Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

Abstract

Three-dimensional Direct Numerical Simulation (DNS) on the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating is presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies by Randriamampianina (2002, 2003), the fluid used is air rather than a liquid as used in all other previous work. This study investigated a bifurcation sequence from the axisymmetric flow to a number of complex flows.The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady, fully-developed baroclinic wave to chaotic flow followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic MAV. In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow.Further transitions of a steady wave with a higher wave number pointed to the possibility that a barotropic instability of the side wall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.

DNS of structural vacillation in the transition to geostrophic turbulence

“…As such, the figure is a bifurcation diagram representing the sequence of three basic solutions, the axisymmetric solution on the x-axis and the m = 2 and m = 3 solutions pro- jected onto the same plane. The instability of the axisymmetric solution and the m = 2 solution branch are reproduced from the earlier study in Randriamampianina et al (2006). The present study focusses on evolution of the m = 3 solution branch when increasing progressively the rotation rate up to T a = 5.…”

Mixed-mode solutions in an air-filled differentially heated rotating annulus

“…The radial, azimuthal and vertical (or axial) coordinates are denoted r, ϕ and z, respectively, with unit vectors e r , e ϕ and e z . This model is essentially the same as those used in many previous numerical studies, including, for example, [23,2,24,17,12]. With air as the working fluid, it is expected that the Boussinesq approximation will lead to quantitatively accurate results for ∆T < 50 [12], while above this, the results may only be qualitatively accurate.…”

“…Although many of the features of the flow transitions are similar to those observed at other Prandtl numbers, it was found that the Prandtl number played an important role in some transitions. In particular, hysteresis is observed along the transition from axisymmetric to steady waves [12], and the onset of amplitude vacillation is found to occur as rotation rate is increased. The experiment also showed the existence of weak (low-amplitude) steady waves over large areas of parameter space.…”

“…Only recently have studies used fluids with Prandtl number near unity. A numerical study that used air as the working fluid (Prandtl number P r ∼ 0.707) has been performed [12], and even more recently, an experimental investigation was carried out [13]. Although many of the features of the flow transitions are similar to those observed at other Prandtl numbers, it was found that the Prandtl number played an important role in some transitions.…”

“…We follow [12,13] and choose the parameters of our fluid to be those of air, which has a Prandtl number P r ≈ 0.707. In the two parameter space defined by the Taylor number and the thermal Rossby number (or equivalently defined by the differential heating ∆T and rotation rate Ω), the transition from axisymmetric to non-axisymmetric flow can be represented by a (one-dimensional) curve.…”