Tracer microstructure in the large-eddy dominated regime

Abstract

Abstract--We have re-examined the inertial range behavior of a passive scalar which is advected by a large-scale velocity field causing a cascade of tracer variance to small scales, where it is dissipated by diffusion. This has been done within the context of an idealized model based on mixing by a 2D area-preserving map alternating with a weak diffusion step; the model is a special case of the general advection-diffusion problem. Both freely decaying and forced equilibrium systems were considered. Our main interest in this concerns the validity of Batchelor's theory predicting a k -I tracer variance spectrum, but the tracer microstructure has been diagnosed in terms of concentration probability distribution functions, generalized dimensions of the dissipation field, structure functions, and cancellation exponents. 2D simulations carried out at 10242 resolution show that i;, the decaying case the evolution settles into a 'fractal eigenmode' in which the variance decays exponentially with time at a rate dependent on the Lyapunov exponent but independent of the diffusion coefficient. Although the concentration pattern is self-similar with time, the power spectrum is not algebraic. Concentration PDFs have exponential tails. The dissipation field is not multifractal, and formally has Dq = 2 for all q. The convergence of the squared-gradient PDFs under coarse-graining indicates some underlying fractal behavior, however, and we have introduced the notion of 'fractal degree of freedom' systems to describe such entities. Cancellation exponents and structure functions were also considered, and have a self-similarity which is compatible with a non-intermittent behavior of the dissipation field. These matters have also been addressed for the equilibrium case. The main difference is that the power spectrum of concentration variance in equilibrium exhibits a power-law inertial range, though it is steeper than k -1, but not as steep as k -2. Other features are similar to the decaying case. Very high resolution simulations of the undiffused problem indicate that a k -1 spectrum is approached asymptotically, but only at resolutions corresponding to 106 x 106. The passive scalar behavior is compared and contrasted with the behavior of scalar pseudo-vorticity (an 'active scalar') in the family of generalized 2D turbulence models introduced elsewhere in this issue.

Passive Fields and Particles in Chaotic Flows

“…It is obvious that stirring enhances mixing, but there will be mixing even without stirring. The persistent patterns proposed in (Pierrehumbert, 1994) and observed experimentally in (Rothstein et al, 1999) and the studies by (Fereday et al, 2002) highlight some of the effects that arise when chaos and diffusion interact to turn stirring into mixing.…”

“…Pierrehumbert, 1994;Fereday et al, 2002). Consider particle spreading in a 2-d flow designed to describe the experiments of (Williams et al, 1997;Rothstein et al, 1999).…”

Scalar Decay in Chaotic Mixing

“…The folding forces them to interact with themselves in a correlated fashion. We enter the regime of the strange eigenmode [27], which has received a lot of attention lately [14,26,[28][29][30][31][32][33][34][35]. Maybe we'll hear more about that in ten years.…”

The strange eigenmode in Lagrangian coordinates

“…The concentration is normalized at each iteration such that the mode with largest concentration has unit magnitude. The iterations plotted are nϭ6, 7,8,9,10,11 ͑circles͒ and nϭ12 ͑black dots͒. Most of the circles appear as large black dots, because all these points lie on top of each other.…”

“…There are powerful theories based on the distribution of Lyapunov exponents [1][2][3] that link the mixing rate of the passive scalar with the chaotic properties of the flow. It has been recently suggested, following earlier work of Pierrehumbert,[4][5][6][7][8][9][10] that the mixing properties of the flow can often be elucidated only by solving a full eigenvalue problem for the advection-diffusion operator, in an analogous manner to what is done for the kinematic dynamo. 11 The resulting eigenfunctions have been dubbed strange eigenmodes by Pierrehumbert, and are closely related to Pollicott-Ruelle resonances in ergodic theory, [12][13][14] which describe the longtime decay of correlations in mixing hyperbolic dynamical systems.…”