View source for Solar dynamo

== The solar activity cycle ==
The solar dynamo yields the theoretical explanation of the basic solar activity cycle which is mainly (but not exclusively) manifested in the periodic appearance of sunspots. Its mean period  is 11 yrs (i.e. 22 yrs for the cycle time of the full magnetic oscillation)  but this value varies between  8  and 14 yrs. The variability  of the activity  period can be expressed by  the  quality <math>\omega_{\rm cyc}/\delta_{\omega_{\rm cyc}}</math> which for the Sun has the rather high value of   5. 

[[Image:butterfly_maunder_1904.jpg|thumb|500px|right|butterfly|
 The butterfly diagram of  sunspot statistics (Maunder 1904). Note in particular
 that the 
         spots only exist very close to the equator; only a few spots appear at  
         latitudes higher than (say) 30<sup>o</sup>. There is a well-established (but not perfect)  symmetry of the phenomenon with respect 
         to the equator.]]
Only the antisymmetric (dipolar) parts  of the solar magnetic field oscillate and reverses its sign during the 22-yrs cycle  (Stenflo & Vogel 1986). 
The magnetic reversal happens at the poles during the maximum
of the sunspot activity.   As the latter  is located rather close to the equator  one can be sure
that  the solar activity phenomenon is a global phenomenon  rather than a local one. 
       
=== The Maunder minimum ===
The amplitude  of the cycle is far from being constant.  The most prominent activity  drop known was the Maunder minimum between 1670 and 1715.  The latitudinal distribution of the few sunspots observed during the end of
the Maunder minimum was highly asymmetric with respect to the equator.  All spots except two or three in this period appeared on the southern
hemisphere (Sp&ouml;rer 1887). The first cycle after the minimum, with maximum in 1706,
existed almost exclusively in
one hemisphere. There is  empirical evidence that the magnetic cycles persisted 
through the solar Maunder minimum,
as first found by Wittmann (1978, old sunspot data), Schr&ouml;der (1992, auroral activity)  and more recently by Beer et al. (1998, Be data in ice cores).

[[Image:maunder.jpg|thumb|650px|left|maunder|
The time series of the sunspot numbers indicate a highly
         nonlinear behavior. Both cycle amplitude and  length are obvious 
         functions of time. From 1670--1715 a grand minimum ('Maunder
         minimum') is seen.]]
Charbonneau (2001) reports an  'odd-even effect' of the cycle amplitudes. The odd-numbered cycles are stronger than their
neighboring  cycles  or -- 
which is the same -- the
sunspot number of the complete 22-year cycle does not fluctuate too much from cycle to cycle.

=== Cycle length variations ===

Measurements of <sup>14</sup>C abundances in sediments and long-lived 
trees provide much longer time series than sunspot datasets.
Vos  et al. (1997) found
a secular periodicity of 80...90 yrs as well as a long-duration 
period of about 210 yrs. The measurements of atmospheric  
<sup>14</sup>C abundances  suggested a periodicity of
2400 yrs which is also associated with a long-term variation of solar
activity. The variety  of frequencies found in solar activity may even
indicate chaotic behavior as discussed by Knobloch & Landsberg (1996) on the basis of nonlinear mode-mixing models (see Brandenburg et al. 1989, Schmitt & Sch&uuml;ssler 1989).

[[Image:cycles.jpg|thumb|650px|right|cycles|
Cycle length (''left'') and cycle amplitudes (''right'')
         for  11 cycles  separated for northern (green) and southern (blue)
         hemispheres.]]

Another question concerns a possible systematic north-south asymmetry of  the  cycle length. If it is true that the  grand minima result from  simultaneous nonlinear  oscillation of modes with equatorial symmetry and antisymmetry then  systematic differences of the cycle length in the two
hemispheres must appear. 
Both cycle lengths and amplitudes in the dataset of Hathaway et al. (2003) are given in  <figref>Cycles.jpg</figref>. There seems to be a slight  anticorrelation  between cycle length and cycle amplitude in the data: short cycles are followed by   stronger cycles (Hoyng  1993, Solanki et al. 2002). 

Any single  cycle is   asymmetric in time.   The strong  cycles are faster  to reach their activity maximum than the weak  cycles.  This  'Waldmeier rule' can be used to produce successful  predictions of the strength and duration of future activity cycles (Sch&uuml;ssler 2007).

=== Solar-stellar relations ===
The activity cycle of the Sun is not exceptional among the stars. The
observation of chromospheric Ca-emission of solar-type stars yields
activity periods between 3 and 30 yrs. Up to 
15% of the solar-type stars, however, do not show any 
significant activity. This suggests that 
even the existence of the grand minima is a typical property of cool main sequence stars like the Sun. From ROSAT X-ray data Hempelmann et al. (1996) find that up to  70% of the stars with a constant level of activity
exhibit a rather low level of coronal X-ray emission.
 
The dependence of the cycle frequency  on the rotation rate <math>\Omega</math> is  rather weak.  Saar & Brandenburg (1999) report  for the  relation 
:<math>\label{nm1}
\omega_{\rm cyc} \propto \Omega^n 
</math>

 
the very small value <math>n \simeq -0.09</math> for their full data-set. 
They  derive values of 1.15, 0.8 and 0.4 for their groupings 'inactive', 'active' and 
'superactive'.  With the same data base B&ouml;hm-Vitense (2007) finds  similar relations with  <math>n \simeq 1\ .</math>
The situation can also be realized in <figref>Active_all_lines.jpg</figref>  where the data of Saar & Brandenburg (1999) and Olah et al. (2000) are collected. The active stars are given in pink while the less active  stars are given in blue and black. The latter really  form an   own  branch with ''n'' of order unity while there 
is no clear rule for the fast-rotating stars. Considering only the shortest cycle for a given rotation rate, one finds the small value ''n'' = 0.18 (Olah et al. 2007).  For fast rotating stars the angular velocity  itself seems to be not  the only  determinant of the cycle time.


[[Image:active_all_lines.jpg|thumb|550px|right|stellarcycles|
Cycle periods in years vs. rotation times in days, compiled 
         from Saar & Brandenburg (1999) and Olah et al. (2000). The linear relations with   <math>n \simeq 1</math> by  B&ouml;hm-Vitense  (2007) are given by straight lines with corresponding colors.]]

== Theory ==

Schwabe's (1844) 11-year sunspot cycle, Carrington's (1863)  differential rotation (of the solar surface)
and Sp&ouml;rer's law (of the equatorward migration of sunspots during the cycle, 1894)  are the basic properties  
of the solar activity that must be explained by the dynamo theory. As these three basics are understandable even  with a ''linear theory'' (which  already  demonstrates the complexity of the problem) only a linear approach is here presented. There are many more nonlinear extensions of the models but the main principles and problems already appear in the given linear approach.

=== Convection zone dynamo ===

In the solar  surface  there is turbulence with velocity fluctuations of 1 km/s and correlation times  of about  10 min. The magnetic diffusivity <math>\eta_{\rm T}</math> formed by  such high-speed  short-living granular motion  is of order  10<sup>12</sup>  cm<sup>2</sup>/s. This value allows a skin  depth of an electromagnetic oscillator with a timescale  of 10 yrs of more  than 100 000 km -- close to the depth of the solar  convection zone (200 000 km). This is the background  for the  common believe that the entire  solar convection zone  should be the site of the solar dynamo. If this is true then also the large ratio of the mean cycle period and the rotation  time of the stars shown in   <figref>Active_all_lines.jpg</figref> becomes understandable. The basic time scale  in convection zones is the correlation time of the turbulent cells which at the bottom of the solar convection zone  reaches the value of the rotation time (25 days).  In a thick
convection shell the number  <math>\tau_{\rm cyc}/\tau_{\rm corr}</math> reflects the square of the
ratio of the stellar radius to the correlation length so that
numbers exceeding 100   are easily possible. 

Most solar dynamo models  also operate in a linear way. Rather weak  seed fields are amplified in the sense  of a self-excitation of large-scale magnetic fields.  The global magnetic field can be described  by  a  superposition of a poloidal field (the field projected in a meridional  plane) and the  toroidal (azimuthal) field component.  Both field components  would independently decay if there  are  no extra  processes which maintain the magnetic  field against the  Ohmic  decay  which would destroy  the fields  after   about 10 yrs.

A dynamo can thus only operate if processes exist  which transform poloidal fields into toroidal fields and v.v. One of the   processes is well-known: if the angular  velocity  in the convection zone is not uniform, then it easily transforms poloidal fields into toroidal fields by wiggling them up. The effectivity of this transformation scales with the magnetic Reynolds number of the differential rotation, i.e <math>B_\phi/B_r\simeq \delta\Omega\  D^2/\eta_{\rm T}</math> (with ''D'' as the depth of the convection zone) which is about 100 for <math>\eta_{\rm T}=5\cdot 10^{12}</math> <math>{\rm cm}^2/{\rm s}</math> and 5000 for <math>\eta_{\rm T}= 10^{11}</math> <math>{\rm cm}^2/{\rm s}\ .</math> It is thus easy to understand that the amplitude of the toroidal field  of the Sun exceeds the amplitude of the poloidal field. Obviously, the observed factor of about 1000 needs  a rather small eddy diffusivity.

=== Electromotive force (EMF) in turbulent plasma ===
More complicated is the regeneration of the poloidal field from the toroidal one. In principle it can be  done by  the action of a meridional flow  but in this case the resulting magnetic field  must be nonaxisymmetric in a large scale (Cowling theorem). One has thus  to look for the hydromagnetic properties of the convection/turbulence in the  solar  convection zone. It is  in particular the structure of the   turbulent electromotive force  <math>\mathbf{\mathcal{E}}=\langle \mathbf{u}'\times \mathbf{B}'\rangle</math> which  might be   important.  It appears  in the Ohm's law for mean-field quantities

:<math>\label{nm2}
\langle\mathbf{J}\rangle=  \sigma \big(\langle\mathbf{E}\rangle+  \langle\mathbf{u}\rangle\times \langle\mathbf{B}\rangle+\mathbf{\mathcal{E}}\big)
</math>


and plays the central role in the mean-field dynamo theory.

The simplest  electrodynamic  effect  of a turbulent field  of electrically conducting matter is its diamagnetism, i.e. the tendency  to transport a mean magnetic field   into regions  of lower  turbulence intensity. It can simply  be written  as

:<math>\label{nm3}
\mathbf{\mathcal{E}}=\mathbf{\gamma} \times \langle \mathbf{B}\rangle + \dots
</math>


with <math>\mathbf{\gamma}=-\frac{1}{2} \nabla  \eta_{\rm T}\ .</math> The field is  thus transported towards  the minimum of the eddy  diffusivity <math>\eta_{\rm T}\ .</math> The effect is also known as turbulent pumping; it is the reason that in dynamo simulations the flux is mainly transported downwards rather than upwards as the various buoyancy effects would suggest. If the downward pumping is too strong  then the dynamo does not operate.
For <math>\eta_{\rm T}\simeq 10^{12}\ {\rm cm}^2/{\rm s}</math> the amplitude of the downward pumping exceeds the value of 1 m/s.

Another turbulence-induced  term  runs with the mean-field  electric current, i.e.

:<math>\label{nm4}
\mathbf{\mathcal{E}}= \dots -\eta_{\rm T}\, {\rm curl}\, \langle\mathbf{B}\rangle,
</math>


which describes  the strongly  enhanced dissipation of the field by the turbulence. The ratio of the eddy  diffusivity <math>\eta_{\rm T}</math> to the microscopic    diffusivity  of the gas is extremely high. If the basic stellar rotation is included into the computations of the EMF the magnetic eddy diffusivity forms a complicated tensor with many components  and consequences also for the dynamo theory (see Kitchatinov et al.  1994).

=== The alpha-effect due to rotation ===
The transformation of the toroidal field to the poloidal  field  needs  another term. It is the alpha-effect  by  Steenbeck,  Krause & R&auml;dler (1966) which is the basis  of almost all of the solar dynamo models  developed so far. After the alpha-effect  concept  in rotating  turbulences of conducting matter  a relation

:<math>\label{nm5}
\mathbf{\mathcal{E}}=\alpha\ \langle\mathbf{B}\rangle+...
</math>


exists so that by the action of the turbulence an electric current flows along the magnetic  field lines. The  current along the toroidal field is always accompanied by a poloidal field, hence the loop can be  closed and the   magnetic field can be maintained  against its Ohmic decay.

An alpha-effect only exists  if  the star rotates. Additionally  there ''must be a  stratification of density and/or  turbulence intensity'' otherwise  the alpha-effect is zero. The following argument describes the situation. The quantity alpha is a pseudo-scalar as it  connects the polar vector <math>{\mathbf{\mathcal{E}}}</math> with the axial vector <math>\langle\mathbf{B}\rangle\ .</math> The simplest pseudoscalar  one can form in turbulent flows  is <math>\langle \mathbf{u}'\cdot {\rm curl}\, \mathbf{u}'\rangle</math> that describes  a  helicity  in the turbulence field.  Helicity means that the turbulence is not mirror-symmetric, i.e. (say) left-handed helical motions are more frequent than right-handed ones or v.v. This is only possible if the star  rotates.

The simplest pseudoscalar one can construct with ''global quantities'' is the scalar product of the vector of the density stratification <math>\mathbf{g}=\nabla {\rm log}\,\rho</math> and the vector of the angular velocity of the basic rotation so that <math>\alpha \propto \mathbf{g}\cdot\mathbf{\Omega}</math> should be true. The sign of this quantity differs in the northern  and  the southern hemisphere and so both the alpha-effect and also the helicity do. At the equator the alpha-effect vanishes, but it exists at the poles. For mesogranulation the helicity can directly be observed at the solar surface. It proves to be negative (positive) at the northern (southern) hemisphere.

There is indeed a  theoretical relation of the alpha-effect and the numerical value of the helicity, i.e.

:<math>\label{nm6}
\alpha \simeq -\tau_{\rm corr} \langle \mathbf{u}' {\rm curl}\, \mathbf{u}'\rangle,
</math>


which has been confirmed by many numerical simulations. Highly nonlinear calculations with   forced turbulence also reveal the  contributions of ''magnetic helicity'' to the EMF which can also be observed at the solar surface (see Brandenburg & Subramanian 2005).

=== Dynamo waves ===
The simplest dynamo models even work in low dimensions  producing by the common action of alpha-effect and differential rotation traveling magnetic waves with a frequency of 

:<math>\label{nm7}
\omega_{\rm cyc}\simeq \left(\frac{\alpha^2  }{\eta}(r\frac{{\rm d}\Omega}{{\rm d}r})^2\right)^{\frac{1}{3}}
</math>


(Tuominen et al. 1988). To obtain a cycle time of about 20 yrs an alpha of order 1 m/s is thus necessary. Simulations of the alpha-effect in the solar convection zone yield  <math>\alpha \simeq 10^{-(1\dots 2)}\ u_{\rm T}</math> with <math>u_{\rm T}</math> as the characteristic turbulence   velocity so that with <math>u_{\rm T}\simeq 100\ {\rm m/s}</math> indeed an alpha-effect of 1 m/s results. The expression \eqref{nm7} also explains the rather weak dependence of the cycle times of stellar dynamos on the rotation time. Theory and observation of the stellar differential rotation lead to the common result that the rotation law in stars hardly depends on their rotation rate (see below). 

Also the alpha-effect does not depend strongly on the rotation rate.  Ossendrijver et al. (2001) report an increase of the alpha-effect (in horizontal direction) by a factor of only 2.5 if the rotation rate varies by two orders of magnitude. Hence, the result \eqref{nm7} complies with the observed small  exponent ''n'' in \eqref{nm1}.

=== The differential solar rotation ===



[[Image:Soho.jpg|thumb|240px|left|Soho|
The internal rotation law of the Sun from the inversion of SOHO data.  The curves are marked with the latitude (Courtesy NSF National Solar Observatory).]]
[[Image:sunrot.jpg|thumb|240px|right|sunrot|
The solar rotation law computed  for 25 days rotation period .  The model  includes two  parameters only  for  the anisotropic correlation tensor of the   turbulence in the convection zone (Kitchatinov & Rüdiger 2005).]]

The rotation law within the solar convection zone is known by the inversion procedures of the helioseismology ( <figref>Soho.jpg</figref>). Formally, the  observed equatorial acceleration at the surface proves to be  the result of a strong polar subrotation (<math>\partial\Omega/\partial r<0</math>) and a  weaker equatorial superrotation (<math>\partial\Omega/\partial r>0</math>) inside the convection zone. The ''strong polar subrotation forms the main problem for the solar dynamo theory'' as naturally  the strongest fields are there induced  which, however, are not observed. The solar core rotates rigidly with  approximately  the same angular velocity as the surface at midlatitudes. The  solution of the   Reynolds equation on the basis of the (also turbulence-induced) Lambda-effect for rotating density-stratified turbulence do indeed explain these observations ( <figref>Sunrot.jpg</figref>).

The stellar-surface rotation laws which are empirically known so far lead to the  result that the absolute equator-pole differences of the solar-type stars hardly differ from star to star.  
[[Image:difrot8.jpg|thumb|420px|right|difrot8|
The sofar known equator-pole differences (in rad/day) of very young stars, the MOST observations and the present-day Sun. For LQ Hya the equatorial acceleration  is reported  as highly  time-dependent. LQ Lup is too fast for present-day theory. Rotation period in days.]]

Data from the MOST satellite recently led to the detection of   rotation laws for two new  young solar-type stars with rotation periods of about 10 days. For <math>\epsilon</math> Eri   it is <math>\delta\Omega=0.061\ {\rm rad/day}</math>  and for <math>\kappa</math><sup>1</sup>Ceti it is <math>\delta\Omega=0.064\ {\rm rad/day}</math> (Walker et al. 2007). Note the almost perfect coincidence of these values with the solar value (see  <figref>Difrot8.jpg</figref>). The two new examples are of extraordinary meaning insofar as their rotational rates start to  fill the gap between the fast rotating very young stars and our (old) present-day Sun. A characteristic value of the equator-pole difference of <math>\Omega</math> for stars (including the Sun) is <math>\delta\Omega= 0.07\ {\rm rad/day}\ .</math> This value excellently  complies with  the results of the  theory of the  rotation laws of stars with  outer  convection zones.

=== Meridional flow ===
All nonuniform rotation laws  exist together with meridional flows  as  the centrifugal force of nonrigid rotation cannot longer be written  as a conservative force. The flow which occurs in the  simulations  at  the bottom of the convection zone drifts always equatorwards with an amplitude of <math>u^{\rm m}\simeq 5\rm  m/s </math> (Küker & Rüdiger 2005). Such a  flow   needs slightly  more than 7 yrs to travel from the pole to the equator.  However, with <math>\eta_{\rm T}= 10^{12}\ {\rm cm}^2/{\rm s}</math> the magnetic Reynolds number <math>{\rm Rm}= u^{\rm m} R/\eta_{\rm T}</math> (R ~ 700.000 km, solar radius) does not strongly exceed unity so that the meridional flow  will not be important   in the dynamo equation. Only if the magnetic eddy diffusivity is reduced by  (say) a factor of 10  the meridional circulation  dramatically changes  the dynamo regime by fixing the cycle time and transporting the azimuthal magnetic field  towards the equator.

== Numerical models ==

The dynamo equation
:<math>\label{nm8}
\frac{\partial \langle\mathbf{B}\rangle}{\partial t}= {\rm curl} \big((\mathbf{u}+\mathbf{\gamma})\times \langle\mathbf{B}\rangle + \alpha \langle\mathbf{B}\rangle -  \ \eta_{\rm T}{\rm curl}\,\langle\mathbf{B}\rangle\big)
</math>


(with the pumping term  included) must be  solved with <math>\mathbf{u}</math> as the given large-scale flow pattern in the solar convection zone. Simple models only apply the solar rotation law ( <figref>Soho.jpg</figref>). If the fields are weak enough then any magnetic feedback  to the terms of the electromotive force  can be ignored. Calculations with this assumption are called as 'kinematic' dynamo models. Nonlinear dynamo models are much more complicated.

=== Distributed dynamos ===

Let  the magnetic eddy diffusivity <math>\eta_{\rm T}</math>  be  homogeneous within the entire convection zone (thickness <math>D\simeq 200.000</math> km) while the solar core is considered as a perfect conductor. The alpha-effect always  runs with <math>\cos\theta</math> so that it peaks at the poles.  If the angular velocity has only a radial gradient, then the cycle  time turns out to be of order <math>\tau_{\rm cyc}\simeq 0.26 DR/\eta_{\rm T}\ .</math> The value <math>\eta_{\rm T} \simeq 10^{12}\ {\rm cm}^2/{\rm s}</math> is thus the reference value  leading to a cycle time of 10 yrs.

[[Image:dynamo1.jpg|thumb|400px|left|dynamo1|
A solar dynamo model with alpha-effect in the entire convection zone and <math>\eta_{\rm T}= 5\cdot 10^{12}\ {\rm cm}^2/{\rm s}\ .</math> Left: The geometry of the induced magnetic field. The colors represent the magnitude of the toroidal field  while the arrows form the field lines of the poloidal field. Note the equatorial asymmetry. Right:  Phase relation of the field components (BT toroidal, BP poloidal). The cycle time is given in units of the diffusion time (here 33 yrs).]]

With  the observed solar rotation law  and positive (in the northern hemisphere) depth-independent alpha-effect  Eq. \eqref{nm8} yields magnetic fields which are represented in  <figref>Dynamo1.jpg</figref>.
One  finds a radial drift of the toroidal field belts but not  an equatorial migration,i.e. the butterfly diagram ( <figref>Butterfly_maunder_1904.jpg</figref>) can not be explained. The parity of the solution is dipolar, the toroidal magnetic amplitudes are much too weak. Also the cycle time is too short.

The model can  be improved if the alpha-effect is concentrated to a thin layer at the bottom of the convection zone. Then the amplitude ratio exceeds values of 200 but one  always gets more magnetic belts than  needed. The butterfly diagram appears at too high latitudes and will never reach the low latitudes which are necessary to reflect the solar observations correctly.

Too simple kinematic <math>\alpha\Omega</math>-dynamos working in the convection zone with positive alpha-effect  and the solar  rotation law are thus not able to fulfill the main observational constraints.

=== Boundary-layer dynamos ===
The idea  of the boundary-layer (BL) dynamo bases on the findings that i) the alpha-effect in the bottom of the solar convection zone, where the convection overshoots into the stably stratified  radiative zone, becomes negative and  ii) the radial gradients of the angular velocity peak there (see Belvedere et al. 1991). One could thus indeed realize  an equatorwards drift of the toroidal field belts in case the alpha-effect vanishes at the poles. If the boundary layer, however, is  too thin then the cycle time becomes too short for  <math>\eta_{\rm T}</math> = 10<sup>12</sup> cm<sup>2</sup>/s   and also the number of magnetic belts becomes too high. Indeed, after current convection zone simulations the overshoot phenomenon restricts to a layer with less than 10.000 km depth. It is hard to believe that in such a thin layer the averaging procedure for the turbulence can be done   leading to  a mean-field formulation of the EMF. Current dynamo models thus prefer to work with the positive alpha-effect dominating  the entire convection zone. In these cases the butterfly diagram can only result from the action of the meridional circulation flowing towards the equator at a radius where the toroidal magnetic field has its maximum.

=== Advection-dominated dynamos===

The inclusion of the meridional flow  has  strong impacts to the mean-field dynamo models only when the eddy diffusivity  is so low that the magnetic Reynolds number of the flow  reaches values of the order of 10<sup>3</sup>. The  consequences for   the dynamo regime are drastic. Models with rather small values of the eddy diffusivity   have  been a subject of intense numerical investigation (Choudhuri et al. 1995,  Charbonneau & Dikpati 2000, Dikpati & Gilman 2006). Models with  a  positive alpha-effect and high magnetic Reynolds number provide correct cycle periods, equatorial parity and  produce rather good butterfly diagrams.

[[Image:dynamo2.jpg|thumb|400px|right|dynamo2|
Advection-dominated solar dynamo with alpha-effect located at the bottom of the convection zone where the magnetic diffusivity has been assumed as  very small ( 2*10<sup>10</sup> cm<sup>2</sup>/s) and with a flow amplitude of 5 m/s. Design as in  <figref>Dynamo1.jpg</figref>.]]

In the simplest model  the  alpha-effect exists in the whole convection zone and the   drift amplitude lies between 2 m/s and 10 m/s.  Almost always solutions with the (wrong) quadrupole solution occur for  
the lowest eigenvalue. Because of the small value of the eddy diffusivity the cycle time becomes too long compared with the 22 yrs of the Sun. The toroidal field belts are migrating equatorwards but the maximum field amplitude  still exists in the polar region.

These results can drastically be improved if the alpha-effect is located mainly at the bottom of the convection zone. The <math>\eta_{\rm T}</math> is 10<sup>12</sup> cm<sup>2</sup>/s in the bulk of the convection zone but it is smaller (We choose the more complex situation the rotation is also unstable
without the density stratification (2*10<sup>10</sup> cm<sup>2</sup>/s) at its bottom. Now the dipolar parity is indeed preferred. The meridional flow advects the field equatorwards producing a butterfly diagram of the observed type 
(see   <figref>Dynamo2.jpg</figref>). For such models one finds that the inclusion of the meridional  circulation into models with low values of the magnetic diffusivity

* improves the cycle time problem 
* transports the fields towards the equator
* favors the dipolar solutions against the quadrupolar solutions
* produces strong enough toroidal fields
* avoids the high number of field belts which often appears in thin layer solutions.

The  needed circulation seems really  to exist. The surface counterpart of this flow   moving towards the poles with an amplitude of about 15 m/s has been observed (Zhao & Kosovichev 2004). However,  the reason for the small value of the eddy diffusivity (i.e. the high magnetic Prandtl number) is still unknown. The dependence of the cycle time on the amplitude of the meridional flow at the bottom of the convection zone is shown in  <figref>Cycletime_drift.jpg</figref> as a summary of various model calculations (see R&uuml;diger & Hollerbach 2004). One can easily see that the cycle time is much too large for too slow circulation but for drift rates in excess of 6 m/s the resulting cycle times well reproduce the observed cycle time of 22 yrs.
[[Image:cycletime_drift.jpg|thumb|500px|left|timedrift|
The cycle times for the dipolar modes of advection-dominated dynamo models of Bonanno et al. (2002). There is a clear anticorrelation of the drift velocity and the cycle time. Note the very long cycle times for too slow meridional circulation which are due to the small values of the used turbulent magnetic diffusivity.]]

===  Simulation models ===

Many of the known mean-field phenomena can also be simulated   with high-developed supercomputer codes which numerically solve the first-principle equations of  magnetohydrodynamics. By heating a convectively unstable spherical shell   from below  rotating magnetoconvection results which transports angular momentum and generates large-scale magnetic fields. Basic details of the mean-field calculations are reproduced. The toroidal magnetic fields in the convection zone appears to be weak and highly nonaxisymmetric but the field in the 'tachocline' layer below the bottom of the convection zone becomes strong (~ 3000 G), axisymmetric and antisymmetric with respect to the equator. The latter result can be described as due to the action of an alpha-effect in the convection zone (Browning et al. 2006). Also the rotation law in the convection zone shows the main properties of the observed pattern  ( <figref>Soho.jpg</figref>). Still some of the results are consequences of special inputs but basic ingredients of the solar dynamo  model -- such as the alpha-effect in the convection zone, differential rotation, turbulent pumping (see Dorch & Nordlund 2001) and  effective toroidal stretching of the magnetic field lines --  can  be obtained by direct numerical simulations. However, as the solar cycle lasts more than  200 rotation times it will be a highly innovative work to reproduce a full solar cycle with the supercomputer.

== References ==
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* [http://adsabs.harvard.edu/abs/2007ApJ...657..486B B&ouml;hm-Vitense, E.: 2007, Chromospheric activity in G and K main-sequence stars, and what it tells us about stellar dynamos, Astrophys. J. 657, 486-493] 
* [http://adsabs.harvard.edu/abs/2002A%26A...390..673B Bonanno, A., Elstner, D., R&uuml;diger, G., & Belvedere, G.: 2002, Parity properties of an advection-dominated solar <math>\alpha^2 \Omega</math>-dynamo, Astron. Astrophys. 390, 673-680]
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<!-- 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>

* Jan A. Sanders (2006) [[Averaging]]. Scholarpedia, 1(11):1760.

* Olaf Sporns (2007) [[Complexity]]. Scholarpedia, 2(10):1623.

* Axel Brandenburg (2007) [[Hydromagnetic dynamo theory]]. Scholarpedia, 2(3):2309.

* Søren Bertil F. Dorch (2007) [[Magnetohydrodynamics]]. Scholarpedia, 2(4):2295.

* Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) [[Periodic orbit]]. Scholarpedia, 1(7):1358.

* Hugh Hudson (2008) [[Solar activity]]. Scholarpedia, 3(3):3967.

* Philip Holmes and Eric T. Shea-Brown (2006) [[Stability]]. Scholarpedia, 1(10):1838.


==See Also==
[[Magneto-Convection]], 
[[Magnetohydrodynamics]],  
[[MHD Turbulence]], 
[[Solar Granulation]], 
[[Sunspots]]

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