Diffusion and dispersion of passive tracers: Navier-Stokes vs. MHD turbulence

W.-C Müller, A Busse
2007 Europhysics letters  
A comparison of turbulent diffusion and pair-dispersion in homogeneous, macroscopically isotropic Navier-Stokes (NS) and nonhelical magnetohydrodynamic (MHD) turbulence based on highresolution direct numerical simulations is presented. Significant differences between MHD and NS systems are observed in the pair-dispersion properties, in particular a strong reduction of the separation velocity in MHD turbulence as compared to the NS case. It is shown that in MHD turbulence the average
more » ... average pair-dispersion is slowed down for τ d t 10τ d , τ d being the Kolmogorov time, due to the alignment of the relative Lagrangian tracer velocity with the local magnetic field. Significant differences in turbulent single-particle diffusion in NS and MHD turbulence are not detected. The fluid particle trajectories in the vicinity of the smallest dissipative structures are found to be characterisically different although these comparably rare events have a negligible influence on the statistics investigated in this work. PACS numbers: 47. 52.30.Cv The diffusive effect of turbulence on contaminants passively advected by the flow is of great practical and fundamental interest. While the study of passive scalars [1] usually reverts to the Eulerian description of the flow, the Lagrangian point of view has proven to be very fruitful regarding investigations of turbulent diffusion and pair-dispersion [2, 3] as well as for the fundamental understanding of turbulence [4] . The three-dimensional dynamics of passive tracers in neutral fluids has been subject of various experimental (for recent works see e.g. [5] [6] [7] ) and numerical, e.g. [8] [9] [10] [11] , investigations. Related problems regarding the turbulent diffusion of magnetic fields and the influence of turbulent magnetic fields on particle diffusion have been studied extensively in space and astrophysics, see e.g. [12] [13] [14] [15] [16] [17] , as well as in the context of magnetically confined nuclear-fusion plasmas, see for example [18] [19] [20] . This Letter reports a first effort to identify differences in the diffusion and dispersion properties of turbulent flows in electrically conducting and in neutral media. To this end the dynamics of fluid particles is studied via highresolution direct numerical simulation of passive tracers immersed in fluids that are described by the incompressible magnetohydrodynamic (MHD) and the Navier-Stokes (NS) approximation. Using the vorticity, ω = ∇ × v, and a uniform mass density, ρ 0 = 1, the non-dimensional incompressible MHD equations are given by The dimensionless molecular diffusivities of momentum and magnetic field are denoted by µ and η, respectively. The magnetic field b is given in Alfvén-speed units. The Navier-Stokes equations which govern the motion of an electrically neutral fluid are obtained by setting b ≡ 0 in Eqs. (1)-(3). The MHD/Navier-Stokes equations are solved by a standard pseudospectral method in a triply periodic cube of linear extent 2π. The velocity field at the position X of a tracer particle is computed via tricubic polynomial interpolation and then used to move the particle according tȯ Eq. 4 is solved by a midpoint method which is straightforwardly integrated into the leapfrog scheme that advances the turbulent fields. Test calculations using Fourier interpolated 'exact' representations of turbulent velocity fields have shown that the chosen tricubic polynomial interpolation delivers sufficient precision with a mean relative error at 512 3 resolution of ∼ O(10 −3 ). In addition tricubic interpolation is numerically much cheaper than the nonlocal spline approach (cf. [21]), especially on computing architectures with distributed memory. The initial particle positions are forming tetrads that are spatially arranged to lie on a randomly deformed cubic super-grid with a maximum perturbation of 25% per super-grid cell. This configuration represents a compromise between statistical independence of particle dynamics and a space-filling particle distribution (cf. [9, 22] ). In addition well-defined initial particle-pair separations ∆ 0 (cf. Table I ) are realized by the tetrad grouping.
doi:10.1209/0295-5075/78/14003 fatcat:5ihuxcuglnbsfllhazmp66fhjy