Alexandre Bouaniche

Nicolas Jaouen

Pascale Domingo

Luc Vervisch
email: vervisch@coria.fr

Vitiated High Karlovitz n-decane/air Turbulent Flames: Scaling Laws and Micro-mixing Modeling Analysis

Keywords: High Karlovitz combustion, Kerosene reduced chemistry, n-decane flame, micro-mixing modeling, Large Eddy Simulation, Spray flame

published or not. The documents may come

Abstract Turbulent flames with high Karlovitz numbers have deserved further attention in the most recent literature. For a fixed value of the Damköhler number (ratio between an integral mechanical time and a chemical time), the increase of the Karlovitz number (ratio between a chemical time and a micro-mixing time) by an order of magnitude implies the increase of the turbulent Reynolds number by two orders of magnitude [START_REF] Bray | The challenge of turbulent combustion[END_REF]. In the practice of real burners featuring a limited range of variation of their turbulent Reynolds number, high Karlovitz combustion actually goes with a drastic reduction of the Damköhler number. Within this context, the relation between the dilution by burnt gases and the apparition of high Karlovitz flames is discussed. Basic scaling laws are reported which suggest that the overall decrease of the burning rate due to very fast mixing can indeed be compensated by the energy brought to the reaction zone by burnt gases. To estimate the validity of these scaling laws, in particular the response of the quenching Karlovitz versus the dilution level with a vitiated stream, the micro-mixing rate is varied in a multiple-inlet canonical turbulent and reactive micro-mixing problem. A reduced n-decane/air chemical kinetics is used, which has been derived from a more detailed scheme using a combination of a directed relation graphs analysis with a Genetic Algorithm. The multiple-inlet canonical micro-mixing problem includes liquid fuel injection and dilution by burnt gases, both calibrated from conditions representative of an aeronautical combustion chamber. The results confirm the possibility of reaching, with the help of a vitiated mixture, very high Karlovitz combustion before quenching occurs.

Introduction

In the quest for combustion systems with high fuel-efficiency, the strong increase of the turbulence intensity has recently seen a renewed interest [1-5, 9, 11, 12, 14, 23-25, 30, 31, 34, 35, 49-54, 57, 58, 61, 65-69]. Higher turbulence intensity is expected to favour enhanced micro-mixing of the reactants, followed by a higher volumetric burning rate. In his seminal works, Karlovitz [START_REF] Karlovitz | Open turbulent flames[END_REF] noted that flames can be 'arrested by high velocity gradients and strong velocity fluctuations'. Such flame quenching by intense turbulence was examined under various conditions. One of the first experimental observation concluded that above a critical value of the number K = (δ L /S L )(u / T ), ratio between a chemical time and a mechanical time, of the order of a few tenth [START_REF] Chomiak | Flame quenching by turbulence[END_REF], combustion propagating in fresh gases cannot be sustained. u denotes the rms turbulent velocity, T is the integral scale of turbulence, δ L the laminar flame thickness and S L the laminar burning velocity. Results which immediately triggered further discussions on the exact definition of the characteristic scales chosen to measure K [START_REF] Karlovitz | Comment on the paper "Flame quenching by turbulence[END_REF].

Along the same lines, the phenomenological analysis of the structure of turbulent flames initiated by Borghi [START_REF] Borghi | Mise au point sur la structure des flammes turbulentes[END_REF], was followed by the development of combustion regime diagrams [START_REF] Borghi | Turbulent combustion modelling[END_REF][START_REF] Peters | Turbulent Combustion[END_REF][START_REF] Peters | Multiscale combustion and turbulence[END_REF], built from Damköhler and Karlovitz numbers and additional ratios of times and length scales characteristic of the turbulence and the reaction zones. The Damköhler number is usually approximated as Da = τ T /τ c , where τ T is an integral mixing time and τ c a chemical time. The Karlovitz number relates to small-scale turbulent mixing, Ka = τ c /τ k , with τ k the small-scale (Kolmogorov) mixing time [START_REF] Libby | Introduction to turbulence[END_REF], leading to the relation [START_REF] Bray | The challenge of turbulent combustion[END_REF] Da

× Ka = τ T τ k ≈ Re 1/2 T , (1)
where Re T = u T /ν is the turbulent Reynolds number, with ν the kinematic viscosity of the fluid. Therefore, for a fixed value of Da at which combustion occurs, Re T ≈ Ka 2 , and increasing the Karlovitz number by, say a factor 10, would mean increasing the turbulent Reynolds number by a factor 100. Such large increase of the turbulent Reynolds number cannot always be achieved. In practice, when Ka goes up, Da does go down in order to fulfil the relation [START_REF] Aspden | Turbulence-chemistry interaction in lean premixed hydrogen combustion[END_REF]. In fact, this approximate relation [START_REF] Aspden | Turbulence-chemistry interaction in lean premixed hydrogen combustion[END_REF] suggests that, at very high Reynolds numbers (Re T → ∞), chemistry can still be fast (Da→ ∞), as long as the chemical time scale is shorter (or of the order of) than the shortest flow time scale, i.e. the amplitude of Ka stays moderate. It is usually believed that this is the operating conditions of most practical combustion systems and it was the state of affair till around the year 2010. Over the last eight years however, studies have been published displaying an exponential growth of the Karlovitz number with years, see Fig. 1. As in the pioneer works reported above, the exact definition of Ka may differ from one study to the other, but the increase in Ka is large enough to be significant, whatever its exact definition. At the same time, the Damköhler number suffered from a drastic decrease of its amplitude (Fig. 2). The flames at the highest Ka examined in the works of Fig. 1 were actually not 'self-sustained', but benefitted from a large reservoir of hot burnt gases, for instance a vitiated co-flow. These flames with very high values of the Karlovitz number (up to more than 1500), and thus very low values of the Damköhler numbers (down to 0.02), may eventually release less heat than they gain from burnt gases to maintain fuel oxidation. Nonetheless, all the fuel injected was burnt and thus high-Karlovitz combustion was achieved. This flame regime may be of interest locally in a combustion chamber, for instance to secure flame stabilisation in a highly turbulent flow zone. It is therefore legitimate to wonder by which mechanisms the dilution by burnt gases makes the flame more robust to high Ka (and low Da) and how the response of the quenching Karlovitz number scales versus the level of dilution by burnt gases. Among the numerous candidates, in the case of premixed combustion, an increase of the flame speed is a potential phenomenon and in the case of non-premixed systems, fast mixing with burnt gases may help local auto-ignition. At least three questions emerge from these preliminary remarks: (i) what is the basic scaling for flame speed including both dilution by burnt gases and Ka effect? (ii) In a non-premixed system, how much of recirculating burnt products must be included to secure high Ka combustion? (iii) What is the global scaling of Ka at quenching versus the dilution level of the fresh mixture with a vitiated stream.

The Damköhler number 'crisis' In an attempt to answer these basic questions, simple scaling laws are proposed in this work combining premixed and non-premixed regimes. Simulations of micro-mixing/chemistry interaction are then performed to verify the trends. If the effect of turbulence is over-simplified through a basic stochastic mixing closure, the impact of spray evaporation and of n-decane detailed chemistry is included. To do so, a set of reduced chemical schemes is first derived for n-decane/air combustion, as reported in the next section. The elementary scaling relations for high Ka flame in vitiated stream are discussed in the subsequent section, before examining the response of n-decane/air combustion at high Ka in the final section.

A fuel-spray stochastic micro-mixing canonical problem for high-Ka analysis

Starting from a given set of initial conditions, representative of the streams feeding the reaction zones in an aeroengine, reference time evolutions of thermochemical quantities are constructed in a pseudo-reactor, combining the approaches discussed in [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF][START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modeling: Impact of the canonical problem[END_REF]. A set of N P = 1000 stochastic particles are distributed at initial time according to the chemical composition of three inlets, containing either liquid kerosene, air or burnt gases. This last inlet mimics recirculation of burnt gases, as it would be observed in a swirled fuel injection. After the initialisation, evaporation of the liquid fuel occurs and the gas phase evolves at about 10 bars according to a stochastic mixing model to which the chemical sources of kerosene/air chemistry are added. The thermochemical properties of the particles vary following a procedure explained below, to generate synthetic turbulent distributions, as the scatter plot of temperature versus mixture fraction seen in Fig. 3. Various distributions will be obtained varying both the initial amont of burnt gases and a Karlovitz number constructed from the control parameters of the canonical problem. The number of stochastic particles initially set at the concentration and temperature of the j-th inlet (j = 1, 2, 3) is proportional to Qmj , the mass flow rate injected by this inlet,

N Pj = N P × Qmj Qm , (2)
where Qm = Qm1 + Qm2 + Qm3 denotes the total mass flow rate. The liquid kerosene inlet at T = 450 K represents 2.3% of Qm ( Qm1 = 0.023 Qm ). The mass flow rate of the air inlet at T = 703 K is Qm2 = 0.644 Qm . To mimic the injection of primary and secondary air, as done in some combustion chambers, 60% of the total mass of air is introduced at the initial time, the rest of air particles are then progressively released over 1 ms. The third and last inlet of burnt gases, taken at chemical equilibrium for the equivalence ratio of the full mixture at T = 1877 K, brings Qm3 = 0.333 Qm . These 33% were estimated from large eddy simulation of a representative aeronautical combustion chamber [START_REF] Jaouen | An automated approach to derive and optimise reduced chemical mechanisms for turbulent combustion[END_REF] and these burnt gases secure ignition in a first set of calculations.

As in Farcy et al. [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF], the elementary mass flow rate qm = qp mL (t) + qp mG (t) carried by every p-th particle (p = 1, • • • , N P ) is decomposed into liquid ( qp mL (t)) and gas ( qp mG (t)) phases. The evolution of the thermochemical property φ p carried by the p-th stochastic particles reads

dφ p (t) dt = MIX p (τ ) + ωp φ + ωp v φ , (3)
where ωp φ is the gaseous phase chemical source obtained from the chemical scheme, ωp v φ relates to the liquid fuel (evaporation or heat), computed from the particle properties. MIX p (τ ) denotes the Curl [START_REF] Curl | Dispersed phase mixing. Theory and effects in simple reactors[END_REF] micro-mixing closure, with τ the characteristic micro-mixing time. Many other micro-mixing models exist in the literature, like modified Curl [START_REF] Dopazo | Relaxation of initial probability density functions in the turbulent convection of scalar fields[END_REF][START_REF] Janicka | Closure of the transport equation for the probability density function of turbulent scalar fields[END_REF], Euclidian Minimum Spanning Tree (EMST) [START_REF] Xu | Pdf calculations of turbulent nonpremixed flames with local extinction[END_REF], Multiple Mapping Conditioning (MMC), which allows for introducing flamelet-like correlations between species [START_REF] Sundaram | Prediction of NOx in premixed high-pressure lean methane flames with an MMC-partially stirred reactor[END_REF], or again the advanced hierarchical parcel-swapping representation of turbulent mixing [START_REF] Kerstein | Hierarchical parcel-swapping representation of turbulent mixing. part 2. application to channel flow[END_REF]. However, for examining the global scaling laws considered in this work, the basic version of the Curl micro-mixing model was found sufficient.

To account for the evaporation of the liquid fuel, the simplified modeling used in [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF] is adopted. The thermal condition is assumed above the liquid boiling point and the droplets diameter follows the so-called D 2 -law [START_REF] Sirignano | Fuel droplet vaporization and spray combustion theory[END_REF][START_REF] Sirignano | Advances in droplet array combustion theory and modeling[END_REF][START_REF] Spalding | The combustion of liquid fuels[END_REF],

d 2 j (t) = d 2 j (t o )(1 -(t/τ v )).
The characteristic evaporation time is fixed at τ v = 0.491 ms, which was estimated from the experimental results by Nomura et al [START_REF] Nomura | Microgravity experiments of fuel droplet evaporation in sub-and supercritical environments[END_REF]. It was thus preferred to fix τ v from experiments, rather than introducing additional uncertainties in its dynamic calculation from approximate liquid and flow properties.

The rate of gaseous mass of the i-th component released by a single droplet reads

Ẇ p i (t) = Y p i,L (t) × ρ L π 6 d j 3 (t) -d j 3 (t + ∆t) ∆t , (4)
where Y p i,L denotes the mass fraction of the i-species in the liquid of the p-th particle. A single stochastic particle issued from the j-th inlet carries a number ṅj of droplets injected per unit time. Then, the balance between liquid and gaseous mass flow rates carried by a given particle issued from the j-th inlet may be written

qp mG (t + ∆t) = qp mG (t) + Ns i=1 ṅj Ẇ p i (t)∆t , (5)
qp mL (t + ∆t) = qp mL (t) - Ns i=1 ṅj Ẇ p i (t)∆t , (6)
where N s is the number of species considered. The net gaseous source of evaporation of the i-th chemical species in the p-th particle is

ωp vi (t) = 1 ∆t qp mG (t)Y p i,G (t) + ṅj Ẇ p i (t)∆t qp mG (t + ∆t) -Y p i,G (t) , (7)
where Y p i,G is the gaseous mass fraction of the i-th species in the p-th particle. The source of the sensible enthalpy h s is

ωp v hs (t) = 1 ∆t qp mG (t)h p s (t) + N i=1 ṅj Ẇ p i (t) [h i (T B ) -L vi ] ∆t qp mG (t + ∆t) -h p s (t) , (8)
in which T B = 750 K is the boiling temperature, h i is the enthalpy of the i-th species and L vi = 251 000 J/kg is the latent heat of evaporation. At injection, a single droplet diameter d = 12.8 µm is imposed, as it was estimated from the Sauter mean diameter of a representative aeronautical injector. This modeling leads to usual and generic scatter plot responses, as seen in Fig. 3.

Transported species Analytically resolved species (QSS)

H 2 , O 2 , CO, CO 2 , CH 4 , C 2 H 6 , CH 2 O, C 2 H 2 , C 2 H 4 , C 3 H 6 , C 4 H 6 , NC 10 H 22 , H, O, OH, HO 2 , H 2 O, CH 3 , C 3 H 3 , AC 3 H 5 , BC 6 H 13 , N 2 HCO, CH 3 OH, C 2 H 5 , CH 3 O, CH 2 OH, CH 2 CO, C 2 H 3 ,

Reduced n-decane chemistry

The Luche et al. [START_REF] Luche | Elaboration of reduced kinetic models of combustion. application to the a kerosene mechanism[END_REF][START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF] chemical kinetics accounts for 91 species and 991 reacting steps, and is derived from the more detailed kerosene-air mechanism by Dagaut [17]. This mechanism was selected because of its reasonable size, at least in comparison to other more detailed schemes for kerosene-air combustion. It is not attempted to provide in this work a reduced chemistry for kerosene that would benefit from the most recent findings in terms of real aeronautical fuels [START_REF] Wang | A physics-based approach to modeling real-fuel combustion chemistry -I: Evidence from experiments, and thermodynamic, chemical kinetic and statistical considerations[END_REF], which may include complex chemical properties distributions across the distillation curve and related complex phenomena. The sole objective is to compare the normalised response of a chemical scheme featuring multiple chemical time scales against the scaling laws discussed thereafter, and using the Luche et al. mechanism as a starting point is sufficient to address this point. Fig. 4 Representative species trajectories (Eq. ( 9)) for the kerosene surrogate and n-decane reduced mechanism. τ = 0.2 ms. Symbols: Reference chemistry [START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF]. Lines: Reduced n-decane mechanism (Table 1). Solid-line: Air inlet trajectory. Dotted-Line: Fuel inlet trajectory.

To further reduce the Luche et al. chemistry so that multiple calculations can be performed, the ORCh approach is followed [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modeling: Impact of the canonical problem[END_REF], which combines the directed graph analysis with error propagation (DRGEP) [START_REF] Lu | A directed relation graph method for mechanism reduction[END_REF][START_REF] Pepiot | An efficient error propagation based reduction method for large chemical kinetic mechanisms[END_REF]. Species and reactions are progressively removed according to their relative importance measured along reference chemical evolutions (φ D (t) trajectories), with the automatic generation of an analytical part using quasi-steady state assumption (QSS) and a subsequent optimisation of the chemical rates with a Genetic Algorithm. All detail concerning this combination of reduction/optimisation methods to generate a series of reduced chemical kinetics can be found in [START_REF] Jaouen | Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes[END_REF][START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modeling: Impact of the canonical problem[END_REF].

To compact the information and ease the analysis, two additional deterministic trajectories are solved, from the air and kerosene inlets, with the linear relaxation deterministic micro-mixing closure (IEM or LMSE) [START_REF] Borghi | Turbulent combustion modelling[END_REF][START_REF] Dopazo | Functional formulation of nonisothermal turbulent reactive flows[END_REF]

dφ D (t) dt = φ (t) -φ D (t) τ + ωD φ . (9)
The statistical mean entering this deterministic mixing model is computed at every instant in time from the full set of stochastic particles,

φ (t) = NP p=1 qp m G (t)φ p (t) N P p=1 qp mG (t) . ( 10
)
These additional deterministic trajectories are easily coupled with a Genetic Algorithm, to optimise chemical rates after chemistry reduction [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modeling: Impact of the canonical problem[END_REF]. Figure 4 shows typical φ D (t) distributions.

The kerosene in the initial mechanism by Luche et al. [START_REF] Luche | Elaboration of reduced kinetic models of combustion. application to the a kerosene mechanism[END_REF][START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF] is a surrogate composed of n-decane (n-C 10 H 22 , 76.7388% in mass), propylbenzene (PHC 3 H 7 , 13.1402%) and propylcyclohexane (CYC 9 H 18 , 10.1210%). In the present reduction, the initial composition of the fuel is even more simplified to 100% n-decane. The species of the reduced mechanism are given in Table 1. This mechanism relies on the transport of 26 chemical species and on the solving of 24 quasi-steady state relations and contains 338 elementary reactions.

The associated trajectories (Eq. ( 9)) for operating conditions reported above are displayed in Figs. 4 and5, in which the symbols denote the detailed scheme. The trajectories from fuel and air inlet differ till a residence time of 1 ms, before the species evolve toward the equilibrium condition. The thermal energy brought by the burnt gases first promote the formation of NO through their mixing with fresh air. At 0.5 ms, the temperature conditions are so that part of this NO is recombined into NO 2 , to then increase again after 1.25 ms through the mixing with the remaining air in a post-flame region. Aside from the unavoidable difference in n-decane (fuel surrogate in the detailed mechanism and only n-decane in the reduced one in Fig. 3), the response of the reduced scheme stays very close to the detailed one, specifically in terms of position of ignition and of the peak of CO (Fig. 3), which will be considered thereafter in the analysis increasing Ka and varying the amount of burnt gases.

The reduced mechanism has been obtained considering only the turbulent micro-mixing problem (Eq. 3), which overall features similarities with non-premixed combustion. It is therefore of interest to verify that a perfectly premixed combustion regime is also reproduced. Figure 6 9)). τ T = 0.2 ms. Symbols: Reference chemistry [START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF]. Lines: Reduced n-decane mechanism (Table 1). Solid-line: Air inlet trajectory. Dottedline: Fuel inlet trajectory.

of the flame velocity, the equilibrium temperature and the equilibrium levels for CO and for NO obtained from perfectly premixed kerosene-air flame computations at a pressure of 9.63 • 10 5 Pa with the Cantera solver [START_REF] Goodwin | Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF]. Some departure is observed on the estimation of the flame speed S L for equivalence ratios between unity and 2, but overall the description is acceptable considering the level of reduction. For comparison, are added the results obtained with a less reduced chemical mechanism (32 transported species, with 27 QSS relations and 419 elementary reactions), obtained keeping the original surrogate fuel composition. Both reduced mechanisms perfectly capture the equilibrium temperature and equilibrium CO level for every equivalence ratio. An over prediction of the NO levels is achieved using the most reduced mechanisms for conditions around the stoichiometric point. However, because NOx species will not be considered in the subsequent analysis, this most reduced mechanism based on n-decane (Table 1) is retained.

Basic scaling for high-Ka flames in vitiated mixture

Oversimplified scaling relations for weakly vitiated flames are recalled in this section. 1).

Vitiated steady premixed flamelet

Consider the steady premixed laminar flamelet equation

ρu dφ dξ = d dξ ρD φ dφ dξ + ωφ -ρ ṠK φ , ( 11
)
where ρ is the density, u is the velocity, D φ is the diffusion coefficient of the scalar φ evolving through the flame front and ωφ its chemical source. The coordinate in the direction normal to the flame front is ξ and ṠK φ is a leakage term, representative of all fluxes occurring along the flame surface [START_REF] De Goey | A flamelet description of premixed laminar flames and the relation with flame stretch[END_REF], as transverse convection and diffusion resulting from straining and curvature of the flamelet surface. The fresh gases condition φ u is expressed from a reference fresh gases unburnt condition φ o and the dilution (or vitiation) factor f b (the subscript 'u' and 'b' denote unburnt and burnt gases respectively)

φ u = φ o (1 -f b ) + f b φ b . ( 12
)
f b = 0 corresponds to fresh mixtures (fuel mixed with oxidiser) and burnt vitiated gases appears for 0 < f b << 1. Integrating ( 11) through an unstrained one-dimensional flame ( ṠK φ = 0) and accounting for mass conservation, the unstrained flame speed reads

S L (f b , ṠK φ = 0) = u u = ρ b ρ u u b = 1 φ b -φ u   1 ρ u +∞ -∞ ρ ωφ dξ   . ( 13
)
The peak reaction rate in the integral in Eq. ( 13) is toward the burnt gaz side, also it is assumed that the density weighted integral of the source does not vary much with dilution by burnt gases, at least for weak levels of f b .

Combining this hypothesis with the relations ( 13) and ( 12) leads to

S L (f b , ṠK φ = 0) = 1 1 -f b × 1 φ b -φ o   1 ρ o +∞ -∞ ρ ωφ dξ   , = 1 1 -f b S o L ( 14
)
with S o L = S L (f b = 0, ṠK φ = 0). Figure 7 shows the evaluation of this relation in the simulation of a freely propagating premixed flame with a single-step chemistry cast as in [START_REF] Liñán | Fundamental aspects of combustion[END_REF], for a Zeldovitch number β = 15. The expected trend is recovered for 0 < f b << 1. In this single-step chemistry simulation, the addition of the vitiated gases is mimicked by preheating the fresh mixture, and the one-dimensional flame is computed with a fully compressible sixthorder flow solver using a PADE scheme [START_REF] Lele | Compact finite difference schemes with spectral like resolution[END_REF]. This hyperbolic behaviour of flame speed versus dilution by burnt gases was also reported in the literature at various places, for instance simulating vitiated premixed flamelets with detailed chemistry (see Fig. 4a in [START_REF] Wang | Self-similar behavior and chemistry tabulation of burnt-gases diluted premixed flamelets including heat-loss[END_REF]). 1Considering now a flamelet submitted to stretch without dilution and assuming that the integral of the source stays close to the unstretched flame burning velocity, Eq. ( 11) leads to

ρ o S L (f b = 0, ṠK φ ) = 1 φ b -φ o   +∞ -∞ ρ ωφ dξ - +∞ -∞ ρ ṠK φ dξ   , = ρ o S o L      1 - +∞ -∞ ρ ṠK φ dξ (φ b -φ o )ρ o S o L      . (15)
The usual linear response of the flame speed to small levels of stretch is found with

[15] S L (f b = 0, ṠK φ ) = S o L (1 -K) , ( 16
) 0 0,1 0,2 fr 0,8 1 1,2 1,4 1,6 1,8 2 S
(φ b -φ o )ρ o S o L × K = +∞ -∞ ρ ṠK φ dξ.
The linear reduction of flame speed by (1-K) (Eq. ( 16)) and the increase of flame speed by (1 -f b ) -1 (Eq. ( 14)), suggest that the effect of flame speed reduction by tangential stretch applied to the flame surface can be compensated by the vitiation of the fresh gases.

Vitiated diffusion flamelet quenching

The relation [START_REF] Cicoria | Large eddy simulation of lean turbulent hydrogen-enriched methane-air premixed flames at high Karlovitz numbers[END_REF] for the unstretched and diluted flame may also be combined with an estimation of the quenching condition in a diffusion flamelet. τ mq the mechanical time at quenching in a non-premixed system is known to relate to the stoichiometric and freely propagating flame speed as [START_REF] Peters | Length scales in laminar and turbulent flames[END_REF][START_REF] Peters | Turbulent Combustion[END_REF] 1

τ mq = Z 2 st (1 -Z st ) 2 a T /S 2 L , (17)
where a T is the thermal diffusivity and Z st the stoichiometric value of the mixture fraction [START_REF] Peters | Turbulent Combustion[END_REF]. Assuming a fixed reference chemical time, Ka q , the Karlovitz number at quenching varies as τ -1 mq . The introduction in [START_REF] Dagaut | On the kinetics of hydrocarbons oxidation from natural gas to kerosene and diesel fuel[END_REF] of the relation [START_REF] Cicoria | Large eddy simulation of lean turbulent hydrogen-enriched methane-air premixed flames at high Karlovitz numbers[END_REF], giving the unstrained flame speed S L = S L (f b , ṠK = 0) in the presence of dilution, provides a scaling relation for a non-premixed system weakly vitiated by burnt gases

Ka q (f b ) ≈ 1 1 -f b 2 . ( 18
)
Hence, the value of the Karlovitz number at the quenching point should increase in the presence of dilution by burnt gases (f b > 0), making the reaction zones evolving in vitiated mixture more robust to intense turbulence. This scaling is now tested against calculations with the reduced kerosene chemistry. 5 Response of Ka at quenching in a vitiated stream

The above dilution factor f b calibrates the relative amount of burnt and fresh gases present in the unburnt mixture (Eq. ( 12)). In the canonical problem used for chemistry reduction (Eqs. ( 3) and ( 9)), the relative amount of burnt gases introduced initially may be varied adjusting the mass flow rates of the three streams injected in the system ( Qm1 : fuel, Qm2 : air and Qm3 : burnt gases), thus varying Y i (t = , the averaged mass fractions injected at t = 0, including liquid fuel.

To study the sensitivity of the pseudo-reactor to the amount of added recirculating burnt gases, f b is defined as

f b = Qm3 Qm -Qm3 . ( 19
)
where Qm = j Qmj is the total mass flow rate injected. With this definition, f b = 0 in the case of no-dilution by burnt gases and f b = 1 in the asymptotic limit where mass flow rates of fresh ( Qm1 + Qm2 ) and burnt gases ( Qm3 ) are equal.

The response of the turbulence/chemistry interaction to variations of τ , the micro-mixing time, is characterized by three stages [START_REF] Liñán | Fundamental aspects of combustion[END_REF]. (i) very small micromixing times, and thus very fast micro-mixing, prevent combustion. (ii) intermediate values of τ lead to ignition and burning with turbulence/chemistry interaction. (iii) above a given threshold, the too large micro-mixing times, associated with very small frequency of micro-mixing, do not allow for sufficient mixing of the fresh and burnt gases to secure ignition within the allowed residence time (here 2 ms). These three stages ((i) no-burning, (ii) burning and (iii) no-burning) are seen in Fig. 8, displaying c , the average progress variable (c = 0 in the fresh gases and c = 1 in fully burnt products) plotted Fig. 9 Illustration of the arbitrary definition of the chemical time from Y D CO (t), the CO mass fraction deterministic trajectory (Eq. ( 9)). Representative case f b = 0.5. at t End = 2 ms for cases featuring a micro-mixing time τ varying between 0.002 ms and 0.8 ms, and for the representative case f b = 0.5. The average progress of reaction c is here calculated from the normalised production of CO 2 by combustion

c (t) = Y CO2 (t) -Y CO2 (t = 0) Y Eq CO2 -Y CO2 (t = 0) , (20)
with Y CO2 calculated from the full mixture at t = 0, including primary and secondary air.

A zoom in the high-Ka zone is given in Fig. 8(b). The decrease of τ down to τ = 0.015 ms (thus the increase of Ka) is followed by a rapid transition from burning to non-burning in the progress variable response. The mid-point of a linear distribution constructed between the last fully burning point and the first very weakly burning condition is chosen to define, τ mq , the mechanical time at quenching. This determines the smallest micro-mixing time that does allow for ignition and burning in the combustion problem defined by Eq (3). The corresponding quenching Karlovitz number can then be related to a limit value that would prevent ignition and thus flame stabilisation in burnt gases vitiated mixture.

The dilution factor f b (Eq. ( 19)) is now varied between 0.25 and 1. For every case, a set of trajectories (Eq. ( 9)) are obtained for given values of τ , to determine the response of τ mq versus f b , according to the procedure shown in Fig. 8(b). To define a Karlovitz number, a chemical time must be chosen. With complex chemistry, in the present case 48 species, many options exist. After considering the evolution of various species, the trajectories of carbon monoxyde were found to provide a similar shape for the various dilution levels, it is also a species which is well captured by the reduced scheme (Fig. 3). The decay of the CO mass fraction after its peak value can be interpolated where Y max CO = Y CO (t peak ) denotes the peak CO mass fraction (Fig. 9). The characteristic relaxation time t o is measured from the observed CO distribution, so that Y CO (t peak + t o ) = Y max CO exp(-1). Twice the chemical time, 2τ c , is arbitrarily chosen as the time duration between the two observations of Y max CO exp(-1) in the CO distributions (Fig. 9). The value of the Karlovitz at quenching for various f b is finally taken as the ratio

Y CO (t) = Y max CO exp(-(t -t peak )/t o ) , (21)
Ka q (f b ) = τ c τ mq . (22)
Obviously, other formulations could be introduced, however this global measure is sufficient to compare against the basic scaling discussed above. The response of Ka q versus the dilution factor f b is given in Fig. 10. The robustness of the n-decane combustion to intense mixing does increase with dilution by burnt gases. A rapid growth of the value of Ka q at quenching is particularly observed after reaching f b = 0.6 (Fig. 10(b)). Interestingly, a zoom in the lower values of f b shows that the response of Ka q follows quite well the scaling given by Eq. [START_REF] Dopazo | Relaxation of initial probability density functions in the turbulent convection of scalar fields[END_REF]. It is worthwhile to note that the relation [START_REF] Dopazo | Relaxation of initial probability density functions in the turbulent convection of scalar fields[END_REF], which was derived for 0 < f b << 1 (thus without matching in the onedimensional approximate analysis the f b definition based on the mass flow rate (19)), seems here reasonably fit for approximating the flame response, even up to large values of f b .

Discussion and summary

A stochastic canonical problem has been constructed to explore turbulence chemistry/interaction with kerosene liquid fuel injection. The information obtained has been used at first to optimise a reduced chemical scheme, whose objective is to predict major species only. Then, the response of the quenching Karlovitz number to the amount of burnt gases mixed with the fresh mixture has been studied. It is found that basic scaling laws derived for burnt-gases vitiated premixed and non-premixed systems agree well with the results, with dilution promoting an inverse squared enhancement of the flame robustness to intense turbulence.

Estimating characteristic length and time scales relevant for turbulent flames in real combustion systems, or in experiments representative of real injectors, is not an easy task [START_REF] Kariuki | Measurements in turbulent premixed bluff body flames close to blow-off[END_REF]. Along these lines, well-resolved Large Eddy Simulation (LES) or quasi Direct Numerical Simulation (DNS) of real burners provide additional information. The Table 6 summarises length and time scales extracted from the quasi-DNS [START_REF] Moureau | From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: Filtered laminar flame-pdf modeling[END_REF] In Fig. 8 for f b = 0.5, combustion is expected for micro-mixing times in the range 0.015 ms < τ < 0.1 ms. Interestingly, τ k in Table 6 lies within this range, close to the lower limit for this swirled flow injector, which was designed to operate with a quite strong precessing vortex core, featuring a characteristic frequency of about 540 Hz with a time scale of the order of 1.9 ms, thus with strong recirculation of burnt gases. Drawing further conclusions on the exact numbers would not be fair at this stage. However, above results open perspectives for advanced injector design, thus considering high Karlovitz combustion aside from toy problems. The opportunity of counterbalancing the negative impact of strong turbulence on the flame-base thanks to entrainment of burnt gases, appears actually stronger than expected from usual combustion regime analysis. At the same times, it confirms that high Karlovitz combustion can hardly be sustained without addition of burnt gases.

  Vitiated high Karlovitz n-decane/air turbulent flames: Scaling laws and micro-mixing modeling analysis Alexandre Bouaniche • Nicolas Jaouen • Pascale Domingo • Luc Vervisch Received: date / Accepted: date

Fig. 1

 1 Fig. 1 Karlovitz numbers considered in a sample of the literature over the past 25 years.

Fig. 2

 2 Fig. 2 Damköhler numbers considered in a sample of the literature over the past 25 years, with a zoom over most recent years.

Fig. 3

 3 Fig. 3 Scatter plot of stochastic particles temperature vs mixture fraction. Particles are coloured depending on their initial condition. Blue: air. Orange: kerosene. Red: burned gases. Zst is the stoichiometric mixture fraction.

Fig. 5

 5 Fig.5Temperature trajectories (Eq. (9)). τ T = 0.2 ms. Symbols: Reference chemistry[START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF]. Lines: Reduced n-decane mechanism (Table1). Solid-line: Air inlet trajectory. Dottedline: Fuel inlet trajectory.

Fig. 6

 6 Fig.6Freely propagating premixed flames. Response versus equivalence ratio, species and temperature taken in burnt gases. Symbols: Reference chemistry[START_REF] Luche | Reduction of large detailed kinetic mechanisms: application to kerosene/air combustion[END_REF]. Solid-line: Reduced surrogate mechanism. Dashed-line: Reduced n-decane mechanism (Table1).

Fig. 7

 7 Fig. 7 Flame speed S L (f b ) versus the dilution factor f b[START_REF] Carlsson | Flame structure analysis for categorization of lean premixed CH4/air and H2/air flames at high Karlovitz numbers: Direct numerical simulation studies[END_REF]. Line: Eq. 14. Symbol: Simulation of 1D freely propagating premixed flame.

Fig. 8

 8 Fig. 8 Response of the progress variable c D (t End ) versus mechanical mixing time. f b = 0.5 and t End = 2 ms.

Fig. 10

 10 Fig. 10 Response of Kaq versus f b . Circle: simulations (Eqs. (3) and (9)). Dashed line: Eq. (18) (b): zoom in moderate Ka level.

  CH 2 HCO, HCCO, NC 3 H 7 , PC 4 H 9 , AC 6 H 13 , AC 8 H 17 , C 10 H 21 (L) NO, HCN, N 2 O, NO 2

HNO, HONO, H 2 CN, NNH, NH 2 , NH, N, CN, NCO, HNCO Table 1 Species of a n-decane (n-C 10 H 22 ) reduced mechanism composed of 26 transported species associated to 24 QSS relations and 338 reactions.

Estimation of a characteristic chemical time

		0.018				32s NC10H22
		0.016
		0.014
		0.012
	Y Y(CO) CO	0.008 0.01
		0.006
		0.004			2⌧ c
		0.002
		0
		0	0.2 0.4 0.6 0.8	1	1.2 1.4 1.6 1.8	2
			Time [ms]
			Time (ms)

Table 2

 2 of the aeronautical swirl burned studied experimentally by Meier et al.[START_REF] Meier | Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame[END_REF]. Characteristic length scales approximated from quasi-DNS of a swirl burner[START_REF] Meier | Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame[END_REF][START_REF] Moureau | From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: Filtered laminar flame-pdf modeling[END_REF].

	Re T	Re λ	T	τ T	η k	τ k
	1480	149	7 mm	2 ms	29 µm	0.051 ms

This scaling may also be retrieved from high-activation energy asymptotic developments after matching fluxes with vitiated fresh gases[START_REF] Boulanger | Asymptotic analysis and direct numerical simulation of partially premixed combustion[END_REF].

Acknowledgements The first author is supported by the European Union under the project SOPRANO, Horizon 2020 Grant Agreement No. 690724. The second author was funded by ANRT (Agence Nationale de la Recherche et de la Technology), SAFRAN-SNECMA and Air Liquide under the CIFRE No 1053/2013. Computing time has been provided by CRIANN (Centre Régional Informatique et d'Applications Numériques de Normandie).

Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


Vincent Jomelli
email: jomelli@cnrs-bellevue.fr

Pierre Pech

EFFECTS OF THE LITTLE ICE AGE ON AVALANCHE BOULDER TONGUES IN THE FRENCH ALPS (MASSIF DES ECRINS)

Keywords: avalanche boulder tongue, lichenometry, Little Ice Age, French Alps

Lichens of the subspecies Rhizocarpon geographicum s.l were measured on 25 avalanche boulder tongues in the Massif des Ecrins to elucidate the Little Ice Age history of avalanche activity. Results show: (1) an increase of lichen size from the median to the distal zone of deposits, and a decrease from the edges to the centre; (2) three types of lichen settlement. From the uppermost to the median zone, lichens are absent, because avalanche activity is very active. Down-slope, lichens occur in two different zones: the median zone is colonized by 5-20 mm size lichens on sides of blocks protected from the abrasional action of avalanches, while in the distal zone lichen diameters are largest (>30 mm) and occur on all sides of the blocks.

The spatial distribution of the lichens and their size according to elevation make it possible to distinguish different phases during which avalanche activity has increased. At high elevation, the avalanche activity was at a maximum before AD 1650 and between AD 1730 and 1830. During these two periods avalanches had sufficient magnitude to reach the basal zone of the deposits. At low elevation since AD 1650 the magnitude and frequency of avalanches have declined.

INTRODUCTION

Increased attention has been paid in the last 20 years to the impacts of the Little Ice Age on different periglacial environments. A very large range of processes and slope deposits have been studied. Previous works on this topic were mostly on debris flows [START_REF] Rapp | Alpine debris flows in northern Scandinavia, morphology and dating by lichenometry[END_REF]Innes, 1985b;[START_REF] Nyberg | Debris flows and slush avalanches in northern Swedish Lappland[END_REF][START_REF] Van Steijn | Debris-flow magnitude-frequency relationships for mountainous regions of Central and Northwest Europe[END_REF], rock glaciers [START_REF] Evin | Nouvelles données sur l'Age de la mise en place et les phases d'activité du glacier rocheux de Marinet 1 (Haute-Ubaye, Alpes du Sud Françaises)[END_REF][START_REF] Haeberli | Creep and Mountain Permafrost: Internal structure and flow of alpine rock glaciers[END_REF][START_REF] Evin | Dynamique, répartition et âge des glaciers rocheux des Alpes du Sud[END_REF][START_REF] Hamilton | Preliminary results from the lichenometric study of the Nautardalur rock glacier, Tröllaskagi, Nothern Iceland[END_REF][START_REF] Humlum | Origin of rock glaciers: observations from Mellemfjors, Disko Island, Central west Greenland[END_REF][START_REF] Humlum | Late-Holocene climate in central west Greenland: meteorological data and rock glacier isotope evidence[END_REF][START_REF] Haeberli | Climate change and its impacts on glaciers and permafrost in the Alps[END_REF][START_REF] Francou | Symptoms of degradation in a tropical rock glacier (Andes of Bolivia, 21°S)[END_REF], and rock avalanches [START_REF] Grove | The incidence of landslides, avalanches and floods in western Norway during the Little Ice Age[END_REF][START_REF] Grove | The Little Ice Age[END_REF][START_REF] Porter | Catastrophic rockfall of September, 12, 1717 on the Italian flank of the Mont Blanc massif[END_REF]. Lichenometric measurements [START_REF] Gray | Geomorphic effects of avalanches and rockfall on steep mountains slopes in the central Yukon Territory[END_REF]Innes, 1985b;[START_REF] Luckman | Estimating long term rockfall accretion rates by lichenometry[END_REF][START_REF] Mccarroll | Spatial and temporal patterns of late Holocene rockfall activity on a Norwegian talus slope: a lichenometric and simulation-modeling approach[END_REF], dendrochronological observations [START_REF] Hétu | Evolution récente d'un talus d'éboulis en milieu forestier, Gaspésie, Quebec[END_REF][START_REF] Lafortune | Dynamique d'un front forestier sur un talus d'éboulis actif en climat tempéré froid (Gaspésie, Quebec)[END_REF] or sedimentation rates [START_REF] Kotarba | High-mountain denudation system of the Polish Tatra mountains[END_REF][START_REF] André | Holocene rockfall retreat in Svalbard: a triple-rate evolution[END_REF][START_REF] Blikra | Climatic signals recorded in snow avalanche-dominated colluvium in western Norway: depositional facies successions and pollen record[END_REF] show that climatic changes, which occurred since the Little Ice Age, induce diachronous variations in slope processes. These variations are highlighted by: (i) the frequency and intensity of processes; and (ii) an initiation or elimination of previously inactive or active processes [START_REF] Kotarba | Transport sorting and deposition processes of alpine debris slope deposits in the Polish Tatra mountains[END_REF][START_REF] Rapp | Mass movement, nivation processes and climatic fluctuations in the northern Scandinavian mountains[END_REF][START_REF] Nyberg | Geomorphic features as indicators of climatic fluctuations in a periglacial environment, Northern Sweden[END_REF][START_REF] Eybergen | Geomorphological processes and climate change[END_REF][START_REF] Schlyster | Geomorphic process studies related to climate change in Northern Sweden status of current research[END_REF][START_REF] André | Holocene climate fluctuations and geomorphic impact of extreme events in Svalbard[END_REF].

However, little attention has been paid to the pattern of snow avalanche processes during and after the Little Ice Age. Until now most studies have been performed in northern Europe [START_REF] Grove | The incidence of landslides, avalanches and floods in western Norway during the Little Ice Age[END_REF][START_REF] Nyberg | Debris flows and slush avalanches in northern Swedish Lappland[END_REF][START_REF] Mccarroll | Modelling late-Holocene snow-avalanche activity; incorporating a new approach to lichenometry[END_REF][START_REF] Mathews | Snow avalanche impact landforms in Breheimen, southern Norway. Origin, Age and Paleoclimatic implications[END_REF][START_REF] Bull | Lichenometric analysis of the Kärkerieppe slush-avalanche fan, Kärkevagge, Sweden[END_REF]Blikra andNemec, 1993, 1998), a region with a special morphoclimatic context. In alpine environments, such observations are still lacking, especially for the French Alps (V. Jomelli, unpublished work, 2000). The aim of this paper is to provide information on the effects of the Little Ice Age on avalanche boulder tongues in part of the French Alps, using an improved lichenometric method performed on a large sample of deposits.

STUDY SITES

The field area is the eastern part of the Massif des Ecrins (French Alps, 45°00′ S, 6°30′ E) (Figure 1). The studied avalanche boulder tongues are situated between 1800 and 2450 m a.s.l., an elevation which is close to the annual 0 °C isotherm. At this altitude, 60 per cent of the precipitation is snow. Snow covers the ground from October to June, with a maximum thickness of 3 m [START_REF] Francou | Géodynamique des dépôts de pied de paroi dans l'étage périglaciaire[END_REF][START_REF] Francou | L'éboulisation en haute montagne[END_REF]. The boulder tongues used in the present study are the result of sedimentation from successive wet snow avalanches [START_REF] Jomelli | Wet snow avalanche deposits in the French Alps: structure and sedimentology[END_REF] (Figure 2). Generally they are located below north-facing cliffs, of varying angles, where debris and snow tend to accumulate on the low-angle cliffs. The occurrence of dirty avalanches is highest in May and June. The debris transported by snow avalanches consists of blocks of granite and gneiss ranging between 10 and 50 cm (a-axis), and matrix including sands and a smaller proportion of silt. Accretion rates, evaluated for at least 3 years, are between 0•02 and 0•35 mm a -1

U N C O R R E C T E D P R O O F Figure 2.
Wet snow avalanche of 450 m length in the Champoleon valley in spring 1998 [START_REF] Jomelli | Géodynamique des dépôts d'avalanches: analyses morphométriques et sédimentologiques[END_REF], with values higher than 2 mm a -1 on the very active north-facing cliffs [START_REF] Francou | 10-years superficial velocities on a rock glacier. Laurichard, French Alps[END_REF].

METHODS

Measurements of Rhizocarpon sp. (ex Rhizocarpon geographicum) lichens were performed during two field campaigns on 25 deposits for which geometrical and sedimentological characteristics have been studied previously [START_REF] Jomelli | Géodynamique des dépôts d'avalanches: analyses morphométriques et sédimentologiques[END_REF][START_REF] Jomelli | Dépôts d'avalanches dans les Alpes françaises: géométrie, sédimentologie et géodynamique depuis le Petit Age Glaciaire[END_REF]. The value of the minimum diameter of the largest lichen was collected with a flexible, transparent plastic ruler on blocks in rectangular sample areas of 5 × 20 m according to methods developed by Innes (1985aInnes ( , 1986b) ) and [START_REF] Mccarroll | Modelling late-Holocene snow-avalanche activity; incorporating a new approach to lichenometry[END_REF][START_REF] Mccarroll | A new approach to lichenometry: dating single-age and diachronous surfaces[END_REF]. These surface areas have been distributed in a draught-board pattern on deposits. The location of lichens on boulders, in relation to the supposed trajectories of avalanches, was also noted. Between 15 and 70 measurements (one measurement on each block) were made within each sample area. This minimum value of 15 measurements in each sample area was used to define the 'green zone' (Innes, 1985a). Sample surfaces with less than 15 blocks colonized were considered as free of lichens. The minimum diameter counted was 5 mm with an accuracy of 1 mm. Anomalous lichen shapes were rejected to reduce the risk of coalescence. Measurements were performed on the 'upper surface' and 'sides' of boulders with an a-axis between 15 cm and 100 cm.

To compare the lichen pattern between deposits, the length of the deposits was measured over a constant segment of 10 m. To facilitate comparison, the distances from the apex of the deposit are divided by the total length of the deposit (fractionated distance from the apex) and converted to percentages [START_REF] Francou | L'éboulisation en haute montagne[END_REF][START_REF] Jomelli | Comparing characteristics of rockfall talus and snow avalanche landforms in an alpine environment using a new methodological approach[END_REF]. Finally, to date this avalanche activity we used a lichenometric growth curve (Figure 3) constituted for the massif des Ecrins (Pech et al., 2002). I). Down-slope, another type of surface is observed on which lichens occur on sides of blocks protected from the abrasional action of avalanches. This surface, named 'type 2', represents between 12 and 26 per cent of the total surface. In the distal zone, we observed a 'type 3' surface on which lichens occur on all faces of blocks, especially those potentially exposed to snow avalanche impacts. The dimensions of this surface are variable and sometimes (three cases out of 20) can be limited to edges along lateral axes (Figure 4a). Locally, we found small surfaces without lichen within zones of types 2 and 3 (arrow in Figure 4a). Along the median axis, type 1 covers about 7/10 of the distance down-slope from the apex while type 3 is found only from 9/10 to the lowermost end of the profiles. Along the sides, however, type 1 is rarely found below 5/10 of the downslope distance and type 3 occurs below 8/10. In other words, along the sides of the deposits types 2 and 3 are present higher up-slope than in their central parts.

U N C O R R E C T E D P R O O F
The extent of the different zones changes with elevation. But, with these data, there seems no simple relation between those two parameters. Nevertheless it is noted that the extent of zone 1 on avalanche boulder tongues is generally smaller at low elevation (<2150 m, corresponding to the altitudinal average of the avalanche boulder tongues selected in this study) compared to higher elevation. In other words, at low elevation the lichens appear up-slope on the profile of the deposits, from 4/10 to 5/10. Finally, for deposits located at equal elevation, the extent of the different zones changes across the slope direction (Figure 4).

Lichen size-frequency distributions were performed for each avalanche boulder tongue (Figure 5a-d). Results show variable characteristics from one to another. Form is rarely symmetric, as confirmed by skewness values between -0•3 and 0•2; only Figure 5b andc Earth Surf. Process. Landforms 29, 000-000 ( 2004) In detail, the distributions show several modes which are frequently between 20 and 27-28 mm (Figure 5b). From these first measurements, some larger lichen have been observed located mainly on blocks in the distal zone. These very large lichens are developed in most cases only on the sides of blocks protected from the abrasional action of avalanches. Figure 6a-d displays the mean size of the five largest diameters on each sample surface along median and lateral axes. Results show a clear down-slope increase of the median size of the five largest lichens and a decrease from the edges to the centre. This pattern is relatively consistent from one deposit to another but differs between median and lateral axes on deposits. Along the median axis, the increase is not linear (Figure 7). It is roughly at 7/10 to 8/10 of the profile then more regular down-slope. Below 8/10, the increase of lichen size is stronger when type 3 exists (Figure 6b,c). Along the edges of the deposits, lichen sizes are larger (usually by more than 0•5 cm) compared to that on the central parts at the same relative distance from the apex. In the distal zone of high elevation deposits, the maximum size is generally around 3•5-4 cm; but in rare cases, values can exceed 5•5 cm. However, the very small number of blocks (specified by the number in parentheses on Figure 6b), on which we observed these very large lichens, does not permit any zones to be distinguished. On the contrary, at low altitude the lichen size frequently exceeds 4•5 cm in the distal zone.

U N C O R R E C T E D P R O O F

DISCUSSION

The lichenometry can be used to analyse the spatial frequency/run-out variation of avalanches and to date these variations. The down-slope change of lichen-colonization types, usually used in relative dating (Innes, 1986a), underlines spatial aspects of snow avalanche sedimentation and allows an identification of variations in the frequency and run-out distance of snow avalanches. This variation is partly due to the magnitude of geomorphic

Copyright © 2004 John Wiley & Sons, Ltd.

Earth Surf. Process. Landforms 29, 000-000 ( 2004)

U N C O R R E C T E D P R O O F Figure 6
. Distribution of the five largest lichens on each of n boulders on the four examples of avalanche boulder tongues. In (b) is an example of a special deposit on which very large lichens have been observed; outlines do not take into account these special extreme values activity controlled by the snow-avalanche processes (erosion, transport and sedimentation). Generally, this geomorphic action decreases irregularly down-slope (Figures 456), as has been observed in previous studies [START_REF] Luckman | Debris accumulation patterns on talus slopes in Surprise Valley, Alberta[END_REF].

From the apex to the upper 7/10, geomorphic activity is high and prevents lichen colonization of the deposit. We deduce that avalanche frequency is high for run-out distances less than 7/10 of the slope. Field observations between 1993 and 2001 indicate that these avalanches have a recurrence time of between 1 and 5 years. However, snow avalanches with larger run-out distance will also occur but at a lower frequency. Parts of the lichen distribution in Figure 5 can thus be attributed to such time-frequency systems. Consequently, the following hypothesis can be formulated: the avalanches can still reach this median zone but they are not very frequent or/and have a lower abrasive power; so lichens can grow on the protected sides of blocks. This hypothesis Copyright © 2004 John Wiley & Sons, Ltd.

Earth Surf. Process. Landforms 29, 000-000 ( 2004)

U N C O R R E C T E D P R O O F
considers that the median zone is still active. Down-slope, the frequency of avalanches reaching 9/10 of the profile is very low, so lichens are older than on the other parts of the deposit. Locally, we observed small surfaces where lichens are absent within zones of types 2 and 3. This demonstrates the spatial heterogeneity of the avalanches' geomorphic action. Nevertheless, this action is stronger on the central axis.

Lichenometry also allows us to show a diachronicity in sedimentation on avalanche boulder tongues during the Little Ice Age (LIA). According to the lichen patterns, an irregular decrease of avalanche run-out distances has occurred for these last centuries (Figures 456). However, a distinction must be made according to the elevation.

At high elevation, we have observed large lichens on four avalanche boulder tongues. However, their position on the blocks and their size in comparison to those observed on the other blocks suggest that they are the remnants of an old phase of colonization. The high erosive potential of avalanches would be responsible for their disappearance on the other fans. If one takes into consideration these largest lichens, it seems that the activity of the avalanches was restricted to the distal zone between 1650 and 1700 (if an extrapolation of the lichen growth curve is made). Before the middle of the seventeenth century, avalanches therefore, had a magnitude sufficient to reach this basal zone and to deposit large blocks. Their frequency was sufficiently high to prevent lichens from colonizing the deposits. This high period of avalanche activity could be viewed as a first maximum during the LIA. However, the exact age of the blocks in the distal zone is difficult to determine because they could also be considerably older than the beginning of the LIA. It is possible that previously deposited boulders were covered by enhanced accumulation during this first LIA maximum. Between 1650 and 1730 avalanche activity decreased, allowing free development of lichens on the surface of snow avalanche fans. Between 1730 and 1830, the activity of the avalanches seems to have increased. Such an evolution could be responsible for the disappearance of most of the lichens which developed during the former period. Only occasional lichens located on protected sides of blocks survived this erosive activity. From about 1830 to 1850, according to the mean of the five largest lichens (Table I), the frequency of the avalanches reaching the distal zone was extremely low. Alternatively, it may reflect a decrease in the quantity of debris transported by avalanches, reducing their erosive power. However, various results in the literature suggest an increase in frost shattering during the LIA. Earth Surf. Process. Landforms 29, 000-000 ( 2004) [START_REF] Jomelli | Géodynamique des dépôts d'avalanches: analyses morphométriques et sédimentologiques[END_REF], from observations carried out on wet snow avalanche deposits, did not observe any relation between the quantity of debris and the length of the wet snow avalanche deposits. On the contrary, a positive linear relationship is observed between the maximum size of rock debris and the volume of wet snow deposits [START_REF] Jomelli | Wet snow avalanche deposits in the French Alps: structure and sedimentology[END_REF]. So, this decrease in run-out distance allowed lichens to colonize the basal zone. Consequently, at this elevation, a second maximum in avalanche activity occurred at the beginning of nineteenth century. Between 1830 and present, two contrasting periods can be distinguished: Because lichen size decreases from the base to the median part of all deposits, the magnitude of avalanches has decreased to 30 per cent between 1830 and 1950. This decrease seems to be due to a lower frequency, and explains the pattern of lichen distribution described earlier. From 1950, magnitude is low but frequency remains high in the upper part of the deposits.

U N C O R R E C T E D P R O O F Moreover,
At low elevation, the variation of the avalanche activity during the LIA is appreciably different. The lack of a clear hiatus in the distribution of the lichens suggests that there was a maximum in avalanche activity around AD 1600-1650 (according to the lichen growth curve extrapolation). Since around 1700, colonization by lichens in the distal zone of avalanche boulder tongues is observed, which is due to a decrease in activity of this process. The phase observed around AD 1830, during which maximum run-out distances at high altitude occurred, also exists at low altitude, but the activity was less intense.

The run-out distance variation of wet snow avalanches in a couloir since AD 1600-1650 could depend on a decrease in the amount of snow mobilized, if we suppose that the coefficients of friction were constant over the whole period [START_REF] Voellmy | Über die Zerstörungskraft von Lawinen[END_REF]. Can we deduce from these observations that precipitation was higher than at present? The answer is complex because until now most work shows no direct relation between the frequency or the intensity of an avalanche and precipitation [START_REF] Föhn | Climate change, snow cover and avalanches[END_REF][START_REF] Schneebeli | Destructive snow avalanches and climate change in the Swiss Alps[END_REF]. Nevertheless, this assumption of a greater quantity of snow mobilized by avalanches during the LIA can be supported by the fact that the two periods, AD 1600-1650 and 1830, during which the run-out distances were maximum at high elevation sites, have corresponded overall to the periods of maximum glacial advances for these last 500 years [START_REF] Roy | Histoire du climat depuis l'an mil[END_REF][START_REF] Reynaud | Histoire des fluctuations des glaciers en remontant le Petit Age de Glace[END_REF]. The first major glacial advance of the LIA ranges between 1600 and 1650 depending on the glaciers, and the second between 1800 and 1850. Since 1850 most French Alpine glaciers have decreased. Moreover, the mass balance of these glaciers is directly correlated with summer temperature and spring precipitation [START_REF] Vincent | Meteorological controls on glacier mass-balance: empirical relations suggested by Sarennes glaciers measurements (France)[END_REF][START_REF] Vincent | Fluctuations des bilans de masse des glaciers des Alpes françaises depuis le début du 20em siècle au regard des variations climatiques[END_REF][START_REF] Vincent | Influence of climate change over the 20th century on four French glacier mass balances[END_REF]. At a larger scale, this relation is confirmed by historical analyses which show a decrease of the Buêch floods at the end of the Little Ice Age [START_REF] Gautier | Recherches sur la morphologie et la dynamique fluviale dans le bassin du Buech (Alpes du Sud)[END_REF], reflecting a decrease of either snow precipitation or snow mantle duration in this part of France.

In other mountainous areas, in particular in northern Europe, the phase of maximum avalanche activity has been frequently recorded around 1850 [START_REF] Grove | The Little Ice Age[END_REF][START_REF] Mathews | Snow avalanche impact landforms in Breheimen, southern Norway. Origin, Age and Paleoclimatic implications[END_REF][START_REF] Bull | Lichenometric analysis of the Kärkerieppe slush-avalanche fan, Kärkevagge, Sweden[END_REF][START_REF] Mccarroll | Late Holocene snow-avalanche activity in Southern Norway: interpreting lichen sizefrequency distributions using an alternative to simulation modelling[END_REF]. More rarely this occurred around AD 1650 [START_REF] Grove | The incidence of landslides, avalanches and floods in western Norway during the Little Ice Age[END_REF][START_REF] Grove | Tax records from western Norway, as an index of Little Ice Age environmental and economic deterioration[END_REF] or at the beginning of the twentieth century [START_REF] Blikra | Climatic signals recorded in snow avalanche-dominated colluvium in western Norway: depositional facies successions and pollen record[END_REF]. The different climatic conditions [START_REF] Jacobeit | Zonal indices for Europe 1780-1995 and running correlation temperature[END_REF] which explain the opposition currently observed between the dynamics of the Scandinavian glaciers and that of the French glaciers [START_REF] Six | Bilans de masse des glaciers alpins et scandinaves, leurs relations avec l'oscillation du climat de l'Atlantique nord[END_REF] could also explain this temporal shift in avalanche activity.

Considering the specific context of this study (avalanches boulder tongues, high elevation, Little Ice Age period), it could be answered that the lichenometric variations simply underline the presence of firns on the avalanche boulder tongues, which would have been responsible for a decrease in the growth rate in the distal zone and the deterioration of the lichens in the proximal and median zones of the deposits? Without excluding a punctually influence, two arguments make it possible to refute this assumption for all of the sites:

Weather data gathered since the 1980s in one of the selected sectors [START_REF] Francou | L'éboulisation en haute montagne[END_REF][START_REF] Bocquet | Quelques indications sur les caractéristiques météorologiques dans le vallon du plan de l'Alpe (Haute Romanche) au cours de ces dernières années (Massif des Ecrins, France)[END_REF] added with field observations carried out annually since 1993 show that up to 2500 m in altitude, the snow does not remain on the ground between the middle of June and the beginning of October on the northern slopes. However these three months without snow cover are sufficient for the normal development of the lichens [START_REF] Benedict | Lichen mortality due to late-lying snow: results of a transplant study[END_REF][START_REF] Benedict | Experiments on lichen growth II. Effects of a seasonal snow cover[END_REF]. If the comparison between climatic conditions of the 1980s and those of the LIA is debatable, it is difficult to understand why no lichen can grow in the proximal and median zones of the deposits since the 1950s without taking into account the action of the avalanches.

Moreover, if climatic conditions during LIA would not permit the development of lichens or their maintenance, it is difficult to explain lichens larger than 5cm on rock glaciers observed in these same valleys above 2400 m in altitude.

Copyright © 2004 John Wiley & Sons, Ltd.

Earth Surf. Process. Landforms 29, 000-000 ( 2004)

U N C O R R E C T E D P R O O F
Some local factors may influence the general tendency explained above. We consider that differences between avalanche boulder tongues on figures 6a, b and c could be explained by morphological factors. As altitude and aspect are similar for these sites, these factors are not pertinent, as has been observed elsewhere [START_REF] Benedict | Experiments on lichen growth II. Effects of a seasonal snow cover[END_REF]. Differences of colonisation and of lichen size observed between deposits could be linked to the slope angle and length of the rock wall above each deposit. It was shown that these variables control snow avalanche run-out distances [START_REF] Voellmy | Über die Zerstörungskraft von Lawinen[END_REF][START_REF] Norem | Simulation of snow avalanche flow in the run-out zone[END_REF][START_REF] Jomelli | Géodynamique des dépôts d'avalanches: analyses morphométriques et sédimentologiques[END_REF]. These variables could explain the fact that for a certain relative position on a profile, the size of the five largest lichens is not equal between two deposits situated in the same valley.

Usually the Little Ice Age impact on periglacial slope deposits can be summarized as either a frequency/ intensity variation of given processes, or as the activation of previously inactive processes (and vice versa). More rarely, morphological and sedimentological modifications on deposits are documented [START_REF] Mathews | Snow avalanche impact landforms in Breheimen, southern Norway. Origin, Age and Paleoclimatic implications[END_REF][START_REF] Humlum | Origin of rock glaciers: observations from Mellemfjors, Disko Island, Central west Greenland[END_REF][START_REF] Humlum | Late-Holocene climate in central west Greenland: meteorological data and rock glacier isotope evidence[END_REF]. In this paper analysis of lichemonetric data confirms an earlier hypothesis which links some special morphological and sedimentological characteristics of avalanche boulder tongues (a break point with relatively constant coordinates on the longitudinal profile and an increase of block size in the distal zone) [START_REF] Jomelli | Dépôts d'avalanches dans les Alpes françaises: géométrie, sédimentologie et géodynamique depuis le Petit Age Glaciaire[END_REF] to a superposition of two deposits linked to dynamic variations since the Little Ice Age. A first generation of wet snow avalanches involving a large amount of snow and coarse debris has developed a long and smooth concavity. Later, less powerful snow avalanches have constructed a newer deposit closer to the proximal zone, the contact between the two deposits being marked by a geometric discontinuity. This change of wet snow avalanche activity may reflect the end of the Little Ice Age, the less powerful snow avalanches belonging to attenuated snow conditions after the Little Ice Age.

CONCLUSION

Analysis of colonization patterns and thallus diameter variation on 25 avalanche boulder tongues in the French Alps illustrates variations in the evidence a frequency/run-out of wet snow avalanches since the Little Ice Age. According to the lichenometric measurements at high elevation, the avalanche activity was at a maximum before AD 1650 and between AD 1730 and 1830. During these two periods avalanches had sufficient basal magnitude to reach the zone of the deposits. At low elevation since AD 1650 the magnitude and frequency of avalanches has declined.

The relatively good correspondence between avalanche boulder tongue and glacier behaviour in the Massif des Ecrins to climatic variation which occurred since the Little Ice Age confirms that avalanche boulder tongues can be viewed as reliable indicators of the activity of certain slope processes within the periglacial zone.

Figure 1 .

 1 Figure 1. Location map of the Massif des Ecrins including the study sites

!

  V. JOMELLI AND P. PECHCopyright © 2004 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 29, 000-000 (2004) tongues lichen settlement changes spatially, with clear boundaries as shown on the four representative examples on Figure4. In proximal and sometimes in median zones thalli are scarce (<15 in each sample area) or absent. This surface, named 'type 1', represents between 40 and 70 per cent of the total surface (Table

Figure 4 .

 4 Figure 4. Lichen colonization types on four examples of avalanche boulder tongues

  show a weak skewness (0•07, 0•09, respectively). Largest diameter values show large variations according to elevation. Below 2150 m the maximum values of thalli are close to 5•8 cm. Largest diameter values remain relatively constant from one deposit to another whether or not located Copyright © 2004 John Wiley & Sons, Ltd.

Figure 5 .

 5 Figure 5. Lichen size-frequency distributions for the four examples of avalanche boulder tonguesin the same valley. Modal values are between 2•3 and 2•5 cm. Above 2150 m the largest thalli diameters show a complex distribution according to avalanche boulder tongues. Maximum values are lower by at least 1•3 cm than those measured at low elevation. Modal values are most often between 21 and 31 mm. In detail, the distributions show several modes which are frequently between 20 and 27-28 mm (Figure5b). From these first measurements, some larger lichen have been observed located mainly on blocks in the distal zone. These very large lichens are developed in most cases only on the sides of blocks protected from the abrasional action of avalanches.Figure6a-d displays the mean size of the five largest diameters on each sample surface along median and lateral axes. Results show a clear down-slope increase of the median size of the five largest lichens and a decrease from the edges to the centre. This pattern is relatively consistent from one deposit to another but differs between median and lateral axes on deposits. Along the median axis, the increase is not linear (Figure7). It is roughly at 7/10 to 8/10 of the profile then more regular down-slope. Below 8/10, the increase of lichen size is stronger when type 3 exists (Figure6b, c). Along the edges of the deposits, lichen sizes are larger (usually by more than 0•5 cm) compared to that on the central parts at the same relative distance from the apex. In the distal zone of high elevation deposits, the maximum size is generally around 3•5-4 cm; but in rare cases, values can exceed 5•5 cm. However, the very small number of blocks (specified by the number in parentheses on Figure6b), on which we observed these very large lichens, does not permit any zones to be distinguished. On the contrary, at low altitude the lichen size frequently exceeds 4•5 cm in the distal zone.

Figure 7 .

 7 Figure 7. Down-slope evolution along the median axis of the deposit of the mean of the five largest diameters on five snow avalanche boulder tongues: dotted lines indicate low elevation boulder tongue; solid lines indicate high elevation boulder tongue

  Copyright © 2004 John Wiley & Sons, Ltd.Earth Surf. Process. Landforms 29, 000-000 (2004) John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 29,000-000 (2004)

Table I .

 I Elevation and lichenometric measurements on avalanche boulder tongues

	Deposit	Type 1	Type 2	Type 3	Mean size of the 5	Elevation	Number of
		(%)	(%)	(%)	largest lichens	(m)	measurements
	1	5 0	1 9	3 1	5•7	1810
	2	5 6	2 1	2 3	5•4	1860
	3	3 5	2 8	3 7	5•9	1920
	4	4 4	2 4	3 2	5•8	1930
	5	5 7	1 8	2 5	5•8	2000
	6	3 8	2 3	3 9	5•6	2010
	7	5 1	2 0	2 9	5•9	2040
	8	4 8	1 3	3 9	5•4	2080
	9	4 5	1 0	4 5	5•7	2130
	10	66	12	18	4•4	2140
	11	65	16	19	4•3	2150
	12	61	22	17	4•2	2180
	13	50	21	29	4•4	2200
	14	52	26	22	4•3	2240
	15	70	16	14	5•4	2250
	16	67	18	15	3•8	2260
	17	66	14	20	5•8	2280
	18	65	17	18	4•5	2300
	19	62	16	22	5•9	2300
	20	54	31	15	4•3	2330
	21	57	26	17	5•8	2420
	22	59	14	27	4•3	2420
	23	54	30	16	4•4	2450
	24	53	25	22	4•4	2460
	25	55	19	26	4•3	2460

Figure 3. Rhizocarpon growth curve in the eastern side of the Massif des Ecrins Copyright © 2004 John Wiley & Sons, Ltd.

Earth Surf. Process. Landforms 29,

000-000 (2004)

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ACKNOWLEDGEMENTS

This study has been conducted as part of the APN 'Impacts des fluctuations climatiques enregistrées depuis le Petit Age Glaciaire sur les versants soumis aux processus périglaciaires et aux mutations du pastoralisme dans le Massif des Ecrins', and of the Eclipse programmes which were supported by the CNRS. Special thanks are due to L. Blikra and D. McCarroll for critically reading the manuscript.