List of Figures

View Image Figure 1:
Parker’s view of cyclonic turbulence twisting a toroidal magnetic field (here ribbons pointing in direction η) into meridional planes [ξ,ζ] (reproduced from Figure 1 of Parker, 1955).
View Image 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).
View Image 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 External Linkhttp://solarscience.msfc.nasa.gov/images/bfly.gif).
View Image Figure 4:
Synoptic magnetogram of the radial component of the solar surface magnetic field. The low-latitude component is associated with sunspots. Note the polarity reversal of the high-latitude magnetic field, occurring approximately at time of sunspot maximum (courtesy of D. Hathaway, NASA/MSFC; see External Linkhttp://solarscience.msfc.nasa.gov/images/magbfly.jpg).
View Image Figure 5:
Isocontours of angular velocity generated by Equation (15View Equation), with parameter values w ∕R = 0.05, ΩC = 0.8752, a2 = 0.1264, a4 = 0.1591 (Panel A). The radial shear changes sign at colatitude 𝜃 = 55 ∘. Panel B shows the corresponding angular velocity gradients, together with the total magnetic diffusivity profile defined by Equation (17View Equation) (dash-dotted line). The core-envelope interface is located at r∕R ⊙ = 0.7 (dotted lines).
View Image Figure 6:
Radial variation of the α-effect for the family of αΩ mean-field models considered in Section 4.2.6. The magnetic diffusivity profile is plotted in red, and the core-envelope interface as a dotted line.
Watch/download Movie Figure 7: (mpg-Movie; 2349 KB)
Movie: Meridional plane animations of various α Ω dynamo solutions using different latitudinal profiles and sign for the α-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 B, yellow to red for positive B), on which poloidal fieldlines are superimposed (blue for clockwise-oriented fieldlines, orange for counter-clockwise orientation). The dashed line is the core-envelope interface at rc∕R = 0.7. Time-latitude “butterfly” diagrams for these three solutions are plotted in Figure 8.
Watch/download Movie Figure 7: (mpg-Movie; 2349 KB)
Movie: Meridional plane animations of various α Ω dynamo solutions using different latitudinal profiles and sign for the α-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 B, yellow to red for positive B), on which poloidal fieldlines are superimposed (blue for clockwise-oriented fieldlines, orange for counter-clockwise orientation). The dashed line is the core-envelope interface at rc∕R = 0.7. Time-latitude “butterfly” diagrams for these three solutions are plotted in Figure 8.
Watch/download Movie Figure 7: (mpg-Movie; 2349 KB)
Movie: Meridional plane animations of various α Ω dynamo solutions using different latitudinal profiles and sign for the α-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 B, yellow to red for positive B), on which poloidal fieldlines are superimposed (blue for clockwise-oriented fieldlines, orange for counter-clockwise orientation). The dashed line is the core-envelope interface at rc∕R = 0.7. Time-latitude “butterfly” diagrams for these three solutions are plotted in Figure 8.
View Image Figure 8:
Northern hemisphere time-latitude (“butterfly”) diagrams for the three αΩ dynamo solutions of Figure 7, constructed at the depth rc∕R ⊙ = 0.7 corresponding to the core-envelope interface. Isocontours of toroidal field are normalized to their peak amplitudes, and plotted for increments ΔB ∕max (B ) = 0.2, with yellow-to-red (green-to-blue) contours corresponding to B > 0 (< 0). The assumed latitudinal dependency of the α-effect is given above each panel. Other model ingredients as in Figure 5. Note the co-existence of two distinct cycles in the solution shown in Panel B.
View Image Figure 9:
A representative interface dynamo model in spherical geometry. This solution has C Ω = 2.5 × 105, Cα = +10, and a core-to-envelope 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, α-effect, and radial shear are also shown, again at latitude 15°. The core-envelope interface is again at r∕R ⊙ = 0.7 (dotted line), where the magnetic diffusivity varies near-discontinuously. Panels C and D show the variations of the core-to-envelope peak toroidal field strength and dynamo period with the diffusivity contrast, for a sequence of otherwise identical dynamo solutions.
View Image Figure 10:
Streamlines of meridional circulation (Panel A), together with the total magnetic diffusivity profile defined by Equation (17View Equation) (dash-dotted line) and a mid-latitude radial cut of u 𝜃 (bottom panel). The dotted line is the core-envelope interface. This is the analytic flow of van Ballegooijen and Choudhuri (1988), with parameter values m = 0.5, p = 0.25, q = 0, and rb = 0.675.
Watch/download Movie Figure 11: (mpg-Movie; 2347 KB)
Movie: Meridional plane animations for an α Ω dynamo solutions including meridional circulation. With Rm = 103, this solution is operating in the advection-dominated regime as a flux-transport dynamo. The corresponding time-latitude “butterfly” diagram is plotted in Figure 12C below. Color-coding of the toroidal magnetic field and poloidal fieldlines as in Figure 7.
View Image Figure 12:
Time-latitude “butterfly” diagrams for the α-quenched αΩ 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, 11 2 −1 ηT = 5 × 10 cm s, Rm = 200 would corresponds to a solar-like circulation speed.
View Image Figure 13:
Time-latitude 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 αΩ with α ∼ cos𝜃 as in Figures 8A and 12. Note the marked increased of the peak surface field strength as Rm exceeds ∼ 100.
View Image Figure 14:
Time-latitude “butterfly” diagrams of the toroidal field at the core-envelope 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 ∼ π, but other parameter settings can bring this lag closer to the observed π ∕2 (diagrams kindly provided by M. Dikpati).
View Image Figure 15:
Stability diagram for toroidal magnetic flux tubes located in the overshoot layer immediately beneath the core-envelope interface. The plot shows contours of growth rates in the latitude-field 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. Ferriz-Mas).
View Image 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 n must first be advected down to the core-envelope interface (dotted line) before production of the toroidal field for cycle n + 1 can take place (B→C). Buoyant rise of flux rope to the surface (C→D) is a process taking place on a much shorter timescale.
Watch/download Movie Figure 17: (mpg-Movie; 6019 KB)
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 advection-dominated αΩ solution of Section 4.4, but now with the non-local surface source term, as formulated in Charbonneau et al. (2005), and parameter values C α = 5, C Ω = 5 × 104, Δη = 0.003, Rm = 840. Note again the strong amplification of the surface polar fields, the latitudinal stretching of poloidal fieldlines by the meridional flow at the core-envelope interface.
View Image Figure 18:
Time-latitude diagrams of the toroidal field at the core-envelope 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 solar-like differential rotation and meridional circulation, the latter here closing at the core-envelope interface. The core-to-envelope contrast in magnetic diffusivity is Δ η = 1∕300, the envelope diffusivity 11 2 −1 ηT = 2.5 × 10 cm s, and the (poleward) mid-latitude surface meridional flow speed is u0 = 16 m s− 1.
View Image Figure 19:
Time-latitude diagrams of the toroidal field at the core-envelope 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 core-to-envelope contrast in magnetic diffusivity is Δ η = 1∕100, the envelope diffusivity ηT = 1011 cm2 s− 1, and the (poleward) mid-latitude surface meridional flow speed is u = 13 m s−1 0 (figure produced from numerical data kindly provided by G. Guerrero).
Watch/download Movie Figure 20: (flv-Movie; 9247 KB)
Movie: Latitude-Longitude Mollweide projection of the toroidal magnetic component at depth r∕R = 0.695 in the 3D MHD simulation of Ghizaru et al. (2010). This large-scale axisymmetric component shows a well-defined overall antisymmetry about the equatorial plane, and undergoes polarity reversals approximately every 30 yr. The animation spans a little over three half-cycles, including three polarity reversals. Time is given in solar days, with 1 s.d. = 30 d.
View Image Figure 21:
(A) Time-latitude diagram of the zonally-averaged toroidal magnetic component the core-envelope interface (r∕R = 0.718) and (B) corresponding time-latitude 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.
View Image 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 13-month running mean in red). Cycles are numbered after the convention introduced in the mid-nineteenth century by Rudolf Wolf. Note how cycles vary significantly in both amplitude and duration. Panel B is a portion of the 10Be 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 higher-than-average and lower-that-average cycle amplitudes (Gnevyshev–Ohl Rule, sometimes also referred to as the “odd-even effect”).
View Image Figure 23:
Amplitude and parity modulation in a 1D slab dynamo model including magnetic backreaction on the differential rotation. These are the usual time-latitude 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).
View Image Figure 24:
Two bifurcation diagrams for a kinematic Babcock–Leighton model, where amplitude fluctuations are produced by time-delay feedback. The top diagram is computed using the one-dimensional iterative map given by Equations (40View Equation, 41View Equation), 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 pn = 0.
View Image Figure 25:
Effect of stochastic fluctuations in the C α dynamo number on an advection-dominated α Ω mean-field dynamo solution including meridional circulation (see Figure 11), here with Rm = 2500, 5 C Ω = 5 × 10, Cα = 0.5, and Δη = 0.1. The fluctuation amplitude is δC α∕C α = 1, and the correlation time of the imposed fluctuations amounts to about 5% of the mean half-cycle 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.
View Image 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 (vp), the coarse time series of vp 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 piecewise-constant 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 pseudo-SSN time series plotted in Panel C (figure produced from numerical data kindly provided by D. Passos).
View Image 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 9.6 ≲ t ≲ 10.2, and a “failed minimum” at t ≃ 11 (figure produced from numerical data kindly provided by M. Ossendrijver).
View Image 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 (r,𝜃) = (0.7,π∕3). The bottom panel is a time-latitude diagram for the toroidal field at the core-envelope interface (numerical data from Charbonneau et al., 2004).