Stability of parallel wake flows in quasigeostrophic and frontal regimes

Abstract

International audienceRecent laboratory experiments [G. Perret, A. Stegner, M. Farge, and T. Pichon, Phys. Fluids 18, 036603 (2006)] have shown that the vortex-street formed in the wake of a towed cylinder in a rotating shallow-water layer could present a strong cyclone-anticyclone asymmetry. In extreme cases, only large-scale anticyclones were observed in the far wake. This asymmetry occurs in the so-called frontal regime when the Rossby number is small and the surface deviation is large. This asymmetry may have various origins and in particular may be attributed to the asymmetry of the flow around the cylinder, to the linear stability property of the wake, or to its nonlinear evolution. To discriminate between these mechanisms, we study the stability of two idealized parallel flows in the quasigeostrophic and in the frontal regimes. These parallel flows correspond to two velocity profiles measured just behind the cylinder in a region where the perturbations are negligible. According to our linear stability analysis, the most unstable mode, in the frontal regime, is localized in the anticyclonic shear region whether the base flow profile is symmetric or not. On a linear basis, it is thus more the instability that imposes the asymmetry than the base flow. Direct numerical simulations of the synthetic parallel wake flows show that nonlinearity exacerbates the dominance of the anticyclonic mode linearly selected. By numerically studying the spatio-temporal evolution of a small perturbation localized in space, we show that, unlike incompressible two-dimensional wake flows and the symmetric wake in the quasigeostrophic regime, the parallel asymmetric wake is strongly convectively unstable in the frontal regime, and not absolutely unstable. When the surface deformation becomes large, the wake instability changes from the absolute instability in the quasi-geostrophic regime to the strongly convective instability of the frontal regime. This explains well the changes. © 2006 American Institute of Physics

Oceanic Island Wake Flows in the Laboratory

“…This is in agreement with the two-dimensional nonrotating flow in the limit of large Reynolds number, for which the Strouhal number, S t = f s D/U 0 = D/L e 0.2 − 0.24, at large Reynolds numbers [Wen and Lin, 2001]. Both experimental [Boyer and Kmetz, 1983;Stegner et al, 2005;Teinturier et al, 2010;Lazar et al, 2013b] and numerical studies [Heywood et al, 1996;Perret et al, 2006a] confirm this result in a deepwater configuration when the bottom friction is negligible. Hence, a quasi-geostrophic wake will have the same pattern as the standard nonrotating Karman street even if the separation and the vortex shedding occur at higher critical Reynolds numbers [Boyer and Kmetz, 1983;Boyer et al, 1984].…”

“…For these experiments, where the drifting cylinder was confined in a thin (δ 0.1) and light (but non-stratified) surface layer, both dye visualization (Figure 14.3) and vorticity measurements (Figure 14.4) show this suprising behavior of large-scale geostrophic wake. This asymmetry was first explained by the linear stability analysis of parallel wake flows in the framework of rotating shallowwater equations [Perret et al, 2006a;Perret et al, 2011]. In the frontal regime, the most unstable mode is fully localized in the anticyclonic shear region.…”

“…Hence, the anticyclonic perturbations, leading to large-scale anticyclones, have the fastest growth rates. Besides, a spatiotemporal analysis of the wake flow behind the cylinder reveals a change in the nature of the instability: For a large-scale cylinder the wake flow is convectively unstable [Perret et al, 2006a] while the quasi-geostrophic wake and the classical Karman street is absolutely unstable [Pier, 2002;Chomaz, 2005]. Since the near wake flow is nearly parallel in the large-scale (frontal) regime, see amplifier, the entire flow being convectively unstable.…”

Transition to turbulence through steep global-modes cascade in an open rotating cavity

“…The DNS code is also used to get local stability properties and in particular the precise location of the transition to absolute instability and the absolute frequency ω 0 and spatial growth rate k 0 there, following the method of Perret et al (2006). This approach yielded results in full agreement with those extrapolated from the single-disc flow analysis by Lingwood (1997) for the position, slope and frequency of the primary front as already shown in VSC.…”

How Large-Scale and Cyclogeostrophic Barotropic Instabilities Favor the Formation of Anticyclonic Vortices in the Ocean

“…However, beyond the quasigeostrophic regime, when the Rossby number becomes finite and induces large isopycnal displacements due to the cyclogeostrophic balance, the nonlinear evolution of the instability leads to asymmetric eddy formation: the cyclones tend to be elongated and stretched in comparison with almost circular anticyclones. This asymmetry was first explained by the linear stability analysis of parallel wake flows in the framework of RSW equations (Perret et al 2006a). For the frontal geostrophic regime, that is, small Rossby number and finite isopycnal displacement (Cushman-Roisin 1986;Cushman-Roisin and Tang 1990), a significant asymmetry occurs in the wake between cyclonic and anticyclonic vortices.…”