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Flux transport dynamos are one specific variety of
[[hydromagnetic dynamos|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 timescales (many months to centuries and millennia).
Their two primary defining features are:

* the observed equatorward migration of sunspot source regions 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 millennial timescales.

==Flux transport dynamos and the solar cycle==

The solar magnetic field is the energy channel and dynamical
engine driving [[solar activity|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|solar cycle]], first noted in
the systematic variations of sunspot counts (see  <figref>Solar_cycle.gif</figref>),
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|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 theory|hydromagnetic dynamo]] models,
of which flux transport dynamos are one particularly promising variety.

[[Image:Solar_cycle.gif|thumb|500px|right|F1|
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  <figref>BL_cartoon.gif</figref>.
Sunspots currently emerging at the solar surface (B) are formed from the
toroidal field ''T<sub>n</sub>'' 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 ''P<sub>n-1</sub>''
having built up during the preceding 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 (''T<sub>n+1</sub>'')
can commence.

[[Image:BL_cartoon.gif|thumb|300px|right|F2|
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  <figref>BL_cartoon.gif</figref> 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 <figref>BL_cartoon.gif</figref>) 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 <figref>BL_cartoon.gif</figref>, assume that
the toroidal field ''T<sub>n</sub>'' is linearly proportional to the
poloidal field strength ''P<sub>n-1</sub>'' of the previous cycle.
Production of the current cycle's poloidal field ''P<sub>n</sub>'', on the
other hand, is understood to be a nonlinear function of the current
toroidal field ''T<sub>n</sub>'', production of ''P<sub>n</sub>'' being
possible only in a finite range of toroidal field strengths. Writing
<math>P_n\propto T_n(1-T_n)\ ,</math> and absorbing the proportionality constants
into scaled forms of the amplitudes, leads to:

:<math>\label{Eq4}
p_{n+1}=a\,p_n^2(1-p_n)
</math>


Equation \eqref{Eq4}
is a one-dimensional
cubic iterative map for the amplitude ''p<sub>n</sub>'' 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  <figref>Bifurc.gif</figref>A).

[[Image:Bifurc.gif|thumb|500px|right|F3|
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|attractor]]
has here a finite-sized [[basin of attraction|basin of attraction]],
indicated by gray shading
on  <figref>Bifurc.gif</figref>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>\label{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 ''a<sub>n</sub>'' and additive noise amplitude
''e<sub>n</sub>'' 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 (<math>p_n\sim 1</math>),
corresponding to "normal" cyclic
behavior, punctuated by quiescent epochs (<math>p_n\ll 1</math>)
of irregular spacing and duration; these are
the map's equivalent
to Maunder Minimum epochs of strongly depressed sunspot counts
(see <figref>Solar_cycle.gif</figref>).

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>\label{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>\label{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>\label{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 <math>\varpi=r\sin\theta\ ,</math> and <math>\eta</math> 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 <math>\Omega(r,\theta)</math>
and a meridional circulation '''u'''<sub>p</sub>.
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. \eqref{nm3} acts as the source
of toroidal field;
the source term <math>S(r,\theta,B(t))</math>
in the poloidal equation \eqref{nm2}
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  <figref>Bifurc.gif</figref>B). Likewise, stochastically-forced version
s
of these models do produce intermittency (see  <figref>Nummod_intermitt.gif</figref>A),
as well as Gnevyshev-Ohl
like pattern of amplitude fluctuations (see 
<figref>Nummod_intermitt.gif</figref>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.

[[Image:Nummod_intermitt.gif|thumb|500px|right|F5|
Intermittency and Ohl-Gnevyshev pattern
in a stochastically-forced 2D kinematic flux transport dynamo model.
The overall format is identical to  <figref>Nummod_intermitt.gif</figref> 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).

==References==

'''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)

[http://solarscience.msfc.nasa.gov 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): [http://solarphysics.livingreviews.org/Articles/lrsp-2005-2 Online Article]

Ossendrijver, M.A.J.H.,
The solar dynamo,
''Astr. & Astrophys. Rev.'', '''11''', 287-367 (2003):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2003A&ARv..11..287O Online Article]

Tobias, S.M.,
The solar dynamo,
''Phil. Trans. A Roy. Soc. London'', '''360''', 2741-2756 (2002):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2002RSPTA.360.2741T 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?1961ApJ...133..572B Online Article]

Bushby, P.J., and Tobias, S.M.,
On predicting the solar cycle using mean-field models,
''Astrophys., J.'', '''661''', 1289-1296 (2007):
[http://adsabs.harvard.edu/abs/2007ApJ...661.1289B Online Article]

Charbonneau, P.,
Multiperiodicity, chaos and intermittency in a reduced model of the solar cycle,
''Solar Phys.'', '''199''', 385-404} (2001):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2001SoPh..199..385C 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2004ApJ...616L.183C 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2005ApJ...619..613C Online Article]

Choudhuri, A.R., Schüssler, M., and Dikpati, M.,
The solar dynamo with meridional circulation,
''Astr. Astrophys.'', '''303''', L29-L32 (1995):
[http://adsabs.harvard.edu/cgi-bin/bib_query?1995A&A...303L..29C 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):
[http://adsabs.harvard.edu/abs/2007PhRvL..98m1103C Online Article]

Dikpati, M., and Gilman, P.A.,
Flux transport dynamos with <math>\alpha</math>-effect
from global instability of tachocline differential rotation: a solution
for magnetic parity selection in the sun,
''Astrophys., J.'', '''559''', 428-442 (2001):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2001ApJ...559..428D 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2006GeoRL..3305102D Online Article]

Durney, B.R.,
On the differences between odd and even solar cycles,
''Solar Phys.'', '''196''', 421-426 (2000):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2000SoPh..196..421 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2001A&A...374..301K Online Article]

Leighton, R.~B.,
A magneto-kinematic model of the solar cycle,
''Astrophys. J.'', '''156''', 1-26 (1969):
[http://adsabs.harvard.edu/cgi-bin/bib_query?1969ApJ...156....1L 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?2006ApJ...647..662R Online Article]

Wang, Y.-M., Nash, A.G., and Sheeley, N.R. Jr,
Magnetic flux transport on the Sun,
''Science'', '''245''', 712-718} (1989):
[http://adsabs.harvard.edu/cgi-bin/bib_query?1989Sci...245..712W 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):
[http://adsabs.harvard.edu/cgi-bin/bib_query?1991ApJ...383..431W Online Article]

<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>

* John W. Milnor (2006) [[Attractor]]. Scholarpedia, 1(11):1815.
* Edward Ott (2006) [[Basin of attraction]]. Scholarpedia, 1(8):1701.
* John Guckenheimer (2007) [[Bifurcation]]. Scholarpedia, 2(6):1517.
* Eugene M. Izhikevich (2006) [[Bursting]]. Scholarpedia, 1(3):1300.
* Axel Brandenburg (2007) [[Hydromagnetic dynamo theory]]. Scholarpedia, 2(3):2309.
* Søren Bertil F. Dorch (2007) [[Magnetohydrodynamics]]. Scholarpedia, 2(4):2295.

==see also==

[[Magnetohydrodynamics]],
[[Hydromagnetic Dynamo Theory]],
[[Solar Dynamo]]
[[Sunspots]]


[[Category:Astrophysics]]
[[Category:Computational Astrophysics]]
[[Category:Dynamical Systems]]