Ondes internes du lac du Bourget: analyse des observations par des modèles linéaires Internal Waves in Lake Bourget: Data Analysis with Linear Models

Résumé

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.

Mots-clés:

• ondes internes,
• lac,
• modélisation mathématique,
• analyse spectrale

Abstract

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.

Keywords:

• internal waves,
• lake,
• mathematical modeling,
• spectral analysis