Each and every one of the mechanisms described in Section 3.2 relies on fundamentally non-axisymmetric physical effects, yet these must be “forced” into axisymmetric dynamo equations for the mean magnetic field. There are a great many different ways of doing so, which explains the wide variety of dynamo models of the solar cycle to be found in the recent literature. The aim of this section is to provide representative examples of various classes of models, to highlight their similarities and differences, and illustrate their successes and failings. In all cases, the model equations are to be understood as describing the evolution of the mean field , namely the large-scale, axisymmetric component of the total solar magnetic field. Those wishing to code up their own versions of these (relatively) simple models should take note of the fact that Jouve et al. (2008) have set up a suite of benchmark calculations against which numerical dynamo solutions can be validated.

4.1 Model ingredients

4.2 mean-field models

4.2.1 Calculating the -effect and turbulent diffusivity

4.2.2 -quenching, diffusivity-quenching, and flux loss through buoyancy

4.2.3 The dynamo equations

4.2.4 Eigenvalue problems and initial value problems

4.2.5 Dynamo waves

4.2.6 Representative results

4.2.7 Critical assessment

4.3 Interface dynamos

4.3.1 Strong -quenching and the saturation problem

4.3.2 Representative results

4.3.3 Critical assessment

4.4 Mean-field models including meridional circulation

4.4.1 Representative results

4.4.2 Critical assessment

4.5 Models based on shear instabilities

4.5.1 Representative solutions

4.5.2 Critical assessment

4.6 Models based on buoyant instabilities of sheared magnetic layers

4.7 Models based on flux tube instabilities

4.7.1 From instability to -effect

4.7.2 Representative solutions

4.7.3 Critical assessment

4.8 Babcock–Leighton models

4.8.1 Formulation of a poloidal source term

4.8.2 Representative results

4.8.3 Critical assessment

4.9 Numerical simulations of solar dynamo action

4.2 mean-field models

4.2.1 Calculating the -effect and turbulent diffusivity

4.2.2 -quenching, diffusivity-quenching, and flux loss through buoyancy

4.2.3 The dynamo equations

4.2.4 Eigenvalue problems and initial value problems

4.2.5 Dynamo waves

4.2.6 Representative results

4.2.7 Critical assessment

4.3 Interface dynamos

4.3.1 Strong -quenching and the saturation problem

4.3.2 Representative results

4.3.3 Critical assessment

4.4 Mean-field models including meridional circulation

4.4.1 Representative results

4.4.2 Critical assessment

4.5 Models based on shear instabilities

4.5.1 Representative solutions

4.5.2 Critical assessment

4.6 Models based on buoyant instabilities of sheared magnetic layers

4.7 Models based on flux tube instabilities

4.7.1 From instability to -effect

4.7.2 Representative solutions

4.7.3 Critical assessment

4.8 Babcock–Leighton models

4.8.1 Formulation of a poloidal source term

4.8.2 Representative results

4.8.3 Critical assessment

4.9 Numerical simulations of solar dynamo action

Living Rev. Solar Phys. 7, (2010), 3
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