Flux transport dynamos

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Flux transport dynamos are one specific variety of hydromagnetic dynamos models for the Sun and stars. They provide a modelling framework for the spatiotemporal evolution of the Sun's large-scale magnetic field (spatial scales commensurate with the solar radius), on long timecales (many months to century and millenia). Their two primary defining features are:

  • the observed equatorward migration of sunspot source region and poleward migration of surface fields are both driven by the "conveyor belt" action of the meridional flow;
  • the cycle period is primarily set by the meridional flow speed.

Flux transport dynamos reproduce fairly well and robustly many observed solar cycle features, and are relatively well-constrained observationally. Their dynamical behavior, driven at least in part by a time-delay in the dynamo regenerative loop, also offers a natural explanation for the significant fluctuations observed in overall activity levels on decadal to millenial timescales.


Flux transport dynamos and the solar cycle

The solar magnetic field is the energy channel and dynamical engine driving solar activity, including all geoeffective solar eruptive phenomena. It is observed to evolve on a wide variety of spatial and temporal scales, the most prominent being the so-called solar cycle, first noted in the systematic variations of sunspot counts (see Figure <ref>F1</ref>), but now understood to be driven by a cyclic variation of the large-scale component of the solar magnetic field, involving polarity reversals occurring approximately every 11 years. This short cycle period points to a contemporaneously regenerated magnetic field, most likely by a hydromagnetic dynamo. This process is described by the equations of magnetohydrodynamics (MHD). Unfortunately, direct numerical simulations of the MHD equations are still a long way from reaching the physical parameter regime characterizing solar interior conditions. Current solar cycle models are therefore based on reduced forms of MHD fluid equations, such as mean-field hydromagnetic dynamo models, of which flux transport dynamos are one particularly promising variety.

Figure 1: The solar cycle, as manifested by variations in the sunspot number (SSN) index, three different reconstructions being plotted here. The 1755-1767 sunspot cycle is traditionally numbered "1", even if reconstructions have now been pushed to the early seventeenth century. While the cycle has a mean period of about 11 yr (22 if magnetic polarity is taken into account), both its amplitude and period are subjected to significant variations. Note in particular that the lack of sunspots in the 1645-1710 time period, known as the Maunder Minimum, is real, rather than an artefact due to incomplete data.

A representative flux transport solar cycle model

Observations suggest that the dynamo loop for the large-scale magnetic field operates as a two-step process. The production of the strong toroidal (i.e. azimuthally-directed) magnetic component ultimately giving rise to sunspots is produced by shearing of the poloidal component by differential rotation. This is now believed to occur primarily deep in the solar interior, at or immediately beneath the core-envelope interface. The second step in the dynamo loop, producing a poloidal component with reversed polarity from the toroidal component, is more uncertain. One promising mechanism is the release into the solar photosphere of magnetic flux liberated by the decay of sunspots and active regions, with subsequent transport and accumulation at high solar latitudes (Babcock 1961; Leighton 1969; Wang et al. 1989). This process is actually observed, so in what follows the focus is placed on dynamo models relying on this poloidal field regeneration mechanism, because they are arguably the best exemplar of flux transport dynamos.

The basic operation of a flux transport dynamo based on the surface decay of sunspots is illustrated schematically on Figure <ref>F2</ref>. Sunspots currently emerging at the solar surface (B) are formed from the toroidal field Tn currently being produced at cycle n in the solar interior at (A). The decay products of these sunspots are carried by the combined diffusion-like action of the turbulent convective flow and the large-scale meridional circulation to high latitudes (C), where they will eventually lead to the reversal of the poloidal field Pn-1 having built up during the preceeding sunspot cycle. Over time, this new poloidal field is transported into the interior, eventually ending up at (A) again, where the production of the subsequent cycle's toroidal field (Tn+1) can commence.

Figure 2: Schematic operation of a flux transport dynamo based on the surface decay of sunspots. The diagram shows a meridional quadrant of the solar interior, with representative meridional flow streamlines plotted in yellow. The flow is poleward at the surface and equatorward at the base of the envelope. The interface between the Sun's convective envelope and underlying radiative core is located a fractional radius 0.7, as indicated by the dotted line.

The form and magnitude of the differential rotation responsible for shearing the poloidal field in a toroidal component at the core-envelope interface is known from helioseismology. The meridional flow is directly measured at the solar surface, detected and measured by helioseismology in the outer half of the solar convective envelope, and relatively well-constrained by theoretical considerations further below. It serves as a "conveyor belt", setting the cycle period by transporting the poloidal component from its source region at the surface to the interior, and gradually displacing equatorward the toroidal field produced at the core-envelope interface (Wang et al. 1991; Choudhuri et al. 1995).

Reduction to a one-dimensional iterative map

One dynamically interesting aspect of the dynamo scenario illustrated on Figure <ref>F2</ref> is that there is an unavoidable time delay built into the dynamo loop, as a consequence of the spatial segregation of the two magnetic source regions. The transport of the surface poloidal field to the core-envelope interface requires a time of the order of the circulation's turnover time, which is of a few decades according to current estimates. The destabilisation and buoyant rise of the toroidal component to the surface (red dashed arrow on Fig.<ref>F2</ref>) is, in comparison, a rapid process. This leads to a time-delay in the dynamo loop, commensurate with the circulation turnover time and thus cycle period: the poloidal field produced at the surface at cycle n is not the source of poloidal field for the current cycle, but will serve as a source of toroidal field for a subsequent cycle a decade or two later.

The dynamical consequences of this long time delay can be explored with the following simple model (Durney 2000; Charbonneau 2001): Going with the scenario illustrated in Fig.<ref>F2</ref>, assume that the toroidal field Tn is linearly proportional to the poloidal field strength Pn-1 of the previous cycle. Production of the current cycle's poloidal field Pn, on the other hand, is understood to be a nonlinear function of the current toroidal field Tn, production of Pn being possible only in a finite range of toroidal field strengths. Writing \(P_n\propto T_n(1-T_n)\), and absorbing the proportionality constants into scaled forms of the amplitudes, leads to:

<math Eq4>

p_{n+1}=a\,p_n^2(1-p_n) </math>

Equation <ref>Eq4</ref> is a one-dimensional cubic iterative map for the amplitude pn of cycle n, where the map parameter a plays the role of a dynamo number. As it is increased, the amplitude iterate undergoes a transition to chaos via a sequence of period-doubling bifurcations characteristic of such single-humped map (see Figure <ref>F3</ref>A).

Figure 3: Part (A) shows a bifurcation diagram for the one-dimensional iterative map embodying the time delay dynamics of flux transport dynamo models based on the surface decay of sunspots and active regions. The gray shading indicates the basin of attraction. Part (B) shows a bifurcation diagram constructed by measuring cycle amplitudes in a sequence of magnetic energy time series reconstructed by numerical 2D kinematic dynamo solutions of a conceptually similar solar cycle model (see Charbonneau et al. 2005). Magnetic energy is used here as a proxy of cycle amplitude.

The attractor has here a finite-sized basin of attraction, indicated by gray shading on Fig. <ref>F3</ref>A. This can lead to intermittency in the presence of low-amplitude stochastic forcing. This is in fact expected in the present context, with (observed) stochastic fluctuations in sunspot emergence rates leading effectively to fluctuations in the map parameter, and small-scale turbulent dynamo action producing small-scale magnetic fields, acting as a form of additive perturbation to the cycle amplitude. Consider then the following stochastically forced map:

<math Eq5>

p_{n+1}=a_n\,p_n^2(1-p_n)+e_n\,\qquad e_n\in [0,E)\,\qquad E\ll 1 </math>

where the map parameter an and additive noise amplitude en are extracted anew at each iteration from some preset distribution. Time series of cycle amplitudes produced with this stochastically forced map are characterized by bursting phase (\(p_n\sim 1\)), corresponding to "normal" cyclic behavior, punctuated by quiescent epochs (\(p_n\ll 1\)) of irregular spacing and duration; these are the map's equivalent to Maunder Minimum epochs of strongly depressed sunspot counts (see Fig.<ref>F1</ref>).

An interesting property of stochastically forced iterative maps such as the ones use here is that, upon being perturbed, relaxation to the attractor is oscillatory over much of the periodic and multiperiodic domain. This translates, in the time series of cycle amplitudes, into more-or-less regular alternance between higher-than-average and lower-than-average cycle amplitudes. Such a pattern is actually observed in the SSN time series, and is known as the Gnevyshev-Ohl Rule in the solar physics literature.

Numerical models

Numerical flux transports models of the solar cycle are typically constructed by solving sets of coupled partial differential equations for the magnetic field's axisymmetric toroidal and poloidal components, the latter being usually represented via a toroidal vector potential; in spherical polar coordinates:

<math nm1>

\mathbf{B}(r,\theta,t)=\nabla\times (A(r,\theta,t){\hat \phi}) +B(r,\theta,t){\hat \phi} </math>

The MHD induction equation can then be separated as:

<math nm2>

{\partial A\over\partial t}= \eta\left(\nabla^2-{1\over\varpi^2}\right)A -{\mathbf{u}_p\over\varpi}\cdot\nabla (\varpi A) +S(r,\theta,B)~, </math>

<math nm3>

{\partial B\over \partial t}= \eta\left(\nabla^2-{1\over\varpi^2}\right)B +{1\over\varpi}{\partial (\varpi B)\over\partial r}{\partial\eta\over\partial r} -\varpi\nabla\cdot \left({B\,\mathbf{u}_p\over\varpi}\right) +\varpi(\nabla\times (A{\hat \phi}))\cdot\nabla\Omega ~. </math>

where \(\varpi=r\sin\theta\), and \(\eta\) is the total magnetic diffusivity, usually taken to be a function of depth in the solar interior. The large-scale flow field u (assumed axisymmetric as well) has been divided into two contributions, namely a (differential) rotation \(\Omega(r,\theta)\) and a meridional circulation up. Kinematic models used a preset flow field (see, e.g., Dikpati et al. 2001; Charbonneau et al. 2005), while non-kinematic models solve simultaneously an evolution equation for u (see, e.g., Bushby and Tobias 2007). Shearing of the poloidal field by differential rotation (last term in eq. (<ref>nm3</ref>) acts as the source of toroidal field; the source term \(S(r,\theta,B(t))\) in the poloidal equation (<ref>nm2</ref>) can take many forms, depending on the poloidal field regeneration mechanism under consideration.

Remarkably, numerical solutions of such axisymmetric kinematic flux transport dynamos can many behaviors "predicted" by the simple iterative map formalism described above. Transition to chaos does occur as the dynamo number is increased, and in portions of parameter space this does take place through a sequence of period-doubling bifurcation (see Figure <ref>F3</ref>B). Likewise, stochastically-forced version s of these models do produce intermittency (see Figure <ref>F5</ref>A), as well as Gnevyshev-Ohl like pattern of amplitude fluctuations (see Fig. <ref>F5</ref>B). This indicates that the time delay inherent to the dynamo loop in this type of model can be a primary driver of amplitude modulation. Note that other classes of flux transport dynamos (see, e.g., Dikpati and Gilman 2001; Küker et al. 2001), where magnetic source regions are spatially coincident, do not have such a built-in time-delay, while remaining bona fide flux transport dynamo models.

Figure 4: Intermittency and Ohl-Gnevyshev pattern in a stochastically-forced 2D kinematic flux transport dynamo model. The overall format is identical to Figure <ref>F4</ref> above, except that The time series in (A) is extracted directly from a 2D dynamo simulation (absolute value of the low-latitude deep toroidal field), covering nearly 90 sunspot cycles. The corresponding peak amplitudes are plotted as solid dots on part (B). The solid blue line is a 1-2-1 running mean of the cycle amplitudes. Note the tendency, during active phases, for cycle amplitudes to alternate between higher-than-average and lower-than-average. Illustrative numerical calculations taken from Charbonneau et al. 2004.

As explanatory constructs applicable to the solar cycle, flux transport dynamos models have met with a number of successes:

  • Robust reproduction of the solar cycle period, as set primarily by the turnover time of the meridional flow
  • Robust reproduction of the observed equatorial drift of sunspot belts, and spatiotemporal evolution of surface magnetic field, as produced by the conveyor belt action of the meridional flow
  • Robustness with respect to noise and stochastic perturbations.
  • Robust reproduction of some observed patterns of solar cycle amplitude fluctuations, such as the Gnevyshev-Ohl rule, and irregularly occurring episodes of suppressed sunspot activity.
  • In models relying on the Babcock-Leighton mechanism, the ability to operate in the strong magnetic field regime, unlike classes of models relying on turbulent electromotive force for poloidal field regeneration.

In the solar context, flux transport dynamos models also face a number of remaining difficulties:

  • A tendency to produce strong toroidal fields at high latitudes; this can be traced to the inductive action of the strong radial shear in the polar region of the tachocline, through which the surface fields is carried by meridional circulation following submergence.
  • A tendency to produce equatorially symmetric dynamo solutions in large portions of parameter space, contrary to the antisymmetric parity characterizing the large-scale solar magnetic field.
  • A difficulty in reproducing robustly the (weak) anticorrelation observed to hold between the amplitudes and durations of sunspot cycles (the so-called Waldmeier Rule). This difficulty is shared with other classes of solar cycle models.
  • The requirement of a rather low magnetic diffusivity for proper operation: while the turbulent magnetic diffusivity in the solar interior is currently impossible to measure or compute from first principles, the values used in most flux transport dynamos in the bulk of the convection zone are smaller by one or two orders of magnitude than that required in the surface layers to yield proper surface evolution.
  • A potentially serious problem with the ability (or rather, lack thereof) of the meridional flow to submerge the surface magnetic fields in polar regions: high resolution magnetographic observations have shown that what looks like a large-scale, diffuse magnetic field on synoptic magnetogram is in fact made up of a large number of small concentrations of magnetic flux. It is now at all clear how the (relatively slow) meridional flow can submerge such magnetized structures, in the face of the opposing upward force provided by magnetic buoyancy.

Nonlinearity and predictivity

Because they can, in principle, be amenable to assimilation of solar surface magnetic data, flux transport dynamos of the type described here may prove useful for long-term prediction of overall solar activity levels, which is of great interest in the context of long-term space weather forecasting, and also for quantifying possible solar influences on terrestrial climate. Recent attempts made at forecasting the amplitude of upcoming sunspot cycle 24 using flux transport dynamos have however yielded conflicting predictions, indicating strong sensitivity to the choice of data assimilation scheme (cf. Dikpati et al. 2006; Choudhuri et al. 2007), as well as model details (see Bushby and Tobias 2007). Even if these difficulties can be sorted out, the existence of a chaotic regime guarantees that useful prediction will be restricted to a finite temporal horizon.

Much remains to be learned on the nonlinear behavior of flux transport dynamos, in particular with regards to the backreaction of the magnetic force on both the differential rotation and meridional circulation. Such non-kinematic models are just starting to be explored (see, e.g., Rempel 2006; Bushby and Tobias 2007).


General references on solar cycle and activity:

Lang, K.R., Sun, Earth and Sky, second ed., Springer (2006)

Foukal, P.~V., Solar Astrophysics, second ed., John Wiley & Sons, New York (2004).

Stix, M., The Sun: an introduction, second ed., Springer, New York (2002)

Solar Physics Web Pages at NASA's Marshall Space Flight Center

Reviews of dynamo models of solar cycle:

Charbonneau, P., Dynamo models of the solar cycle, Liv. Rev. Solar Phys., 2, 2 (2005): Online Article

Ossendrijver, M.A.J.H., The solar dynamo, Astr. & Astrophys. Rev., 11, 287-367 (2003): Online Article

Tobias, S.M., The solar dynamo, Phil. Trans. A Roy. Soc. London, 360, 2741-2756 (2002): Online Article

Technical references cited in the text:

Babcock, H.W., The topology of the sun's magnetic field and the 22-year cycle, Astrophys. J., 133, 572-587 (1961): Online Article

Bushby, P.J., and Tobias, S.M., On predicting the solar cycle using mean-field models, Astrophys., J., 661, 1289-1296 (2007): Online Article

Charbonneau, P., Multiperiodicity, chaos and intermittency in a reduced model of the solar cycle, Solar Phys., 199, 385-404} (2001): Online Article

Charbonneau, P., Blais-Laurier, G., and St-Jean-Leblanc, C., Intermittency and phase persistence in a Babcock-Leighton model of the solar cycle, Astrophys., J. Lett., 616, L183-L186 (2004): Online Article

Charbonneau, P., St-Jean-Leblanc, C., and Zacharias, P., Fluctuations in Babcock-Leighton models of the solar cycle. I. period doubling and transition to chaos, Astrophys., J., 619, 613-622 (2005): Online Article

Choudhuri, A.R., Schüssler, M., and Dikpati, M., The solar dynamo with meridional circulation, Astr. Astrophys., 303, L29-L32 (1995): Online Article

Choudhuri, A.R., Chatterjee, P., and Jiang, J., Predicting solar cycle 24 with a solar dynamo model, Phys. Rev. Lett., 98, id 131103 (2007): Online Article

Dikpati, M., and Gilman, P.A., Flux transport dynamos with \(\alpha\)-effect from global instability of tachocline differential rotation: a solution for magnetic parity selection in the sun, Astrophys., J., 559, 428-442 (2001): Online Article

Dikpati, M., deToma, G., and Gilman, P.A., Predicting the strength of solar cycle 24 using a flux transport dynamo-based to ol, Geophys. Res. Lett., 33, L05102 (2006): Online Article

Durney, B.R., On the differences between odd and even solar cycles, Solar Phys., 196, 421-426 (2000): Online Article

Küker, M., Rüdiger, G., and Schulz, M., Circulation-dominated solar shell dynamo models with positive alpha effect, Astr. Astrophys., 374, 301-308 (2001): Online Article

Leighton, R.~B., A magneto-kinematic model of the solar cycle, Astrophys. J., 156, 1-26 (1969): Online Article

Rempel, M., Flux-Transport Dynamos with Lorentz Force Feedback on Differential Rotation and Meridional Flow: Saturation Mechanism and Torsional Oscillations, Astrophys. J., 647, 662-675 (2006): Online Article

Wang, Y.-M., Nash, A.G., and Sheeley, N.R. Jr, Magnetic flux transport on the Sun, Science, 245, 712-718} (1989): Online Article

Wang, Y.-M., Sheeley, N.R. Jr, and Nash, A.G., A new cycle model including meridional circulation, Astrophys. J., 383, 431-442 (1991): Online Article

see also

Magnetohydrodynamics, Hydromagnetic Dynamo Theory, Solar Dynamo Sunspots

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