Route to chaos in the weakly stratified Kolmogorov flow

We consider a two-dimensional fluid exposed to Kolmogorov’s forcing cos(ny) and heated from above. The stabilizing effects of temperature are taken into account using the Boussinesq approximation. The fluid with no temperature stratification has been widely studied and, although relying on strong simplifications, it is considered an important tool for the theoretical and experimental study of transition to turbulence. In this paper, we are interested in the set of transitions leading the temperature stratified fluid from the laminar solution [U∝cos(ny),0, T ∝ y] to more complex states until the onset of chaotic states. We will consider Reynolds numbers 0 < Re ≤ 30, while the Richardson numbers shall be kept in the regime of weak stratifications (Ri ≤ 5 × 10 −3 ). We shall first review the non-stratified Kolmogorov flow and find a new period-tripling bifurcation as the precursor of chaotic states. Introducing the stabilizing temperature gradient, we shall observe that higher Re are required to trigger instabilities. More importantly, we shall see new states and phenomena: the newly discovered period-tripling bifurcation is supercritical or subcritical according to Ri; more period-tripling and doubling bifurcations may depart from this new state; strong enough stratifications trigger new regions of chaotic solutions and, on the drifting solution branch, non-chaotic bursting solutions

Transitions in a stratified Kolmogorov flow

We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is $Ri=10^-5$, and restrict our analysis to the range $0<Re<30$. The effect of the stabilizing temperature is to shift bifurcation points and to reduce the region of existence of stable drifting states. The flow reaches chaotic configurations through two different routes, one involving drifting states, the other involving a gluing bifurcation. Along the latter route we observe, as the precursor to chaotic states, a period tripling bifurcation

Formation of Coherent Structures in Kolmogorov Flow with Stratification and Drag

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures

Complex singularities in KdV solutions

In the small dispersion regime, the KdV solution exhibits rapid oscillations in its spatio-temporal dependence. We show that these oscillations are caused by the presence of complex singularities that approach the real axis. We give a numerical estimate of the asymptotic dynamics of the poles