A coalescence model for freely decaying two-dimensional turbulence

– We propose a ballistic coalescence model (punctuated-Hamiltonian approach) mimicking the fusion of vortices in freely decaying two-dimensional turbulence. A temporal scaling behaviour is reached where the vortex density evolves like t. A mean-field analytical argument yielding the approximation ξ = 4/5 is shown to slightly overestimate the decay exponent ξ whereas Molecular Dynamics simulations give ξ = 0.71 ± 0.01, in agreement with recent laboratory experiments and simulations of Navier-Stokes equation. Two-dimensional turbulence has the fascinating property of organizing into coherent structures from a disordered background. This feature has been observed in laboratory experiments [1, 2] and in numerical simulations, as first emphasized and investigated by McWilliams [3]. It can account for the robustness of Jupiter’s Great Red Spot, a large scale vortex which has been persisting in a turbulent shear for more than three centuries [4]. We shall address the issue of freely decaying two-dimensional turbulence, for which the decay from random initial conditions can be divided into three stages. In the first stage, the system self-organizes into a set of coherent vortices containing most of the flow vorticity. The evolution is then dominated by the mutual advection of these structures, punctuated by dissipative events: whenever two like-sign vortices come closer than a critical distance, they merge (coalesce) to form a bigger vortex, while dipoles form a very stable state. Finally, the third stage starts when there are very few dipoles left, which decay diffusively. In this Letter, we shall concentrate on the second stage of the evolution, during which the average extension of the vortices and their relative distance grow whereas their number density n decreases in time. Numerical simulations of the Navier-Stokes equation [5, 6] have shown that this decay is algebraic: n(t) ∝ t with ξ ≃ 0.72 ± 0.03. On the experimental (∗) Present address: FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam (The Netherlands). E-mail: [email protected] Typeset using EURO-TEX 2 EUROPHYSICS LETTERS side, recent investigations in thin stratified layers of electrolyte emphasize the importance of coherent vortex dynamics for the decay of turbulence in two dimensions [7, 8]. Assuming a self-similar evolution of the vortex system, one can infer from dimensional grounds that n(t) ∝ t, provided the energy is the only conserved quantity [6, 9]. However, the emergence of coherent structures significantly slows the density decay (ξ < 2). In addition, numerical solutions [5, 6] indicate the appearance of a second conserved quantity: the vorticity amplitude ω inside the vortex cores. On the assumption that all the vorticity is concentrated in the vortices and given the two above mentioned invariants, the authors of [5, 6] derived a non-conservative scaling theory expressing all statistical properties in terms of the scaling exponent ξ, and constructed the simplest model capturing the essential features of the selfsimilar stage of freely decaying two-dimensional turbulence. As no inviscid invariants other than the energy and the typical vorticity are preserved, their theory is non-conservative. The vortices are modelized by discs of radii σi, uniform vorticity ω and thus circulation Γi = π σ 2 i ω. The equation of motion of vortex i, having position ri(t) = {xi(t), yi(t)}, is assumed to be governed by the Hamiltonian dynamics of point-vortices (the so-called Kirchoff’s laws [10]):