How rapidly is a passive scalar mixed within closed streamlines?

Abstract

The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈P1/3(L/U), where the Péclet number P=LU/κ (L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

“…the streaky flow essentially behaves as a passive scalar field at large effective 'Peclet' number. Crucially, numerous studies of two-dimensional scalar advection by a counter-rotating cellular velocity field in the weak diffusion (large Peclet number) limit confirm that the scalar field is strongly homogenized [25][26][27], thereby providing a possible mechanism for the observed quasi-uniformity of the streamwise momentum within UMZs. Empirical support for this proposition is provided by the DNS results of Papavassiliou & Hanratty [28], which confirm that low-momentum regions of large-scale (inertial layer) structures in turbulent plane Couette flow are separated by the vortex cores of nearly inviscid streamwise roll modes, implying a large effective Peclet number for the advected streamwise flow.…”

“…The counter-rotating cellular velocity field associated with the rolls then may be expected to largely homogenize the streamwise-averaged streamwise flow, thereby creating a UMZ. For this same reason, however, the leading-order mean momentum equations within the UMZs must be regularized by retaining ⊥-Laplacian diffusion: through a shear dispersion mechanism, large gradients in the y-z plane are generated, first along mean ⊥-streamlines and, ultimately, across them [26]. To properly capture this phenomenon, the regularization terms (āRe) −1 ∇ 2 ⊥ (ū 0 ,v 3/5 ,w 3/5 ), respectively, should be included on the right-hand sides of the first three equations in system (3.3), where the perpendicular Laplacian operator ∇ 2 ⊥ ≡ (∂ 2 y + ∂ 2 z ).…”

Scales and dynamics of Submesoscale Coherent Vortices formed by deep convection in the northwestern Mediterranean Sea

“…They are isolated and coherent circulation features characterized by a small radius (about 5 km in the northwestern Mediterranean Sea [Testor and Gascard, 2006]), an extended lifetime (>1 year [Testor and Gascard, 2003;Ronski and Budeus, 2006]). Their rotation sets transport barriers that drastically reduce the lateral exchanges between their core and the surrounding waters [Rhines and Young, 1983;Provenzale, 1999]. They are, therefore, extremely efficient in transporting physical and biogeochemical tracers characteristics of their generation site over long distances [D'Asaro, 1988a;Testor and Gascard, 2003;Bower et al, 2013;L'Hegaret et al, 2016].…”

Vortex motion in a weak background shear flow

“…The flow field of the vortex wraps up the passive scalar to form a spiral structure, leading to the diffusive decay of scalar fluctuations in the vicinity of the vortex. Several time scales are involved: in particular there is an enhanced shear-diffusion time scale for the destruction of scalar fluctuations on given closed streamlines (Moffatt & Kamkar 1983;Rhines & Young 1983;Bajer, Bassom & Gilbert 2001). The spiral distribution of the passive scalar also has a fractal nature, with a non-trivial box-counting dimension which can determine spectral power laws and anomalous diffusion properties (for example , Gilbert 1988;Vassilicos 1995).…”