Evolution of non-linear α <sup>2</sup>-dynamos and taylor's constraint

Fearn, David R.; Rahman, M.M.

doi:10.1080/03091920410001724124

Cited by 11 publications

(11 citation statements)

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“…There are few exceptions of the rule that α 2 -dynamos with scalar α-effect are stationary. Fearn & Rahman (2004) solved the Navier-Stokes equation and a mean-field induction equation for an α 2 -dynamo model with a radial dependence of the α-effect given by α ∝ sin π(r − R in ). This radial profile leads to a vanishing α-effect at the boundaries of the fluid outer core but the α-effect does not show any zero within the interior.…”

Section: Introductionmentioning

confidence: 99%

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

2005

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Abstract. A geodynamo-model based on an α-effect which has been computed under conditions suitable for the geodynamo is constructed. For a highly restricted class of radial α-profiles the linear α 2 -model exhibits oscillating solutions on a timescale given by the turbulent diffusion time. The basic properties of the periodic solutions are presented and the influence of the inner core size on the characteristics of the critical range that allows for oscillating solutions is shown. Reversals are interpreted as half of such an oscillation. They are rather seldom events because they can only occur if the α-profile exists long enough within the small critical range that allows for periodic solutions. Due to strong fluctuations on the convective timescale the probability of such a reversal is very small. Finally, a simple non-linear mean-field model with reasonable input parameters based on simulations of Giesecke et al. (2005) demonstrates the plausibility of the presented theory with a long-time series of a (geo-)dynamo reversal sequence.

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Section: Introductionmentioning

confidence: 99%

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

2005

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“…energy has dropped to approximately one hundredth of its peak value). We still find that the flow is dominated by the geostrophic component and varies together with the magnetic field, although we note that the relative role of the geostrophic flow (compared with the ageostrophic flow) is time dependent, for example, taking a more dominant role in the minimal energy state (see also Fearn & Rahman 2004). The flow is enslaved to the magnetic field, where the energy of u m and u g and the energy of B reaches maxima and minima simultaneously.…”

Section: A Taylor State Dynamo Model Driven By the α 2 L -Effectmentioning

confidence: 65%

“…This can be easily understood: a finite time step ∆t cannot be used together with Taylor's solution that considers ∂T /∂t, just as Newton's method does not find the root of an equation in one step unless the equation is linear. In this vein, to quote a 2004 review, "as elegant as this [Taylor's] prescription undoubtedly is, no-one has ever succeeded in following it" (Rüdiger & Hollerbach 2004), although convincing demonstrations of viscosity-independent solutions (the approach to the Taylor state) have been made using increasingly weak viscosity (Hollerbach & Ierley 1991;Jault 1995;Fearn & Rahman 2004). It is our contention that central to any robust algorithm for time stepping the magnetostrophic equations must be the concept of optimal control, a technique we shall now go on to describe.…”

Section: Introductionmentioning

confidence: 99%

Taylor state dynamos found by optimal control: axisymmetric examples

Jackson

Livermore

2018

J. Fluid Mech.

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Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

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“…It is known that a model of α 2 -dynamo for scalar α-effect have difficulties to explain the reversals. However, there are series of works in which it is shown that for a some special radial inhomogeneity of the α-effect, reversals in α 2 -dynamo can arise [3][4][5]. Therefore, we can say that in order to reproduce the reversal phenomenon in α 2 -dynamo, it is necessary to take into account spatial inhomogeneities.…”

Section: α 2 -And α 2 ω-Dynamomentioning

confidence: 99%

“…In this case, the mean-field models and the concept of the alpha-effect become indispensable. These models make it possible to obtain long-term realizations of the field dynamics, but the solutions depend very much on the predetermined spatial structure of the alpha-effect in both scalar and anisotropic cases [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

The Lorenz system and its generalizations as dynamo models with memory

Vodinchar

Казаков

2018

E3S Web Conf.

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The one of the known applications of the classical Lorenz system is an axisymmetric αω-dynamo with a dynamical quenching of the α-effect by the helicity. In this paper we consider generalizations of the Lorentz system, which are the models of α2 - and α2ω-dynamo. The cases of finite and infinite memory in the quenching functional are considered. The conditions for the existence of stationary dynamo regimes and regimes of regular and chaotic inversions are analytically and numerically studded.

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Evolution of non-linear α ²-dynamos and taylor's constraint

Cited by 11 publications

References 20 publications

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

Taylor state dynamos found by optimal control: axisymmetric examples

The Lorenz system and its generalizations as dynamo models with memory

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Evolution of non-linear α 2-dynamos and taylor's constraint

Cited by 11 publications

References 20 publications

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

Oscillatingα 2-dynamos and the reversal phenomenon of the global geodynamo

Taylor state dynamos found by optimal control: axisymmetric examples

The Lorenz system and its generalizations as dynamo models with memory

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Evolution of non-linear α ²-dynamos and taylor's constraint