François Girard

Michel Vennetier
email: michel.vennetier@cemagref.fr

Frédéric Guibal

Christophe Corona

Samira Ouarmim

Asier Herrero

Girard François

Vennetier Michel

Guibal Frederic

Corona Christophe

Ouarmim Samira

Herrero Asier

Pinus halepensis Mill. crown development and fruiting declined with repeated drought in Mediterranean France

Keywords: Pinus halepensis, Primary growth, Shoot length, Branching rate, Needle number, Needle length, Polycyclism, Fruiting, Ring width, Tree architecture, Climate change, Mediterranean forest, Drought

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-Introduction

In the Mediterranean basin, summer drought is the main constraint on tree growth with great interannual variations in duration and intensity [START_REF] Houerou | The Isoclimatic Mediterranean Biomes: Bioclimatology, Diversity and Phytogeography[END_REF]. For the 21 st century, climate models predict in the Mediterranean basin a faster warming than in most other areas in the world, associated with a reduction of rainfall during the growth season [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF][START_REF] Hesselbjerg-Christiansen | Regional Climate Projection[END_REF]. In addition, extreme climatic events, such the 2003 scorching heat, are prone to be recurrent [START_REF] Meehl | More intense, more frequent, and longer lasting heat waves in the 21st century[END_REF]. The response of forests to the forecasted increase in drought occurrence is considered a key issue in climate change scenarios [START_REF] Hesselbjerg-Christiansen | Regional Climate Projection[END_REF]. Rapid decline in precipitation and higher temperatures are already noticeable in some areas in the Mediterranean basin giving a foretaste of the mid 21 st century climate. In Southeastern France, very low rainfall were observed each year from [START_REF] Damesin | Respiration and photosynthesis characteristics of current-year stems of Fagus sylvatica: from the seasonal pattern to an annual balance[END_REF]to 2007(Météo-France, 2011). Moreover, ten of the twelve hottest years since 1850 were recorded between 1997 and 2008 [START_REF] Vennetier | Forest flora turnover with climate change in the Mediterranean region: A case study in Southeastern France[END_REF]. The mean annual and mean summer temperatures of years 1998-2007 were respectively 0.9°C and 1.3°C over the average of the previous 30 years, while spring and summer rainfall were reduced from 18% and 32% respectively during this decade.

Common in the Mediterranean areas with nearly 3.5 millions hectares, Aleppo pine (Pinus halepensis Mill.) shows an exceptional plasticity, growing on any substrate and on poor soils. Being thermophilious and heliophilous, it tolerates high temperatures but fears excessive humidity, frost and snow [START_REF] Nahal | Relationships between tree size, crown shape, gender segregation and sex allocation in Pinus halepensis, a Mediterranean pine tree[END_REF]Vennetier et al., 2010). Aleppo pine strategy towards drought is to avoid water stress by closing precociously its stomata [START_REF] Borghetti | Impact of long-term drought on xylem embolism and growth in Pinus halepensis Mill[END_REF]. Despite their adaptation to unstable climatic and edaphic conditions, some of Aleppo pine growth processes may have been negatively impacted by repeated dry and warm years. In Tunisia, El [START_REF] El Khorchani | The impact of drought on the growth of three forest species in Tunisia (Pinus halepensis Mill., Pinus pinea L. et Pinus pinaster Sol.)[END_REF] showed that, after 1978, frequent droughts significantly reduced its radial growth and increased mortality. Similar observations were made after year 2003 in southern France (Vennetier and Ripert, 2010). These results are consistent with models and long term observations showing that Aleppo pine growth is expected to suffer from changes in timing and duration of drought, particularly in spring and summer [START_REF] Borghetti | Facing drought in a Mediterranean post-fire community: tissue water relations in species with different life traits[END_REF][START_REF] Ogaya | Effect of drought on diameter increment of Quercus ilex, Phillyrea latifolia, and Arbutus unedo in a holm oak forest of NE Spain, For[END_REF][START_REF] Rathgeber | Simulated responses of Pinus halepensis forest productivity to climatic change and CO2 increase using a statistical model[END_REF][START_REF] Sardans | Drought changes the dynamics of trace element accumulation in a Mediterranean Quercus ilex forest[END_REF][START_REF] Vila | Has global change induced opposite trends in radial growth of Pinus sylvestris and Pinus halepensis at their bioclimatic limit? The example of the Sainte-Baume forest (south-east France)[END_REF].

Relationship between Aleppo pine radial growth and climate variability were well established in Mediterranean France [START_REF] Nicault | Intra-annual variations of radial growth and ring structure[END_REF]Rathgeberet al., 2000;[START_REF] Rathgeber | Bioclimatic model of tree radial growth: application to the French Mediterranean Aleppo pine forests[END_REF]. On the other hand, except for tree height, scant literature exists on the relationship between primary growth processes and climate, whatever the species. Primary growth corresponds to the creation of new tissues outside existing organs, and includes bole, branch and root length growth, branching processes (birth of new branches or roots), creation and growth of leaves or needles and rootlets, flowering and fruiting [START_REF] Barthelemy | Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny[END_REF][START_REF] Halle | Tropical trees and forests: an architectural analysis, Tropical trees and forests: an architectural analysis[END_REF]. By contrast, secondary growth for trees corresponds to the radial growth of existing branches and roots. Aleppo pine is commonly polycyclic [START_REF] Girard | Polycyclism, a fundamental tree growth process, decline with recent climate change. The example of Pinus halepensis Mill. in Mediterranean France[END_REF]: when adult, it is known to produce up to four annual growth units on vigorous branches: one or two, more rarely three, from late winter to the beginning of summer and sometimes one after summer drought [START_REF] Debazac | Morphologie et sexualité chez les pins[END_REF][START_REF] Serre | Les rapports de la croissance et du climat chez le pin d'Alep (Pinus halepensis (Mill)) II L'allongement des pousses et des aiguilles, et le climat Discussion Générale[END_REF]. An increased number of annual growth flushes means more ramifications and needles, longer shoots and higher primary production [START_REF] Barthelemy | Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny[END_REF]. Impact of successive droughts such as the period 2003-2007 in southern France had radical impact on Scots pine (Pinus sylvestris L.) primary and secondary growth [START_REF] Thabeet | Response of Pinus sylvestris L. to recent climate change in the French Mediterranean region[END_REF]. Indeed, a sharp fall of ring-width, shoot length, branching rate, fructification and needle length was observed.

Aleppo Pines have serotinous cones that take three years to mature. Seeds can be stored and protected in the canopy for as long as 15-30 years [START_REF] Nahal | Relationships between tree size, crown shape, gender segregation and sex allocation in Pinus halepensis, a Mediterranean pine tree[END_REF]. Such a long lasting aerial seed bank is an adaptation to the pioneer status of this species, and is devoted to regeneration after wildfires. But the long maturation process makes each annual cone generation dependant on several years, and a cone deficit during one or several years have long lasting delayed effects on the reproduction potential. Climate change and a higher frequency of climate accidents should make Aleppo pine regeneration more uncertain, The objectives of this study were (1) to describe Aleppo pine aerial primary growth processes (including cone production) and (2) to compute relationships between climate and tree primary and secondary growth over the last 15 years in Mediterranean France.

-Materials and methods

-Study site

The study took place in the Departmental Forest of Font-Blanche (43 o 14' 25'' N, 5 o 40' 40'' E, altitude 425m a.s.l.) in Provence, South-eastern France (Figure 1). Climate data were interpolated as a function of altitude (0.6°C/100 m) and distance from four stations of French National Meteorological Survey Network [START_REF] Météofrance | Données météorologiques des stations d'Aubagne, Gémenos, Cuges-les-pins[END_REF], located near the experimental site (5-10 km, altitude difference less than 200 m). The climate is Mediterranean, characterized by a severe drought lasting usually two months in summer, but regularly three to four months, mild and humid winters, and a very low cloudiness. At this site, mean annual temperature and precipitations are 13.2 o C and 695 mm . They were respectively 12.7°C and 745 mm for the period 1961-94 (Figure 2). Years 2003 to 2007 were characterized by the longest drought ever recorded (since 1860), with annual rainfall 20 to 60% under the average, and particularly a 50 to 60% yearly deficit of cumulated rainfall from January to August. Moreover, this period started with the 2003 scorching heat and ended in 2007 by the driest 12-month period ever recorded (281mm). At the end of 2003 summer, trees were severely defoliated and in very bad health conditions [START_REF] Vennetier | Impact of climate change on pine forest productivity and on the shift of a bioclimatic limit in a Mediterranean area[END_REF] as in most European forests [START_REF] Zaitchik | Europe's 2003 heat wave: A satellite view of impacts and land-atmosphere feedbacks[END_REF][START_REF] Zaitchik | Europe's 2003 heat wave: A satellite view of impacts and land-atmosphere feedbacks[END_REF](Zaitchik et al., and 2004 were marked by a terrible spring drought, lasting from February to July. Annual rainfalls of years 2008 and 2009 were normal but the summer drought of these years lasted four months. 2010 rainfall were 10% under the average but the summer drought was limited to 2 months and significant rains occurred up to the end of June. 2010 can be considered as the only normal year since 2003. [START_REF] Vennetier | A new bioclimatic model calibrated with flora for Mediterranean forested areas[END_REF], site conditions and water balance at Font-Blanche (BI = 8) are very close to the mean value for the French Mediterranean area (BI span = -80 to +80, mean = 0). Accordingly, the site index for Aleppo pine calculated from tree height/age ratio is exactly in the middle of the average fertility class for this species in France (Vennetieret al., 2010). Font-Blanche site can therefore be considered as highly representative of French Aleppo pine stands. Morphological markers like size and density of scales at the base of the growth units, lateral axes, and position of cones were used to reconstruct tree primary growth for the last 15 years on each branch, following methods described by [START_REF] Sabatier | Intra-and interspecific variations of polycyclism in young trees of Cedrus atlantica (Endl.) Manetti ex. Carriere and Cedrus libani A. Rich (Pinaceae)[END_REF] and [START_REF] Pardos | Shoot growth components and flowering phenology in grafted Pinus halepensis Mill[END_REF] which proved to be relevant for Aleppo pine. Crossdating of several secondary axes, and in some cases annual growth rings, were used to confirm the age of segments. In addition to measurements on branch principal axis, two pairs of secondary axes (one strong and one weak in each pair) were sampled: one pair 10 yearold and the other pair 5 year-old. At first, twigs were classified within each branch as function of their status: principal (P) (main axis), strong secondary axis (S-Strong) and weak secondary axis (S-Weak) (Figure 3). The status is based on relative diameter and length growth on each branch and not related to absolute vigour. Then, we classified twigs according to their absolute vigour, independently of their origin, status and position, To do so, twig length of years 2003-2005 (period common to all samples of the first period) was calculated and values were split into three groups of equal span (Figure 5C): frail twigs (annual growth ≤70 mm), medium vigour (71-140 mm) and vigorous (≥141 mm). Twigs were further distributed into theses groups according to the mean length of years 2003-2005 [START_REF] Girard | Polycyclism, a fundamental tree growth process, decline with recent climate change. The example of Pinus halepensis Mill. in Mediterranean France[END_REF].

A weekly follow-up of branch primary growth was performed from scaffoldings for 16 additional trees from January 2008 to December 2010, with exactly the same protocols than on cut branches. On these trees and on cut branches of 6 other trees, the proportion of shoots bearing male flowers as well as the length of male flowers on these shoots were measured retrospectively for years 2007-2010. On young shoots, the scars of male flowers, characterized by a prominent curved scale, are easy to differentiate from flat needle scars.

-Growth reconstruction and measures

Analysis of the number and length of flushes per annual shoot and total shoot length were used to elaborate master chronologies between 1995 and 2010 for each crown level and axes status. The 16year period for retrospective analysis was chosen because structural information disappears with time due to (1) the natural fading of morphological markers and/or (2) the natural self-pruning of twigs. Length of each growth units was measured using a calliper electronic ruler (± 0.1 mm). Along with morphological markers identification, number of ramifications, number of living and aborted cones and number of needles were counted per growth unit and then bulked per year. Only fully developed cones were considered for fruiting rate, aborted cones being analyses separately. As Aleppo pine usually keeps its needles between two and four years on adult trees, living needles were directly counted on each growth unit of the two last years. For older years, fascicles scars were counted following the spiral arrangement when visible, and living needles were counted when present to assess the proportion of survival in the third and fourth year. As for other pines species, needle fascicles are generally arranged in three to five spirals along the shoot. Counting needles on one or two spirals and then bulking the count for the number of spirals proved to be a very reliable assessment (error < 2%) [START_REF] Jalkanen | The Use of Needle-Trace Method (Ntm) in Retrospectively Detecting Lophodermella Needle-Cast Epidemic[END_REF][START_REF] Jalkanen | Revealing past needle density in Pinus spp[END_REF]. Needle length was measured with the same electronic ruler than shoots (± 0.1 mm) and averaged on 10 needles per growth unit, regularly sampled from its base to its end all around the twig. In case of polycyclism, for a given axis, annual needle length was obtained by the mean of all growth units weighed by the number of needles per unit. We also assessed the normal needle length of Aleppo pine in the study area. We averaged the values of all specimens collected close to the study area in similar site conditions and preserved in the herbarium of Provence University at Marseille (L. Samat; H. Roux, P. Blancand general herbarium): from 8 sites sampled from 1834 to 1921, 26 twigs and 41 growth units were measured, all of the last two years before the sampling date which was always in April to collect flowers.

A weak tendency to branch growth slowing down with time corresponds to the normal ageing of branches and the progressive decline with increased competition by the top crown and neighbouring trees. This natural trend is described in literature [START_REF] Barthelemy | Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny[END_REF][START_REF] Caraglio | Pinus halepensis architectural analysis for fuel modelling[END_REF][START_REF] Masotti | Study on the effect of the environment on the growth, branching and architecture of the atlas cedar, Cedrus atlantica (Endl) Manetti ex Carriere[END_REF] and can be assessed thanks to the successive branch sampling (2005)(2006)(2007)(2008)(2009) and the sampling at three levels in the crown. As middle crown branches were chosen to be 10 years older than top branches, we can compare the growth of shoots of the extremity of top branches with 10 years older shoots of middle branches: they were, with a ten years interval, exactly in the same relative position in tree crown and in the stand. The same reasoning allows comparing the extremity of middle crown branches (25 years old) and 10 years older shoots of low crown branches (35 years old). As Aleppo pine height growth models for Southern France [START_REF] Rathgeber | Modélisation de la croissance en hauteur du pin d'Alep (Pinus halepensis Mill.) en région méditerranéenne française[END_REF][START_REF] Vennetier | Short and long term evolution of Pinus halepensis (Mill.) height growth in Provence (France) and its consequences for timber production[END_REF] indicate a linear trend and no change in tree height growth between 35 and 55 years, one can consider that the same position in the crown should lead to the same primary growth rate. This trend was calculated for each parameter.

-Ring-width chronologies

Two cores per tree were collected from all sampled trees at 1.3 m height. To construct radial growth curves by tree and for the Font-Blanche stand, ring widths of each core were measured using a Velmex micrometer (Velmex Inc., Bloomfield, N.Y.) (± 0.002 mm). Ring-width series were firstly crossdated visually and a second check was made using the statistical software COFECHA [START_REF] Holmes | Tree-ring chronologies of western North America: California, eastern Oregon and northern Great Bassin, with procedures used in the chronology development work, including user manuals for computer program COFECHA and ARSTAN[END_REF]. Each ring-width series was standardized using a 32-year cubic spline using the software package ARSTAN [START_REF] Cook | A time series analysis approach to tree ring standardization[END_REF]. The program was used with a double detrending; (1) a negative exponential or a straight line and (2) a cubic spline function with a 50% cutoff wavelength. The standardization process allows age related tendencies to be removed from the growth curve to obtain a homogenized variance. The resulting data consists of a dimensionless, standardized index with variations representing the effect of stress factors (climate, insect herbivory, etc.) on tree growth. Once ringwidth series were accurately crossdated and standardized, elementary series were averaged for each tree and master chronologies were constructed for Font-Blanche by averaging indexed tree series. Only the 1995-2010 period from these indexed series were used for the purpose of this study (see complete chronology in Figure 4).

-Statistical analysis

Statistical analyses were computed with ADE4 [START_REF] Thioulouse | ADE-4: A multivariate analysis and graphical display software[END_REF] and R software (R Development Core Team, 2004). At first, a principal components analysis (PCA) was performed to provide an overall assessment of the role of time (year of measurement) and the correlation between ramifications (Ra), fructification (Fr), shoot length (SL), number of needles, number of growth units = number of cycles per year (GU) and ring width (RW) for the period 1995-2010 and for strong axes. Then, relationships between each of these variables and climate were investigated using a partial least square (PLS) regression. PLS regression was chosen because it handles many variables with relatively few observations [START_REF] Cramer | Cross-validation, bootstrapping, and partial leastsquares compared with multiple-regression in conventional qsar studies[END_REF] and deals with correlated variables [START_REF] Wold | PLS for multivariate linear modelling[END_REF]. The number of significant PLS components was chosen by a permutation test [START_REF] Good | Permutation tests[END_REF] with a 5% threshold for the explained variance. Variables were tested with a 1000-step cross-validation [START_REF] Amato | Bootstrap-based (Q)over-cap(kh)(2) for the selection of components and variables in PLS regression[END_REF]: they were retained only when the confidence interval (p<5%) for their partial correlation coefficient excluded zero. Climatic monthly parameters tested in the regression were rainfall (P), maximum temperature (MaxT), minimum temperatures (MinT) and mean temperatures (MT) from January of year (n-1) to November of year (n) over the period 1995-2010, according to the phenology of this species in South-eastern France [START_REF] Orshan | Plant pheno-morphological studies in Mediterranean type ecosystems[END_REF][START_REF] Serre | Les rapports de la croissance et du climat chez le pin d'Alep (Pinus halepensis (Mill)) II L'allongement des pousses et des aiguilles, et le climat Discussion Générale[END_REF]. The low number of observations ( 16) and high inter-annual variability of climate made grouping monthly climatic parameters necessary to obtain significant variables. We grouped the successive months having same signs for their individual partial correlation coefficients (sum of precipitations, average temperature). To compare exposition, position, status and vigour classes for each growth variable, normality was checked using a standard Shapiro-Wilks test. When the distribution was normal, a variance analysis and a multiple comparison test were performed to look for significant differences globally and further compare the different classes two by two. When the distribution was non-normal, these comparisons were performed respectively by a Kruskall-Wallis test and a Neymenyi test [START_REF] Nemenyi | Distribution-free multiple comparisons[END_REF]

-Results

For the period 1995-2010, the first PCA axis explained 58% of the variance and was strongly correlated with the number of growth units, ramifications, shoot length, ring width and the number of needles (Figure 5). This axis was clearly related to primary and secondary growth. The second PCA axis explained 17% of the total variance and was correlated positively with fruiting and negatively with needle number. In the PCA plane, years were represented as observations which allowed discriminating three groups. Top-level, south-exposed, principal and vigorous axes were more ramified than low, north-exposed, weak and frail axes respectively. (Table 1 and Figure 6a). The number of ramifications peaked in 1995, 1997, 2003 and 2009 for vigorous axes, but these peaks did not appear on medium vigour and frail chronologies which interannual variations were far smaller. However, the low value of 1999 as well as the break after 2003 appeared for each vigour class. Vigorous axes branching rate progressively recovered from 2007 to 2010. In opposition, medium axes, showed no sign of recovery before 2009 and frail axes no increasing trend after 2004. A significant decreasing trend was observed from 1995 to 2010 for vigorous, medium vigour and frail axes (r 2 = 0.34, 0.60 and 0.71 respectively for p < 0.01). But there were no trend before 2004 nor from 1999 to 2010 for strong and medium vigour axes, showing that the global trend was mainly due to the break after 2003. Total annual shoot length followed globally the same patterns than the branching rate and showed the same differences between classes of position, status and vigour (Table 1 andFigure 6b). However, medium vigour axes did not recover after 2004 and remained at a very low level close to that of frail axes up to 2010. A significant decrease occurred since 1995 whatever the position, the exposition or the status. When standardized, all shoot length chronologies followed a very similar pattern, with maximal values in 1996, 1998, 2000 and 2003, and The number of needles (Table 1 andFigure 6c) was different between all vigour classes, and also between position and status classes except for low vs middle crown and S-Stong vs Principal axes. At the opposite of all other primary growth variables, it was significantly different between north and south exposed branches. Needle number was stable up to 2003 for the three vigour classes. After a peak in 2003, particularly pronounced on strong axes, it dropped sharply to reach the lowest value in 2008. It increased for the three vigour classes in 2009 and 2010, but rose again to pre-2003 values only for strong axes. Needle life-span was longer on weak and medium vigour axes (respectively 45% and 25% of shoots with 3 year-old needles) than on vigorous axes (10%), the proportion of surviving needles for weak and medium vigour twigs in the third year being highly variable (average 40%, median 30%, range 5-100%). A small proportion of frail axes (8%) kept part of their needles 4 years.

The number of growth units was significantly different at each position, for each status and for each vigour class: high, principal and vigorous axes produced more cycles than low, weak and frail axes respectively (Table 1 andFigure 6d). For the period 1995-2010, the number of growth units followed exactly the same trends as the number of branches and peaked at the same time, with only two slight differences: (i) the recovery was a bit faster for the number of growth units on vigorous axes, reaching pre-2003 values since 2008 and a peak equivalent to the maximum values of 2003 and 1997 in 2009;

(ii) the drop of the number of growth units after 2003 on medium vigour axes was more abrupt and the loss more important. Since 2005, frail axes are almost only monocyclic, and each year only two fifth of medium vigour axes are polycyclic (maximum 2 cycles). The increasing number of growth units from 2008 to 2010 come mainly from late polycyclism appearing in October or November, and giving very short shoots (5-20 mm) often needleless or with abnormally short needles (4-20 mm).

Fructification was found on the most vigorous axes and particularly on principal axes (Table 1 Figure 7 shows the relationships between shoot length and climatic variables (total monthly precipitation and mean monthly temperature) for vigorous shoots. Medium vigour and frail shoots as well as other position and status classes where investigated and showed very similar patterns. However, PLS regression models and climatic variables were sometimes less significant.

Figure 7, Significant climate parameters R 2 and p values in PLS regression (one significant component) for shoot length on vigorous axes.

Successive months having same signs for their individual partial correlation coefficients were grouped (sum of precipitations, average temperature). Minus signs (-) stand for months of the preceding year and the '+' stands for months of the current growing year. Prec: monthly precipitations; MT: mean monthly temperatures.

Figure 8 presents the significant explanatory climate variables in the PLS regressions for all studied parameters. Globally, except partially fructification, the response of secondary and primary growth parameters is always positive to rainfall and negative to temperature. Significant climatic variables can be grouped into five periods: rainfall and temperature preceding and accompanying bud formation before the summer drought of previous year (year [n-1]; period 1) are critical for all parameters but fructification, as are rainfall of august and fall [n-1] (period 2). Temperatures of this second period influence mainly growth parameters and less branching rate. Winter rainfalls as a whole (period 3) increase the number of growth units of the following year. February rainfalls influence ring width and fructification. Spring rainfalls of current year (period 4) are important for ring width and fructification, and spring temperature for shoot elongation. Finally, rainfall of period 5 including august and fall of year [n] act upon all growth parameters. Ring width responds to temperatures of this last period. Overall, the dominant role of climatic conditions of the previous year [n -1] for primary growth, particularly rainfall but also mean temperature, must be underlined. Previous year is as important as present year for crown development. The rate of male flowering from 2007 to 2010 was respectively 32%, 42%, 60% and 41% of all measured twigs. The length of male flowers on twigs was respectively 3.7 (±1.6), 5.3 (±2.0), 7.1 (±2.6) and 7.7 (±3.8) mm. It was each year 15% longer for S-strong and principal axes than on S-Weak and frail twigs, reaching up to 25 mm on secondary axes of the middle and top crown. In the low crown, male flowers appeared mainly on the principal axes and on vigorous secondary twigs. In the top crown, male flowers developed mainly on secondary and medium vigour twigs. In the middle of the crown, male flowers appeared on any shoot except the most vigorous, generally devoted to female flowers. The less vigorous twigs rarely gave male flowers, except in 2009. The exceptional male flowering rate in 2009 was due to the fact that many S-strong shoots of the top crown flowered, as well as frail shoots of the low and middle crown. The natural trend of decrease of the parameters with age proved to be weak for branch length growth (30 mm/10 years) and branching rate (0.45/10 years), and non significant for the other variables. For the 16 studied years, all observed trends were sharper than the natural one. For shoot length, observed values were 15-40% under the natural trend during the 2004-2007 period

-Discussion

-Relationship between primary growth and climate

The PCA plane showed that Aleppo pine primary and secondary growth severely suffered from an exceptionally dry and hot period from 2003 to 2007 in southern France. Shoot length, number of growth units and branching rate are closely interrelated explaining partly why they appeared clustered on the PCA plane. They are also correlated with needle number. Indeed, each new cycle for a given year on a branch adds length, new branches and usually needles. These four parameters showed the same decreasing patterns after 2003 and partial recovery from 2007 to 2010. Although the series is shorter, the trend of needle length is consistent with the general trend of previous parameters, with low values from 2003 to 2006 and a significant recovery from 2007 to 2010. Our PLS results suggest that high water availability whatever the season in previous and present year improves tree health and vigour and therefore primary growth, through the induction of many cycles in buds or their elongation. Abundant winter rainfall specifically improves tree health condition at the beginning of the following growth season [START_REF] Borchert | Induction of rehydration and bud break by irrigation or rain in deciduous trees of a tropical dry forest in Costa-Rica[END_REF] and helps the development of spring flushes, and probably the induction of later flushes. The earlier water stress is released at the end of summer, the longer and more developed the late flush. In the last 12 years, drought generally lasted up to the middle or the end of September. Thanks to very high temperatures in fall, late polycyclism occurred after 2007 but rarely before mid or end of October [START_REF] Vennetier | Adaptation phénologique du pin d'Alep au changement climatique[END_REF], too late for the shoots to fully develop and bear significant needles. From 1995 to 2000 and in herbarium samples, most of late flushes are short but bear normal needles. At Font-Blanche, the higher number of growth units after 2008 compared to previous years comes from such late flushes, increasing at the same time branching rate. Despite an additional flush, the 2008 total shoot length is shorter than in 2007, as this late flush was unfinished. This poor growth performance in 2008 is probably the delayed impact of 2007 drought, added to a severe (four months) and early summer drought in 2008. The correlation between number of growth units and annual shoot length which was good in the past should decrease in the future due to recurrent delayed late flushes, as already observed in 2009-2010, shoot length increasing slower than the number of growth units. The slow recovery or stagnation of needle number and shoot length for medium vigour axes in 2007-2010, compared to strong axes, may be partly due to the increasing male flowering rate and length. At the base of the first annual flush, each male flower takes the place of a needle, so that the higher the flowering rate and the flowering length, the fewer needles are formed on a given tree or shoot. Moreover, male flowering uses a lot of resources, so that flowering shoots are usually monocyclic [START_REF] Vennetier | Adaptation phénologique du pin d'Alep au changement climatique[END_REF].

High temperature in all seasons and whatever the year are detrimental to primary growth. In the context of global warming, higher temperatures mean higher water stress throughout the growth season. In previous studies in the same area (Thabeetet al., 2009;Vilaet al., 2008), high temperatures in fall and winter had positive coefficients in climate/ring width relationship models for the 20 th century, which was interpreted as the favourable effect of an earlier and longer growing season. In this research, the negative correlation or non significant coefficients for temperature in these periods show that critical thresholds were probably passed: average temperatures were always high enough at the end of winter and fall to induce an early budburst or a late flush, but in fact too high and led to unfavourable stress conditions. Moreover, high temperatures at the very end of fall induce fragile very late flushes, increasing branching rate but also the risk of abortion due to frost in winter [START_REF] Vennetier | Adaptation phénologique du pin d'Alep au changement climatique[END_REF]) and therefore to a waste of resources. A short photoperiod and low solar energy limit the potentially favourable effects of higher temperature in winter: whereas CO 2 net uptake could occur in winter with mild weather conditions [START_REF] Holst | Impacts of summer water limitation on the carbon balance of a Scots pine forest in the southern upper Rhine plain[END_REF], the reserves used in this period to maintain a physiological activity [START_REF] Damesin | Respiration and photosynthesis characteristics of current-year stems of Fagus sylvatica: from the seasonal pattern to an annual balance[END_REF] are no longer available in spring. At the same time, the succession of hot periods and frost leads to reversible cavitation in the vessels, decreasing their conductivity [START_REF] Cruiziat | Hydraulic architecture of trees: main concepts and results[END_REF]. This may explain why the lack of cold in winter can delay budbreak [START_REF] Falusi | Geographic variation and bud dormancy in beech seedlings (Fagus sylvatica L)[END_REF].

Pre-induction of the primary growth characteristics of several flushes in buds makes primary growth mainly dependant on the global health status of trees and branch vigour at the time of bud formation. Tree vigour and health are rarely changing annually, except in case of extreme events or accidents. A healthy tree has reserves to withstand a difficult period, and a weak tree recovers only slowly even in case of good climate conditions. Such an autocorrelation is well known in ring width series [START_REF] Bigler | Drought as an inciting mortality factor in Scots pine stands of the Valais, Switzerland[END_REF][START_REF] Dekort | Ring Width, Density and Wood Anatomy of Douglas-Fir with Different Crown Vitality[END_REF][START_REF] Schweingruber | Tree rings and environment: Dendroecology[END_REF] and is logically found for primary growth in this study. Year 2001 low values for all variables correspond to the snow break added to a long summer drought: at Font-Blanche more than 30 cm of heavy snow overloaded branches and damaged most of the crowns, slowing spring development. The summer drought prevented a late polycyclism to compensate for the reduced leaf area. The delayed effects of this event probably limited the crown development in year 2002 which was climatically favourable (nearly no summer drought), but shows only medium values.

Even if 2003 was the most severe heat wave ever recorded, it was quite a good year for Aleppo pine primary growth because (1) 2002 was very favourable, preparing trees to a good following year, (2) the growth season started early thanks to a warm end of winter, and (3) the extreme temperatures started in May, when all spring flushes were already growing, water reserve in the soil allowing trees to develop normally these shoots. New needles remained very short in 2003 as half of their length is gained during summer [START_REF] Vennetier | Adaptation phénologique du pin d'Alep au changement climatique[END_REF]), but trees lost most of 2 year-old needles. The impact of 2003 heat wave was mainly visible from 2004 on primary and secondary growth, due to the delayed impact of a reduced leaf area (few and short needles) and bad health condition. As a whole, the partial or total recovery of all primary growth parameters from 2007 to 2010, as well as the very abrupt fall after 2003, clearly showed that this trough was not due to branch ageing.

-Fruiting

For Aleppo pine, cone production is strongly climate-and vigour-dependent; as no cycle of intense mast years was ever documented in literature. Our PLS models were not significant for frail and medium vigour axes. On such twigs, fruiting was too rare or absent added to high cone abortion rates, reproduction effort being mainly devoted to male flowers as already described by [START_REF] Thabeet | Response of Pinus sylvestris L. to recent climate change in the French Mediterranean region[END_REF] for Pinus silvestris and confirmed by Ne'eman et al. (2011) for Aleppo pine in Israel. Fructification success depends on the number of cones initially formed, and the development and survival rates of these cones. Cone initiation is supposed to be entirely predetermined in the terminal bud during the previous year, and then depends solely on climatic conditions and tree health status in year [n-1]. Cone development and survival depend on current year conditions and on twig ability to feed it. On the PCA plane, years 2002 and 2007 are marked out by abundant fruiting because timing was excellent: abundant precipitations before and after flowering (2002, 60% more rainfall) and low during flowering (2007, 65% less rainfall). According to [START_REF] Serre | Les rapports de la croissance et du climat chez le pin d'Alep (Pinus halepensis (Mill)) II L'allongement des pousses et des aiguilles, et le climat Discussion Générale[END_REF], male flowering begins between mid-March and April, whereas female flowering is delayed two to three weeks later in South-eastern France. Low rainfall and higher temperatures during male flowering (March-April) may allow better pollen dispersion (Pardoset al., 2003) and then, higher fructification. Year 2003 was a good illustration of excellent female flowering due to the favourable climate of previous year, followed by high abortion rates because of current year's bad conditions. Fructification collapsed from 2004 to 2005, due to both low female flowering rates induced by year [n-1] climate and abortion due to the lack of available resources [START_REF] Climent | To grow or to seed: Ecotypic variation in reproductive allocation and cone production by young female Aleppo pine (Pinus halepensis, Pinaceae)[END_REF]. This reduction of the reproduction potential of Aleppo pine with severe drought is consistent with a the reduction of this potential along a spatial aridity gradient observed for this species in Tunisia by [START_REF] Ayari | Geographical variation on cone and seed production of natural Pinus halepensis Mill. forests in Tunisia[END_REF]. A high acorn abortion rate was also observed with decreasing rainfall for Quercus ilex in a Mediterranean area [START_REF] Misson | Phenological responses to extreme droughts in a Mediterranean forest[END_REF]

-Ring width

When the 16-year long ring width chronology is placed in the whole chronology for Font-Blanche (1890-2008), the impact of the 2003-2007 episode is marked out by the worse trough by far ever caused by heat or drought. Whatever the study (Nicaultet al., 2001;Vilaet al., 2008) no accident except deep frosts ever led to so low indices or such a long lasting depression in Aleppo pine ring width. The impact of the most severe droughts of the 20 th century was generally rubbed out after only one year. This confirms that critical thresholds were recently passed in terms of drought intensity and repetition, and should be again at stake in the future. This threat is illustrated by the unprecedented decrease of ring width observed in Greek islands in the two last decades after repeated drought and heat waves [START_REF] Sarris | Impact of recent climatic change on growth of low elevation eastern Mediterranean forest trees[END_REF] .

-Primary growth, global leaf area and physiology

Globally, from 2003 to 2008, Aleppo pines branching rate was low, limiting the number of active axes, each axis bore less needles and needles were abnormally short. The potential photosynthesis was drastically cut and may be significantly hold back for several following years due to (1) the 2-3 year life span of these short needles (2) the impossibility for pines to develop new twigs on 2 and more year-old axes, so that the lack of active twigs cannot be rapidly compensated, (3) primary growth is related to tree health so that weak trees recover only progressively a normal crown development. Such a reduction of the leaf area (LAI) at stand scale with increasing aridity is confirmed for Aleppo pine by [START_REF] Vicente-Serrano | Aridification determines changes in forest growth in Pinus halepensis forests under semiarid Mediterranean climate conditions[END_REF] in Spain during the last decades, and for Quercus ilex (L.) in a rain exclusion experiment in Southern France [START_REF] Misson | Leaf physiological responses to extreme droughts in Mediterranean Quercus ilex forest[END_REF]. This last study showed that repeated and severe drought may changes in the relationship between photosynthetic parameters and water stress, decreasing photosynthesis efficiency. This low photosynthetic potential may contribute to a slower than expected recovery of tree growth, particularly if trees allocate most of their rare resources to recover an adequate photosynthetic biomass, and to reproduction, as observed from 2008 to 2010 in our plot. A high level of LAI deficit may also contribute to delayed forest die-back through carbon starvation after climate accidents [START_REF] Allen | A global overview of drought and heat-induced tree mortality reveals emerging climate change risks for forests[END_REF]. During repeated drought and heat, Aleppo pine showed a reduced cone production for a few years. Such periods are favourable to wildfires. As Aleppo pine depends entirely on it cones to regenerate after fire, climate change may hamper its regeneration potential in the mediterranean area.

-Conclusion and prospects

Similar studies on other tree species are required to confirm the coherent results obtained on the impact of climate change on Pinus halepensis and P. sylvestris primary growth and crown development. Such results should be considered to improve functional models using leaf area indices and carbon allocation to various tree compartments (primary/secondary growth) and tree architectural development models, integrating crown development interannual variability.

- ---------

Figure 1 ,

 1 Figure 1, Location of the study area in southern France. Grayscale colors represent altitude whereas black zones represent main towns.

Figure 2 ,

 2 Figure 2, Average annual precipitations (bars) and temperatures (black line) in Font-Blanche since 1995. Horizontal lines are 1961-2008 averages: doted line = mean temperature, grey line = mean rainfall).

Figure 3 ,

 3 Figure 3, Schematic representation of shoot sampling. Principal axis, S-Weak and S-Strong axes of five years-old and those of 10 years-old are represented[START_REF] Girard | Polycyclism, a fundamental tree growth process, decline with recent climate change. The example of Pinus halepensis Mill. in Mediterranean France[END_REF].

Figure 4 ,

 4 Figure 4, Aleppo pine tree-ring indexed chronology at Font-Blanche for the period 1961-2008. Gray periods represented severe drought and snowflakes represents severe frosts.

  a) Years 2001 and 2004 to 2008 were characterized by poor radial and shoot growth and crown development, 2005 and 2006 being the worst. b) At the opposite, 1995 to 1998, 2000, 2002 and 2003, were good years for primary and secondary growth, 1995 and 2003 being the best. c) 2009, followed by 1999 and 2010, allowed intermediate growth and crown development. Among these groups, variations along axis 2 were due mainly to the differences in fruiting rate and secondarily to contrasted needle number. 2010 was very close to the middle of the plane.

Figure 5 ,

 5 Figure5, PCA main plane (axes 1 and 2), 1995-2010 period, for every axes using six variables: ramifications (Ra), fructifications (Fr), shoot length (SL), number of needles (NN), number of growth units (GU) and ring-width (RW). The first two axes explained respectively 58 and 17% of the total variance.

Figure 6 ,

 6 Figure 6, Annual number of branches (a), number of needles (b), fructification (c), shoot length (d) number of cycles (e) and needles length (f) as function of axe vigour (frail, medium, vigorous) at Font-Blanche for the period 1995-2010.

  an abrupt fall after 2003 with a minimum in 2005 and 2006. Except for frail axes, victims of a first break in 1998, no trend could be seen before 2003.

Figure 8 ,

 8 Figure 8, Significant climate parameters R 2 and p values in PLS regression (one significant component) for shoot length, ramifications, number of needles, number of growth units, fructifications and ring-width on frail, medium vigour and vigorous axes. Black squares are significant at p=0.001, dark grey at p=0.01 and light grey at p=0.05. Signs (+ or -) indicate a positive or a negative relationship. P, precipitation, MT, mean temperature.

Table 1 :

 1 Differences in the number of branches, number of needles, fructification, shoot length, and number of cycles computed using a Neymeni test. Results are shown for axe position(low, middle, top), axe exposition (north vs south), axe status (principal, S-Strong, S-Weak) and axe vigour(frail, medium and Vigorous). Figures in bold correspond to significant differences (value superior to the Nemenyi threshold).

		Number of branches	Number of needles	Number of fructifications	Shoot length	Number of cycles
	Axe position
	Low < middle	3.281	0.224	0.608	3.867	2.782
	Low < top	5.866	4.267	0.736	7.205	6.592
	Middle < top	2.994	4.727	0.286	3.929	4.491
	Neymeni threshold	2.394	2.394	2.394	2.394	2.394
	Axe exposition
	North < South	0.001	3.912	0.343	0.521	0.262
	Neymeni threshold	2.638	2.638	0.96	2.638	2.638
	Axe status
	S-Weak < S-Strong	2.416	8.835	3.095	6.592	6.514
	S-Weak < Principal	6.968	6.211	5.147	4.061	4.205
	S-Strong < Principal	10.159	2.225	2.555	11.249	11.33
	Neymeni threshold	2.394	2.394	2.394	2.394	2.394
	Axe vigour
	Frail < Medium	6.684	4.89	1.423	4.638	4.881
	Frail <Vigorous	9.52	8.458	5.633	9.867	8.36
	Medium <Vigorous	3.867	4.138	4.183	12.831	11.922
	Neymeni threshold	2.394	2.394	2.394	2.394	2.394

Table 2 : Mean needles length per year at Font-Blanche between 2003 and 2010.

 2 and Figure6c), mainly in the top and middle of the crown up to 2003 and after 2008, but only in the top canopy from 2004 to 2008. Medium vigour and S-strong axes give three to four times less cones per year than vigorous and principal ones, and nearly only in the top crown. However, due to the high variability and generally low values, no significant difference was found between top, middle and low crown whatever axes vigour and status. Frail axes nearly never give female flowers. In many cases from 2003 to 2007, medium vigour axes which gave a cone became frail as they were unable to feed both the cone and the shoot. No general pattern of fruiting was found at Font-Blanche. On vigorous axes, minimum values were observed in 1999 and 2005, with breaks following 1998 and 2003, lasting one and two years respectively, and peaks in1997, 2000, 2002 and 2007 and 2009. Medium vigour and frail axes follow the same pattern. Cone abortion rate reached its maximum during the2003-2006 period (40-60%), and remained high (20-40%) except on the most vigorous axes up to 2008. annual shoot length = 105 ± 33 mm) and 65 ± 11 mm for all other twigs, bearing male flowers and generally frail (no polycyclism, mean annual shoot length = 36 ± 24 mm). In Font-Blanche, average needle length for vigorous and frail axes with similar shoot length were respectively 35% and 25 % smaller in the last six years, and even for the longest ones and the best years hardly reached the minimum value cited in all plant guides (60 mm) or measured in botanical collections (65 mm).

	The average needle length at Font-Blanche since 2003 was 57.6 mm, (SD 9.1 mm) in the best case i.e.
	vigorous, principal and top crown axes (Table 2 and figure 6f). Maximum annual values (67.5 mm)
	were reached on these axes in 2010, slightly above the levels of 2004 and 2008 (65 mm). For medium
	and frail axes, needle length was shorter: 50.5 ± 3.8 mm and 49.3 ± 4.1 mm respectively. No
	retrospective study before 2003 was possible due to the short needle life-span on adult trees. This life
	span was particularly reduced for 2003 needles (<2 years): none of them were found in spring 2005 on

vigorous and medium vigour axes, a few ones remaining only on frail axes. Nevertheless, several observations were possible. The average needle length slowly increased from 2003 to 2010, with a relapse in 2006, this fall being more drastic on vigorous than on other axes. Samples from the herbarium of Provence University showed an average length of 73 ±17 mm as a whole. It ranged from 91 ± 20 mm for main, vigorous and generally polycyclic twigs bearing female flowers (7 samples, mean

Acknowledgments

We would like to thank Christian Ripert, Roland Estève, Willy Martin, Aminata N'Diaye Boucabar, Frederic Faure-Brac and Maël Grauer for their assistance in the field and laboratory work, as well as Régine Verlaque and Bruno Vila, curators from Marseille University herbarium, who contributed material to control needle length. This research was funded by the French National Research Agency (DROUGHT+ project, N° ANR-06-VULN-003-04), the French Ministry for Ecology, Energy and Sustainable Development (GICC -REFORME project, n° MEED D4E CV05000007), the Conseil Général des Bouches-du-Rhône (CG13), ECCOREV Research Federation (FR3098), the "F-ORE-T" LTER network and Cemagref.


Róbert Busa-Fekete

György Szarvas

Tamás Éltető
email: tamas.elteto@ericsson.com

Balázs Kégl
email: bal-azs.kegl@gmail.com

An apple-to-apple comparison of Learning-to-rank algorithms in terms of Normalized Discounted Cumulative Gain

The Normalized Discounted Cumulative Gain (NDCG) is a widely used evaluation metric for learning-to-rank (LTR) systems. NDCG is designed for ranking tasks with more than one relevance levels. There are many freely available, open source tools for computing the NDCG score for a ranked result list. Even though the definition of NDCG is unambiguous, the various tools can produce different scores for ranked lists with certain properties, deteriorating the empirical tests in many published papers and thereby making the comparison of empirical results published in different studies difficult to compare. In this study, first, we identify the major differences between the various publicly available NDCG evaluation tools. Second, based on a set of comparative experiments using a common benchmark dataset in LTR research and 6 different LTR algorithms, we demonstrate how these differences affect the overall performance of different algorithms and the final scores that are used to compare different systems.

Introduction

In subset ranking [START_REF] Cossock | Statistical analysis of Bayes optimal subset ranking[END_REF] (or web page ranking), the goal is to learn a ranking function that approximates the ideal partial ordering of a set of objects (or documents retrieved for the same query). The partial ordering is provided by relevance labels representing the relevance of documents with respect to the query on an absolute scale.

In the past, manually designed ranking functions, such as BM25 [START_REF] Robertson | The probabilistic relevance framework: BM25 and beyond[END_REF], were used to rank the retrieved documents in web page ranking. More recently, this problem is tackled as a machine learning task, where the training data is given in the form of (query, document, relevance label) triplets. These machine learning based ranking approaches are referred to as learning-to-rank (LTR) systems.

The Normalized Discounted Cumulative Gain (NDCG) is a widely used evaluation metric for learning-to-rank systems. NDCG is designed for ranking tasks with more than one relevance levels. There are many freely available, open source tools for computing the NDCG score for a ranked result list (for full list of these tools see Section 3). Even though the definition of NDCG is unambiguous 6 , the various tools can produce different scores for ranked lists with certain properties, deteriorating the empirical tests in many published papers and thereby making the comparison of empirical results published in different studies difficult to compare.

We found that for certain benchmark datasets, the relative order of the performance of different LTR methods can change depending on which evaluation tool was used. The reason for this can be two-fold. First, the implemented NDCG calculation can itself result in different scores, and in some cases, a different order for the same ranked result lists. Second and more importantly, we found some of the LTR algorithms that compute the NDCG scores during the training phase to be sensitive to the different ways of computing the NDCG score, i.e. depending on which NDCG implementation is used, some LTR methods can produce different models that provide significantly different overall performance. In previous work [START_REF] Busa-Fekete | Tune and mix: learning to rank using ensembles of calibrated multi-class classifiers[END_REF], we recognized this, here we investigate the effect of using different evaluation tools more detailed.

The contribution of this study is two-fold. First, we identify the major differences between the various publicly available NDCG evaluation tools. Second, based on a set of comparative experiments using a common benchmark dataset in LTR research and 6 different LTR algorithms, we demonstrate how these differences affect the overall performance of different learning methods and the final scores that are used to compare different systems. We analyze these differences and also draw conclusions on which metric should be used and why. This is not only an important step towards making the empirical research results reported in the literature uniformly comparable, but it also has implications on how benchmark datasets should be designed in order to be equally suited to train and evaluate any particular learning to rank algorithm.

The rest of this paper is organized as follows. In the next section we will describe the formal setup, and then, in Section 3 we list all the evaluation tools that are available freely to compute NDCG and, in addition, we identify precisely the differences in the way they compute the NDCG score. Next, in Section 4 we briefly overview the LTR algorithms we are used for comparison in the experiments in Section 5. Finally, based on the observed differences in experimental results, we draw our conclusions in Section 6.

In this section we formally define the learning-to-rank problem and introduce the notation that will be used in the rest of the paper. Let us assume that we are given a set of query objects D = {D (1) , . . . ,

D (M ) }. Each query object D (k) consists of a set of n (k) pairs D (k) = n `x(k) 1 , ℓ (k)
1 ´, . . . ,

`x(k) n (k) , ℓ (k)
n (k) ´o.
The real-valued feature vectors x (k) i represent the kth query and the ith document received as a potential hit for the query. 7 The label index ℓ (k) i of the query-document pair x (k) i is an integer between 1 and K. We assume that we are given a set of numerical relevance grades Z = {z1, . . . , zK }.

The relevance grade z

(k) i = z ℓ (k) i
expresses the relevance of the ith document to the kth query on a numerical scale. A popular choice for the numerical relevance grades is

z ℓ = 2 ℓ-1 -1 (1)
for all ℓ = 1, . . . , K.

The goal of the ranker is to output a permutation j (k) = (j1, . . . , j n (k) ) over the integers (1, . . . , n (k) ) for each query object D (k) . A widely used empirical measure of the quality of the permutation j (k) is the Discounted Cumulative Gain (DCG)

DCG `j(k) , D (k) ´= n (k) X i=1 ciz (k) j i , (2)
where ci is the discount factor of the i th document in the permutation. The most common discount factor is

ci = 1 log(1 + i) . (3)
The rationale of this formula is that a user will be particularly satisfied if he/she finds relevant documents early in the permutation.

To normalize DCG between 0 and 1, (2) is usually divided with the DCG score of the best permutation (NDCG) which can be computed as IDCG `D(k) ´= max j DCG `j, D (k) This score referred to as the ideal DCG score. It is also a common practice to truncate the sum (2) at nmax, defining the DCGn max and NDCGn max scores. The reason for this is that a user would rarely browse beyond the first page of search results containing the first nmax hits.

Evaluation tools

In this subsection we briefly describe and compare the various tools available to compute NDCG scores. The definition (2) is unambiguous, nevertheless, the tools can differ in the definition of the discount factor ci (3). More importantly, there can be an important difference in the way the DCG score is normalized when i) there is no relevant document for a query (zi = 0 for all i), or ii) when the number of documents is less then the truncation level nmax. Although this seems to be a technical subtlety, it turns out that the confusion between the different tools can significantly alter the numerical scores and in some case can even change the relative ordering of the algorithms on benchmark datasets.

We compared six evaluation tools computing the NDCG scores:

1. The LETOR 3.0 script implemented in Perl 8 2. The LETOR 4.0 script implemented in Perl 9 3. The MS script implemented in Perl10 4. The YAHOO script implemented in Python11 5. The RANKLIB package implemented in Java12 6. The TREC evaluation tool v8.1 implemented in C13

The evaluation tools can be divided into three groups. The tools of the first group compute DCGn max according to the definition (2) described in Section 2. The LETOR 3.0, YAHOO, and TREC tools belong to this group. All of these tools assign zero score to a query if it is empty, that is, zi = 0 for all i which means that there are no relevant documents. The TREC tool uses the labels of documents given in the input file as relevance grades by default. From this point of view, this is the most flexible implementation, since arbitrary relevance grades can be defined. For example, in the case of MQ2008 dataset, the labels 0, 1 and 2 should be simply replaced by 0, 1 and 3 respectively, to have the commonly used exponential grades [START_REF] Burges | Learning to rank using gradient descent[END_REF].

The second group is composed of the YAHOO tool alone. It also computes the DCGn max according to the definition ( 2), but it assigns 1.0 to the empty queries. This is a minor difference that generates an additive bias between the NDCGn max computed by YAHOO tool and the three tools of the first group.

The third group consists of the LETOR 4.0 and MS tools. Except for a small technical difference (the LETOR 4.0 tool can be used for up to three relevance labels, whereas the MS tool can handle up to five relevance labels), they compute the same score. As the RANKLIB and LETOR 3.0 tools, they assign zero to a query where the ideal DCGn max is zero. Their rather strange feature is that they also assign zero DCGn max score to a query with less then nmax documents in it, even if these documents are highly relevant. So, formally, they compute the DCGn max score as DCGn max `j(k) , D (k) ´= (

P nmax i=1 ciz (k) j i if nmax ≤ n (k) 0 otherwise. (4)
This truncation does not only distort the test score, but it can also alter the training of such algorithms that depend directly on the NDCG score. Indeed, for example, in ADARANK [START_REF] Xu | AdaRank: a boosting algorithm for information retrieval[END_REF] which optimizes the NDCG10 evaluation metric, a query containing less than 10 documents does not influence the computation of the coefficient of the weak ranker at all, and the the weight of such queries converge to zero over the boosting iterations.

The only source of difference between the tools that have not been identified yet, is the way they sort the objects of interest based on the scores. All tools except TREC tool, make use of the default builtin, programing language dependent sorting function which mainly implement the quick sort algorithm. Therefore, the order of two elements with the same score depends on the implementation of the sorting algorithm used. Whereas the TREC tool uses the lexicographic ordering based on document ID given in the input file if the systempredicted scores are equal for two documents.

We believe that none of these two ways of handling equal scores are desirable. On the one hand, the sorting algorithm should not have an effect on the evaluation itself. On the other hand, the document ID normally does not bear any useful information regarding the content of a document, so in this sense, the use of the ordering based on document ID corresponds to a particular random ordering which is also not a desired feature in an evaluation tool.

Learning-to-rank algorithms

In our comparison study, we used six state-of-the-art ranking methods. Here we briefly summarize them.

1. ADARANK [START_REF] Xu | AdaRank: a boosting algorithm for information retrieval[END_REF] is a listwise boosting approach aiming to optimize an arbitrary listwise IR metrics, such as the Mean Average Precision (MAP), ERR, or NDCG. Inspired by ADABOOST, it uses a stepwise greedy optimization technique for maximizing the chosen IR metrics. In every boosting iteration, ADARANK reweights the queries based on their scores obtained by the evaluation metrics: it up-weights the query having lower score and down-weights high-scoring queries. The weak learner is chosen by optimizing the listwise evaluation metrics of interest which is usually hard to optimize except for very simple weak classifiers. This can be viewed as a handicap of this method. According to the original implementation of ADARANK, we used the best feature ranker (BF) described above as base ranker taking into account the weighting of queries. The only hyperparameter of ADARANK is the number of boosting iterations which we optimized by using early-stopping on the validation set. We refer to this method as ADARANK.{NDCG}. 2. RANKNET [START_REF] Burges | Learning to rank using gradient descent[END_REF] is a neural-net-based method which employs a loss based on pairwise cross entropy as its objective function. The neural net with one output node is trained to optimize directly the differentiable probabilistic pairwise loss instead of the common squared loss. We validated the number of hidden layers ranging from 1 to 3 and the number of neurons in the hidden layers ranging from 10 to 500. For the number of training epochs we applied early stopping. 3. RANKBOOST [START_REF] Freund | An efficient boosting algorithm for combining preferences[END_REF] is a pairwise boosting approach. The objective function is the rank loss (as opposed to ADABOOST which optimizes the exponential loss). In each boosting iteration the weak classifier is chosen by maximizing the weighted rank loss. For the weak learner we used decision stumps and a variant of the single decision stump described in [START_REF] Freund | An efficient boosting algorithm for combining preferences[END_REF] which is able to optimize the rank loss in an efficient way. 4. RANKSVM [START_REF] Herbrich | Large margin rank boundaries for ordinal regression[END_REF] is a pairwise method based on SVM, formulating the ranking task as a binary classification. We used linear kernel because the optimization using non-linear kernels cannot be carried out in reasonable time. The tolerance level of the optimization was set to 0.001 and the regularization parameter was validated in the interval [10 -6 , 10 4 ] with a logarithmically increasing step 5. COORDINATEASCENT (CA) [START_REF] Metzler | Linear feature-based models for information retrieval[END_REF] is a linear listwise model where the scores of the query-document pairs are calculated as weighted combinations of the feature values. The weights are tuned by using a coordinate ascent optimization method where the objective function is an arbitrary evaluation metrics given by the user. The coordinate ascent optimization method itself has two hyperparameters to be tuned: the number of restarts R from random initial weights, and the number of iterations T taken after each restart. We used R = 30 and T = 100. We did not validate these hyperparameters, but using the validation set we evaluated every model obtained due to restarting the optimization process, and we kept the one having highest performance. 6. LAMBDAMART [START_REF] Wu | Adapting boosting for information retrieval measures[END_REF] is a boosted regression tree model. Since it handles the LTR problem as a regression task, it could be classified as pointwise method, but during the training phase, it adjust the parameters of the regression trees based on the derivative estimate of NDCG, therefore it is considered as a listwise approach. We validated the number of boosting iterations. The number of leaves were set to 10 and the learning rate to 0.1.

Experiments

We identified two major differences in the way the DCG score is normalized in publicly available NDCG evaluation scripts:

1. When there is no relevant document for a query (zi = 0 for all i), evaluation scripts conforming to the definition assign a 0.0 NDCG value to the query. Optionally, some scripts may assign a different value as default for such queries (namely, 1.0 for the Yahoo metric). 2. When the number of documents is less then the truncation level nmax, evaluation scripts conforming to the definition assign an NDCG n (k) value equal to the number of documents n (k) available for that query (thereby assuming that it is always possible to fill in a result list with irrelevant documents, and at the same time assuming that the provided relevance labeling is exhaustive (i.e. there are no further, unseen relevant documents and thus the use of IDCG n (k) score in the normalization step is a realistic estimate. 14 ) Optionally, some scripts may assign a different value as default for such queries (namely, 0.0 for the LETOR metric).

We consider the difference stemming from the different strategy to order results with equal predicted scores less crucial (in practice two different documents seldom get the same predicted score, i.e. ties are rare) and we do not assess its impact, and we use all metrics with exponential relevance grades (the TREC tool can be parameterized to use exponential gain, while the other tools are implemented to use this). That is, in our experiments we consider the Ranklib and TREC evaluation tools to be equivalent and compare their scores to those provided by the YAHOO and LETOR metrics, which are representative examples for the two major differences we found between the various tools.

In Figure 1, we plot the NDCG1-10 scores for 4 characteristic learning to rank algorithms (ADARANK, LAMBDAMART, RANKNET and CA ) using the three different evaluation formulas to evaluate the models: in each row, the left, middle and right plots show the values provided by the RANKLIB/TREC, the YAHOO and the LETOR tools, respectively. All models were trained to optimize the NDCG10 scores, and the plots show the NDCG scores of these models for cut-off values from 1 to 10. Each plot shows 3 different curves that correspond to models trained using a specific metric during the training of the model: the blue, black and red lines indicate models learned using the YAHOO, RANKLIB/TREC and LETOR metrics, respectively. This way we can visually compare the effects (k) CA/TREC of the different evaluation tools both on on the numeric outcomes (comparing the curves with same color, denoting the same models, across the 3 plots horizontally) and on the learnt model (contrasting the 3 curves on the same plots, denoting models learnt using different measures).

We can make several observations based on figure1. First, for all the algorithms in figure 1, the NDCG scores provided by the YAHOO tool are on average around 0.25 higher than the scores provided by the RANKLIB/TREC tool which conforms the definition of NDCG. This difference comes solely from the presence of queries with 0 relevant documents in the dataset: for those queries, the YAHOO tool assign 1.0 as NDCG which results in an additive bias compared to the official NDCG scores. On the other hand, the scores provided by the LETOR tool degrade rapidly for cutoff values of 8 --10 for all algorithms, with a difference around or above 0.25 compared to the official NDCG scores. As opposed to the former additive difference this behavior might result in a different ordering of the algorithms: a method that is on average superior to others on larger sets but is weaker than the competitors on small sets is preferred by the Letor metric, as this tool assigns zero to small sets (smaller than the NDCG cut-off).

Second, and more interestingly, we can observe that it is by no means irrelevant which metric was used during the training of the algorithms: for those algorithms that make use of the NDCG scores in some way in the training process (namely, the ADARANK, LAMB-DAMART and CA methods), we see differences in the performance of the learnt models. We observe that in general, the LETOR tool is not suited for training the algorithms -its property to assign 0.0 score to small sets causes these small sets to be useless for training (to optimize NDCG10), i.e. the algorithms can exploit less data in a meaningful way to learn patterns. This results in a significantly 15 worse performance for these algorithms, when trained using the LETOR metric (instead of the official NDCG scores). On the other hand, for some algorithms, it is worth to assign a non-zero score to sets which contain no relevant documents at all: doing so, the algorithms do not increase the weight of such queries similarly to other low-performing queries where there is hope to improve performance. This can result in better learning rates and in some cases slightly better performance (see e.g. the blue vs. black curves for the ADARANK and LAMB-DAMART methods).

Overall, we can observe that the best metric to train the algorithms is the one provided by YAHOO: in general this results in equal or better learnt models than the other evaluation tools. To shed light on how these characteristics of the evaluation tools affect the rela- 15 The error bars on the plots correspond to the standard errors of querywise NDCG scores averaged out in quadrature over the folds.

tive performance of benchmark LTR algorithms, in figure 2, we plot the NDCG scores for 6 different learning algorithms (trained using the official NDCG metric) with all the evaluation tools surveyed in this study. As can be seen, the performance of the best algorithms is very close to each other, with Coordinate Ascent having a slight advantage regardless which metric is used for evaluation. Slightly worse than CA, the three algorithms RANKBOOST, RANKSVM and LAMBDAMART perform very close to each other. If we compare the results obtained with different evaluation measures, the relative order of these (otherwise very similar) models change depending on the metric: apparently the RANKBOOST algorithm has a slight advantage over the other two for short sets, which advantage disappears when the LETOR metric is used for evaluation. In general, we were unable to reproduce the competitive results of ADARANK [START_REF] Xu | AdaRank: a boosting algorithm for information retrieval[END_REF], using the reimplementation of the algorithm in the Ranklib package16 ,17 .

Conclusion

In this study, we reviewed the publicly available NDCG evaluation scripts, identified and compared the differences between them and systematically analyzed how these differences affect the numeric results and the training processes of various learning-to-rank algorithms. It is reasonable to assume that most previous studies use one of the assessed evaluation tools, and at the same time it seems likely that most measures are used at least in some studies to evaluate the performance of machine learnt ranking systems. We found that there indeed are differences between tools despite the relative simplicity of the popular NDCG evaluation metric, and our experiments demonstrate that these differences can easily lead to non-trivial differences between research results. Since most studies do not discuss such details as the evaluation script used, this fact makes previous studies very difficult to compare and might lead to the misinterpretation of results and false conclusions. We identified that the two key points of difference between different tools are i) the way how small queries (queries with less than nmax documents in the case of NDCGn max ) and ii) the way how queries with no relevant document (i.e. when all documents have the same, zero relevance score and therefore the ideal DCG is zero) are handled by the tools. These two factors can lead to different overall scores (for the same model) and also to inherently different learnt models, depending on which learning algorithm was used. Some divergences from the definition of the NDCG measure might be jus-tified from an ML perspective though: for example, if the measure assigns a perfect score to queries with zero relevant documents, these queries are not treated like other queries that can be improved, and do not distort the model. On the other hand, we could not identify any benefit of other modifications, such as zeroing out the NDCGn max score for queries with less than nmax documents in total.

To summarize our findings, we suggest the following protocols to make learning to rank results more comparable and benchmark datasets more suited to training and evaluating systems:

1. The use of tools that give zero score to small queries should be avoided, as these can negatively impact certain algorithms (while others are unaffected) and thus the reported results can represent a false relative order of the algorithms. 2. If possible, benchmark datasets should be free of not meaningful queries. Queries that does not have any relevant documents do not play a role in learning to rank -they cannot be used to learn meaningful patterns and they do not have an impact on evaluation. At the same time, such queries can negatively impact the performance of some algorithms, and can motivate non-standard evaluation tools (c.f. the YAHOO metric). 3. Regardless which evaluation script is used while training the systems (c.f. the YAHOO tool which is reasonable in the presence of queries without relevant documents in the data), the final evaluation should be carried out using a tool fully conforming the official NDCG definition. This way machine learnt performance scores would become directly comparable to non-machine learnt ones, such as those coming from TREC and other evaluation exercises.

8

  http://research.microsoft.com/en-us/um/beijing/ projects/letor/LETOR3.0/EvaluationTool.zip http://research.microsoft.com/en-us/um/beijing/ projects/letor/LETOR4.0/Evaluation/Eval-Score-4. 0.pl.txt

Figure 1 .

 1 Figure 1. The dependence of NDCG 10 scores on NDCG method used on MQ2008 dataset.

Figure 2 .

 2 Figure 2. The dependence of NDCG 10 scores on NDCG method used on MQ2008 dataset.

See,for example http://nlp.stanford. edu/IR-book/html/htmledition/ evaluation-of-ranked-retrieval-results-1.html

When it is not confusing, we will omit the query index and simply write x i for the ith document of a given query.

http://research.microsoft.com/en-us/projects/ mslr/eval-score-mslr.pl.txt

http://learningtorankchallenge.yahoo.com/ evaluate.py.txt

http://www.cs.umass.edu/ ∼ vdang/ranklib.html

http://trec.nist.gov/trec eval/

Note that relevance labels are many times pooled in IR datasets and there is no guarantee that no relevant, but unlabeled documents exist.

http://people.cs.umass.edu/ ∼ vdang/ranklib.html

We found no clear indication in the original article what stopping criterion was used to terminate the iteration in the training process, so we tried to use various stopping strategies and provide the best results we obtained. These are nevertheless lower than those reported at the letor website.


F Ravelet

B Dubrulle

F Daviaud

P-A Ratié

Kinematic Alpha Tensors and dynamo mechanisms in a von Kármán swirling flow

Keywords: numbers: 47.20.Ky, 47.27.-i, 47.27.Cn

We provide experimental and numerical evidence of in-blades vortices in the von Kármán swirling flow. We estimate the associated kinematic α-effect tensor and show that it is compatible with recent models of the von Karman Sodium (VKS) dynamo. We further show that depending on the relative frequency of the two impellers, the dominant dynamo mechanism may switch from α 2 to α -Ω dynamo. We discuss some implications of these results for VKS experiments.

The dynamo effect is the conversion from mechanical energy to magnetic energy. It is at the origin of most magnetic fields in the Universe (Earth, stars, galaxies,..) and therefore deserves special attention. The von Kármán sodium experiment (VKS) is at present time one of the three succesfull liquid metal dynamo experiments. In this experiment, the mechanical energy is provided by two iron counter-rotating impellers in a cylindrical vessel, and the spontaneous conversion into magnetic energy is obtained as soon as one of the impellers rotates faster than 16 Hz [1][2][3]. The process of how this conversion occurs is however still a matter of debate. Indeed, a quasi axial axisymmetric mean magnetic dynamo field is observed in VKS experiment, while an equatorial dipole is expected if the dynamo process only involves the time averaged axisymmetric mean flow [4]. Several models have recently been suggested to explain this observation, based upon the now-classical α-effect in mean field theory of dynamo [START_REF] Krause | Mean-Field Magnetohydrodynamics and Dynamo Theory[END_REF][START_REF] Moffat | Magnetic Field Generation in Electrically Conducting Fluids[END_REF]: in the presence of a non-axisymmetric fluctuating velocity field, the mean VKS magnetic field satisfies:

∂ t B = ∇ × u × B + α B -β∇ × B , (1)
where u i and B i are the velocity and magnetic field, denotes time-azimuthal average and α and β are secondorder tensors depending on u and u ′ = u-< u >. Specifically, it was suggested that vortical coherent structures in between the impeller blades such as pictured in Fig. 1c generate higly-correlated non-axisymmetric velocity-vorticity fluctuations that would result in a nonnegligible α-tensor able to produce an axial dipole [START_REF] Petrelis | [END_REF]8]. Kinematic simulations using a numerical model of such vortices indeed reproduce the growth of an axisymmetric axial field as observed in VKS experiments [9]. Numerical simulations [8,10] using ad-hoc α tensors in the induction equation and different boundary conditions satisfactorily reproduce the large scale structure of the dynamo magnetic field, provided the non dimensional α parameter is of the order of 2 (ferromagnetic boundary conditions, [8]) or 0.02 if variations of the permeability due to soft iron are taken into account [10]. However, no measure-ments of the velocity structure in between the blades has been avalaible so far, preventing check of the plausibility of these estimates, and, thus, the soundness of these α-models. In the present Letter, we first provide some experimental evidence of these in-blades vortical structures and give more quantitative information on their averaged properties using Reynolds Averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) calculations. We then use these results to compute the resulting kinematical α effect. Some implications of these results for VKS experiments are finally discussed. Depending on the relative frequency of the two impellers, the dominant dynamo mechanism may switch from α 2 to α -Ω dynamo, resulting in onset of dynamical regimes.

To observe the vortices potentially responsible for the α-effect, we use a von Kármán water experiment which is a 1/2 scale reproduction of the first sucessful VKS dynamo experiment described in [1]. Its geometry is summarized in Fig. 1: the fluid contained in a Plexiglas cylinder of radius R = 100 mm is stirred by a pair of coaxial impellers that can be rotated independently at rotation frequencies f 1 and f 2 ranging from 1 to 12 Hz. The impellers are flat disks of diameter 2R p = 150 mm fitted with either straight blades or radial blades of constant height h b = 20 mm and curvature radius 92.5 mm. The inner faces of the discs are H = 180 mm apart. Velocity measurements are performed with a LDV and a stereoscopic PIV system provided by DANTEC Dynamics. In the sequel, we use R and T = 1/π(f 1 + f 2 ) as units of length and time.

In the VKS experiment, dynamo regimes have been observed for a wide variety of rotating frequencies (f 1 , f 2 ) [2]. For simplification, we focus here on the simplest case where the top disk is fixed and the bottom disk is rotating, with the convex face of the blades pushing the fluid forward: (f 1 , f 2 ) = (1/(πT ), 0). In such a case, dynamo action is observed for f 1 > 16 Hz through a nearly axisymmetric dipolar dynamo field, with azimutal component localized at each impeller ( [3,11] and Fig. 1b). Fig. 2a shows the mean time-averaged non-dimensional velocity on a half meridional plane measured by PIV in the experiment. It consists in a fully non-symmetric flow, with the outside radial part rotating as a block almost at the impeller's velocity. Due to the observing angle, this representation does not allow for velocity measurements inside the blades. Additional measurements using a LDV system and impellers with straight blades allow the measurement of the instantaneous azimutal velocity component at one point located inside a blade (Fig. 2b).

In this measurement, one observes a modulation of the azimutal velocity, over a period of 1/8f 1 sec corresponding to one blade crossing. During a period, the azimutal velocity first drops, and then catches again with the velocity of the blade. This drop results in a smaller averaged velocity (black curve) than the impeller velocity (blue curve) and is an experimental evidence of the presence of in-blades vortical movements.

To characterize further these motions, we now perform numerical experiments with the CFD finite-volume solver code FLUENT 6.3. The model that has been used is the stationary realizable k -ǫ RANS model [START_REF]FLUENT 6.3 User's Guide[END_REF]. We ran several different mesh configurations before reaching a satisfactory converged solution. It corresponds to a case where the fluid volume is divided into three parts: one central part with a typical element size of 3 mm and two volumes surrounding the impellers 15 mm and 15 mm apart from the blade tips in resp. the axial and radial direction, with conformal interfaces. The mesh is refined to 1 mm close to the blades. The equations are solved in a Moving Reference Frame in one of these volumes to simulate the impeller's rotation. A grid convergence check has been performed and a stable converged solution with one rotating disk was found, with torques at both impellers equal to those measured in the experimental flow within 10 percent. The result is presented in Fig. 2c for the azimutally averaged flow. One observes a one cell circulation close to the experimental one. To quantify this, we present in Fig. 3 a comparison between the numerical and the experimental velocity radial profiles at three locations: z = ±0.53; z = 0. The agreement is within 5% in most of the flow. The CFD solution enables the description of the in-blades velocity. On Fig. 2c, one observes an intensification of the azimutal velocity at the edge of the rotating impeller, as well as some edge vortices near the top and the bottom edge. In the stationary impeller, the azimutal velocity is slightly negative, so there is a huge differential rotation at the blades shrouds and tips. The streamlines displayed in Fig. 2d enables the clear visualization of in-blades radial trailing vortices, that are located behind the blades. Such vortices can be responsible for a deceleration of the azimutal velocity observed in Fig. 2b.

Using the CFD velocity field, one can compute the (non-dimensional) azimutally averaged helicity tensor

h ij = ǫ ikn u ′ k ∂ j u ′ n , (2)
where u ′ = u -u is the fluctuating velocity field,

h ii = T r(h) = -u ′ i ω ′
i is minus the helicity of the fluctuations, ω is the vorticity and the symbol denotes azimuthal average. The result is displayed in Fig. 4. One sees that the helicity is mainly concentrated in the impellers, with a maximum absolute non-dimensional value around 1 (corresponding to 100 ms -2 for impeller rotating at 10 Hz). In the rotating impeller, the largest helicity component is h φφ with a value of the order of -0.5. Its value in the stationary impeller is of opposite sign and slightly larger, 1. This difference may be explained by the existence of a very large vertical differential rotation at the stationary impeller, which can generate a large toroidal vorticity. In the rotating impeller, other non-negligible helicity components are found as h φr and h rz . In the stationary impeller, other fairly large helicity components are h rr , h rφ , h rz ,h zφ and h zr (of the order of ±1). We also computed the (non-dimensional) azimutally averaged Reynolds tensor r ij = u ′ i u ′ j (not shown). We observed a similar localization of the tensor in the impellers, with a much smaller value, of the order of 5×10 -2 (corresponding to 0.5 m 2 s -2 for impeller rotating at 10 Hz). The exact dynamo properties of such a velocity field would require solving the full kinematic problem, like in [9]. To get an order of magnitude estimates of the dynamo efficiency, we may however approximate the fluctuations by a short in time field. In such a case, one use the computation of [START_REF] Rädler | [END_REF] equation (29), see also [14] to link the helicity and Reynolds tensor to α and β as:

α ij = τ h ij , β ij = τ r ij , ( 3
)
where τ is the correlation time of the non-axisymmetric velocity perturbations. The CFD simulation provides a time-independent solution and cannot be used to compute this correlation time. However, we can use the LDV measurements in water (see Fig. 2) to estimate that this time is at most of the order of 1/8f 1 . Therefore, in non-dimensional units, τ ≤ π/8. This estimate leads to maximal non-dimensional α tensors of the order of ±0.4 (resp. -0.2) in the stationary (resp. rotating) impeller, and maximal non-dimensional β tensors of the order of 2 × 10 -2 . Our estimate of α using CFD simu- lations and experiments can be compared with existing models of the VKS dynamo. First, we observe that the largest component of the tensor in the rotating impeller is α φφ , as assumed in [8,10]. It is negative, as required for instability [8]. However, its maximal value is an order of magnitude smaller than the value |α c | = 2.1 needed to obtain dynamo action with ferromagnetic boundary conditions at magnetic Reynolds number reached in the VKS experiment [8]. On the other hand, it is one order of magnitudes larger than the value needed to obtain a dynamo action in VKS according to the model with localized permeability distribution [10]. Given the uncertainty in evaluation of the correlation time, we may say that our findings clearly validates the hypothese of model of [10], but not rule those of the model of [8]. Dynamo action may be impeded by a large enough turbulent resistivity, parametrized by β. This effect has been discarded in the numerical models so far [8,10]. Our estimate shows that this approximation is marginally legitimate in VKS, where the critical magnetic Reynolds number Rm c ∼ 50 in our time and length units, resulting in a non-dimensional molecular magnetic diffusivity of the order of 1/Rm c ∼ 0.02, i.e. of the order of the maximal value for the turbulent magnetic diffusivity, resulting in a 100 percent increase of the total magnetic diffusivity at Rm = 50. Our finding is in agreement with a recent estimate of [15] who observed a 50 percent increase of magnetic diffusivity at Rm = 30, in a nonstationary turbulent flow of liquid sodium, generated in a closed toroidal channel.

The present computation shows that the in-blade velocity is the VKS impellers is able to generate sufficient α-effect to produce a dynamo mechanism, provided ad-equate boundary conditions [8] or localized permeability distributions [10] are taken into account. The special localization of this α-effect found in the present study how calls for further discussion about a qualitative picture of the VKS one disk dynamo. Indeed, using Eq. 1, one can show that the poloidal field (r, z component) is generated from the toroidal field (φcomponent) through the α-effect. In contrast, there are two possibilities (possibly concomitant) to generate toroidal field from poloidal field: either through the mean differential rotation Ω ′ = (1/r)d u φ /dz (Ω-effect) -possibly reinforced by the jump of permeability or conductivity at the impellers [17] and/or through α effect. The first case corresponds to a α -Ω dynamo, while the second corresponds to an α 2 dynamo. Since the α tensor varies over a region of size h b , the crossover in between the two mechanism lies at the critical value R p Ω ′ ∼ α rr /h b . This induces a dissymetry in between the two impellers: at the rotating one, the differential rotation is almost zero Ω ′ ∼ 0 while at the stationary one, the differential rotation occurs over the height of the blades, so that in non-dimensional shape R p Ω ′ ∼ R p /h b ∼ 4, i.e. is of the order of α rr R/h b ∼ 4. Therefore, the dynamo generation is α 2 at the rotating impeller, while it can be a combination of an α 2 and an α -Ω at the stationary one (an α 2 -Ω) dynamo). Such a dissymetry may explain generically the field dissymetry observed in the VKS configuration with one rotating disk, where the field is much more intense near the rotating impeller, than near the stationary impeller cf Fig. 1.

If the α-effect computed in the present case is not strongly modified when the two impellers are rotating, we may draw interesting implications about the VKS dynamo in more general rotating regimes. Indeed, the location of the azimuthal velocity shear layer strongly depends on the relative velocity of the two impellers θ = (f 1 -f 2 )/(f 1 + f 2 ) [16]: as long as |θ| ≤ θ c , the azimuthal velocity shear layer is in-between the two impellers, and there is a constant shear at the two impellers, of the order of Ω ′ ≈ π(f 1 -f 2 )/H. Once |θ| ≥ θ c , the shear layer is at the slowest impeller, like in the one rotating disk case explored in the present paper: the differential rotation is almost zero at the fastest impeller, and very strong at the slowest one[21]. This peculiar property however offers possible very different generic dynamo behavior under and above θ c . For |θ| < θ c , the Ω-effect is of the order O(1/H), so that it is too weak to superseed the α effect, of the order 1/h b (since h b /H ≪ 1). In such case, the dominant dynamo mecanisl is α 2 at each impellers. For |θ| > θ c , the differential rotation switches to O(1/h b ) at the slowest impeller, and the Ω effect becomes comparable to the α effect. There is therefore the possibility of an α 2 -Ω mechanism at the slowliest impeller, with an α 2 dynamo mechanism at the fastest impeller. This natural switch, induced by the behavior of the shear layer, opens interesting perspectives in terms of possible dynamical regimes in the VKS experiments. Indeed, an homogeneous α 2 dynamo is generically steady, with dynamo threshold Rm c = O(k/α), while an homegeneous α -Ω dynamo is generically oscillatory with dynamo threshold Rm c = O(k 2 (2/αkR p Ω ′ )), with a rotating frequency bifurcating from a finite value at the dynamo onset ω osc = (αkR p Ω ′ /2) [18]. (Meridional circulation and/or parity properties with respect to the equator of α can steady α -Ω dynamos.) In our setting, this translates in Rm c = O(R/h b ) for α 2 dynamos, and Rm c = O(R/h b )/ θ c /θ and ω osc = O(R/h b ) θ c /θ for the α -Ω dynamo. Due to inhomogeneities and possible non-linear coupling in between the two dynamo mecanism at each impeller [3], it is not garanteed that these generic features will persist. However, we observe that for θ < θ c , dynamos observed in VKS are indeed steady, while both steady and dynamical regimes with oscillations appear for θ ≥ θ c [19], with oscillation frequency bifurcating from finite value [2]. The scale separation in the VKS experiment, with α and Ω-effect occuring in region of size h b << R is in any case a very interesting result, since it validates the main hypotheses underlining the α-effect and opens the way to homogenization techniques [20]. It would also be interesting in the VKS experiment to vary the blade's height to see how it affects both the threshold and the oscillation frequencies and check their potential R/h b scaling.

FIG. 1 :

 1 FIG.1: a) Geometry of the experiment. The shaded blue zone denote the rotating fluid volume used in the numerical simulation. b) Reconstruction of the dynamo magnetic field in a meridional plane in a VKS experiment with rotating bottom disk and stationary upper disk (after[3]). The poloidal (resp. toroidal) component is coded with arrows (resp. color). Measuring probes are in white. c) Putative in-blades vortice creating the α-effect.

  FIG. 2: a) Non-dimensional experimental mean velocity in a meridional plane. b) non-dimensional azimuthal velocity in the blades synchronized on the blade crossing vs non dimensional time. Green: LDV measurements; red: running average; black: time average; blue: blade velocity. c) Nondimensional numerical mean velocity. d) Radial vortices as observed with streamlines in the CFD simulation.

FIG. 3 :

 3 FIG. 3: Comparison between numerical (line) and experimental (symbols) profiles at z = -0.53 (red), z = 0 (green) and z = 0.53 (blue). Errors bars are of the order of the size of the symbols.

FIG. 4 :

 4 FIG. 4: Components of the (non-dimensional) azimutally averaged helicity tensor hij in numerics. The figure in between parenthesis is the volume averaged value.

We thank A. Chiffaudel and F. Bakir for fruitful discussions, P-P. Cortet for PIV measurements of Fig. 2 and the VKS team for magnetic field measurements used for Fig. 1.