Crisis-induced intermittency in truncated mean field dynamos

Covas, Eurico; Tavakol, Reza

doi:10.1103/physreve.55.6641

Cited by 16 publications

(7 citation statements)

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“…This type of intermittency has been found in several experimental and numerical studies [11]. Other types of crisis [10] have also been concretely demonstrated to exist in dynamo models [12,13]. …”

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confidence: 87%

Crisis-Induced Intermittency Due to Attractor-Widening in a Buoyancy-Driven Solar Dynamo

Ossendrijver

Covas

2003

Int. J. Bifurcation Chaos

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In a recent paper [M. A. J. H. Ossendrijver, A&A 359, 364 (2000)] numerical simulations of a 2D mean-field model where shown to produce grand minima, typical of the long-term behavior of solar magnetic activity. The model consisted of dynamo that features an α effect based on the buoyancy instability of magnetic fluxtubes, which gives rise to the switching back and forth from grand minima to "regular" solar behavior. In this Letter, we report evidence from a time-series analysis of the model for the presence of crisis-induced intermittency due to attractor-widening. We support this finding by showing that the average duration of the minima, τ , follows the theoretically predicted scaling τ ∼ (C δα − C * δα )−γ , where C δα is the bifurcation parameter of interest, together with other statistical evidence. As far as we are aware, this is the first time concrete and detailed evidence has been produced for the occurrence of this type of crisis-induced intermittency -due to attractor widening -for such dynamo models.The records of past solar magnetic activity reveal an outstanding phenomenon referred to as grand minima. During the latest of such intervals, the so-called Maunder minimum (1645-1715), sunspots were virtually absent. This and earlier grand minima are clearly visible in the records of cosmogenic isotopes, e.g.14 C and 10 Be [2]). Timing and duration of the known grand minima are irregular. Solar variability is also apparent e.g. in length and amplitude variations of the 11-year sunspot cycle. The spectral and statistical properties of solar variability are still not well-known. Although there is some evidence for various modulations of the solar cycle [3], it remains unclear, due to the lack of a sufficiently long and accurate data set, whether they are truly periodic rather than chaotic or even purely stochastic [4].Intermittency has been proposed [5] as one of the possible scenarios for the underlying mechanism behind the occurrence of grand minima. Several dynamo models have been show to exhibit quasiperiodic or chaotic intermittent behavior [6]. In [1] a case was made for a stochastically driven 2D mean-field dynamo model showing grand minima [7]. Earlier, the authors in [8] produced grand minima using a 1D model, in which the flux injections were treated as an additive random source term. The main advantage of a 2D model is that the radial structure is resolved, so that the flux injections can be treated more realistically through the advection term. Of course, mean-field theory cannot replace full MHD calculations [9] and is at best capable of capturing the most important aspects of solar dynamo action. The purpose of the calculations was to illustrate that grand minima are an inherent feature of a solar dynamo based on magnetic flux tubes.In this Letter, we show concrete evidence that the switching back and forth from grand minima to the "regular" 22-year cycle is a manifestation of a known dynamical process called attractor-widening, resulting in crisis-induced intermittency [10]. This type of in...

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confidence: 87%

Crisis-Induced Intermittency Due to Attractor-Widening in a Buoyancy-Driven Solar Dynamo

Ossendrijver

Covas

2003

Int. J. Bifurcation Chaos

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“…Such models may be derived using normal form theory of dynamical systems (Tobias et al 1995;Wilmot-Smith et al 2005) or by truncating at various orders a suitable form of the mean-field dynamo equations (Roald & Thomas 1997;Schmalz & Stix 1991;Weiss et al 1984). A wide variety of dynamical behaviors occur in these low-dimensional models, including various types of intermittency that may account for events such as grand minima (Covas & Tavakol 1997). However, the removal of all spatial dependence in low-order models' descriptions of the field evolution gives an implied instantaneous communication between the two field components (toroidal and poloidal) that would not occur in spatially segregated models.…”

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confidence: 99%

A Time Delay Model for Solar and Stellar Dynamos

Wilmot-Smith

Nandy

Hornig

et al. 2006

ApJ

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Magnetohydrodynamic dynamos operating in stellar interiors produce the diverse range of magnetic activity observed in solar-like stars. Sophisticated dynamo models including realistic physics of convection zone flows and flux tube dynamics have been built for the Sun, for which appropriate observations exist to constrain such models. Nevertheless, significant differences exist in the physics that the models invoke, the most important being the nature and location of the dynamo-effect and whether it is spatially segregated from the location of the-effect. Spatial segregation of these source layers necessitates a physical mechanism for communication between them, involving unavoidable time delays. We construct a physically motivated reduced dynamo model in which, through the use of time delays, we mimic the generation of field components in spatially segregated layers and the communication between them. The model can be adapted to examine the underlying structures of more complicated and spatially extended numerical dynamo models with diverse-effect mechanisms. A variety of dynamic behaviors arise as a direct consequence of the introduction of time delays in the system. Various parameter regimes give rise to periodic and aperiodic oscillations. Amplitude modulation leads to episodes of reduced activity, such as that observed during the Maunder minima, the length and duration of which depend on the dynamo number. Regular activity is more easily excited in the flux transportYdominated regime (when the time delay is smaller than the dissipative timescale), whereas irregular activity characterizes solutions in the diffusion-dominated regime (when the time delay is larger than the dissipative timescale).

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“…The construction of low‐order models of the solar dynamo has traditionally utilized one of two alternative approaches. The first is to derive sets of ordinary differential equations (ODEs) via a truncation of the partial differential equations (PDEs) of mean‐field electrodynamics (Priest 1982; Zeldovich, Ruzmaikin & Sokolov 1983; Martens 1984; Weiss, Cattaneo & Jones 1984; Jones, Weiss & Cattaneo 1985; Schmalz & Stix 1991; Covas & Tavakol 1997; Roald & Thomas 1997). This approach has the advantage that each term in the truncated set of ODEs has an obvious physical interpretation, as it has been derived from an analogous term in the PDEs.…”

Section: Introductionmentioning

confidence: 99%

Low-order stellar dynamo models

Wilmot-Smith

Martens

Nandy

et al. 2005

Monthly Notices of the Royal Astronomical Society

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Stellar magnetic activity -which has been observed in a diverse set of stars including the Sunoriginates via a magnetohydrodynamic dynamo mechanism working in stellar interiors. The full set of magnetohydrodynamic equations governing stellar dynamos is highly complex, and so direct numerical simulation is currently out of reach computationally. An understanding of the bifurcation structure, likely to be found in the partial differential equations governing such dynamos, is vital if we are to understand the activity of solar-like stars and its evolution with varying stellar parameters such as rotation rate. Low-order models are an important aid to this understanding, and can be derived either as approximations of the governing equations themselves or by using bifurcation theory to obtain systems with the desired structure. We use normal-form theory to derive a third-order model with robust behaviour. The model is able to reproduce many of the basic types of behaviour found in observations of solar-type stars. In the appropriate parameter regime, a chaotic modulation of the basic cycle is present, together with varying periods of low activity such as that observed during the solar Maunder minima.

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Crisis-induced intermittency in truncated mean field dynamos

Cited by 16 publications

References 15 publications

Crisis-Induced Intermittency Due to Attractor-Widening in a Buoyancy-Driven Solar Dynamo

Crisis-Induced Intermittency Due to Attractor-Widening in a Buoyancy-Driven Solar Dynamo

A Time Delay Model for Solar and Stellar Dynamos

Low-order stellar dynamo models

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