Determination of humidity and temperature ¯uctuations based onMOZAIC data and parametrisation of persistent contrail coveragefor general circulation models

K. M. Gierens1, U. Schumann1, H. G. J. Smit2, M. Helten2, G. ZaÈ ngl1

1 DLR, Institut fuÈ r Physik der AtmosphaÈ re, Oberpfa�enhofen, D-82230 Weûling, Germany2 Forschungszentrum JuÈ lich, Institut fuÈ r Chemie und Dynamik (ICG2), Postfach 1913, D-52425 JuÈ lich, Germany

Received: 21 October 1996 / Revised: 7 February 1997 /Accepted: 24 February 1997

Abstract. Humidity and temperature ¯uctuations atpressure levels between 166 and 290 hPa on the gridscale of general circulation models for a region coveredby the routes of airliners, mainly over the Atlantic, havebeen determined by evaluation of the data obtained withalmost 2000 ¯ights within the MOZAIC programme. Itis found that the distributions of the ¯uctuations cannotbe modelled by Gaussian distributions, because large¯uctuations appear with a relatively high frequency.Lorentz distributions were used for the analyticalrepresentation of the ¯uctuation distributions. Fromthese a joint probability distribution has been derivedfor simultaneous temperature and humidity ¯uctua-tions. This function together with the criteria for theformation and persistence of contrails are used to derivethe maximum possible fractional coverage of persistentcontrails in a grid cell of a GCM. This can be employedin a statistical formulation of contrail appearance in aclimate model.

1 Introduction

Contrails, like natural clouds, modify the radiationbudget of the Earth-atmosphere system. The questionwhether contrails may have an in¯uence on the climatehas so far been addressed by means of principle studies(Graûl, 1990) or 1-D radiation-convection models (e.g.Fortuin et al., 1995). In such studies only the directe�ect of local radiative and convective processes can beinvestigated, whereas other feedback mechanisms mustbe neglected. Ponater et al. (1996) ®rst estimated thecontrail e�ect on climate using a 3-D global circulationmodel (GCM), namely the German climate communitymodel ECHAM (Roeckner et al., 1992). It has been

demonstrated that contrails have the potential for aclimate change. However, Ponater et al.'s (1996) studydid not provide a quantitative assessment becausecontrails were inserted into the climate model withoutregard to the meteorological conditions. Therefore, it isdesirable to put the contrail parametrisation in GCMsonto a sounder footing by taking into account thethermodynamic criteria for the formation and persist-ence of contrails (Schmidt, 1941; Appleman, 1953;Schumann, 1996).

The Schmidt-Appleman criterion states that anaircraft produces a contrail only when a supersaturatedstate (with respect to the liquid phase) is reached duringthe mixing of the exhaust gas with the ambient air. Thiscriterion is a local one, i.e. it relates the possibility ofcontrail formation to the local thermodynamic state. Ona scale that is resolved by a GCM, however, such a localcriterion would be of questionable applicability becausethe GCM provides only grid cell mean values of thethermodynamic state. A statistically more satisfyingtreatment of contrail formation can be obtained whenthe ¯uctuations of the state variables around their grid-scale mean values are known. Then the rigid yes/no-alternative of the Schmidt-Appleman criterion can bereplaced by a probability for contrail formation, whichcan be translated into a fractional coverage with respectto one grid cell of a GCM. It is one aim of this study todetermine the ¯uctuations of temperature and relativehumidity on a scale that is resolved by a GCM. This willbe achieved by an evaluation of the data obtainedduring the MOZAIC programme (Marenco et al., 1994).

The objective of the MOZAIC programme (Mea-surement of OZone on Airbus In-service airCraft) is toperform large scale and continuous measurements ofozone and water vapour aboard in-service aircraft inorder to realise a sound data base of atmosphericcomposition and trace species chemistry at currentaircraft cruise altitudes. For this purpose ®ve automaticmeasuring units have been installed on commercialairliners. These units not only measure ozone and watervapour concentration, but also record the meteorologi-Correspondence to: K. M. Gierens

Ann. Geophysicae 15, 1057±1066 (1997) Ó EGS ± Springer-Verlag 1997

cal quantities temperature and pressure. MOZAICtherefore provides a dataset of meteorological variablesthat is unique with respect to spatial and temporalcoverage of the major ¯ight routes.

In this study a parametrisation of persistent contrailcoverage for GCMs is developed. We will carry out theparametrisation for a speci®c GCM, namely ECHAM inT42 resolution. The setup of the parametrisation forother GCMs or other resolutions is then straightfor-ward. In order to achieve this objective, we ®rst use theMOZAIC data to determine the spatial distribution oftemperature and humidity ¯uctuations on a 300 � 300km2 scale which corresponds to the size of one grid cellin T42 resolution (Sect. 2). The ¯uctuation distributionstogether with the local criteria for the formation andpersistence of contrails then allow us to compute themaximum fractional coverage of persistent contrails inan ECHAM grid cell (Sect. 3). In this fraction of theconsidered cell the air is cold and moist enough to allowcontrails to persist. These areas can be interpreted as themoist lenses where contrails appear in groups (Bakanet al., 1994). The methods and results are discussed inSect. 4 and summarized in Sect. 5.

2 Evaluation of MOZAIC data

2.1 General matters

MOZAIC covers the time from July 1994 untilDecember 1995, however we constrain our analysis tothe year 1995 in order to have a complete annual cycle.The data bank contains data from almost 2000 ¯ights in1995. From the data base we obtain 1 min averages ofpressure p, temperature T , and relative humidity RH atthe current ¯ight position (longitude k, latitude b,altitude z). Each ¯ight position (k; b) belongs to acertain grid cell in the ECHAM general circulationmodel (Roeckner et al., 1992). In T42 resolution thezonal direction is divided into 128 sectors, themeridional direction into 64 segments, giving128� 64 � 8192 grid cells. The data points are sortedinto the ECHAM altitude layers by means of themeasured static pressures p, since ECHAM uses ahybrid r-p altitude coordiante which at cruise ¯ightlevels is nearly equivalent to a pure pressure coordinate.We constrain the evaluation on the altitude range 8±14km, since this range covers the usual cruise ¯ight levelsof civil aviation. The corresponding pressures inECHAM range from 122 to 367.5 hPa, which arerepresented by levels 6 to 9 (the counting starts at thetop of the atmosphere). In level 6, there were no data atall. In the lowest considered level 9, the aircraft usuallywere in ascent after take o� or in descent before landing.Our analysis is therefore e�ectively constrained to levels7 and 8, which corresponds to a pressure range 166±290hPa. The mid-level pressures are 190 hPa for level 7 and250 hPa for level 8. The mean accuracy of the 1995MOZAIC RH measurements is �6% at 290 hPa and�35% at 166 hPa relative to the measured value. Theaccuracy of the temperature measurements is �0.5 K.

2.2 Indicator variables

From the physical variables p; T , and RH , we computeindicator variables, viz.

C � 1 if the Schmidt-Appleman criterion is ful®lled,otherwise C � 0

IC � 1 if ice saturation is reached, otherwise IC � 0WC � 1 if water saturation is reached, otherwise WC � 0PC � 1 if contrails can persist, otherwise PC � 0

There are the obvious logical relations:

IC � 0) WC � 0; WC � 1) IC � 1;

and

PC � �1ÿ WC� � IC � C;since contrail persistence requires ice saturation and theSchmidt-Appleman criterion to be ful®lled. Situationswith RH exceeding water saturation usually imply ¯ightin clouds and are not considered contrail cases.

For the evaluation of the Schmidt-Appleman criter-ion we need some assumptions about properties of thekerosene burned in the engines and on the propulsione�ciency g of an average (present day) aircraft (seeBusen and Schumann, 1995; Schumann, 1996). Weassume that the water vapour emission indexEIH2O � 1:25 kg/kg fuel and that the combustion heatis 43 MJ/kg fuel. Furthermore, we assume g � 0:3 whichis a typical value for modern bypass engines (Schumann,1996). The critical temperature of the Schmidt-Apple-man criterion varies by 0.7 K for a variation ofDg � 0:05 (Schumann, 1996).

The indicator variable PC is a function of onlytemperature and relative humidity if the pressure is held®xed, as e.g. on a constant ¯ight level. We may thusconsider a �T ; RH�-diagram and determine the part ofthe �T ; RH�-space where PC � 1. This area has a rathersimple shape: it is bounded on three sides by nearlystraight lines (Fig. 1). The two boundaries at low RH(about 60%) and at RH=100% are due to the ice andwater saturation conditions. The boundary at aboutT � ÿ45 �C is due to the Schmidt-Appleman criterion(see the logical relation for PC already mentioned). Onlythe latter boundary depends on the chosen pressure. InFig. 1 we have plotted the Schmidt-Appleman boundaryfor the two mean pressures of the ECHAM levels 7 and8, i.e. 190 and 250 hPa. The variation of the Schmidt-Appleman boundary with pressure within one ECHAMlayer is of the order 1 K.

Although it would be interesting to show statistics of,e.g. ice saturation or conditions leading to persistentcontrails, we dispense with it for the present study, sinceit is not necessary for the purpose of the desiredparametrisation.

2.3 Fluctuations in ECHAM grid cells

A single ECHAM-T42 grid cell is about 300 � 300 km2

in size. The prognostic physical variables must thereforebe regarded as mean values for such a cell and for the

1058 K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations

time step (21 min). For the simulation of subgrid-scalephenomena like clouds (including contrails), the ¯uctua-tions of physical variables around their mean valuesshould be known. Here we can determine such¯uctuations from the data on the MOZAIC data bank.Since the pressure is mostly more or less constant during¯ights, it is not possible to determine pressure¯uctuations. However, since the vertical ECHAMcoordinate assumes constant pressure and in view ofthe small vertical pressure variation within one ECHAMaltitude layer, the determination of horizontal pressurevariations does not seem necessary. Only ¯uctuations oftemperature and relative humidity are thus considered.

As an illustration of such ¯uctuations consider Fig. 2and 3 where we have plotted on the world map the``mean standard deviation'' of temperature and relativehumidity, as obtained from the data bank. The termmean standard deviation means that we have ®rstdetermined standard deviations for each pixel and each¯ight separately. This has been done only when eight ormore data points (i.e. one minute average values ofhumidity and temperature) were obtained in the grid cellunder consideration. Eight data points correspond to adistance of about 100 km which we considered minimumto be representative for grid cells with 300 km sidelength. Measurements in cloudy cells �RH � 100%)have been ignored. After that, we have averaged theindividual standard deviations when more than 10¯ights crossed the pixel in 1995. Averaging this wayavoids contamination of the results with annual or dayby day variations. The mean standard deviation is,therefore, a measure of instantaneous ¯uctuations oftemperature and relative humidity. The maps show that

instantaneous temperature ¯uctuations on the 300� 300km2 scale are less than 1 K almost everywhere. They areslightly larger (0.4±0.8 K) over the Atlantic than in thetropics (0±0.4 K), presumably because the meridionalgradient of the zonal mean temperature is much largerin mid-latitudes than in tropical regions (see, e.g. Fig.10.11 in Holton, 1979). Accordingly, an aircraft ¯ying ata constant pressure level and with a meridionalcomponent in its ¯ight route passes through a largertemperature change over the Atlantic than in the tropics.Apparent maxima of temperature ¯uctuations at theendpoints of ¯ight routes are caused by ascent anddescent near airports. For the evaluation only datapertaining to constant pressure levels are used. Fluctua-tions of relative humidity are typically of the order 4±6% over the North Atlantic, but in tropical regions theyreach 10%, presumably because of the strong convec-tion there.

We can determine the distribution law for instanta-neous temperature and relative humidity ¯uctuations.Again, we ®rst consider each ¯ight separately. Wecollect all data points that pertain to one ECHAM gridcell and compute mean temperature hT i and meanrelative humidity hRHi for this cell. The number of datapoints for the calculation of these mean values istypically around 20. Grid cells with fewer than six datapoints and data points inside clouds �RH � 100%) areignored. Furthermore, we record the individual devia-tions dTm � Tm ÿ hT i and dRHm � RHm ÿ hRHi (m is thedata point number). Strictly speaking, the computedmean values and ¯uctuations refer to a line through acell, not to the cell as an area or a volume. This maycause an underestimation of the width of the ¯uctuationdistribution within the grid cell. Ideally, we would like tohave (nearly) simultaneous measurements at manylocations throughout a grid cell. Since only ®ve aircraftwere involved in the MOZAIC programme, the database does not contain such details. So there is no otherchoice than to relate the computed mean values to theprognostic ECHAM variables and to consider therecorded ¯uctuations as characteristic for the grid cellas a volume. The ¯uctuations are sorted in bins of 0.04 Kand 0.2%, respectively; the mean values are not usedfurther.

The ¯uctuation data for all grid cells and all 1924¯ights in 1995 are then collected. Thus, we do not retainany information about spatial and temporal variation ofthe ¯uctuation distributions. This is not necessary sinceGCM modellers prefer to have one parametrisation forthe whole globe and the whole year instead of dealingwith many parametrisations for the same purpose. Thereason is, that generally a parametrisation involvestuning factors, and the number of tuning factors shouldbe as small as possible. Therefore, we have only retainedthe altitude information in the data and analysed the¯uctuations for ECHAM levels 7 and 8 separately. Thedistributions of temperature and relative humidity¯uctuations are displayed in Fig. 4 (upper and middlepanel) as dots. It can be seen that these distributionscannot be modelled by a Gaussian distribution becauselarge ¯uctuations have a relatively high probability. The

Water saturation

Ice saturation

Schmidt-

Appleman-

criterion

PC = 1

110

PC = 0

T (°C)

RH

(%

)

100

90

80

70

60

50

40

30

20

10

0

-70 -60 -50 -40 -30 -20

Fig. 1. For a constant pressure (as e.g. on a constant ¯ight level) thepossibility for persistent contrails development depends only ontemperature and relative humidity. Persistent contrails are possible inthe area indicated by PC � 1. This area is bounded on three sides: bylow RH because of the ice saturation condition, at RH � 100%because of the water saturation condition, and at high T because ofthe Schmidt-Appleman criterion. The latter criterion depends on thepressure. The two boundaries are drawn for 190 (left curve) and250 hPa (right curve)

K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations 1059

``wings'' of the distribution are much broader than thoseof a Gaussian distribution. This ®nding con®rms aconjecture that Sommeria and Deardor� (1977) made 20years ago, namely that a Gaussian distribution is notappropriate to model ¯uctuations in a grid cell of ageneral circulation model. A much better ®t to the¯uctuation data is obtained with Lorentz curves (solidlines) Kc�d�, i.e.

Kc�d� � 1

pc

d2 � c2: �1�

The Lorentz curve is known from spectroscopy to be theline shape of pressure broadened spectral lines. Math-ematicians call it the Cauchy distribution. The para-meter, c, is the half width at half maximum (HWHM) ofthe distribution. By least squares ®tting we found thefollowing set of HWHMs: for level 7, cT � 0:362 K;cRH � 1:69%; for level 8, cT � 0:287 K; cRH � 2:48%. Itmust be admitted, that the ®t using a Lorentz curve is

not very good in the case of the ¯uctuations of relativehumidities. Other distributions, such as a two-sidedexponential distribution, could give better ®ts. However,in view of the derivation that follows, we decided tokeep to the Lorentz distribution because of its highlyadvantageous analytical properties, which most otherdistributions do not share.

3 Construction of the parametrisation

3.1 The joint probability for temperature and humidity¯uctuations

We derive an expression for the joint probability density( jpd), fdT dRH , where

fdT dRH �dT ; dRH�ddT ddRH

is the probability for a simultaneous occurrence (in thesame ECHAM-T42 grid cell) of a temperature and

Level 7

60

30

0

-30

-60

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Level 8

60

30

0

-30

-60

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

0 0.2 0.4 0.6 0.8 1.0 1.5 2.0 3.0 4.0

Fig. 2.Mean standard deviationof temperature in K. Mid-levelpressures are 190 hPa for level 7and 250 hPa for level 8

1060 K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations

relative humidity ¯uctuation in the ranges�dT ; dT � ddT � and �dRH ; dRH � ddRH�, respectively.Given are the (one dimensional) probability densityfunctions (pdf) udT and udRH which are both Lorentzdistributions. If these distributions were statisticallyindependent, we would just have to multiply them toobtain the jpd. However, RH � e=es�T �, so that therelative humidity ¯uctuation depends on the tempera-ture ¯uctuation via the temperature dependence of thewater vapour saturation pressure es�T �. For small¯uctuations we may write:

dRH � @RH@T

dT � @RH@e

de � adT � bde; �2�with

a � ÿ�RH � L�=�RV T 2�; b � 1=es�T �: �3�For the derivation, the Clausius-Clapeyron equationand the ideal gas law for water vapour have been used.

From this we ®nd that the pdf, udRH , is a convolution(indicated by ?) of the following form:

udRH ���1=a�udT �dT=a�� ? ��1=b�ude�de=b����1=b�

ZudT �dT � � ude��dRH ÿ adT �=b�ddT : �4�

Here we have to consider the ¯uctuations of the watervapour partial pressure. These have been determined inthe same way as before for the ¯uctuations of the othervariables. The distributions of de (dots) are displayed inFig. 4 (lower panel) together with ®tting Lorentz curves(ce � 0:068 Pa and 0:124 Pa for ECHAM levels 7 and 8,respectively). Since the convolution of two Lorentziansyields another Lorentzian (this is the desired analyticalproperty), viz.:

Kc1 ? Kc2 � K�c1�c2�; �5�we have a consistent model insofar as we have proposeda Lorentzian distribution for the relative humidity. If we

Level 7

60

30

0

-30

-60

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Level 8

60

30

0

-30

-60

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

0 1 2 3 4 5 6 8 10 15

Fig. 3.Mean standard deviationof relative humidity in %. Mid-level pressures are 190 hPa forlevel 7 and 250 hPa for level 8

K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations 1061

had chosen another type of distribution (such as a two-sided exponential), we would have to resort now tonumerical methods in order to perform the convolution.For the distribution in Eq. (4) we have simply:

�1=a�udT �dT=a� � KacT�dT �;

�1=b�ude�de=b� � Kbce�de�: �6�

The desired jpd can be written in the following form:

fdT dRH �dT ; dRH�ddT ddRH � udT �dT �ddT �� ude��1=b��dRH ÿ adT ���ddRH=b�: �7�

Inserting the Lorentz distributions for the pdf's u�, andnoting the relation ce � �1=b��cRH ÿ acT �, which followsfrom Eqs. (4) and (5), we ®nd:

fdT dRH � 1

p2cT

dT 2 � c2T� cRH � AcT

�dRH � AdT �2 � �cRH � AcT �2�KcT

�dT � � K�cRH�AcT ��dRH � AdT � �8�with A � ÿa. Please note that this function depends onthe state variables T and RH (via A). The jpd isillustrated in Fig. 5 for two states �T ;RH�. The shape ofthe jpd is discussed next.

3.2 Formulation of the parametrisation

Consider a certain state �T0;RH0� in Fig. 1 to representthe actual mean values of T and RH in an ECHAM gridcell. The joint temperature and humidity ¯uctuationsaround the mean state �T0;RH0� are described by thejpd, fdT dRH , that we have derived in the previous section.A certain ¯uctuation �dT ; dRH� leads to a state in whichthe air can carry persistent contrails, if

PC�T0 � dT ;RH0 � dRH� � 1:

The total probability for such ¯uctuations is

bPC�T0;RH0� �Z Z

PC�T0 � dT ;RH0 � dRH�fdT dRH �dT ; dRH�ddT ddRH ; �9�

where the factor A in the de®nition of fdT dRH has to beevaluated at �T0;RH0�. In nature, ¯uctuations leading toeven slight water supersaturation would initiate rapidcloud formation, thereby keeping RH at 100% . This hasbeen accounted for in the parametrisation by usingPC�T0 � dT ; fRH� under the integral sign, with fRH �min�RH0 � dRH ; 99:9% �. The resulting quantity, bPC ,can be interpreted to be the fraction of the cell area inwhich a contrail (if existent) would persist. This is themaximum possible contrail coverage. The actual contrailcoverage is usually less than bPC. For its determinationmeasures of air-tra�c density and spatial expansion ofthe contrails are needed. Such data cannot be obtainedfrom the MOZAIC data bank. Hence, the task ofdetermination of the actual contrail coverage is beyondthe scope of the present study. For the calculation ofbPC�T ;RH� we have evaluated the Schmidt-Applemancriterion for pressures 190 and 250 hPa, i.e. at the meanpressures of the two considered ECHAM altitude layers.

0

1000

2000

3000

4000

5000

6000

7000

8000

-20 -15 -10 -5 0 5 10 15 20

Num

ber

ofev

ents

δRH (%)

0

1000

2000

3000

4000

5000

6000

7000

8000

-4 -3 -2 -1 0 1 2 3 4

Num

ber

ofev

ents

δT (°C)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

-2.0 -1.5 -1.0 -0.5 0

δe (Pa)

0.5 1.0 1.5 2.0

Num

ber

ofev

ents

Fig. 4. Distributions (dotted ) of the temperature (upper), relativehumidity (middle), and water vapour partial pressure ¯uctuations(lower panel ) for ECHAM level 7 (166±222 hPa) over areas thatcorrespond to the size of an ECHAM-T42 grid cell (i.e. about300� 300 km2). The solid lines are Lorentz curves ®tted to thedistributions. The corresponding distributions for ECHAM level 8(222±290 hPa) are very similar

1062 K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations

It has been plotted in Fig. 6. Comparison of Fig. 6 withFig. 1 shows that bPC > 0 under conditions where themean grid cell properties indicate no persistent contrails.It may also be noted that in the limiting cases of PC � 1(i.e. the lines in Fig. 1, except for the water saturationboundary) the corresponding bPC is 0:5 in Fig. 6, since the¯uctuations are symmetrically distributed around zero.

4 Discussion

4.1 Origin of the ``wings'' in the ¯uctuation distributions

The ``wings'' in the ¯uctuation distributions mean thatlarge deviations from the mean values of the statevariables occur with a relatively high frequency. We nowdiscuss some possible origins of such large ¯uctuations.In particular, we investigate the role of (1) ¯ight levelchanges, (2) tropopause penetrations, and (3) deep levelconvection in the tropics. Finally we show a surprising

link between the large-scale meridional temperaturegradient and the Lorentz distribution.

Aircraft sometimes change their ¯ight level (i.e.altitude) while cruising. This pressure change (aircraftmeasure their altitude via the pressure) is usuallyaccompanied by distinct changes in temperature andhumidity. Therefore, if such a manoeuvre takes place inthe middle of an ECHAM grid cell, large ¯uctuationsshould arise. Then, the determined mean values refer tosome level in between the two assumed ¯ight levels, andthe actually measured values of temperature andhumidity all are at a relatively large variation with thesemean values. We found that ¯ight level changes are onlya minor source of large ¯uctuation events, both for thehumidity and the temperature. Since ¯uctuations due tochanges of the ¯ight level are arti®cial and not caused bymeteorological or climatic circumstances, they havebeen ignored in the evaluation of the MOZAIC data.The distributions presented in Sect. 2 are based on datacollected at constant ¯ight levels.

Another source of large ¯uctuations is penetrationsof the tropopause during ¯ight. Flight in the strato-sphere is indicated in the MOZAIC data by high ozonemixing ratio and very low humidity. The temperaturechange during a tropopause penetration can reach 10 Kover a distance of 200 km. We have checked thein¯uence of tropopause crossing on the shape of the¯uctuation distributions by ignoring all data with atemperature change of more than 3 K within one gridcell. We found that tropopause crossings are a majorcause of large humidity ¯uctuations (jdRH jJ5%) and ahighly dominant cause of large (i.e. jdT jJ1 K) tem-perature distributions. Fluctuations due to penetrationsof the tropopause are real as they can be traced back tometeorological phenomena (e.g. upper level lows). Thus,they are contained in the distributions presented inSect. 2.

T = -30 °C,RH = 20 %

T= -50 °C,RH = 60 %

δRH

(%)

25

20

15

10

5

0

-5

-10

-15

-20

0.001

0.0010.003

0.1

0.030.01

0.01

0.001

0.001

0.001

0.001

0.001

0.0030.003

δRH

(%)

25

20

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0

-5

-10

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-20

-5 -4 -3 -2 -1 0

δT (°C)

-5-5 1 2 3 4 5

0.1

0.3

0.03

0.01

0.0010.003

Fig. 5. Contours of the theoretical joint probability density (jpd)fdTdRH for common ¯uctuations �dT ; dRH� around a mean state�T0;RH0�. The jpds shown refer to �ÿ50 �C; 60%) and �ÿ30 �C; 20%)for ECHAM level 7. The corresponding jpds for ECHAM level 8 arevery similar

RH

(%)

100

90

80

70

60

0.01

0.1

0.99 0.9

0.5

0.1

0.01

50

40

30

20

10

0

-70 -60 -50

T (°C)

-40 -30 -20

Fig. 6. Maximum possible fractional coverage of persistent contrails,bPC , for ECHAM level 7, as a function of the state variablestemperature, T , and relative humidity, RH . The correspondingfunction for ECHAM level 8 is very similar

K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations 1063

Restriction of the data evaluation to the tropics (i.e.south of 30�N) shows that these regions contributesubstantially (almost as much as tropopause crossings)to large humidity ¯uctuations, probably because of deepconvection (see Fig. 3). On the contrary, large tempera-ture ¯uctuations are caused almost exclusively outsidethe tropics. The reason for that is the very smallmeridional gradient of the zonal mean temperature overthe tropics. Furthermore, the troposphere reaches veryhigh altitudes over the tropics. Therefore tropopausepenetrations rarely occur in this region.

It was shown in Sect. 2 that Lorentz distributions arewell suited to ®t the measured distributions of humidityand temperature ¯uctuations. In particular for the latter,the Lorentz formula worked very well. If data wereignored that are recorded during tropopause crossings,we still would ®nd a Lorentz pro®le, though a narrowerone. It is interesting to note that there is a link betweenthe Lorentz distribution and the large-scale meridionaltemperature gradient. A typical meridional temperaturepro®le of the upper troposphere of the middle latitudesand the polar front region displays a northwardtemperature gradient of about )0.5 to )1 K/100 kmover a region of about 1000±1500 km width on thesouthern side of the jet axis. Since most of the MOZAIC¯ights took place over the North Atlantic a considerablenumber of ¯ights were su�ciently close to the polarfront jet to be inside the large-scale temperaturegradient. Considering a certain ECHAM grid cell thatwas crossed by a large ensemble of ¯ights during 1995,we ®nd a corresponding number of angles, 90� ÿ a,between the actual ¯ight heading and the direction ofthe temperature gradient. As the ¯ights considered haddi�erent starting positions and destinations, we mayconsider the angle a a random variable. In the simplestcase of a linear N-S temperature pro®le, the totaltemperature variation measured while crossing the gridcell under consideration is proportional to tan a. If a isuniformly distributed between ÿ90� and �90�, then thetemperature variation will be distributed like tan a, andthis is a Lorentz (or Cauchy) distribution (see, e.g.Goodman, 1988, p. 62). In spite of the somewhatidealistic assumptions in these arguments we think thatthe large-scale meridional temperature gradient at leastestablishes a strong tendency towards a Lorentzdistribution.

4.2 Check of the joint probability density

In Sect. 2c we derived a theoretical expression for thejoint probability density (jpd) of common temperatureand humidity ¯uctuations. This function depends on themean state, say �T0;RH0�, via the factor A which dependson the state variables. Next we need to check whetherthe theoretical jpd is a good model for the ¯uctuations�dT ; dRH� that can be obtained from the data bank. Tothis end, we determine the frequencies of ¯uctuations in®nite ranges DT � DRH , with bin widths DT � 0:1 Kand DRH � 0:5% and plot the result in Fig. 7.

Comparison between Figs. 5 (theoretical jpds) and 7(empirical jpd) shows that the empirical jpd is broaderand rounder than the theoretical jpds. Whereas in thetheoretical jpds the outer contours are strongly curved,this is not so in the empirical jpd. In both jpds weobserve two ``main axes'', which correspond to dT � 0(vertical axis) and de � 0 (the oblique axis, i.e. ¯uctua-tion of RH induced by a pure temperature ¯uctuation,without a ¯uctuation of the absolute humidity). Theangle between the main axes depends on the mean state�T0;RH0�. The ®rst impression is that the angle betweenthe main axes is rather large in the empirical jpd, whichresembles rather the shape of the theoretical jpd for�ÿ30 �C; 20%) than that for �ÿ50 �C; 60%) where theoblique axis is much steeper. The ®rst of these stateswould, however, imply a much lower mean contrailfrequency than is actually observed. Do these di�erencesmean that the theoretical jpd is a poor model of theactual ¯uctuations? We don't think so. We rather thinkthat the empirical jpd is a weak representation of the

δRH

(%)

25

20

15

10

Level 7

5

0

-5

-10

-15

-20

δRH

(%)

δT (°C)

25

20

15

10

5

0

-5

-10

-15

-20

-5 -4 -3 -2 -1 0-5-5 1 2 3 4 5

0.010.1

1

0.3

1

0.3

0.03

0.01

0.010.03

0.03

0.01

0.1

0.01

Level 8

Fig. 7. Relative frequency of common ¯uctuations of relativehumidity, dRH , and temperature, dT . This frequency distributionmay be conceived an empirical jpd of common temperature andhumidity ¯uctuations as obtained from the data from the MOZAICdata bank. The normalisation is as in Fig. 5.

1064 K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations

truth, because for its determination we had to pile up alldata, disregarding the di�erent mean state variables.Thus, the empirical jpd of Fig.7 is a mixture of jpds for alarge number of mean states. The mixing obviously leadsto rounder contours. Remember the central limittheorem: in the limit of mixing more and moredistributions of the same type one ends up with aGaussian distribution. The product of two Gaussiandistributions possesses elliptic contours. The anglebetween the two main axes of the jpds is a function ofA which, in turn, depends on the mean state �T0;RH0�.Mixing of the states thus yields a mixture of A-values,which makes the angle between the two axes unclear (i.e.the oblique axis is not well de®ned, which alsocorresponds to the rounding of the contours). As afurther test we picked out all ¯uctuation data thatbelong to mean values in a small interval�T0 � 0:5 K;RH0 � 2:5% �, where T0 and RH0 werechosen such that as large as possible number of eventscould be obtained. Although the resulting empirical jpdis rather noisy in this constrained ensemble, it agreesreasonably well with the corresponding theoretical jpd,or at least no striking di�erences are visible (Fig. 8).

Furthermore, there is a weak positive correlationbetween ¯uctuations of water vapour partial pressure,de, and temperature, dT (Fig. 9). This has not beentaken into account in the derivation of the theoreticaljpd. Such a correlation can turn the de � 0 axis towardsthe horizontal direction. (A positive correlation betweende and dT would lead to a reduction of the factor A inthe theoretical derivation.) The correlation between deand dT can probably be explained using the following

meteorological argument: often it gets warmer when airis advected from the south. Air from southern regions isusually moister than air from other directions. Hence itusually becomes moister when it gets warmer (at least ona synoptic scale). We have neglected this weak correla-tion in the derivation of the jpd because there is nophysical relation between de and dT and we did not wantto introduce empirical relations. Anyway, the possibleerror that the neglect of this correlation causes in the®nal result, viz. the maximum contrail coverage bPC , isprobably small when compared to the errors that arisein the determination of the actual contrail coverage fromuncertainties in air tra�c densities and contrail spread-ing rates.

4.3 In¯uence of data quality on the parametrisation

Finally, let us discuss the in¯uence of possible systematicerrors in the humidity data on the derived parametrisa-tion. If such a bias were constant on time scales longerthan 30 min, it would not in¯uence the parametrisation

T = -52.2 ± 0.5 °C,RH = 45.5 ± 2.5%

δRH

(%)

25

20

15

10

5

0

-5

-10

-15

-20

-5 -4 -3 -2 -1 0δT (°C)

-5-5 1 2 3 4 5

Fig. 8. Direct comparison of a (non-normalized) empirical jpd (solid)with a corresponding theoretical jpd (dashed) for a constrained database. The ¯uctuation data in the empirical jpd refer to mean values inthe small intervals indicated along the top border of the Fig., thetheoretical jpd is drawn for the central state in this interval. Theempirical jpd is rather noisy, because the data base is small. Contourvalues are 1, 3 and 10 events for the empirical jpd, and 0.001, 0.003,0.01, and 0.03 for the theoretical jpd. The empirical jpd has not beennormalised because the statistical errors are large and a normalisationwould pretend only small statistical errors

Level 7

Level 8

δe(P

a)

5

4

3

2

1

0

-1

-2

-3

-4

-5 -4 -3 -2 -1 0δT (°C)

-5-5 1 2 3 4 5

0.01

1

0.3

0.03

0.010.03

0.03

0.01

1

0.01

0.01

0.030.01

0.1

0.10.01

0.03

-5

δe(P

a)

5

4

3

2

1

0

-1

-2

-3

-4

-5

Fig. 9. Relative frequency of common ¯uctuations of water vapourpartial pressure, de, and temperature, dT based on the MOZAIC databank. The horizontal ``axis'' of the frequency distribution is slightlytilted which means a weak positive correlation between de and dT

K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations 1065

because it would then have no e�ect on the ¯uctuationdata. Such a bias would only yield incorrect meanvalues, yet these are not used for the parametrisation. If,on the other hand, such an error possessed a morerandom nature, i.e. if its time scale were less than 30min,it would corrupt the ¯uctuation data and producebroader distributions.

However, it turned out that the widths of thedistributions only weakly a�ect the resulting parame-trisation, viz. the function bPC. This has been checked asfollows: instead of ®tting the ¯uctuation data with asimple least squares ®t, we ®tted them also using aweighted least squares criterion, with weights 1=ni whereni is the number of events. This ®tting yieldedLorentzians with a di�erent set of half widths: for level7, cT � 0:292 K, cRH � 2:07% ; for level 8, cT � 0:216 K,cRH � 3:18% . The resulting function bPC (not shown)for this set of HWHMs is little di�erent from the oneshown in Fig. 6. Thus, we may conclude that ourparametrisation is rather stable against small errors inthe determination of the ¯uctuation distributions.

5 Summary

We have evaluated humidity and temperature data fromMOZAIC for the aim of a parametrisation of persistentcontrail coverage for global circulation models likeECHAM. The derived parametrisation is the functionbPC�T ;RH� (Sect. 3) as plotted in Fig. 6, which dependson the prognostic state variables temperature andrelative humidity and which can be implemented intoGCMs, e.g. as a table or an approximative function. bPCis the fraction of the volume of a grid cell in whichcontrails, if present, would persist. How much of thisfraction actually contains contrails depends on the airtra�c density and on the spatial expansion and durationof the contrails. Generally, the actual contrail coveragewill be less than bPC. Thus, bPC primarily characterisesthe existence and expansion of the ``moist lenses'' inwhich contrails occur in groups (Bakan et al., 1994).

As a further result of the MOZAIC data evaluationwe could determine quantitatively temperature andhumidity ¯uctuations on a 300� 300 km2 scale. As faras we are aware, this has been done here for the ®rsttime, since data sets with such an extensive spatial andtemporal coverage like MOZAIC were not availablebefore. We showed that ¯uctuations on a large scalecannot be modelled by Gaussian distributions. Insteadthey must be modelled using distributions with extendedwings, because large deviations of a quantity from itslarge-scale average have a relatively high probability. Sofar, cloudiness in general circulation models has beenmodelled using ad hoc assumptions about the distribu-tion of temperature and humidity ¯uctuations (e.g.Smith 1990; Ricard and Royer 1993). Here we o�er the

possibility to set up such a parametrisation on anobservationally justi®ed basis.

Acknowledgement. The MOZAIC project is funded by theCommission of the European Union (contract AERO-CT92-0052) and coordinated by A. Marenco of CNRS in Toulouse.Part of the work has been supported within the BMBF-project``Schadsto�e in der Luftfahrt'' (contract 01LL9602-9). We thanktwo anonymous reviewers for their useful criticism.

Topical Editor L. Eymard thanks S. Bakan and another refereefor their help in evaluating this paper.

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1066 K. M. Gierens et al.: Determination of humidity and temperature ¯uctuations