Y Long

Nadine Chaumerliac
email: lampn.chaumerliac@opgc.univ-bpclermont.fr

L Deguillaume

M Leriche

F Champeau

"Effect of mixed-phase cloud on the chemical budget of trace gases: a modeling approach"

Keywords: multiphase chemistry, cloud modeling, mixed-phase microphysics

A multiphase cloud chemistry model coupling a detailed chemical reactivity mechanism in gas phase and aqueous phase to a cloud parcel model with a two-moment microphysical scheme has been extended to include ice phase processes. This newly developed model is used to study the influence of the ice phase on HCOOH, HNO 3 , H 2 O 2 and CH 2 O in a mixedphase cloud. Microphysical processes are describing the interactions between the water vapor phase, the liquid phase (cloud and rain water) and the ice phase (pristine ice, snow and graupel) in the cloud and for soluble chemical species, their transfer by mixed-phase microphysical processes has been included. In addition to microphysical transfer between iced hydrometeors, the probable two main processes incorporating soluble chemical species in iced hydrometeors are the retention in ice phase as riming or freezing occurs and the burial in the ice crystal as the crystal grows by vapor diffusion.

The model is applied to a cloud event describing a moderate precipitating mixed-phase cloud forming in a continental air mass in winter. The main features of the cloud are described and the evolution of key chemical species as function of time and temperature is discussed.

Sensitivity tests are performed: a run without ice to highlight the influence of ice phase on the chemical gas phase composition of the cloud, a run without burial showing that it is a negligible process, a run assuming full retention in ice for all species and a run varying the ice crystal shapes. A detailed analysis of the microphysical rates and chemical rates linked to retention and burial effects show that for this cloud event, the effect of the ice phase on gas phase composition is driven by riming of cloud droplets onto graupels, which leads to retention or not of soluble chemical species in the ice phase. Finally, the impact of crystal geometry on the efficiency of collection is studied together with its impact on the riming of cloud droplets on graupels and also on the retention of chemical species in ice phase.

INTRODUCTION

Clouds, which cover 50% of terrestrial surface, influence the redistribution of traces gases between the different layers of atmosphere and modify the Earth's radiative budget. Among those, mixed-phase convective clouds constitute 60 % of all clouds [START_REF] Hegg | The impact of clouds on aerosols population[END_REF]. As ice develops in clouds, it influences all major cloud characteristics including precipitation formation [START_REF] Tao | Goddard cumulus ensemble (GCE) model: Application for understanding precipitation processes[END_REF], interactions with radiation [START_REF] Ackerman | Heating rates in tropical anvils[END_REF][START_REF] Liu | A numerical study of cirrus clouds. Part II: Effects of ambient temperature, stability, radiation, ice microphysics, and microdynamics on cirrus evolution[END_REF], latent heat release and cloud dynamics [START_REF] Willoughby | Project STORMFURY: a scientific chronicle 1962-1983[END_REF], and water vapor content [START_REF] Heymsfield | Upper tropospheric relative humidity observations and implications for cirrus ice nucleation[END_REF].

The presence of ice favors tracer vertical transport due to intensified latent heat effects, modifies the exchange of matter among the various water phases, and perturbs photochemistry through retroaction between ice and radiation and reactivity pathways. Also, the terminal fall speed of the solid hydrometeors is significantly reduced compared to that of the liquid drops of the same weight. A possible consequence of these different aerodynamical properties is that partially glaciated clouds can be expected to provide a larger residence time of solid hydrometeors that can favor gas and aerosol scavenging [START_REF] Lawrence | The impact of cloud particle gravitational settling on soluble trace gas distributions[END_REF].

Due to the variety of effects of clouds on tropospheric chemistry, there have been considerable efforts to understand the complex processes of clouds through in-situ measurements, laboratory experiments and modeling studies. These have included numerous studies focused on the effects of ice clouds on dynamics and precipitation [START_REF] Rutledge | The Mesoscale and Microscale Structure and Organization of Clouds and Precipitation in Midlatitude Cyclones. XII: A Diagnostic Modeling Study of Precipitation Development in Narrow Cold-Frontal Rainbands[END_REF][START_REF] Reisner | Explicit forecasting of supercooled liquid water in winter storms using the MM5 mesoscale model[END_REF][START_REF] Thompson | A New Bulk Microphysical Parameterization in WRF[END_REF]. There have been also many studies on gaseous tracer cloud venting [START_REF] Flossmann | Venting of gases by convective clouds[END_REF][START_REF] Cotton | Cloud venting -A review and some new global annual estimates[END_REF][START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF] and of cloud chemistry pertinent to specific impacts such as acidic precipitation [START_REF] Barth | Effect of marine boundary layer clouds on tropospheric chemistry as analyzed in a regional chemistry transport model[END_REF][START_REF] Leriche | Numerical quantification of sources and phase partitioning of chemical species in cloud: application to wintertime anthropogenic air masses at the Puy de Dôme station[END_REF].

One area of cloud chemical processing that is still not well understood involves interactions of ice-phase cloud hydrometeors with chemical species. There are various iced hydrometeors (pristine, snow, graupel…) with a large range of sizes, shapes and characteristics like the level of riming for instance. Ice-chemical interactions include gas-ice transfer, partitioning during freezing of liquid hydrometeors, surface and bulk reactions in/on ice hydrometeors.

Experimental studies have shown that the amount of gas taken up by an ice surface is dependent on the type of gas, the temperature, the crystalline structure of the ice and on whether the uptake takes place on a growing or non-growing ice surface [START_REF] Pruppacher | Microphysics of clouds and precipitation[END_REF][START_REF] Huthwelker | The uptake of acidic gases on ice[END_REF], which is the relevant atmospheric case. The recent paper from [START_REF] Kärcher | Trapping of trace gases by growing ice surfaces including surface-saturated adsorption[END_REF] presents a theoretical approach of the uptake of gases by ice crystals, which considers the combined effect of mass accommodation and net adsorption of trace gases on the surfaces of growing ice particles. However, this approach is limited to nitric acid and is not directly applicable to other species. An alternative approach is to use a single parameter, the burial coefficient that describes different efficiencies of gas trapping in growing ice hydrometeors [START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF]. The partitioning of soluble gases during the freezing of liquid hydrometeors is classically obtained by a retention coefficient describing the fraction of a dissolved trace gas which is retained in hydrometeors during freezing. The information is almost non-existent for the surface of bulk reactivity of chemical species in the ice crystals and concerns mainly stratospheric conditions [START_REF] Sander | Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies[END_REF].

Previous modeling studies that have included some interactions between volatile chemicals and ice phase particles have found that they may significantly impact chemical distributions and deposition (Rutledge et al., 1986;[START_REF] Chen | The role of precipitation microphysics in the selective filtration of air entering the upper troposphere[END_REF][START_REF] Wang | A Three-Dimensional Numerical Model of Cloud Dynamics, Microphysics, and Chemistry 4. Cloud Chemistry and Precipitation Chemistry[END_REF][START_REF] Audiffren | Deviations from the Henry's law equilibrium during cloud events: a numerical approach of the mass transfer between phases and its specific numerical effects[END_REF][START_REF] Barth | Numerical simulations of the July 10, 1996, Stratospheric-Tropospheric Experiment: Radiations, Aerosols, and Ozone (STERAO)-Deep Convection experiment storm: Redistribution of soluble tracers[END_REF][START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF][START_REF] Salzmann | Model sensitivity studies regarding the role of the retention coefficient for the scavenging and redistribution of highly soluble trace gases by deep convective cloud systems[END_REF]. Simulated impacts have included enhanced scavenging of intermediate to highly soluble species [START_REF] Barth | Numerical simulations of the July 10, 1996, Stratospheric-Tropospheric Experiment: Radiations, Aerosols, and Ozone (STERAO)-Deep Convection experiment storm: Redistribution of soluble tracers[END_REF], higher amounts of H 2 O 2 and sulphates in precipitation [START_REF] Audiffren | A Modeling Study of the Influence of Ice Scavenging on the Chemical Composition of Liquid-Phase Precipitation of a Cumulonimbus Cloud[END_REF], suppression of H 2 O 2 scavenging in the glaciated part of the cloud that might explain the observations of enhanced H 2 O 2 in outflow from tropical deep convection [START_REF] Mari | Transport and scavenging of soluble gases in a deep convective cloud[END_REF][START_REF] Salzmann | Model sensitivity studies regarding the role of the retention coefficient for the scavenging and redistribution of highly soluble trace gases by deep convective cloud systems[END_REF]. Modelling results also suggest that the parameterization of partitioning can strongly affect the simulated outcome [START_REF] Chen | The role of precipitation microphysics in the selective filtration of air entering the upper troposphere[END_REF][START_REF] Barth | Numerical simulations of the July 10, 1996, Stratospheric-Tropospheric Experiment: Radiations, Aerosols, and Ozone (STERAO)-Deep Convection experiment storm: Redistribution of soluble tracers[END_REF], though results may also depend on the cloud type [START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF] and the convection initiation representation [START_REF] Salzmann | Model sensitivity studies regarding the role of the retention coefficient for the scavenging and redistribution of highly soluble trace gases by deep convective cloud systems[END_REF]. However, most of them limit their studies to soluble tracers assuming Henry's law equilibrium and did not discuss in details solubility, retention coefficients and uptake efficiencies onto ice particles for gases described with explicit chemistry fully coupled with microphysics. They mainly focused on cloud vertical transport of species rather than studying in details the partitioning of species among gas, liquid and solid phases by microphysical processes.

In this paper, we investigate the role of ice-phase processes on chemistry in the case of an isolated cloud, even if cloud dynamics is not fully represented. Exchanges of chemical species between gas phase, cloud droplets, raindrops and ice particles are studied in details with considering three ice categories (pristine, snow and graupel) distributed with a gamma function. The study is based upon a 1D model with entrainment coupling two-moment microphysics derived from [START_REF] Thompson | Explicit Forecasts of Winter Precipitation Using an Improved Bulk Microphysics Scheme. Part I: Description and Sensitivity Analysis[END_REF] with an explicit multiphase chemistry model [START_REF] Leriche | A model for tropospheric multiphase chemistry: application to one cloudy event during the CIME experiment[END_REF]2007).

In the first part of this paper, the microphysical scheme extension to ice phase is briefly described since it is based upon the works of [START_REF] Thompson | Explicit Forecasts of Winter Precipitation Using an Improved Bulk Microphysics Scheme. Part I: Description and Sensitivity Analysis[END_REF] and [START_REF] Morrison | Modeling Supersaturation and Subgrid-Scale Mixing with Two-Moment Bulk Warm Microphysics[END_REF]. Then, results are presented from a simulation case derived from [START_REF] Audiffren | Deviations from the Henry's law equilibrium during cloud events: a numerical approach of the mass transfer between phases and its specific numerical effects[END_REF] that considered very limited chemistry compared to the one presented in this paper. Indeed, for this modeling study, the chemistry includes 282 reactions with 180 reactions in the aqueous phase, 102 reactions in the gas phase and 54 mass transfer exchanges between gas and aqueous phases and fully coupled scavenging processes. Results are presented pertaining to the partitioning of soluble gas phase compounds among the various phases and discuss the potential importance of mixed-phase processes compared to only warm ones. Finally, sensitivity tests address the question of the influence of crystal shape on gas scavenging by mixed-phase microphysical processes.

MODEL FORMULATION

Description of the M2C2 model

The Model of Multiphase Cloud Chemistry (M2C2) has been developed to simulate tropospheric cloud events and results from the coupling [START_REF] Leriche | Coupling quasi-spectral microphysics with multiphase chemistry : case study of a polluted air mass at the top of the Puy de Dôme mountain (France)[END_REF] between a multiphase chemistry model with explicit multiphase chemical mechanisms [START_REF] Leriche | A model for tropospheric multiphase chemistry: application to one cloudy event during the CIME experiment[END_REF][START_REF] Leriche | Modeling study of strong acids formation and partitioning in a polluted cloud during wintertime[END_REF][START_REF] Deguillaume | The role of transition metal ions on HO x radicals in clouds: a numerical evaluation of its impact on multiphase chemistry[END_REF]) and a two moment microphysical scheme [START_REF] Caro | A two-moment parameterization of aerosol nucleation and impaction scavenging for a warm cloud microphysics: description and results from a two-dimensional simulation[END_REF]. Initially, the dynamical framework of this model was an air parcel [START_REF] Gérémy | A modelling study on the activation of small Aitken-mode aerosol particles during CIME 97[END_REF] that has been recently improved with entrainment/detrainment processes.

The chemistry included in the chemical module is explicit and describes the exchanges of chemical species between the gas phase and the aqueous phase, which is parameterized following the mass transfer kinetic formulation developed by [START_REF] Schwartz | Mass-Transport considerations pertinent to aqueous phase reactions of gases in liquid water cloud[END_REF]. In the full mechanism, the aqueous phase chemistry includes the detailed chemistry of H x O y , chlorine, carbonates, NO y , sulfur, the oxidation of organic volatile compounds (VOCs) with two carbon atom [START_REF] Leriche | Modeling study of strong acids formation and partitioning in a polluted cloud during wintertime[END_REF]2007), the chemistry of transition metal ions for iron, manganese and copper [START_REF] Deguillaume | The role of transition metal ions on HO x radicals in clouds: a numerical evaluation of its impact on multiphase chemistry[END_REF]2005). Aerosol particles temporal evolution and their activation in cloud droplets can also be described using the parameterization of droplets nucleation from [START_REF] Abdul-Razzak | A parameterization of aerosol activation. II. Multiple aerosol types[END_REF] since aerosol particles contribute to the chemical composition of cloud droplets by nucleation and impaction scavenging followed by dissolution of soluble compounds [START_REF] Leriche | Numerical quantification of sources and phase partitioning of chemical species in cloud: application to wintertime anthropogenic air masses at the Puy de Dôme station[END_REF]. The model considers calculations of photolysis frequencies both in the gas phase and in cloud droplets. The pH is calculated at each time step by solving the electroneutrality equation.

However, in the atmospheric environment cloud evolution and scavenging of gases and aerosol particles are also modified by the presence of ice phase [START_REF] Willoughby | Project STORMFURY: a scientific chronicle 1962-1983[END_REF][START_REF] Michael | The fate of volatile chemicals during wet growth of a hailstone[END_REF]. Special attention is given in the next sections to the extension of the actual microphysical scheme to ice phase and mixed-phase processes, based on the approach of [START_REF] Thompson | Explicit Forecasts of Winter Precipitation Using an Improved Bulk Microphysics Scheme. Part I: Description and Sensitivity Analysis[END_REF].

Description of the warm microphysical module

First, the "warm" microphysical module presented in detail in [START_REF] Leriche | Coupling quasi-spectral microphysics with multiphase chemistry : case study of a polluted air mass at the top of the Puy de Dôme mountain (France)[END_REF] and [START_REF] Leriche | Numerical quantification of sources and phase partitioning of chemical species in cloud: application to wintertime anthropogenic air masses at the Puy de Dôme station[END_REF] is briefly recalled here. The microphysical module derives from the parameterization of [START_REF] Chaumerliac | Sulfur scavenging in a mesoscale model with quasi-spectral microphysics: Two dimensional results for continental and maritime clouds[END_REF] that considers two liquid water categories (cloud water and rain) distributed according to log-normal distributions, ( ) The considered microphysical processes are droplet nucleation, condensation / evaporation, collision / coalescence and sedimentation (Berry and Reinhardt, 1974a, b, c, d). Cloud water and rainwater number concentration and mixing ratios are calculated in a prognostic way at each microphysical time step (∆t = 5s).

D G : ()                 σ - πσ =
Nucleation processes are parameterized following [START_REF] Chaumerliac | Sulfur scavenging in a mesoscale model with quasi-spectral microphysics: Two dimensional results for continental and maritime clouds[END_REF] who uses an activation spectrum to form cloud droplets:

k CS N = ( 2 )
where S is the supersaturation and C and k empirical constants related to the origin of the air mass (moderate continental in this paper, with C=600 cm -3 and k=0.7).

New developments -mixed-phase cloud microphysics

This part of the manuscript describes the recent developments of the M2C2 model for simulating mixed-phase clouds with a two-moment scheme. The introduction of these new processes in the model are going first to feedback on the temperature through latent heat release effects which can be expressed with the net latent heat release rate (LHR) through

where Lv, Ls, and Lf are latent heat of vaporization, sublimation, and fusion (J/kg) at T 0 =0°C, respectively, Ls=Lv+Lf; T 00 =-35°C; LHR is the net latent heat release rate (J/kg/s).

The various microphysical rates involved in the calculation of net latent heat release term are reported in the list of symbols at the end of the manuscript and in the scheme in Figure 1 which summarizes the mixed-phase microphysics, considered in the M2C2 model. Mixedphase processes include pristine formation, vapor deposition and collision / coalescence growth, evaporation, sedimentation and complex conversions between hydrometeors categories, such as aggregation, accretion, riming, freezing, melting and sublimation.

The introduction of the ice phase implies interactions between ice crystals and raindrops which will be easier to represent if both kind of hydrometeors are distributed with a similar distribution function; therefore, a generalized gamma distribution [START_REF] Meyers | New RAMS cloud microphysics parametrization Part II: the two-moment scheme[END_REF] is used with the consideration of the shape parameter,ν :

() ()                 -         ν Γ α = α - αν n 1 n n D D exp D D D 1 D G (4)
where A log-normal distribution is maintained for cloud water resulting only from aerosol activation and melting of ice crystals.

The mixed-phase scheme predicts the evolution of six water species: q v (vapor), q cw and q rw (cloud droplets and raindrops) and q ice , q snow and q graup (pristine ice, snow and graupel defined by an increasing degree of riming) for the mixing ratios and N cw , N rw for the total number concentrations of cloud droplets and raindrops respectively and N ice , N snow and N graup for the total number concentrations of pristine ice, snow and graupel respectively. The use of two classes of liquid hydrometeors together with three classes of iced hydrometeors is required to properly represent ice phase in mixed-phase clouds (Mc Cumber et al., 1991).

The mixed-phase scheme used in this study corresponds to the updated version of the Reisner's scheme by [START_REF] Thompson | Explicit Forecasts of Winter Precipitation Using an Improved Bulk Microphysics Scheme. Part I: Description and Sensitivity Analysis[END_REF], for the prediction of mixing ratios for ice pristine, snow and graupel. Ice number concentrations are a total of three prognostic variables described in the scheme of [START_REF] Morrison | Modeling Supersaturation and Subgrid-Scale Mixing with Two-Moment Bulk Warm Microphysics[END_REF] and the reader is referred to these publications for further details.

The pristine ice category is initiated by homogeneous nucleation when the temperature is less than -35°C or more likely by heterogeneous nucleation and the number of ice crystals newly formed follow the [START_REF] Cooper | Ice initiation in natural clouds. Precipitation Enhancement-A Scientific Challenge[END_REF] empirical parameterization based on direct ice crystal measurements.

Ice crystal multiplication in the model can increase the crystal concentration through rime splintering [START_REF] Hallet | Production of secondary ice particles during the riming process[END_REF]. However, this case study mostly lies outside the assumed temperature range for rime splintering to occur. These crystals grow by water vapor deposition and/or at the expense of cloud droplets by the Bergeron-Findeisen effect. The snow phase is initiated by autoconversion of the primary crystals; it grows by deposition of water, by aggregation through small crystals collection and by the light riming produced by impaction of cloud droplets and of raindrops. Graupels are produced by the heavy riming of snow and by rain freezing when supercooled raindrops get in contact with pristine ice crystals.

According to the heat balance equation and to the efficiency of their collecting capacity, graupels can grow either in the dry mode or in the wet mode when riming is very intense (as for hailstone embryos). In the latter case, the excess of non-freezable liquid water at the surface of the graupels is shed and evacuated to form raindrops. When the temperature becomes positive, pristine crystals immediately melt into cloud droplets, while snowflakes are progressively converted into graupels that melt as they fall.

Furthermore, the scheme also predicts number concentration as in [START_REF] Morrison | Modeling Supersaturation and Subgrid-Scale Mixing with Two-Moment Bulk Warm Microphysics[END_REF]; processes that involve particle concentration change have to be considered. The change in snow and graupel number concentration due t o m e l t i n g a n d s u b l i m a t i o n i s calculated by assuming that the relative change in number concentration is the same as the change in mixing ratio due to these processes following [START_REF] Ferrier | A double-moment multiple-phase four-class bulk ice scheme. Part I: Description[END_REF]. It is also assumed that the decrease of number concentration during sublimation of snow and graupel is the same as mixing ratio, which implies that the mean size does not change. The increase in rain number concentration due to melting is equal to the decrease of graupel and snow number concentration. In the case of evaporation/sublimation shrinking, the number is only adjusted upon complete evaporation/sublimation.

Size and fall velocity of the various hydrometeors are provided in Table 1 since they determine the collection and the growth of the various ice categories thereby influencing the partitioning of gases among them as detailed later on in section 4.

Two cases are considered, depending on whether snow and graupel are spherical particles or not. We assume that main differences are for snow and graupel categories. For snow, we chose 2D plate cold-type particles [START_REF] Woods | The IMPROVE-1 storm of 1-2 February 2001. Part III: Sensitivity of a mesoscale model simulation to the representation of snow particle types and testing of a bulk microphysical scheme with snow habit prediction[END_REF], which are typical at temperature range from -8°C to -12°C and non-spherical for graupel. Crystal shape influences the fall velocity which determines the residence time of hydrometeors during which they can scavenge chemicals. Also, in the two cases, we can expect different levels of riming that will perturb gas scavenging.

New developments: gas scavenging by ice hydrometeors

The effects of clouds on chemistry through microphysical processes are treated as transfer rates of the gases dissolved in cloud droplets or raindrops that are proportional to the microphysical rates of autoconversion, accretion and sedimentation. Gases dissolved and scavenged by the liquid hydrometeors after mass transfer are released back to the gas phase in case of evaporation. Same treatment is applied for gases attached to the ice-phase hydrometeors and exchanges between pristine, snow and graupel follow the microphysical processes describing the autoconversion, the aggregation and collection of crystals among the three ice categories. Concerning microphysical processes between liquid and ice or between vapor and ice, a different treatment is applied for the volatile chemical phase partitioning during freezing and riming and also during the growth of ice crystals by vapor diffusion respectively.

During riming and freezing, only a fraction of the gas R c is retained by the ice crystal and (1-R c ) is released to the gas phase [START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF].

Given the uncertainties in the physical processes driving gas uptake by ice crystals, the growth of ice by diffusion of water vapor is assumed to lead to the burial of the trace gas molecules following [START_REF] Yin | Redistribution of trace gases by convective clouds mixedphase processes[END_REF]. One gas molecule that becomes attached to the surface of a growing crystal rapidly becomes entrapped under several layers of ice before they have a chance to desorb. To represent such a process, a burial efficiency β is defined with a value of 1 if all the trace gas molecules that diffuse on the growing crystal surface become buried, while a smaller value of β implies that a weak interaction with the ice surface so that gases are not efficiently captured in the ice crystal.

The main limitation for describing retention (R c ) and burial (β) is that they are poorly known even if several laboratory and experimental studies have studied phase partitioning during the liquid-to-solid freezing and riming (Iribarne and Pyshnov, 1990a,b;[START_REF] Lamb | Measurements of the entrapment of sulphur dioxide by rime ice[END_REF][START_REF] Snider | Hydrogen peroxide retention in rime ice[END_REF][START_REF] Voisin | Scavenging of acidic gases (HCOOH, CH 3 COOH, HNO 3 , HCl, and SO 2 ) and ammonia in mixed liquid-solid water clouds at the Puy de Dôme mountain (France)[END_REF]. Some values of the retention coefficient R c are available in the literature for species such as SO 2 , H 2 O 2 , HCHO, HNO 3 , HCOOH and are reported in Figure 2. The most common formulation for the retention coefficient is the one from [START_REF] Lamb | Measurements of the entrapment of sulphur dioxide by rime ice[END_REF]:

( ) 0 T T 0058 .0 012 .0 RET - + = ( 5 )
which has been adopted for almost all species, like for instance HCOOH. Exceptions are for H 2 O 2 , CH 3 OOH, SO 2 and CH 2 O, HNO 3 , HCl. A value of 0.5 is taken for H 2 O 2 [START_REF] Snider | Hydrogen peroxide retention in rime ice[END_REF], a value of 0.02 is taken for CH 2 O, CH 3 OOH [START_REF] Mari | Transport and scavenging of soluble gases in a deep convective cloud[END_REF] and for SO 2 (Voisin et al., 2000), and a value of 1 is applied for strong acids: nitric, sulfuric and chloridric acid (Iribarne and Pyshnov, 1990a).

Figure 2 shows the retention coefficient of several species as a function of their effective Henry's law constant (see [START_REF] Sander | Compilation of Henry's Law Constants for Inorganic and Organic Species of Potential Importance in Environmental Chemistry[END_REF] for a range of temperatures between -20°C and 0°C.

This figure illustrates the contrasted behavior for a panel of gases more or less soluble and more or less sensitive to uptake in ice phase. These values and variations of both retention coefficients and Henry's law constant will help us in interpreting further results in the next sections. For instance, nitric acid is very soluble and has a retention coefficient of 1.

However no single factor controls the extremes of the retention coefficient which varies between 0 and 1, rather combination of factors such as solubility, but also ice-liquid interface supercooling, wet or dry growth conditions, as discussed by [START_REF] Michael | The fate of volatile chemicals during wet growth of a hailstone[END_REF].

The situation is even worse for the burial coefficient which has never been documented. In this paper, the burial coefficient is set constant, equal to 1, which means that the gas is retained in the crystal when it growths by vapor diffusion.

In the model, the processes of burial and retention are represented as transfer between two phases and are written with proportionality relationships.

So in the gas phase, during riming and freezing processes, the retention coefficient RET is used to calculate the concentration change rates as follows:

rim _ freez liq liq liq rim _ freez g t q q C ) RET 1( t C         ∂ ∂ - - =         ∂ ∂ ( 6 )
which is a source for gas phase, while in the liquid and ice phase:

rim _ freez liq liq liq rim _ freez liq t q q C t C         ∂ ∂ =         ∂ ∂ ( 7 ) rim _ freez p _ ice liq liq rim _ freez p _ ice t q q C ) RET ( t C         ∂ ∂ =         ∂ ∂ ( 8 )
The tendency of liquid water (cloud or rain) by freezing and riming processes is negative and is a sink for liquid phase.

During the water vapor deposition on ice hydrometeors, the gaseous species can be partly buried into the growing crystal.

The burial coefficient β transfers material from gas to ice phase with the following rates:

ice _ vap _ dep v v g dep g t q q C t C       ∂ ∂ β =         ∂ ∂ ( 9 ) dep ice dep g t C t C       ∂ ∂ - =         ∂ ∂ ( 10
)
where in equations ( 6), ( 7), ( 8), ( 9), ( 10 (molec.cm -3 ). liq q, v q, p ice q _ are respectively the mixing ratio of water cloud or rain, vapor and graupel or snow (g/kg).

Model set-up and results

Cloud features

In M2C2, the mixed-phase cloud is simulated using an entrainment rate of 4.7•10 -4 m -1 , which causes the air parcel to reach the altitude of 5 km in 10 minutes. This dynamical framework in favor of ice formation faithfully reproduces the one previously described by [START_REF] Audiffren | Deviations from the Henry's law equilibrium during cloud events: a numerical approach of the mass transfer between phases and its specific numerical effects[END_REF] with temperature and dew point profiles from [START_REF] Reisin | Rain production in convective clouds as simulated in an axisymmetric model with detailed microphysics. Part II: effect of varying drops and ice initiation[END_REF]. The cloud event is characterized with specific CCN spectrum at 1% supersaturation which is representative of a moderate continental case, as described in [START_REF] Reisin | Rain production in convective clouds as simulated in an axisymmetric model with detailed microphysics. Part II: effect of varying drops and ice initiation[END_REF] and furthermore will not precipitate very much. This case was selected to limit the wet scavenging of species by precipitation and emphasize the uptake of trace gases in mixed-phase cloud.

The 0°C isotherm is located around 2600 m. The convective air parcel starts at 600 m above ground level with temperature equal to 15°C. Cloud base and top are, respectively at the altitudes of 600 m and 5100 m where the temperature reaches -18°C. The main results for the moderate continental cloud are presented in Table 2. Compared to the model of [START_REF] Reisin | Rain production in convective clouds as simulated in an axisymmetric model with detailed microphysics. Part II: effect of varying drops and ice initiation[END_REF], in our model, snow is considered and the hydrometeor number concentration is predicted. Despite these differences, general features are retrieved with a maximum updraft of 13 m/s, a maximum water content of 6 g/kg, 0.05 g/kg, and 4.3 g/kg respectively, for the maxima of cloud, ice pristine + snow and graupel. The run lasts approximately 20 min with the mature stage of the cloud between 5 min and 15 min followed by a decaying stage the last five minutes where a net decrease in the cloud water content is observed, together with a slowing down of the riming process.

Figure 3a illustrates the relative water fractions for different phases as a function of time, normalized with respect to maximum liquid water content. The total liquid water content reached its maximum value five minutes from model initialization, at which time the graupel fraction was less than 0.1%. At the same time, the total water fraction (water + ice + snow + graupel) of the cloud reached its maximum.

Then, the graupel fraction rapidly increases as a result of graupel production by riming and drop freezing. Ten minutes later, most of the cloud fraction is in the form of graupel, while the drops fraction composed only 10% and the ice crystal a few percent. Snow fraction driven by ice growth, riming of raindrops on ice and of cloud droplets on snow steadily increases up to around 1% of the maximum liquid water content and decreases at the end of the simulation through precipitation and evaporation.

In Figure 3a, dotted curves represent the total liquid water content and the water fractions obtained for the same cloud than before but when neglecting ice phase processes. The total liquid water content is decreasing faster and the cloud is lasting longer than in the reference case when mixed-phase processes are activated.

In the same manner, the total water content and the water fractions are drawn in Figure 3b, for a case that will be studied in section 3.3 involving spherical crystals. Already, we can observe that the cloud duration is longer in the case of spherical crystals and this discrepancy will probably have some impact on gas scavenging (see section 4.2).

Chemical features

Initial data for the chemical species considered in the M2C2 model are given in Table 3. The gas phase chemical species are initialized using measurements from [START_REF] Voisin | Scavenging of acidic gases (HCOOH, CH 3 COOH, HNO 3 , HCl, and SO 2 ) and ammonia in mixed liquid-solid water clouds at the Puy de Dôme mountain (France)[END_REF] who characterized a winter continental cloud at mid-latitudes and performed experimental studies on retention coefficient for inorganic and organic chemical species during mixed-phase processes.

Figure 4a presents the time evolution of gas phase concentrations for specific chemical species, which have been selected to illustrate a representative range of chemical reactivities, retention efficiencies and solubility rates (Figure 2). CH 2 O is a stable terminal organic compound in gas phase, revealing the level of atmospheric oxidation capacity, H 2 O 2 is one of the most reactive oxidant in aqueous phase [START_REF] Daum | Measurement and interpretation of concentrations of H 2 0 2 and r ela ted spe cie s in the up per Midwest during summer[END_REF]) and HCOOH, HNO 3 are respectively weak and strong acids which influence pH of cloud water [START_REF] Chameides | The coupled gas phase/aqueous phase free radical chemistry of a cloud[END_REF][START_REF] Schwartz | Cloud Chemistry[END_REF]. The species exhibit similar evolution with a strong decrease at the beginning of the run, when cloud appears and the various curves are organized with respect to solubility levels. CH 2 O is not scavenged as efficiently as H 2 O 2 , HCOOH and HNO 3 . From 12:05 to 12:10, nitric acid concentration suddenly increases because at that time cloud droplets are converted into raindrops, leading to a less efficient mass transfer from gas phase to rain phase together with efficient gas phase production of nitric acid by the reaction with OH and NO 2 . At those times, the cloud is supercooled with liquid phase still present.

Later on, the temperature and the total condensed water decrease and ice forms and a global increase of gas concentrations is observed for all species. However, only CH 2 O and HCOOH almost retrieve their initial concentration levels due to their low retention in the ice phase.

Several factors are of importance in driving the final composition of cloud ice and include the partitioning of species between gaseous and supercooled liquid phases, the amount of rimed ice collected by snowflakes, and the retention of gas during shock freezing of supercooled droplets onto ice particles. Strong acids such as HNO 3 , being sufficiently soluble in water, are mainly partitioned into supercooled water droplets. Furthermore, being subsaturated in liquid droplets, these species are well retained in rimed ice. For these species, riming is found to be the main process driving the final composition of snowflakes, direct incorporation from the gas phase during growth of snowflakes remaining insignificant because of low concentrations in the gas phase [START_REF] Voisin | Scavenging of acidic gases (HCOOH, CH 3 COOH, HNO 3 , HCl, and SO 2 ) and ammonia in mixed liquid-solid water clouds at the Puy de Dôme mountain (France)[END_REF].

Figure 4b presents the evolution of gaseous CH 2 O, H 2 O 2 , HNO 3 and HCOOH but this time as a function of temperature along the whole simulation. According to temperature, two regimes can be differentiated with liquid phase processes for temperatures greater than about -5°C and with ice phase processes otherwise. The evolution of concentration in Figure 4b reflects the temperature-dependent solubility of chemical species for temperatures comprised between -5°C and 15°C. For negative temperatures (below -5°C) the amount of gaseous chemical species is modified by the presence of ice phase: chemical species depending on retention and burial coefficients are partially retained in the ice (more particularly the acidic species in Figure 2). The warming of the descending parcel creates a kind of hysteresis, which occurs when temperature starts to increase (around -18°C) and is directly linked to the efficiency of chemical species retention by ice for temperatures ranging from -18°C to -5°C.

Sensitivity tests

To get insights in the relative effects of ice processes, retention, burial, crystal shapes on gas scavenging, it is interesting to proceed to sensitivity tests. Four simulations have been performed in addition to the reference case which considers complex crystal geometry, a burial efficiency of 1, a variable retention coefficient as defined in Figure 2 and Section 2.4, mixed-phase processes and is fully described in sections 3.1 and 3.2. Those sensitivity tests include the following departures from the reference case: 1) a warm simulation where ice processes are switched off;

2) a case where the burial efficiency is set to 0;

3) a case where the retention efficiency is set equal to 1 for all chemical species; 4) a case where ice crystals are considered as spheres.

Figure 5 shows the time evolution of gas concentrations for (a) CH 2 O, (b) H 2 O 2 , (c) HNO 3 and (d) HCOOH as in Figure 4a, but zooming on the end of the simulation when ice processes can occur. One point that has to be raised is the abrupt changes in gas concentration that are observed at times denoted by the arrows on Figure 5, and that corresponds to cloud water decrease observed in every run (reference case, case without ice and case with spherical crystals). The first main result is that burial is ineffective to scavenge all gases considered in our study by comparing case 2 with the reference case. Considering a full retention in ice for all species, obviously does not affect nitric acid but leads to large discrepancies for HCHO, H 2 O 2 and HCOOH compared to their original retention coefficients respectively equal to 0.02, 0.5 and to 0.012+0.0058(T-T 0 ), as drawn in Figure 2. Their evolution with time before the decrease of cloud water amount just reflects that, when graupels form, they are fully retained in ice phase. Then, the time evolution is different for these three species due to the competition between gas phase reactivity, exchanges between gas and liquid phases and aqueous phase reactivity. Large deviations are observed between the run with and without ice for all compounds. For chemical species with a small retention coefficient ( H CH O , H 2 O 2 a nd H CO O H ) , the ir gaseous concentrations are smaller when no ice is considered because, in this case, soluble species are scavenged by cloud water whereas in the reference case, no more cloud water is available and species are released back to the gas phase when liquid water freezes. H 2 O 2 has a contrasted behavior than the other compounds, at the end of the reference simulation. Its concentration increases as the total condensed water is depleted and then decreases when ice stabilizes in the form of graupels. The only possibility for H 2 O 2 concentration to decrease is its reactivity with OH and its direct photolysis. In the meantime, nitric acid is produced from the gas phase reaction between NO 2 with OH radical. This continuous production explains the difference between the runs with and without ice for nitric acid: neglecting ice, nitric acid produced in gas phase is scavenged at once by cloud droplets whereas when ice is considered, burial is inefficient to scavenge nitric acid from the ice phase.

The influence of crystal shape is similar for all gases with a shift in time for the gas content evolution, due to different riming intensity of the regular versus complex crystals, as will be discussed in Section 4.2.

Impacts of ice microphysics on chemistry

Retention and burial processes

As discussed previously (section 3), two processes have been highlighted in the treatment of gas uptake by ice hydrometeors: gas retention which occurs for retaining a fraction of gas during the freezing/riming processes and burial which depicts the entrapment of gas in ice crystals during their growth by vapor diffusion. For this reason, special attention is now given to these particular microphysical pathways.

Figure 6a presents the main microphysical processes i n v o l v e d i n t h e e v o l u t i o n o f i c e

hydrometeors, which influence burial and retention and determines gas partitioning among phases as a function of temperature. Freezing is negligible over the whole simulation because of small continental cloud droplets.

Water vapor deposition on graupel, which drives the burial process, initially dominates for temperatures below -10°C. Riming causes ice crystals to grow by accretion (collection growth) and is represented in Figure 6a. Collection g r o w t h b y c l o u d w a t e r o n i c e hydrometeors is the most efficient process when temperature is below -8°C, around -14°C with a predominant importance of riming on graupel (COLLQCG). This will strongly influence retention of gases by ice crystals. These processes are effective when the mixing ratio of graupel, snow and ice become important i.e. after 10 min of simulation time (Figure 3). Figure 6b illustrates the influence of retention and burial on different chemical species with various solubilities and retention coefficients (see Figure 2), through the production or destruction rates in s -1 , which exchange chemical matter between liquid phase, ice phase and gas phase and expressed by

BUR _ RET g g ) C t ( C 1 ∂ ∂ where g C

i s t h e g a s c o n c e n t r a t i o n ( m o l e c . c m

-3 ) ; BUR RET g C t _ ) ( ∂ ∂
is the temporal variation of gas concentration related to burial and retention effects (molec.cm -3 .s -1 ).

Since nitric acid has retention coefficient equal to one, it is totally transferred from liquid water to ice when riming occurs, and the only effect that could be seen on Figure 6a is the effect of burial. This effect of burial on nitric acid is insignificant as seen in Figure 6a, on the right hand scale: this confirms that burial effect is negligible. For others species, significant contribution can only come from the retention effect, which occurs for temperatures less than -10°C and leads to a maximum value for temperatures between -15°C and -13°C. Degassing is larger for H 2 O 2 and HCOOH than for CH 2 O. The shift and the intensity for each chemical species peaks depend on solubility and retention coefficient.

Influence of crystal shape on gas scavenging

In the previous section, retention has been identified as the main process that drives the partitioning of gases between gas, liquid and ice phases. By definition, retention is strongly linked to riming and freezing processes.

So it is important to consider the complex structures of ice crystals since they have in reality a variety of habits, while riming changes and modifies the properties of ice particles. This has been shown by [START_REF] Heymsfield | Refinements to ice particle mass dimensional and terminal velocity relationships for ice clouds: Part I: Temperature dependence[END_REF] who collected observations sampled for a wide range of temperatures and by [START_REF] Barthazy | Fall velocity of snowflakes of different riming degree and crystal types[END_REF] who measured fall velocities of graupel and of snowflakes of different riming degree and consisting of different crystal types.

Also, various habits for ice particles lead to fall velocity differences that directly impact on the residence time and fallout precipitation which both regulate the gas scavenging efficiency.

One recent study from [START_REF] Michael | The fate of volatile chemicals during wet growth of a hailstone[END_REF] concludes that no clear effect was observed on retention in case of hail probably due to very efficient ice-liquid interface supercooling.

The uptake of inorganic compounds like HCl, HNO 3 , and SO 2 by planar ice surfaces, ice spheres, and dendritic ice crystals during crystal growth have been already reported [START_REF] Mitra | A laboratory study on the scavenging of SO 2 by snow crystals[END_REF][START_REF] Diehl | A laboratory study of the uptake of HNO 3 and HCl vapor by snow crystals and ice spheres at temperatures between 0 and -40°C[END_REF][START_REF] Conklin | Sulphur dioxide reactions on ice surface: implications for dry deposition to snow[END_REF]Domine and Thibert, 1996;[START_REF] Santachiara | Scavenging of SO 2 and NH 3 during growth of ice[END_REF]. However, data on the uptake of organic compounds during crystal growth are rather limited [START_REF] Fries | Laboratory studies on the uptake of aromatic hydrocarbons by ice crystals during vapor depositional crystal growth[END_REF] and all these experimental studies do not allow for quantifying the effect of crystal shape on the efficiency of gas uptake.

This section is devoted to sensitivity tests that highlight the importance of riming and consequently of gas retention as a function of crystal shape.

For this, we compared the reference case described in the previous section and assuming nonspherical particles (Table 1) with a new simulation performed with hypothesis of spherical snow and graupel particles. Time evolution of the various hydrometeors contents is presented in Figure 3b, for an easier comparison with the reference case.

As seen before in Figure 6a, retention is driven by the growth of graupel by cloud water collection on graupel (COLLQCG) and this is still true for ice crystals with spherical geometry. To evaluate the influence of crystal shape on riming and thereby retention processes, evolution of graupel mixing ratio is followed by looking at the rimed mass fraction of all particles defined as: cw ice graup snow graup q q q q q RI + + + =

( 1 2 )

where q graup , q snow , q ice and q cw are the mixing ratios for graupel, snow, ice and cloud water respectively (g.kg -1 ).

This ratio defines the riming intensity as described in [START_REF] Lin | Development of a new bulk microphysical parameterization with varying riming intensity and its evaluation within WRF[END_REF] and ranges from 0 (pristine ice particles) to 1 (graupel). Figure 7 displays the evolution of this ratio as a function of the temperature for the two simulations with non-spherical and spherical hypothesis for the crystal shapes. The growth of graupel by cloud water collection on graupel (COLLQCG) is also shown in both cases. In the non-spherical case, riming is much more important when the temperature gets colder (-17°C à -12°C) while in the spherical case COLLQCG is smaller but is spread over a larger range of temperatures (from -17°C to -9°C).

The riming intensity is larger in the non-spherical case vs. the spherical one because it is directly determined by the mean size and the terminal velocity of graupels. Figure 8 presents the evolution of graupel mean sizes and terminal velocities in the two simulations with spherical and non-spherical assumptions for graupels. The double prediction of mixing ratios and number concentrations of ice particles, allows for the knowledge of the crystal mean diameter at every time step in the simulation. This mean diameter is used to calculate the terminal fall velocities for the various ice particles as defined in Table 1. Graupels with complex shape are larger for temperatures above -16°C, they become heavier and have larger terminal velocities since their mass and velocity are directly related to the size. Due to these properties, the collection growth of non-spherical graupel by cloud water on graupel is more efficient and quicker over that range of temperatures than in the case with spherical particles.

The strong dependency of the retention on the collection growth of graupel by cloud water on graupel (COLLQCG) is now illustrated. Figure 6b shows the evolution of the retention and burial rates for different chemical species with various solubilities, for spherical ice particles with dotted lines and can be compared with solid lines showing the equivalent rates for nonspherical ice particles. The production rates in s -1 describing the exchanges of chemical species between ice phase and gas phase are larger and spread over a smaller range of temperatures in the non-spherical reference case (Figure 6b, solid lines) than in the spherical case (Figure 6b dotted lines). They reflect the behavior of the collection growth of graupel by cloud water on graupel (COLLQCG) previously discussed. Comparing the two figures, for all species considered, degassing is larger for non-spherical crystals than for spherical ones. For temperatures ranging from -18°C to -11°C, non-spherical graupels are growing faster than spherical ones as seen in Figure 6b. Also, the riming intensity in Figure 7 peaks at colder temperatures for non-spherical crystals than spherical ones. This results in a more efficient degassing for non-spherical graupels.

The uptake of nitric acid in ice is more efficient when the riming intensity reaches its maximum (Figure 6) and when the graupel diameter does not increase anymore (Figure 7). It is also greater for non-spherical crystals that are much larger in size than spherical ones.

These findings are in agreement with experimental observations done by [START_REF] Hoog | Laboratory studies about the interaction of ammonia with ice crystals at temperatures between 0 and -20°C[END_REF] who concluded that uptake of gases by ice is the most efficient for nitric acid for non-growing crystals, even if it is rather small compared to uptake by water droplets.

Conclusion and discussion

In this paper, a multiphase cloud chemistry model M2C2 is presented with new developments including two-moment mixed-phase microphysics coupled with explicit chemistry.

Partitioning of chemical species among gas, cloud, rain, and ice, snow and graupel is investigated to document gas-ice transfer, redistribution of chemical species by microphysical processes.

First, the importance of ice phase processes against liquid only processes leads to a very different gas budget for all species considered (HCOOH, HNO 3 , H 2 O 2 , CH 2 O). Several factors have been studied as possible contribution of the final composition of mixed-phase clouds: temperature dependency of the chemical rates (reactivity and solubility), entrapment (retention and burial) of gases in ice particles, retention coefficient and solubilities of gases,

The main conclusions of the paper are: retention is the important process for all species, retention depends on the riming intensity which is determined by the collection growth of graupel by cloud droplets on rimed ice particles. Additionally, it is found that burial is negligible. Collection growth is found to be very sensitive to crystal shape, because varying mass-diameter relationships and terminal velocity dependencies on diameter is allowed with a two-moment microphysical parameterization. From our study which considers a mixed-phase cloud, it can be concluded that not the direct uptake by ice crystals but rather riming seems to be the major process for the scavenging of gases from the atmosphere via the ice phase. This work confirms the primordial role of retention in the chemical budget of soluble chemical species in the mixed-phase clouds. Further experimental efforts are needed to document the retention coefficients of other compounds apart from acids.

Even if our study shows a negligible effect of direct uptake of gas by crystals, these conclusions are not valid for the only glaciated part of mixed-phase clouds (anvil of convective clouds for instance) and for cirrus. Processes of gas-ice mass transfer include adsorption on non-growing ice hydrometeors surfaces [START_REF] Tabazadeh | A surface chemistry model for nonreactive trace gas adsorption on ice: Implications for nitric acid scavenging by cirrus[END_REF], co-deposition with water vapor on growing ice [START_REF] Fries | Laboratory studies on the uptake of aromatic hydrocarbons by ice crystals during vapor depositional crystal growth[END_REF], dissolution in a quasi-liquid layer on ice surface [START_REF] Michael | The fate of volatile chemicals during wet growth of a hailstone[END_REF][START_REF] Abbatt | Interactions of atmospheric trace gases with ice surfaces: Adsorption and reaction[END_REF], diffusion in the bulk ice and incorporation into the crystal structure [START_REF] Kärcher | Trapping of trace gases in growing ice crystals[END_REF]. These processes are poorly understood or have been studied in laboratories with artificial ice and for temperatures that are far from real tropospheric conditions. A quantitative physical understanding of the interactions of snow and ice with trace gases is critical for predicting the effects of climate change on atmospheric composition, and for modeling atmospheric chemistry [START_REF] Mcneill | Ice in the environment: connections to atmospheric chemistry[END_REF]. Additional laboratory investigations are urgently needed to provide elements for

LIST OF SYMBOLS

LIST OF TABLES

• Table 1 : Values of the a,b,c,d coefficients used in the formula of the crystal mass, given by aD b (kg) and of the fall velocity given by cD d (ρ 0 / ρ a ) 0,4 (m/s), as a function of crystal diameter. These data are provided by [START_REF] Lafore | The Meso-NH Atmospheric Simulation System. Part I: Adiabatic formulation and control simulations[END_REF] and [START_REF] Barthe | Description and first results of an explicit electrical scheme in a 3D cloud resolving model[END_REF].

• Table 2: Main results for the moderate continental case simulation.

• Table 3: Chemical initial concentrations for gas phase compounds in the M2C2 model.

LIST OF FIGURES

• Figure 1: Microphysical exchanges between various phases considered in the model.

• • Figure 3: a) Time evolution of the various hydrometeors fractions normalized by the maximum of the liquid water mixing ratio (q i = mixing ratio of the "i" hydrometeor).

The total mixing ratio of condensed water expressed by the sum of graupel snow ice cloud rain q q q q q + + + + i s i n d i c a t e d o n t h e r i g h t Y -a x i s . D o t t e d l i n e s represents the time evolution of the total mixing ratio and the water fraction when ice is neglected.

b) same as in Figure 3a but for spherical crystals.

• Figure 4: a) Time evolution of gas phase concentrations in the reference case. Time evolution of the temperature is also drawn in black.

b) Evolution of gas phase concentration with temperature. The vertical dashed line materializes the separation between supercooled and ice processes.

The beginning and the end of the simulation are indicated with t 0 and t end respectively.

• Figure 5: Time evolution of gas concentrations for CH 2 O, HCOOH, H 2 O 2 , HNO 3 for several simulations: the reference case (solid line), a case without ice, a case considering spherical crystals, a case neglecting burial, a case assuming full retention.

• Figure 6: a) Microphysical rates related to the riming, freezing and deposition processes versus temperature (note that collection growth of graupels by cloud droplets on graupel is one order of magnitude larger than the other rates on the right-hand side scale).

b) Evolution of gas species destruction or production rates

BUR _ RET g g ) C t ( C 1 ∂ ∂ due to
retention and burial effects as a function of temperature. Those rates correspond to the microphysical conversion rates shown in Figure 6a. Also, they are superimposed with the rates obtained for the simulation with spherical crystals (dashed lines). 1 : Values of the a,b,c,d coefficients used in the formula of the crystal mass, given by aD b (kg) and of the fall velocity given by cD d (ρ 0 / ρ a ) 0,4 (m/s), as a function of crystal diameter.

These data are provided by [START_REF] Lafore | The Meso-NH Atmospheric Simulation System. Part I: Adiabatic formulation and control simulations[END_REF] and [START_REF] Barthe | Description and first results of an explicit electrical scheme in a 3D cloud resolving model[END_REF].

Parameters Values

C (cm The beginning and the end of the simulation are indicated with t 0 and t end respectively.

  e d i a n g e o m e t r i c d i a m e t e r ; σ : geometric standard deviation.

  phase changes among different cloud species as: LHR = Lv [CONDQCW + CONDQRW -EVAPMQS -EVAPMQG] + Ls [XNUCQI (T<T 0 ) + DEPVQI(T<T 0 ) + DEPQSNOW(T<T 0 ) + DEPQGRAUP(T<T 0 )] + Lf [FREZQR(T 00 <T<T 0 )] + FREZQI(T 00 <T<T 0 ) + RIMQI(T<T 0 ) + COLLQCS(T<T 0 ) + COLLQRIG(T<T 0 ) + COLLQCG(T<T 0 ) + COLLQRS(T<T 0 ) + COLLQRG(T<T 0 ) -XMELTQS(T>T 0 ) -XMELTRG(T>T 0 ) -XMELTCG(T>T 0 ) -XMELTQG(T>T 0 )]

Figure 2 :

 2 Values of retention coefficients and effective Henry's law constants (pH is varying around a value of 5) for different temperatures (0, -5, -10, -15 and -20°C) and for several chemical species (SO 2 , CH 2 O, H 2 O 2 , HNO 3 , HCOOH).

Figure 2 :

 2 Figure 2: Values of retention coefficients and effective Henry's law constants (pH is varying

Figure 3

 3 Figure 3: a) Time evolution of the various hydrometeors fractions normalized by the

Figure 4

 4 Figure 4: a) Time evolution of gas phase concentrations in the reference case. Time evolution

Figure 6

 6 Figure 6: a) Microphysical rates related to the riming, freezing and deposition processes

Figure 8 :

 8 Figure 8: Graupel diameter and terminal velocity for non-spherical (in black) and spherical

Table 2 :

 2 Main results for the moderate continental case simulation.

	CCN spectrum : N = C.S k	k	-3 )	600 0.7
	Maximum updraft (m/s)			13
	Location (m)			2500
	Cloud duration (min)			14
		Cloud	5.6
	Maximum mixing ratio	Ice		7.4.10 -3
	(g/kg)	Snow	4.2.10 -2
		G r a u p e l	4.3
		Cloud	6.00.10 9
	Maximum number concentration	Ice		7.77.10 4
	(cm -3 )	Snow	4.05.10 4
		G r a u p e l	4.12.10 4
		Cloud	2.28.10 -5
	Mean diameter	Ice		4.10.10 -5
	(m)	Snow	5.7.10 -5
		G r a u p e l	3.77.10 -4

  Figure 5: Time evolution of gas concentrations for CH 2 O, HCOOH, H 2 O 2 , HNO 3 for several

								Time (hh:mm:ss)
		12:10:55	12:11:10	12:11:30	12:11:50	12:12:10	12:12:30	12:12:50	12:13:10	12:13:30	12:13:50	12:14:10	12:14:30	12:14:50	12:15:10	12:15:30	12:15:50
		1,0x10 11
		1,0x10 10	CH 2 O
	Concentration (molec.cm -3 )	1,0x10 7 1,0x10 8 1,0x10 9
		1,0x10 6
		1,0x10 5
		1,0x10 11
		1,0x10 10	HCOOH
	Concentration (molec.cm -3 )	1,0x10 7 1,0x10 8 1,0x10 9
		1,0x10 6
		1,0x10 5
		1,0x10 11
		1,0x10 10	H 2 O 2
	Concentration (molec.cm -3 )	1,0x10 7 1,0x10 8 1,0x10 9
		1,0x10 6
		1,0x10 5
		1,0x10 11
		1,0x10 10	HNO 3
	Concentration (molec.cm -3 )	1,0x10 7 1,0x10 8 1,0x10 9
		1,0x10 6
		1,0x10 5
		12:10:55	12:11:10	12:11:30	12:11:50	12:12:10	12:12:30	12:12:50	12:13:10	12:13:30	12:13:50	12:14:10	12:14:30	12:14:50	12:15:10	12:15:30	12:15:50
								Time (hh:mm:ss)
				Reference							Spherical crystals
				No-ice								No-burial
													Retention 100%

simulations: the reference case (solid line), a case without ice, a case considering spherical crystals, a case neglecting burial, a case assuming full retention.

validating parameterizations of gas scavenging by growing ice particles [START_REF] Kärcher | Trapping of trace gases by growing ice surfaces including surface-saturated adsorption[END_REF][START_REF] Marécal | Modelling the reversible uptake of chemical species in the gas phase by ice particles formed in a convective cloud[END_REF]. They could include experiments with different temperatures in the range of tropospheric conditions to reproduce different conditions of crystal growth and could concern a wider range of species, possibly being able to react between them. Cloud water q c ,N c Snow q s ,N s q r ,N r Graupel q g ,N g

Acknowledgments

Ice cloud q i ,N i -6,0x10

COLLQCG

-5

-5,0x10

-5

-4,0x10

-5

-3,0x10

-5

-2,0x10

-5

-1,0x10


Jean-Pierre P Massat
email: jean-pierre.massat@sncf.fr

T M L Nguyen-Tajan

H Maitournam

E Balmès

Fatigue analysis of catenary contact wires for high speed trains

Keywords: Overhead Contact Line (OCL), hard drawn copper contact wire, fatigue failure, Dang Van's criterion, Finite Element volume mesh

The fatigue fracture is one of the most critical failures which may occur on the high speed network because it is undetectable and it has a huge impact on traffic disruption. The contact wire lifespan of a high speed line is estimated at more than 50 years and thus it is necessary to consider the risk of fatigue. The Railway Technical Research Institute in Japan studied this phenomenon for a long time and performed experimental tests. Using these results and by comparing with failures occurred in France, a preliminary analysis is carried out to identify parameters which significantly influence the fatigue phenomenon. This analysis consists in using the numerical software OSCAR © to evaluate the loads, perform a fatigue assessment of the contact wire. The procedure, using a onedimensional and a three-dimensional model, is described in this article.

1

Introduction Nowadays, the main criterion used to determine the contact wire replacement of the Overhead Contact Line (OCL) is wear but, on high speed lines, the measured wear rate is very small. In this way, the contact wire lifespan of a high speed line is estimated at more than 50 years and thus it is necessary to consider the risk of having other types of system failure.

The fatigue fracture is one of the most critical failures which may occur on the high speed network because it is undetectable and it has a huge impact on traffic disruption, client discomfort (delays, speed slowdowns) and cost explosion. Unfortunately, conditions of use increase the risk of fatigue failure:

with overloaded traffic, each year more than 220 000 pantographs run under the catenary between Paris and Lyon, the tensile load applied to the contact wire is very high particularly on high speed lines and may increase in the future, the damping ratio of the catenary is so low that a loading cycle is composed of several bending waves preceding and following the pantograph passage, there is a lot of defects in the contact wire surface due to pantograph friction, electrical arcs and corrosion which can cause crack initiation (see. Figure 1a, b &c). place under the junction claws. This component, commonly used in maintenance on classical lines, appeared to be inadequate when used on high speed lines. Its heavy weight and its high stiffness produce important dynamic loads and arcing with the pantograph passage. A modification of the maintenance procedure limits the number of settled claws and their duration, but the fatigue could take place on other lines. Indeed, the French network speed rising tendency requires an increase of the mechanical tension in the contact wire and this parameter heightens the fatigue failure phenomenon.

The RTRI 1 , one of the research departments of Japan railways, studied the fatigue failure of the contact wire for a long time, see [START_REF] Yamashita | Influence of Mean Stress on Contact Wire Fatigue[END_REF], [START_REF] Sugahara | Preventing Fatigue Breakage of Contact Wires[END_REF] and [START_REF] Sugahara | Applicability of Rain Flow Method to Fatigue Life Span Estimation of Overhead Contact Wire[END_REF]. They developed a specific test bench and performed inline tests in order to measure and study the stress cycles in the contact wire. A comparison of the different approaches is led over the partnership between RTRI and SNCF.

In this paper, we show the numerical study developed by SNCF using experimental results of RTRI as reference. The numerical study was carried out using OSCAR ©2 , the pantograph-catenary dynamic interaction simulation software developed at SNCF and based on Finite Elements (FE) method [START_REF] Cléon | OSCAR © : La caténaire en 3D[END_REF]. This tool is used to compute dynamic loads in the catenary using 1D finite element and to deduce the zone of maximum uniaxial stresses along the catenary. A fully 3D contact wire model of this selected zone is then built to study the multiaxial stress history in the contact wire section [START_REF] Daouk | Analyse en fatigue du fil de contact de la caténaire[END_REF].

2

Preliminary analysis In this section, results of inline fatigue failure obtained on the French network and on the test bench of RTRI are compared. Using this preliminary analysis, it appears that the phenomena are complex and difficult to predict. Indeed, contact wire failure which occurred inline (next to a claw) are different from those obtained using the test bench of RTRI. In both cases, the fatigue failure process follows three stages: crack initiation, crack propagation and failure. This last step occurs when the material that has not been affected by the crack cannot withstand the applied stress. One can determine material failed by fatigue by examining the appearance of the fracture. A fatigue fracture will have two distinct regions; one being smooth or burnished as a result of the rubbing of the bottom and top of the crack; the second is granular, due to the rapid failure of the material. These visual clues may be seen in Figure 3 and Figure 5a but these figures illustrate significant differences between test bench and inline failure.

2.1

Contact wire fatigue fracture on test bench The RTRI research department built a specific test bench to study fatigue breakage phenomenon [START_REF] Sugahara | Preventing Fatigue Breakage of Contact Wires[END_REF]. This test bench described in the Figure 2 was designed to reproduce realistic operation conditions by using a sample of real contact wire, by adding a tensile load on the contact wire to obtain the relevant mean stress and by applying a representative bending stress controlled by deformation. Considering the test is performed on a sample of a real contact wire, without any machining and thus with a constant section, a three bending point bench was chosen to force the fatigue failure to take place under the oscillating roller as described in Figure 2. In this way, it is easier to stick sensors close to the crack initiation point. Some tests were made on this bench using samples of OCL contact wire provided by SNCF. This was a hard drawn copper contact wire used on high speed lines and its section was 150mm The fatigue crack appearance is shown in Figure 3. The three steps of the fatigue process can be identified: crack initiation, crack propagation and failure. The origin of the fatigue crack occurred on the top of the contact wire. Indeed, a dark area corresponding to crack initiation begins on the upper edge of the wire and the crack propagates to failure by necking of the wire. One can note that these tests are reproducible and evolution of the fatigue fracture appearance remains independent of the bending strain amplitude.

2.2

Inline fatigue failure analysis The behaviour described above is different from that obtained inline though contact wire is similar. This may be explained because all inline fatigue failures on the French network occurred under a junction claw (see Figure 4). This component is used to connect two contact wires because of a breakage or to reinforce a weak point of the wire. Its mass is greater than the apparent mass of the catenary. When the pantograph passes under this component, it sets off a significant dynamic loading associated to high level of stress.

Figure 5a illustrates the contact wire fatigue fracture appearance obtained inline under a junction claw and Figure 5b shows that the contact wire rupture occurred at the boundary of the junction claw. However, the fatigue crack appearance is different from test bench: the origin of the crack failure took place under a junction claw on the lower edge of the contact wire. In other words, the crack initiation occurred on the sliding part.

Failure zone Necking

In conclusion, the phenomena involved in the fatigue failure of the contact wire are very sensitive to environmental conditions. For instance under a junction claw, the stress fields are different:

First, the clamping force applied by the junction claw to the wire exceeds 2kN/cm that has the effect of locally, but significantly, increasing the mean stress. Second, the dynamic stresses due to pantograph passage are strongly increased with the inertial effects added by the junction claw which generate a discontinuity of the apparent mass, The coupling of these phenomena makes the analysis of catenary fatigue complex.

Since it is impossible to perform tests on a bench with a junction claw, we attempted to simulate this phenomenon using a numerical model. Given the importance of having realistic stresses and the high difficulty of simulating the dynamic interaction between pantograph and catenary, OSCAR © is used because it is already validated and certificated.

OSCAR © : pantograph-catenary dynamic simulation

In the present section, we will first explain the complexity of the overhead line physics (also see [START_REF] Bobillot | Pantograph-Catenary: a High-Speed European couple[END_REF], [START_REF] Massat | Modélisation du comportement dynamique du couple pantographe-caténaire. Application à la détection de défauts dans la caténaire[END_REF]), and then, how they are handled in the OSCAR © simulation software.

Firstly, the dynamic behaviour of the two flexible structures (the pantograph and the catenary) in sliding contact is very different in spite of having comparable stiffness. On one hand, the overhead line is a very long structure of wires, strongly prestressed in which bending waves propagate and, on the other hand, the pantograph is an articulated frame excited on two sides, by the train and by the overhead line, subjected to a disturbed aerodynamic environment. The sliding contact between these two flexible structures is the main difficulty in the modelling of the pantograph catenary dynamics. In OSCAR © , the overhead line is modelled using the Finite Element method and the contact with the pantograph is managed using the penalty method.

Secondly, pantograph-overhead line dynamics is particularly sensitive to the catenary static state. The computed static state is used to initialise the dynamic calculation and has consequently a very large impact on results. A realistic static state is however very complex to compute since it is highly non-linear. Indeed, the mechanical tension on the overhead line components involves high levels of stress and geometrical deflexions. In OSCAR © , the final geometry is obtained through a strongly non linear procedure which consists of solving iteratively ( )

i i i F q q K = ⋅
, where i is the iteration number, q is the displacement vector, K the stiffness matrix and F the force vector.

This procedure uses a progressive loading considering both the mechanical tension and the gravity. At each iteration i, the stiffness matrix is updated to consider the stress evolution in the whole overhead line and the convergence of displacements is finally ensured.

Prestressed beam formulations are used in order to take into account the additional stiffness due to mechanical tension. The equation below shows that the tension T has a big impact on the stiffness elementary matrix:

[ ] ∑         ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = j w T w w T w u T u j r N r N EI L r N r N T L r N r N ES L K 2 2 2 2 3 1 1 1 ω
This step plays a major role in the calculation of the mean stress in the contact wire. A validated computation is mandatory to obtain good results.

Other non-linearities appear in the system dynamics. First of all, the pantograph-catenary contact is unilateral, which can generate electrical arcs. The second non-linearity of the catenary is due to the droppers' unilaterality, as illustrated in the figure below, where one can notice that droppers do slacken at the pantograph passage.

Lastly, the overhead line is a wire structure of very big dimensions, strongly prestressed and very slightly damped. Bending waves propagate in the whole structure, reflect on different elements of the overhead line, and remain some minutes after the pantograph passage. Damping modelling quality thus plays an important role in the software accuracy. OSCAR © uses a damping model based on Rayleigh hypothesis with different coefficients for each part of the overhead line (contact wire, messenger wire, droppers, steady arm, stitch wire):

∑ + = i i i i i K M C β α
where i is the index of the overhead line component group. Figure 7 presents the comparison between simulated contact force of OSCAR © (blue) and measurements (red). The good correlation of the both signals of contact force between the pantograph and the catenary confirms that loadings computed in OSCAR © can be used to study fatigue phenomena. The Figure 8 illustrates the strong effect of junction claws on pantograph catenary dynamics. Indeed, the two curves are calculated with exactly same models. Only a junction was added in the black curve. Big fluctuations due to claw can be observed with the maximum value and maximum amplitude of the contact force. This modifications begins before the passage of the pantograph under the claw, this shows that it is mandatory to calculate stress cycle directly in OSCAR © because loading cycles are strongly related to operational conditions.

Fatigue analysis

This section presents the whole procedure built to perform a fatigue analysis.

The loading path is a very sensitive parameter in fatigue study. For instance, the bending waves in catenary may influence the fatigue phenomenon. Therefore, a new functionality to study stresses in the contact wire was included in OSCAR © because it provides very predictive loading path.

It is necessary to study the multiaxial stresses because friction between the pantograph and the contact wire generates stresses upon plans xz and yz, the clamping force imposed by the junction claw may exceed 2kN/cm (the average stress in the contact wire is then heavily modified near this component) and last the junction claw applies a torsional stress caused by the swaying of the claw around the contact wire axis as drawn in the Figure 9. Therefore, it is necessary to work with a volume mesh of the contact wire to take into account all these phenomena. However, a volume mesh of the complete contact wire is impossible because the mesh refinement required here is very high. OSCAR © uses one-dimensional (1D) finite elements. Thus, as a first step, longitudinal stress is studied using the 1D model to identify where the greatest stress is obtained along the OCL contact wire, for instance, near a support -where the stiffness is the most important -or rather at mid-spanwhere movements are important. This critical area in the catenary is studied using a volume mesh of the contact wire, like a zoom (see Figure 10), in order to limit required resources. The dynamic interaction between pantograph and catenary is computed using only the 1D mesh that is fully validated. The volume mesh is used to obtain a fully three-dimensional representation of stresses corresponding to a multiaxial loading. The kinematics fields and strain fields from 1D calculation are imposed at boundaries and the contact force history is applied to the volume mesh to calculate multiaxial stresses. For instance, the dropper claw requires an additional element corresponding to the dropper to take into account all boundary conditions as shown in the Figure 11.

The contact wire wear increasing with the number of cycle (corresponding to pantograph passage), fatigue and wear are strongly coupled [START_REF] Sugahara | Applicability of Rain Flow Method to Fatigue Life Span Estimation of Overhead Contact Wire[END_REF]. Therefore, it is necessary to update the model because the mean stress, which is a critical parameter, changes with the wire section. First, the dynamic study made in OSCAR © takes into account the beam element properties changes due to mass and stiffness reduction. Second, the volume mesh is changed to obtain a realistic stress values.

4.1

One dimensional analysis The RTRI carried out inline tests to measure the bending strain amplitudes [START_REF] Sugahara | Preventing Fatigue Breakage of Contact Wires[END_REF]. The strain waveform, given in Figure 12, is composed of a steep peak due to pantograph passage, preceded and followed by smaller amplitudes waves. Sugahara and all estimated the effect of accompanying waves on the fatigue life by using a waveform counting method described in [START_REF] Sugahara | Applicability of Rain Flow Method to Fatigue Life Span Estimation of Overhead Contact Wire[END_REF]. The measured average is null because sensors were stuck on the upper surface of the inline contact wire. Thus the mean tensile strain exerted by the tensile load is not measured.

A simulation using OSCAR © was done to compare numerical results to this measurement. To reproduce this test conditions in OSCAR © , the stress σ xx was calculated and the strain ε a was deduced as

E xx a 0 σ σ ε - =
where σ 0 is the mean stress due to tension and E is the hard copper Young modulus.

OSCAR © being a finite element model using pretensionned Euler-Bernoulli beams and using generalized constraints, the longitudinal stress σ xx is given by ( ) ( )

y z y y z z xx z y E I M z I M y S N t z y x κ κ σ + - ∈ = + - = , , ,
, at any time t and for each point of the contact wire section using the coordinates (y, z). In this way, as sensors were stuck on the upper surface of the contact wire, the study was performed for the point C represented in Figure 13. The numerical result, illustrated in Figure 14, is different from measurements for waves which precede and follow the pantograph passage. It can be explained by differences of French and Japanese catenaries and pantographs or by different damping ratio in catenary or by differences of frequency bandpass between measurements and software [START_REF] Bianchi | Using modal damping for full model transient analysis. Application to pantograph/catenary vibration[END_REF]. Nevertheless, the maximum and minimum bending strain amplitudes due to peak are similar and thus these results are good enough to make a preliminary fatigue analysis.

A criterion based on the Goodman's theory [START_REF] Van | Fatigue Polycyclique des Structures[END_REF] is used, assuming an uniaxial loading. Although this assumption is not verified, it is applied in order to identify the most critical zone in the catenary as a preliminary study.

To apply this criterion, the maximum value and the minimum value of the longitudinal stress is calculated for each node of the mesh, using respectively

( ) σ = Σ and ( ) σ = Σ .
Secondly, the criterion is applied and the point P(x,y,z), corresponding to the maximum value of where a is the tensile strength and b is the fatigue limit.

The result of this study, applied on French catenary and shown in Figure 15, illustrates that the highest values of the criterion are closed to support represented by vertical dotted lines on Figure 15. As a conclusion, the main sensitive part of the catenary, regarding fatigue and using a 1D criterion, is close to supports.

4.2

Three dimensional analysis In this section, the 3D fatigue study is described. As it is explained before, the pantograph passage corresponds to one loading cycle and the related stress cycle is multiaxial because of the clamping force of claw junction and friction at the contact point. Therefore, the multiaxial Dang Van's criterion [START_REF] Van | High-Cycle Metal Fatigue in the Context of Mechanical Design[END_REF], which is a generalization of the Goodman's criterion to a multiaxial stress cycle, is used. It is written as

If ( ) { } b t p a t t < ⋅ +
) ( max τ there will not take place a crack initiation. To apply the Dang Van's criterion to catenary, it was necessary to build volume meshes for each configuration including different types of claws. In order to limit the size of claw meshes, their geometry was simplified and a mechanically equivalent component is meshed, as shown in Figure 17. The image at the bottom in Figure 17 represents isovalues of stress obtained in the contact wire, using FE analysis. The junction claw is considered as a non-deforming component. The stress tensor obtained can be used to perform a fatigue analysis by making a cumulative damage.

( ) ( ) ( ) ( ) ( ) t t t

Conclusion

The proposed article deals with fatigue in the Overhead Contact Line (OCL) applied to a typical French catenary. This system failure is critical for railways operators. An analysis of the different parameters that significantly influence the fatigue phenomenon is carried out in this study. First, a preliminary analysis based on experimental results shows that involved phenomena are complex, particularly because of coupling of dynamics and wear, and because they are very sensitive to condition of use. Then, to ease understanding of these phenomena, a numerical study was made. The followed procedure is presented here. In order to use the very accurate results of dynamic interaction given by software OSCAR © , new functionalities were developed to determine stresses fields in the catenary. The fatigue study is divided into two steps, first a 1D study is carried out to localise the critical area in catenary and then a volume mesh of the contact wire is built to particularly study junction claws, a component used in catenary maintenance which has triggered some incidents in France. Indeed, it is shown in this paper that the loading path is multiaxial and thus three dimensional models are necessary. Finally, an introduction to fatigue cycling based on the Dang Van's criterion is presented.

Figure 1a .

 1a Figure 1a. (left) Crack initiations and surface defects in the contact wire; Figure 1b. (centre) micrographic section of the contact wire: specks due to electrical arcs; Figure 1c. (right) micrographic section of the contact wire: surface defects due to friction.

Figure 2 .

 2 Figure 2. RTRI test bench for OCL contact wire study [2].

Figure 3 .

 3 Figure 3. Fatigue fracture obtained on the RTRI's bench.

Figure 4 .

 4 Figure 4. Photos of a junction claw.

Figure 5a .

 5a Figure 5a. Inline CW fatigue fracture appearance under a junction claw; Figure 5b. Side view of a failure under a junction claw.

Figure 6a .

 6a Figure 6a. (left) Finite Element Catenary; Figure 6b. (right) Multibody pantograph.

Figure 7 .

 7 Figure 7. Contact force correlation between simulation and measurements.

Figure 8 .

 8 Figure 8. Illustration of the contact force fluctuations due to junction claw using simulation tools.

Figure 9 .

 9 Figure 9. Swaying of the claw around the contact wire axis.

Figure 10 .

 10 Figure 10. Zoom on contact wire using a volume mesh.

Figure 11 .

 11 Figure 11. Dropper claw and boundary conditions.

Figure 12 .

 12 Figure 12. Contact wire bending stress amplitude measured inline in Japan [3].

Figure 13 .

 13 Figure 13. Virtual contact wire section.

6 )Figure 14 .

 614 Figure 14. Contact wire bending stress amplitude computed with OSCAR © .

  be expressed in a better known form as:

Figure 15 .

 15 Figure 15. Fatigue analysis using 1D criterion based on Goodman theory.

  pressure τ is the shear stress amplitude at mesoscopic scale a and b are two material parameters determined from the endurance limits This criterion is generally represented in the (τ ( t),  p(t)) diagram given in Figure16.

Figure 16 .

 16 Figure 16. Dang Van Diagram theory description.It is applied at each node of the volume mesh in order to identify which are critical for fatigue. All loading path are plotted in the diagram and if one loading path exceeds the threshold given by the material line (a, b), a crack inititation may occur. Otherwise, with none loading path over the material line, the component has a unlimited lifetime.

Figure 17 .

 17 Figure 17. Procedure to build volume mesh with a junction claw.

. The tensile load applied was 19.6kN. 1 Railway Technical Research Institute 2 OSCAR : Outil de Simulation du CAptage pour la Reconnaissance des défauts

Acknowledgments

This paper describes work undertaken in the context of the partnership between SNCF and RTRI. These fruitful exchanges have allowed discussions on methods and different approaches. We particularly emphasize the useful tests conducted by the RTRI to characterize the SNCF contact wire. Moreover, the authors thank S. Daouk for the good work he did during his internship.