Jean-Alix Barrat
email: barrat@univ-brest.fr.

Germain Bayon

Stefan Lalonde

Calculation of cerium and lanthanum anomalies in geological and environmental samples by

Keywords: Rare earth elements, La anomaly, Ce anomaly, linear extrapolation, geometric extrapolation

The determination of La and Ce anomalies in natural waters, biological samples and sedimentary rocks can provide unique information on biogeochemical processes in Earth surface environments. Over the last decades, several approaches have been used for calculating La/La* and Ce/Ce*, based on the comparison between measured and theoretical abundances (La* and Ce*) extrapolated from neighboring rare earth element concentrations normalized to chondritic or shale reference values. These extrapolations can be achieved either linearly or semi-logarithmically ("geometrically"), both methods being used in the literature in the absence of any consensus. We show here, using a database of rocks exhibiting no La and Ce anomaly, that the linear extrapolation of La and Ce abundances can result in markedly different results depending on whether chondritic or shale values are used for normalization. The geometric extrapolation allows consistent calculation of La and Ce anomalies for the entire compositional range tested in this study, regardless of whether data are normalized to chondritic or shale reference values. The differences between linear and geometric extrapolations are illustrated by a few selected examples from the literature, including various carbonate rock and seawater samples, further demonstrating that linear extrapolation can result in erroneous estimates of La and Ce anomalies. We thus propose that La/La* and Ce/Ce* ratios in all geological and environmental samples should be determined using the geometric extrapolation only.

1/ Introduction

Over the last sixty years, rare earth elements (REE) have become one of the most studied groups of elements in Earth sciences. These elements have the particularity of having an extremely coherent geochemical behavior that can be classically described with the famous "Masuda-Coryell plots" [START_REF] Masuda | Regularities in variation of relative abundances of Lanthanide elements and an attempt to analyse separation index patterns of some minerals[END_REF][START_REF] Coryell | A procedure for geochemical interpretation of terrestrial rare-earth abundance patterns[END_REF], known today as "rare earth patterns". The principle of these diagrams is simple. Rare earth elements are ordered by increasing atomic number, and their abundances in any given sample are normalized to a set of reference values that generally correspond to average concentrations for chondrites (e.g., [START_REF] Anders | Abundances of the elements: meteoritic and solar[END_REF][START_REF] Pourmand | A novel extraction chromatography and MC-ICP-MS technique for rapid analysis of REE, Sc and Y: Revising CI-chondrite and Post-Archean Australian Shale (PAAS) abundances[END_REF][START_REF] Barrat | Geochemistry of CI chondrites: Major and trace elements, and Cu and Zn isotopes[END_REF][START_REF] Palme | Solar System Abundances of the Elements[END_REF] or shales (e.g., [START_REF] Nance | Rare earth element patterns and crustal evolution-I. Australian post-Archean sedimentary rocks[END_REF][START_REF] Pourmand | A novel extraction chromatography and MC-ICP-MS technique for rapid analysis of REE, Sc and Y: Revising CI-chondrite and Post-Archean Australian Shale (PAAS) abundances[END_REF][START_REF] Bau | The European Shale: an improved data set for normalization of rare earth elements and yttrium concentrations in environmental and biological samples from Europe[END_REF]. The first advantage of these diagrams is that the Oddo-Harkins effect (i.e. chemical elements with even atomic numbers are more abundant than adjacent odd atomic number elements) disappears with normalization. Smooth REE patterns are generally obtained for most geological and environmental samples, mostly because all REE are trivalent and not significantly decoupled from each other under relevant physico-chemical conditions. There are notable exceptions, however. For instance, europium (Eu 2+ and Eu 3+ ) and cerium (Ce 3+ and Ce 4+ ) can be found in two valence states in geological samples. During particular magmatic or environmental processes, these two elements can be significantly decoupled from their neighboring REE, thereby producing specific elemental depletion or enrichment in normalized REE patterns.

Additionally, significant anomalies in La, Sm, Gd and Tm can be also found in terrestrial rocks or natural waters, unrelated to any effect of valence. The occurrence of Tm anomalies in terrestrial rocks are thought to be inherited from the building blocks that formed our planet [START_REF] Dauphas | Thulium anomalies and rare earth element patterns in meteorites and Earth: Nebular fractionation and the nugget effect[END_REF][START_REF] Barrat | Evidence from Tm anomalies for non-CI refractory lithophile element proportions in terrestrial planets and achondrites[END_REF]. The origin of other anomalies is not always well understood, except when they are the result of environmental pollution (e.g., Bau and Dulski, 1996b;Kulaksiz andBau, 2007, 2013;[START_REF] Merschel | Rare earth elements in the aragonitic shell of freshwater mussel Corbicula fluminea and the bioavailability of anthropogenic lanthanum, samarium and gadolinium in river water[END_REF][START_REF] Ma | Rare earth elements in the Pearl River Delta of China: Potential impacts of the REE industry on water, suspended particles and oysters[END_REF][START_REF] Goff | Compound-specific recording of gadolinium pollution in coastal waters by great scallops[END_REF][START_REF] Valdés-Vilchis | Rare Earth Elements and Yttrium (REE+Y) patterns in recent Anadara brasiliana shells from Playa Norte, Barra de Cazones (Veracruz, Mexico): Evidence of anthropogenic contamination linked to river output?[END_REF]. Among these anomalies, Ce anomalies have received considerable interest. The distinctive behavior of Ce in the marine environment was discovered more than 50 years ago (e.g., [START_REF] Goldberg | Rare earth distributions in the marine environment[END_REF]. The first reliable REE analyses of seawater, various authigenic phases, and ichtyoliths showed very early on that aqueous phases, marine carbonates and other seawater archives could display pronounced negative Ce anomalies (e.g., Piper, 1974;Elderfield and Pagett, 1986 and references therein). The role of Fe and Mn oxides in the development of these anomalies was subsequently proposed because Mn-rich nodules and crusts displayed complementary positive anomalies in Ce (e.g., [START_REF] Goldberg | Rare earth distributions in the marine environment[END_REF]Piper, 1974). Today, it is generally well accepted that the decoupling of Ce from other REEs in the oceans, or more generally in aqueous environments, mainly results from oxidative scavenging of Ce by Fe and Mn hydroxides (e.g., [START_REF] Bau | Oxidative scavenging of cerium on hydrous Fe oxide: Evidence from the distribution of rare earth elements and yttrium between Fe oxides and Mn oxides in hydrogenetic ferromanganese crusts[END_REF]. The occurrence of Ce anomalies in natural waters and in the biogenic or authigenic phases that precipitate from them is used as a proxy for oxidative conditions in Earth surface environments (e.g., [START_REF] German | Application of the Ce anomaly as a paleoredox indicator: The ground rules[END_REF][START_REF] German | Redox cycling of rare earth elements in the suboxic zone of the Black Sea[END_REF]Bau et al., 1997;[START_REF] Wallace | Oxygenation history of the Neoproterozoic to early Phanerozoic and the rise of land plants[END_REF][START_REF] Bellefroid | Constraints on Paleoproterozoic atmospheric oxygen levels[END_REF]. Cerium anomalies are also of interest for magma-related studies, although their application to basalts and other igneous rocks remains limited. Lavas from subduction zones and oceanic islands occasionally show negative Ce anomalies, interpreted as reflecting a recycled sedimentary component in their mantle sources (e.g., [START_REF] Shimizu | Ce and Nd isotope geochemistry on island arc volcanic rocks with negative Ce anomaly: existence of sources with concave REE patterns in the mantle beneath the Solomon and Bonin island arcs[END_REF][START_REF] Class | Ce anomalies in Gough Island lavas -Trace element characteristics of a recycled sediment component[END_REF]. In contrast, zircons frequently show excess Ce (Ce 4+ having the same ionic radius as Zr 4+ ), indicating preferential selective incorporation of Ce relative to neighboring REE during crystal growth. The resulting positive Ce anomalies could reflect the Ce 4+ /Ce 3+ ratios of their parental magmas, and therefore provide constraints on oxygen fugacity (e.g., [START_REF] Burnham | An experimental study of trace element partitioning between zircon and melt as a function of oxygen fugacity[END_REF][START_REF] Trail | Ce and Eu anomalies in zircon as proxies for the oxidation state of magmas[END_REF][START_REF] Smythe | Magmatic oxygen fugacity estimated using zircon-melt partitioning of cerium[END_REF].

Unlike Ce, the La anomaly in geological and environmental samples has received much less attention. Lanthanum anomalies represent a common feature in seawater and marine authigenic phases (e.g., [START_REF] Elderfield | The oceanic chemistry of the rare-earth elements[END_REF][START_REF] Tostevin | Effective use of cerium anomalies as a redox proxy in carbonate-dominated marine settings[END_REF], but remains to date undocumented in igneous rocks. The cause of these anomalies is still poorly understood but could relate to the higher stability of La in solution relative to the other light REE [START_REF] Baar | Rare earth elements in the Pacific and Atlantic oceans[END_REF][START_REF] Byrne | Rare earth element scavenging in seawater[END_REF][START_REF] Byrne | The influence of phosphate coprecipitation on rare earth distributions in natural waters[END_REF]. Previous studies have suggested that marine barite could play a role in the development of La anomalies in seawater [START_REF] Grenier | Differentiating lithogenic supplies, water mass transport, and biological processes on and off the Kerguelen Plateau using rare earth element concentrations and neodymium isotopic compositions[END_REF]. Indeed, [START_REF] Hein | Multifarious barite-forming environments along a rifted continental margin, Southern California Borderland[END_REF] have reported positive La anomalies in this phase. It is likely that the La excesses they measured, however, are only analytical artefacts generated in the plasma, notably isobaric interferences from Ba (e.g., 138 BaH + on 139 La + ?), and need to be confirmed. Recently, a number of pioneering studies have demonstrated that biological activity can also fractionate light-REE (e.g., [START_REF] Pol | Rare earth metals are essential for methanotrophic life in volcanic mudpots[END_REF][START_REF] Semrau | Metals and Methanotrophy[END_REF]Bayon et al., 2020a). [START_REF] Wang | Lanthanum anomalies as fingerprints of methanotrophy[END_REF] reported positive La anomalies in methanotrophic mussels at submarine methane seeps, interpreted as resulting from microbial enzymatic activity related to aerobic methane oxidation. These results have suggested that the La anomaly could be serve as a diagnostic tool for tracing past biological activity related to aerobic methanotrophy. Moreover, La also represents an emerging pollutant in modern environments due to its widespread industrial use in the industry in magnetic alloys and or catalysts for gasoline engines [START_REF] Kulaksiz | Anthropogenic dissolved and colloid/nanoparticle-bound samarium, lanthanum and gadolinium in the Rhine River and the impending destruction of the natural rare earth element distribution in rivers[END_REF][START_REF] Merschel | Rare earth elements in the aragonitic shell of freshwater mussel Corbicula fluminea and the bioavailability of anthropogenic lanthanum, samarium and gadolinium in river water[END_REF]. The emergence of anthropogenic La issues in Earth surface environments calls for a better understanding of the mechanisms that control the decoupling of La from neighboring REE in aquatic environments.

Over the last few decades, different methods have been proposed for calculating La and Ce anomalies in geological and environmental samples, yet no consensus exists among geochemists on best practices for these calculations. In the literature, historical conventions usages or habits prevail. These anomalies are calculated from normalized concentrations and by interpolating or extrapolating La and Ce concentrations (La* and Ce*) assuming smooth REE patterns with linear or logarithmic scales. This results in anomaly values that can be very different from one study to another, and cannot be compared. Here we show that that some commonly used approaches for calculating La and Ce anomalies can lead to aberrant results, hence our call for a standardization of the calculation.

2/ The different ways to calculate La and Ce anomalies

2.1/ Normalization values

The patterns of average CI chondrites and average shales do not show significant anomalies in La and Ce when normalized to each other. Therefore, one would expect to calculate similar La/La* or Ce/Ce* values in any given sample following normalization to either chondritic or shale reference values. We will see below that this is not necessarily the case. In this work, we used the average of the Orgueil chondrite concentrations measured by [START_REF] Barrat | Geochemistry of CI chondrites: Major and trace elements, and Cu and Zn isotopes[END_REF], and the Post Archean Australian Shale (PAAS) average obtained by [START_REF] Pourmand | A novel extraction chromatography and MC-ICP-MS technique for rapid analysis of REE, Sc and Y: Revising CI-chondrite and Post-Archean Australian Shale (PAAS) abundances[END_REF], which was recalculated relative to our standard values to correct for a slight calibration bias [START_REF] Barrat | A new chemical separation procedure for the determination of rare earth elements and yttrium abundances in carbonates by ICP-MS[END_REF]. These preferred normalization values are given in Table 1. In the following, XCI and XSN corresponds to the element X concentrations normalized to chondritic or shales values, respectively.

2.2/ The calculation of Ce and La anomalies

By definition, an anomaly visualized in a normalized-REE diagram for an element (X) can be quantified by dividing its measured abundance by its theoretical concentration in the absence of any anomaly (X*). This latter can be calculated by interpolation or extrapolation using the normalized abundances of for neighboring elements, assuming a smooth REE pattern.

The measured/theoretical ratio elemental ratio (X/X*) thus makes it possible to quantitatively measure a positive (X/X*>1) or a negative (X/X*<1) anomaly for this element.

Many approaches to calculating Ce* and La* have been proposed in the last few decades. In the case of magmatic rocks or zircons, which are devoid of La anomalies, Ce* is given by the geometric mean of the normalized concentrations of La and Pr, or interpolated "semi-logarithmically" between La and Nd when Pr abundances are not determined:

Ce/Ce*= CeCI/(LaCI x PrCI) 1/2 (1) or Ce/Ce*= CeCI/(LaCI 2/3 x NdCI 1/3 ) (2)
The choices of a geometric mean or the semi-log interpolation are justified here by the fact that the REE patterns are plotted in semi-log diagrams, where linear abscissa corresponds to the number of protons (Z) of REE and ordinates refer to normalized concentrations conventionally displayed in logarithmic scale.

During the 1970s, new developments in sedimentary geochemistry were accompanied by with increasing use of reference shale values for normalizing measured REE abundances (e.g., Piper, 1974). In these early studies, Ce* was interpolated as described above (e.g., Piper (1974) with Equation 2) but also linearly, hence considering that shale-normalized REE patterns were sometimes plotted using linear scales for both ordinates and abscissae. Additionally, prior to the 1990s and the advent of ICP-MS, REE abundances were mostly measured by INAA and Pr concentrations were rarely determined, meaning that Ce* was generally interpolated between La and Nd (e.g., [START_REF] Elderfield | Negative cerium anomalies in the rare earth element patterns of oceanic ferromanganese nodules[END_REF], and occasionally between La and Sm [START_REF] Toyoda | Rare earth elements of Pacific pelagic sediments[END_REF]:

Ce/Ce*= 3 CeSN/(2 LaSN + NdSN) (3) Ce/Ce*= 5 CeSN/(4 LaSN + SmSN) (4)
It was only with the development of plasma source mass spectrometry in the mid-1980's that the simultaneous determination of the concentrations of all the REE's in the a given samples was greatly facilitated became routine, and the number of analyses that also includeding Pr increased considerably. The calculation of the Ce anomaly with Ce* linearly interpolated between La and Pr, is given by the following equation (e.g., [START_REF] Baar | Rare-earth distributions with a positive Ce anomaly in the Western North Atlantic Ocean[END_REF]:

Ce/Ce*= 2 CeSN/(LaSN + PrSN) (5)
However, many natural waters, biogenic or authigenic precipitates, and sediments display La anomalies, thereby biasing the determination of the Ce anomaly using the above formula. With this approach, the calculation of Ce* calculated with La abundances using either linear or geometrical interpolations (Fig. 1) resulted in biased results, generating for instance Ce/Ce* ratios < 1 even in the case of samples devoid of any Ce anomaly. For this reason, Bau and Dulski (1996a) developed a Ce/Ce* vs. Pr/Pr* diagram (Fig. 2, with Ce* linearly interpolated between La and Pr (equation 5), and Pr* linearly interpolated between Ce and Nd, Pr*SN=(CeSN +NdSN)/2). This plot can be used to identify whether La anomalies are present or not, and whether Ce/Ce* ratios < 1 correspond to true negative Ce anomalies or not. This diagram, which was also popularized by [START_REF] Webb | Rare earth elements in Holocene reefal microbialites: a new shallow seawater proxy[END_REF], is frequently used today.

We will come back to it later.

Since one cannot properly interpolate Ce* using La with Pr or Nd abundances, one can instead extrapolate Ce* and La* with Pr and Nd abundances (Fig. 1). To avoid ambiguity, we use here the symbols X* g and X* l for the geometric (semi-log) and linear extrapolations, respectively, of the theoretical concentrations of X. The following equations give LaSN *g , CeSN *g , LaSN *l , and CeSN *l [e.g., [START_REF] Bolhar | Characterisation of early Archaean chemical sediments by trace element signatures[END_REF] for the linear interpolations and [START_REF] Lawrence | Rare Earth Element and Yttrium variability in South East QueenslandWaterways[END_REF] for the geometric interpolations], and of course similar equations can be written for the chondritic normalization:

LaSN *g = PrSN 3 /NdSN 2 (6) CeSN *g = PrSN 2 /NdSN (7) LaSN *l = 3 PrSN -2 NdSN (8) CeSN *l = 2 PrSN -NdSN (9)
These are the equations used by most teams working today on sedimentary rocks or natural waters, without consensus on whether geometric or linear interpolations should be used or not. Questions arise about whether linear or geometric extrapolation allows for a better We have built a database including a total of 286 magmatic rocks, covering a large range of light-REE depletion or enrichment, in order to evaluate the utility ability of both linear and geometric extrapolations to meet these requirements [START_REF] Hamelin | A low δ 7 Li lower crustal component: evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Hamelin | Sr-Nd-Hf isotopes along the Pacific Antarctic Ridge from 41 to 53[END_REF][START_REF] Cordier | Time scales of melt extraction revealed by distribution of lava composition across a ridge axis[END_REF][START_REF] Daoud | A LREEdepleted component in the Afar plume: further evidence from Quaternary Djibouti basalts[END_REF][START_REF] Pelleter | Melilite-bearing lavas in Mayotte (France): An insight into the mantle source below the Comores[END_REF][START_REF] Barrat | Evidence from Tm anomalies for non-CI refractory lithophile element proportions in terrestrial planets and achondrites[END_REF][START_REF] Caroff | Kersantites and associated intrusives from the type locality (Kersanton), Variscan Belt of Western Armorica[END_REF][START_REF] Pelleter | Melilite-bearing lavas in Mayotte (France): An insight into the mantle source below the Comores[END_REF]. The chosen suite of igneous rocks ranges from highly-depleted MORBs to highlyenriched alkaline rocks through lamprophyres, and some evolved rocks [(La/Sm)CI = 0.24 -13.4]. All of these rocks were analyzed using the same procedure and calibration to avoid any potential analytical bias (e.g., [START_REF] Barrat | Geochemistry of CI chondrites: Major and trace elements, and Cu and Zn isotopes[END_REF]. Finally, and obviously most importantly, these rocks do not have Ce, nor La anomalies: their "correct" La/La* or Ce/Ce* ratios are hence ~1.

We calculated the Ce/Ce* ratio with Ce* interpolated geometrically between La and Pr (equation 1) for the rocks in our database. This ratio varies from 0.92 to 1.11 only, with most of the samples exhibiting Ce/Ce* between 0.95 and 1.05, confirming the lack of significant Ce anomaly (Fig. 2a). The curvature of some of the patterns explains mostly is largely responsible for this range. In the CeSN/CeSN* vs. PrSN/PrSN* plot of Bau and Dulski (1996), where Ce* and Pr* are linearly interpolated, only half of studied rock samples is are located in the panel attributed to samples having no Ce and La anomalies; the other half being scattered in an area corresponding to positive anomalies in Ce with either positive or negative La anomaly. This discrepancy is somewhat surprising, as one would have expected the vast majority of studied rocks to fall within the area corresponding to samples having no anomalies. This can be explained by the fact that the rocks used in our database display a much greater LREE compositional range relative to those used by Bau and Dulski (1996).

Next, we use Ce/Nd vs. Pr/Nd plots to compare linear or semi log extrapolations for Ce, normalizing ratios to either chondritic (Fig. 3a) or PAAS values (Fig. 3b). Indeed, the Ce* g /Nd and Ce* l /Nd ratios define parabolas and straight lines in these plots, respectively: In eq. 10 and 11, I just removed some extra spaces.

The rocks of our database having no Ce anomaly, they can be directly compared to these curves or lines. Figure 3 shows directly that the parabolas calculated using a geometric extrapolation reproduce very satisfactorily the Ce/Nd ratios of the rocks over the whole range of Pr/Nd ratios considered in this study. On the other hand, the lines corresponding to the linear extrapolation allow for an acceptable approximation of Ce* in a given range of values only.

Note that in each diagram the parabola and the line are tangent to the point (1,1). If we consider that both interpolations give acceptable results when Ce* g /Ce* l is between 1 and 1. We followed the same approach for La anomalies. In La/Nd vs. Pr/Nd plots, the La* g /Nd and La* l /Nd ratios define cubic curves and straight lines, respectively: LaCI* g /NdCI = (PrCI/NdCI) 3 and LaSN* g /NdSN = (PrSN/NdSN) 3 (12) LaCI* l /NdCI = 3 (PrCI/NdCI) -2 and LaSN* l /NdSN = 3 (PrSN/NdSN) -2 (13)

As shown above, the curves corresponding to the geometric extrapolation superimpose well on the correlation trends displayed by the rock data used in this study (Fig. 5). The straight lines corresponding to the linear extrapolation only allow a good estimation of the La/Nd ratios for the patterns exhibit little REE decoupling (i.e., when (Pr/Nd)CI or (Pr/Nd)SN close to 1).

Calculation of La anomalies using the linear extrapolation even appears to be inconsistent for many samples. When the data are normalized to chondritic reference values, the linear extrapolation underestimates considerably La* values, leading to anomalously high La/La* l ratios. The situation is much more problematic when data are normalized to PAAS values, especially for the most light-REE depleted samples. Not only does Equation 8underestimate LaSN*, but when (Pr/Nd)SN ≤ 2/3, LaSN* l ≤ 0 and the LaSN/LaSN* l ratio becomes negative or tends toward -∞ when (Pr/Nd)SN is just below 2/3. The LaSN/LaSN* l ratio also tends toward +∞ when (Pr/Nd)SN is just above 2/3. These cases are not uncommon on Earth: the LaSN/LaSN* l ratios calculated with our database range from -216.3 to 103.5, while much smaller ranges around 1 are obtained with other La/La* calculations (Fig. 6). As for the Ce anomaly, the ratios LaSN/LaSN* g and LaCI/LaCI* g are perfectly proportional, and the La/La* g and La/La* l ratios are not strongly correlated, even when the latter ratios are not aberrant. The range of (Pr/Nd)SN ratios for which La anomaly calculations are similar for linear or geometric interpolations (La* g /La* l <1.05) is narrow and only between 0.9 and 1.16.

These results demonstrate that La and Ce anomalies calculated using linear extrapolation can lead to biased or even aberrant values in many cases. Moreover, the calculated anomalies are very dependent on the type of normalization used. On the other hand, the results obtained with our database indicate that the use of geometric extrapolation results in more reliable estimates of La and Ce anomalies, whatever regardless of whether chondrite or shale are used for normalization, and this applies for the whole range of compositions tested in this study.

3/ Some examples

The theoretical ground discussed above is illustrated below for a few examples taken from the literature (Table 1). It is not our goal here to discuss the differences obtained between the anomalies calculated for each type of rocks or waters, normalizations and extrapolations.

We have chosen to discuss in more detail the case study specific cases of seawater samples and biogenic carbonates, in order to show how the choice of extrapolation can affect data interpretation.

Carbonates

We selected 5 series of carbonate samples corresponding to microbialites or stromatolites of different ages: the 3.45-Ga-old Strelley Pool stromatolites [START_REF] Van Kranendonk | New geological and trace element evidence from 3.45 Ga stromatolitic carbonates in the Pilbara Craton: support of a marine, biogenic origin and for a reducing Archaean ocean[END_REF]; the 2.84-Ga-old Mushandike stromatolites [START_REF] Kamber | Geochemistry of late Archaean stromatolites from Zimbabwe: evidence for microbial life in restricted epicontinental seas[END_REF]; the 2.52-Ga-old

Campbellrand stromatolites [START_REF] Kamber | The geochemistry of late Archaean microbial carbonate: Implications for ocean chemistry and continental erosion history[END_REF]; the late Devonian reefal carbonates from the Lennard Shelf (Nothdurft et al., 2004); and the Holocene microbialites from the Heron Reef [START_REF] Webb | Rare earth elements in Holocene reefal microbialites: a new shallow seawater proxy[END_REF]. All data used here were obtained in the same laboratory (ACQUIRE, Brisbane) following similar analytical procedures, and are of excellent quality.

These series of samples do not of course cover the full compositional range existing for such these types of carbonates, but nevertheless display important variations in various REE signatures.

We plotted these analyses in the Ce/Ce* vs. Pr/Pr* diagram of Bau and Dulski (1996a), which can be used, as discussed above, to identify the presence of Ce and La anomalies (Fig. 7a). In this diagram, Ce* and Pr* are linearly interpolated between La and Pr and Ce and Nd respectively. All but 2 samples have a Ce/Ce* ratio<1 when calculated in that this way, but only those with a Pr/Pr* ratio >1 have a true negative Ce anomaly, due to the bias introduced by the La anomalies. We calculated the Ce and La anomalies using both geometrical and linear extrapolation from PAAS-normalized concentrations. The obtained CeSN/CeSN* g and CeSN/CeSN* l ratios are very similar (Fig. 8a) for most samples. Only 4 samples from Strelley

Pool deviate significantly from the trend, their CeSN* l certainly being underestimated. The LaSN/LaSN* g and LaSN/LaSN* l ratios are for most samples reasonably well correlated, but the Holocene samples markedly deviate from the trend with LaSN/LaSN* l ratios greater than La SN /LaSN* g , and 4 Strelley Pool samples display outlier LaSN/LaSN* l values (=-27.7 to -4.6).

These calculations show that in many cases, one can use either linear or geometric extrapolations to estimate La or Ce anomalies, without detecting an anomaly. This is particularly the case when the patterns are little or not fractionated, but linear extrapolation can nevertheless generate artifacts or even aberrant values. To avoid the latter, we recommend using only the geometrical extrapolation to calculate La or Ce anomalies.

Although the Ce/Ce* vs. Pr/Pr* diagram proposed by Bau and Dulski (1996a) can bring useful insights for discussing REE patterns and the origin of La and Ce anomalies, it also suffers from inherent drawbacks that are briefly described below:

-The Ce/Ce* ratio used in this diagram depends on a Ce* value linearly interpolated between La and Pr. This ratio does not allow a correct quantification of the anomaly, because in addition to the problems related to the linear interpolation, it can be largely biased for the samples with an anomaly in La, as already reported by these authors -the Pr/Pr* ratio allows for the detection of samples with positive or negative anomalies in La, but the diagram does not allow the quantification of the latter.

For all the above-mentioned reasons, we propose using instead the Ce/Ce* g vs. La/La* g diagram, which is best better suited for illustrating whether any given sample displays La or Ce anomalies, and which also provides direct quantification of these anomalies. Additionally, the use of the geometric extrapolation ensures that calculated anomalies in this diagram are independent of the type of reference used for normalization, but and above all it avoids the calculation of erroneous Ce/Ce* or La/La* ratios due to artifacts linked to linear extrapolation.

For the carbonate samples selected here, the Ce/Ce* g vs. La/La* g diagram shows that these carbonates exhibit a wide range of positive La anomalies, but also allows one to identify a clear distinction between those Archean samples characterized by the absence of marked negative Ce anomalies (Ce/Ce* g ≥ 1), in contrast with the Devonian or Holocene carbonates.

Seawater

We employ used here a previously published REE database for seawater samples (n=1649; Bayon et al., 2020b). We normalized the concentrations with PAAS and examined reported (Ce/Nd)SN and (La/Nd)SN vs. (Pr/Nd) SN plots systematics (Fig. 9). The (Pr/Nd)SN ratios range from 0.49 to 1.19: 62% of the analyses have (Pr/Nd)SN ratios <0.82, and thus have Ce/Ce* l and Ce/Ce* g ratios that differ by more than 5%; 97% of the analyses have (Pr/Nd)SN ratios<0.9, and thus have La/La* l and La/La* g ratios that differ by more than 5%. The position of the points with respect to the calculated curves and lines shows unambiguously that the choice of the extrapolation method is critical here, as the La* l or Ce* l concentrations are most often underestimated by the calculation, or even aberrant (La* l <0 for many samples).

In order to illustrate the pitfalls of using the linear extrapolation for the case of seawater samples, we chose two hydrographic stations from the China Sea [START_REF] Alibo | Dissolved rare earth elements in the South China Sea: geochemical characterization of the water masses[END_REF] and the Kerguelen Plateau [START_REF] Grenier | Differentiating lithogenic supplies, water mass transport, and biological processes on and off the Kerguelen Plateau using rare earth element concentrations and neodymium isotopic compositions[END_REF], and calculated examined the vertical profiles of La anomalies calculated by normalizing the data with respect to both CI-chondrite and PAAS, extrapolating La* linearly or geometrically (Fig. 10). For both stations, LaSN/LaSN* l ratios are always much larger than those estimated geometrically by normalizing with PAAS or with CIchondrite: for the first station, LaSN/LaSN* l ratios are 1.30 to 1.42 times larger than LaSN/LaSN* g ratios, and for the second station they are 1.54 to 3.71 times larger. The La anomalies estimated with the LaSN/LaSN* l ratios are obviously strongly exaggerated, and these calculations must be rejected. Note that the LaCI/LaCI* l ratios are very close to the LaCI/LaCI* g or LaSN/LaSN* g ratios, and similar or identical profiles are obtained with these three ratios. These results adds further support that the use of geometric extrapolation is best suited for calculating La and Ce anomalies relative to the linear extrapolation. The fact that the LaCI/LaCI* l ratios are correct here is fortuitous, and is easily explained. The (Pr/Nd)CI ratios are higher than the (Pr/Nd)SN ratios, and are then in the range of values for which geometric and linear extrapolations give equivalent results.

4/ Conclusion

  CeCI* g /NdCI = (PrCI/NdCI) 2 and CeSN* g /NdSN = (PrSN/NdSN) 2 (10) CeCI* l /NdCI = 2 (PrCI/NdCI) -1 and CeSN* l /NdSN = 2 (PrSN/NdSN) -1 (11)

  598

Figure 1 .

 1 Figure 1. The different ways of calculating La* and Ce*.

Figure 2 .

 2 Figure 2. The 286 samples devoid of true La and Ce anomalies of the database used to compare linear and geometric extrapolations for calculating La* and Ce*, are plotted in a Ce/Ce* vs. La/Sm plot (a) and in the Ce/Ce* vs. Pr/Pr* plot of Bau and Dulski (1996a) (b). Ce* is geometrically interpolated in (a) and linearly interpolated in (b).

Figure 3 .

 3 Figure 3. Ce/Nd vs. Pr/Nd plots for the samples devoid of true La and Ce anomalies of our database, used here to compare linear (blue line) and geometric (red parabola) extrapolations for Ce*. The data are normalized with CI chondrite (a) or with PAAS (b).

Figure 4 .

 4 Figure 4. CeCI/CeCI* l vs. CeCI/CeCI* g (a) and CeSN/CeSN* l vs. CeSN/CeSN* g (b) plots for the samples devoid of true Ce anomalies used to test the different extrapolations. Notice the ranges of values obtained.

Figure 5 .

 5 Figure 5. La/Nd vs. Pr/Nd plots for the samples devoid of true La and Ce anomalies of our database used to compare linear (blue line) and geometric (red cubic) extrapolations for Ce*. The data are normalized with CI chondrite (a) or with PAAS (b).

Figure 6 .

 6 Figure 6. LaCI/LaCI* l vs. LaCI/LaCI* g (a) and LaSN/LaSN* l vs. LaSN/LaSN* g (b) plots for the samples devoid of true La anomalies used to test the different extrapolations. Notice the ranges of values obtained.

Figure 7 .

 7 Figure 7. Selected carbonates are plotted in the Ce/Ce* vs. Pr/Pr* plot of Bau and Dulski (1996a) (a) where Ce* and Pr* are linearly interpolated, and in a Ce/Ce* vs. La/La* where Ce* and La* are geometrically interpolated. Notice the different Ce/Ce* ranges obtained. See text for more details.

Figure 8 .

 8 Figure 8. CeSN/CeSN* l vs. CeSN/CeSN* g (a) and LaSN/LaSN* l vs. LaSN/LaSN* g (b) plots for the selected carbonates used to test the different extrapolations (same caption as Fig. 7).

Figure 9 .

 9 Figure 9. (Ce/Nd)SN (a) and (La/Nd)SN (b) vs. (Pr/Nd)SN plots for seawater. The shaded areas correspond to the range in Pr/Nd ratios for which the linear and geometric extrapolations are similar (1<X*l/X*g<1.05), correspond to the areas that have been shaded. A large proportion of the samples are outside the ranges where the linearly extrapolated Ce* or La* are equivalent to the geometrically extrapolated ones.

Figure 10 .

 10 Figure 10. Vertical profiles of La anomaly (La/La*) at (a) station PA-11, South China Sea (February 11 and 12, 1997; 15°22'N, 115°17'E; depth: 4240 m ; Alibo and Nozaki, 2000) and at (b) the meander core station E1, Kerguelen Plateau (October 30, 2011; 72.178°E, 48.498°S; depth: 2058 m, Grenier et al., 2018). La/La* was calculated linearly and geometrically with data normalized with chondrite or with PAAS. The LaSN/LaSN* l values are always much larger than the other La/La* estimates, and is an artifact.

  estimation of the anomalies. Additionally, do the normalization values (chondrite or shale) lead to different estimates of La and Ce anomalies?

	2.3/ Which type of extrapolation to select?
	An ideal extrapolation should meet the following criteria:
	-To allow for the best estimation of La* and Ce* concentrations;
	-To be universal, i.e. to correctly estimate La* and Ce* over the whole entire compositional
	range encountered in Earth systems;

-To be independent of normalization values; in other words, the type of extrapolation must be able to calculate consistent La/La* or Ce/Ce* ratios even if the data are normalized to chondritic or shale values.

  CeSN* l ratios may be very different depending on the Pr/Nd ratios. In this study, this is strikingly illustrated by the fact that the CeCI/CeCI* l ratio varies from 0.91 to 1.24 only, while CeSN/CeSN* l varies from 0.96 to 5.11. On the other hand, the CeSN/CeSN* g and CeCI/CeCI* g ratios

	are perfectly proportional, with (CeSN/CeSN* g )/(CeCI/CeCI* g ) being equal to the PAAS
	CeCI/CeCI* g ratio.
	In Fig. 4, we show the Ce/Ce*

05, we can easily calculate that we can then use indifferently one or the other extrapolation only when (Pr/Nd)CI or (Pr/Nd)SN are between 0.82 and 1.28. Whenever (Pr/Nd)CI or (Pr/Nd)SN plot outside this range of values, the Ce/Ce* l and Ce/Ce* g ratios diverge, due to a clear underestimation of Ce* l . The choice of normalization values is not without consequence if linear extrapolation is used. The use of shales instead of chondritic reference values results in a shift towards the left of the diagram [because (Pr/Nd)SN<(Pr/Nd)CI], and, as a consequence, the CeCI/CeCI* l and CeSN/g and Ce/Ce* l ratios obtained after normalization to either chondritic or PAAS values. The diagrams not only indicate that the ranges of values obtained are different, but that the correlations are poor: Ce anomalies calculated in different ways are not comparable.

Table 1 .

 1 An extended REE database for a suite of igneous rocks devoid of La and Ce anomalies, was used to investigate the effects of linear and geometric extrapolations for calculating La/La* and Ce/Ce* ratios in geological and environmental samples, based on Pr and Nd concentrations.We show that the linear extrapolation only provides reliable estimates of La* and Ce* for a limited range of REE compositions. These calculations can lead in many cases to biased La/La* and Ce/Ce* values, which are also critically dependent on the type of normalization used (i.e. chondritic versus shale reference values). The artifacts generated when using the linear extrapolation are illustrated with examples from the literature for seawater and biogenic carbonates, demonstrating that it can lead to misleading interpretations regarding the presence and/or significance of La and Ce anomalies. Finally, we show that the use of geometric extrapolation ensures reliable quantitative calculation of Ce and La anomalies in all samples, which remain unaffected by the type of normalization used. We propose that linear extrapolations be discontinued here, and instead recommend the exclusive use of geometric extrapolations to quantify La and Ce anomalies. Preferred normalization values and examples of calculation. (References: 1:[START_REF] Barrat | Geochemistry of CI chondrites: Major and trace elements, and Cu and Zn isotopes[END_REF] 2: Pourmand et al., 2012; 3: Barrat et al., 2016 ; 4 : 596 Bau et al., 2003 ; 5 : Charles et al., 2021 ; 6 : Van Kranendonk et al., 2003). 597

		CI-chondrite CI-chondrite	PAAS	PAAS	MORB	fluorite	Mn-nodule	stromatolite	BIF
	ref.	1	1	2	2	3	4	5	6	6
	#					PI 18-06	CT2a	GSMC-1	2-9-11a	IF-G
	unit	µg/g	µmol/kg	µg/g	µmol/kg	µg/g	µg/g	µg/g	ng/g	ng/g
	Y	1.56	17.55	32.2	362		35.6	259	1014.2
	La	0.235	1.692	44.75	322.2	0.504	0.54	326	56.3
	Ce	0.600	4.28	87.29	623.0	2.13	0.89	1246	85.9
	Pr	0.091	0.646	10.1	71.68	0.464	0.23	68.74	14	430
	Nd	0.464	3.22	36.98	256.4	3.00	1.42	283	77.7
	Sm	0.153	1.018	6.908	45.94	1.30	0.57	58.4	45.9	399
	Eu	0.0586	0.386	1.188	7.818	0.582	0.18	14.36	27.1	362
	Gd	0.206	1.31	5.958	37.89	2.18	1.33	61.68	89.6	667
	Tb	0.0375	0.236	0.894	5.625	0.432	0.23	9.53	13.6	112
	Dy	0.254	1.563	5.272	32.44	3.16	1.71	56.48	70	791
	Ho	0.0566	0.343	1.078	6.536	0.743	0.4	11.58	17.7	207
	Er	0.166	0.992	3.094	18.50	2.23	1.15	31.93	54.8	619
	Tm	0.0262	0.155	0.468	2.770		0.13
	Yb	0.168	0.971	3.028	17.50	2.28	0.61	29.2	48	580
	Lu	0.0246	0.141	0.438	2.503	0.34	0.07	4.26	8.6	90.4
	LaCI/LaCI* g	1	1	0.88	0.88	0.68	1.33	1.20	1.84	1.52
	CeCI/CeCI* g	1	1	0.94	0.94	0.88	0.71	2.22	1.01	1.09
	LaCI/LaCI* l	1	1	1.10	1.10	0.91	1.57	1.33	1.89	1.71
	CeCI/CeCI* l	1	1	1.02	1.02	0.95	0.74	2.31	1.02	1.14
	LaSN/LaSN* g	1.13	1.13	1	1	0.76	1.51	1.35	2.09	1.72
	CeSN/CeSN* g	1.06	1.06	1	1	0.94	0.75	2.36	1.08	1.15
	LaSN/LaSN* l	2.71	2.71	1	1	-0.46	-1.42	1.42	-28.69	1.77
	CeSN/CeSN* l	1.26	1.26	1	1	2.27	1.43	2.40	1.47	1.17

Acknowledgements.

We thank X for the editorial handling, and Y1, Y2, Yi+1 for their constructive reviews.


Hiba Hamdi Kermap

Laure Roupioz

Thomas Corpetti

Xavier Briottet

Antoine Lefebvre

Evaluation of Urban Weather Generator for air temperature and urban heat islands simulation over Toulouse (France)

Keywords: Urban Heat Island, Local Climate Zones, Remote Sensing, Urban Weather Generator Model, Air temperature

In recent years, the significant increase both in terms of frequency and intensity of heat waves has dramatic consequences in urban areas. The amplification of the urban heat island (that can exceed 10 0 C) has indeed hazardous impacts on health or biodiversity and it becomes urgent to set up early warning systems. Understanding and simulating the Urban Heat Island (UHI) is therefore of prime importance to help city planners in the conception of more resilient cities. In practice, because of the complexity of this phenomenon, no trivial model exists and number of contributions have been proposed (mostly based on physics or statistics), all of them requiring a variety of input parameters related to urban morphology, land use, material properties or meteorological conditions. In this paper, we analyze the performances of the open source Urban Weather Generator (UWG) model in the context of Toulouse, one of the biggest French cities. Morphological and land cover parameters are derived from remote sensing and performances of UWG are analyzed on various Local Climate Zones (LCZs). Simulated air temperatures showed good consistency on different LCZs with a mean bias error (MBE) ranging between ´0.51 0 C to 0.37 0 C and a root mean square error (RMSE) between 1.73 0 C to 2.35 0 C. We also show that best performances are achieved on dense urbanized areas.

I. INTRODUCTION

Since two decades, the urban heat island (UHI) gathers many actors (researchers, city planners, etc.) as this phenomenon has become drastically sound with negative impacts mostly in terms of health, biodiversity or global warming. It is now well understood that UHI is due to artificialisation, which has replaced natural surfaces with heat-retaining materials (buildings, pavements, . . . ). In a context of dense urbanization (it is expected that 70% of the world population will live in cities by 2050) and global warming, helping city planners in modifying/constructing new areas able to limit UHI is of prime importance.

Because of the complexity of UHI phenomenon, the definition of physical models is still an open issue for which several techniques have been proposed [START_REF] Shahraiyni | High-resolution air temperature mapping in urban areas a review on different modelling techniques[END_REF], each of them having their pros and cons. Complementary, some statistical models have also been defined to link land surface to air temperatures measured from meteorological networks or even crowd sourcing [START_REF] Marquès | Urban heat island estimation from crowdsensing thermometers embedded in personal cars[END_REF], [START_REF] Zhou | Satellite remote sensing of surface urban heat islands: Progress, challenges, and perspectives[END_REF]. In a prospective context, the use of consistent simulation tools to generate air temperatures on the basis of urban morphology and land cover is crucial to evaluate the impact of an urban development plan or to help decisions regarding city management.

Among existing models, Urban Weather Generator [START_REF] Bueno | The urban weather generator[END_REF] is appealing since it is open source, easy to operate and applicable at city scale. Therefore this paper presents an evaluation of its performances to simulate urban air temperatures and to derive UHI over Toulouse. To this end we use remote sensing data to extract the required surface parameters used in UWG. The analysis is performed on various LCZs in the city of Toulouse, France, for which a large meteorological network is available to evaluate UWG simulations.

II. MATERIALS AND METHODS

A. Study Area

Toulouse is the fourth largest city of France with almost 500.000 inhabitants. Located in the southwest of France, its climate is generally characterized by warm and clear summers and cold and partly cloudy winters. In the past years, longer periods with extreme temperatures, above 35 0 C during daytime and 25 0 C during nighttime, were observed leading to urban heat island effects. A large network of meteorological stations monitoring mainly air temperature and humidity was set through the city and its surroundings, providing a unique dataset to investigate their variability at city scale.

Toulouse meteorological network is composed of 76 weather stations representing a range of LCZs [START_REF] Stewart | Local Climate Zones for Urban Temperature Studies[END_REF]. All the stations are showed in Fig. 1. However, all of them were not simultaneously active and some had to be removed for consistency reasons (see [START_REF]Map stations in[END_REF] for more information). The stations used in this study were chosen along a North/South transect, yielding different LCZs ranging from heavily wooded landscape (urban park) to area with tightly packed buildings of 3 to 9 stories tall and little or no green space.

B. Urban Weather Generator

Urban Weather Generator (UWG) is a computationally efficient tool simulating hourly canopy level air temperature from meteorological information acquired at a rural reference weather station located close to the city [START_REF] Bueno | The urban weather generator[END_REF]. Based on the Town Energy Balance (TEB) scheme, UWG offers an easy to use alternative to computationally expensive physical models. Along with the meteorological data provided by the rural reference station, it requires a parametric description of the urban surface to characterise the studied area. The fact that the source code is open1 is also a major asset for an easy use of the model as well as for future improvements.

C. Remote sensing for urban surface parametric

UWG simulates the canyon air temperature for a single location corresponding to a specific urban configuration described by several parameters such as building density and height or land cover distribution. These parameters are provided for a given spatial resolution, being usually a few hundred meters in coherence with the physical concepts of the modeling approach. When working at city scale, it is of interest to derive the surface parameters from remote sensing data to properly apprehend the urban spatial variability.

In this study, the surface parameters are first generated at the native spatial resolution of the initial dataset and then aggregated at 200m (using a convolution kernel). Two main families of parameters are computed from remote sensing sources:

' Topological parameters. In practice we rely on the BD TOPO 2021, from the french National Institute of Geographic and Forest Information (IGN) [START_REF] Bd Topo | [END_REF]. It represents Digital Surface Models (DSM) at 1m spatial resolution, computed in practice from ortho-images [START_REF] Bd | Institut national de l'information géographique et forestière[END_REF]. From DSM, we derive 3 main geomorphological indicators: build density, build height and vertical to horizontal height ratio. An illustration of the two first indicators is visible in Fig. 2; ' Vegetation parameters. From the same ortho-images, we derive the grass and tree cover information at 50cm resolution using specific vegetation classification criteria mixed with deep neural networks [START_REF] Kermap | Tree and Grass Cover[END_REF]- [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]. Results have been refined by Computer-Assisted Photo Interpretation (CAPI). From such data we compute grass and tree cover percentage, as illustrated in Fig. 3.

D. Validation approach

The validation is performed on specific days in 2020 year favorable for UHI: no clouds, low wind, high air temperatures, atmospheric pressure and humidity. The 5, 6, 19 and 20 of August 2020 are selected based on these criteria.

As for the Rural Reference station, we rely on Centre équestre (in green in Fig. (1)) as it meets the constraints set in the bibliography: it is neither close to urban land use, nor mountains or large water bodies [START_REF] Bueno | Computationally efficient prediction of canopy level urban air temperature at the neighbourhood scale[END_REF]. The Carmes station (in blue in Fig. (1)) is selected as urban reference for the first validation of the estimated air temperature as it is located in one of the most urbanized areas and is usually the one recording the highest air temperature values. Then, the validation was extended to five other stations from different LCZs (visible in the Tab. I) in order to compare the performance of UWG for difference spatial context. Finally, the urban heat island was estimated for every station.

III. RESULTS

In this section, we first analyze the performances of UWG for the 6 chosen LCZs and in section III-B we carry out UHI simulations for 3 different LCZs.

A. Validation of UWG simulated air temperature

The air temperature simulations for the Carmes station are visible on the Fig. 4. As shown in this figure, there is an overall good agreement between the simulated air temperature and the values measured at Carmes station, with a MBE of 0.02 0 C and a RMSE of 1.73 0 C. Some differences are observed on the 5 th , 6 th and 19 th of August with an overestimation of the air temperature starting in the middle of the night until early morning followed by a slight underestimation until sunset. At the beginning of night, the simulation sticks well to the measured values. The day of the 20 th has a different behaviour, with very low differences between rural, urban station and simulated values. Except on the last day, anomaly peaks occur at around 4 am, which corresponds to the sunrise UTC hour and this regardless the station used for validation. The absence of anomaly on the 20 th of August may be due to the fact that the night before was the hottest of the studied period (minimum 22 0 C) and, as mentioned previously, that there is low difference between the rural and in city air temperature values that day. This point would require further investigations. In the meantime, it appears relevant to automatically detect this anomaly at 4am, to filter it out and to linearly interpolate it. This process is applied for all validation stations.

A comparative table of UWG performances on the different LCZs (before and after removing the 4am outlier) is presented in Tab. I, as well as MBE and RMSE for each station. On this table, one can observe a global consistency of UWG simulations, both in terms on RMSE and bias. It is also interesting to outline that best estimations are obtained for the Carmes station, i.e. the most urbanized one, and that the performance slightly decreases when the urbanisation decreases.

B. Computation of UHI from UWG simulations

The hourly UHI simulated by UWG for the Carmes station is presented in Fig. 5. The other stations showed similar patterns with, as expected from the previous results, an overestimation of UHI during the day, slight underestimation at night and the best results obtained over the most urbanized LCZs. As for the air temperature, the UHI pattern is different for the 20 th of August. The values of humidity, wind speed, atmospheric pressure, downward short-wave and long-wave radiations were checked over the simulation period. They are all stable over the entire period except the downward long-wave radiation that shows higher values during the night between the 19 th and the 20 th of August. This could explain the different behaviour observed that day, with rural air temperature close to the city centre ones, as compared to the three other days. Even though UHI simulations from UWG are globally coherent with ground truth, the absence of spatial gradient between the air temperature values simulated for the different LCZs tend to homogenize the UHI effect over the city and to mask the differences between the various urban configurations. The root mean pairwise square differences of simulated temperatures (or UHI) is about 0.14 0 C (the root median pairwise square differences is about 0.1 0 C). Thus, in this context, UWG appears to be limited to generate UHI maps and to detect the spatial variability over Toulouse. IV. CONCLUSION In this paper, Urban Weather Generator, an accessible and computationally efficient tool, has been evaluated for the simulation of air temperatures on different LCZs over the city of Toulouse. The simulated air temperature values are then used to derive UHI. While it provides reliable results over densely urbanised areas, the performances decrease when the rate of urbanization decreases and the model does not appear to be so sensitive to variations in the urban surface such as different building density or the presence of vegetation. Consequently, better results were also obtained for UHI derived for the most urbanized areas. In the context of this study on the city of Toulouse, the lack of spatial variability in the values simulated by UWG did not allow to investigate the patterns of the UHI phenomenon over the different LCZs, despite very efficient results and consistent UHI patterns in dense areas. Future works should concern the adaptation of UWG on various spatial contexts.

Fig. 1 .

 1 Fig.1. Toulouse weather stations map.[START_REF]Map stations in[END_REF],[START_REF] Bd | Institut national de l'information géographique et forestière[END_REF]

Fig. 2 .Fig. 3 .

 23 Fig. 2. UWG inputs related to built area issued from DSM and filtered at the scale of 200m: building density (a) and building height (b).

Fig. 4 .

 4 Fig. 4. Comparison of hourly (at Universal Time Coordinated (UTC)) air temperature measured at the rural reference station, the Carmes station (Groundtruth) and simulated by UWG for selected UHI favorable days.

Fig. 5 .

 5 Fig. 5. Comparison of hourly UHI (in °C) : true (top) vs simulated (bottom) at different stations in Toulouse on sunny and hot days in August 2020

TABLE I

 I PERFORMANCE INDICATORS OF THE 6 CHOSEN STATIONS BETWEEN SIMULATED AIR TEMPERATURES AND GROUND TRUTH (VALUES BEFORE REMOVING ANOMALIES ARE IN BRACKETS)

	Station	RMSE (°C)	MBE (°C)	LCZ [13]
	C-Cafarelli	2.01 p2.35q	-0.37 p´0.51q	11-Dense trees
	P. Sabatier	1.98 (2.2 )	-0.26 (´0.39)	5-Open mid-rise
	St-Exupery	1.94 (2.2)	0.02 (0.11)	6-Open low-rise
	Busca	1.77 (2.1)	0.5 (0.37)	3-Compact low-rise
	Thibaud	1.77(2.06)	-0.03(´0.16)	8-Large low-rise
	Carmes	1.41 (1.73)	0.14 (0.02)	2-Compact mid-rise

see: https://github.com/ladybug-tools/uwg

ACKNOWLEDGMENT

We thank Meteofrance [14] and Toulouse Métropole for provided data and UWG team for their help to correctly parameterize UWG. We thank Auline Rodler for the enriching discussions. We thank also Kermap team for the support especially Alexandre Rouault and Maxime Knibbe.


Nikhil Mahalingesh
email: nikhil.mahalingesh@univ-amu.fr

Pierre Dupont
email: pierre.dupont@univ-amu.fr

TRANSITIONAL SHOCK-WAVE BOUNDARY LAYER INTERACTION OVER A COMPRESSION RAMP

Experiments are performed to study the interaction between shock-waves and a laminar boundary layer on a compression ramp at Mach 1.66. Pitot probe measurements and hot-wire anemometry are complemented with Schlieren visualisation, while Reynolds number is varied by a factor of two. The results suggest a "fully laminar" type of interaction where transition of the boundary layer occurs downstream of reattachment. On the contrary, oblique shock-wave laminar boundary layer interactions resulted in a "transitional" type of interaction with transition occurring at reattachment, for similar freestream conditions and equivalent shock strength. This difference in the type of interaction resulted in subtle deviations in the relative length and time scales of the shockwave boundary layer interaction. The non-dimensional lengths of interactions, based on mass balance across the interaction, were similar for the two geometrical configurations, but the evolution of these length scales with increasing Reynolds number were found to be different. The Strouhal number associated with low frequency unsteadiness matched across both the configurations, with typical time scales associated with transitional mechanisms in oblique shock reflection, "delayed" in the compression ramp measurements.

INTRODUCTION

The interaction between shock-waves and boundary layers (SBLI) is a classical flow phenomenon in the field of aerodynamics. It is found in a number of real-world applications such as transonic aerofoils, rocket nozzles, supersonic inlets, high-speed engines, etc. and is usually characterised by a number of unfavourable effects including separation of the boundary layer, intense thermal loading, structural fatigue, leading to unsteadiness of the associated shock-waves. Consequently, SBLIs have been a subject of research since the 1940s [START_REF] Ackeret | Investigations of compression shocks and boundary layers in gases moving at high speed[END_REF]. Experiments and numerical simulations have been performed for various geometrical configurations (such as oblique shock impingement, compression ramps, aerofoils, compressor and/or turbine blades, etc.) and for various flow parameters, including different Mach and Reynolds numbers. These studies have shed light on the spatial scales [START_REF] Dupont | Space and time organization in a shock-induced separated boundary layer[END_REF][START_REF] Souverein | A scaling analysis for turbulent shock-wave/boundary-layer interactions[END_REF] and characteristic frequen-cies [START_REF] Erengil | Unsteady wave structure near separation in a mach 5 compression rampinteraction[END_REF][START_REF] Piponniau | A simple model for lowfrequency unsteadiness in shock-induced separation[END_REF][START_REF] Touber | Large-eddy simulation of low-frequency unsteadiness in a turbulent shockinduced separation bubble[END_REF] of the flow. [START_REF] Délery | Shockwave boundary layer interactions[END_REF] reviewed some of the early studies on the topic, while [START_REF] Dolling | Fifty years of shock-wave/boundarylayer interaction research: what next[END_REF] provided some perspectives for future research. [START_REF] Chapman | Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition[END_REF] recently provided a detailed retrospection of the prevailing debate on the origins of the low frequency unsteadiness. Comprehensive reviews on the subject include books by [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF], [START_REF] Doerffer | Unsteady effects of shock wave induced separation[END_REF] and [START_REF] Babinsky | Shock waveboundary-layer interactions[END_REF].

However, limitations from an experimental and computational point of view had restricted much of the focus on turbulent interactions. And hence, attention to the interaction between shock-waves and laminar boundary layers had been relatively lower. Recent advancements in experimental techniques and computational resources, together with emerging focus on sustainability and increased efficiency (Commission et al., 2011), have renewed interest on laminar interactions. As part of the European TFAST project, experiments were performed to study such interactions at the IUSTI laboratory of Aix-Marseille University, involving an oblique shock-wave impinging on a laminar boundary layer. The results showed a relatively long length of interaction with large aspect ratios compared to turbulent interactions [START_REF] Diop | Transition mechanism in a shock wave boundary layer interaction[END_REF]. Further, temporal measurements revealed several frequencies being amplified along the interaction, with the low frequency unsteadiness of the separation region being much higher than that of turbulent interactions [START_REF] Diop | On the length and time scales of a laminar shock wave boundary layer interaction[END_REF]. Several open questions were left unanswered, including the specific effects of transitional mechanisms on the spatial and temporal scales of the interaction [START_REF] Diop | High resolution lda measurements in transitional oblique shock wave boundary layer interaction[END_REF].

The current work is an extension of those experiments, being performed with a compression ramp geometry as opposed to an oblique shock reflection case. The paper discusses the interaction between a laminar boundary layer and a 6 • compression ramp at Mach 1.66. Reynolds number is varied by a factor of two to investigate its effect on the interaction. The experimental setup and methodology are detailed in the following section, describing the Pitot probe, hot-wire anemometry and Schlieren imaging. The initial part of the results focuses on the validation of the laminar boundary layer and canonical nature of the ramp SBLI. The latter part highlights the differences in length and time scales between the 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

Osaka, Japan (Online), July [19][20][21][22]2022 compression ramp and oblique SBLI. These differences are addressed in the discussion section and conclusions are drawn.

EXPERIMENTAL METHODOLOGY

The experiments are conducted in the supersonic wind tunnel of IUSTI laboratory of Aix-Marseille University. It is a closed-loop hypo-turbulent facility which can be continuously operated up to four hours without significant changes in freestream properties. The freestream Mach number is 1.66 for a test-section of 170 mm in spanwise width and a height of 105 mm. The compression ramp consists of an initial flat section with a sharp leading edge and a 6 • deflection at 115 mm from the leading edge. This geometry is placed at a height of 25 mm from the floor of the wind tunnel as shown in Figure 1. Experiments are repeated for different freestream total pressures, corresponding to different unit Reynolds numbers. Table 1 describes the equivalent unit Reynolds number (Re u ) and the Reynolds number based on the location of the corner (Re c ) for different freestream total pressures (P 0 ). Pitot probe and hot-wire measurements are made in the vertical plane containing the streamwise and wall-normal directions, at the mid-span of the wind tunnel. The supersonic Pitot probe has a flat tip of 0.15 mm in height and is used for measurements of the laminar boundary layer profile, with a spatial resolution of 0.05 mm in the wall-normal direction. Longitudinal explorations are also made with the Pitot probe outside the boundary layer at a height of 5 mm from the wall with a spatial resolution of 1 mm in the streamwise direction.

External measurements of mass-flux fluctuations are also made using hot-wire anemometry with a tungsten and platinum wire of 2.5 µm diameter. The Streamline amplifier by Dantec Dynamics is used in symmetrical bridge configuration, operating in constant temperature mode with an overheat ratio of 0.8. The effective frequency bandwidth of the anemometer is approximately 150 kHz (depending on the unit Reynolds number). The calibration of the anemometer is not performed and hence the resulting spectral data are analysed in arbitrary units of the measured voltage. Temporal acquisition is obtained using National Instruments system (NI6133) at a sampling frequency of 400 kHz for 10 seconds, resulting in four million points of acquisition.

Flow visualisation is carried out with a conical Schlieren setup with a low-speed acquisition system. A classical fullframe digital SLR camera (Nikon D700) is used, having a CMOS sensor with a pixel size of 8.45 µm and a maximum resolution of 4256 × 2832 pixels with a exposure time of 150 µs.

RESULTS

Corner

LE

Shock

Mach Wave

Reattachment Shock

Figure 2: Schlieren visualisation at Re c = 0.65 × 10 6 .

Figure 2 shows the Schlieren image for the lowest Reynolds number. A weak shock-wave from the leading edge of the ramp (annotated by LE Shock) can be seen. A weak Mach wave is seen originating from the roof of the wind tunnel upstream of the leading edge. This Mach wave is due to a microscopic structural discontinuity between the end of the diverging section of the nozzle and the start of the test section. Looking closely near the corner, it can be seen that the boundary layer separates a few millimetres upstream of the corner and consequently re-attaches downstream of the corner. A separation bubble is visible between the points of separation and reattachment (identified by the small bright white region). Separation of the boundary layer is not characterised by a distinct shock-wave, but rather weak compression waves which cannot be seen by Schlieren imaging. However diffuse compression waves are observed at reattachment, which coalesce further away from the wall and merge in to a shock-wave.

Measurements of the laminar boundary layer are made over a simple flat plate using the Pitot probe. and setup of the flat plate is identical to the compression ramp, is used mainly to study the supersonic laminar boundary layer. As the geometry is placed at a height from the floor of the wind tunnel, a new natural laminar boundary layer develops from the leading edge. These measurements are repeated for the same three Reynolds numbers of the compression ramp and are shown in similarity coordinates (Figure 3). The boundary layer has a distinct linear section and having a thickness of less than 1 mm. The measurements collapse on top of each other and agree well with the Blasius boundary layer profile (the compressible Blasius boundary layer equations are solved numerically to obtain the theoretical results). These measurements confirm the laminar nature of the boundary layer for the three Reynolds numbers for the flat plate. The nature of the boundary layer is expected to be same over the compression ramp given the similarity of two geometrical setups. Longitudinal measurements using the Pitot probe in the external flow are plotted in terms of the free-interaction coefficient [START_REF] Chapman | Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition[END_REF] in Figure 4. The coefficient of free-interaction (F(X * )) is determined according to Equation 1, which takes into account the change in Reynolds number through the skin-friction coefficient.

p -p 0 q 0 = F(X * ) 2c f M 2 -1 (1/2) (1)
Standard values of the coefficient of free-interaction from various studies of laminar SBLIs [START_REF] Babinsky | Shock waveboundary-layer interactions[END_REF] at separation and plateau are also plotted in Figure 4. The streamwise coordinates are normalised with respect to the location of the corner (X * = 1) and the mean location of boundary layer separation (X * = 0). The mean location of separation is determined by identifying the inflection point in the gradient of pressure measurements. The measurements show a classical two-step pressure rise characteristic of separated SBLIs for the three Reynolds numbers. The values predicted by freeinteraction theory at separation and over the plateau region are well reproduced by the experiments, validating the canonical nature of the measurements. Additionally, Pitot measurements provide quantitative insight into the upstream influence of the SBLI through the length of interaction, which is defined as the streamwise distance between the compression corner and the mean location of boundary layer separation. Figure 5 shows the evolution of this non-dimensional length of interaction (L) (normalised with the compressible displacement thickness of the boundary layer at the point of separation) for the three Reynolds numbers. It seems that the non-dimensional length of interaction is nearly constant for the three Reynolds numbers. Furthermore, Figure 5 compares the lengths of interaction to the oblique shock reflection of [START_REF] Diop | Transition à la turbulence en écoulements compressibles décollés[END_REF] for a flow deflection of 3 • . The inviscid location of the impingement shock-wave is used as the length scale for the Reynolds number for the oblique SBLI results. The lengths of interaction in the compression ramp cases are lower by 47% on average, compared to the oblique shock reflection cases. [START_REF] Souverein | A scaling analysis for turbulent shock-wave/boundary-layer interactions[END_REF] proposed a scaling model for the length scales of shock-wave boundary layer interactions taking into account the effect of different geometrical configurations. This non-dimensional length scale is defined according to Equation 2, where δ * is the compressible displacement thickness at separation, β is the inviscid shock-wave angle and ϕ is the imposed flow deflection.

L * = L sin (β ) sin (ϕ) sin (β -ϕ) = L δ * sin (β ) sin (ϕ) sin (β -ϕ) (2)
The variation of this length scale (L * ) for the three Reynolds numbers is shown in Figure 6. It can be seen that the scaling significantly reduces the difference in lengths between the two geometrical configurations, with the mean difference being 12%.

Additionally, it is of interest to note that while the length scales are nearly constant for the compression ramp configuration, the same cannot be said for the oblique SBLI. Figures 5 and6 show that the length of interaction is decreasing with increasing Reynolds numbers. This difference is addressed in the discussion section, after presenting the temporal results. Moving to the time scales of the SBLI, hot-wire anemometry gives insight into the temporal aspects of the flow. External longitudinal measurements are made over the interaction at 23 mm from the wall. This height is chosen such that the interaction of the probe with the SBLI is minimal, verified by Schlieren imaging. Generally, the hot-wire anemometer is sensitive to fluctuations of total temperature, mass-flux and Mach number of the flow. In the potential region of the flow (23 mm from the wall), where the Mach number is greater than 1.4 and for a high overheat of 0.8, voltage fluctuations of the anemometer can be directly related to mass-flux fluctuations through the King's law coefficient [START_REF] Morkovin | Fluctuations and hot-wire anemometry in compressible flows[END_REF][START_REF] Dupont | Etude expérimentale des champs turbulents dans une couche limite supersonique fortement chauffée[END_REF]. The longitudinal evolution of the non-dimensional root-mean square (RMS) of these fluctuations are shown in Figure 7 for the three Reynolds numbers. The streamwise coordinates are normalised similar to Figure 4.

The first point measured upstream of the interaction is an indication of the freestream noise of the wind tunnel and is around 0.3% of the freestream mass-flux. This low value ensures that the noise is not too high to trigger bypass transition of the laminar boundary layer [START_REF] Van Ingen | The en method for transition prediction. historical review of work at tu delft[END_REF]). An overshoot of RMS can be observed before separation (X * ≈ -0.35) and is due to the Mach wave observed from the Schlieren visualisation (Figure 2). This Mach wave is considered to be quite weak as it has no significant effect on the mean flow (Figure 4) and the RMS returns to its original value after crossing this wave.

It can be seen that the shape of the curves are similar for the three Reynolds numbers, suggesting that the type of interaction is similar. A clear local overshoot of RMS can be seen around separation and reattachment, with the RMS gradually returning to its original value after both events. No exponential increase of RMS is observed after reattachment for the three Reynolds numbers. Looking at the spectral space for the three Reynolds numbers, Figure 8 shows the estimates of the power spectral density measured at the reattachment shock-wave in the external flow. The spectral data is represented in normalised premultiplied form to highlight the contribution of different frequency bands to the total energy content. Frequency is shown in the classical non-dimensional form of Strouhal number based on the length of interaction and the freestream velocity. Probe vibrations and support oscillations are magnified at the reattachment shock-wave, characterised by sharp localisation of energy at very low frequency and their harmonics (St L < 10 -2 ). The shape of the spectra is similar for the three Reynolds numbers with two major time scales. A low frequency scale around 0.07 ≤ St L ≤ 0.09 and a high frequency 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

Osaka, Japan (Online), July 19-22, 2022 scale around 0.5 ≤ St L ≤ 0.7, which are more clearly visible at lower Reynolds numbers.

Figure 9: Longitudinal evolution of normalised premultiplied power spectral density at Re c = 1.27 × 10 6 for the compression ramp.

To understand the evolution of dominant frequency scales over the interaction, Figure 9 shows the contours of the normalised pre-multiplied power spectral density for the highest Reynolds number. The Mach wave observed in Figure 7 is highlighted by the dashed white line. Probe vibrations and their harmonics observed in Figure 8 can be identified here through sharp red spots at very low frequencies at the reattachment shock-wave. The overall organisation is similar for the three Reynolds numbers and hence the results for only one case is shown here.

It can be seen that most of the energy is concentrated at high frequencies before separation (notwithstanding the Mach wave). Separation of the boundary layer is associated with the same low frequency observed in Figure 8 (0.07 ≤ St L ≤ 0.09, annotated by the black circle). On the contrary, the region over the separation bubble is associated with high frequency scales centred around St L ≈ 1. Low frequencies are again found at the reattachment shock-wave of the boundary layer. However, further downstream, intermediate frequency scales around 0.1 ≤ St L ≤ 0.2 are most dominant. The high frequency scales observed over the separation bubble are also found after reattachment, although not the dominant scales.

Figure 10 is a similar visualisation of the power spectral density from the hot-wire measurements of [START_REF] Diop | Transition à la turbulence en écoulements compressibles décollés[END_REF] for the interaction between an oblique shock-wave and a laminar boundary layer. These experiments were performed at the same Mach number for a effective flow deflection of 3.7 • and Reynolds number of 0.65 million. The very low frequency content (St L < 10 -2 ) around separation (X * = 0) corresponds to an artefact of the experiment, possibly due to probe vibrations and support oscillations.

Comparison with Figure 9 shows that similar frequencies are involved at separation and over the mixing layer, with St L ≈ 0.09 (annotated by the black circle at the bottom left) and St L ≈ 1.5 respectively. The major difference is the concentration of most of the energy at high frequencies after reattachment (annotated by the black circle at the top right). This is associated with breakdown of coherent structures in the boundary layer leading to transition to full turbulence [START_REF] Diop | High resolution lda measurements in transitional oblique shock wave boundary layer interaction[END_REF]. [START_REF] Diop | Transition à la turbulence en écoulements compressibles décollés[END_REF].

DISCUSSION

The experiments of the oblique SBLI from [START_REF] Diop | Transition à la turbulence en écoulements compressibles décollés[END_REF] showed that transitional mechanisms were strongly amplified over the interaction, evidenced by exponential amplification of characteristic frequency scales associated with the mixing layer. Particularly, intermediate frequency scales (St L ≈ 0.5) were significantly amplified over the separation bubble. These scales were typically associated with the non-linear phase of the transition process [START_REF] Mayer | Direct numerical simulation of complete transition to turbulence via oblique breakdown at mach 3[END_REF]. This led to transition of the laminar boundary layer at reattachment, featured by the concentration of energy at high frequencies near this region. On the contrary, there is no clear evidence of transition at reattachment in the current experiments of the compression ramp (Figures 7 and9), even though the freestream conditions and the equivalent shock strength is the same. Correspondingly, the intermediate scales, thought to be essential for transition, are found after reattachment. Therefore, it seems that the transition to turbulence occurs at some point far downstream of reattachment. Of course, the boundary layer after reattachment is definitely altered and might not follow the shape of a canonical Blasius profile, but it is not fully turbulent either. This suggests that the compression ramp has a "fully laminar" interaction as opposed to a "transitional" type for the oblique SBLI.

This fundamental difference in the type of interaction has subtle effects on the length and time scales between the two geometrical configurations. It is known that the length scales of the compression ramp are relatively smaller compared to oblique SBLIs, due to the geometrical constraints imposed by the ramp on the reattachment of the boundary layer, as highlighted in Figure 5 . While scaling the length scales based on the respective geometrical configurations significantly reduces the large differences (Figure 6), a non-trivial difference is still observed in the slope of the curves between the two geometrical configurations. For a fully laminar interaction, as transition occurs far downstream of reattachment, the length of interaction is thought to be governed solely from a mass balance point of view [START_REF] Souverein | A scaling analysis for turbulent shock-wave/boundary-layer interactions[END_REF]. For a transitional interaction, the length scale is governed by a combination of mass balance and transitional mechanisms, which tend to reduce the length of interaction when Reynolds number is increased. The presence of these transitional mechanisms can be characterised by the dominance of intermediate frequency scales, which can be found over the separation bubble in the 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

Osaka, Japan (Online), July [19][20][21][22]2022 oblique SBLI experiments and after reattachment for the compression ramp. Perhaps, the length scales of the compression ramp might show the same behaviour at higher Reynolds numbers, where transition could occur at reattachment. But this is outside the feasibility of the current experiments and hence could not be explored. With regard to the time scales, the low-frequency unsteadiness associated with separation are mostly similar between the two configurations, even though the type of interaction is different. This suggests that the intermediate frequency scales do not play a significant role in the mechanisms of the low-frequency unsteadiness. The other part of the picture is the amplitude of unsteadiness, as [START_REF] Chapman | Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition[END_REF] suggested that fully laminar interactions are "steady" in nature while transitional interactions are unsteady. Further investigations are necessary to understand this effect in more detail.

These results highlight the delicate effects of transitional mechanisms on the type of interaction, and consequently affecting the associated length and time scales.

CONCLUSIONS

The effect of Reynolds number on the interaction between a laminar boundary layer and shock-waves are investigated experimentally over a compression ramp. Pitot probe, hotwire anemometry and Schlieren visualisation are performed for three Reynolds numbers, varying by a factor of two between the extreme cases. The results suggest that the interaction is of the "fully laminar" type, with transition occurring far downstream of reattachment. This is in contrast to what was found in experiments of oblique SBLIs for the same freestream conditions and shock strength, where transition occurred at reattachment. Further comparisons with oblique shock reflection cases show reduced lengths of interaction for the compression ramp and different evolutions of the length scale with increasing Reynolds numbers. However, low-frequency unsteadiness associated with separation are mostly identical for both configurations.

  (a) Schematic of Geometry. (b) Picture of Geometry.

Figure 1 :

 1 Figure 1: Compression Ramp.

Figure 4 :

 4 Figure 4: Free interaction theory.

Figure 5 :

 5 Figure 5: Variation of non-dimensional length of interaction for the three Reynolds numbers.

Figure 6 :

 6 Figure 6: Scaling of length of interaction using Souverein et al. (2013) model.

Figure 7 :

 7 Figure 7: Longitudinal evolution of non-dimensional root-mean square of mass-flux for the three Reynolds numbers.

Figure 8 :

 8 Figure 8: Pre-multiplied power spectral density at reattachment shock-wave for the three Reynolds numbers.

Figure 10 :

 10 Figure 10: Longitudinal evolution of normalised premultiplied power spectral density at Re c = 0.65 × 10 6 from oblique SBLI (Diop, 2017).

Table 1 :

 1 Operating conditions of the experiments.

	P 0 [atm] Re u m -1 × 10 6	Re c × 10 6
	0.4	5.61	0.65
	0.6	8.37	0.96
	0.8	11.01	1.27

  The geometry 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)Osaka, Japan (Online), July[19][20][21][22] 2022

		1.2
		1
		0.8
	e
	u/u	0.6
		0.4						Re x ( 10 6 )
								0.65
		0.2						0.96 1.27
								Blasius BL
		0
		0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
						y/ 99
	Figure 3: Boundary layer velocity profiles for various streamwise locations and comparison with Blasius pro-file.

This work is part of the European TEAMAero project (Towards Effective Flow Control and Mitigation of Shock Effects in Aeronautical Applications). The authors would like to acknowledge the support from the European Union's Horizon 2020 research and innovation programme under grant agreement No. EC grant 860909.