Marginal and weakly nonlinear stability in spatially developing flows

AbstractThis work is devoted to revealing the essence of near-critical phenomena in nonlinear problems with nonparallel effects. As a generalization of the well-known concept of linear stability in Fourier space for a parallel basic state, we introduce a new concept valid for nonparallel flows as well. The new picture allows one to demonstrate the possible singular limit to the parallel case. Also, on its basis we derive a weakly nonlinear model valid near criticality. The damped Kuramoto-Sivashinsky equation with variable coefficients is used to illustrate the application of the theory

Pressure-anisotropy-induced nonlinearities in the kinetic magnetorotational instability

In collisionless and weakly collisional plasmas, such as hot accretion flows onto compact objects, the magnetorotational instability (MRI) can differ significantly from the standard (collisional) MRI. In particular, pressure anisotropy with respect to the local magnetic-field direction can both change the linear MRI dispersion relation and cause nonlinear modifications to the mode structure and growth rate, even when the field and flow perturbations are small. This work studies these pressure-anisotropy-induced nonlinearities in the weakly nonlinear, high-ion-beta regime, before the MRI saturates into strong turbulence. Our goal is to better understand how the saturation of the MRI in a low collisionality plasma might differ from that in the collisional regime. We focus on two key effects: (i) the direct impact of self-induced pressure-anisotropy nonlinearities on the evolution of an MRI mode, and (ii) the influence of pressure anisotropy on the "parasitic instabilities" that are suspected to cause the mode to break up into turbulence. Our main conclusions are: (i) The mirror instability regulates the pressure anisotropy in such a way that the linear MRI in a collisionless plasma is an approximate nonlinear solution once the mode amplitude becomes larger than the background field (just as in MHD). This implies that differences between the collisionless and collisional MRI become unimportant at large amplitudes. (ii) The break up of large amplitude MRI modes into turbulence via parasitic instabilities is similar in collisionless and collisional plasmas. Together, these conclusions suggest that the route to magnetorotational turbulence in a collisionless plasma may well be similar to that in a collisional plasma, as suggested by recent kinetic simulations. As a supplement to these findings, we offer guidance for the design of future kinetic simulations of magnetorotational turbulence.Comment: Submitted to Journal of Plasma Physic

Nonlinear self-sustained structures and fronts in spatially developing wake flows

International audienceA family of slowly spatially developing wakes with variable pressure gradient is numerically demonstrated to sustain a synchronized finite-amplitude vortex street tuned at a well defined frequency. This oscillating state is shown to be described by a steep global mode exhibiting a sharp Dee--Langer type front at the streamwise station of marginal absolute instability. The front acts as a wavemaker which sends out nonlinear travelling waves in the downstream direction, the global frequency being imposed by the real absolute frequency prevailing at the front station. The nonlinear travelling waves are determined to be governed by the local nonlinear dispersion relation resulting from a temporal evolution problem on a local wake profile considered as parallel. Although the vortex street is fully nonlinear, its frequency is dictated by a purely linear marginal absolute instability criterion applied to the local linear dispersion relation

Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur

Elephant modes and low frequency unsteadiness in a high Reynolds number, transonic afterbody wake

International audienceExperiments and large eddy numerical simulation of a fully turbulent afterbody flow in the high subsonic regime, typical of that developing in the wake of a space launcher, exhibit a large-scale low frequency oscillation of the wake. In the present paper, we investigate to what extent the existence of the synchronized oscillations can be interpreted, at the high Reynolds numbers prevailing in this class of flows, by a local stability analysis of the mean flow, as measured in experiments or computed in numerical simulations. This analysis shows the presence of a pocket of absolute instability in the near wake, slightly detached from the body. The global frequency is strikingly well predicted by the absolute frequency at the upstream station of marginal absolute instability, this frequency selection being in agreement with the theory of nonlinear global modes. This result strongly suggests that a so-called elephant mode is responsible for the intense oscillations observed in the lee of space launcher configurations. © 2009 American Institute of Physics