Clustering of passive tracers in free-surface flows

Jörg Schumacher

Fachbereich Physik
Philipps-Universität Marburg, Germany

Experimental and numerical studies of turbulent fluid motion in a free surface are presented. The flow is realized experimentally on the surface of a tank filled with water stirred by a vertically oscillating grid positioned well below the surface (Goldburg et al 2001). The effect of surface waves appears to be negligible so that the flow can numerically be realized with a flat surface and stress-free boundary conditions above three-dimensional volume turbulence (Eckhardt and Schumacher 2001).

The two-dimensional free surface flow, ${\bf v}(x,y,t)$, is unconventional: it is not incompressible, i.e. $\partial_x v_x+\partial_y v_y\ne 0$, and neither kinetic energy, nor squared vorticity (enstrophy) are conserved in the limit of zero fluid viscosity and of absence of external driving as it is the case for ``usual'' two-dimensional turbulent flows (Lesieur 1990). According to both experiment and numerical simulation, statistical properties of the surface flow are closer to those of three-dimensional turbulence.

The dynamics of passive Lagrangian tracers that are advected in such flows is dominated by rapidly changing patches of the surface flow divergence. Single particle and pair dispersion show different behavior for short and large times: on short times particles cluster exponentially rapidly until patches of the size of the divergence correlation function are depleted; on larger times the pair dispersion is dominated by subdiffusive hopping between clusters. We also find that the distribution of particle density is algebraic, and not lognormal as predicted for flows that are delta-correlated in time (Klyatskin and Saichev 1997). The latter so-called Kraichnan flows are rather synthetic but allow for making analytical progress. Our results can be traced back to the exponential distribution of the divergence field of the surface flow. Very recently, physical mechanisms for the formation of rain drops were discussed by Balkovsky et al (2001). They considered the motion of tracers that have inertia (Maxey and Riley 1983) but are advected in an incompressible turbulent Kraichnan flow. The relation of our findings to this problem is discussed.

References
Balkovsky E, Falkovich G and Fouxon A 2001 Phys. Rev. Lett. 86 2790
Goldburg W I, Cressman J R, Vörös Z, Eckhardt B and Schumacher J 2001 Phys. Rev. E 63 065303(R)
Eckhardt B and Schumacher J 2001 Phys. Rev. E 64 016314
Klyatskin V I and Saichev A I 1997 JETP 84 716
Lesieur M 1990 Turbulence in Fluids (Dordrecht: Kluwer)
Maxey M and Riley J 1983 Phys. Fluids 26 883