Length-scale dependence of horizontal dispersion in the surface water of lakes

Horizontal dispersion in the surface waters of a medium-sized lake was investigated based on four experiments with ensembles of 14 to 17 drifters deployed in Lake Constance during the winter season. The experiments cover length scales between 30 and 3000 m which are typical for the grid length scales employed in 3-D models. Horizontal dispersion coefficients Kdisp were estimated to range from 0.01 to 0.03 m s 1 at 100 m and from 0.1 to 0.7 m s 1 at 1000 m length scale L of the drifter distributions. In all experiments Kdisp increased with L and in three of the four experiments this increase was about linear in L. However, although a linear increase of Kdisp with L is consistent with dispersion by shear diffusion in flow fields with constant current shear, numerical modelling of the paths of the drifters, and the resulting spreading of the drifter ensembles indicates that horizontal shear diffusion is not sufficient to explain the increase of Kdisp with L. The model results suggest that not shear diffusion due to large-scale shear across the entire drifter distribution, but scale-dependent turbulent diffusion is dominating the scale-dependence of Kdisp at length scales between 100 and 1000 m. According to the model results, effects due to horizontal shear contribute less than 40% to the overall dispersion coefficient at length scales between 100 and 1000 m. However, horizontal shear is required to explain shape and orientation of the drifter distributions. Spreading and dilution of dissolved substances in lakes result from the interplay of small-scale molecular and turbulent motions with the large-scale flow field. Mixing due to turbulence is typically described in analogy to molecular diffusion as turbulent diffusion and is characterized by a eddy diffusivity. At sufficiently long time scales, dispersion of dissolved substances resulting from the combined effect of the large-scale flow field and turbulent diffusion can also be approximated as a diffusive type process, which is commonly characterized by a dispersion coefficient (Fischer et al. 1979). Although turbulent diffusion and dispersion are central for estimating the spread and dilution of dissolved substances, field measurements of horizontal eddy diffusivities and dispersion coefficients in lakes are scarce. Horizontal dispersion coefficients in lakes and oceans have been derived from the temporal change in the horizontal size of tracer distributions deployed far from the boundaries. Tracer experiments investigating horizontal dispersion in the surface water of lakes were conducted e.g., in Lake Ontario (Murthy 1976), the very small Twin West Lake (Lawrence et al. 1995) and the very narrow fjord like Kootenay Lake (Stephens et al. 2004). Horizontal dispersion coefficients determined from tracer experiments in oceanic surface water have been compiled by Okubo (1971). Horizontal dispersion in the thermocline of several lakes has been investigated by Peeters et al. (1996). In all these studies the horizontal dispersion coefficients increased with the length scale of the tracer distribution. An alternative to the assessment of horizontal dispersion from the spreading of dissolved substances are dispersion experiments with ensembles of Lagrangian drifters. In such experiments drifters are typically deployed at a fixed depth and designed to remain at this depth. In this case, the horizontal spreading of the drifter ensemble is not affected by shear dispersion due to vertical current shear but only by the horizontal component of the turbulent motions and of the large-scale flow field. Drifter experiments investigating the horizontal dispersion in the surface water of a lake were conducted in Lake Kinneret during stratified conditions (Stocker and Imberger 2003). From the spreading of four drifters deployed in the surface water a time series of a horizontal dispersion coefficient was estimated. This dispersion coefficient showed large almost periodic oscillations between positive and negative values most likely resulting from the periodic convergence and divergence in the horizontal component of the large-scale flow field associated with basinscale seiching (Stocker and Imberger 2003). Because negative *Correspondence: Frank.Peeters@uni konstanz.de