Dynamo Theory

Roberts, Paul; Soward, A. M.

doi:10.1146/annurev.fl.24.010192.002331

Cited by 227 publications

(71 citation statements)

References 61 publications

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“…First, according to boundary layer theory results [16,17] and for a stationary flow, in the limit of large R m the magnetic field which has the highest growth rate satisfies µ ≈ 0. This resonant condition means that the pitch of the magnetic field is roughly equal to the pitch of the flow.…”

Section: A Case (I): Solid Body Flowmentioning

confidence: 99%

“…Such helical flows have been identified to produce dynamo action [14,15]. Their efficiency has been studied in the context of fast dynamo theory [16,17,18,19,20,21] and they have led to the realization of several dynamo experiments [3,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

“…In both cases some resonant conditions leading to dynamo action have been derived [16,17,18,20,21]. Such resonant conditions can be achieved by choosing an appropriate geometry of the helical flow, like changing its geometrical pitch.…”

Section: Introductionmentioning

confidence: 99%

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Parametric instability of the helical dynamo

2007

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We study the dynamo threshold of a helical flow made of a mean (stationary) plus a fluctuating part. Two flow geometries are studied, either (i) solid body or (ii) smooth. Two well-known resonant dynamo conditions, elaborated for stationary helical flows in the limit of large magnetic Reynolds numbers, are tested against lower magnetic Reynolds numbers and for fluctuating flows (zero mean). For a flow made of a mean plus a fluctuating part the dynamo threshold depends on the frequency and the strength of the fluctuation. The resonant dynamo conditions applied on the fluctuating (resp. mean) part seems to be a good diagnostic to predict the existence of a dynamo threshold when the fluctuation level is high (resp. low).PACS numbers: 47.65.+a arXiv:physics/0703216v1 [physics.flu-dyn]

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Section: A Case (I): Solid Body Flowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parametric instability of the helical dynamo

2007

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“…the low-order ODE alpha-omega model of Parker 1955). There is a long history of dynamo theory (Moffatt 1978;Krause & Radler 1980;Roberts & Soward 1992;Brandenburg & Subramanian 2005), but much of it is comprised of a closure or parameterization ansatz for how fluctuating velocity and magnetic fields act through the mean electromotive force curl to amplify the large-scale magnetic field. In general the mean-field equations are not derived from fundamental principles.…”

Section: Introductionmentioning

confidence: 99%

The elemental shear dynamo

McWilliams

2012

J. Fluid Mech.

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A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, ${\mathit{Re}}_{\eta } \ll 1$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma $ can occur for arbitrary values of ${\mathit{Re}}_{\eta } $, viscous Reynolds number ${\mathit{Re}}_{\nu } $, and random-force correlation time ${t}_{f} $ and orientation angle ${\theta }_{f} $ in the shearing plane. The value of $\gamma $ is independent of the domain size. The shear dynamo is ‘fast’, with finite $\gamma \gt 0$ in the limit of ${\mathit{Re}}_{\eta } \gg 1$. Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small ${\mathit{Re}}_{\eta } $ and ${\mathit{Re}}_{\nu } $, the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying ($\gamma {t}_{f} \gg 1$) and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

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“…Moffatt (1978) gives an accessible outline of the theory. Nevertheless, subtle issues emerge from the two rotor system, which stimulated the further analytic studies of Gibson (1968a) and Gailitis (1973) and was the focus of discussion in the reviews of Gibson & Roberts (1967), Roberts (1967a, b), Gibson (1969) and Roberts (1971). Generalizations include multiple rotors like the three-rotor system of Gibson (1968b), although the idea of image rotors to accommodate boundary conditions in electromagnetic induction problems had earlier been introduced by Herzenberg & Lowes (1957).…”

Section: Introductionmentioning

confidence: 99%

New results for the Herzenberg dynamo: steady and oscillatory solutions

Brandenburg

Moss

Soward

1998

Proc. R. Soc. Lond. A

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The Herzenberg dynamo, consisting of two rotating electrically conducting spheres with non-parallel spin axes, immersed in a finite spherical conducting medium, is simulated numerically for a variety of parameters not accessible to the original asymptotic theory. Our model places the spheres in a spatially periodic box. The largest growth rate is obtained when the angle, ϕ, between the spin axes is somewhat larger than 125 • . In agreement with the asymptotic analysis, it is found that the critical dynamo number is approximately proportional to the cube of the ratio of the common radius of the spheres and their separation. The asymptotic prediction, strictly valid only in the limit of small spheres, remains approximately valid even when the diameter of the spheres becomes comparable to their separation. For |ϕ| < 90 • we also find oscillatory solutions, which were not predicted by Herzenberg's analysis. To understand such solutions we present a modified asymptotic analysis in which the separation of the two spheres is essentially replaced by the skin depth which, in turn, depends on the diameter of the spheres. The magnetic field consists of magnetic flux rings wrapped around the two spheres. Applications to local models of turbulent dynamos and to dynamo action in binary stars are discussed.

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Dynamo Theory

Cited by 227 publications

References 61 publications

Parametric instability of the helical dynamo

Parametric instability of the helical dynamo

The elemental shear dynamo

New results for the Herzenberg dynamo: steady and oscillatory solutions

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