Strain-vorticity induced secondary motion in shallow flows

Abstract

Deviations from two-dimensionality of a shallow flow that is dominated by bottom friction are quantified in terms of the spatial distribution of strain and vorticity as described by the Okubo-Weiss function. This result is based on a Poisson equation for the pressure in a quasi-horizontal (primary) flow. It is shown that the Okubo-Weiss function specifies vertical pressure gradients, which for their part drive vertical (secondary) motion. An asymptotic expansion of these gradients based on the smallness of the vertical to horizontal scale ratio demonstrates that the sign and magnitude of secondary circulation inside the fluid layer is dictated by the signs and magnitude of the Okubo-Weiss function. As a consequence of this, secondary motion as well as nonzero horizontal divergence do also depend on the strength, i.e., the Reynolds number of the primary flow. The theory is exemplified by two generic vortical structures (monopolar and dipolar structures). Most importantly, the theory can be applied to more complicated turbulent shallow flows in order to assess the degree of two-dimensionality using measurements of the free-surface flow only.

Clustering of floaters on the free surface of a turbulent flow: An experimental study

“…However, the global correlator, t i = ∆Ω i • ∆q i /(σ(q i ) • σ(Ω i )) between the vorticity and the horizontal divergence is 0.83, which is large. This strong correlation (much larger than the one between Ω i and ρ i ) may be explained by secondary flows that are induced in a shallow fluid layer around vertical vortices [48]: upwelling flows merge at the vortex core whereas downwelling flows dive at the vortex edge. Therefore, we propose that the apparent correlation between vertical vorticity and the particles concentration is only the result of the correlation between Ω i and q i .…”

The Okubo-Weiss Criteria in Two-Dimensional Hydrodynamic and Magnetohydrodynamic Flows

“…2 The Okubo-Weiss parameter describing the local strain-vorticity balance in the horizontal flow field of a shallow fluid layer turns out also to quantify the deviations from two-dimensionality of this flow (Balkovsky et al [8], Cieslik et al [9]). More specifically, the Okubo-Weiss parameter turns out to be the source term in the Poisson equation for the pressure Hessian matrix (Kamp [10]).…”

“…9 Quasi-geostrophic dynamics refers to the nonlinear dynamics governed by the first-order departure from the linear geostrophic balance between the Coriolis force and pressure gradient transverse to the rotation axis of a rapidly rotating fluid (Charney [21]). 10 The β-plane approximation corresponds to replacing the curved surface of the earth locally by a tangent plane, but allowing the Coriolis parameter f to vary linearly with latitude (the y-direction).…”

The Okubo–Weiss criterion in hydrodynamic flows: geometric aspects and further extension

“…It is of interest to note that, in considering the shape of a material curve in periodic 2D incompressible flows,Berry et al (1979) distinguished the elliptic and hyperbolic regions via the wrapping-around action (termed a 'whorl') in the former and a stretching-compressing action (termed a 'tendril') in the latter.3 The Okubo-Weiss parameter describing the local strain-vorticity balance in the horizontal flow field of a shallow fluid layer turns out also to quantify the deviations from two-dimensionality of this flow(Balkovsky et al 2001, Cieslik et al 2009, Kamp 2012). 4 Lagrangian coherent structures (LCS) are material lines with locally maximum attracting/repelling/shearing effects on neighboring material lines and hence collectively organize the flow into ordered patterns showing global flow coherence properties associated with material transport(Haller 2015).…”