Figure 1:
Parker’s view of cyclonic turbulence twisting a toroidal magnetic field (here ribbons pointing in direction ) into meridional planes (reproduced from Figure 1 of Parker, 1955). 

Figure 2:
The Babcock–Leighton mechanism of poloidal field production from the decay of bipolar active regions showing opposite polarity patterns in each solar hemisphere (reproduced from Figure 8 of Babcock, 1961). 

Figure 3:
The sunspot “butterfly diagram”, showing the fractional coverage of sunspots as a function of solar latitude and time (courtesy of D. Hathaway, NASA/MSFC; see http://solarscience.msfc.nasa.gov/images/bfly.gif). 

Figure 4:
Synoptic magnetogram of the radial component of the solar surface magnetic field. The lowlatitude component is associated with sunspots. Note the polarity reversal of the highlatitude magnetic field, occurring approximately at time of sunspot maximum (courtesy of D. Hathaway, NASA/MSFC; see http://solarscience.msfc.nasa.gov/images/magbfly.jpg). 

Figure 5:
Isocontours of angular velocity generated by Equation (15), with parameter values , , , (Panel A). The radial shear changes sign at colatitude . Panel B shows the corresponding angular velocity gradients, together with the total magnetic diffusivity profile defined by Equation (17) (dashdotted line). The coreenvelope interface is located at (dotted lines). 

Figure 6:
Radial variation of the effect for the family of meanfield models considered in Section 4.2.6. The magnetic diffusivity profile is plotted in red, and the coreenvelope interface as a dotted line. 

Figure 7:
Movie: Meridional plane animations of various dynamo solutions using different latitudinal profiles and sign for the effect, as labeled. The polar axis coincides with the left quadrant boundary. The toroidal field is plotted as filled contours (constant increments, green to blue for negative , yellow to red for positive ), on which poloidal fieldlines are superimposed (blue for clockwiseoriented fieldlines, orange for counterclockwise orientation). The dashed line is the coreenvelope interface at . Timelatitude “butterfly” diagrams for these three solutions are plotted in Figure 8. 

Figure 7:
Movie: Meridional plane animations of various dynamo solutions using different latitudinal profiles and sign for the effect, as labeled. The polar axis coincides with the left quadrant boundary. The toroidal field is plotted as filled contours (constant increments, green to blue for negative , yellow to red for positive ), on which poloidal fieldlines are superimposed (blue for clockwiseoriented fieldlines, orange for counterclockwise orientation). The dashed line is the coreenvelope interface at . Timelatitude “butterfly” diagrams for these three solutions are plotted in Figure 8. 

Figure 7:
Movie: Meridional plane animations of various dynamo solutions using different latitudinal profiles and sign for the effect, as labeled. The polar axis coincides with the left quadrant boundary. The toroidal field is plotted as filled contours (constant increments, green to blue for negative , yellow to red for positive ), on which poloidal fieldlines are superimposed (blue for clockwiseoriented fieldlines, orange for counterclockwise orientation). The dashed line is the coreenvelope interface at . Timelatitude “butterfly” diagrams for these three solutions are plotted in Figure 8. 

Figure 8:
Northern hemisphere timelatitude (“butterfly”) diagrams for the three dynamo solutions of Figure 7, constructed at the depth corresponding to the coreenvelope interface. Isocontours of toroidal field are normalized to their peak amplitudes, and plotted for increments , with yellowtored (greentoblue) contours corresponding to (). The assumed latitudinal dependency of the effect is given above each panel. Other model ingredients as in Figure 5. Note the coexistence of two distinct cycles in the solution shown in Panel B. 

Figure 9:
A representative interface dynamo model in spherical geometry. This solution has , , and a coretoenvelope diffusivity contrast of 10^{–2}. Panel A shows a sunspot butterfly diagram, and Panel B a series of radial cuts of the toroidal field at latitude 15°. The (normalized) radial profiles of magnetic diffusivity, effect, and radial shear are also shown, again at latitude 15°. The coreenvelope interface is again at (dotted line), where the magnetic diffusivity varies neardiscontinuously. Panels C and D show the variations of the coretoenvelope peak toroidal field strength and dynamo period with the diffusivity contrast, for a sequence of otherwise identical dynamo solutions. 

Figure 10:
Streamlines of meridional circulation (Panel A), together with the total magnetic diffusivity profile defined by Equation (17) (dashdotted line) and a midlatitude radial cut of (bottom panel). The dotted line is the coreenvelope interface. This is the analytic flow of van Ballegooijen and Choudhuri (1988), with parameter values , , , and . 

Figure 11:
Movie: Meridional plane animations for an dynamo solutions including meridional circulation. With Rm = 10^{3}, this solution is operating in the advectiondominated regime as a fluxtransport dynamo. The corresponding timelatitude “butterfly” diagram is plotted in Figure 12C below. Colorcoding of the toroidal magnetic field and poloidal fieldlines as in Figure 7. 

Figure 12:
Timelatitude “butterfly” diagrams for the quenched solutions depicted earlier in Panel A of Figure 8, except that meridional circulation is now included, with (A) Rm = 50, (B) Rm = 100, (C) Rm = 1000, and (D) Rm = 2000 For the turbulent diffusivity value adopted here, , Rm = 200 would corresponds to a solarlike circulation speed. 

Figure 13:
Timelatitude diagrams of the surface radial magnetic field, for increasing values of the circulation speed, as measured by the Reynolds number Rm. This is for the same reference with as in Figures 8A and 12. Note the marked increased of the peak surface field strength as Rm exceeds 100. 

Figure 14:
Timelatitude “butterfly” diagrams of the toroidal field at the coreenvelope interface (top), and surface radial field (bottom) for a representative dynamo solution computed using the model of Dikpati and Gilman (2001). Note how the deep toroidal field peaks at very low latitudes, in good agreement with the sunspot butterfly diagram. For this solution the equatorial deep toroidal field and polar surface radial field lag each other by , but other parameter settings can bring this lag closer to the observed (diagrams kindly provided by M. Dikpati). 

Figure 15:
Stability diagram for toroidal magnetic flux tubes located in the overshoot layer immediately beneath the coreenvelope interface. The plot shows contours of growth rates in the latitudefield strength plane. The gray scale encodes the azimuthal wavenumber of the mode with largest growth rate, and regions left in white are stable. Dynamo action is associated with the regions with growth rates 1 yr, here labeled I and II (diagram kindly provided by A. FerrizMas). 

Figure 16:
Operation of a solar cycle model based on the Babcock–Leighton mechanism. The diagram is drawn in a meridional quadrant of the Sun, with streamlines of meridional circulation plotted in blue. Poloidal field having accumulated in the surface polar regions (“A”) at cycle must first be advected down to the coreenvelope interface (dotted line) before production of the toroidal field for cycle can take place (BC). Buoyant rise of flux rope to the surface (CD) is a process taking place on a much shorter timescale. 

Figure 17:
Movie: Meridional plane animation of a representative Babcock–Leighton dynamo solution from Charbonneau et al. (2005). Color coding of the toroidal field and poloidal fieldlines as in Figure 7. This solution uses the same differential rotation, magnetic diffusivity, and meridional circulation profile as for the advectiondominated solution of Section 4.4, but now with the nonlocal surface source term, as formulated in Charbonneau et al. (2005), and parameter values , , , . Note again the strong amplification of the surface polar fields, the latitudinal stretching of poloidal fieldlines by the meridional flow at the coreenvelope interface. 

Figure 18:
Timelatitude diagrams of the toroidal field at the coreenvelope interface (Panel A), and radial component of the surface magnetic field (Panel B) in a Babcock–Leighton model of the solar cycle. This solution is computed for solarlike differential rotation and meridional circulation, the latter here closing at the coreenvelope interface. The coretoenvelope contrast in magnetic diffusivity is , the envelope diffusivity , and the (poleward) midlatitude surface meridional flow speed is . 

Figure 19:
Timelatitude diagrams of the toroidal field at the coreenvelope interface (Panel A), and radial component of the surface magnetic field (Panel B) in a Babcock–Leighton model of the solar cycle with a meridional flow restricted to the upper half of the convective envelope, and including (parametrized) radial and latitudinal turbulent pumping. This is a solution from Guerrero and de Gouveia Dal Pino (2008) (see their Section 3.3 and Figure 5), but the overall modelling framework is almost identical to that described earlier, and used to generate Figure 18. The coretoenvelope contrast in magnetic diffusivity is , the envelope diffusivity , and the (poleward) midlatitude surface meridional flow speed is (figure produced from numerical data kindly provided by G. Guerrero). 

Figure 20:
Movie: LatitudeLongitude Mollweide projection of the toroidal magnetic component at depth in the 3D MHD simulation of Ghizaru et al. (2010). This largescale axisymmetric component shows a welldefined overall antisymmetry about the equatorial plane, and undergoes polarity reversals approximately every 30 yr. The animation spans a little over three halfcycles, including three polarity reversals. Time is given in solar days, with 1 s.d. = 30 d. 

Figure 21:
(A) Timelatitude diagram of the zonallyaveraged toroidal magnetic component the coreenvelope interface () and (B) corresponding timelatitude diagram of the surface radial field, in the 3D MHD simulations presented in Ghizaru et al. (2010). Note the regular polarity reversals, the weak but clear tendency towards equatorial migration of the deep toroidal magnetic component, and the good coupling between the two hemispheres despite marked fluctuations in successive cycles. The color scale codes the magnetic field strength, in Tesla. 

Figure 22:
Fluctuations of the solar cycle, as measured by the sunspot number. Panel A is a time series of the Zürich monthly sunspot number (with a 13month running mean in red). Cycles are numbered after the convention introduced in the midnineteenth century by Rudolf Wolf. Note how cycles vary significantly in both amplitude and duration. Panel B is a portion of the ^{10}Be time series spanning the Maunder Minimum (data courtesy of J. Beer). Panel C shows a time series of the yearly group sunspot number of Hoyt and Schatten (1998) (see also Hathaway et al., 2002) over the same time interval, together with the yearly Zürich sunspot number (purple) and auroral counts (green). Panels D and E illustrate the pronounced anticorrelation between cycle amplitude and rise time (Waldmeier Rule), and alternation of higherthanaverage and lowerthataverage cycle amplitudes (Gnevyshev–Ohl Rule, sometimes also referred to as the “oddeven effect”). 

Figure 23:
Amplitude and parity modulation in a 1D slab dynamo model including magnetic backreaction on the differential rotation. These are the usual timelatitude diagrams for the toroidal magnetic field, now covering both solar hemispheres, and exemplify the two basic types of modulation arising in nonlinear dynamo models with backreaction on the differential rotation (see text; figure kindly provided by S.M. Tobias). 

Figure 24:
Two bifurcation diagrams for a kinematic Babcock–Leighton model, where amplitude fluctuations are produced by timedelay feedback. The top diagram is computed using the onedimensional iterative map given by Equations (40, 41), while the bottom diagram is reconstructed from numerical solutions in spherical geometry, of the type discussed in Section 4.8. The shaded area in Panel A maps the attraction basin for the cyclic solutions, with initial conditions located outside of this basin converging to the trivial solution . 

Figure 25:
Effect of stochastic fluctuations in the dynamo number on an advectiondominated meanfield dynamo solution including meridional circulation (see Figure 11), here with , , , and . The fluctuation amplitude is , and the correlation time of the imposed fluctuations amounts to about 5% of the mean halfcycle period. Panel A shows a portion of the time series of total magnetic energy (red), used here as a proxy for cycle amplitude, and of the surface polar field strength (green), both scaled to their peak value over the full simulation run. Panel B shows a correlation plot of cycle amplitude and duration, both now normalized to their respective means over the simulation interval. Panel C snows a correlation plot of cycle amplitude versus the preceding peak value of the surface polar field. 

Figure 26:
Effect of persistent variations in meridional circulation on the amplitude of the solar cycle, as modeled by Lopes and Passos (2009). Panel A shows the signed square root of the sunspot number (gray), here used as a proxy of the solar internal magnetic field. A smoothed version of this time series (black) is fitted, one magnetic cycle at a time (green), with the equilibrium solution of the truncated dynamo model of Passos and Lopes (2008); assuming that variations in the fitting parameters are due to variations in the meridional flow speed (), the coarse time series of of panel B (in green) is obtained, scaled to the magnetic cycle 1 value and with error bars from the fitting procedure. Input of this piecewiseconstant meridional flow variation (scaled down by a factor of two, in red in panel B) in the 2D Babcock–Leighton dynamo model of Chatterjee et al. (2004) yields the pseudoSSN time series plotted in Panel C (figure produced from numerical data kindly provided by D. Passos). 

Figure 27:
Intermittency in a dynamo model based on flux tube instabilities (cf. Sections 3.2.3 and 4.7). The top panel shows a trace of the toroidal field, and the bottom panel is a butterfly diagram covering a shorter time span including a quiescent phase at , and a “failed minimum” at (figure produced from numerical data kindly provided by M. Ossendrijver). 

Figure 28:
Intermittency in a dynamo model based on the Babcock–Leighton mechanism (cf. Sections 3.2.4 and 4.8). The top panel shows a trace of the toroidal field sampled at . The bottom panel is a timelatitude diagram for the toroidal field at the coreenvelope interface (numerical data from Charbonneau et al., 2004). 
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Living Rev. Solar Phys. 7, (2010), 3
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