domain: nlin spi

text:
de la démarche L'objet de ce livrable est de présenter les bases méthodologiques et les outils utilisés pour conduire une analyse multi-échelle d'un flux géophysique : géométrie fractale, statistique de la turbulence et analyse multi-fractale. Ces outils ont été mis en oeuvre sur les données scintillométriques (Cn 2 ) et de température collectées en continu pendant 5 semaines sur le campus de l'ENPC. Ce livrable a été réalisé sur la base des travaux de thèse d'Alejandra Castellanos-Diaz.

L'analyse spectrale a montré une invariance d'échelle des deux flux sur une large gamme d'échelle. Les fonctions structures, quant à elles, ont démontré que ces deux séries temporelles sont affectées par un phénomène d'intermittence et qu'elles sont multi-fractales. L'analyse multi-fractale qui en résulte a montré que le Cn 2 est fortement affecté par les fluctuations de température, mais que d'autres champs géophysiques interviennent dans la genèse du flux de chaleur sensible. Ce qui reste à étudier.

Par conséquent, la variabilité spatio-temporelle des flux thermiques sur la Vague Bleue Verte nécessite une analyse plus approfondie, nécessitant davantage de données géophysiques, telles que la vitesse du vent. Cela permettrait de mieux caractériser l'invariance d'échelle ainsi que les propriétés statistiques présentées dans ce travail.

Statistics of turbulence

In complement of the fractal geometry presented in Deliverable 4.2, here are introduced the basic statistics of turbulence. Based on space and time changes of convective and mechanical turbulence, the eddies observed in a variety of turbulent atmospheric fluxes have different sizes and lifetime. This leads to atmospheric fluxes appear stochastic with structure with structure at all scales, and its behaviour difficult to predict. In fact, the chaotic nature of atmospheric turbulence is governed by the non-linearity of Navier-Stokes's equations, for which no formal analytical solution has been found.

The scientific community has been interested by this issue for years and has proposed statistical methods as an appropriate approach to describe turbulence. In this way, statistical properties such as the Reynolds decomposition, the statistical moments, the structure function, and the Fourier series are used to characterise turbulent fluxes. These statistical properties are quite useful for the analysis of spatial and temporal variability of an atmospheric variable observed at some fixed point over long periods, as well as the underlying physical processes. The spatial approach of the structure function will serve in this section to derive the structure function parameter of a scalar, which is the basis for scintillometry technique; while the temporal approach of the structure function will be used in Section 4 to analyse the statistical properties and the scaling behaviour of thermal fluxes measured at a fixed point.

Reynolds' decomposition

A random atmospheric variable can be generally described into two parts: the mean and the fluctuating about the mean , as follows:

(1) The mean of turbulent fluctuations is zero (

). This decomposition is applied for every turbulent flux in the atmosphere.

Statistical moments

The random variable at any position of a four-dimensional space (time or space), can be described with the statistical additive moments (Moene et al., 2004). The -th order moments were defined by [START_REF] Monin | Statistical Fluid Mechanics: Mechanics of Turbulence[END_REF], as the mean values of products of values of the field:

(2) where are integers of which the sum gives the order of the moment. Following the Reynolds' decomposition of the variable :

(3) The first central moment is zero when defined the probability distribution with reference to the mean, and the second central moment is the variance. The general second order moment corresponds to the covariance ( ) and it is generally used to define three cases of a field (Kesteren, 2012): stationary, homogeneous and isotropic turbulence [START_REF] Monin | Statistical Fluid Mechanics: Mechanics of Turbulence[END_REF]):

• For a stationary turbulent field, the mean value is constant and invariant under a translation in time . Thus, the covariance does not depend on and :

(4)

• For a homogeneous turbulent field, the mean value is constant and invariant under a spatial translation.

only depends on :

(5)

• For the isotropic turbulent field, the mean value is constant in all directions and only depends on the distance between two points and not on its direction or position:

(6)

Structure function

Since atmospheric fields are hardly stationary when the time interval between two measurements is too large (the mean value is not constant), Kolmogorov introduced in 1941 the principle of stationary increments. He argued the use of the difference is more practical than . Then, the second order moment of (Eq.7) becomes:

(7) (8) Thus, statistical moments applied to increments of a stochastic process are called structure function. Then, as only depends on the time interval , the structure function, between and , can be introduced as the mean of the difference of the variations of the flux between and (Tatarski et al., 1961), as follows:

(9) In conditions of stationary increments, the second order moment or covariance is related to the second order structure function by: (10) where, and correspond to the statistical moments at any time and at the time interval , respectively. Apart from stationary increments, the principles of locally homogeneous and isotropic fields, for analysing the spatial evolution of the flux between two points of the space can be assumed too. Then, the structure function can be expressed as:

(11) In this way, the structure function is a statistical property than only depends on the distance .

Fourier Analysis

The Fourier analysis is a widely known method used to decompose the time-domain series of turbulence, into trigonometric functions such as sines and cosines. Hence, the Fourier series of a periodic function can be expressed as:

(12)
where, is the fundamental frequency ( ), and , and , the corresponding Fourier coefficients. Because of the periodicity of events in nature, the Fourier analysis is used with the object to study these amplitudes, frequencies and cycles present on the signal of the turbulent field, as well as its noise. For this purpose, the Fourier transform is used to express a non-periodic function in time as a continuous series of harmonics, as follows:

(13)
where, is the imaginary unit, is the frequency, and corresponds to the Fourier transform of . Hence, converts the signal to its transform:

(14)
Let's now consider a discrete version of the Fourier transform known as Discrete Fourier Transform (DTF), where the time series is measured at just a finite number of times . So, the DTF results from evaluating the integral from 0 to time over which the signal is measured:

(15)
where, is the index of the data array of length , and the harmonic number. The inverse transform is given by:

(16)
The applicability of DFT was globally extended because of its usability and the development of the Fast Fourier Transform (FFT) algorithm, for which the DFT has been factored and restructured to take advantage of the binary computation processes of the digital computer (Stull, 1988).

Based on the Fourier transform, the power spectral density (or energy spectrum) enables to examine the energy distribution of a turbulent time series data over the frequencies:

(17)

The FFT algorithm is usually used to compute . It is important to note that the FFT is restricted to data sizes , where and is the sample size.

The spectral analysis is an approach used to verify the scale invariant properties of a field. In the case of a scaling field, the power spectral density has a power law relationship with a range of frequency (temporal analysis) or wave number (spatial analysis) as:

(18) When this relationship is represented in a log-log scale, the spectral exponent is deduced from the (absolute) spectral slope.

Scaling invariance laws for describing fluxes

In the previous section, fractals were introduced as self-similar complex patterns across different scales, which follows a power law and are characterised by the fractal dimension . Fractals have not only been useful in the study of scaling laws of urban morphology, but also in the analysis of scaling properties of dynamic and non-linear geophysical fields.

Geophysical fields usually exhibit fluctuations that follow scaling power laws. Hence, they were reduced to a fractal set. However, it was found they did not have a unique scaling exponent. When different thresholds (singularities) were set to estimate the distribution pattern of a field, for each singularity a different fractal dimension appeared. This led to define multifractals and their properties, which serves for the analysis and modelling of fields.

Since the measure of a field is generally conducted at a unique scale, the properties at other scales must necessarily be deduced from the measurement scale. Thus, a time series can be decomposed into frequencies (or wavenumber, by involving the Taylor's hypothesis 1 ), in order to analyse the distribution of energy, and the scale invariance of the atmospheric flux can be detected though the spectral exponent .

To better understand the statistical properties of the fluxes, let's see now a power law that serves for the generalisation of the structure function of an arbitrary moment at any point (i.e., integer and non-integer moments , apart than 2), for analysing of scaling multiplicative processes. Thus, it can be written as: (19) Under the hypothesis of constant energy flux density and self-similar turbulent flux at small scales, the structure function follows a power law as follows:

(20

)
where, is the scaling moment function, related to the similarity exponent by the Kolmogorov theory. In the absence of corrections because turbulent intermittency of the flux, there is a perfect linear relation . Then, for any order moments :

(21)

For , the second-order moment structure function is related to the scaling exponent of the spectral analysis in the following manner:

(22)
The intermittency is responsible of the extreme variability of the turbulent fluxes in the PBL. Hence, for models considering intermittency, there may be a variation of , hence the spectral exponent is greater than 5/3. The intermittency could be characterised by the fully knowledge of and . The relationship between and on logarithmic scale, allows to deduce the scaling exponent as the slope of the linear regression. Hence, the characterization of over different moments order enables to determinate the scaling behaviour of the flux [START_REF] Calif | Multiscaling and joint multiscaling description of the atmospheric wind speed and the aggregate power output from a wind farm[END_REF]. Mono-scaling (or monofractal) processes are described by a perfect linear relation . By contrast, the deviation of the linear trend indicates intermittent multifractal fluctuations, where is no longer linear, but concave.

Despite K41 demonstrated its utility under homogeneous and isotropic turbulence conditions, some works have shown the limits of these scaling laws. The intermittency of turbulent fluxes is characteristic of strong fluctuations at all scales. Hence, the mean value energy dissipation rate is not constant and insufficient to characterize the turbulence.

This variability motivated Kolmogorov and Obukhov (Kolmogorov, 1962;[START_REF] Obukhov | Some specific features of atmospheric turbulence[END_REF] to purpose a new theory that considers the intermittency of nonlinear dynamical systems. This

1 The Taylor's hypothesis of frozen turbulence states that temporal and spatial fluctuations of a flux can be related by a constant velocity. If the turbulence intensity is small compared to the mean flux speed , the temporal response at a fixed point in space can be view as the result of a spatial pattern advected by the mean velocity .

theory known as K62 considers the structure function of velocity increments is a function of locally averaged energy flux density over a volume of linear dimension ( ). In addition, the energy dissipation rate follows a lognormal distribution, with the variance of given by: (23) where, is a constant associated to the macrostructure of the flux, is a universal constant, and the largest external scale. The probability density function over a range of scales can be written as follows:

(24) This law is known as the Refined Similarity Hypothesis (RSH) of Kolmogorov. By considering from the Eq. ( 24), the scaling exponent function is given by:

(25) [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF] also proposed a model including the intermittency, where the scaling exponent function involves the scaling moment function :

(26) The function could be characterised by only three parameters in the Universal Multifractal framework.

Self-similarity extension: from fractal sets to multifractals

Based on the fractal geometry, [START_REF] Parisi | A multifractal model of intermittency[END_REF] introduced the term "multifractal" to explain the nonlinearity of the scaling exponent function. They considered a turbulent field (e.g., the velocity field) on a space dimension [START_REF] Lovejoy | The Weather and Climate: Emergent Laws and Multifractal Cascades[END_REF]. The points of the velocity field have a singularity of order at the point if:

(27)

This leads to say that: (28) In this way, is the set of points for which the field has a singularity of order , called the "Hölder" exponent, or singularity strength. Hence , and the set has a fractal dimension . Thus, different singularities (defined as ) are associated with fractal sets having different dimensions , which leads to say that is multi-fractal.

From the definition of fractal dimension and box-counting, is nonnegative and its link with is found: (29) where, is the codimension function [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF] bounded by . Hence, the events with low occurrence have a dimension of 0 and those with more frequency fill the space, the .

Statistical properties of multifractal fields

The statistical properties of a multifractal field at a given resolution are defined by two scaling laws [START_REF] Lovejoy | The Weather and Climate: Emergent Laws and Multifractal Cascades[END_REF][START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF], 1991): the probability distribution of the energy flux density and the statistical moments; involving the codimension function and the scaling moment function , respectively.

The measure of the probability that is greater than a given scaling threshold (singularity) , is described by the relation [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF]):

(30) Being the renormalized intensity of the field at the scale-ratio (defined in Eq. Error! Reference source not found.) and the codimension function associated to a given singularity .

Via the Mellin transform, [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF] demonstrated the probability distribution of Eq.33 is equivalent to the statistical moments. Hence, the scaling moment function was introduced, to characterize the multifractal field : (31) where, is the statistical moments order and is the moment mean of the intensity at the scale ratio .

3.2 Legendre Transform [START_REF] Parisi | A multifractal model of intermittency[END_REF] demonstrated for deterministic multifractals, the statistical moments as well as the probability distribution of singularities, are equivalents through the involutive Legendre Transform between and :

(32) and functions are convex (see Figure ) and differentiable, Eq.35 becomes:

(33)

These relations demonstrate every moment of order is related to only one singularity .

Because of the conservation of flux from one scale to another, implies for : (34) is the "co-dimension of the mean" and it represents a characterization of the variability near the mean.

Universal Multifractals (UM)

As previously demonstrated, because of convexity of and , both functions depend on an infinity of parameters. Nevertheless, through a multiplicative central limit theorem, [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF] proposed the Universal Multifractal model, for which only few parameters are necessary to characterize and :

(35

) (36)
where , and are the three basic universal multifractal parameters:

• The mean singularity , measure the clustering of the mean intensity (i.e., the fractality/sparseness of the mean field), and at the same time it corresponds to the codimension of the mean field:

(37)

Note that for the field is homogeneous, and for greater values of the mean singularity of the field is sparseness.

• The Lévy index , which indicates the extent of multifractality. This measures the mean intermittency evolve when considering singularity. The values of varies between 0 and 2, being characteristic of monofractal fields, and characteristic of maximal occurrence of extremes (log-normal fields).

• The Hurst exponent , that measure the degree of non-conservation of the mean field (for conservative field ):

(38)

After to prove the scaling behaviour of the field and to determinate the values of universal parameters, the Trace Moment (TM) and Double Trace Moment (DTM) techniques based on the function are generally used.

Trace Moments (TM)

The Trace Moment technique allows to calculate the function by means of the scale invariance moments of order of the conservative field [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF]:

(39) where, is the moment mean of the intensities at the scale and denotes the asymptotic equivalence. The function is convex and it characterises the scaling invariance of the field (multifractal behaviour), showing the asymptotic behaviour of moment of order .

To apply the TM method, the field must be considerate approximately uniform, and it is normalized assuming a mean of one, as follows:

(40)

The normalized temporal data series should be divided into non-overlapping intervals of a certain resolution . At each resolution , the sample is up-scaling independently and the moment of order is calculated .The average moments of order is calculated over all resolutions , as follows:

(41)

Then, the scaling behaviour of the field can be validated as a function , by plotting the logarithm of versus the logarithm of for different values. The slopes of the linear regressions correspond to the estimations of , and the value of r² is used to validate the accuracy of the straight-line approximation. Then, the different values and the associated allows the estimation of the empirical function.

Then, the UM parameters and can be estimated using the properties of the curve, by estimating the first two derivates of :

(42)

The appearance of the function specifies the type of scaling involving: a straight line implies a monofractal structure whereas a convex curve implies a multifractal structure.

Double Trace Moments (DTM)

This technique is a direct method to obtain UM parameters, for a conservative multifractal field . DTM generalise the TM technique based on a unique exponent and the introduction of a second exponent . This method consists of performing a TM analysis to a renormalized -power of . For this purpose, firstly the field must be normalized, by arising the conservative field to the power at the highest resolution, in the following way:

(43) Then, the average statistical moments also scales with the resolution :

(44)

In this manner, the original field is transformed into and the function into . Then, the th-order moments of the renormalized field remain scale invariant:

(45) when , is reduced to . The expression above can be solved from the universality of function:

(46) With this expression, UM parameters can be directly estimated. In a log-log plot, is plotted as a function of , for a fixed value of . The diagram allows to estimate as the slope of the linear part of and the intercept is used to determinate the parameter .

A divergence of empirical functions of , deduced from the slope of and in logarithmic scale for different values, and theoretical functions from the UM parameters with TM and DTM methods can occur. Then, after a given value of the theoretical function of becomes linear, leading to a departure from universal behaviour. This phenomenon is known as the multifractal phase transitions [START_REF] Lavallée | On the Determination of the Codimension Function[END_REF]Schertzer & Lovejoy, 1989), and it could be caused by the critical moment order of divergence and the undersampling . According to [START_REF] Lavallée | On the Determination of the Codimension Function[END_REF], in the case of one-dimensional field (

) and the use one data sample , the value of is given by the formula:

(47)

The critical order of divergence of moments represents value for which the extreme values of the field are dominant. Thus, the average statistical moment or order tend to infinity.

Scaling invariance behaviour of the BGW's thermal fluxes

In order to explore the scaling behaviour of micrometeorological measures carried out over the Blue Green Wave, the power spectral density and the structure function will be investigated. As presented in Deliverable 1.1, some data collected from techniques of ET measure were not exploitable and because of frequency of measure and length of campaigns, the number of datapoints were limited (maximum 32 data points). However, as the application of statistical analysis requires sufficient data, a new campaign of measurements was performed with the LAS MKI and the CNR4 for a longer period. Due to irregularity of measure with the Ch and missing values of CWS665 sensors (which is not favourable to the application of the FFT), the Ch and WB were not analysed. Despite the main interest of this research concerns BGW's fluxes during summer months, particularly in UHI events, this long campaign with the LAS MKI scintillometer was conducted during the winter months of December 2019 and January 2020 because logistic convenience. Hence, this is considered a first approximation of the variability of thermal fluxes in the BGW.

Data set

The LAS MKI transmitter and receiver units were mounted over the roofs of the Carnot and Bienvenüe buildings, at the core of the Cité Descartes (See Figure 6). The CNR4 was located next to the LAS receiver, 1.5 m above the ground, to monitor radiation components and approximate the conditions of atmospheric stability. Finally, as optical LAS MKI is more sensitive to temperature fluctuations, the air temperature was monitored too. The elevations (including the height of buildings) are 119.15 m and 117 m for the transmitter and the receiver, respectively. Just like the summer set-up campaign, the effective beam height was calculated through EVATION software from the elevation of units and the surface topography of the transect profile (see Figure 7). The path length between both units was 190 m, hence diaphragms for short-range applications were necessary. The EM alignment pass over the BGW, stone floor and buildings roofs. (upper) and (middle) and calculations (bottom). The QC parameters are marked with the grey dotted line.

To understand potential causes of intensity reduction in LAS MKI measurements, the rainfall data were analysed in Figure 9. The rainfall intensity was collected every 5 minutes by three optical disdrometers of the Fresnel Platform from HM&Co Laboratory (see more details in [START_REF] Gires | Two months of disdrometer data in the Paris area[END_REF], located just next to the LAS MKI receiver unit on the roof of the Carnot building. A significant event of rainfall of 30 mm/h was recorded on December 27 th , which caused slight alterations on the signal, but that did not affect the QC parameter. The rest of the rainfall events were lightly or moderated without significant consequence on the scintillometer measure.

Regarding the atmospheric conditions during the campaign, most of the days in December were sunny and cloudy, while in January low rates of solar radiation were more frequent. Daily fluctuations of temperature were large during cloudless days (12/29 to 01/01), ranging from 0 to 10 °C. However, there was no specific temperature variation having an impact on the LAS measurements. Hence, meteorological conditions can be discarded from LAS signal disruption.

The values of were used to determinate the stability of the atmosphere since there are no measures of thermal gradient between the ground and the upper atmosphere. Thus, just like works of [START_REF] Han | Using Large-Aperture Scintillometer to Estimate Lake-Water Evaporation and Heat Fluxes in the Badain Jaran Desert, China[END_REF] and [START_REF] Kohsiek | Saturation of the Large Aperture Scintillometer[END_REF], measurements taken under conditions of greater than 10 W/m² were considered as unstable atmosphere.

Figure 9. Meteorological conditions during winter months.

Spectral analysis

To avoid errors in the statistical analysis due to inaccuracy of SEB fluxes measures that may be transferred into , as well uncertainty of parameters involved in the deduction of SEB parameters, the spectral analysis was applied to raw data of from LAS MKI, through the FFT algorithm.

The three methods to estimate from the output signals of LAS, discussed in Deliverale 1.1, can be appreciated at the bottom of Figure 9. There is a good fit between from (green) and

(yellow) ranging around 10 -14 and 10 -13 . Concerning from (red), there are more significant fluctuations producing lower (10 -15 ) or higher (10 -12 ) turbulence intensity compared to the other estimations of . and was used in the analysis.

The original data sets of in Figure 8 was divided into two periods (displayed in Figure 10) to avoid the signal disruption of January 1 st of 2020 and rainfall event on December 27 th of 2019. Since the computation of FFT uses sample length power of 2, both series of are composed of 1024 data points, equivalent to 7 continuous days of measure.

A good linear scaling behaviour is observed for the first period in Figure 10.c. This scaling of ranges from log(f)=2.5 up to 5.5, corresponding to time-scale of 14 h and 41 min, respectively. The spectral slope is 1.649, comparable to the slope 5/3~1.66 in the inertial range of isotropic and homogeneous turbulence from K41. Regarding the second period in Figure 10.d, a similar behaviour is observed from log(f)=3 up to 5.5, corresponding to timescale of 8.5 h and 41 min, slighted lower that the 5/3 of Kolmogorov.

The time scale of 8.5 h represented by the dotted vertical line is associated with the period of daylight time or solar forcing during a winter day. The radiation from the sun is fundamental, as it creates buoyancy heterogeneities and unstable atmospheric conditions when MOST theory is fully valid to deduce . Regarding the 41 min, this is the time-scale of transition between unstable and stable atmosphere, this means the sunrise and sunset duration.

The spectral slopes of for both periods coincide with the assumption of [START_REF] Monin | Statistical Fluid Mechanics: Mechanics of Turbulence[END_REF] that power spectral density of turbulent fluctuations for a scalar follows the 5/3 law.

These results also agrees with the investigation of Maronga et al. (2013), where the power spectral density of the structure parameters of temperature and humidity from Large Eddy Simulations in the convective boundary layer (where LAS is typically operated) followed a power law of 5/3 in the inertial range.

December January a) b) c) d)
Figure 10. Two series of 10 minutes average of in December and January and their respective power spectra density.

Since the fluctuations of the refractive index are caused by eddies with different temperature and humidity in the PBL, and the LAS's wavelength is mainly sensitive to temperature fluctuations, we will perform a spectral analysis of temperature measurements from the CNR4 (bottom of Figure 9). The same periods of analysis that were considered and the size of the data sets was the double that as the time period of measurement was 5 min, which corresponds to 2048 data points represented in Figure 11. Time-scales related to solar forcing are marked in the energy spectrum of temperature: 24 h of diurnal cycle and 8.5 h of solar radiation. A single scaling range is observed after the 8.5 h up to 30 min (log (f)= 6.5). The spectral slope was 2.083 in December and 1.969 in January. This behaviour is associated with buoyancy effects and convective process in the atmosphere. In fact, these slopes are near to 11/5 scaling law predicted by Bolgiano-Oboukhov (BO) spectra, for a scaling stable stratified turbulence. Similar values of the spectral exponent were obtained by [START_REF] Fitton | Analyse multifractale et simulation des fluctuations de l'énergie éolienne[END_REF][START_REF] Karatasou | Multifractal Analysis of High-Frequency Temperature Time Series in the Urban Environment[END_REF] when analysing the scaling behaviour of air temperature and water temperature in a lake [START_REF] Mezemate | Analyse et modélisation multifractales des interactions ondesturbulence-biologie dans un lac urbain[END_REF]. The flattering of spectrum of both series at highest frequencies (after 30 min), is related to white noise as the intensity of the spectrum is constant.

December January a) b) c) d)
Figure 11. Two series of 5 minutes average of the air temperature in December and January and their respective power spectra density.

In conclusion, the scaling invariance of and temperature were verified through this spectral analysis. The spectral exponents are close to 5/3 for and 2 for the temperature, which indicates the fluxes is non-stationary ( , the process is nonstationary). In addition, the spectral analysis proves to be a statistical tool of analysis to identify main frequencies influencing measurements, as well as potential sources of error. Nevertheless, the power spectral density only represents the second-order moment statistics, and its slope is not enough to fully describe the scaling behaviour of the data.

Structure function scaling exponent

The structure function introduced earlier allows to characterize the fluctuations of the flux, through the estimation of the scaling exponent function over several order moments . As we are dealing with time series, the structure function is written for time series fluctuations as . The structure function of was estimated over 1024 data points, for both periods of analysis, with and the statistical order moments with increments of 0.1. Figure shows the results of the structure function obtained of December and January, for and . A single linear behaviour was observed in the range up to . This confirms the scale invariance behaviour of for both periods of analysis observed previously with the power spectral density in the inertial range. The scaling exponent was estimated from the slopes of linear regressions in Figure 12 and was plotted in Figure 13. From the relations presented in Eq.24 and Eq.25, the estimates of from and were compared. The results presented in the Table 2 demonstrate the values of from were higher than those obtained from the spectral exponent . The discrepancy between both methods is kept for the second-order moment in both periods. The average of was predicted from the relation of the scaling exponent evaluated for all the moments through Eq.24. For the period of December and for January . The non-linearity and concavity of in Figure 13 is undisputed for both periods, indicating the multifractal nature of the structure parameter Cn 2 . This agrees with the research of [START_REF] Pérez | Multifractal characteristics of optical turbulence measured through a single beam holographic process[END_REF], which showed fractal dimension of surface patterns of a wavefront, measured through the box counting dimension, is not unique and it changed as convective turbulence is developed. Hence, fluctuations of Cn 2 from the BGW are not monofractal but multifractal.

December January As , the structure function of temperature series was evaluated, with 2048 data points, for the statistical order moments with increments of 0.1 and for each 300s interval, . Figure 14 plots the straight lines of the linear regression, which indicates the scale invariance is well respected over the range of time up to . This behaviour is associated with buoyancy effects and convective processes in the atmosphere.

December January The empirical scaling exponent of the temperature for both periods of measure, estimated from the slopes in Figure 14, is shown in Figure . In comparison with the monofractal process from the classical BO theory of buoyancy-driven turbulence, it is demonstrated the deviation of and the intermittent character of the temperature. For December period, the estimates of from the spectral exponent (

) and the first-order structure function ( ) are 0.541 and 0.674 respectively. However, for the period of January there is a discrepancy of 0.3 between and .

December January From the results of the spectral density was proved the experimental data from the BGW are scaling and strongly non-conservative. The premised hypothesis the structure function follows a power law given by and represents a (mono)-fractal behaviour, is not respected by and air temperature. This means a unique self-similarity law does not describe the fluctuations of and temperature because of intermittency, which is characterised by many fractal dimensions. Therefore, to fully characterize this non-linearity and scaling properties, the correction of intermittency given by should be introduced, which will be characterised using the Universal Multifractal (UM) framework.

Characterisation of the scale invariance properties

The TM and the DTM technique are now applied to the experimental data from the BGW. The spectral slope of and temperature, in both periods of analysis, indicates fluctuations are non-conservative since . Therefore, the approach of [START_REF] Lavallée | Nonlinear variability and landscape topography: Analysis and simulation[END_REF] from the absolute values of data increments was used to transform the non-conservative fields into conservative (which implies

). This method was successfully applied by Stanic, (2020), to characterise UM parameters of spatial and temporal variability of the water balance components of the BGW: soil water content, rainfall intensity and drained discharge.

The statistical moments of fluctuations, were evaluated for different values of , for a series length of 1024 data points. Then, we processed to display at the top of Figure to check scaling. The fluctuations of shows a good scaling given by the value of r², mainly for the period of December. However, for in both periods, the value of r² is the lowest.

The Figure b) and c) shows the average moments as function of the resolution in logarithmic scale, with different values of and . The accuracy of the linear regressions for any is higher than 0.9 (rather better than TM), which demonstrates the scaling is well respected when varies from up to .

The slope of the linear regression is obtained for every value of and presented in Figure d) and e), from which and parameters were deduced in the range of the market with the red line. In December, universality is observed in the range , while in January the range varies to . By estimating the linear regression over each in Figure 16 a) and b), the empirical function of can be deduced. As well, from UM parameter estimation the theoretical function of is computed. The results in Figure 17 shows there is a good agreement between empirical and theoretical functions for lower than 2.5 in December, and lower than 3 in January, approximatively. This divergence was assessed though Eq. ( 50) from the UM parameters deduced with DTM method. Thus, in December and January are 3.06 and 3.43 respectively, which is close to the values in Figure 17.

The characterisation of the intermittency of temperature fluctuations, implies following the same procedure applied previously for . The estimates of TM and DTM over the whole range of scales, for December and January periods, are presented in Figure 18 and the UM parameters summarized in Table 4.

December January In both techniques, the accuracy of the linear regression over all the scale-ratio at order is not so good compared with higher values of . However, the scale invariance behaviour of temperature fluctuations is well observed over all the range of scales in the results of DTM. The curves of as function of in logarithmic scale, with is presented in Figure 18 e) and f). The slope was evaluated in the range of between 0.81 and 1.87 for December, and between 0.65 and 1.52 for January. The values of and , are slightly different from those obtained of turbulent velocity measurements in the atmospheric boundary layer in Paris,  = 1.45 and C1 = 0.29 [START_REF] Schmitt | Estimation of universal for atmospheric turbulent multifractal indices velocity fields[END_REF]. However, the estimates of are similar to the values found by [START_REF] Mezemate | Analyse et modélisation multifractales des interactions ondesturbulence-biologie dans un lac urbain[END_REF] from water temperature data measured at different depths in the Creteil lake (Paris region). varies between 0.085 and 0.016, for the nearest depth to the surface of the lake (0.5m) and the deepest (4.5 m), respectively. In addition, the multifractal analysis conducted by [START_REF] Bodri | Multifractal analysis of temperature time series: Data from boreholes in kamchatka[END_REF] for two temperature series measured in boreholes in Kamchatka Peninsula It appears reasonable to expect variability of is close to that of the temperature as LAS MKI is mainly sensitive to temperature fluctuations in the horizontal path-length between transmitter and receiver. The empirical functions of and temperature in December and January, displayed in Figure 20, partially validate this assumption, as of in both periods is much more convex than the temperature. This may suggest measurements are the results of turbulent processes additional to the heat convection, such as the advection from the surrounded atmosphere. Therefore, it may be useful some additional experimentation, that includes wind speed, to analyse the variability and temperature fluctuations.

Through the codimension function , the fractal dimension value can be deduced. Figure 21.a shows the corresponding estimates of in the framework of the codimension multifractal formalism, using UM parameters and Eq.39. Then, the latter is related to the geometrical dimension multifractal formalism [START_REF] Parisi | A multifractal model of intermittency[END_REF]. In the dimension multifractal formalism, for each singularity the function is linked to the codimensions as , and . The subscript of and represents the dimension of the space on which the process is observed, thus for temporal series. The fractal dimension was plotted as a function of the order of singularity in Figure 21 for and temperature in December and January. This plot is known as the "singularity spectrum" or "multifractal spectrum". Thus, multifractality is corroborated since spectrums of both temporal series, and temperature, exhibit a distribution of singularities (Hölder exponents) and its corresponding fractal dimension.

The spectrums of all temporal series in both periods are asymmetrical between range going from 0.6 up to 1.1. All spectrums are left-skewed as only large fluctuations ( ) are evaluated. The spectrum of in both periods has a quite similar multifractal behaviour, while that of the temperature is lower in December 2019 than in January 2020. Nevertheless, when the spectrum of and temperature in both periods converges to the same value, which not exceed the dimension of the observing space.

Conclusions and Perspectives

In this chapter statistical tools to address the scaling invariance of measurements of scintillometry and temperature over the BGW were used. The spectral analysis, as well as the scaling exponent of the parameter of refraction index of air , demonstrated its scale invariance behaviour. The spectral analysis indicated a single scale-invariance in the timescale of 14-8h to 41 min with a slope close to that of K41, 1.64 in December and 1.57 in January. Regarding the temperature, a linear behaviour for the whole range of scales was observed and the spectral exponent is close to that of BO (2.09 and 1.97).

More precisely, the structure function that characterises the statistical properties of the fluctuations demonstrated a single scaling power law over 300s and 8.5h. In addition, the strong nonlinearity of the scaling exponent function shows that and temperature fluctuations are affected by intermittency and they are multifractal.

The UM parameters were characterised to consider the corrections of intermittency . The obtained values of Lévy index and the mean codimension demonstrate, fluctuations of ( and from DTM) and temperature ( and from DTM) measured over the BGW in both periods of analysis, are multifractal and intermittent. The estimates of the UM parameters of the and temperature remain close to those obtained in the turbulent atmospheric layer.

The multifractal structure of and temperature increments was proved through the estimation of , which in both cases is convex and curved. In addition, the empirical function of and temperature increments have some differences and diverge, indicating that although is mainly affected by temperature fluctuations, additional geophysical fields would impact heat convection in the surrounded atmosphere of the BGW.

Therefore, the temporal and spatial variability of thermal fluxes in the BGW requires a further analysis with more geophysical data, such as wind speed. This would allow to better characterise the scaling invariance as well as the statistical properties presented in this work.

Finally, through the codimension function and the UM parameters, the singularity spectrum of and temperature was characterised, to deduce the fractal dimension of their fluctuations that exceed different order of singularity.

Figure

  Figure 5. Legendre transform between and functions. Adapted from Lovejoy & Schertzer (2013).

Figure 6 .

 6 Figure 6. Plan view of beam in winter months.

Figure 7 .

 7 Figure 7. Horizontal path length and effective height of LAS during winter months.

  /18 -2020/01/17The sampling frequency was 1 Hz with 10 minutes average for the LAS and 5 minutes average for the CNR4. The output LAS signals were controlled via the QC parameter as explained in Deliverable 1.1. The measurements represented in the Figure8shows two periods of time where the values of and exceed -50 mV and 0 V respectively: • 2020/01/01 03:00 to 2020/01/01 19:00• 2020/01/16 02:00 to 2020/01/16 10:30 Without these disruptions, the values of ranges between -1.5 and -2V, while values do not go beyond -350 mV.

Figure 8 .

 8 Figure 8. LAS MKI signals:(upper) and (middle) and calculations (bottom). The QC parameters are marked with the grey dotted line.

Figure 12 .

 12 Figure 12. Structure function of increments in December and January.

Figure 13 .

 13 Figure 13. Scaling exponent of increments in December and January. The dotted line corresponds to the structure function with for linear model of turbulence K41.

Figure 14 .

 14 Figure 14. Structure function of temperature increments in December and January.

Figure 15 .

 15 Figure 15. Scaling exponent of temperature increments in December and January. The dotted line corresponds to the structure function with for BO theory.

Figure 16 .

 16 Figure 16. Characterisation of UM parameters to experimental data; a) and b) TM technique for different values; c) and d) DTM technique with and ; e) and f) Statistical moment function for different values of , the red line represents the linear regression from which UM parameters are determined.

Figure 17 .

 17 Figure 17. Comparison between curves of of , from the empirical value of the TM regression in Figure 16 a) and b), and from UM parameters and the Eq. (38).

Figure 18 .

 18 Figure 18. Characterisation of UM parameters to experimental temperature data; a) and b) TM technique for different values; c) and d) DTM technique with and ; e)

(

  Figure 19. Comparison between curves of of temperature, from the empirical value of the TM regression in Figure 16 a) and b), and from UM parameters and the Eq. (38). The curves of in Figure 19 are convex in both periods which proves the statistics of the temperature in the BGW are multifractals. In addition, from TM and DTM have a good agreement with the empirical curve of A slight deviation apparently occurs over in December and in January. According to estimation of in December and January, the divergence occurs in values of 5.86 and 5.10, respectively.December January

Figure 21

 21 Figure 21. a) The codimension function of and temperature. b) Singularity spectrum, as a function of singularities .

Table 1 .

 1 LAS MKI Setting Experiment on winter months.

Table 2

 2

	.	estimates of	following K41.
			December	January
			0.51	0.424
			0.324	0.289
			0.822	0.665
			0.649	0.578
			0.248±0.1	0.139±0.1

Table 3

 3 presents the UM parameters ( and ) estimated through TM and DTM techniques. The values of and are similar between both periods and techniques, which suggest the fluctuations have comparable features. The estimates of in both periods, demonstrate strong multifractality of fluctuations, as is nearly its maximum 2 (with both techniques). With regards to , the estimates are not very large, which indicates mean intensity is not very sparse.

Table 3 .

 3 UM parameters estimates of .

	Technique	TM		DTM
	Period
	December/2019	1.839	0.106	1.97	0.11
	January/2020	1.879	0.10	1.85	0.10

Table 4 .

 4 UM parameters estimates of temperature.

	Technique	TM		DTM
	Period
	December/2019	1.702	0.055	1.625	0.056
	January/2020	1.680	0.073	1.587	0.075


domain: sde spi

text:
Chloé

Versini

Report on the relationship between functional traits and ecosystem services of urban vegetation: Inventory and recommendations

de la démarche

Ce livrable s'intéresse aux Solutions Fondées sur la Nature (SFN) dans le cadre d'une approche d'ingénierie écologique. Il a pour objectif d'une part de faire le lien entre les traits fonctionnels des espèces végétales mises en oeuvre dans les SFN, et les fonctions et services éco-systémiques qu'elles peuvent prodiguer. Sur la base d'une revue de la littérature et d'entretiens, les espèces les plus couramment utilisées ont été listées et caractérisées à l'aide de leurs traits fonctionnels. Afin d'optimiser les services éco-systémiques auxquels s'intéresse le projet EVNATURB (gestion des eaux pluviales, atténuation des îlots de chaleur, préservation de la biodiversité), les traits favorisant ces services ont été mis en avant.

Dans un second temps, ce travail s'interroge sur les conséquences des changements globaux sur cette nature en ville, sur l'évolution de ses traits fonctionnels et sur ses conséquences en termes de durabilité tant des espèces que des services éco-systémiques qu'elle peut rendre.

Enfin, le livrable se conclut par des recommandations en termes d'espèces et de pratiques pour s'assurer de la durabilité de ces solutions dans le temps dans un tel contexte.

Introduction

In the urban environment, Nature-Based Solutions (NBS) can take various forms [START_REF] Dorst | Urban greening through naturebased solutions -Key characteristics of an emerging concept[END_REF] such as urban forests [START_REF] Tomao | Resilient landscapes in Mediterranean urban areas: Understanding factors influencing forest trends[END_REF][START_REF] Yao | What Causal Drivers Influence Carbon Storage in Shanghai, China's Urban and Peri-Urban Forests?[END_REF], green roofs and facades [START_REF] Xing | Characterisation of nature-based solutions for the built environment[END_REF][START_REF] Frantzeskaki | Seven lessons for planning nature-based solutions in cities[END_REF], parks and street trees [START_REF] Fink | Human-nature for climate action: Nature-based solutions for urban sustainability[END_REF][START_REF] Giannakis | Linear parks along urban rivers: Perceptions of thermal comfort and climate change adaptation in Cyprus[END_REF], ecological corridors [START_REF] Giannakis | Linear parks along urban rivers: Perceptions of thermal comfort and climate change adaptation in Cyprus[END_REF], ponds, natural water retention areas or vegetated swales [START_REF] Scott | Nature-based solutions for the contemporary city/Re-naturing the city/Reflections on urban landscapes, ecosystems services and nature-based solutions in cities/Multifunctional green infrastructure and climate change adaptation: brownfield greening as an adaptation strategy for vulnerable communities?/Delivering green infrastructure through planning: insights from practice in Fingal, Ireland/Planning for biophilic cities: from theory to practice[END_REF][START_REF] Xing | Characterisation of nature-based solutions for the built environment[END_REF], pervious pavements [START_REF] Fini | Nature based solutions to mitigate soil sealing in urban areas: Results from a 4-year study comparing permeable, porous, and impermeable pavements[END_REF], urban gardens [START_REF] Cabral | Urban gardens as multifunctional nature-based solutions for societal goals in a changing climate[END_REF][START_REF] Van Der Jagt | Cultivating nature-based solutions: The governance of communal urban gardens in the European Union[END_REF] including rain gardens [START_REF] Scott | Nature-based solutions for the contemporary city/Re-naturing the city/Reflections on urban landscapes, ecosystems services and nature-based solutions in cities/Multifunctional green infrastructure and climate change adaptation: brownfield greening as an adaptation strategy for vulnerable communities?/Delivering green infrastructure through planning: insights from practice in Fingal, Ireland/Planning for biophilic cities: from theory to practice[END_REF], or urban agriculture [START_REF] Artmann | The role of urban agriculture as a nature-based solution: A review for developing a systemic assessment framework[END_REF].

The integration of such NBS in cities, in a context of global change, raises many questions on the species that are usually used, the way they are managed and on the practices that are used for their maintenance. Consequently, practices vary accordingly: human interventions on a site can range from very limited modifications to major ones, such as the creation of ecosystems [START_REF] Gómez Martín | An operationalized classification of Nature Based Solutions for water-related hazards: From theory to practice[END_REF]. NBS requiring greater management efforts are usually implemented in the downtown areas of cities, while in the peripheries, NBS that require less management are often present (Krauze & Wagner, 2019). This deliverable aims to address these questions and attempts to provide some recommendations for a sustainable implementation of urban NBS. After a first section introducing the main ecological properties of NBS, the second one present the current species and management practices implementing in cities. Based on this analysis, the third section focus on the ways to overcome the disruptions due to global change and their consequences. The rationale behind this approach is to provide a detailed representation of NBS challenges and potential solutions to ensure the continuity of these solutions in the cities of tomorrow.

NBS properties: the functional traits behind the ecosystem services

The urban forms of NBS mentioned above are often aimed at flood prevention, urban heat islands mitigation, biodiversity conservation, etc. [START_REF] Frantzeskaki | Seven lessons for planning nature-based solutions in cities[END_REF][START_REF] Hobbie | Nature-based approaches to managing climate change impacts in cities[END_REF]. Today, NBS are being used as tool for local adaptation and for immediate results, as part of pragmatic responses to manage emergencies. They are a direct answer to climate change and landscape fragmentation as they restore biomass in the city. The benefits they provide, such as flood prevention or heat control, are called ecosystem services. Ecosystem services are usually classified into four main categories: supporting (providing habitats for species, and maintaining genetic diversity), provisioning (provision of food, freshwater and raw material), regulating (air filtering, microclimate regulation, noise reduction, and water drainage/filtering) and cultural services (physical-mental health and recreation). These services are of particular interest in urban environments [START_REF] Bolund | Ecosystem services in urban areas[END_REF][START_REF] Maas | Green spaces, urbanity and health: how strong is the relationship?[END_REF][START_REF] Tzoulas | Promoting ecosystem and human health in urban areas using green infrastructure: A literature review[END_REF].

Ecosystem services (an anthropocentric concept) can be seen as the result of ecological functions (a concept centred on ecosystems and their components) provided by living beings. An ecological function is best described by the interaction of species or through their ecological role where a species or group of species maintain a biogeochemical flux or pool, and/or support ecosystem productivity [START_REF] Brodie | Ecological function analysis: incorporating species roles into conservation[END_REF].

In turn, these functions are influenced by individual characteristics often referred to as functional traits. A functional trait is a morphological, physiological or phenological characteristic of an organism measured at the level of individuals and which affects its individual performance (Violle et al., 2007).

For instance, in the case of a plant, these traits would be its leaf shape or persistence, type of pollination or dispersal means [START_REF] Knapp | Changes in the functional composition of a Central European urban flora over three centuries[END_REF]. As such, the air purification/regulation as an ecosystem service is linked to the ecological function of particle trapping, which in turn depends on the functional trait of the leaf surface [larger leaf area gives a greater particle capture potential] (Commissariat Général au Développement Durable, 2010; [START_REF] Chen | Accumulation of three different sizes of particulate matter on plant leaf surfaces: effect on leaf traits[END_REF]. Furthermore, the pollination service depends on the insect attraction ecological function, which is related to the entomophilous trait.

Here are the main functional traits that are usually considered: root system, type of dispersion, type of pollination, phenology (e.g. time of reproduction), leaf anatomy, leaf persistence, leaf area, etc… Databases summarizing known traits for certain species exist. Here are the main ones:

-The LEDA Traitbase (https://uol.de/en/landeco/research/leda): The LEDA Traitbase provides information on plant traits that describe three key features of plant dynamics: persistence, regeneration and dispersal. It is focussed on the Northwest European flora.

-BiolFlor (https://www.ufz.de/index.php?en=38567): The database BiolFlor on biological and ecological traits of the flora of Germany covers almost 3660 species of vascular species including established and frequent casual alien plant species.

-Baseco (https://baseco.imbe.fr/): BASECO is a floristic and ecological database focused on the Mediterranean French flora.

-TRY (https://www.try-db.org/TryWeb/Home.php): TRY represents the largest database of plant traits that exists (11 850 781 trait records, 279 875 plant taxa, and 214 publications).

In this study, the TRY database has been mainly used to make a link between the functional traits and the ecosystem services on which the EVNATURB project is focused on (stormwater management, urban heat island mitigation and biodiversity preservation essentially).

Overview of current practices, disturbances and possible consequences 3.1 Analysis of existing species

A text-mining analysis was conducted to study the current NBS implemented in urban environments. Carried out using the SCOPUS database for peer-reviewed literature, the occurrence of some keywords used in publications on urban NBS from the appearance of the concept in 2016 until December 13, 2021 was studied. The keywords used for this search were: nature-based AND solution AND city. The analysis was then carried out for the titles, abstracts and keywords of the studied corpus (453 documents). The most publishing journals on this topic were found to be: Sustainability (MDPI), Urban Forestry and Urban Greening (Elsevier), Environmental Research (Elsevier), Science of The Total Environment (Elsevier) and Water (MDPI). Only the keywords appearing at least 8 times were conserved. Accordingly, the main types of NBS described in literature were found to be: green spaces (82), urban forests (81), trees (25), green roofs (22), urban agriculture (19), recreational parks (18) and wetlands (8).

In addition, an individual analysis of some papers dedicated to green roofs was carried out. These documents were retained because they provided an exhaustive list of used/studied species with their relative quantities (e.g. the three species most present on the roofs were kept, or, the three species present on the greatest number of roofs). As a result, a list of 27 dominant (planted and spontaneous) plant species on green roofs used in temperate environments (Germany, Belgium, Canada and Switzerland) was obtained. Some plant functional traits which seem essential for ecological functions (water retention, evapotranspiration, gas exchange, particle trapping, pollen transfer) and ecosystem services (regulation of hydrological cycles, flood risks and climate regulation, air and water purification, pollination and support of biodiversity) were particularly studied. With the help of the TRY plant traits database (https://www.try-db.org; [START_REF] Kattge | TRY plant trait database-enhanced coverage and open access[END_REF], a composite portrait of a typical plant has been created based on the 27 inventoried species (by averaging the numerical variables, and retaining the most common value for the qualitative variables; Table 1). As this analysis was carried out on a relatively limited sample, it would be interesting to do further research on a larger sample of rooftops or even with other types of NBS present in the city. Although this is an important result, it is not a panacea. The plant species are rather small and lack heterogeneity. Pollination of this typical plant is dependent on insects, especially bees, which makes it more vulnerable to disturbance. The modelled plant has a rather long leaf persistence time and average leaf surface and stomatal conductance. These trait values allow an important performance of water retention, evapotranspiration, gas exchange and particle trapping (among other ecosystem services). However, these functions and services can be influenced by the many disturbances that characterize the urban environment.

3-2 Disturbances due to global change

Often, ecosystem hyper-disruption is overlooked, especially in cities. Even when ecosystems are left to self-organize, there are always drivers of change such as climate change, land-use change, pollution, overexploitation and invasive alien species [START_REF] Pereira | Global biodiversity change: the bad, the good, and the unknown[END_REF]. In the urban environment, disturbances are even more numerous, given the strong anthropogenic influence. Indeed, the world's urban population is steadily increasing; 34% of humans lived in cities in 1960 and have increased to 56% in 2020 (World Bank, 2020). Horticultural plants, which are very present in cities, are often the source of invasive plant spreading [START_REF] Reichard | Horticulture as a pathway of invasive plant introductions in the United States: most invasive plants have been introduced for horticultural use by nurseries, botanical gardens, and individuals[END_REF]. Human-induced drivers of change can then affect evolutionary trajectories [START_REF] Palumbi | Humans as the world's greatest evolutionary force[END_REF]. For example, heavy metal exposures have caused changes in the evolutionary properties of (resistance) plants living in contaminated sites [START_REF] Wu | Copper tolerance in two legume species from a copper mine habitat[END_REF]. Another example is that in habitats disturbed by humans, plants interbreed more than in an undisturbed environment [START_REF] Eckert | Plant mating systems in a changing world[END_REF]. This is mainly due to the lack of pollinators. Indeed, even if humans do not directly act on an ecosystem, they still have an indirect impact via climate change or various pollution sources.

However, disruptions are not always sudden. Some of them can occur continuously like climate change. As already mentioned, the IPCC reports that "Cities intensify human-induced warming locally, and further urbanization together with more frequent hot extremes will increase the severity of heatwaves" and "Urbanization also increases mean and heavy precipitation over and/or downwind of cities" (IPCC, 2021). The increase of extreme events (drought, rainfall) in terms of intensity and frequency can clearly have consequences on vegetation [START_REF] Zhang | Socioeconomic and climatic changes lead to contrasting global urban vegetation trends[END_REF]. Therefore, the selection of plants species that can survive water-deficits, especially on dry soil as green roof substrates is crucial (see [START_REF] Du | Is plant survival on green roofs related to their drought response, water use or climate of origin?[END_REF].

The consequences of climate change can also be temporary positive as global warming can advance the phenological phases of leaf development and delay the phenological phases of leaf senescence [START_REF] Wohlfahrt | The urban imprint on plant phenology[END_REF].

3-3 Possible consequences of these disturbances on NBS

The multitude of disturbances present in the city can have multiple consequences on urban NBS, ranging from simple modifications of functional traits and/or populations of living beings to their destruction. Indeed, disturbances also called drivers of change can modify the functional traits of plants, create new ones or eliminate them. Some of these traits named "keystones" are very important for humans, as they interfere with many ecosystem services [START_REF] Hevia | Trait-based approaches to analyze links between the drivers of change and ecosystem services: Synthesizing existing evidence and future challenges[END_REF]. The modification or disappearance of such traits can therefore mean the decrease or the disappearance of many of their related services. The leaf surface of a plant is one of these "keystones" traits, as it is linked to many services such as the regulation of hydrological cycles, flood risk, micro-climate and the purification and maintenance of air quality. This trait appears to be most affected by land use change [START_REF] Hevia | Trait-based approaches to analyze links between the drivers of change and ecosystem services: Synthesizing existing evidence and future challenges[END_REF].

The above-mentioned drivers of change can lead to the selection of plants that have certain functional traits. For example, urbanization can lead to the selection plants that are more tolerant to nitrogen and heat [START_REF] Knapp | Changes in the functional composition of a Central European urban flora over three centuries[END_REF]. Human disturbances can also destroy parts of plant population or even the whole, even the specimen present on a site, thus reducing sitebiodiversity. Indeed, a study on a Chinese island showed that the direct factors making small plant populations susceptible to extinction were human disturbances [START_REF] Chen | Factors affecting the distribution pattern of wild plants with extremely small populations in Hainan Island, China[END_REF].

According to [START_REF] Köhler | Long-term Vegetation Research on Two Extensive Green Roofs in Berlin[END_REF], floristic diversity of green roofs is mainly influenced by weather conditions (temperature and rainfall), so climate change can have a serious impact on this diversity in the future. The change of land use in the environment of a given NBS can alter the necessary conditions necessary for solutions' life cycle or for the delivery of its purposes, according to the complementary land use concept of [START_REF] Colding | Ecological land-use complementation" for building resilience in urban ecosystems[END_REF]. This concept is built "on the idea that land uses in urban green areas could synergistically interact to support biodiversity when clustered together in different combinations" [START_REF] Colding | Ecological land-use complementation" for building resilience in urban ecosystems[END_REF]. For example, private gardens adjacent to urban public parks would enhance bird diversity through the landscape complementation functions provided by native tree cover, berry bushes, ponds, or freshwater springs, which increase the likelihood of attracting species [START_REF] Blair | Land use and avian species diversity along an urban gradient[END_REF][START_REF] Melles | Urban bird diversity and landscape complexity: species-environment associations along a multiscale habitat gradient[END_REF].

Destruction of populations and/or changes in environmental conditions can lead to the creation of new ecological niches, and thus the colonization of a NBS by new species. It can also change the dominant species that are most likely to provide ecosystem services. Indeed, according to [START_REF] Xie | Phylogenetic diversity and plant trait composition predict multiple ecosystem functions in green roofs[END_REF], the functional characteristics of dominant plant species on green roofs are important to provide several services. Drivers of change can therefore destroy natural equilibrium and can also change evolutionary trajectories through adaptation. Accordingly, the question of impoverishing the evolutionary trajectories of the species present can be raised. It would also be important to refocus human/non-human relationships on maintaining evolutionary potential [START_REF] Lecomte | Repenser I'innovation dans une perspective évocentrée de la biodiversité[END_REF]. Therefore, the risk of reducing or losing ecological functions and ecosystem services of urban NBS through human disturbances exists. For example, in non-urban environments, land use intensification is linked to the loss of functional traits (such as thin bark and large size for trees in [START_REF] Brown | Use of provisioning ecosystem services drives loss of functional traits across land use intensification gradients in tropical forests in Madagascar[END_REF] and the erosion of several ecosystem services (such as the supply of firewood and construction wood in [START_REF] Brown | Use of provisioning ecosystem services drives loss of functional traits across land use intensification gradients in tropical forests in Madagascar[END_REF][START_REF] García-Llorente | Biophysical and sociocultural factors underlying spatial trade-offs of ecosystem services in semiarid watersheds[END_REF][START_REF] Laliberté | Land-use intensification reduces functional redundancy and response diversity in plant communities[END_REF]. This raises the question of whether the targeted ecosystem services are still being provided, and whether the original problems are still being solved despite the anthropogenic disturbances present in the city and their consequences. Accordingly, one can also question if we are still dealing with nature-based solutions at this level.

Recommandations

4-1 Species and soil/vegetation complex to optimize ecosystem services

Based on the analysis of some papers dedicated to green roofs (Catalano et al., 2016 ;Rivière, 2019 ;Lundholm et al., 2010 ;[START_REF] Köhler | Long-term Vegetation Research on Two Extensive Green Roofs in Berlin[END_REF]Rochefort et al., 2016. Among others) presented in Section III-1 and the list of 27 identified species, traits have been related to some particularly ecological functions (and ecosystem service) for which we are interested in. Traits values were extracted from the TRY database.

Water retention:

Water retention allows the regulation of hydrological cycles and the risk of flooding.

It seems that taproots tend to favor the development of preferential water flows due to the formation of macropores. These macropores, potentially associated with preferential flows, would therefore promote an increase in hydraulic conductivity. Large and flat (non-curved) leaf, characterized by a hydrophilic tendency (contact angle less than 110°), as opposed to long and oval leaf would improve water retention.

Considering the other variables, the most interesting species in the list from a water retention point of view are Festuca ovina and Poa compressa, which have a large leaf area per plant (respectively 864 and 704 cm²), a high stomatal conductance per leaf area (0.68 and 0.57 mol/m²/s) and long leaf persistence (6 and 18 months).

Evapotranspiration:

Evapotranspiration allows the regulation of hydrological cycles, flood risks and the local climate (especially temperature).

Canopies with large leaves have a higher surface temperature compared to smaller ones. It is due to the higher thickness of the boundary layer (which would therefore limit the effects of surface cooling by convection). Physiologically, this reflects a higher transpiration (and therefore a loss of water) for the plant with large leave surfaces. Small leaf, especially those characterized by low width, seem suitable for hot environments.

The most significant traits favoring evapotranspiration are then: LAI (Leaf Area Index, characterizing the size of assimilatory surface of a crop), leaf persistence and stomatal density (characterizing the interaction between the plant and the atmosphere).

The most interesting species inventoried into the list from evapotranspiration point of view are Festuca ovina and Poa compressa, which have a large surface area of leaves per plant (respectively 864 and 704 cm²), a large conductance of stomata per leaf area (0.68 and 0.57 mol/m²/s) and long leaf persistence (6 and 18 months).

Particle trapping:

The trapping of particles allows the purification and maintenance of air and water quality. This function is mainly conditioned by the leaf surface.

The species that have the largest leaf areas per plant are in descending order: Trifolium repens, Trifolium arvense, Vicia sativa, Plantago lanceolata, Festuca ovina, Taraxacum campylodes (officinale), Poa pratensis, Trifolium campestre, Poa annua, Poa compressa, Bromus tectorum.

N.B.:

There are lot of missing data in the TRY database for the leaf area per plant variable, and many species have not been taken into consideration.

Pollen and seeds transfer:

The transfer of pollen represents a vector of biodiversity through the dissemination it generates, but also through its contribution to insect populations.

The traits identified in relation to pollen transfer are: the means of pollen transfer (by wind, insects, etc.), the lifespan of the plant (an annual plant will produce pollen every year, a biennial every 2 years, a perennial it depends), and the flowering period. Note that there no information about the flowering period in the TRY database.

The most interesting species for pollination and support for biodiversity via pollen are (in alphabetical order, in bold the annual species which therefore necessarily flower every year): Allium schoenoprasum, Erigeron annuus, Geranium pusillum, Lactuca serriola, Medicago lupulina, Petrorhagia saxifraga, Polygonum aviculare, Taraxacum campylodes (officinale), Trifolium campestre. Indeed, these species can be pollinated by many species of insects.

To a lesser extent, the following plant species can also support pollination via insects (in alphabetical order, annual species in bold): Arabidopsis thaliana, Cerastium semidecandrum, Plantago lanceolata, Sedum acre, Sedum album, Sedum sexangulare, Sedum spurium (hybridum), Trifolium arvense, Vicia sativa, Viola arvensis. Indeed, these species are pollinated by fewer kinds of insects than the previous ones or may have other means of pollination (autogamous or by the wind). 4-2 How to choose the species regarding the disturbances?

One of the objectives of NBS is to provide environmental benefits (see IUCN and European Commission definitions). Nonetheless, the main goal is to the conservation of biodiversity.

For this purpose, it may be a good idea to use local plant species, with species already present in the city where the NBS is implemented. As a result, vegetation populations will be larger and therefore more viable. This is important as small and range-restricted populations are highly vulnerable to extinction [START_REF] Terborgh | Some causes of extinction[END_REF][START_REF] Gilpin | Minimal viable populations: processes of species extinction[END_REF]. The establishment of NBS in the city can also be an opportunity to introduce species that were not initially present in the urban environment, but in its surroundings, hence increasing biodiversity in the city. There are already initiatives that promote and facilitate the use of local wild species such as the French label "Végétal local" (Végétal local; https://www.vegetal-local.fr/). However, will these local species be able to cope with disturbances like global warming? Perhaps it would be better to plant exotic (non-invasive) species that can withstand droughts for example or rising temperatures? In support of this argument, one study showed that in North America, the phenology of exotic species adapted better to climate change than native species [START_REF] Wolkovich | Temperature-dependent shifts in phenology contribute to the success of exotic species with climate change[END_REF].

Another solution is to find and install plant species that can cope with the disturbances present in the city, such as "resistant" or ruderal species, or species that have been proven to work in a given context in the face of a given disturbance. The rationale behind this approach is the concepts of ecological resilience, which is the ability of a living system to recover the structures and functions of its baseline state after a disturbance [START_REF] Holling | Resilience and stability of ecological systems[END_REF], and resistance, which is the ability of a system to remain fundamentally unchanged when subjected to a disturbance [START_REF] Grimm | Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion[END_REF]. The idea is to return to the initial trajectories, before disturbances (or new favourable ones). However, this aspirational target is not always guaranteed. In the case of new evolutionary trajectories, one can speak about transformability, which is the ability to create a fundamentally new system when ecological, economic, or social structures make the existing system unsustainable [START_REF] Walker | Resilience, adaptability and transformability in social-ecological systems[END_REF].

In this sense, anticipating changes in eco-evolutionary trajectories by experimenting or studying correlations in urban environments could be an interesting platform to explore.

4-3 What about future practices and maintenance?

The previous section raised the question of how to deal with these disruptions and their consequences for NBS in the city. It can also relate to the question of ensuring the sustainability of the sought solutions. Therefore, it could be interesting to have ecosystems that are well-maintained over time and which are autonomous with planted and spontaneous species that reappear from one year to another or reproduce on site. Eventually, new colonisations that replace plants that have disappeared from the ecosystem can appear. However, the question of how to achieve this sustainability is raised. The idea is to work on the long term, to tap the full potential of the species' life cycles, and to anticipate the drivers of change and their consequences. For this purpose, urban NBS could be eco-designed, hence accommodating for life cycles and natural adaptation. Eco-design is based on the fact that: "the environment helps to define the direction of design decisions and the environment becomes a co-pilot in product [here the NBS] development" [START_REF] Brezet | Dynamics in ecodesign practice[END_REF]. There is also the question of NBS-derived ecosystem service sustainability. But, is there a will to maintain all of these services as much as possible on the long term? Or target one or more in particular?

The purpose is to move towards a dynamic ecological balance that provides ecosystem services, hence maintaining the durability of the designed/implemented NBS.

Regarding the management of NBS in cities for coping with disruptions, a legitimate question arises: do we let "nature" take over or do we accompany it? It would be probably a good idea to let nature take over more often in urban NBS, without direct human intervention or just the bare minimum, such as an annual maintenance of green roofs, the removal of invasive plants, etc. This type of laissez-faire management encourages the development of spontaneous species that are important for the functioning of ecosystems [START_REF] Couvet | Des solutions fondées sur la nature[END_REF] and allows ecosystems to become somewhat self-organized. Ecosystems are thus closer to a functional natural state It also reduces the financial, labour and time costs associated with NBS maintenance. [START_REF] Nesshöver | The science, policy and practice of nature-based solutions: An interdisciplinary perspective[END_REF] stress the fact that NBS governance must be adaptive. Indeed, management must respond to the "reality of the field" and react to unexpected events. Lambert and Donihue (2020) go further to stress that evolving urban biodiversity requires incorporating evolutionary perspectives into management for these efforts to succeed given the dynamic nature of the urban environment. It is also important to take into consideration the landscape's heterogeneity, the ecological dynamics and the evolutionary consequences on biodiversity, i.e. to have an evo-centric approach [START_REF] Lecomte | Repenser I'innovation dans une perspective évocentrée de la biodiversité[END_REF]. Thus, solid progress towards socio-eco-evolution in the city is needed [START_REF] Des Roches | Socio-eco-evolutionary dynamics in cities[END_REF]. Accordingly, human societies, non-human living beings and their ecology and evolution, as well as their interactions should be better accounted for. Whether in eco-design or management, it could be interesting to mimic nature and its ecosystems in an effort to understand their behaviour in the face of various disturbances.

Conclusions

Urban environments are very complex and challenging ecosystems. Drivers of change influencing this particular context are numerous. On one hand, the increase of urban populations and the resulting intensifications of urban cover can modify the water cycle (by inducing imperviousness and heat storage), facilitate invasive species spreads and create various types of pollution. On the other, climate change can intensify the occurrence of extreme events like flooding or heat waves.

The use of NBS is therefore one of the main means to help cities adapt to these constraints and to make them more resilient. For this reason, NBS for climate change adaptation (called Nature-Based adaptation Solutions, NBaS) are starting to gain prominence in scientific discourses (IUCN French Committee, 2019). The increase in related scientific literature supports this statement.

Through the (re)introduction of nature, NBS also represent a good solution for biodiversity and for its effective return to the city. However, if the constraints of global changes on societies are now obvious and well documented, the same is probably also true for the nonhuman living world. Nevertheless, the latter's response to these constraints remains uncertain.

To our knowledge, scientific literature has rarely addressed this question. Thus, the use of NBS for mitigation or adaptation purposes in cities has to be examined with great caution. NBS are usually implemented to mitigate or to adapt the environment to the current situation, often leaving aside the need to forecast the evolution of future constraints and how NBS will respond to them. This stationary point of view has obviously some limitations.

Indeed, resistance to strong constraints also has its limits for city biodiversity. Accordingly, it is difficult to predict whether introduced or local species will survive extreme conditions which they may face. It is also unclear whether this newly introduced nature will be able to provide the ecosystem services for which it was initially planned for. Climatic hazards, such as various types of pollution, can indeed alter some of these "keystones" traits.

This clearly questions the way of approaching the introduction of biodiversity in cities. The time has come to question whether Nature-Based Solutions should be considered as a way to rethink our cities in their entirety. A paradigm shift is surely necessary to move from business-as-usual approach "adapting to what?" all the time, to rather "adapting what to what?", taking into account the possibilities of the territory and its limits.

This approach requires the adoption of a systemic and dynamic approach that gives the diversity, complementarity and renewal of species an important role to ensure services, their sustainability and their adaptability. A stronger involvement of ecologists is required to question these points at scales that go beyond that of the infrastructure on which NBS are often installed. It is about understanding better understanding how a particular NBS fits its surrounding environment, and how its complementarity with the whole system can benefit everyone.

The acquisition of knowledge in these fields should be fed by past and current experiences.

This requires experiments and capitalizing on feedbacks from existing NBS. A recent initiative, the Life ARTISAN project (Increasing the Resilience of Territories to Climate Change by Encouraging Nature-Based Adaptation Solutions, www.life-artisan.fr), can contribute to this task. ARTISAN aims to identify the obstacles and levers related to the implementation of NBS. By following several demonstrator NBS sites, it will provide information on the sustainability of the expected benefits.

In addition, a more global reflection integrating local authorities, engineering design offices and urban planners could be beneficial. The objective is to link aspects of biodiversity with planning, public policies, and climate, in order to share the knowledge with more technical and decision-making organizations. This framework should allow a more successful and cautious reintroduction of nature into cities hence avoiding "greenwashing"

Table 1 :

 1 Characteristics of the typical dominant plant on green roof in cities of temperate climate using data from the TRY database and a sample of roofs.

	Characteristics (27)	Lifespan (27)	Pollination (27)	Root system (27)	Persistence of leaves (24)	Leaf surface per plant (15)	Stomatal conductance per leaf area* (26)
	Typical plant	Perennial	By insects	Non-	11 months 878 cm²	0.24 mol/m².s
		(14)	(20), mainly	pivoting
			bees (19)	(14)

* Transfer of water vapor or CO₂ through stomata. The number in parenthesis corresponds to the number of plants concerned (total number for the first line, number of the dominant trait for the second).

Table 2 :

 2 Relationship between traits, ecological functions and ecosystem services

	TRAITS	FUNCTIONS	SERVICES
	Leaf Area Index (LAI)		Regulation of hydrological cycles
	Leaf persisitence	Water retention	Regulation of the risk of flooding
	Leaf shape
	Root system
	LAI		Regulation of hydrological cycles
	Leaf persisitence	Evapotranspiration Regulation of the risk of flooding
			Regulation of micro-climate
	Stomatal density		(temperature)
			Purification and maintenance of air
	LAI	Particles trapping	quality
			Purification and maintenance of water
			quality
	Pollen transfer medium
	Life time	Pollen and seed
	Flowering period	transfer	Pollination
			Support for biodiversity (insects, etc.)

DUFFAUT (HM&Co-ENPC)

Pierre-Antoine