An efficient method for computing the eigenfunctions of the dynamo equation

In this study we address the question under which conditions a saturated velocity field stemming from geodynamo simulations leads to an exponential growth of the magnetic field in a corresponding kinematic calculation. We perform global self-consistent geodynamo simulations and calculate the evolution of a kinematically advanced tracer field. The self-consistent velocity field enters the induction equation in each time step, but the tracer field does not contribute to the Lorentz force. This experiment has been performed by Cattaneo & Tobias (2009) and is closely related to the test field method by Schrinner et al. (2005, 2007). We find two dynamo regimes in which the tracer field either grows exponentially or approaches a state aligned with the actual self-consistent magnetic field after an initial transition period. Both regimes can be distinguished by the Rossby number and coincide with the dipolar and multipolar dynamo regimes identified by Christensen & Aubert (2006). Dipolar dynamos with low Rossby number are kinematically stable whereas the tracer field grows exponentially in the multipolar dynamo regime. This difference in the saturation process for dynamos in both regimes comes along with differences in their time variability. Within our sample of 20 models, solely kinematically unstable dynamos show dipole reversals and large excursions. The complicated time behaviour of these dynamos presumably relates to the alternating growth of several competing dynamo modes. On the other hand, dynamos in the low Rossby number regime exhibit a rather simple time dependence and their saturation merely results in a fluctuation of the fundamental dynamo mode about its critical state.Comment: 6 pages, 8 figure

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“…The integration is carried out over the whole fluid domain V . For a derivation of we refer to Hoyng (2009) and Schrinner et al (2010).…”

Section: Discussionmentioning

confidence: 99%

“…Eigenvalues and eigenfunctions of D have been computed as reported by Schrinner et al (2010) for models 2-4. In Fig.…”

Section: Discussionmentioning

confidence: 99%

Saturation and time dependence of geodynamo models

Schrinner

Schmitt

Cameron

et al. 2010

Geophysical Journal International

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“…Roberts andStix 1972, Dudley andJames 1989). Here we solve (8) in a sphere surrounded by a non-conducting vacuum applying the method presented in Schrinner et al (2010). The approach is based on the biorthogonality of the electric current j = µ −1 0 ∇ × B and the vector potential A and explicitly utilizes an expansion of the magnetic field B into (analytical known) free decay modes.…”

Section: Equations and Methodsmentioning

confidence: 99%

Spectral properties of oscillatory and non-oscillatory α²-dynamos

Giesecke

Stefani

Gerbeth

2012

Geophysical & Astrophysical Fluid Dynamics

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The eigenvalues and eigenfunctions of a linear 2 -dynamo have been computed for different spatial distributions of an isotropic -effect. Oscillatory solutions are obtained when exhibits a sign change in the radial direction. The time-dependent solutions arise at the so-called exceptional points where two stationary modes merge and continue as an oscillatory eigenfunction with conjugate complex eigenvalues. The close proximity of oscillatory and non-oscillatory solutions may serve as the basic ingredient for reversal models that describe abrupt polarity switches of a dipole induced by noise. Whereas the presence of an inner core with different magnetic diffusivity has remarkable little impact on the character of the dominating dynamo eigenmodes, the introduction of equatorial symmetry breaking considerably changes the geometric character of the solutions. Around the dynamo threshold, the leading modes correspond to hemispherical dynamos even when the symmetry breaking is small. This behavior can be explained by an approximate dipole-quadrupole degeneration for the unperturbed problem. More complicated scenarios may occur in the case of more realistic anisotropies of -and -effect or through nonlinearities caused by the back-reaction of the magnetic field (magnetic quenching).

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“…For more details concerning the eigenvalue calculation, we refer to Schrinner et al (2010b). We also consider the evolution of a kinematically advanced magnetic field, B Tr , governed by a second induction equation…”

Section: Dynamo Calculationsmentioning

confidence: 99%

Oscillatory dynamos and their induction mechanisms

Schrinner

Petitdemange

Dormy

2011

A&A

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Context. Large-scale magnetic fields resulting from hydromagnetic dynamo action may differ substantially in their time dependence. Cyclic field variations, characteristic for the solar magnetic field, are often explained by an important Ω-effect, i.e., by the stretching of field lines because of strong differential rotation. Aims. The dynamo mechanism of a convective, oscillatory dynamo model is investigated. Methods. We solve the MHD-equations for a conducting Boussinesq fluid in a rotating spherical shell. We computed the dynamo coefficients for the resulting oscillatory model with the help of the so-called test-field method. Subsequently, these coefficients were used in a mean-field calculation to explore the underlying dynamo mechanism. Results. The oscillatory dynamo model we consider is an α 2 Ω one. Although the fairly strong differential rotation of this model influences the magnetic field, the Ω-effect alone is not responsible for its cyclic time variation. If the Ω-effect is suppressed, the resulting α 2 -dynamo remains oscillatory. Surprisingly, the corresponding αΩ-dynamo leads to a non-oscillatory magnetic field. Conclusions. The assumption of an αΩ-mechanism does not explain the occurrence of magnetic cycles satisfactorily.

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An efficient method for computing the eigenfunctions of the dynamo equation

Cited by 6 publications

References 36 publications

Saturation and time dependence of geodynamo models

Saturation and time dependence of geodynamo models

Spectral properties of oscillatory and non-oscillatory α²-dynamos

Oscillatory dynamos and their induction mechanisms

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An efficient method for computing the eigenfunctions of the dynamo equation

Cited by 6 publications

References 36 publications

Saturation and time dependence of geodynamo models

Saturation and time dependence of geodynamo models

Spectral properties of oscillatory and non-oscillatory α2-dynamos

Oscillatory dynamos and their induction mechanisms

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Product

Resources

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Spectral properties of oscillatory and non-oscillatory α²-dynamos