Hyperbolic Instability Research Articles

Coherence dynamics of kicked Bose-Hubbard dimers: Interferometric signatures of chaos

We study the coherence dynamics of a kicked two-mode Bose-Hubbard model starting with an arbitrary coherent spin preparation. For preparations in the chaotic regions of phase space we find a generic behavior with Floquet participation numbers that scale as the entire N-particle Hilbert space, leading to a rapid loss of single-particle coherence. However, the chaotic behavior is not uniform throughout the chaotic sea and unique statistics is found for preparations at the vicinity of hyperbolic points that are embedded in it. This is contrasted with the low log(N) participation that is responsible for the revivals in the vicinity of isolated hyperbolic instabilities.

Squeezing in driven bimodal Bose-Einstein condensates: Erratic driving versus noise

We study the interplay of squeezing and phase randomization near the hyperbolic instability of a two-site Bose-Hubbard model in the Josephson interaction regime. We obtain results for the quantum Zeno suppression of squeezing, far beyond the previously found short time behavior. More importantly, we contrast the expected outcome with the case where randomization is induced by erratic driving with the same fluctuations as the quantum noise source, finding significant differences. These are related to the distribution of the squeezing factor, which has log-normal characteristics: hence its average is significantly different from its median due to the occurrence of rare events.

Transient growth of secondary instabilities in parallel wakes: Anti lift-up mechanism and hyperbolic instability

This paper investigates the three-dimensional temporal instabilities and the transient growth of perturbations on a Von Kármán vortex street, issuing from the development of the primary instability of a parallel Bickley velocity profile typical of a wake forming behind a thin flat plate. By solving iteratively the linearized direct Navier Stokes equations and its adjoint equations, we compute the optimal perturbations that exhibit the largest transient growth of energy between the initial instant and different time horizons. At short time horizons, optimal initial perturbations are concentrated on the points of maximal strain of the base flow. The optimal gain of energy and the mechanism of instability are well predicted by local theories that describe the lagrangian evolution of a perturbation wave packet. At time of order unity, hyperbolic region leads the dynamics. Only at large time (t ≥ 20), the growth is led by the most amplified eigenmode. This eigenmode evolves, when the wavenumber increases, from perturbation centred in the core of the vortices, to perturbations localised on the stretching manifold of the hyperbolic points. At finite and large time, the gain in energy is initially associated with a mechanism reminiscent to the anti lift-up mechanism described by Antkowiak and Brancher [J. Fluid Mech. 578, 295 (2007)] in the context of an axisymmetric vortex. Presently, the optimal initial condition (the adjoint modes at large time) corresponding to streamwise streaks localised on the contracting manifold of the hyperbolic point induces streamwise vortices aligned with the stretching manifold of the hyperbolic point (the direct modes). The localisation on distinct manifolds of direct and adjoint eigenmodes is more pronounced when the Reynolds number is increased. An interpretation is proposed based on a balance between diffusion and stretching effects that predicts the thickness of the energy containing region for the adjoint and the direct mode decreasing as 1/Re. The extra gain of energy due to non normal effects grows, since direct and adjoint modes are localised in different regions of space, i.e., the stretching and contracting manifold, a novel effect of the so called convective non normality associated with the transport of the perturbation by the base flow.

Three-dimensional instabilities and transient growth of a counter-rotating vortex pair

This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb–Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, ReΓ=Γ/ν, with Γ the circulation of one vortex and ν the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier–Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non- or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t=2, during which the dipole has traveled twice the separation distance between vortices b. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds number ReΓ=2000. For time larger than t=2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers, ReΓ=2000 and ReΓ=105, different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory [S. C. Crow, AIAA J. 8, 2172 (1970); S. Le Dizes and F. Laporte, J. Fluid Mech. 471, 120 (2002)] and numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids 15, 1861 (2003)]. A novel oscillatory elliptic instability involving Kelvin waves with azimuthal wavenumbers m=0 and |m|=2, that was missed in previous numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids 15, 1861 (2003)] is found. For the stationary elliptic instability, we show that viscous effect may be estimated using the large Reynolds number direct and adjoint eigenmodes. This asymptotically exact estimate of the viscous damping of elliptic instability mode agrees with our direct numerical computation of instability branches at moderate Reynolds number and demonstrates that formula proposed by Le Dizes and Laporte [J. Fluid Mech. 471, 120 (2002)] strongly over estimated the viscous correction.

Exponential dichotomy and admissibility of linearized skew-product semiflows defined on a compact positively invariant subset of semiflows

We study exponential dichotomy of linear skew-product semiflows which come from linearizing skew-product semiflows on a compact positively invariant subset M of semiflows and construct the relationship between continuous separation and exponential dichotomy under assumptions that skew-product semiflows are eventually strongly monotone. In addition, we deduce that the exponential dichotomy is trivial when M is hyperbolically stable, and the hyperbolic instability of M is the necessary condition of the state space admitting a trivial separation in another forms. Simultaneously, we list some conditions for hyperbolic stability and instability of M . At last, we construct a sufficient and necessary condition for exponential dichotomy of linear skew-product semiflows in terms of the admissibility of the pair ( B ( R + , X ) , B 0 ( R + , X ) ) .

Three-dimensionalization of freely-decaying two-dimensional turbulence

The three-dimensionalization of freely-decaying two-dimensional turbulence is studied numerically. Using two different basic states, it is shown that the hyperbolic instability of steady mixing layers generalizes to unsteady flow. The growth of three-dimensional (3D) perturbations is interpreted in terms of a “pressureless” analysis in which horizontal pressure gradients are neglected and the underlying physical mechanism is advective in nature. For a steady, pressureless base flow the (asymptotic) growth rate is exactly determined, in a few special cases, by variants of the so-called Weiss criterion; more generally, local growth rate estimates are obtained. For freely-decaying two-dimensional turbulence there is analogous behavior: while the 3D perturbation remains small the pressureless approximation holds at large horizontal scales and local growth rate estimates apply. The pressureless modes have a determining influence on the growth (the damping effect of the horizontal pressure gradient is small).

Secondary instability mechanisms in the wake of a flat plate

We extend the work of Sutherland and Peltier (Phys. Fluids 6 (1994) 1267) and investigate numerically the three-dimensional (3D) secondary stability of a wake forming behind a thin flat plate. The primary flow is a Kármán vortex street numerically computed from the two-dimensional (2D) even instability of a parallel wake based on the Bickley velocity profile. Considering the symmetries of the von Kármán Street, the 3D modes are classified into two families, whether symmetric or antisymmetric. For each family, we determine the leading eigenmodes using a Krylov method. The growth rate curves show that both the most unstable symmetric and antisymmetric modes are stationary and present a maximum of amplification for a wavelength of the order of the primary vortex spacing. The maximum growth rate, corresponding wavelength and cutoff wavelength are well predicted by the elliptic instability of the vortex core. The eigenmode structure of the most unstable wavenumber is centered in the core and is typical of the elliptic instability. The hyperbolic instability of the braid region gives a growth rate five times larger and a cutoff two times higher than the ones computed. As recently discussed for mixing layers by Caulfield and Kerswell (Phys. Fluids 12 (5) (2000) 1032), this is not surprising since the hyperbolic instability applies for an unbounded hyperbolic flow. When the region of hyperbolic flow is bounded, intense transient growth is generated, but when time goes to infinity, the instability becomes small or even dies out. Finally, good qualitative and quantitative agreement is found with the experiments previously done by Julien, Lasheras and Chomaz (J. Fluid Mech. 479 (2003) 155) on the secondary instability in the wake of a flat plate for the symmetry selection, the most amplified wavenumber and growth rate.

Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with external rotation

The stability analysis of a street of Stuart vortices in a rotating frame is performed by integrating the Kelvin–Townsend equations along the mean flow trajectories, using the geometrical optics technique (Lifschitz & Hameiri 1991) for short-wave perturbations. A parallel is drawn between the formulations of this zonal approach and that of rapid distortion theory, better known to the turbulence community. The results presented confirm those obtained by the standard stability analysis based on normal-mode decomposition: depending on the rotation parameter and the oblique mode considered, three unstable zones are identified, related to the centrifugal, elliptic and hyperbolic instabilities, as observed for Taylor–Green cells (Sipp et al. 1999). Anticyclonic rotation is shown to destabilize Stuart vortices through a combination of the elliptical and centrifugal instability mechanisms, depending on the ratio of its rate to the structure core vorticity. Available stability criteria are discussed in the general case of two-dimensional rotating flows, in relation to their streamline topology and the values of the local Rossby number or vorticity.

The nonlinear development of three-dimensional disturbances at hyperbolic stagnation points: A model of the braid region in mixing layers

The properties of steady, two-dimensional flows with spatially uniform strain rates ε and rotation rates γ where ε2⩾γ2, and hence open, hyperbolic, streamlines are investigated. By comparison with a high resolution numerical simulation of a free shear layer, such a quadratic flow is an idealized local model of the “braid” region which develops between neighboring saturated Kelvin–Helmholtz billows in an unstable free shear layer. A class of exact three-dimensional nonlinear solutions for spatially periodic perturbations is derived. These solutions satisfy the condition that the amplitude of the time-varying wave number of the perturbation remains bounded in time, and hence that pressure plays an asymptotically small role in their dynamics. In the limit of long time, the energy of such perturbations in an inviscid flow grows exponentially, with growth rate 2ε2−γ2, and the perturbation pressure plays no significant role in the dynamic evolution. This asymptotic growth rate is not the maximal growth rate accessible to general perturbations, which may grow transiently at rate 2ε, independently of γ. However, almost all initial conditions lead to, at most, transient growth and hence finite asymptotic perturbation energy in an inviscid flow as time increases, due to the finite amplitude effects of pressure perturbations. Perturbations which do undergo significant transient growth take the form of streamwise-aligned perturbation vorticity which varies periodically in the spanwise direction. By comparison of this local model with a numerically simulated mixing layer, appropriately initialized “hyperbolic instabilities” appear to have significantly larger transient growth rates than an “elliptical instability” of the primary billow core. These hyperbolic instabilities appear to be a simple model for the spanwise periodic perturbations which are known to lead to the nucleation of secondary rib vortices in the braid region between adjacent billow cores.

Vortices in rotating systems: Centrifugal, elliptic and hyperbolic type instabilities

This paper is devoted to the effects of rotation on the linear dynamics of two-dimensional vortices. The asymmetric behavior of cyclones and anticyclones, a basic problem with respect to the dynamics of rotating flows, is particularly addressed. This problem is investigated by means of linear stability analyses of flattened Taylor–Green vortices in a rotating system. This flow constitutes an infinite array of contra-rotating one-signed nonaxisymmetric vorticity structures. We address the stability of this flow with respect to three-dimensional short-wave perturbations via both the geometrical optics method and via a classical normal mode analysis, based on a matrix eigenvalue method. From a physical point of view, we show that vortices are affected by elliptic, hyperbolic and centrifugal instabilities. A complete picture of the short-wave stability properties of the flow is given for various levels of the background rotation. For Taylor–Green cells with aspect ratio E=2, we show that anticyclones undergo centrifugal instability if the Rossby number verifies Ro>1, elliptic instability for all values of Ro except 0.75<Ro<1.25 and hyperbolic instability. The Rossby number is here defined as the ratio of the maximum amplitude of vorticity to twice the background rotation. On the other hand, cyclones bear elliptic and hyperbolic instabilities whatever the Rossby number. Besides, depending on the Rossby number, rotation can either strengthen (anticyclonic vortices) or weaken elliptic instability. From a technical point of view, in this article we bring an assessment of the links between the short-wave asymptotics and the normal mode analysis. Normal modes are exhibited which are in complete agreement with the short-wave asymptotics both with respect to the amplification rate and with respect to the structure of the eigenmode. For example, we show centrifugal eigenmodes which are localized in the vicinity of closed streamlines in the anticyclones; elliptical eigenmodes which are concentrated in the center of the cyclones or anticyclones; hyperbolic eigenmodes which are localized in the neighborhood of closed streamlines in cyclones.

An illustration of the link between ribs and hyperbolic instability

The hyperbolic instability in planar Taylor–Green cells is investigated by direct numerical simulations at Reynolds number 400. The mechanism of this three-dimensional instability is a stretching of the vorticity perturbation along the principal axis of strain in accordance with well-known results of rapid distortion theory and stability analysis, leading to the formation of spanwise counter-rotating vortices (or ribs) in the irrotational hyperbolic stagnation point regions, as described by Kerr and Dold [J. Fluid Mech. 276, 307 (1994)].

Three-dimensional instabilities in wake transition

It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability, modes “A” and “B”, occurring in different Reynolds number ranges. These modes are quite distinct in spanwise lengthscale and in symmetry, and they are found to scale on different physical features of the flow. Mode A has a large spanwise wavelength of around 3–4 cylinder diameters, and scales on the larger physical structure in the flow, namely the core of the primary Kármán vortices. The feedback from one vortex to the next gives an out-of-phase streamwise vortex pattern for this mode. In contrast, the mode B instability has a distinctly smaller spanwise wevelength (1 diameter) which scales on the smaller physical structure in the flow, namely the braid shear layer. The symmetry of mode B is determined by the reverse flow behind the bluff cylinder, leading to a system of streamwise vortices which are in phase between successive half cycles. The symmetries of both modes are the same as the ones found in the vortex system evolving from perturbed plane wakes studied by Meiburg and Lasheras (1988) and Lasheras and Meiburg (1990). Furthermore, the question of the physical origin of these three-dimensional instabilities is addressed. We present evidence that they are linked to general instability mechanisms found in two-dimensional linear flows. In particular, mode A seems to be a result of an elliptic instability of the near-wake vortex cores; predictions based on elliptic instability theory concerning the initial perturbation shape and the spanwise wevelength are in good agreement with experimental observations. For the mode B instability, it is suggested that it is a manifestation of a hyperbolic instability of the stagnation point flow found in the braid shear layer linking the primary vortices.

Quantum dynamical system with hyperbolic instabilities

An analysis of a quantum counterpart of a parametrically kicked nonlinear oscillator is given. The method, using as a basic criterion the recently introduced quantum characteristic exponents, is analogous to the technique developed in classical dynamical system theory. However, our approach to the characterization of the stability of an observable's evolution is done in pure quantum terms.

Molecular-dynamics approach: From chaotic to statistical properties of compound nuclei.

Statistical aspects of the dynamics of chaotic scattering in the classical model of \ensuremath{\alpha}-cluster nuclei are studied. It is found that the dynamics governed by hyperbolic instabilities, characterized by the positive Lyapunov exponents and the fractal dimensions, which results in an exponential decay of the survival probability, evolves to a limiting energy distribution whose density develops the Boltzmann form. The angular distribution of the corresponding decay products shows symmetry with respect to the \ensuremath{\pi}/2 angle. The time estimated for the compound nucleus formation ranges within the order of ${10}^{\mathrm{\ensuremath{-}}21}$ s.

A theory of period-doubling bifurcations in two-dimensional reversible area preserving mappings

For a class of two-dimensional reversible area preserving mappings, a theory is developed which describes how the period doubling bifurcation entails the onset of hyperbolic instability with reflection. Henon's quadratic mapping, Chirikov's standard mapping and De Vogelaere's mapping all belong to this class.

An equation for hyperchaos

A simple four-variable oscillator containing but one quadratic term produces a higher form of chaos with two (rather than one) directions of hyperbolic instability on the attractor. The topology is simple, a model difference equation complicated. Turbulence may be hyperchaos of high order.