Unbounded Energy Growth Research Articles

Dynamics of the nonholonomic Suslov problem under periodic control: unbounded speedup and strange attractors

This paper is concerned with the study of a nonholonomic system with parametric excitation, the Suslov problem with variable gyrostatic momentum. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analog of Fermi’s acceleration). The existence of trajectories is shown for which the angular velocity of the rigid body increases indefinitely and has the asymptotics .

Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver

Solving the Navier–Stokes equations when the density field contains a large and sharp discontinuity—such as a water/air free surface—is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass-conservation properties. The equations are discretised using a symmetric interior-penalty discontinuous Galerkin finite element method that is divergence-free to machine precision.A slope limiter made specifically for exactly divergence-free (solenoidal) fields is presented and used to illustrate the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component.We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density-ratios and high Reynolds numbers—typical for marine and offshore water/air simulations—in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.

Fermi-like acceleration and power-law energy growth in nonholonomic systems

This paper is concerned with a nonholonomic system with parametric excitation—the Chaplygin sleigh with time-varying mass distribution. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analogue of Fermi’s acceleration) in the case where excitation is achieved by means of a rotor with variable angular momentum. The existence of trajectories for which the translational velocity of the sleigh increases indefinitely and has the asymptotics is proved. In addition, it is shown that, when viscous friction with a nondegenerate Rayleigh function is added, unbounded speed-up disappears and the trajectories of the reduced system asymptotically tend to a limit cycle.

From localization to anomalous diffusion in the dynamics of coupled kicked rotors.

We study the effect of many-body quantum interference on the dynamics of coupled periodically kicked systems whose classical dynamics is chaotic and shows an unbounded energy increase. We specifically focus on an N-coupled kicked rotors model: We find that the interplay of quantumness and interactions dramatically modifies the system dynamics, inducing a transition between energy saturation and unbounded energy increase. We discuss this phenomenon both numerically and analytically through a mapping onto an N-dimensional Anderson model. The thermodynamic limit N→∞, in particular, always shows unbounded energy growth. This dynamical delocalization is genuinely quantum and very different from the classical one: Using a mean-field approximation, we see that the system self-organizes so that the energy per site increases in time as a power law with exponent smaller than 1. This wealth of phenomena is a genuine effect of quantum interference: The classical system for N≥2 always behaves ergodically with an energy per site linearly increasing in time. Our results show that quantum mechanics can deeply alter the regularity or ergodicity properties of a many-body-driven system.

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

Energy growth rate in smoothly oscillating billiards

Particle motion in a smoothly oscillating nonintegrable billiard is known to result in unbounded energy growth. The asymptotic energy growth rate of an ensemble of particles in a smoothly oscillating chaotic billiard is known to be quadratic in time. In this paper, it is shown that although this holds for the case of small oscillations, large oscillations in chaotic billiards can lead to faster than quadratic-in-time energy growth. It is also shown that the energy growth rate in oscillating pseudo-integrable billiards could be exponential in time. This is important for applications since slower than exponential-in-time energy growth can be annihilated by dissipation.

Exponential energy growth in a Fermi accelerator

An unbounded energy growth of particles bouncing off two-dimensional (2D) smoothly oscillating polygons is observed. Notably, such billiards have zero Lyapunov exponents in the static case. For a special 2D polygon geometry--a rectangle with a vertically oscillating horizontal bar--we show that this energy growth is not only unbounded but also exponential in time. For the energy averaged over an ensemble of initial conditions, we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type. We demonstrate numerically that the ensemble averaged energy indeed grows exponentially, at a close to the analytically predicted rate-namely, the process is controllable.

Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.

Unbounded growth of energy in nonautonomousHamiltonian systems

The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincaré-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian. AMS classification scheme number: 58F05

On the Closure Problem in the Reduction of Complex Atmospheric Models by PIPs and EOFs: A Comparison for the Case of a Two-Layer Model with Zonally Symmetric Forcing

The reduction of complex systems to their essential degrees of freedom (e.g., patterns spanning that part of state space that the system’s trajectory passes in time) and development of reduced models based on these might be one tool for improving basic understanding of simulation results obtained from current, increasingly more complex, meteorological circulation models. Some successful work on the reduction of complex systems by principal interaction patterns (PIPs) or empirical orthogonal functions (EOFs) has already been done. However, the parameterization of the influence of unresolved modes onto the selected patterns, called closure in this context, is still an outstanding problem. Nonlinear closure schemes, although greatly improving prediction on shorter timescales, have so far been observed to lead to absolute instability of the reduced model, that is, explosive unbounded energy growth after some integration time. In this work a method is outlined for circumventing this problem. Energy conservation constraints are formulated that can be used in the extraction of a useful empirical closure from synthetic model data by minimizing the error between tendencies in the full and the reduced model with the closure parameters as variables. In the present context the computational size of the associated minimization calculations could be reduced by utilizing the zonal symmetry in the full model’s forcing and boundary conditions. So it could be shown that each EOF or PIP must be an element of the subspace of a single zonal wavenumber. Coupling conditions for the closure coefficients are derived that further decrease the dimensionality of the problem. The method is tested by reducing multiple baroclinic wave life cycles in a quasigeostrophic two-layer model on the basis of both EOFs and PIPs. It is shown that the aforementioned stability problems connected with the nonlinearity of a closure are indeed avoided by the method. Furthermore, the closure greatly improves the simulation capabilities of the reduced model both on short and long timescales. In contrast to previous results on linear closure schemes, the authors find that in nonlinear closure schemes also the linear terms have to be handled carefully in order to ensure realistic behavior of the reduced model on longer timescales. In a comparison between reductions based on EOFs and PIPs substantial superiority of the latter in effectively extracting the essential degrees of freedom is demonstrated.

Bounded Energy States in Homogeneous Turbulent Shear Flow—An Alternative View

The equilibrium structure of homogeneous turbulent shear flow is investigated from a theoretical standpoint. Existing turbulence models, in apparent agreement with physical and numerical experiments, predict an unbounded exponential time growth of the turbulent kinetic energy and dissipation rate; only the anisotropy tensor and turbulent time scale reach a structural equilibrium. It is shown that if a residual vortex stretching term is maintained in the dissipation rate transport equation, then there can exist equilibrium solutions, with bounded energy states, where the turbulence production is balanced by its dissipation. Illustrative calculations are presented for a k–ε model modified to account for net vortex stretching. The calculations indicate an initial exponential time growth of the turbulent kinetic energy and dissipation rate for elapsed times that are as large as those considered in any of the previously conducted physical or numerical experiments on homogeneous shear flow. However, vortex stretching eventually takes over and forces a production-equals-dissipation equilibrium with bounded energy states. The plausibility of this result is further supported by independent calculations of isotropic turbulence which show that when this vortex stretching effect is accounted for, a much more complete physical description of isotropic decay is obtained. It is thus argued that the inclusion of vortex stretching as an identifiable process may have greater significance in turbulence modeling than has previously been thought and that the generally accepted structural equilibrium for homogeneous shear flow, with unbounded energy growth, could be in need of re-examination.