change from water droplets to ice crystals will occur immediately.
The temperature increase, DT , of the affected ambient air due to the combustion of one
mass unit of fuel is calculated according to:
DT =
Q
k ⋅ N ⋅ cp
(1)
†
where Q denotes the liberated heat by the combustion of one mass unit of fuel
10000cal
† = 41.84 ⋅ MJ ), and k is the ratio of exhaust gas to the mass of fuel ( k =12
(Q ª
g
kg
kg/kg). The mass†ratio of entrained environmental air to exhaust gas is denoted by N;
thus k ⋅ N characterizes the mass of environmental air, which is affected by the
† ratio N depends strongly on the distance
combustion of one mass unit of fuel. The mass
†
of the considered mixing parcel behind the aircraft, the combustion efficiency of the
engines, and the density and stability of the atmosphere controlling the spreading of the
†
exhaust gases. The value of N ranges from 0 immediately behind the exhaust to infinity.
The temperature increase DT further depends on the specific heat of air ( c p ); the specific
heat of dry air considering constant pressure is used for c p (=0.24 cal/gºC =1004 J/kgK).
Estimating the emission index for water vapor as amount of water vapor produced by the
†
†
†
19
†
combustion of 1 kg of fuel, k H 2O (ª1.4 kg/kg), the increase of mixing ratio Drf (g/kg) is
derived:
k ⋅1000
(2)
Drf = H 2O
k⋅N
†
†
The combination of equations (1) and (2) leads to:
Drf k H 2O†⋅ c p ⋅1000
=
DT
Q
(3)
The increase of the mixing ratio per degree temperature increase due to the entrained
Drf
exhaust is† independent of N. The ratio
is called the contrail factor (CF), it is
DT
characteristic of aircraft engine combustion; an original value of 0.0336 g/kgK has been
derived. Appleman compared the mixing ratio deficit (difference between saturation
mixing ratio res and actual atmospheric mixing ratio re ) with the contrail factor to obtain
†
the ambient temperature T and relative humidity fw thresholds
for certain pressure levels p
and mixing states N, which are necessary for saturation. Using equation 2, the amount of
water vapor Dr (g/kg) necessary for saturation with respect to water in an air-parcel
†
†
which is mixed with aircraft
exhaust is given by:
Dr = Drf - (res - re ) =
k H 2O ⋅1000
f
- res ⋅ (1- w )
†k ⋅ N
100
(4)
As the affected air is undergoing heating due to the exhaust, saturation requires moisture
excess (to avoid a lowering of f w below 100%). This moisture excess due to the heating
was formulated
as the difference of the saturation mixing ratios:
†
Drh = rs (T + DT) - rs (T)
(5)
† as DT(N) using equation 1. Thus for the
The temperature difference DT is calculated
maintenance of saturation equation 4 becomes:
Dr = Dr -†(r - r ) - Dr
(6)
f
es
e
h
†
†
Appleman (1953) calculated critical temperatures ( Tcrit ) based on equation 6 for certain
pressure levels and relative humidity values for Tcrit = T(Dr = 0) , and for a range of N
values from
† 58 to 7000. Using the stipulation that a faint visible and a distinct visible
contrail require solid water contents of 0.004 and 0.010 g/m3 respectively, critical
†
temperatures were also calculated for saturation with respect to ice taking into account
†
the excess moisture content for the production of visible contrails.
Because the saturation
vapor pressure is higher over water than over ice at an identical temperature, the
comparison of the critical temperatures for saturation with respect to water and for
saturation with respect to ice showed that the critical temperatures for water trails must be
colder. Thus, taking into account the excess moisture content, which is necessary for the
formation of a visible contrail, saturation with respect to water satisfies the requirement
for the formation of contrails. For given environmental pressure and relative humidity
values, critical temperatures were calculated for various mixing states between the
20
environmental air and the exhaust (N). These critical temperatures show a maximum at a
certain mixing state, which can be used as a general threshold for contrail-formation.
Appleman showed with his algorithm that no contrails are likely to form at temperatures
above -29ºC. He further compared his results with the U.S. Standard Atmosphere, and
found a favorable pressure range for the formation of contrails to be between 240 and 205
hPa.
Hanson Algorithm
Hanson and Hanson (1995) published an algorithm, which is based on Appleman's
concept. The development of different aircrafts and different engines required changes in
the estimation of the characteristic values defining the fuel combustion and thus in the
contrail factor derived originally by Appleman (1953). They also calculated critical
temperatures for the formation of contrails. Visible contrails are likely to be formed if the
air is at or below a critical temperature, formulated as a function of pressure, humidity,
and an aircraft specific contrail factor. The formation of contrails requires enough
moisture, supplied by the exhaust and is mixed with the ambient air, in order to maintain
Dr
saturation with respect to water. Instead of a linear ratio
, the curvature in a vapor
DT
pressure versus temperature curve for saturated air is considered. In order to specify the
temperature dependence of the saturation vapor pressure, Hanson has used the approach
of Goff and Gratch (1946) instead of the Clausius-Clapeyron equation. Goff & Gratch
†
obtained more accurate pressure values, especially for low temperatures.
The Goff &
Gratch formulation is included in the programming script in Appendix A.
With the pressure for dry air, p d, the pressure for humid air, p (p= pd +e), and the gas
constants for dry air, Rd, and for vapor, R v, the mixing ratio (in g/kg) is converted into
terms of vapor pressure by:
r=
Rd ⋅ e
Rd ⋅ e
⋅1000 =
⋅1000
Rv ⋅ pd
Rv ⋅ ( p - e)
(7)
For p>>e a good approximation is:
[email protected]
Rd ⋅ e †
622 ⋅ e
⋅1000 =
Rv ⋅ p
p
(8)
The amount of moisture released during the mixing between exhaust and environment,
drf
which was
= CF ), is
† estimated by Appleman (equation 3) as contrail factor (
dT
converted into terms of vapor pressure change using relation (8):
de p ⋅ CF
@
dT
622
(9)†
†
21
An isobaric mixing process is assumed. Hanson used contrail factors defined by Peters
(1993); the values were for the non-bypass engine type 0.030 g(kgK)-1, the low-bypass
engine 0.034 g(kgK)-1, and for the high-bypass engine 0.039 g(kgK)-1. By replacing the
vapor pressure, e, with the saturation vapor pressure, es, in equation 9, Hanson calculates
a tangent to the saturation vapor pressure curve (formulated by Goff & Gratch) in order
to get a stipulation for critical temperatures. The critical temperature, Tcrit, for contrail
formation is derived, where the slope of the saturation vapor pressure versus temperature
de
p ⋅ CF
curve coincides with the mixing line from the exhaust ( s =
). This formulation is
dT
622
valid for initially saturated conditions. Relative humidity values less than 100%
(f<100%) are taken into account by the product of the saturation vapor slope with a value
100
:
†
f
100 des p ⋅ CF
⋅
=
f dT
622
†
(10)
In accordance with observations, decreasing critical temperatures are resulting for
decreasing relative humidity values; nevertheless a physical explanation of the direct
†
implementation
of the relative humidity, f, as a factor in equation 10 would fail.
Schumann Algorithm
In analogy to former studies, Schumann's algorithm (1996) is based on the calculation of
a critical slope in order to derive a threshold temperature for contrail formation. Besides
the effects of the phase changes due to mixing of exhaust fumes with ambient air,
Schumann also considers the transformation of combustion heat into kinetic energy of the
aircraft's wake motion. Appleman described the contrail factor, CF, as the ratio of the
change in the mixing ratio, r, to the temperature change due to the mixing of exhaust
gases to the ambient air. According to equation 9, the contrail factor can be considered as
ratio of changes of the partial pressure of the vapor, e, to temperature changes:
CF =
Drf 622De 622G
@
=
DT
pDT
p
(11)
Considering the propulsion efficiency, h, of an aircraft's engine, Schumann defined the
De
parameter†G (=
) as:
DT
G=
k H 2O c p p
0.622Q(1- h)
(12)
†
The combination of equations 11 and 12 leads to:
†
22
CF =
k H 2O c p 1000
(13)
Q(1- h)
This equation differs from equation 3, derived by Appleman (1953), only by the factor
( 1- h ) in the denominator, which accounts for the amount of work performed for the
†
aircraft drag. The propulsion efficiency, h, is defined as:
Fv
(14)
†
Qm f
The parameter F denotes the aircraft thrust, v denotes the true air speed, and m f , the rate
of fuel flow. Typical values of propulsion efficiencies h for modern bypass turbofan
engines are
† between 0.3 and 0.4 (Schumann, 1996). He obtained critical temperatures Tcrit
for contrail formation for a saturated ambient air (relative humidity f=100%) in analogy
†
to Hanson (1995) by calculating the temperatures, where the slope of the saturation vapor
de
pressure equals the slope G in a vapor pressure e versus temperature T curve ( s = G ).
dT
The non-saturated conditions (f<100%) are derived according to:
h=
Tcrit, f = Tcrit,100 -
(es T
crit,100
-eT
crit , f
)
G
= Tcrit,100 -
(es T
crit,100
- f es T
crit , f
G
)
†
(15)
The index 100 in equation 14 accounts for the previously calculated saturated threshold
condition, and the index f considers the values for the actual relative humidity in the
†
environmental
air. Schumann calculated critical temperatures in equation 15 with
Newton’s iterations. Besides calculating the critical temperatures for saturation over
water, ice saturation was also considered. Due to the larger contrail factors, the threshold
temperatures derived by Schumann are significantly higher than Appleman's
temperatures.
A similar method to calculate critical temperatures for contrail formation and a brief
explanation of the basic physics involved in the mixing of aircraft exhaust gases with the
ambient air was given by Schrader (1997). He pointed out errors in the physics of the
algorithms by Hanson (1995) and Peters (1993). Schrader's derivation of critical
temperatures as threshold for contrail formation coincides for initially saturated air with
the Hanson method. For a relative humidity less than 100%, critical temperatures are
derived as intersections of the mixing line with the respective vapor pressure curves (see
Figure 13). Schrader's solution is in general equivalent to Schumann's method; equation
15 is solved for the vapor pressure, e, instead of the temperature, T. Relations 11 and 15,
and the same indices as before yield to:
eT
crit , f
= es T
crit ,100
- (Tcrit,100 - Tcrit, f )(
pCF
)
622
(16)
†
23
With f =
eT
crit , f
es T
Schrader showed a possibility to calculate Tcrit, f iteratively:
crit , f
es T
crit ,100
pCF
- (Tcrit,100 - Tcrit, f )(
) = f es T
crit, f
622
†
†
(17)
Critical temperatures were calculated for the contrail factors also used by Hanson (1995)
and Appleman (1953, CF=0.0336 g(kgK)-1).
†
Figure 13: Water vapor partial pressure as function of temperature (Mollier-Schmidt
diagram) for relative humidity values of 100 % (saturation – dark blue) and 60 % and
30%, respectively (light blue). The red straight line represents the threshold line for
isobaric mixing of exhaust with environmental air (p=200 hPa and CF=0.036 g(kgK)-1.
Summarizing, it can be stated that the algorithms are based on the following assumptions
and have the following similarities:
• A critical temperature can be calculated as a threshold, whether a contrail will be
formed or not.
• Condensation starts to form water droplets instead ice crystals. Below the freezing
point, the water is super-cooled.
• Contrails originate from emitted water vapor and subsequent condensation on preexisting nuclei in the environment or on nuclei from the exhaust.
24
•
•
•
•
During the cooling of the exhaust due to the mixing with the ambient air, the
decrease of the absolute humidity in the exhaust is directly proportional to the
decrease of the temperature.
Hanson, Schumann and Schrader used ‘Mollier– Schmidt’ diagrams (water vapor
partial pressure versus temperature). For saturated air their algorithms yield the
same results.
Combustion specific contrail factors, CF, define the mixing between aircraft plume
and ambient air.
The critical temperatures are functions of the contrail factors, the atmospheric air
pressure, and the relative humidity.
There are, however, some differences in the algorithms:
•
•
•
•
Appleman used a linear extrapolation from saturated to the non-saturated condition in
the ‘Mollier- Schmidt’ diagram for describing the mixing of the exhaust plume with
the ambient air.
Hanson made an error in the physics. He calculated the slope of the vapor pressure
versus temperature during the mixing process proportional to the relative humidity.
The contrail factors used by the different authors vary significantly due to different
engine types. Schrader compiled published contrail factors ranging from values of
0.0295 g(kgK)-1 (Pilié and Jiusto 1958) to 0.049 g(kgK)-1 (Peters 1993).
Busen and Schumann (1995) and Schumann (1996) discussed contrail factors; they
considered aircraft aerodynamics and flight-parameters besides combustion specific
parameters. Their formulation of contrail factors yielded minimum values of 0.028
g(kgK)-1.
Model verification
The critical temperatures were calculated using the different algorithms and were
compared to our database (see Figure 6). The vapor pressure-temperature relationship
after Mollier-Schmidt and the water vapor pressure, which were derived from the radiosonde measurements, are presented in Figure 14. Vapor pressures close to a saturated
state characterized the long lasting contrails. In contrast, 65% of our no-contrails showed
vapor pressures below 0.005 hPa. In order to obtain appropriate contrail factors for the
separation of contrail from no-contrail cases, we applied a wide range of contrail factors
from 0.02 g(kgK)-1 to 0.05 g(kgK)-1 to each algorithm.
Using the different algorithm and contrail factors, we could use our data set to test the
quality of the prediction. A successful algorithm is characterized by a large number of
forecasted and observed contrails as well as predicted no-contrails, which were observed.
In Figure 15 the results are presented for various contrail factors. Applemann’s and
Schumann’s algorithms coincided and gave a hit rate of 92% for contrail factors of 0.035
and 0.036 g(kgK)-1 , corresponding to mean values between low-bypass and high-bypass
engine types.
25
Figure 14: Mollier-Schmidt diagram with radio-sounding measurements corresponding to
the different observation classes. The lines indicate relative humidity of 20% and 60%
and the saturation vapor pressure over water.
95%
Appleman
Schumann
Hanson
Hitrate (%)
90%
85%
80%
75%
70%
65%
0.020
0.025
0.030
0.035
0.040
0.045
0.050
-1 -1
Contrail Factor (gkg K )
Figure 15: Validation of different contrail forecasting algorithms using different contrail
factors (CF)
26
Hanson's method resulted in a less perfect agreement, 84% and 85%, respectively were
found using contrail factors of 0.039 g(kgK)-1 (high bypass) and 0.047/ 0.048 g(kgK)-1.
The high and not realistic contrail factor of 0.047 g(kgK)-1 indicates the errors in the
physics for non-saturated air in the Hanson algorithm. A maximum hit rate of 90% was
obtained without the consideration of the relative humidity and comparing only the
critical temperatures for saturation over water (Tcrit,100) with the measured temperatures.
Figure 16 shows a scatter plot of critical temperatures versus measured temperatures
derived from the Schumann algorithm with a contrail factor of 0.036 g(kgK)-1. The range
of the critical temperatures found is from 218.5 K to 225.7 K. Ninety percent of the nocontrail temperatures correspond to correct algorithm results ( Tmeasured > Tc ). The overall
numbers of the agreement between prediction and observation for all contrails and nocontrails are given in Table 6. Fifteen observed contrails are not correctly classified as
contrails by the algorithm. The majority of these algorithm errors occur due to fast
dissolving, short contrails; Figure 19 shows an example-image of a short †
lasting contrail.
The critical temperatures of 11 short contrails are below the corresponding temperatures
at the aircraft altitudes; these cases would produce incorrect no-contrail predictions,
although contrails occurred (Figure 17). Nevertheless a high number of 93.3% of all
contrail cases is classified correctly by the Schumann algorithm (CF=0.036 g(kgK)-1).
Almost no errors occur for the medium (1 case) and for long lasting contrails (3 cases).
Table 6: Contingency table including the total algorithm agreement of observed and
predicted contrails and no-contrail cases.
Contrail
Prediction
Contrail Observation
yes
no
yes
208
2
no
15
18
Discussion
Radiosonde data allow the prediction of the layers in the atmosphere at which contrails
are most likely to form. An example is given in Figure 18. The contrail layers are mostly
situated close to the tropopause, which is defined by a temperature minimum and is often
characterized by relatively high humidity values. In the lower stratosphere the air is
already substantially drier, and hence the likelihood of formation of contrails is reduced.
The height as well as the thickness of the layer varies seasonally. During the winter
months the contrail layers overhead Fairbanks are more than twice as thick than during
summer; the layer base altitudes are significantly lower in winter (compare Table 7).
27
Table 7: Seasonal characteristics of contrail layers. The mean values refer to the 0:00h
GMT radiosonde ascents at Fairbanks Airport from January 2000 to July 2002.
Dec-Jan-Feb
Mar-Apr-May
Jun-Jul-Aug
Sep-Oct-Nov
Altitude of layer Thickness below Thickness above
base
tropopause
tropopause
(m a.s.l.)
(m)
(m)
7606
1861
987
8095
1427
811
9718
962
299
8141
1540
835
Total layer
thickness
(m)
2849
2238
1261
2374
U.S. Standard Atmospheres for latitudes of 60º North are available for winter (January)
and summer (July), (U.S. Standard Atmosphere 1966). The comparison of these
standards with critical temperatures for different relative humidity values reveal an
atmospheric pressure layer between 180 hPa und 350 hPa, where contrails are most likely
to form in winter even at very dry conditions. Most of our observed over-flights are
situated within this pressure layer, which corresponds to altitudes between 7,500m and
12,500m. In contrast to the winter situation the likelihood of contrail formation seems to
decrease significantly in summer. The U.S. Standard Atmosphere is significantly warmer
in summer than in winter at most pressure levels; also below 300 hPa the seasonal
temperature differences are pronounced. These warm temperatures in summer would
require relative humidity values above 78% in order to form contrails (see Appleman
type chart: Figure 20). From atmospheric sounding measurements at Fairbanks we have
observed only minor seasonal temperature differences above the 300 hPa level. In 2001,
for instance, the mean temperature differences between summer (June-July-August) and
winter (January-February-December) were 1.3ºC and 0.4ºC at pressure levels of 250 hPa
and 200 hPa respectively. These cold summer-temperatures observed at high levels favor
contrail formation also during the warm season at certain heights.
28
Temperature at Aircraft Altitude (K)
230
no contrail
short duration
medium duration
long duration
-
CF=0.036 g/(kgK)
225
220
215
210
210
215
220
225
230
Critical Temperature (K)
Figure 16: Scatter plot of critical temperatures for the observation classes for CF=0.036
g(kgK)-1.
100%
wrong prediction
agreement
80%
60%
40%
20%
long
duration
medium
duration
short
duration
no
contrail
0%
Figure 17: Hit rate of contrail prediction (blue columns) and wrong prediction
percentages (red columns) using the Schumann algorithm with a contrail factor
CF=0.036 g(kgK)-1.
29
Figure 18: Temperature and dew-point sounding of 22 June 2000. The critical
temperature derived from the Schumann (CF=0.036 g/kgK) algorithm is included.
Contrails are likely to be formed at levels, where Tcrit > T (yellow marked level from
8500 m to 10250 m).
30
Figure 19: Example of fast dissolving contrail (~ 4 sec) produced on 22 March 2002 by a
Boeing 747-200 jet in 10,700 m (Japan Airline Flight 6422 from Frankfurt to
Anchorage) overhead the International Arctic Research Center (UAF).
31
Figure 20: Critical Temperatures for specific relative humidity values and various
atmospheric pressure, and the U.S. Standard Atmosphere 1966 in 60ºN for winter
(blue) and summer (red) respectively. The critical temperatures were calculated
according to Schumann with a contrail factor of 0.036 g(kgK)-1. The ordinate is in
logarithmic scale.
Future Work
• We have been successful in predicting most of the observed contrails. For further
prediction-improvement the significance of radiosonde data will be analyzed in
more detail. Frequently a time lag between aircraft passages and radiosonde
ascents exists. Algorithm errors may occur especially in cases of fast changing
meteorological conditions at flight level (frontal passages). Furthermore the
quality of radiosonde data will be checked by comparison with re-analysis data
from meteorological models.
32
•
The evaluation of the characteristics of engine combustion from different aircraft
types will be possible with an extended database. An appropriate contrail factor
for each aircraft engine might reduce errors in the prediction of contrails.
•
At present, we are verifying the occurrence/ non-occurrence of contrails with the
radiosonde ascents from the Fairbanks International Airport. However, the goal is
the prediction, for which a contrail forecast model will be needed, most likely
MM5. Forecast charts with favorable altitude levels for the formation of contrails
will be produced. This should lead to closer cooperation with the US Air Force
Weather Agency.
•
So far our emphasis has been on the occurrence/ non-occurrence of contrails. We
distinguished 3 different types, short, medium and long lasting ones. Spreading of
contrails, even observed, was not systematically analyzed. Parameters, which
were of minor importance for the formation of contrails, such as the wind speed
aloft, wind shear, and atmospheric stability of flight level seem to be of primary
importance for spreading and dissolving of contrails. We plan to investigate this
in greater detail, including the radiative effects of contrails on the atmosphere and
surface.
References
Appleman, H. S. 1953: The Formation of Exhaust Condensation Trails by Jet Aircraft,
Bulletin American Meteorological Society, 34, p 14-20.
Busen, R. and Schumann, U., 1995: Visible contrail formation from fuels with different
sulfur content. Geophysic. Res. Letters, 22, p 1357-1360.
Carleton, A. and P. Lamb 1986: Jet contrails and cirrus clouds: a feasibility study
employing high resolution satellite imagery. BAMS, 67, 301-309
Gayet, J., G. Febvre, G. Brogniez, H. Chepfer, W. Renger and P. Wendling 1996.
Microphysical and optical properties of cirrus and contrails. J. Atmos. Sci. 53,
126-138
Goff, J. A. and Gratch, S., 1946: Low pressure properties of water from -160 to 212 F.
Trans. Amer. Soc. Heat. Vent. Eng., 52, 95
Grassl, H., 1990. The climate at maximum entropy production by meridional and
atmospheric heat fluxes. Quarterly Journal of the Royal Meteorological Society
107: 153-166.
Hanson, H. M. and Hanson, D. M., 1995: A Reexamination of the Formation of Exhaust
Condensation Trails by Jet Aircrafts, Journal of Applied Meteorology, p 24002405.
33
Iribarne, J. V. and Godson, W. L., 1981: Atmosperic Thermodynamics, 2nd ed., D.
Reidel Publishing, 128 pp.
IPCC 1999: Aviation and the global atmosphere. A special report of IPCC working
groups I and III, Cambridge University Press, 373pp
Kuhn, P. 1970: Airborne observations of contrail effects on the thermal radiation budget.
J. Atmos. Sci. 27, 937-942
Nakanishi, S., Curtis, J. and Wendler, G., 2001: The influence of increased jet airline
traffic on the amount of high level cloudiness in Alaska. Theor. Appl. Climatol.
68, 197-205
Peters, J. L. 1993: New techniques for contrail forecasting. AWS/TR-93/001, 26 pp.
Pilié, R. J., Jiusto, J. E., 1958: A laboratory study of contrails. J. Meteor, 15, p 149-154.
Sassen, K., J.M. Comstock, Z. Wang, and G.G. Mace 2001: Cloud and Aerosol Research
Capabilities at FARS: The Facility for Atmospheric Remote Sensing. Bull. of the
Americ. Meteorol. Soc., Vol. 82, Nr. 6, 1119-1138
Schmidt, E. 1941: Die Entstehung von Eisnebel aus den Auspuffgasen von Flugmotoren.
Schriften der Deutschen Akademie der Luftfahrtforschung, Verlag R. Oldenbourg,
Muenchen und Berlin, Heft 44, p 1-15.
Schrader, M.L. 1997: Calculations of aircraft contrail formation critical temperatures. J.
Appl. Meteorol. ,36, 1725-1729
Schumann, U. 1996: On conditions for contrail formation from aircraft exhausts,
Meteorol. Zeitschrift, 5, p 4-23.
Seinfeld, J. 1998: Clouds, contrails and climate. Nature, 391, 837-838
Shull, J., D., 1998: A validation study of the air force weather agency (AFWA) jetrax
contrail forecast algorithm. Thesis, 118 p.
Smith,W., S. Ackerman, H. Rivercomb, H. Huang, D. DeSlover, W. Feltz, L. Gumley
and A. Collard 1998: Infrared spectral absorption of nearly invisible cirrus clouds.
Geophys. Res. Lett. 25, 1137-1140
Travis, D.J., A.M. Carlton and S.A. Changnon 1997: An empirical model to predict
widespread occurrences of contrails J. Appl. Meteorol. 36, 1211-1220
US Standard Atmosphere, Supplements, 1966. ESSA, NASA, US-Airforce, Washington
D.C. 289 p.
34
Appendix A
Scripting
The Perl script calculates critical temperatures with the mixing cloud algorithm described
by Schumann (1996) and Schrader (1997). Air temperature, humidity and pressure at
flight level and an aircraft specific contrail factor have to be entered as input data.
#!/usr/bin/perl -w
#
#
#
#
#
#
#
#
#
#
Perl script for the calculation of jet contrails formation.
Author: Stuefer
August 2002
Input: Air Pressure, Relative Humidity, Temperature, Contrail Factor
Critical temperatures for contrail formation decision are calculated
formation (saturation stipulation).
The saturation vapor pressure for low temperatures is derived using a
Goff Gratch Formulation.
(Goff, J.A. and S. Gratch, 1946, 'Low pressure properties of water
from -160 to 212 F.', Trans. Americ. Soc. Heat. Vent. Eng., 52, 95).
#==============================================================================
# Initialisation of parameters
#==============================================================================
$ks = 0;
$ka = 0;
$esat100 = 0;
$esatf = 0;
$h9 = 1;
#
#
#
#
#
#
#
#
derivative of saturation vapor pressure
versus temperature, $ks = 10**($h1-$h2-$h3-$h4) *
(log(10)) * ($h5-$h6+$h7+$h8);
conversion of vapor pressure per kelvin from aircraft
specific contrail factors.
saturation vapor pressure at Tc100
saturation vapor pressure at Tc
help variable
#==============================================================================
# Input of the atmospheric parameters and the contrail factor
#==============================================================================
print
chomp
print
chomp
print
chomp
print
chomp
'Enter the air temperature at flight level in deg C: ';
($Ta = <STDIN>);
'Enter the atmospheric pressure at flight level in hPa: ';
($p = <STDIN>);
'Enter the relative humidity at flight level in %: ';
($rf = <STDIN>);
'Enter the Contrail Factor in g/kgK: ';
($CF = <STDIN>);
#==============================================================================
# Calculation Routine
#==============================================================================
$Tc100 = 253.16;
$Tc = 253.16;
#
#
#
#
Initial critical temperature in saturated state used
as limit value for contrail formation algorithm
initial critical temperature at relative humidity f
used as limit value for contrail formation algorithm
$ka = ($p*$CF/622);
35
Goff_Gratch_slope_comparison();
#=> Tc100 calculated
$esat100 = 10**(23.832241-5.02808*(log($Tc100)/log(10))-(1.3816*10**(7))*10**(11.334-0.0303998*$Tc100)+8.1328*10**(-3)*10**(3.491491302.884/$Tc100)-2949.076/$Tc100);
$Tc = $Tc100;
while (0 <= $h9) {
$Tc -= 0.01;
$esatf = 10**(23.832241-5.02808*(log($Tc)/log(10))-(1.3816*10**(7))*10**(11.334-0.0303998*$Tc)+8.1328*10**(-3)*10**(3.491491302.884/$Tc)-2949.076/$Tc);
$h9 = $esat100-($Tc100-$Tc)*$ka-($rf/100)*$esatf;
}
$Tc=$Tc-273.16;
$Tc100=$Tc100-273.16;
printf ("CF = %.4f g/kgK, p = %.0f hPa, Tc100 = %.4f deg C, Tc(f=%.0f percent)
= %.4f deg C\n", $CF, $p, $Tc100, $rf, $Tc);
if ($Ta < $Tc) {
print "Warning: Contrails are expected to form!\n";
} else {
print "No contrails are expected to form.\n";
}
#==============================================================================
# sub program
sub Goff_Gratch_slope_comparison {
#==============================================================================
# Goff Gratch Formulation for the calculation of the temperature derivative of
# the saturation vapor pressure
#==============================================================================
$h1
$h2
$h3
$h4
$h5
$h6
$h7
$h8
$s1
=
=
=
=
=
=
=
=
=
'';
'';
'';
'';
'';
'';
'';
'';
0;
#
#
#
#
#
#
#
#
#
help variable
help variable
help variable
help variable
help variable
help variable
help variable
help variable
slope difference 1, => help variable
#==============================================================================
# Calculation of the critical temperature representing a limit for contrail
# formation
#==============================================================================
while (0 >= $s1) {
$Tc100 -= 0.01;
$h1 = 23.832241+8.1328*10**(-3)*10**(3.49149-1302.8844/$Tc100);
$h2 = 1.3816*10**(-7)*10**(11.334-0.0303998*$Tc100);
$h3 = 2949.076/$Tc100 ;
$h4 = 5.02808*(log($Tc100)/log(10));
$h5 = 2949.076/$Tc100**2;
$h6 = 5.02808/$Tc100/log(10);
$h7 = 1.3816*3.03998*10**(-9)*10**(11.3340.0303998*$Tc100)*log(10);
$h8 = 8.1328*log(10)*10**(-3)*10**(3.491491302.8844/$Tc100)*1302.8844/$Tc100**2;
$ks = 10**($h1-$h2-$h3-$h4) * (log(10)) * ($h5-$h6+$h7+$h8);
$s1 = $ka-$ks
}}
36