List Of Figures

View Image Figure 1:
Parker’s view of cyclonic turbulence twisting a toroidal magnetic field (here ribbons pointing in direction j) into meridional planes [q,z] (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).
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 weaker, high-latitude magnetic field, occurring approximately at time of sunspot maximum (courtesy of D. Hathaway).
View Image Figure 5:
Isocontours of angular velocity generated by Equation (17View Equation), with parameter values w/R = 0.05, _O_C = 0.8752, a2 = 0.1264, a4 = 0.1591 (Panel A). The radial shear changes sign at colatitude h = 55o. Panel B shows the corresponding angular velocity gradients, together with the total magnetic diffusivity profile defined by Equation (19View Equation) (dash-dotted line). The core-envelope interface is located at r/Ro . = 0.7 (dotted lines).
View Image Figure 6:
Radial variations of the a-effect for the two classes of mean-field models considered in Section 4.2.5. The magnetic diffusivity profile is again indicated by a dash-dotted line, and the core-envelope interface by a dotted line.
View Image Figure 7:
Northern hemisphere time-latitude (“butterfly”) diagrams for a selection of a_O_ dynamo solutions, constructed at the depth rc/Ro . = 0.7 corresponding to the core-envelope interface. Isocontours of toroidal field are normalized to their peak amplitudes, and plotted for increments DB/ max(B) = 0.2, with yellow-to-red (green-to-blue) contours corresponding to B > 0 (< 0). All but the first solution have the a-effect concentrated at the base of the envelope, with a latitude dependence as 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. Four corresponding animations are available in Resource 1.
View Image Figure 8:
Time series of magnetic energy in four mean-field a_O_ dynamo solutions. Panels A and B show the time series associated with the solutions shown in Panels B and D of Figure 7 (base CZ, 2 a ~ sin h cosh, Ca = ± 10). Note the initial phase of exponential growth of the magnetic field, followed by a saturation phase characterized here by an (atypical) periodic modulation in the case of the solution in Panel A. Panels C and D show time series for two interface dynamo solutions (see Section 4.3 and Figure 10) for two diffusivity ratios. The energy scale is expressed in arbitrary units, but is consistent across all four panels.
View Image Figure 9:
Three interface dynamo models in semi-infinite Cartesian geometry. In Parker’s original model, the a-effect occupies the space z > 0 and the radial shear z < 0, with the magnetic diffusivity varying discontinuously from (low) j 1 to (high) j 2 across the z = 0 surface. The MacGregor and Charbonneau model is similar, except that the shear and a-effect are spatially localized as d-functions, located a finite distance away from z = 0 in their respective halves of the domain. The Zhang et al. model moves one step closer to the Sun by truncating vertically the Parker model, and sandwiching the dynamo layers between a vacuum layer (top) and very low diffusivity layer (bottom); this bottom layer represents the radiative core, while the shear region is then associated with the tachocline. Each one of these models supports travelling waves solutions of the form exp(ikx - wt), vertically localized about the core-envelope interface (z = 0).
View Image Figure 10:
A representative interface dynamo model in spherical geometry. This solution has C_O_ = 2.5× 105, Ca = +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 o 15. The (normalized) radial profiles of magnetic diffusivity, a-effect, and radial shear are also shown, again at latitude o 15. The core-envelope interface is again at r/Ro . = 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. Corresponding animations are available in Resource 2.
View Image Figure 11:
Streamlines of meridional circulation (Panel A), together with the total magnetic diffusivity profile defined by Equation (19View Equation) (dash-dotted line) and a mid-latitude radial cut of uh (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.
View Image Figure 12:
Time-latitude diagrams for three of the a_O_ solutions depicted earlier in Panels B to D of Figure 7, except that meridional circulation is now included, with Rm = 50 (top row), Rm = 200 (middle row), and Rm = 103 (bottom row). For the turbulent diffusivity value adopted here, jT = 5 × 1011 cm2 s-1, Rm = 200 corresponds to a solar-like circulation speed. Corresponding animations are available in Resource 3.
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 an a_O_ solution with the a-effect concentrated at low-latitude and at the base of the convective envelope (see Section 4.2.5 and Panel B of Figure 7). Recall that the Rm = 0 solution in Panel A exhibits amplitude modulation (cf. Panel B of Figure 7 and Panel A of Figure 8).
View Image Figure 14:
A sample of longitudinally-averaged kinetic helicity profiles associated with the linearly unstable horizontal planforms of azimuthal order m (as indicated) in the shallow-water model of Dikpati and Gilman (2001). The parameters s2 and s4 control the form of the latitudinal differential rotation, and are equivalent to the parameters a2 and a4 in Equation (18View Equation) herein. The parameter G is a measure of stratification in the shallow-water model, with larger values of G corresponding to stronger stratification (and thus a stronger restoring buoyancy force); G -~ 0.1 is equivalent to a subadiabaticity of - 4 ~ 10 (diagram kindly provided by M. Dikpati).
View Image Figure 15:
Time-latitude “butterfly” diagrams of the toroidal field at the core-envelope interface (left), and surface radial field (right) 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 ~ p, but other parameter settings can bring this lag closer to the observed p/2 (diagrams kindly provided by M. Dikpati).
View Image Figure 16:
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 17:
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.
View Image Figure 18:
Time-latitude diagrams of the surface 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 Dj = 1/300, the envelope diffusivity 11 2 -1 jT = 2.5 × 10 cm s, and the (poleward) mid-latitude surface meridional flow speed is u0 = 16 m s-1.
View Image Figure 19:
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 alternance 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 20:
Amplitude and parity modulation in a 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 21:
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 22:
Stochastic fluctuations of the dynamo number in an a_O_ mean-field dynamo solution. The reference, unperturbed solution is the same as that plotted in Panel D of Figure 7, except that is uses a lower value for the dynamo number, Ca = -5. Panel A shows magnetic energy time series for three solutions with increasing fluctuation amplitudes, while Panel B shows a correlation plot of cycle amplitude and duration, as extracted from a time series of the toroidal field at the core-envelope interface (r/R = 0.7) in the model. Solutions are color-coded according to the relative amplitude dCa/Ca of the fluctuations in the dynamo number. Line segments in Panel B indicate the mean cycle amplitudes and durations for the three solutions. The correlation time of the noise amounts here to about 5% of the mean half-cycle period in all cases.
View Image Figure 23:
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 24:
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,h) = (0.7,p/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)).