Modélisation de l'évolution thermique saisonnière du lac du Bourget

Le lac du Bourget, l'un des principaux lacs alpins situé en France, a fait l'objet en 1981 d'importants travaux d'aménagement en vue de ta restauration de la qualité de ses eaux. Une campagne de mesure portant sur les années 1988-89 a été mise en place afin de faire le point sur l'évolution du lac depuis la fin des travaux.Un modèle thermique et biogéochimique (phosphore, oxygène, azote) sera utilisé pour synthétiser les connaissances, prévoir l'évolution de la qualité des eaux du lac ainsi que l'influence d'éventuels aménagements complémentaires. Les résultats présentés ici concernent la première étape du projet d'études, la modélisation thermique du lac du Bourget.Le modèle utilisé est un modèle unidimensionnel, vertical, basé sur l'équation d'advection-diffusion.L'expression des coefficients de dispersion selon la profondeur reprend celle d'un modèle du lac Léman (Tassin, 1986). Les équations utilisées distinguent l'épilimnion, le métalimnion et l'hypolimnion.Les résultats présentés montrent que le modèle décrit de façon satisfaisante le cycle thermique annuel et l'évolution inter-annuelle des températures dans le lac du Bourget.Les profils et les valeurs des coefficients de dispersion calculés sur le lac du Bourget sont proches de ceux obtenus sur d'autres lacs à partir de mesures fines de température ou de concentrations d'isotopes naturels.Les coefficients de dispersion obtenus pourront donc être utilisés dans la modélisation des substances dissoutes dans le lac.In 1981, important works including the diversion of the main sewers entering Lake Bourget (one of the largest French alpine lakes) were undertaken in order to restore acceptable water quality standards. A detailed water quality survey will be performed in 1988-89 to complete the data base which already covers ten years. It should help in quantifying the evolution of the lake since the 1981 restoration works. The following activities will be undertaken as part of the survey : measurements within the water column, a study of the bottom sediments, the setting-up of sediment traps and the coupling of Landsat satellite images with measurements performed at some stations at the lake surface. A thermal and biogeochemical (phosphorus, oxygen, nitrogen) model will be used to summarise the information available and to forecast the evolution of water quality and the effect of other restoration measures. This paper reports on the first part of the study : the thermal modelling of Lake Bourget. A one-dimensional, vertical model based on the advection-diffusion equation is used. This equation is solved using a finite difference, semi-implicit method. The resolution grid has 145 layers (1 meter high) and the time step is 3 hours.The heat fluxes at the air-water interface are computed with the three-hourly meteorological data collected at the Chambéry airport station, at the south end of the lake. The formulas are empirical and well established in the literature.The expression for the eddy diffusion coefficients is based on a model of the lake of Geneva (Tassin, 1986). Different equations are used for the epilimnion, metalimnion and hypolimnion. In the epilimnion, the eddy diffusion coefficient expresses, by the Richardson gradients number, the interaction between the shear stress of the wind and the water column stability. It depends on the value of the eddy diffusion in neutral conditions and on a stability function which includes the Richardson number. The vertical profile of the horizontal number. The vertical profile of the horizontal currents is computed following Ekman (1905) and Simons (1981) and is approximated by a gradient which decays exponentially.In the metalimnion and the hypolimnion, the water layer stability is characterised by the Brünt-Väisälä frequency. In the hypolimnion, the eddy diffusion coefficient includes a corrective term, which is a function of depth; this term serves to reduce the dispersion near the bottom of the lake.The thermal model also includes the mixing of the first layers by waves, the vertical advective transport induced by rivers inflows into the lake and by surface water withdrawal, as well as the thermal convection induced by local instabilities of the water column.The eight parameters occurring in the heat flux equations and the eddy diffusion coefficients were estimated using data from 1981. The model has been validated over an 8 years period 1976-1983).Results of the model agree with the observed seasonal and long-term evolutions of temperature in Lake Bourget. The main characteristics of the annual cycle are reproduced : set-up of the thermocline in spring, depth of the epilimnion, thermocline and metalimnion, value of the temperature gradient in the metalimnion, deepening of the thermocline in fall and winter overturn. Significant differences between observed and simulated temperatures occur in the metalimnion and they may be partially explained by internal waves which are dominant features at this level during summer stratification. This kind of mechanism cannot be accounted for in a one-dimensional vertical model.Profiles and values of eddy diffusion coefficients calculated for Lake Bourget show a good agreement with those obtained in other large takes from measurements of temperature or natural isotopes concentrations.In general, the thermal model gives a good account of heat transport mechanisms in Lake Bourget. Accordingly, it provides acceptable dispersion coefficients which can be used to model the distribution of dissolved species in the lake

Ondes internes du lac du Bourget: analyse des observations par des modèles linéaires

Deux campagnes de mesures effectuées sur le lac du Bourget en période de faible stratification (avril et décembre 1994) ont mis en évidence des oscillations de la thermocline de période comprise entre deux et trois jours. Ces oscillations atteignent 40 m d'amplitude pour une profondeur maximale de 145 m. Elles ont probablement un impact sur les processus biologiques et physico-chimiques qui gouvernent l'évolution de la qualité des eaux du lac.L'analyse des données brutes des températures révèle une corrélation étroite entre la génération des ondes internes et les événements de vents importants. Il apparaît en particulier que seuls les vents violents (< 8 m/s) affectent la stratification thermique de manière significative.Deux approches sont utilisées pour caractériser ces ondes :- une approche par traitement du signal qui donne accès aux périodes d'oscillations prédominantes ainsi qu'à la répartition de l'énergie dans la colonne d'eau en fonction de la fréquence.- une approche par modélisation mathématique au cours de laquelle les résultats obtenus par tjjois techniques distinctes utilisant plusieurs degrés de représentation de la bathymétrie du lac sont comparés. Ces modèles permettent de calculer les périodes d'oscillations ainsi que les déplacements de l'interface de densité et les vitesses dans chaque couche. A partir des valeurs des amplitudes d'oscillation obtenues expérimentalement, des vitesses maximales de l'ordre de 7 cm/s dans l'épîlîmnion et 3 cm/s dans l'hypolimnion ont pu être estimées pour les deux épisodes considérés.On montre que les modèles mathématiques et l'analyse spectrale corroborent les observations.There is a great concern about the understanding of water mass movements in lakes as they play a crucial role in the way nutrients and pollutants are trans-ported. This work brings new insights to the study of internal waves as it compares field data to various mathematical approaches. During the months of April and December 1994, a thermistor chain was deployed in Lake Bourget, France (length 18 km, width 3 km, maximum depth 145 m) to record temperature every 10 min, over nine unevenly spaced depths, from 10 to 51m. The time series of measurements provide a detailed picture of the characteristics and dynamics of internal waves. Records are discussed in view of the wind data observed at a meteorological station located at the south-end of the lake. According to the intensity of the wind forcing, the thermal structure is altered in different ways. When the winds are weak, the thermo-cline tilts and sets up a hydrostatic pressure gradient which balances the wind stress (TURNER, 1973). When the wind stops, the density interface oscillates until buoyancy is strong enough to balance the baroclinic pressure field. Strong winds, however, enhance large amplitude nonlinear waves which may break (Kelvin-Helmoltz instabilities), and therefore give rise to vertical mixing in the hypolimnion. During the recording periods wind stress in general is low, but occasional bursts of energy generate internal waves (fig. 2 and 3). In this paper, we focus on linear internal waves as records display a dominant response of the first longitudinal mode. The internal seiche continues to oscillate with decreasing amplitude after the wind has ceased. Two approaches have been implemented in order to characterize the internal waves. One consiste of signal treatment through spectral analysis and the second one involves mathematical modeling.Spectral analysis discloses responses of the first mode with periods of about 80h and 40h, respectively, for the April and December fleld survey (fig. 4 and 5). Further analysis of the April spectra shows that winds generate highly non-linear waves with high energy levels in a large band located in the first 30m. This band results from the mergence of two peaks of high energy at 80 and 40h respectively which probably correspond to the fundamental and second harmonie of a nonlinear wave. Then, as the wind stops, internal seiche of the first mode develops in the layer located between 30 and 50m indicating a deepening of the thermocline.Those fluctuations and their energy spectra are compared with the prédictions of three methods which are based on linear théories and consequently are not valid when the magnitude of oscillations is too high. The full phenomenon of wind-forced motion in a lake is not treated here. However, the analysis of postforcing phase is undertaken to charaterize free internai waves. One method is the Merian formula, which considers the lake as a two-layer system of constant properties and assumes the lake as a rectangular box. Another is a modified version of the Defant procédure (MORTIMER, 1979) which again assumes two layers but solves the momentum and mass équations with a varying cross section. The third method is the two layered variable depth model (TVDM) deve-loped by Schwab (HORN et aL, 1986), fitted to the basin topography and inclu-ding the free surface displacement It is expressed here through a one dimensional version directed along the main axis of the lake (i.e. the lateral variations of depth are not considered).The models display pattems of thermocline displacements (illustrated in fig. 7 and 8) which, in periodicity, are closely similar to those observed. Moreover, they give estimates of the maximum velocity induced by the seiche. Values of the order of 7 cm/s and 3 cm/s are found in the epilimnion and hypolimnion respectively. Finally, the influence exerted by the morphometry on the wave shape and associated field velocity is emphasized. In particular, the difference in the maximum speed calculated in the hypolimnion probably stems from the lateral contraction of the lake (and thus increasing speed) near Aix-les-Bains which is not taken into account in the TVD Model. The validity of the models implemented here is thrown back into question when the magnitude of the oscillations is sufficiently high to steepen the thermocline and in this particular case, a nonlinear theory (Korteweg-de Vries Equation) would be appropriate.The importance of a better knowledge of internal seiches goes beyond the field of physics. Through their influence on mixing and dispersal, those motions profoundly affect the chemical and biological economies of many lakes. Internal waves are responsible for periodic vertical displacement of the resuspended biomass, and consequently for variation in the light intensity to which algal cells are exposed. Furthermore, associated bottom currents can enhance dissolution and remobilization of nutrients by transporting the products of bacterial decomposition away from the sediment-water interface into the water column

Tunable hybrid surface waves supported by a graphene layer

We study surface waves localized near a surface of a semi-infinite dielectric medium covered by a layer of graphene in the presence of a strong external magnetic field. We demonstrate that both TE-TM hybrid surface plasmons can propagate along the graphene surface. We analyze the effect of the Hall conductivity on the disper- sion of hybrid surface waves and suggest a possibility to tune the plasmon dispersion by the magnetic field.Comment: 3 pages, 3 figure

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