Transition to the dynamical chaos and anomalous transport of a passive scalar in the annular Kolmogorov flow

Abstract

In this paper, we are concerned with the transition to dynamical chaos and related anomalous transport of a passive scalar in the annular Kolmogorov flow, which is considered as a model of the barotropic zonal flows in the Earth’s atmosphere and ocean or their laboratory analogs. The investigation of the anomalous transport is conducted within a dynamically consistent flow model describing the saturation of barotropic instability. The analysis is based on the numerical solution of equations of a quasi-two-dimensional flow in an annular channel with rigid walls taking into account the beta-effect and external (bottom) friction. It is supposed that the sinusoidal velocity profile of the Kolmogorov flow has three periods inside a channel and the sticking condition on the channel walls is satisfied. Four basic regimes arising with increasing flow supercriticality, the last of which corresponds to dynamical chaos, are distinguished. It is found that five modulated chains of wave-vortex structures with closed streamlines are formed in the channel and their temporal behavior is studied by making videos. The frequency–wavenumber spectra of the longitudinal velocity at certain values of radial coordinates are drawn and the largest Lyapunov exponent is determined in the regime of dynamical chaos. The relationship between the streamlines behavior and the discrete peaks of frequency–wavenumber spectra is elucidated. The occurrence of anomalous transport of a passive scalar is confirmed by drawing trajectories of tracer particles, as well as by determining exponents of the time dependence of mean particle displacement and its variance.

Dynamical Chaos and Lateral Transport of a Passive Scalar in the Annular Reverse Jet Flow

“…where K is the number of "nonzero" harmonics, ζ m and ψ m are the complex profiles of the mth harmonic of vorticity and streamfunction, respectively. The simultaneous system of equations governing the profiles ψ m , ζ m and mean velocity v θ including the boundary conditions were derived in [12]. We utilize the velocity profile v 0 (y) of the equilibrium reverse jet flow specified by formulae (2.4) of [11] replacing the Cartesian coordinate y by r. The numerical coefficients of the profile approximation are also taken the same as in [11].…”

“…We introduce the quantity Ω = = Re ω/m that is an angular phase velocity of the wave with wavenumber m coinciding with the angular frequency of rotation of the chain produced by this wave. The solution of the eigenvalue problem was discussed in detail in [10,11] for the plane-parallel flows and in [12] for the annular flow. Following [10,12], we introduce the critical velocity U c , at which the instability occurs first, and the associated supercriticality γ = U/U c .…”

“…The solution of the eigenvalue problem was discussed in detail in [10,11] for the plane-parallel flows and in [12] for the annular flow. Following [10,12], we introduce the critical velocity U c , at which the instability occurs first, and the associated supercriticality γ = U/U c . The dimensionless parameters corresponding to the instability threshold are designated by R c , β c , and λ c , while the critical wavenumber and angular velocity are m c and Ω c , respectively [12].…”

“…Following [10,12], we introduce the critical velocity U c , at which the instability occurs first, and the associated supercriticality γ = U/U c . The dimensionless parameters corresponding to the instability threshold are designated by R c , β c , and λ c , while the critical wavenumber and angular velocity are m c and Ω c , respectively [12]. An increase in γ at constant λ * * , β * * , and ν leads to the variation of R, β, and λ in accordance with the relations γ = R/R c = λ c /λ = = β c /β.…”

“…The walls of the annular channel are supposed to be rigid and nonpercolating. The quasi-geostrophic equations for a barotropic (quasi-two-dimensional) flow with allowance for the beta-effect and external friction written in polar coordinates are solved by the pseudospectral method borrowed from the work [12]. To study the transport phenomena, the equations for the tracer particles are introduced in the numerical scheme.…”