Abstract

This paper reviews recent advances and current debates in modeling the solar cycle as a hydromagnetic dynamo process. Emphasis is placed on (relatively) simple dynamo models that are nonetheless detailed enough to be comparable to solar cycle observations. After a brief overview of the dynamo problem and of key observational constraints, I begin by reviewing the various magnetic field regeneration mechanisms that have been proposed in the solar context. I move on to a presentation and critical discussion of extant solar cycle models based on these mechanisms, followed by a discussion of recent magnetohydrodynamical simulations of solar convection generating solar-like large-scale magnetic cycles. I then turn to the origin and consequences of fluctuations in these models and simulations, including amplitude and parity modulation, chaotic behavior, and intermittency. The paper concludes with a discussion of our current state of ignorance regarding various key questions relating to the explanatory framework offered by dynamo models of the solar cycle.

Highlights

  • 1.1 Scope of reviewThe cyclic regeneration of the Sun’s large-scale magnetic field is at the root of all phenomena collectively known as ‘‘solar activity’’

  • The Dikpati and Gilman (2001) dynamo model is of the flux transport variety, with the advective action of the deep meridional flow setting equatorward propagation of the deep toroidal field; it uses a solar-like differential rotation, depth-dependent magnetic diffusivity and meridional circulation pattern much similar to those shown in Fig. 4

  • An important feature of solar cycle models based on the Babcock–Leighton mechanism is that the two substeps of the dynamo loop are segregated spatially; the P ! T step is driven by rotational shear somewhere within the solar convection zone, as in the mean-field models considered in Sect. 4; whereas the T ! P step

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Summary

Scope of review

The cyclic regeneration of the Sun’s large-scale magnetic field is at the root of all phenomena collectively known as ‘‘solar activity’’. A near-consensus exists to the effect that this magnetic cycle is to be ascribed to the inductive action of fluid motions pervading the solar interior. At this writing nothing resembling consensus exists regarding the detailed nature and relative importance of various possible inductive flow contributions. To review ‘‘dynamo models of the solar cycle’’, is daunting. This review will not discuss in any detail solar magnetic field observations, the physics of magnetic flux tubes and ropes, the generation of small-scale magnetic field in the Sun’s nearsurface layers, solar cycle prediction, or magnetic field generation in stars other than

Page 4 of 104
What is a ‘‘model’’?
A brief historical survey
Page 6 of 104
Sunspots and the butterfly diagram
Organization of review
Magnetized fluids and the MHD induction equation
Page 10 of 104
Kinematic models
Axisymmetric formulation
Page 12 of 104
Mechanisms of magnetic field generation
Page 14 of 104
Turbulence and mean-field electrodynamics
Page 16 of 104
Hydrodynamical and magnetohydrodynamical instabilities
A selection of representative mean-field models
Common model ingredients
Calculating the a-effect and turbulent diffusivity
Page 20 of 104
Algebraic a-quenching
Dynamical a-quenching
Page 22 of 104
Diffusivity quenching
Backreaction on large-scale flows
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Flux loss through magnetic buoyancy
The aX dynamo equations
Eigenvalue problems and initial value problems
Dynamo waves and cycle period
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4.2.10 Representative results
4.2.11 Critical assessment
Strong a-quenching and the saturation problem
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Representative results
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Critical assessment
Including meridional circulation: flux transport dynamos
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Models based on HD and MHD instabilities
Hydrodynamical shear instabilities
Instability of sheared magnetic layers
Buoyant instability of magnetic flux tubes
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Babcock–Leighton models
The tilts of bipolar active regions
Page 42 of 104
Surface magnetic flux transport and the Babcock–Leighton mechanism
Magnetic flux transport
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Axisymmetric kinematic mean-field-like models
Formulation of a poloidal source term
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Beyond 2D: non-axisymmetric models
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Modelling flux emergence
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The surface dipole as precursor
Global MHD numerical simulations
Convection and large-scale flows
Polarity reversals and large-scale magnetic cycles
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Turbulent induction and mean-field coefficients
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Magnetic quenching of the a-effect and turbulent diffusivity
Cyclic magnetic modulation of large-scale flows
Formation of buoyant magnetic structures
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Lessons learned
The observational evidence: an overview
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Cycle modulation: generic behaviors
Going critical and Hopfing along
Stochastic forcing and the art of noise
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Nonlinear modulation: surfing the wave
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Time delays: lagging behind
Rattling the conveyor belt
Page 80 of 104
Grand minima in MHD simulations
Fossil fields and the 22-years cycle
Page 82 of 104
What is the primary poloidal field regeneration mechanism?
What limits the amplitude of the solar magnetic field?
How constraining is the sunspot butterfly diagram?
Page 84 of 104
Is meridional circulation crucial?
Is the mean solar magnetic field really axisymmetric?
What causes Maunder-type Grand Minima?
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Where do we go from here?
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Findings
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Full Text
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