In-out intermittency in PDE and ODE models of axisymmetric mean-field dynamos (original) (raw)

In–out intermittency in partial differential equation and ordinary differential equation models

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2001

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation ͑PDE͒ and ordinary differential equation ͑ODE͒ models of mean field dynamos. This type of intermittency ͓introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 ͑1999͔͒ occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities.

Multiple forms of intermittency in PDE dynamo models

Phys Rev E, 1999

We find concrete evidence for the presence of crisis-induced and Pomeau-Manneville Type-I intermittencies in an axisymmetric PDE mean-field dynamo model. These findings are of potential importance for two different reasons. Firstly, as far as we are aware, this is the first time detailed evidence has been produced for the occurrence of these types of intermittency for such deterministic PDE models. And secondly, despite the rather idealised nature of these models, the concrete evidence for the occurrence of more than one type of intermittency in such models makes it in principle possible that different types of intermittency may occur in different solar-type stars or even in the same star over different epochs. In this way a multiple intermittency framework may turn out to be of importance in understanding the mechanisms responsible for grand-minima type behaviour in the Sun and solar-type stars and in particular in the interpretation of the corresponding observational and proxy evidence.

In--out intermittency in PDE and ODE models

2001

The models employed here to demonstrate in--out intermittency are axisymmetric mean--field dynamo models which are often used to study the observed large scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities.

Non-normal parameter blowout bifurcation: An example in a truncated mean-field dynamo model

Physical Review E, 1997

We examine global dynamics and bifurcations occurring in a truncated model of a stellar mean field dynamo. This model has symmetry-forced invariant subspaces for the dynamics and we find examples of transient type I intermittency and blowout bifurcations to transient on-off intermittency, involving laminar phases in the invariant submanifold. In particular, our model provides examples of blowout bifurcations that occur on varying a non-normal parameter; that is, the parameter varies the dynamics within the invariant subspace at the same time as the dynamics normal to it. As a consequence of this we find that the Lyapunov exponents do not vary smoothly and the blowout bifurcation occurs over a range of parameter values rather than a point in the parameter space. *

Crisis-induced intermittency in truncated mean field dynamos

Physical Review E, 1997

We investigate the detailed dynamics of a truncated αω dynamo model with a dynamic α effect. We find the presence of multiple attractors, including two chaotic attractors with a fractal basin boundary which merge to form a single attractor as the control parameter is increased. By considering phase portraits and the scaling of averaged times of transitions between the two attractors, we demonstrate that this merging is accompanied by a crisis-induced intermittency. We also find a range of parameter values over which the system has a fractal parameter dependence for fixed initial conditions. This is the first time this type of intermittency has been observed in a dynamo model and it could be of potential importance in accounting for some forms of intermittency in the solar and stellar output.

Dynamics of Axisymmetric Truncated Dynamo Models

Studia Geophysica Et Geodaetica - STUD GEOPHYS GEOD, 1998

An important question regarding the study of mean field dynamo models is how to make precise the nature of their underlying dynamics. This is difficult both because relatively little is known about the dynamical behaviour of infinite dimensional systems and also due to the numerical cost of studying the related partial differential equations.

Axisymmetric mean field dynamos with dynamic and algebraic alpha\alphaalpha--quenchings

1997

We study axisymmetric mean field spherical and spherical shell dynamo models, with both dynamic and algebraic alpha\alphaalpha--quenchings. Our results show that there are qualitative as well as quantitative differences and similarities between these models. Regarding similarities, both groups of models exhibit symmetric, antisymmetric and mixed modes of behaviour. As regards differences, the important feature in the full sphere models is

Axisymmetric mean eld dynamos with dynamic and algebraic -quenchings

1998

We study axisymmetric mean eld spherical and spherical shell dynamo models, with both dynamic and alge- braic -quenchings. Our results show that there are qualita- tive as well as quantitative differences and similarities between thesemodels.Regardingsimilarities,bothgroupsofmodelsex- hibit symmetric, antisymmetric and mixed modes of behaviour. As regards differences, the important feature in the full sphere modelsistheoccurrenceofchaoticbehaviourinthealgebraic- quenching models. For the spherical shell models with dynamic the main features include the possibility of multi-attractor regimeswithnalstatesensitivitywithrespecttosmallchanges in the magnitude of and the initial parity. We nd the effect of introducing a dynamic is likely to be complicated and depend on the region of the parameter space considered, rather than a uniform change towards simplicity or complexity.

Grand minima and equatorial symmetry breaking in axisymmetric dynamo models

Astron Astrophys, 1998

We consider the manner in which time-periodic solutions of an axisymmetric dynamo model can undergo breaking of equatorial symmetry, i.e. loss of pure dipolar or quadrupolar symmetry. By considering the symmetry group underlying the solutions, we show that the fluctuations responsible for the symmetry-breaking can be constrained such that they are in resonance with the former solution. They can then be amplified until they are comparable in magnitude to the former solution. If the bifurcation is supercritical, the amplitude of the fluctuation is stabilised and a stable mixed parity limit cycle is formed. If it is subcritical it gives rise to a recently identified form of intermittency, called icicle intermittency. This produces episodes in which the original solution and the fluctuation are almost exactly synchronised and the fluctuation grows exponentially in amplitude, interrupted by brief episodes where synchronicity is lost and the amplitude of the fluctuation declines rapidly by several orders of magnitude. During these latter episodes there is a significant dip in the amplitude of the total magnetic field. This model-independent analysis can produce quantitative predictions for the behaviour of this bifurcation and we provide evidence for this behaviour by analysing timeseries from four different mean-field dynamo models, where intermittency is observed without the need for stochastic, or chaotically driven, forcing terms in the dynamo equations. We compare these results with recent work on intermittency in dynamo models and consider their relevance to the intermittency present in solar and stellar cycles.

Asymptotic Methods in the Nonlinear Mean-field Dynamo

International Astronomical Union Colloquium, 1991

We discuss the methods and results of analysis of nonlinear mean-field dynamo models based on α-quenching in two asymptotic regimes, namely for weakly and highly supercritical excitation. In the former case the spatial distribution of the steady-state magnetic field is close to that given by the neutrally stable eigenfunction of the corresponding kinematic dynamo. In the latter case the magnetic field distribution within the main part of the dynamo volume is presumably determined by the balance between the Lorentz and Coriolis forces while near the boundaries boundary layers arise in which the field adjusts itself to the boundary conditions. The asymptotic behaviour of the highly supercritical αω-dynamos is sensitive to the particular form of dependence of the mean helicity on magnetic field while α2-dynamos are less sensitive to this dependence.