5,416 research outputs found
Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles
International audienceThe chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed
The Effect of Resistivity on the Nonlinear Stage of the Magnetorotational Instability in Accretion Disks
We present three-dimensional magnetohydrodynamic simulations of the nonlinear
evolution of the magnetorotational instability (MRI) with a non-zero Ohmic
resistivity. The properties of the saturated state depend on the initial
magnetic field configuration. In simulations with an initial uniform vertical
field, the MRI is able to support angular momentum transport even for large
resistivities through the quasi-periodic generation of axisymmetric radial
channel solutions rather than through the maintenance of anisotropic
turbulence. Simulations with zero net flux show that the angular momentum
transport and the amplitude of magnetic energy after saturation are
significantly reduced by finite resistivity, even at levels where the linear
modes are only slightly affected. This occurs at magnetic Reynolds numbers
expected in low, cool states of dwarf novae, these results suggest that finite
resistivity may account for the low and high angular momentum transport rates
inferred for these systems.Comment: 8 figures, accepted for publication in Ap
Evolution of self-gravitating magnetized disks. II- Interaction between MHD turbulence and gravitational instabilities
We present 3D magnetohydrodynamic (MHD) numerical simulations of the
evolution of self--gravitating and weakly magnetized disks with an adiabatic
equation of state. Such disks are subject to the development of both the
magnetorotational and gravitational instabilities, which transport angular
momentum outward. As in previous studies, our hydrodynamical simulations show
the growth of strong m=2 spiral structure. This spiral disturbance drives
matter toward the central object and disappears when the Toomre parameter Q has
increased well above unity. When a weak magnetic field is present as well, the
magnetorotational instability grows and leads to turbulence. In that case, the
strength of the gravitational stress tensor is lowered by a factor of about~2
compared to the hydrodynamical run and oscillates periodically, reaching very
small values at its minimum. We attribute this behavior to the presence of a
second spiral mode with higher pattern speed than the one which dominates in
the hydrodynamical simulations. It is apparently excited by the high frequency
motions associated with MHD turbulence. The nonlinear coupling between these
two spiral modes gives rise to a stress tensor that oscillates with a frequency
which is a combination of the frequencies of each of the modes. This
interaction between MHD turbulence and gravitational instabilities therefore
results in a smaller mass accretion rate onto the central object.Comment: 31 pages, 19 figures, accepted for publication in ApJ, animation
avalaible at http://www2.iap.fr/users/fromang/simu3d/simu3d.htm
Transitions in large eddy simulation of box turbulence
One promising decomposition of turbulent dynamics is that into building
blocks such as equilibrium and periodic solutions and orbits connecting these.
While the numerical approximation of such building blocks is feasible for flows
in small domains and at low Reynolds numbers, computations in developed
turbulence are currently out of reach because of the large number of degrees of
freedom necessary to represent Navier-Stokes flow on all relevant spatial
scales. We mitigate this problem by applying large eddy simulation (LES), which
aims to model, rather than resolve, motion on scales below the filter length,
which is fixed by a model parameter. By considering a periodic spatial domain,
we avoid complications that arise in LES modelling in the presence of boundary
layers. We consider the motion of an LES fluid subject to a constant body force
of the Taylor-Green type as the separation between the forcing length scale and
the filter length is increased. In particular, we discuss the transition from
laminar to weakly turbulent motion, regulated by simple invariant solution, on
a grid of points
The quasi-periodic doubling cascade in the transition to weak turbulence
The quasi-periodic doubling cascade is shown to occur in the transition from
regular to weakly turbulent behaviour in simulations of incompressible
Navier-Stokes flow on a three-periodic domain. Special symmetries are imposed
on the flow field in order to reduce the computational effort. Thus we can
apply tools from dynamical systems theory such as continuation of periodic
orbits and computation of Lyapunov exponents. We propose a model ODE for the
quasi-period doubling cascade which, in a limit of a perturbation parameter to
zero, avoids resonance related problems. The cascade we observe in the
simulations is then compared to the perturbed case, in which resonances
complicate the bifurcation scenario. In particular, we compare the frequency
spectrum and the Lyapunov exponents. The perturbed model ODE is shown to be in
good agreement with the simulations of weak turbulence. The scaling of the
observed cascade is shown to resemble the unperturbed case, which is directly
related to the well known doubling cascade of periodic orbits
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional, non-linear magnetohydrodynamic process whose study is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics of transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to turbulent injection of both kinetic and magnetic energy in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to better understand the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows
Resistivity-driven State Changes in Vertically Stratified Accretion Disks
We investigate the effect of shear viscosity and Ohmic resistivity on the
magnetorotational instability (MRI) in vertically stratified accretion disks
through a series of local simulations with the Athena code. First, we use a
series of unstratified simulations to calibrate physical dissipation as a
function of resolution and background field strength; the effect of the
magnetic Prandtl number, Pm = viscosity/resistivity, on the turbulence is
captured by ~32 grid zones per disk scale height, H. In agreement with previous
results, our stratified disk calculations are characterized by a subthermal,
predominately toroidal magnetic field that produces MRI-driven turbulence for
|z| < 2 H. Above |z| = 2 H, magnetic pressure dominates and the field is
buoyantly unstable. Large scale radial and toroidal fields are also generated
near the mid-plane and subsequently rise through the disk. The polarity of this
mean field switches on a roughly 10 orbit period in a process that is
well-modeled by an alpha-omega dynamo. Turbulent stress increases with Pm but
with a shallower dependence compared to unstratified simulations. For
sufficiently large resistivity, on the order of cs H/1000, where cs is the
sound speed, MRI turbulence within 2 H of the mid-plane undergoes periods of
resistive decay followed by regrowth. This regrowth is caused by amplification
of toroidal field via the dynamo. This process results in large amplitude
variability in the stress on 10 to 100 orbital timescales, which may have
relevance for partially ionized disks that are observed to have high and low
accretion states.Comment: very minor changes, accepted to Ap
Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
Motivated by recent success in the dynamical systems approach to transitional
flow, we study the efficiency and effectiveness of extracting simple invariant
sets (recurrent flows) directly from chaotic/turbulent flows and the potential
of these sets for providing predictions of certain statistics of the flow.
Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a
sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a
rectangular torus extended in the forcing direction. In the former case, an
order of magnitude more recurrent flows are found than previously (Chandler &
Kerswell 2013) and shown to give improved predictions for the dissipation and
energy pdfs of the chaos via periodic orbit theory. Over the extended torus at
low forcing amplitudes, some extracted states mimick the statistics of the
spatially-localised chaos present surprisingly well recalling the striking
finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At
higher forcing amplitudes, however, success is limited highlighting the
increased dimensionality of the chaos and the need for larger data sets.
Algorithmic developments to improve the extraction procedure are discussed
Global MHD simulations of stratified and turbulent protoplanetary discs. I. Model properties
We present the results of global 3-D MHD simulations of stratified and
turbulent protoplanetary disc models. The aim of this work is to develop thin
disc models capable of sustaining turbulence for long run times, which can be
used for on-going studies of planet formation in turbulent discs. The results
are obtained using two codes written in spherical coordinates: GLOBAL and
NIRVANA. Both are time--explicit and use finite differences along with the
Constrained Transport algorithm to evolve the equations of MHD. In the presence
of a weak toroidal magnetic field, a thin protoplanetary disc in hydrostatic
equilibrium is destabilised by the magnetorotational instability (MRI). When
the resolution is large enough (25 vertical grid cells per scale height), the
entire disc settles into a turbulent quasi steady-state after about 300 orbits.
Angular momentum is transported outward such that the standard alpha parameter
is roughly 4-6*10^{-3}. We find that the initial toroidal flux is expelled from
the disc midplane and that the disc behaves essentially as a quasi-zero net
flux disc for the remainder of the simulation. As in previous studies, the disc
develops a dual structure composed of an MRI--driven turbulent core around its
midplane, and a magnetised corona stable to the MRI near its surface. By
varying disc parameters and boundary conditions, we show that these basic
properties of the models are robust. The high resolution disc models we present
in this paper achieve a quasi--steady state and sustain turbulence for hundreds
of orbits. As such, they are ideally suited to the study of outstanding
problems in planet formation such as disc--planet interactions and dust
dynamics.Comment: 19 pages, 29 figures, accepted in Astronomy & Astrophysic
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