Destabilization by noise of transverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory

Oprea, Iuliana; Proctor, M. R. E.; Rucklidge, Alastair M.

doi:10.1098/rspa.1999.0498

Cited by 13 publications

(15 citation statements)

References 23 publications

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“…This suggests the core flow is also continually changing. Kinematic studies show that steady flows make poor dynamos because they contain stagnation points; it is also worth noting that some of the dynamos driven by time-dependent, oscillatory flows studied by Willis & Gubbins (2004) had lower critical magnetic Reynolds numbers than any dynamo driven by a stationary, instantaneous flow in the cycle [see also Gog et al (1999)]. Most dynamic dynamo models have chaotic flows that are highly time dependent and, therefore, avoid this problem: it seems that any sufficiently chaotic and vigorous flow will generate a magnetic field.…”

Section: Main Fieldmentioning

confidence: 99%

Implication of kinematic dynamo studies for the geodynamo

Gubbins¹

2008

Geophysical Journal International

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SUMMARY In the kinematic dynamo problem Maxwell's equations are solved for the magnetic field given a prescribed fluid velocity. Although no dynamic equations are involved, it does provide an accurate link between the magnetic field and fluid velocity and can therefore be used to infer something about the flow underlying the observed geomagnetic field. In this sense it complements the commonly used frozen‐flux theory for inverting secular variation for core flow, in which electrical diffusion is neglected, and can be used to show up some of the strengths and weaknesses of the frozen‐flux approximation. It might be thought that kinematic models have been superceded by dynamic models that include the momentum and heat equations, but this is not the case. Even the biggest numerical simulation cannot approach the correct parameters for the Earth's core, and the classes of flows that result are, in fact, quite restricted as well as being too complex for simple physical interpretation. A variety of simple flows have been studied for dynamo action; of particular interest here are a broad class of flows, based loosely on extensions of the early simple choice of Bullard & Gellman and to some extent representative of what might be generated by convection in the Earth's rapidly rotating core: this paper reviews the implications of the solutions for geomagnetism. The non‐axisymmetric flows mimic convection rolls in a rotating sphere, the axisymmetric poloidal flows describe meridional circulation, a likely secondary flow, and the axisymmetric toroidal flow is a simple differential rotation. Helicity, which seems to be important for dynamo action, is related to spiralling of the rolls. Dipole, rather than quadrupole, fields are preferred when spiralling is eastward and differential rotation westward at the surface. Magnetic flux tends to be concentrated at stagnation points of the flow, and dynamo action fails when this concentration becomes so intense that steep gradients develop so as to enhance energy loss by diffusion. On the surface these stagnation points are centres of downwelling that concentrate vertical flux. Flux concentration plays an important role in determining the magnetic field's symmetry: for example, concentration of vertical flux in high latitudes favours dipolar fields while concentration near the equator, where dipole symmetry requires a change of sign, favours quadrupole symmetry. Steady fluid flow can only produce steady or oscillatory solutions to the kinematic problem at onset of instability. Steady solutions are preferred in three dimensions, in contrast to predictions of axisymmetric mean‐field dynamo theory, where oscillatory solutions are the most common. Oscillatory solutions are preferred when poloidal and toroidal field coincide, which is unlikely to happen in the Earth because poloidal field is concentrated at high latitudes around the tangent cylinder while the toroidal field is probably strongest in mid‐latitudes. Geomagnetic reversals are not oscillatory in nature and, therefore, require t...

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Section: Main Fieldmentioning

confidence: 99%

Implication of kinematic dynamo studies for the geodynamo

Gubbins¹

2008

Geophysical Journal International

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“…The efficiency of some time dependent flows in the dynamo process has been recognised in various studies [4][5][6][7][8]. These studies focused on leading order time dependence of the flow (either analytical, or in the form of an heteroclinic cycle), whereas we want to study the effects of vanishing perturbations on steady flows.…”

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confidence: 99%

Time scales separation for dynamo action

Dormy¹,

Gérard-Varet²

2008

Europhys. Lett.

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Abstract. -The study of dynamo action in astrophysical objects classically involves two timescales: the slow diffusive one and the fast advective one. We investigate the possibility of field amplification on an intermediate timescale associated with time dependent modulations of the flow. We consider a simple steady configuration for which dynamo action is not realised. We study the effect of time dependent perturbations of the flow. We show that some vanishing low frequency perturbations can yield exponential growth of the magnetic field on the typical time scale of oscillation. The dynamo mechanism relies here on a parametric instability associated with transient amplification by shear flows. Consequences on natural dynamos are discussed.

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“…However, even if the biomass B(t) remains oscillatory on the attractor, one can also measure the nonequilibrium stability and resilience in such an ecosystem. The GLV equations are popular for describing the dynamics of n competing populations; they consist of n first-order differential equations, and are usually ameliorated by introducing noise [35][36][37] or diffusion terms [38][39][40][41] in order to avoid a slowing down state:…”

Section: Stability and Resilience In An N-competitor Heteroclinic Cycmentioning

confidence: 99%

Measuring nonequilibrium stability and resilience in an -competitor system

Zheng

2010

Nonlinear Analysis: Real World Applications

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Destabilization by noise of transverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory

Cited by 13 publications

References 23 publications

Implication of kinematic dynamo studies for the geodynamo

Implication of kinematic dynamo studies for the geodynamo

Time scales separation for dynamo action

Measuring nonequilibrium stability and resilience in an -competitor system

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