Analyse et modélisation multifractales des interactions ondes-turbulence-biologie dans un lac urbain

Research in limnology has generally focused on natural lakes and dams. Moreover, the scientific challenges posed by urban lakes are numerous because of their large width and shallow depth resulting in complex problems that involve. This work is part of the project Petits Lacs Urbains Mesures Modèles Multi-Echelles (PLUMMME), founded by Région Île-de-France (DIM R2DS program). The project provides the high-resolution equipment measuring equipment used at Lake Créteil. Physical and biological measurements are the primary scope of this thesis, with some comparisons of measurements done between Lake Creteil and Lake Bourget. With this in mind, the first step of the thesis was to enhance currently available data. In order to do this we installed a station for the continuous measurement of: temperature, chlorophyll and light, also meteorological quantities such as wind speed, and air temperature were measured. The quantities were measured using two measuring chains comprising different sensors. Measurements were conducted in order to characterise hydrodynamics, using a current meter of type Acoustic Doppler Velocimeter (ADV) and a profiler of type Acoustic Doppler Current Profilers (ADCP) positioned at different points in the lake (centrally, and at the point of stormwater discharge). The analysis of the various measured fields shows that the lake is periodically stratified with a period of one week. We also identify the different modes of the internal waves that occur due to the effects of the wind. Spectral analysis was used to highlight the first scale invariant propriety of the different measured fields. Various physical processes (turbulence, stratification, near-wall flow) operating along the water column were also characterised using this method. The spectral analysis, however, does not provide information about the intermittency of the fluctuations of the measured fields, this can only be done using multifractal techniques. In this thesis, we have shown that when there is a dependence between two fields, the dependence is multiscale. The use of the Universal Multifractal (UM) model, allows one to quantify the degree of this dependence. The quality of the estimation of the UM parameters depends strongly on that of the observed scale: time-series with trends are not scaling. The effects of the latter on the estimation of UM parameters can be improved by the application of the Empirical Modal Decomposition method. The measured velocity data from the ADCP shows that the slope of the power spectra density follow a logarithmic profile along the depth of the lake, it shows that different physical processes operate along the water column. We also show that the hydrodynamics of the lake at small scale are strongly perturbed at the point of waterstorm discharge. The last part of the thesis is focused on what our analysis at small-scales brings to numerical models. We show that, if the deterministic models are able to reproduce some phenomena at large scale, they fail to describe the small-scale variability. the small scale variability and the physical processes involved. A multifractal analysis showed that the small-scale variability of the physical fields displays a strong intermittency, an extremely important feature for biological or chemical reactions and therefore for biological scenarios. Knowing that the majority of biological/chemical interactions occur at smaller scales, this result underlines the necessity to greatly improve the closure of the Navier-Stokes equations. Finally, we show that the structure function, a frequently used statistical tool in turbulence, do not uniquely characterize non-conservative fields, i.e., they do not correspond to identical simulationsLes recherches en milieu lacustre ont généralement porté sur les grands lacs naturels et les barrages. Or les défis scientifiques que présentent les lacs urbains sont nombreux du fait de leur faible profondeur et de leur extension .Le travail que nous présentons ici s'inscrit dans le cadre du projet Petits Lacs Urbains Mesures Modèles Multi-Echelles (PLUMMME), projet financé par la région Île-de-France (programme DIM R2DS).Ce projet a permis l'équipement du lac de Créteil en mesures à haute résolution. Ce sont ces mesures physiques et biologiques qui sont essentiellement exploitées dans cette thèse, parfois en comparaison avec celles du lac du Bourget. La première étape de cette thèse a été ainsi d'enrichir la base de données déjà disponible. Pour cela, nous avons mis en place une station de mesure en continu (capteurs de température, chlorophylle et luminosité, données météorologique telles que vitesse du vent et température atmosphérique), ainsi que deux chaînes de mesures comportant différents capteurs. Des campagnes de mesures spécifiques ont également été réalisées pour l'étude de hydrodynamique à l'aide d'un courantomètre de type Acoustic Doppler Velocimeter (ADV), ainsi qu'un profileur de type Acoustic Doppler Current Profilers (ADCP) à deux points différents du lac (point central et point du rejet d'eaux pluviales). L'analyse des différents champs montre que le lac présente des stratifications avec une période d'une semaine. Nous avons également identifié les modes d'oscillations des ondes internes lorsque ces dernières se produisent sous l'effet du vent. L'analyse spectrale a permis de mettre en évidence une première propriété d'invariance d'échelle des différents champs mesurés. Les différents processus physique (turbulence, stratification, écoulement proche de la paroi) opérant le long de la colonne d'eau ont également été caractérisés. L'analyse spectrale ne permet pas de mettre en évidence le caractère intermittent des fluctuations des champs mesurés, ces dernières peuvent par contre être analysées à l'aide des techniques multifractales. Dans cette thèse nous avons montré que lorsqu'il existe une dépendance entre deux champs, celle ci est multi-échelle. L'utilisation du modèle multifractal universel (UM) a permis de quantifier le degré de cette dépendance. La qualité de l'estimation des paramètres UM dépend fortement de celle de l'invariance d'échelle observée: des séries temporelles présentant des tendances ne respectent pas cette invariance. Les effets de ces dernières sur l'estimation des paramètres UM peuvent être éliminés en utilisant la méthode de décomposition empirique modale dans l'espace physique. L'utilisation des données du champ de vitesse mesuré à l'aide de l'ADCP montre que, les pentes des spectres suivent un profil logarithmique selon la verticale, cela met en évidence que différents processus physiques opèrent le long de la colonne d'eau. Nous montrons également que l'hydrodynamique du lac est fortement perturbée à petite échelle au point du rejet d'eau pluviale. La dernière partie de la thèse examine ce que notre analyse à petites échelles apporte aux modèles numériques. Nous montrons que si les modèles déterministes arrivent à reproduire certains phénomènes à grande échelle, ils sont dans l'incapacité de représenter correctement les fluctuations à petites échelles, donc les processus physiques correspondants. L'analyse multifractale montre que la variabilité des champs physiques présente une forte intermittence. Le fait que la majorité des interactions biologiques/chimiques se produisent à petites échelles souligne la nécessité d'améliorer qualitativement les modèles de fermeture des équations de Navier-Stokes. Enfin, nous montrons que les fonctions de structures, observables statistiques de base en turbulence, ne permettent pas de caractériser de façon unique les champs non conservatifs et donc correspondent à des simulations multifractales non identique

Multifractal analysis and simulation of turbulence, waves and biochemistry interactions in small urban lakes

Les recherches en milieu lacustre ont généralement porté sur les grands lacs naturels et les barrages. Or les défis scientifiques que présentent les lacs urbains sont nombreux du fait de leur faible profondeur et de leur extension .Le travail que nous présentons ici s'inscrit dans le cadre du projet Petits Lacs Urbains Mesures Modèles Multi-Echelles (PLUMMME), projet financé par la région Île-de-France (programme DIM R2DS).Ce projet a permis l'équipement du lac de Créteil en mesures à haute résolution. Ce sont ces mesures physiques et biologiques qui sont essentiellement exploitées dans cette thèse, parfois en comparaison avec celles du lac du Bourget. La première étape de cette thèse a été ainsi d'enrichir la base de données déjà disponible. Pour cela, nous avons mis en place une station de mesure en continu (capteurs de température, chlorophylle et luminosité, données météorologique telles que vitesse du vent et température atmosphérique), ainsi que deux chaînes de mesures comportant différents capteurs. Des campagnes de mesures spécifiques ont également été réalisées pour l'étude de hydrodynamique à l'aide d'un courantomètre de type Acoustic Doppler Velocimeter (ADV), ainsi qu'un profileur de type Acoustic Doppler Current Profilers (ADCP) à deux points différents du lac (point central et point du rejet d'eaux pluviales). L'analyse des différents champs montre que le lac présente des stratifications avec une période d'une semaine. Nous avons également identifié les modes d'oscillations des ondes internes lorsque ces dernières se produisent sous l'effet du vent. L'analyse spectrale a permis de mettre en évidence une première propriété d'invariance d'échelle des différents champs mesurés. Les différents processus physique (turbulence, stratification, écoulement proche de la paroi) opérant le long de la colonne d'eau ont également été caractérisés. L'analyse spectrale ne permet pas de mettre en évidence le caractère intermittent des fluctuations des champs mesurés, ces dernières peuvent par contre être analysées à l'aide des techniques multifractales. Dans cette thèse nous avons montré que lorsqu'il existe une dépendance entre deux champs, celle ci est multi-échelle. L'utilisation du modèle multifractal universel (UM) a permis de quantifier le degré de cette dépendance. La qualité de l'estimation des paramètres UM dépend fortement de celle de l'invariance d'échelle observée: des séries temporelles présentant des tendances ne respectent pas cette invariance. Les effets de ces dernières sur l'estimation des paramètres UM peuvent être éliminés en utilisant la méthode de décomposition empirique modale dans l'espace physique. L'utilisation des données du champ de vitesse mesuré à l'aide de l'ADCP montre que, les pentes des spectres suivent un profil logarithmique selon la verticale, cela met en évidence que différents processus physiques opèrent le long de la colonne d'eau. Nous montrons également que l'hydrodynamique du lac est fortement perturbée à petite échelle au point du rejet d'eau pluviale. La dernière partie de la thèse examine ce que notre analyse à petites échelles apporte aux modèles numériques. Nous montrons que si les modèles déterministes arrivent à reproduire certains phénomènes à grande échelle, ils sont dans l'incapacité de représenter correctement les fluctuations à petites échelles, donc les processus physiques correspondants. L'analyse multifractale montre que la variabilité des champs physiques présente une forte intermittence. Le fait que la majorité des interactions biologiques/chimiques se produisent à petites échelles souligne la nécessité d'améliorer qualitativement les modèles de fermeture des équations de Navier-Stokes. Enfin, nous montrons que les fonctions de structures, observables statistiques de base en turbulence, ne permettent pas de caractériser de façon unique les champs non conservatifs et donc correspondent à des simulations multifractales non identiquesResearch in limnology has generally focused on natural lakes and dams. Moreover, the scientific challenges posed by urban lakes are numerous because of their large width and shallow depth resulting in complex problems that involve. This work is part of the project Petits Lacs Urbains Mesures Modèles Multi-Echelles (PLUMMME), founded by Région Île-de-France (DIM R2DS program). The project provides the high-resolution equipment measuring equipment used at Lake Créteil. Physical and biological measurements are the primary scope of this thesis, with some comparisons of measurements done between Lake Creteil and Lake Bourget. With this in mind, the first step of the thesis was to enhance currently available data. In order to do this we installed a station for the continuous measurement of: temperature, chlorophyll and light, also meteorological quantities such as wind speed, and air temperature were measured. The quantities were measured using two measuring chains comprising different sensors. Measurements were conducted in order to characterise hydrodynamics, using a current meter of type Acoustic Doppler Velocimeter (ADV) and a profiler of type Acoustic Doppler Current Profilers (ADCP) positioned at different points in the lake (centrally, and at the point of stormwater discharge). The analysis of the various measured fields shows that the lake is periodically stratified with a period of one week. We also identify the different modes of the internal waves that occur due to the effects of the wind. Spectral analysis was used to highlight the first scale invariant propriety of the different measured fields. Various physical processes (turbulence, stratification, near-wall flow) operating along the water column were also characterised using this method. The spectral analysis, however, does not provide information about the intermittency of the fluctuations of the measured fields, this can only be done using multifractal techniques. In this thesis, we have shown that when there is a dependence between two fields, the dependence is multiscale. The use of the Universal Multifractal (UM) model, allows one to quantify the degree of this dependence. The quality of the estimation of the UM parameters depends strongly on that of the observed scale: time-series with trends are not scaling. The effects of the latter on the estimation of UM parameters can be improved by the application of the Empirical Modal Decomposition method. The measured velocity data from the ADCP shows that the slope of the power spectra density follow a logarithmic profile along the depth of the lake, it shows that different physical processes operate along the water column. We also show that the hydrodynamics of the lake at small scale are strongly perturbed at the point of waterstorm discharge. The last part of the thesis is focused on what our analysis at small-scales brings to numerical models. We show that, if the deterministic models are able to reproduce some phenomena at large scale, they fail to describe the small-scale variability. the small scale variability and the physical processes involved. A multifractal analysis showed that the small-scale variability of the physical fields displays a strong intermittency, an extremely important feature for biological or chemical reactions and therefore for biological scenarios. Knowing that the majority of biological/chemical interactions occur at smaller scales, this result underlines the necessity to greatly improve the closure of the Navier-Stokes equations. Finally, we show that the structure function, a frequently used statistical tool in turbulence, do not uniquely characterize non-conservative fields, i.e., they do not correspond to identical simulation

Searching for a multifractal signature of the lake algal proliferation, a multifractal correlation

Green algae proliferations affect water bodies such as the Lake Bourget (France). They are an environmental issue as well as a mater of public health. In the framework of the PROLIPHYC project a system based on temperature and chlorophyll measurements coupled to a lake model was implemented to predict sudden algal blooms. This classical approach relies on the analysis of large scale trends of the measured fields and does not take into account small scale fluctuations. A more innovative approach has been developed by the R2DS PLUMMME project to investigate the correlation between environmental fields across the full range of space-time scales, down to the smallest scale of observations. The first results of the project demonstrate that multi-scaling behaviour of environmental fields, such as temperature and chlorophyll, becomes evident only after the removal of the large-scale data trends that otherwise induce biases to the multifractal parameter estimates. First, a spectral analysis of temperature and chlorophyll data is performed on sub-samples of the time series to investigate the scaling behaviour. The multifractal analysis (Trace Moment, Double Trace Moment) directly applied on each sub-sample shows unsatisfying results on some sub-samples, in particular on those having a strong gradient compared with the amplitude of the fluctuations. Hence, non-stationary and seasonal effects should be first removed from the time series. To put on evidence a good scaling of the analysed data, we choose the Hilbert-Huang transform to de-trend the data. This method has been widely used for different fields (see F.G.Schmitt et al, 2009 for review). After having applied this method, the K(q) function shows that the investigated fields are indeed multifractal and the determination of their multifractal parameters becomes robust. Then, we proceed to a multifractal correlation analysis between the fields. In conclusion, we discuss the prediction of algal blooms based on multifractal cross-correlations with temperature across scales

Multifractal analysis of geophysical time series in the urban lake of Créteil (France).

Urban water bodies take part in the environmental quality of the cities. They regulate heat, contribute to the beauty of landscape and give some space for leisure activities (aquatic sports, swimming). As they are often artificial they are only a few meters deep. It confers them some specific properties. Indeed, they are particularly sensitive to global environmental changes, including climate change, eutrophication and contamination by micro-pollutants due to the urbanization of the watershed. Monitoring their quality has become a major challenge for urban areas. The need for a tool for predicting short-term proliferation of potentially toxic phytoplankton therefore arises. In lakes, the behavior of biological and physical (temperature) fields is mainly driven by the turbulence regime in the water. Turbulence is highly non linear, nonstationary and intermittent. This is why statistical tools are needed to characterize the evolution of the fields. The knowledge of the probability distribution of all the statistical moments of a given field is necessary to fully characterize it. This possibility is offered by the multifractal analysis based on the assumption of scale invariance. To investigate the effect of space-time variability of temperature, chlorophyll and dissolved oxygen on the cyanobacteria proliferation in the urban lake of Creteil (France), a spectral analysis is first performed on each time series (or on subsamples) to have an overall estimate of their scaling behaviors. Then a multifractal analysis (Trace Moment, Double Trace Moment) estimates the statistical moments of different orders. This analysis is adapted to the specific properties of the studied time series, i. e. the presence of large scale gradients. The nonlinear behavior of the scaling functions K(q) confirms that the investigated aquatic time series are indeed multifractal and highly intermittent .The knowledge of the universal multifractal parameters is the key to calculate the different statistical moments and thus make some predictions on the fields. As a conclusion, the relationships between the fields will be highlighted with a discussion on the cross predictability of the different fields. This draws a prospective for the use of this kind of time series analysis in the field of limnology. The authors acknowledge the financial support from the R2DS-PLUMMME and Climate-KIC BlueGreenDream projects

Performance Assessment of a 3D Hydrodynamic Model Using High Temporal Resolution Measurements in a Shallow Urban Lake

International audienceUrban lakes provide many ecosystem services, e.g., flood control, nature protection, coolness island, recreation. Hydrodynamic models will increasingly be used to enhance these benefits. We present the first validation of a three-dimensional (3D) hydrodynamic model on a small shallow lake with high resolution and high frequency measurements. Lake Créteil, France (area 0.4 km2, mean depth 4.5 m, and catchment area 1 km2) is a former gravel pit and now part of a regional park. The model Delft3D-FLOW was calibrated on a one-month period, with continuous measurements of temperature at five depths at the center of the lake and at three depths at two other stations, and with current speed profiles at the centre of the lake. The model was then verified on 18 1-month periods with similar temperature measurements. The model reproduced very well the temperature dynamics, including the alternation between mixing and stratification periods and internal wave patterns. The mean absolute errors over the five depths at the central point remained below 0.55∘C in spring and summer, the most favorable seasons for phytoplankton growth. Horizontal temperature differences, which rose up to 3∘C at the beginning of stratification periods, were also well reproduced, as well as current speeds. These results are very promising for assessing nutrient and pollutant diffusion, settling and resuspension, as well as for understanding how phytoplankton blooms start in small shallow lakes