Dispersion and Mixing in Quasigeostrophic Turbulence

Les Houches - Ecole d’Ete de Physique Theorique, 2001

To describe transport of scalar and vector fields by a random flow one needs to apply the methods of statistical mechanics to the motion of fluid particles, i.e. to the Lagrangian dynamics. We first present the propagators describing evolving probability distributions of different configurations of fluid particles. We then use those propagators to describe growth, decay and steady states of different scalar and vector quantities transported by random flows. We discuss both practical questions like mixing and segregation and fundamental problems like symmetry breaking in turbulence. Contents I. Introduction A. Propagators B. Kraichnan model C. Large Deviation Approach II. Particles in fluid turbulence A. Single-particle diffusion B. Two-particle dispersion in a spatially smooth velocity C. Two-particle dispersion in a non-smooth incompressible flow D. Two-particle dispersion in a compressible flow E. Multi-particle configurations and zero modes III. Unforced evolution of passive fields A. Decay of tracer fluctuations B. Growth of density fluctuations in compressible flow C. Vector fields in a smooth velocity IV. Cascades of a passive tracer A. Direct cascade B. Inverse cascade in a compressible flow V. Active tracers A. Activity changing cascade direction B. Two-dimensional incompressible turbulence VI. Conclusion References

Journal of Fluid Mechanics, 2014

In this work, the scaling statistics of the dissipation along Lagrangian trajectories are investigated by using fluid tracer particles obtained from a high-resolution direct numerical simulation with $\mathit{Re}_{\lambda }=400$. Both the energy dissipation rate $\epsilon $ and the local time-averaged $\epsilon _{\tau }$ agree rather well with the lognormal distribution hypothesis. Several statistics are then examined. It is found that the autocorrelation function $\rho (\tau )$ of $\ln (\epsilon (t))$ and variance $\sigma ^2(\tau )$ of $\ln (\epsilon _{\tau }(t))$ obey a log-law with scaling exponent $\beta '=\beta =0.30$ compatible with the intermittency parameter $\mu =0.30$. The $q{\rm th}$-order moment of $\epsilon _{\tau }$ has a clear power law on the inertial range $10<\tau /\tau _{\eta }<100$. The measured scaling exponent $K_L(q)$ agrees remarkably with $q-\zeta _L(2q)$ where $\zeta _L(2q)$ is the scaling exponent estimated using the Hilbert methodology. All of t...

Physics-uspekhi, 2000

We review the results of numerical and experimental studies in quasi-two-dimensional (Q2D) turbulence. We demonstrate that theoretical energy spectra with slopes -5/3 and -3 (Kraichnan-Batchelor-Leith) can be observed only for a special set of external parameters. The bottom drag, beta effect, finite Rossby-Obukhov radius or vertical stratification, which distinguish geophysical Q2D turbulence from its purely 2D counterpart, determine the organization

Physical Review Letters, 2002

Using a new experimental technique, based on the scattering of ultrasounds, we perform a direct measurement of particle velocities, in a fully turbulent flow. This allows us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that is extremely long range correlated in time. Theoretically, a Langevin equation is proposed and shown to account for the observed one-and two-point statistics. This approach connects intermittency to the dynamics of the flow.

Physics of Fluids, 2000

We study the statistical properties of the velocity and velocity gradient distributions in barotropic turbulence. At large enough Reynolds number, the velocity distribution becomes non-Gaussian outside the vortex cores, and its characteristics are completely determined by the properties of the far field induced by the coherent vortices. The velocity gradients are always non-Gaussian inside coherent vortices, due to the spatial velocity correlations associated with the ordered flow in the vortex cores, and become non-Gaussian also in the background turbulence at large enough Reynolds number.

Physics of Fluids, 1998

2007

Abstract. We review the results of numerical and experimental studies in quasi-two-dimensional (Q2D) turbulence. We demonstrate that theoretical energy spectra with slopes À5/3 and À3 (Kraichnan±Batchelor±Leith) can be observed only for a special set of external parameters. The bottom drag, beta effect, finite Rossby±Obukhov radius or vertical stratification, which distinguish geophysical Q2D turbulence from its purely 2D counterpart, determine the organization of a Q2D flow on a large scale.

Journal of Fluid Mechanics, 1984

We investigate the interaction of concentration fields of passive tracer with velocity fields characterizing geostrophic turbulence. We develop and compare results from equilibrium statistical mechanics, from turbulence-closure theory and from numerical simulation. A consistent account emerges. Among the results we show (1) that velocity fields efficiently scatter tracer variance to all scales, (2) that tracer variance evolves toward an equilibrium spectrum which is different from the equilibrium spectrum for vorticity variance, and (3) that intermittency of the tracer field is characteristic of a cascade of tracer variance across wavenumber space. The greater efficiency of the cascade of tracer variance relative to a vorticity cascade is due to wavenumber-local advective terms which affect tracer but not vorticity. We suggest that the more efficient tracer cascade results in shorter Lagrangian autocorrelation times for tracer than for vorticity.We investigate the spatial flux of tr...

Nonlinear Processes in Geophysics

We study numerically the comparison between Lagrangian experiments on turbulent particle dispersion in 2-D turbulent flows performed, on the one hand, on the basis of direct numerical simulations (DNS) and, on the other hand, using kinematic simulations (KS). Eulerian space-time structure of both DNS and KS dynamics are not comparable, mostly due to the absence of strong coherent vortices and advection processes in the KS fields. The comparison allows to refine past studies about the contribution of nonhomogeneous space-time 2-D Eulerian structure on the turbulent absolute and relative particle dispersion processes. We particularly focus our discussion on the Richardson's regime for relative dispersion.

Journal of Marine Research, 1999

We consider the dispersion of particles in potential vorticity (PV)-conserving ows. Because particle drift is preferentially along the mean PV contours, Lagrangian dispersion is strongly anisotropic. If the mean PV eld moreover is spatially variable, as when there is topography, the anisotropy is more clearly visible in the dispersion of displacements along and across the mean PV eld itself. We examine several numerical examples of unforced barotropic ows; in all cases, this ''projected'' dispersion is more anisotropicthan that in cartesian (x, y) coordinates.What differs is the rate at which spreading occurs, both along and across contours. The method is applicable to real data, as is illustrated with oat data from the deep North Atlantic. The results suggest a preferential spreading along contours of (barotropic) f/H.

Physics of Fluids, 2005

We present a detailed investigation of the particle pair separation process in homogeneous isotropic turbulence. We use data from direct numerical simulations up to R λ ∼ 280 following the evolution of about two million passive tracers advected by the flow over a time span of about three decades. We present data for both the separation distance and the relative velocity statistics. Statistics are measured along the particle pair trajectories both as a function of time and as a function of their separation, i.e. at fixed scales. We compare and contrast both sets of statistics in order to gain an insight into the mechanisms governing the separation process. We find very high levels of intermittency in the early stages, that is, for travel times up to order ten Kolmogorov time scales. The fixed scale statistics allow us to quantify anomalous corrections to Richardson diffusion in the inertial range of scales for those pairs that separate rapidly. It also allows a quantitative analysis of intermittency corrections for the relative velocity statistics.

Science, 1994

High-resolution numerical simulations were made of unforced, planetary-scale fluid dynamics. In particular, the simulation was based on the quasi-geostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment, which is an idealization for large regions of either the atmosphere or ocean. The solutions show significant discrepancies from the long-standing theoretical prediction of isotropy. The discrepancies are associated with the self-organization of the flow into a large population of coherent vortices. Their chaotic interactions govern the subsequent evolution of the flow toward a final configuration that is nonturbulent.

Nonlinear Processes in Geophysics, 2007

We study numerically the comparison between Lagrangian experiments on turbulent particle dispersion in 2-D turbulent flows performed, on the one hand, on the basis of direct numerical simulations (DNS) and, on the other hand, using kinematic simulations (KS). Eulerian space-time structure of both DNS and KS dynamics are not comparable, mostly due to the absence of strong coherent vortices and advection processes in the KS fields. The comparison allows to refine past studies about the contribution of nonhomogeneous space-time 2-D Eulerian structure on the turbulent absolute and relative particle dispersion processes. We particularly focus our discussion on the Richardson's regime for relative dispersion.

2013

1999