Julien Ruffault
email: julien.ruff@gmail.com

Florent Mouillot

Contribution of human and biophysical factors to the spatial distribution of forest fire ignitions and large wildfires in a French Mediterranean region

Contribution of human and biophysical factors to the spatial distribution of forest fire ignitions and large wildfires in a French Mediterranean region

Julien Ruffault, Florent Mouillot

Introduction

Wildfire is a widespread ecological disturbance, but one with a heterogeneous spatial distribution (Krawchuck et al. 2009;[START_REF] Bowman | The human dimension of fire regimes on Earth[END_REF]. Identifying the drivers of the distribution of fires is crucial for the design of appropriate fire policies, especially in regions where fire regimes are being altered by global changes as is the case in the Mediterranean area [START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF].

Regional fire distribution is the result of interactions between three environmental conditions: ignitions patterns, fuel availability and atmospheric conditions that favor combustion [START_REF] Moritz | Wildfires, complexity, and highly optimized tolerance[END_REF][START_REF] Archibald | What limits fire? An examination of drivers of burnt area in Southern Africa[END_REF]. Biophysical factors, including fuel type and continuity, topography and weather conditions play an important role in determining the spatial patterns of fires [START_REF] Parks | Spatial bottom-up controls on fire likelihood vary across western North America[END_REF][START_REF] Parisien | An analysis of controls on fire activity in boreal Canada: Comparing models built with different temporal resolutions[END_REF]. Humans also affects regional fire distribution through three main mechanisms: by igniting fires, by limiting the spread of fires and by modifying land cover [START_REF] Bowman | The human dimension of fire regimes on Earth[END_REF].

Because of the complex interactions between biophysical and human influences, the extent to which they determine the spatial patterns of fires varies geographically, according to ecosystem characteristics [START_REF] Parisien | Environmental controls on the distribution of wildfire at multiple spatial scales[END_REF], orographic and synoptic conditions [START_REF] Heyerdahl | Spatial controls of historical fire regimes: A multiscale example from the interior west, USA[END_REF] or to the level and nature of the human imprint [START_REF] Parisien | The spatially varying influence of humans on fire probability in North America[END_REF]).

In the Euro-Mediterranean region, evidence is accumulating that the spatial pattern of ignitions is mainly determined by human settlements and activities, while the continuity and type of fuels largely control fire spread probabilities, and therefore the location of the largest fires [START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF][START_REF] Duane | Predictive modelling of fire occurrences from different fire spread patterns in Mediterranean landscapes[END_REF][START_REF] Fernandes | Bottom-Up Variables Govern Large-Fire Size in Portugal[END_REF].

However, there is no consensus on the extent to which spatial variations in weather control the spatial patterns of fires at landscape to regional scales [START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF].

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Yet, in regions of complex orography like the Mediterranean, we observe steep weather gradients [START_REF] Lionello | The Mediterranean climate: An overview of the main characteristics and issues[END_REF]) that could help define areas where wildfires are most likely to occur [START_REF] Moritz | Spatial variation in extreme winds predicts large wildfire locations in chaparral ecosystems[END_REF]). In the Mediterranean, this knowledge gap is often mentioned as one of the major source of uncertainty when projecting future fire likelihood at the regional level [START_REF] Piñol | Climate warming, wildfire hazard, and wilfire occurence in coastal eastern Spain[END_REF][START_REF] Martínez | Human-caused wildfire risk rating for prevention planning in Spain[END_REF][START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF]) and part of the problem is the difficulty in identifying and mapping wildfire danger.

First, quantifying the contribution of weather requires unravelling the multiscale temporal relationships between climate and fire activity. Climate affects fire probability both indirectly, through its long term control of vegetation, and directly through what is termed fire weather, i.e. the weather conditions that influence fire ignition, fire behavior and suppression [START_REF] Bradstock | A biogeographic model of fire regimes in Australia: Current and future implications[END_REF]. In turn, fire weather is the result of atmospheric processes at different time scales: fuel production and desiccation in the medium term and instantaneous conducive atmospheric conditions in the short term [START_REF] Ruffault | Objective identification of multiple large fire climatologies: an application to a Mediterranean ecosystem[END_REF]. Hence, linking fire activity to mean climatic conditions helps determine the biophysical niche of fires (e.g. [START_REF] Parisien | Environmental controls on the distribution of wildfire at multiple spatial scales[END_REF][START_REF] Whitman | The climate space of fire regimes in north-western North America[END_REF]. These indices nevertheless cannot separate the indirect from the direct effects of climate on the spatial distribution of fires, nor do they represent the co-occurrence of several multiscale weather events associated with fires.

Second, in regions with marked climate seasonality like the Mediterranean, variations in climate over time have a much greater influence than spatial variations, making it difficult to extract the impact of its spatial component on fire activity. Finally, as the fire-weather relationship is shaped by both the vegetation (Pausas and Paula 2012) and human practices [START_REF] Marlon | Climate and human influences on global biomass burning over the past two millennia[END_REF][START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF], the weather conditions that drive the fire occurrence and fire spread processes can vary considerably in space and over time [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF][START_REF] Higuera | The changing strength and nature of fire-climate relationships in the northern Rocky Mountains, U.S.A[END_REF].

In this context, we conducted a comprehensive evaluation of the effects of weather, fuel and human variables on the spatial distribution of fires in a French Mediterranean area (Fig. 1). In this area, the fire regime is dominated by typical Mediterranean crown fires in shrublands and in mixed oak-pine woodlands, which are mostly ignited by humans and usually last less than a day [START_REF] Fréjaville | Spatiotemporal patterns of changes in fire regime and climate: defining the pyroclimates of south-eastern France (Mediterranean Basin)[END_REF]. Fire activity is "droughtlimited", i.e. fires are most likely to occur when two conditions occur simultaneously: vegetation drought and meteorological fire prone days [START_REF] Ruffault | Objective identification of multiple large fire climatologies: an application to a Mediterranean ecosystem[END_REF][START_REF] Ruffault | Daily synoptic conditions associated with large fire occurrence in Mediterranean France: evidence for a wind-driven fire regime[END_REF]. We used a set of seven explanatory variables describing the spatial patterns of human, fuel and fire weather factors in the study area. We assessed the relative importance of environmental controls on the spatial distribution of fires for individual variables as well as for the grouped variables humans, land cover and weather. Our objectives were (i) to identify the spatial structure of fire weather, (ii) to quantify the contribution of the human, land cover and weather variables to the spatial distribution of fires, and (iii) to assess whether the relative contribution of these factors varies with increasing fire size.

Materials and Methods

Study area

The study area covers four French administrative districts (total area 21,637 km 2 ) in southern France (Fig. 1). The climate is Mediterranean with hot dry summers, cool wet winters and high inter-annual variability. The rainfall gradient depends on the topography and ranges from 550 mm at the coast (elevation = 0 m) to 1,630 mm in the central foothills (elevation = 1,450 m). Closed forests dominated by Mediterranean evergreen tree species (Quercus ilex, Pinus halepensis) and shrublands (Cistus 6 monspeliensis, Quercus coccifera) cover 65% of the study area (French national forest inventory, 2006). Agricultural land use (mainly vineyards) accounts for 28% and urban areas the remaining 7%.

General method and selection of the variables

We assessed the drivers of the spatial distribution of fires for a range of final fire size classes by modeling the relationship between fire presence as the response variable and a set of potential human, weather and land cover factors as predictor variables. The spatial partitioning of the study area followed the French fire management agency's 4km 2 grid system for the French territory. We limited our analysis to the most recent period (1990-2006) to avoid biases resulting from the shift in fire activity observed at the end of the 1980s in the study area [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF]. For each class of final size of fire, a binary variable was generated to represent its corresponding spatial fire distribution for each grid cell, positive if at least one fire had occurred over the 17year study period. The decisions to select a variable (or not) were based on both objective analysis (ease of interpretation and a range of final fire size classes) and the results of previous studies conducted in the Euro-Mediterranean region [START_REF] Martínez | Human-caused wildfire risk rating for prevention planning in Spain[END_REF][START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF]2011;[START_REF] Oliveira | Forest Ecology and Management Modeling spatial patterns of fire occurrence in Mediterranean Europe using Multiple Regression and Random Forest[END_REF][START_REF] Duane | Predictive modelling of fire occurrences from different fire spread patterns in Mediterranean landscapes[END_REF][START_REF] Fernandes | Bottom-Up Variables Govern Large-Fire Size in Portugal[END_REF]. Seven variables (Table 1) were selected to build a model by iteratively eliminating correlated and non-informative variables from an original set of 21 variables (Table S1). To this end, the correlation between all explanatory variables was assessed to identity variables correlated above a threshold of |ρ|=0.7. Then the fits of spatial fire distributions were assessed against these two correlated variables to decide which variable to retain. Variables with spurious effects were also discarded. Regional maps of fire distribution, weather, human and land cover variables were then collected at the 4-km 2 grid scale. For land cover and human factors, all the explanatory variables were mapped to reflect the average conditions over the study period (1990-2006) as closely as possible, but due to the lack of historical data for some variables, the conditions prevailing on a single date had to be chosen as representative of the 17-year period (see details below). However, as no major economic or societal changes occurred during this period, we did not expect this simplification to be a major limitation.

Fire data

The location, date and size of fires in the period 1990 to 2006 were extracted from the PROMETHEE fire database (available on line at www.promethee.com). This database is managed by the French forestry services and includes the final size, the day and location of ignition for each registered fire on a 4-km 2 reference grid. A total of 6,381 fires occurred in the study area between 1990 and 2006. Most were small (median = 1 ha; mean = 6.5 ha), and because only 16 fires spread beyond the area of the reference grid cell (400 ha), we assumed that fires did not spread beyond their ignition grid cell.

Each fire was then attributed to one or more of the five classes of final fire size burned area: > 0 ha (all fires); > 1 ha; > 6 ha; > 15 ha; > 30 ha (large fires); representing respectively the 0 th , 50 th , 90 th , 95 th , 97 th , 98 th percentile of the fire size distribution over the study period. A binary response variable was then selected to describe the presence or absence of fire in each grid cell. Forty-six percent of the grid cells experienced at least one fire over the 1990 to 2006 period, but this percentage decreased sharply with an increase in fire size. For instance, fires covering an area of more than 6 ha and 30 ha were observed in 10% and 3.3% of grid cells, respectively (Fig. 1).

Fire weather maps

Spatial variations in fire weather were described by using a single and integrative variable and expressed for each grid cell as the fire weather season length (hereafter FWSL, i.e. the annual number of days suitable for burning). This approach is therefore similar to that successfully used by [START_REF] Jolly | Climate-induced variations in global wildfire danger from 1979 to 2013[END_REF] on a global scale, but in our case, was based on the prior identification of the fire-weather relationship. In addition, as the conditions controlling the probabilities of fire start differ from those driving fire spread in Mediterranean France [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF][START_REF] Ruffault | Objective identification of multiple large fire climatologies: an application to a Mediterranean ecosystem[END_REF][START_REF] Ruffault | Daily synoptic conditions associated with large fire occurrence in Mediterranean France: evidence for a wind-driven fire regime[END_REF], the FWSL was computed for each fire size class. We used the following two-step methodology to map the FWSL:

(i) First, to determine the fire-weather relationship, we assessed the relative importance of several key weather variables in the probability of fire occurrence using the spatiotemporal framework of [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF]. For each fire, we extracted the associated weather conditions in our basic spatio-temporal unit (or voxel, on 4-km 2 grid cells and at a daily time step). Boosted regression tree (BRT, see description below) models were then used to describe the relationship between fire occurrence and the weather by comparing the conditions associated with fire voxels (positive elements) with those associated with a sample of non-fire voxels (absence elements). Five variables were used to capture the weather conditions driving fire occurrence and fire spread probabilities: a proxy of fuel moisture content of litter (Surface drought), a proxy of fuel moisture content of living biomass (Vegetation drought), relative air humidity, temperature and wind speed. The two proxies of fuel moisture content were derived from a daily process based water-budget model, validated in forest stands in southern France by [START_REF] Ruffault | Differential regional responses in drought length, intensity and timing to recent climate changes in a Mediterranean forested ecosystem[END_REF], and computed here for a single plant functional type (PFT) representative of the deep-rooted evergreen woody species (trees and large shrubs) encountered in our study area. Surface drought is expressed as the relative soil water content of the uppermost soil layer. Vegetation drought is expressed as the ratio of actual evapotranspiration to maximum transpiration (transpiration without water stress).

(ii) In the second step, we used the regional statistical models between weather variables and fire occurrence to determine the daily probabilities of fire occurrence in each grid cell between 1990 and 2006. The FWSL was then determined as the averaged annual number of days when their fire probability was higher than a given probability threshold. To facilitate comparisons among fire size classes, this threshold value was set as equal to the prevalence (percentage of positive fire voxels) observed in the original sampling dataset.

Daily historical observations of precipitation, relative humidity, temperature, and global solar radiation, used as inputs in the process based water budget model and in the BRT analyses, were obtained from the SAFRAN dataset [START_REF] Vidal | A 50-year high-resolution atmospheric reanalysis over France with the Safran system[END_REF]. SAFRAN is a reanalysis of daily surface observations on an 8-km resolution grid of France. To match the resolution of our spatial sampling unit (national reference grid, 2x2 km), these daily variables were previously re-interpolated using altitude-dependent methods described and validated over the region by [START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF]. Soil data used as inputs for simulations of the water budget were extracted from the regional DONESOL database (1/250000; INRA; [START_REF] Gaultier | L'organisation et la gestion des données pédologiques spatialisées: Le projet DONESOL[END_REF].

Human variables

Three variables were selected to represent the pattern of human influence over the study area: road density, housing density, and the percentage of wildland urban interface (WUI). Vectorial information related to roads and human habitats was extracted from 10 the BD-Topo database (IGN 2007). The density of roads is expressed as the total road length per unit land area. The density of houses was calculated as the fraction of unit area covered by urban polygons. The WUI percentage was calculated as the distance to aggregated and open human habitat according to the method proposed by [START_REF] Lampin-Maillet | Mapping wildland-urban interfaces at large scales integrating housing density and vegetation aggregation for fire prevention in the South of France[END_REF] and is expressed as the percentage of WUI in each grid cell. Fig. S1 shows a map of these variables.

Land cover variables

Three variables were selected to represent the patterns of land cover in the study area: percentage of shrubland area, percentage of forest area, and the diversity of vegetation types. All these variables were derived from the Corine land cover database 2000 (EEA 1994), which was previously reclassified into four main categories (shrublands, forests, grasslands and other non-fire prone areas: urban area, crops, bare ground) according to the method described in [START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF]. Shrubland and forest areas were calculated as the percentage area of the grid cell covered by forest and shrubland type vegetation, respectively. The landscape diversity represents the diversity in the fractional distribution of land cover types that make up the landscape and was calculated with the Shannon diversity index with a 10-km wide moving window using FRAGSTATS V3 [START_REF] Mcgarigal | FRAGSTATS: Spatial Pattern Analysis Program for Categorical Maps. Analysis 3.3[END_REF]. Fig. S1 shows a map of these variables.

Fire spatial model

We used a machine-learning algorithm, boosted regression trees (BRT; [START_REF] De | Boosted trees for ecological modeling and prediction[END_REF][START_REF] Elith | A working guide to boosted regression trees[END_REF] to predict the spatial pattern of fires. BRT uses the iterative partitioning approach of regression trees, but reduces predictive error by "boosting" initial models with additional, sequential trees that model the residuals in randomized subsets of the data [START_REF] De | Boosted trees for ecological modeling and prediction[END_REF][START_REF] Elith | A working guide to boosted regression trees[END_REF]. BRT methods have been increasingly recommended for ecological analyses because of their flexibility in modeling complex nonlinear relationships and interactions without the restrictive assumptions of parametric statistics [START_REF] Olden | Machine learning methods without tears: a primer for ecologists[END_REF].

BRT models need information about the presence and absence of fires in the grid cells to be able to determine the conditions associated with fire. To be able to compare the models more easily, the prevalence (the percentage of fire presence grid cells in the sample dataset) was fixed at 0.1. The absence data for each fire size class was randomly selected from the pool of absence grid cells. The learning rate (lr), the tree size or tree complexity (tc) and the number of trees (nt) are the main parameters of BRT models and were set according the procedure recommended by [START_REF] Elith | A working guide to boosted regression trees[END_REF]. For all models, a bag fraction of 0.5 was used, meaning that, at each step, 50% of the data were randomly drawn from the training dataset. As the number of samples could subsequently vary between models, we set tc to 4 (based on preliminary analyses) and then determined lr as a value that resulted in the average test error being minimized between approximately 1,000 and 2,000 trees [START_REF] Elith | A working guide to boosted regression trees[END_REF]. The nt in each BRT model was selected automatically using 10 fold cross-validation to avoid model overfitting. BRT models were computed in R with the gbm package [START_REF] Ridgeway | Generalized boosted regression models. Documentation on the R Package 'gbm[END_REF] and custom functions created by [START_REF] Elith | A working guide to boosted regression trees[END_REF] computed using a Bernoulli (logistic) error structure. We used the area under the receiver operating characteristics (ROC) curve (AUC) to evaluate the suitability of the models. For each model, 70% of the observations were randomly selected from the complete dataset to build the statistical model (training dataset). The remaining observations (30%) were used to evaluate the accuracy of model classification (validation dataset). We also report the commission error (false 12 positive rate) and omission error (false negative rate) at the probability threshold that maximizes the sum of sensitivity (the fraction of true positives) and specificity (the fraction of false positives) values (Lobo et al. 2007). To limit the stochasticity in model outcomes caused by the subsampling and bagging, we created an ensemble of 25 BRT models and then averaged the results.

We interpreted the BRT models by first looking at the relative contribution of the variables to the predictive models. The contribution of the different predictors was estimated from the sum of squared improvements associated with that variable and averaged across all trees in the boosted model [START_REF] De | Boosted trees for ecological modeling and prediction[END_REF][START_REF] Elith | A working guide to boosted regression trees[END_REF]. The relative importance of environmental factors controlling the spatial distribution of fires was assessed for individual variables as well as for the grouped variables for humans, land cover and weather (Table 1). The contributions of the grouped variables were determined by adding the percentage contributions of their constituent variables. To estimate the degree of similarity among variable contributions among different classes of fire size classes, a rank (Spearman) correlation was performed on the mean contribution of the variables in pair-wise combinations of fire size classes. A lack of significance (p < 0.05) in a correlation indicates that the relative importance of environmental controls differs among sizes. We also examined the relationship between the dependent and independent variables by plotting the partial dependencies of responses to individual predictors. Finally, we computed the fire hazard maps, which were derived from statistical models between the spatial distribution of fires and its environmental drivers.

International Journal of Wildland Fires 2017 13

Results

Regional patterns of fire weather season length

Fire activity in southern France was shown to be mainly linked to drought conditions but with an increasing contribution of wind speed with increasing fire size (Fig. S2, see also [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF]. A significant spatial gradient was observed in the fire weather season length (FWSL), as shown for two contrasted sizes of fires: fires > 0 ha (hereafter all fires) and fires > 30 ha only (hereafter large fires) (Fig. 2). For all fires, the FWSL ranged from 40 to180 days.year -1 (Fig. 2a). The lowest values were observed in the western part of the region where rainfall and mean temperature are respectively, higher and lower than the regional means [START_REF] Ruffault | Differential regional responses in drought length, intensity and timing to recent climate changes in a Mediterranean forested ecosystem[END_REF]. By contrast, a longer FWSL was observed in the drier coastal areas [START_REF] Ruffault | Differential regional responses in drought length, intensity and timing to recent climate changes in a Mediterranean forested ecosystem[END_REF]. Some marked local variations were also detected according to the variations in soil water holding capacity that influenced the variations in live fuel moisture content, and in turn, the FWSL. For large fires, the regional pattern of fire weather was similar to the one observed for all fires (Fig. 2b) but with a coherently shorter FWSL (ranging from about 10 days.year -1 in the western part of the region to 100 days.year -1 in the coastal area).

Relative contribution of climate, land-cover and human variables to the spatial distribution of fires.

The BRT fire spatial models performed well (AUC≥ 0.72) and similarly in the different fire size classes (Table 2), with commission and omission errors of about 30%. The performance metrics variability was higher for larger fire size classes, most likely due to the exponential reduction in the number of "presence" grid cells with increasing fire size.

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The relative contribution of the explanatory variables to the spatial fire distribution models showed a size-dependent pattern (Fig. 3). The distribution of all fires was largely controlled by the percentage of wildland urban interface (WUI; 26.0%), the FWSL (18.8%) and road density (16.4%). The relative importance and ranking of the explanatory variables changed significantly with an increase in fire size: a major shift was observed between fires > 1 ha and fires > 6 ha (Table 3, Spearman test, p-value<0.1). When the individual contributions of all fires and large fires were compared, there was a significant increase in shrubland area (from 6% to 24% and from 18.4% to 27%, respectively) (Fig. 2c) and in the FWSL (Fig. 2a) and a significant decrease (from 26% to 9%) in WUI (Fig. 2f). When these contributions were grouped, we observed marked control of human variables over spatial fire patterns, but increasing influence of climate and land cover factors with increasing fire size (summarized in Fig. 3e)

Individual relationships between each explanatory variable and fire hazard showed different patterns (Fig. 4). Fire probabilities were higher under intermediate levels of human pressure (Fig. 4b,d,f) but this pattern tended to disappear with increasing fire size. As one might expect, the FWSL positively and monotonically affected fire probability, regardless of the final fire size (Fig. 4a). Finally, vegetation and land cover variables showed some contrasting patterns. The probability of large fire occurrence increased with an increase in the shrubland area (Fig. 4c) but decreased with increasing forest area. The probabilities of ignition and large fires both increased with higher landscape diversity but the impact of this variable was greater on fire ignitions (Fig. 4e).

Fire hazard maps

Despite the variability of fire hazard at local scale, some regional patterns emerged with, on average, less relative probability of fire in the western part than in the southern and eastern parts of the region (Fig. 5a). A very similar pattern was observed for larger fires, but with more contrasted differences (Fig. 5b).

Discussion

The peculiar Euro-Mediterranean context points to an advanced stage of anthropogenic fire-regime transformation [START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF]. Accordingly, in southern France, where the majority of fires are caused by humans [START_REF] Curt | Modelling the spatial patterns of ignition causes and fire regime features in southern France: Implications for fire prevention policy[END_REF], the distribution of fires was mainly driven by variables that reflect the level and nature of human pressure (Fig. 3). This is further evidence for the overwhelming impact of anthropogenic factors on the spatial patterns of fire ignitions in the Euro-Mediterranean area (Martinez et al. 2009;[START_REF] Oliveira | Forest Ecology and Management Modeling spatial patterns of fire occurrence in Mediterranean Europe using Multiple Regression and Random Forest[END_REF], as also observed in a number of fire prone areas where anthropic pressure is high and can override the effects of biophysical factors [START_REF] Cardille | Environmental and Social Factors Influencing Wildfires in the Upper Midwest, United States[END_REF][START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF][START_REF] Hawbaker | Human and biophysical influences on fire occurrence in the United States[END_REF][START_REF] Faivre | Controls on the spatial pattern of wildfire ignitions in Southern California[END_REF][START_REF] Mann | Incorporating Anthropogenic Influences into Fire Probability Models: Effects of Human Activity and Climate Change on Fire Activity in California[END_REF]. Fire-start hazard was higher under intermediate levels of anthropogenic pressure (Fig. 4), a pattern that has been reported in other anthropogenic fire regimes [START_REF] Syphard | Human influence on California fire regimes[END_REF][START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF], Parisien et al. 2012).

In southern France, the greater influence of human-related variables on the spatial pattern of fires tended to decrease with an increase in fire size, with a concomitant increase in the importance of fuel characteristics and land cover (Fig. 3). This result strengthens the fire size dependence hypothesis that was previously proposed to explain the spatial patterns of fires in Western Europe [START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF]. It also suggests that fuel fragmentation is one the most important factors limiting the occurrence of large fires in the Mediterranean. A direct outcome of this phenomenon is that the areas where most wildfires occur did not match those where the largest fires occur (Fig. 5).

Most small fires typically occurred in locations where human development and natural vegetation intermingle, in the most complex landscape mosaics and with frequent or long lasting droughts. By contrast, large fires are most likely to occur in landscapes characterized by a dense shrubland cover, low but still significant human pressure to enable frequent ignitions, and by frequent severe fire weather conditions favoring extreme fire behavior and hence ineffective fire suppression operations. The greater probability of large fires in shrublands compared to forested ecosystems is common in the Mediterranean basin [START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF] and was particularly high in Mediterranean France although we were unable to conclude whether it is due to the higher flammability of this type of fuel or less intense fire suppression efforts in shrublands than in forested ecosystems. Finally, we did not observe a higher probability of large fires in continuous and homogeneous landscapes (Fig. 4), in contrast to what is generally observed [START_REF] Cardille | Environmental and Social Factors Influencing Wildfires in the Upper Midwest, United States[END_REF][START_REF] Heyerdahl | Spatial controls of historical fire regimes: A multiscale example from the interior west, USA[END_REF][START_REF] Viedma | Landscape structural features control fire size in a Mediterranean forested area of central Spain[END_REF][START_REF] Loepfe | Feedbacks between fuel reduction and landscape homogenisation determine fire regimes in three Mediterranean areas[END_REF][START_REF] Fernandes | Bottom-Up Variables Govern Large-Fire Size in Portugal[END_REF] or simulated from landscape fire succession models [START_REF] Hargrove | Simulating fire patterns in heterogeneous landscapes[END_REF][START_REF] Cary | Comparison of the sensitivity of landscape-fire-succession models to variation in terrain, fuel pattern, climate and weather[END_REF]. One possible explanation is the need for a balance between a complex urban/wildland interface and homogeneous landscape pattern for fires to start and spread, but generally leading to a weak effect of landscape diversity on large fire hazard. In addition, our large fire threshold (> 30 ha) is low compared to other regions and ecosystems [START_REF] Hantson | Global fire size distribution is driven by human impact and climate[END_REF] and it is therefore also possible that the relative contribution of weather, human and land-cover factors is quite different when larger fires are considered [START_REF] Liu | Identifying the threshold of dominant controls on fire spread in a boreal forest landscape of Northeast China[END_REF][START_REF] Fernandes | Bottom-Up Variables Govern Large-Fire Size in Portugal[END_REF].

One interesting finding of our study is that the spatial variations in weather conditions also largely influenced the location of fires in Mediterranean France. We estimated that the FWSL accounted for between 20% and 30% in the BRT models predicting the spatial distribution of fires. These figures are not surprising given the marked regional and local variability in drought and wind conditions [START_REF] Ruffault | Differential regional responses in drought length, intensity and timing to recent climate changes in a Mediterranean forested ecosystem[END_REF], two of the critical variables for the probability of large fires occurring in this region [START_REF] Ruffault | Daily synoptic conditions associated with large fire occurrence in Mediterranean France: evidence for a wind-driven fire regime[END_REF]. Given the rapid and non-uniform changes towards hotter and drier conditions that are projected in the Mediterranean area in the coming decades [START_REF] López-Moreno | Climate change prediction over complex areas: Spatial variability of uncertainties and predictions over the Pyrenees from a set of regional climate models[END_REF], the spatial patterns of fire hazard might be modified through some changes in fire weather. Of particular concern are the western and northern parts of our study area both located at the edge of the Mediterranean bioclimatic area. In these peculiar locations, the FWSL is still short (Fig. 3) but these areas are expected to undergo the most intense changes [START_REF] Ruffault | Projecting future drought in Mediterranean forests: bias correction of climate models matters![END_REF].

A similar hypothesis about the control of weather over the spatial patterns of fires could apply to several parts of the Euro-Mediterranean region. Unfortunately, it is difficult to compare our figures with the results of similar studies, as most spatial models of wildfires did not include the impact of fire weather in a meaningful way (e.g. Martínes et al. 2009;[START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF][START_REF] Duane | Predictive modelling of fire occurrences from different fire spread patterns in Mediterranean landscapes[END_REF]. However, the 30% reported here for large fires is in the same order of magnitude as the 29.2% found by Parisien et al.

(2011a) in a flat boreal region with a relatively low fire suppression. One might have expected the influence of fire weather to be higher in our study area because the topography generally increases the spatial variations in weather. But many other factors can also influence the importance of fire weather in controlling the spatial patterns of fires, and in many different ways. For instance, evaluating the net effect of suppression policies is challenging because they may simultaneously have opposite influences. On the one hand, fire suppression policies are likely to reduce the importance of the spatial structure of weather on wildfire probabilities. On the other hand, fire suppression also distorts the relationship between fires and the weather and can increase the role of specific combinations of synergic weather conditions (or fire weather types, [START_REF] Ruffault | Objective identification of multiple large fire climatologies: an application to a Mediterranean ecosystem[END_REF] for which fire suppression operations are not effective. For instance, it is likely that the implementation of a new fire policy in southern France resulted in an increase in the relative probability of fires in the windiest areas [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF].

Here our aim was to provide a comprehensive understanding of the regional spatial pattern of wildfires and we consequently limited our analyses to a small number of relevant explanatory variables (Table 1). Nevertheless, we obtained AUC values that are in the same range as those observed by other authors predicting the spatial distribution of fires at a regional scale in the Mediterranean [START_REF] Moreira | Size-dependent pattern of wildfire ignitions in Portugal: When do ignitions turn into big fires?[END_REF][START_REF] Duane | Predictive modelling of fire occurrences from different fire spread patterns in Mediterranean landscapes[END_REF] and elsewhere in fire prone areas with fragmented landscapes [START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF][START_REF] Bar-Massada | Wildfire ignition-distribution modelling: a comparative study in the Huron-Manistee National Forest, Michigan, USA[END_REF][START_REF] Hawbaker | Human and biophysical influences on fire occurrence in the United States[END_REF][START_REF] Faivre | Controls on the spatial pattern of wildfire ignitions in Southern California[END_REF]. Three important methodological choices are worth mentioning here. First, a variable that estimated the distance to the nearest fire station (Table S1) had to be discarded from our statistical models despite its relative contribution in BRT models (up to 14% for large fires, not shown). Indeed, we observed that fire hazard was higher in areas located close to the fire stations, surely because they were intentionally located to maximize suppression, as also shown by [START_REF] Robinne | Anthropogenic influence on wildfire activity in Alberta, Canada[END_REF] in a region in Canada. This is a good example of the complex impact of humans on the spatial patterns of fires, with both direct and indirect impacts that challenge our assessment of their net effect [START_REF] Brotons | How Fire History, Fire Suppression Practices and Climate Change Affect Wildfire Regimes in Mediterranean Landscapes[END_REF][START_REF] Parisien | The spatially varying influence of humans on fire probability in North America[END_REF]. In our study, we separated groups of human and land cover variables, which was helpful from a methodological point of view (Fig. 3, Tables 1 and2) but this boundary is very blurred in Mediterranean landscapes that are largely shaped by the history of human activities. Our second important methodological choice was not to include topographical variables. There is compelling evidence that topography directly influences fire spread through slope steepness and local atmospheric air movements [START_REF] Rothermel | Predicting behavior and size of crown fires in the northern Rocky Mountains[END_REF] and is related to several spatial fire metrics [START_REF] Parks | Spatial bottom-up controls on fire likelihood vary across western North America[END_REF][START_REF] Duane | Predictive modelling of fire occurrences from different fire spread patterns in Mediterranean landscapes[END_REF]Liu and Wimberley 2016) but the indirect effects of topography, expressed through contrasting vegetation types, fuel amounts or fuel moisture are often more relevant [START_REF] Mouillot | Some determinants of the spatio-temporal fire cycle in a mediterranean landscape (Corsica, France)[END_REF]. In fact, the indirect effects of topography on vegetation were partially taken into consideration in our study, through our altitude dependent methods for the interpolation of daily precipitation and temperature data, and the impact of weather on LAI estimations and functional drought indices. Third, we paid particular attention to providing some realistic estimations of the spatial structure of fire weather. To this end, we based our approach on the prior identification of the fire-weather relationship using a few relevant weather variables and drought indices, thereby avoiding generic fire danger indices whose validity may be limited in areas, ecosystems or anthropogenic contexts for which they were not designed. In addition, our drought indices were estimated with a process based water balance model and therefore have the advantage of accounting for weather, local soil conditions and vegetation functioning in a single metric. The use of such functional indices as proxies of fuel dryness is a step forward in our understanding of the fuelweather interactions [START_REF] Ruffault | How a new fire-suppression policy can abruptly reshape the fire-weather relationship[END_REF][START_REF] Williams | Correlations between components of the water balance and burned area reveal new insights for predicting forest fire area in the southwest United States[END_REF][START_REF] Boer | Future changes in climatic water balance determine potential for transformational shifts in Australian fire regimes[END_REF].

Conclusion

Wildfire is a highly scale-dependent process [START_REF] Moritz | Wildfires, complexity, and highly optimized tolerance[END_REF], as are the relative contributions of the drivers of wildfire distribution [START_REF] Heyerdahl | Spatial controls of historical fire regimes: A multiscale example from the interior west, USA[END_REF][START_REF] Parisien | Environmental controls on the distribution of wildfire at multiple spatial scales[END_REF]Parisien et al. 2011b) but the implementation of effective fire policies and landscape management relies on the identification of the drivers of wildfire at landscape

Ruffault and Mouillot

International Journal of Wildland Fires 2017 20 to regional scales. Here, we provide evidence that, in the French Mediterranean area, where important funds are dedicated to fire prevention and suppression, the spatial pattern of fire likelihood remains extremely uneven (Fig. 5) due to the spatial interactions between top-down (climate) and bottom-up (fuel, ignition patterns) factors.

While the projected extent of WUI and vegetation shifts due to climate change has been identified as one of the critical factors that will determine future shifts in wildfire hazard [START_REF] Moreira | Landscape--wildfire interactions in southern Europe: implications for landscape management[END_REF][START_REF] Batllori | Climate change-induced shifts in fire for Mediterranean ecosystems[END_REF]Liu and Wimberley 2016), our results also suggest that local changes in fire weather should be taken into account when projecting fire hazard in the Mediterranean region. Relative probability

Figure 1 Figure 1 :

 11 Figure 1

Figure 2 Figure 2 .Figure 3 Figure 3 .Figure 4 Figure 4 .Figure 5 Figure 5 .

 22334455 Figure 2

Table 1 .

 1 Selected explanatory variables used as inputs in boosted regression tree (BRT) models for predicting the spatial distribution of fires in Mediterranean France. Maps of the fire weather season length (FWSL) are shown in Fig. 2. Maps of the human and fuel-related variables are shown in Fig. S1. Human and land cover variables were derived from the BD Topo (IGN 2007) and Corine land cover (EEA 1994) databases, respectively.

	Variable	Unit	Data source
	Humans
	Road density	km.km -2	BD Topo
	Housing density	nb.km -2	BD Topo
	Wildland Urban interface (WUI)	km 2 .km -2	BD Topo
	Fuel/Land cover
	Shrubland area	%	Corine land cover
	Forest area	%	Corine land cover
	Landscape diversity	-	Corine land cover
	Weather
	Fire weather season length (FWSL)	days

Table 2 .

 2 Performance of spatial boosted regression tree (BRT) models in predicting the spatial distribution of fires in southern France between 1990 and 2006 and for different final fire size classes. For each model, 70% of the grid cells were used to build models and the remaining 30% were used for validation. Means and standard deviations of an ensemble of 25 BRT models are given. AUC is the area under the receiving operator curve (ROC) curve. The commission error (false positives) is the percentage of fire events misclassified as absences. The omission error is the percentage of non-fire events misclassified as presences (false negatives). The probability threshold is minimized according to the sum of these two values.

	Fire size	Omission	Commission	AUC
		error (%)	error (%)
	> 0 ha	34.0. (3.5)	31.0 (2.8)	0.73 (1.2)
	> 1 ha	33.8 (3.6)	32.6 (3.7)	0.72 (1.6)
	> 6 ha	34.7 (3.3)	31.9 (5.3)	0.72 (2.7)
	> 15 ha	35.0 (5.4)	32.7 (6.4)	0.72 (3.6)
	> 30 ha	34.7 (8.3)	32.0 (8.9)	0.72 (4.3)

Table 3 .

 3 Correlation among the mean contributions of each variable to boosted regression tree (BRT) models predicting the spatial distribution of fires for different final fire size classes. Spearman correlation coefficient (ρ) and its associated P-value are reported (*P< 0.1; **P< 0.01).

	Fire size > 0 ha	> 1 ha	> 6 ha	> 15 ha	> 30 ha
	> 0 ha	1
	> 1 ha	0.92 **	1
	> 6 ha	0.17	0.50	1
	> 15 ha	0.14	0.46	0.75	1
	> 30 ha	0.32	0.59	0.89 *	0.89 *	1

Acknowledgments

This work is a contribution to the UE 7th FP Env.1.3.1.1 FUME ''Forest fires under climate, social and economic changes in Europe, the Mediterranean and other fireaffected areas of the world,'' Grant agreement no. 243888. A doctoral research Grant to J. Ruffault was provided by the Languedoc-Roussillon (LR) region and the Centre National de la Recherche Scientifique (CNRS). This work is also a contribution to the Labex OT-Med (ANR-11-LABEX-0061) funded by the ''Investissements d'Avenir,'' French Government program of the French National Research Agency (ANR) through the A*Midex project (ANR-11-IDEX-0001-02). We thank the associate editor and two anonymous reviewers for their constructive comments, which helped us improve our paper.


Robert Chahine

Wouter J T Bos

On the role and value of β in incompressible MHD simulations

The parameter β, defined as the ratio of the pressure to the square of the magnetic field, is widely used to characterize astrophysical and fusion plasmas. However, in the dynamics of a plasma flow it is the pressure gradient which is important rather than the value of the pressure itself. It is shown that if one is interested in the influence of the pressure gradient on the dynamics of a plasma, it is not the quantity β which should be considered, but a similar quantity, depending on the pressure gradient. The scaling of this newly defined quantity is investigated using incompressible magnetohydrodynamic simulations in a periodic cylinder in the Reversed Field Pinch flow regime.

I. INTRODUCTION

In conducting fluids and plasmas, the force-balance on a fluid element contains, in addition to the pressuregradient, also a magnetic pressure gradient. To measure their relative importance, one generally considers the ratio of the hydrodynamic to magnetic pressure, also called β. In astrophysical applications, the value of β is used to characterize the plasma which is considered. Typically, low values of β are observed in the solar wind, e.g. β ≪ 1 in the mid-corona [START_REF] Gary | Plasma beta above a solar active region: rethinking the paradigm[END_REF] . The value of β in compressible magnetohydrodynamics (MHD) determines the precise behaviour and dimensionality of the system 2 . Furthermore, since β is directly related to the magnetic Mach number, its value will determine the importance of magnetosonic compared to Alfvén wave propagation in a plasma [START_REF] Andrés | Interplay between alfvén and magnetosonic waves in compressible magnetohydrodynamics turbulence[END_REF] . In the remainder of this investigation we will focus on the use of β in the magnetically confined fusion community, but the results and ideas are, we think, of broader relevance.

Let us first define β. Using Alfvénic units to normalize the plasma pressure and magnetic field (i.e. the plasma velocity is normalized by the Alfvén velocity v A ≡ B 0 / √ ρµ 0 , where B 0 is a characteristic value of the magnetic field, ρ the plasma density and µ 0 the magnetic permeability), β is given by

β = 2 P B 2 . ( 1
)
In magnetic fusion devices such as tokamaks or Reversed Field Pinches (RFPs), obtaining a high temperature and pressure in the center of the plasma is one of the indicators of good confinement. Indeed, the challenge is to confine a plasma as well as possible for a given magnetic field. The quantity β gives then a rough measure of the reactor's efficiency, since it compares the pressure P in the plasma, which indicates how high the temperature is, to the strength of the imposed magnetic field (characterized by B 2 ).

One particular definition of β, β, is defined here as

β = 2 ⟨P ⟩ ⟨B 2 ⟩ , ( 2
)
where the brackets indicate a volume average over the plasma volume of interest. Other definitions are also used [START_REF] Ja Wesson | Hydromagnetic stability of tokamaks[END_REF] based on the value of B at the wall of the plasma for instance. The quantity β is thus a key parameter in fusion design and operation, and further plays an important role in investigations of the stability of confined plasmas [START_REF] Paccagnella | Ideal toroidal stability beta limits and shaping effect for reversed field pinch configurations[END_REF] and the dynamics of magnetic islands in tokamaks [START_REF] Muraglia | Effect of the curvature and the β parameter on the nonlinear dynamics of a drift tearing magnetic island[END_REF] .

In many fusion related flows the value of β is low, of the order of a few percent or even less, and as such, the effects of pressure are therefore sometimes neglected. This approximation was for instance used in investigations of the single-helicity states in the RFX reactor [START_REF] Cappello | Bifurcation in viscoresistive MHD: The hartmann number and the reversed field pinch[END_REF][START_REF] Df Escande | Chaos healing by separatrix disappearance and quasisingle helicity states of the reversed field pinch[END_REF] , and in studies on helical modulation of the magnetic field at the boundary of the plasma 9,10 and on sawtooth mitigation [START_REF] Bonfiglio | Sawtooth mitigation in 3d mhd tokamak modelling with applied magnetic perturbations[END_REF] . Also in the prediction of the relaxed Taylor-state [START_REF] Taylor | Relaxation and magnetic reconnection in plasmas[END_REF] , the influence of β was considered negligible. These are only few of the studies where the low-β approximation is invoked to neglect the influence of pressure. This approximation, is in most of the more recent works not used, and it seems to be recognized by now that even when β is small, the pressure cannot be neglected. Indeed, in most of the more recent numerical investigations of both tokamaks and RFPs the pressure is retained in the dynamics (see for instance in references [START_REF] Glasser | The nimrod code: a new approach to numerical plasma physics[END_REF][START_REF] Stephen C Jardin | A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions[END_REF][START_REF] Huysmans | Mhd stability in x-point geometry: simulation of elms[END_REF][START_REF] Bonfiglio | Nonlinear three-dimensional verification of the specyl and pixie3d magnetohydrodynamics codes for fusion plasmas[END_REF][START_REF] Lütjens | Xtor-2f: a fully implicit newton-krylov solver applied to nonlinear 3d extended mhd in tokamaks[END_REF] reporting on sophisticated MHD codes for magnetically confined fusion applications).

The present study is not dedicated to the question whether finite β effects are important or not, but how to efficiently quantify their importance. Hereby we contribute to the understanding of the counter-intuitive observation that, even for very small values of β, the pressure still is important in determining the precise dynamics. In other terms, we illustrate why the zero-β (or constant pressure) description is not a good approximation of low-β incompressible MHD simulations, no matter how small the value of β is.

In the next section we will propose a new quantity, which we will call β ′ ∇ . In section III we will present numerical results which show that the new quantity is intrinsically different from the classical definition of β, since it has a radically different dependence on the Lundquist number. To further illustrate the importance of the new parameter, we show that finite β simulations do not converge towards the zero-β case for decreasing β. Indeed, decreasing β does not necessarily imply decreasing β ′ ∇ .

II. IS β A RELEVANT QUANTITY IN INCOMPRESSIBLE MHD SIMULATIONS?

In the MHD description, the Navier-Stokes equations are modified by the presence of the Lorentz force F L . In Alfvénic units, this gives

d t u = ν∆u -∇P + F L . ( 3
)
where u(x, t) is the velocity, P (x, t) the pressure and ν the viscosity. We consider for simplicity the incompressible isothermal formulation, where ∇ • u = 0 and the density ρ is uniform. The left hand side of this equation represents the acceleration of a plasma-fluid element. Through Newton's second law the right-hand-side gives the force-balance per unit mass. We see the contributions of three forces: viscous friction, the pressure gradient and the Lorentz-force. The Lorentz-force is given by

F L = J × B, ( 4
)
where the current density is given by the curl of the magnetic field J (x, t) = ∇ × B(x, t). Using vector identities, part of the Lorentz force can be written as the gradient of a scalar quantity, so that it can be regrouped with the pressure term. Doing so, equation ( 3) becomes

d t u -ν∆u = - 1 2 ∇(B 2 (1 + β)) + B • ∇B, ( 5
)
where β is given by expression (1). It seems from this expression that if P ≪ B 2 , the hydrodynamic pressure drops out of the equation. This leads to the intuitive result that if in a plasma we measure a low value of β, the hydrodynamic pressure can be neglected. In this limit, (5) does not need the restriction ∇ • u = 0 anymore and the description thereby simplifies.

We will here illustrate that this approximation is not straightforward. The important point is that in equation ( 5), it is the gradient of the pressure and of the magnetic pressure which play a role, and not the values of the pressures themselves. If one would like to measure the influence of the hydrodynamic pressure gradient compared to the gradient of the magnetic pressure, one needs to compare the norms of these vectors. Introducing the unit vectors

e P = ∇P ∥∇P ∥ and e B = ∇B 2 ∥∇B 2 ∥ (6)
one can write equation ( 5) as

d t u -ν∆u = - 1 2 ∥∇B 2 ∥ (e B + e P β ∇ ) + B • ∇B. (7)
In this expression β ∇ is defined by

β ∇ ≡ 2 ∥∇P ∥ ∥∇B 2 ∥ . ( 8
)
The difference between expressions ( 5) and ( 7) is obviously the presence of the gradients in the definition of β ∇ , which is essential in measuring the effects of pressure on the dynamics. Evidently a large uniform pressure will not induce any acceleration of the plasma and one can thus always change the value of the pressure without changing its gradient by adding a uniform pressure field. Furthermore, the separation of the Lorentz-force into a gradient ∇(B 2 /2) and B • ∇B seems natural when we want to consider the influence of the pressure, and it is so in the absence of motion, when the left-hand-side of equation ( 3) is zero. Computing subsequently an equilibrium field by considering a force-free magnetic state is a classical way to determine tokamak equilibria, already widely used in the early days of fusion research [START_REF] Grad | Hydromagnetic equilibria and force-free fields[END_REF][START_REF] Vd Shafranov | Plasma equilibrium in a magnetic field[END_REF] , and plasma physics in general [START_REF] Woltjer | A theorem on force-free magnetic fields[END_REF] . For this static case we have thus

1 2 ∇(B 2 (1 + β)) = B • ∇B, ( 9
)
and for small values of β, only the magnetic pressure can balance B • ∇B. However, if one focuses on the importance of the pressure on the nonlinear dynamics in equation ( 7), i.e. when u ̸ = 0, there is no reason, other than esthetics, to separate the Lorentz-force in two parts. Indeed, as soon as the plasma will start to move, the acceleration du/dt of the plasma will be governed by the pressure gradient, the Lorentz-force and the viscous stress, and the pressure gradient is no longer a priori negligible.

Note that we can ignore the contribution of the viscous diffusion of momentum on the acceleration when the dynamics become turbulent [START_REF] Tsinober | Random taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence[END_REF][START_REF] Nelkin | Universality and scaling in fully developed turbulence[END_REF] . This is plausible to hold in the present case for large values of the Lundquist number. Indeed, viscous dissipation is an essential feature in the eventual dissipation of kinetic energy of turbulent flows, but its influence on the acceleration is small, when eddy diffusion becomes more important than molecular viscous diffusion. We will not further consider it here.

What we will consider is the comparison of the forces due to pressure, and the Lorentz-force. Therefore we rewrite the momentum equation as,

d t u -ν∆u = (-e P β ′ ∇ + e F ) ∥J × B∥, ( 10
)
where e F = F L /∥F L ∥, and

β ′ ∇ ≡ ∥∇P ∥ ∥J × B∥ . ( 11
)
In order to have global measures of the importance of the different terms, we introduce, as in expression (2), the volume averaged quantities,

β ∇ ≡ 2 ⟨∥∇P ∥⟩ ⟨∥∇B 2 ∥⟩ and β ′ ∇ ≡ ⟨∥∇P ∥⟩ ⟨∥J × B∥⟩ . ( 12
)
It is only when β ′ ∇ is small that the pressure term might be negligible in the dynamics. However, if this quantity is of order unity, there is no reasonable indication that the pressure gradient can be safely ignored. We will now assess this in the context of RFP simulations and we will show that β and β ′ ∇ behave differently.

III. RESULTS FOR β AND β∇ IN STRAIGHT-CYLINDER RFP SIMULATIONS

A. Numerical set-up

In this section we will evaluate how β, β ∇ and β ′ ∇ scale in incompressible MHD simulations in a periodic cylinder with aspect ratio Γ = L z /2πr = 4, where L z and r = 1 are the cylinder length and radius, respectively. Equation ( 7) is solved together with the induction equation,

∂B ∂t = ∇ × (u × B) + λ∇ 2 B, (13)
and the condition ∇ • B = 0. Computations are performed using a pseudo-spectral solver on a Cartesian mesh of 512 × 128 × 128 gridpoints, combined with a volume-penalization technique to enforce the boundary conditions. The boundary conditions are no-slip on the cylinder wall for the velocity. The magnetic field is enforced by imposing the poloidal component of the magnetic field B p at the wall, which thereby determines the global axial current. The radial magnetic field vanishes at the wall and the axial component B z consists of a uniform, fixed contribution plus a variable part which is left free to evolve, so that the magnetic field can self-organize towards a typical RFP dynamic equilibrium. Details on the numerical method can be found in [START_REF] Morales | Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization[END_REF] . The pressure field is calculated using the incompressibility condition by solving the Poisson equation obtained from taking divergence of the momentum equation, expression (3),

∇ 2 P = ∇ • (-u • ∇u + J × B) , ( 14
)
where the time-derivative and viscous term vanish due to incompressibility. In a pseudo-spectral code, this Poisson equation can be easily solved using Fourier transforms. The boundary condition for this resolution is zero pressure on the plasma boundaries. In all simulations, we will consider the RFP regime where the magnetic geometry is self-organized into a helical structure. Typical similations using the same code and set-up are reported in [START_REF] Morales | On the effect of toroidicity on reversed field pinch dynamics[END_REF] .

In Figure 1 we show visualizations of the pressure field in cross-sections of the cylinder for a typical flow.

B. Influence of the pinch-ratio

We consider first the case of a fixed Lundquist number, S ≡ B p r/λ = 2.8×10 4 , P m ≡ ν/λ = 1 for different values of the pinch Θ = B p /⟨B z ⟩, where the overline indicates the surface average on the edge of the plasma and the brackets indicate a volume average. Error-bars here and in the following measure the temporal fluctuations of the evaluated quantities during the statistically stationary state, during which the quantities are evaluated.

In figure 2 the evolution of the different definitions of β is shown to stay constant with respect to the pinch ratio. Furthermore, β takes a value less than 20% while the quantity β ′ ∇ is greater than 80%. For the parameters considered here, the value of β, β ∇ and β ′ ∇ all three seem to be roughly independent of the pinch-ratio. Considering only this result, one might be tempted to conclude that all three quantities play a similar role, so that low β dynamics are equivalent to low β ∇ dynamics. That this is not so will now be shown.

C. Influence of the Lundquist number

We now turn to the dependence of the different βs on the Lundquist number. In Figure 3 it is observed that increasing S, β drops, approximately following a powerlaw proportional to S -1/2 . Judging from this dependence only and extrapolating, one would be tempted to say that in realistic regimes, where S is at least an order of magnitude higher, the pressure is negligible. However, considering the influence of the pressure gradient on the force balance by comparing the pressure-gradient to the Lorentz-force, it is observed that the influence of the pressure force remains comparable to the Lorentzforce for all values of S. Indeed β ′ ∇ remains order unity when S is increased, while β ∇ remains between the two other quantities.

The pressure gradient is thus not negligible in the dynamics, and this is the main result of the present work: observation of a low value of β does not mean that the pressure gradient can be neglected in the dynamics. A question is now how the results change if the we neglect the pressure anyway. This will be illustrated in the following.

IV. ON THE DIFFERENCE BETWEEN INCOMPRESSIBLE AND ZERO-β SIMULATIONS

To illustrate the importance of the foregoing we will show here that finite β simulations are radically different from zero-β simulations. Indeed, this is implicitly accepted in the fusion community and the pressure is retained in most of the recent investigations of tokamak and RFP dynamics even when β is small. However the novelty here is that this difference can be directly measured using the quantity β ′ ∇ . In order to illustrate this we compare the foregoing results with results obtained in the absence of pressure. The system we consider is thus

d t u -ν∆u = J × B, (15)
combined with equation ( 13). A similar comparison was reported in [START_REF] Ay Aydemir | Compressibility as a feature of field reversal maintenance in the reversed-field pinch[END_REF] . A theoretical study 26 explained some of the differences between the results. More recently zero-β and finite-β simulations were compared in the context of RFP relaxation in reference [START_REF] King | First-order finite-larmorradius fluid modeling of tearing and relaxation in a plasma pinch a[END_REF] .

In Figure 4 we show the evolution of the kinetic energy (E kin = (|u| 2 )/2) for two simulations at S = 22400, Θ = 7. It is observed that the dynamics are more violently fluctuating in the pressureless simulations. This is somehow expected, since pressure is known to redistribute momentum over different velocity components, thereby smoothing the flow with respect to pressureless dynamics. In the absence of a magnetic field this is a well-known difference between Navier-Stokes dynamics and the dynamics of Burgers' equation, associated with the generation of shocks. Indeed, the behavior of the pressureless equations should be very similar to the limit of very high compressibility [START_REF] Tatsumi | One-dimensional shock turbulence in a compressible fluid[END_REF] .

In Figure 5 we show the average axial magnetic field as a function of the radial coordinate. We compare the results from incompressible and zero-β simulations for Lundquist numbers ranging from S = 11200 upto S = 33600. The main observation is that we have a dichotomy where the profiles of the simulations without pressure are qualitatively different from the full incompressible dynamics. For both cases the profiles become roughly independent of S for S > 10 4 , but the asymptotic profiles are different with a deeper reversal for the pressureless dynamics.

For the incompressible runs we have a β which decreases as a function of S, as shown in Figure 3. We observe that between the lowest and the highest values of the Lundquist number, β has decreased by an order of magnitude. However, even at the highest value of S, the radial profile of the axial magnetic field has not approached the incompressible solution towards the zero-β solution. This clearly illustrates that the pressure plays an important role and that it is not β which can measure the importance of the pressure force on the dynamics. Indeed the value of β ′ ∇ is of order unity for all the incompressible simulations. Therefore, considering β ′ ∇ one would be less tempted to ignore the pressure term, since it remains of order unity, even when β decreases.

V. CONCLUSION

We have argued that the quantity which measures the importance of the pressure on the dynamics in incompressible MHD simulations is β ′ ∇ = ⟨∥∇P ∥⟩/⟨∥J × B∥⟩. Indeed, it is shown that even when β might vary considerably, in particular when increasing the Lundquist number, the value of β ′ ∇ remains of order unity and the influence of the pressure gradient cannot be neglected. Our results considered the simplest case of incompressible isothermal flow. We have not carried out simulations of compressible MHD, but we expect that the coupling between the β quantities and the Lundquist number will be affected and that their dependence wil be somewhat weaker. Furthermore, we expect that the general behaviour in compressible similations will qualitatively be intermediate between those of the pressureless and the incompressible cases considered in the previous section. Assessing this by complexifying the description is however an interesting issue to be further explored.

These results do not completely invalidate the insights obtained in the zero-β simulations. Indeed, a lot of qualitative features observed without pressure can carry over to the actual incompressible MHD dynamics. Some of this coincidence is fortunate and it would be wise to validate these results using more a more complete MHD description involving the pressure term. However, MHD is already a very simplified description of the actual plasma and ignoring the pressure gradient is only one approximation among many others (for instance, non-uniform resistivity influences the RFP dynamics importantly [START_REF] Onofri | Effects of the resistivity profile on the formation of a reversed configuration and single helicity states in compressible simulations of the reversed-field pinch[END_REF][START_REF] Futatani | Dynamic equilibria and magnetohydrodynamic instabilities in toroidal plasmas with non-uniform transport coefficients[END_REF] ). It is our viewpoint that MHD simulations should be considered as tools to discover possible physical mechanisms, which can be assessed in realistic experiments. Furthermore, neglecting the pressure in MHD is less dramatic than in hydrodynamics. Indeed, the presence of the magnetic pressure will take over part of the role normally played by the hydrodynamic pressure. This is not so in the absence of a magnetic field, where the pressureless equations (Burger's equation) will lead to a far more extreme dynamics with large gradients and shocks which change completely the flow-dynamics. However even in the present case the pressureless dynamics are more violent than the full incompressible system, as illustrated in Figure 4.

We conclude by stressing that, by no means, we claim that β is an unimportant parameter to characterize a plasma, neither do we criticise the zero-β assumption as a simplifying hypothesis. What should be retained from this study is that the importance of the pressure forces can directly be measured by comparing the pressure gra-dient term to the Lorentz-force. The resulting quantity β ′ ∇ which measures this is intrinsically different from β, which is for instance illustrated by a markedly different dependence on the Lundquist number.

FIG. 1 .

 1 FIG. 1. Visualization of (a) z = 0 (b) z = L/2 cross-sections of the pressure field in the RFP in the turbulent state S = 33600 and Θ = 7, in the statistically steady state.

FIG. 2 .FIG. 3 .

 23 FIG. 2. Importance of the pressure force on the dynamics as measured by β, β∇ and β ′ ∇ for different values of the pinch ratio with S = 2.8 × 10 4 .

FIG. 4 .

 4 FIG.4. Time evolution of the kinetic energy for simulations with S = 22400 Θ = 7, in the presence (blue) and absence (red) of the pressure gradient.

8 FIG. 5 .

 85 FIG.5. Importance of the pressure force to determine the magnetic field for different values of the Lundquist number.

ACKNOWLEDGMENTS

Simulations were carried out using the facilities of the PMCS2I ( École Centrale de Lyon), IDRIS (Project No. 22206) and P2CHPD. Support of the French Research Federation for Fusion Studies (FR-FCM) is acknowledged.