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Степанов, Г. В.; Rasulov, A. R.; Shakhbanov, K. A.; Radzhabova, L. M.

doi:10.1023/a:1017595421248

Cited by 4 publications

(6 citation statements)

References 2 publications

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“…The magnetohydrodynamic (MHD) version of this model, also introduced in PS07, permits to study turbulent dynamo action for arbitrary low or high values of P m. In PS07 we calculated the energy transfer functions of the MHD system in a (statistically stationary) saturated state. We found that for P m ≤ 1 the energy transfers are mainly local, eventually strengthening our previous results obtained with a local shell model of MHD turbulence (Stepanov & Plunian 2006). For P m ≫ 1 the dominant transfers are also mainly local except the ones from the flow scales lying in the inertial range to the magnetic scales smaller than the viscous scale.…”

Section: Introductionsupporting

confidence: 89%

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Phenomenology of Turbulent Dynamo Growth and Saturation

Stepanov¹,

Plunian²

2008

ApJ

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International audienceWith a nonlocal shell model of magnetohydrodynamic turbulence we investigate numerically the turbulent dynamo action for low and high magnetic Prandtl numbers (Prm). The results obtained in the kinematic regime and along the way to dynamo saturation are understood in terms of a phenomenological approach based on the local ( ) or nonlocal ( ) nature of the energy transfers. In both cases, the magnetic energy grows at a small scale and saturates as an inverse “cascade.

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Section: Introductionsupporting

confidence: 89%

“…This is also consistent with the fact that the dynamo threshold should not depend on P m in the limit of low P m as shown previously (Ponty etal. 2005;Stepanov & Plunian 2006;Schekochihin etal. 2007).…”

Section: Kinematic Regimementioning

confidence: 99%

Phenomenology of Turbulent Dynamo Growth and Saturation

Stepanov¹,

Plunian²

2008

ApJ

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“…Let us remark that the statistical properties of intermittent fluctuations, computed either using shell variables or the in-stantaneous rate of energy dissipation, are in close qualitative and quantitative agreement with those measured in laboratory experiments, for homogeneous and isotropic turbulence [14]. MHD shell model -introduced in [15]allow a description of turbulence at low magnetic Prandtl number since the steps of both cascades can be freely adjusted [16,17]. Although geometrical features are lost, this is a clear advantage over 3D simulations [18,19].…”

supporting

confidence: 52%

Magnetic Reversals in a Simple Model of Magnetohydrodynamics

Benzi

Pinton

2010

Phys. Rev. Lett.

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We study a simple magnetohydrodynamical approach in which hydrodynamics and MHD turbulence are coupled in a shell model, with given dynamo constrains in the large scales. We consider the case of a low Prandtl number fluid for which the inertial range of the velocity field is much wider than that of the magnetic field. Random reversals of the magnetic field are observed and it shown that the magnetic field has a non trivial evolution -linked to the nature of the hydrodynamics turbulence.PACS numbers: 47. 91.25.Cw,47.27.Ak Observations show that natural dynamos are intrinsically dynamical. Complex magnetic field evolutions have been reported for many systems, including the Sun and the Earth [1]. Formally, the coupled set of momentum and induction equations are invariant under the transform: (u, B) → (u, −B) so that states with opposite polarities can be generated from the same velocity field (u and B are respectively the velocity and magnetic fields). In the case of the geodynamo, polarity switches are called reversals [1] and occur at very irregular time intervals [2]. Such reversals have been observed recently in laboratory experiments using liquid metals, in arrangements where the dynamo cycle is either favored artificially [3] or stems entirely from the fluid motions [4,5]. In these laboratory experiments, as also presumably in the Earth core, the ratio of the magnetic diffusivity to the viscosity of the fluid (magnetic Prandtl number P M ) is quite small. As a result, the kinetic Reynolds number R V of the flow is very high because its magnetic Reynolds number R M = R V P M needs to be large enough so that the stretching of magnetic fields lines balances the Joule dissipation. Hence, the dynamo process develops over a turbulent background and in this context, it is often considered as a problem of 'bifurcation in the presence of noise'. For the dynamo instability, the effect of noise enters both additively and multiplicatively, a situation for which a complete theory is not currently available. Some specific features have been ascribed to its onset (e.g. bifurcation via an on-off scenario [6]) and to its dynamics [7]. Turbulence also implies that processes occur over an extended range of scales; however, in a low magnetic Prandtl number fluid the hydrodynamic range of scales is much wider than the magnetic one. In laboratory experiments, the induction processes that participate in the dynamo cycle involve the action of large scale velocity gradients [4,8,9], with also possible contributions of velocity fluctuations at small scales [10,11,12].Building upon the above observations, we propose here a simple model which incorporates hydromagnetic turbulent fluctuations (as opposed to 'noise') in a dynamo instability. The models stems from the approach introduced in [13] for the hydrodynamic studies. Magnetic field reversals are observed above onset and we detail their characteristics.

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“…Here we choose the one referred to as Sabra shell model. MHD shell model -introduced in [15] -allow a description of turbulence at low magnetic Prandtl number since the steps of both cascades can be freely adjusted [16,17]. We consider a formulation extended from the Sabra hydrodynamic shell model:…”

Section: Simple Model Of Magnetic Reversalsmentioning

confidence: 99%

Stochastic Resonance in a Simple Model of Magnetic Reversals

Benzi¹,

Pinton

2011

Int. J. Bifurcation Chaos

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We discuss the effect of stochastic resonance in a simple model of magnetic reversals. The model exhibits statistically stationary solutions and bimodal distribution of the large scale magnetic field. We observe a non trivial amplification of stochastic resonance induced by turbulent fluctuations, i.e. the amplitude of the external periodic perturbation needed for stochastic resonance to occur is much smaller than the one estimated by the equilibrium probability distribution of the unperturbed system. We argue that similar amplifications can be observed in many physical systems where turbulent fluctuations are needed to maintain large scale equilibria. 91.25.Cw,47.27.Ak THE PROBLEMIn this paper we discuss the effect of Stochastic Resonance (SR) for a simple model of magnetic reversals. As we will discuss later on, we show that the effect of SR can be amplified for systems where statistically stationary states are observed, i.e. where metastable equilibria are due to non linear equilibration between turbulent fluctuations and non linear large scale effect. We argue that this effect is relevant in many physical systems and could be eventually observed experimentally.Before defining more precisely the problem we want to focus, it is worthwhile to review shortly the basic idea behind SR. SR was introduced almost 30 years ago in [1] and [2] within the framework of long term climate theory. One can be understood the mechanism of SR in the simple case of the stochastic differential equation:where dW (t) is gaussian white noise δ−correlated in time. Because of the noise, φ shows a bimodal probability distribution peaked around ±φ m where φ 2 m = m/g. The average transition time τ between the two peaks is proportional toIt is well known that the transition time is a random variable exponentially distributed for small σ. If we add on the r.h.s of (1) a periodic perturbation Asin(ωt), something interesting can happen. For π/ω ∼ τ the behavior of φ becomes nearly periodic, i.e. φ "jumps" between the two states ±φ m periodically with period 2π/ω. This behavior can be understood in a number of different ways and we refer the reader to the original papers [1] and [2], see also [3] for a review on SR. For ω small and the deterministic time scale 1/m much shorter than τ , the condition for SR to occur [1] can be written asIn the simplified model (1), the equilibrium probability distribution P (φ) is peaked around ±φ m which are stable stationary solutions of (1) for σ = 0. In many physical systems, however, the probability distribution of the relevant order parameter (let us still call it φ ) is bimodal although there exists no stable stationary states: the peaks in the probability distribution arise because non trivial equilibration of internal dynamics which on the average can be approximated by an effective equation similar to (1). This implies that the peaks in the bimodal probability distribution should correspond to statistically stationary solutions. A particular interesting case is the one where statistically stationary s...

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Cited by 4 publications

References 2 publications

Phenomenology of Turbulent Dynamo Growth and Saturation

Phenomenology of Turbulent Dynamo Growth and Saturation

Magnetic Reversals in a Simple Model of Magnetohydrodynamics

Stochastic Resonance in a Simple Model of Magnetic Reversals

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