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The Solar Dynamo M.R.E.Proctor D


  • pg 1
									The Solar Dynamo I
  DAMTP, University of Cambridge

 Leeds, 6 September 2005
                Indicators of the Solar Cycle:
•Cyclical behaviour of the Sun is
shown by the evolution of sunspots,
observed since time of Galileo.
•Sunspots appear in pairs of opposite
                                                                                   QuickTime™ and a
polarity, with leader spots of opposite         QuickTime™ and a
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polarity in the two hemispheres.
(Hale’s Law)
•Butterfly diagram shows a basic 11y
cycle, with long period modulation of
cycles (Grand Minima) over times of
order 200y. Also evidence of shorter
modulation period (Glassberg cycle)

Sunspot Structure and
     evolution in polar pairs with opposite

                   •Dark in central umbra -
                   cooler than surroundings ~3700K.

                   •Last for several days
                   (large ones for weeks)

                   •Sites of strong magnetic field

                   •Axes of bipolar spots tilted by
                   ~4 deg with respect to equator

                   •Part of the solar cycle

                   •Fine structure in sunspot
                   umbra and penumbra                 3
        Solar activity through the cycle

•Solar cycle not just visible in sunspots.
•Solar corona also modified as cycle progresses.
•Weak polar magnetic field has mainly one polarity at each pole and two poles
have opposite polarities.
•Polar field reverses every 11 years – but out of phase with the sunspot field.
•Global Magnetic field reversal.                                                4
                      Modulations of the Cycle
•   Grand minimum (hardly any sunspots: cold climate in
    N Europe (“little Ice Age”) can be seen in early sunspot
    record. (Maunder Minimum).
•   Proxy data provided by 14C (tree rings) and 10Be (ice
    cores). Intensities reflect cosmic ray abundance - varies
    inversely with global solar field. Shows regular
    modulations with period ~200y.
•   Cyclic behaviour apparently persisted through Maunder
•   Shorter modulation periods can be found (e.g.
    Glassberg 88y cycle)

The Solar interior and surface
                   Solar Interior

                    •   Core
                    •   Radiative Interior
                    •   (Tachocline)
                    •   Convection Zone

                    Visible Sun
                    •   Photosphere
                    •   Chromosphere
                    •   Transition Region
                    •   Corona
                    •   (Solar Wind)
      Magnetic activity in other stars

•Late-type stars of solar type also
exhibit magnetic activity

• Can be detected by Ca II HK emission

•Mount Wilson survey shows a wide
variety of behaviour

•Activity and modulation increases with
rotation rate (decreases with Rossby no.

Basic equations of solar
•Solar convection zone governed by
  equations of compressible MHD

 Solar Parameters (Ossendrijver 2003)
                       BASE OF CZ   PHOTOSPHERE

Ra  gd 4
               H P     1020           1016

Re  UL                  1013           1012
Rm  UL                  1010           106
Pr                     10-7
                                       10-7
  2 0 p         2     105             1
 Pm  
                        10-3            10-6
 M U                    10-4             1
 Ro  U                 0.1-1           10-3-0.4
          2L                                      9
                  Magnetic fields and flows
                                    Yeah, like,
                Welcome to          what makes
                Basic Solar         Astronomy          Lots
                Astronomy.             from          and lots
               Before we start,     Astrology?
                are there any                        of Maths

•   Interaction of magnetic fields and flows due to induction (kinematics) and body forces (dynamics).
•   Recall induction equation (from Faraday‟s Law, Ampère‟s Law and Ohm‟s Law)

                              Induction        Dissipation
         • Induction – leads to growth of energy through extension of field lines
         • Dissipation – leads to decay of energy into heat through Ohmic loss.
         • Sufficiently vigorous flows convert mechanical into magnetic energy if
           Magnetic Reynolds number                           large enough.       10
                Kinematic Dynamos 1
• For large Rm energy grows on
  advective time but is
  accompanied by folding of field
  lines. Large gradients appear
  down to dissipation scale
  ℓ~Rm-1/2 .

• Simple situations
  (axisymmetric, slow flows…)       STF mechanism    Folded fields
  cannot lead to growth.

• Chirality of flow can be an

                                           Disc dynamo

            Kinematic Dynamos 2
• “Anti dynamo theorems” rule out many simple situations

• Cowling‟s theorem : no axisymmetric magnetic field can be a dynamo.

• Poloidal field decays: ultimately zonal field also.
• Backus‟ necessary condition: Dynamo not possible unless max strain
  Σ>π2η/a2 so Rm= Σa2/η cannot be small.

      Large and small scale Dynamos 1

• Small scale fields
•    Appear on Sun at small scales:
    large Ro: no connexion with
•   Seen well away from active
•   Could be relic of old active
    regions; but little or no net flux.
•   Unsigned flux appears continually
    in mini bipolar pairs.                      QuickTime™ an d a
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•   Even slowly rotating stars with no    are need ed to see this p icture .
    obvious cycle have non zero
    “basal flux”.
•   Suggests small-scale dynamo
                Large and small scale Dynamos 2
• Small scale dynamos
•      Fields and flows on same
•      Broken mirror symmetry not
       essential                                         QuickTime™ and a
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       energies comparable

• Dynamo produced by
     Boussinesq convection
    (Emonet & Cattaneo)
    (t   2 )   i (ui )  u3
    (t  P2 )ui   j ( Bi B j  uiu j  p ij )  RaP i 3
    (t  P / Pm2 ) Bi   j (ui B j  u j Bi )

                                                                                      Ra (1e6)     P    Pm         S=P/Pm            Re          Rm       Em/Ek
                                                                                        0.5        1     5           0.2            200          1000       0.20
                                                                                         2         1     1            1             428          428       0.045
                                                                                         4         1     1            1             550          550      0.06 (+)
                                                                                         16        1    0.2           5             1200         240     1e-06 (-)
                                                                                         16        1    0.5           2             1200         600     1e-06 (+)
                                                                                         16       0.5   0.5           1             1900         950    14
                                                                                                                                                         1e-05 (+)
                                                                                         2       0.25   0.5          0.5            1100         550        0.01
     The Solar cycle is due to a
       large scale dynamo
Sun‟s natural decay time τη=R2/η is very long (~1010 y) but
  cycle time is much less than τη :
Coherence of sunspot record suggests global mechanism
  operating at all longitudes. Polarity of leading spots and
  dipole moment changes every 11y.

Two possibilities:
• Nonlinear oscillator involving torsional waves: Oscillatory zonal flow
  produces toroidal magnetic field from poloidal field. In this case zonal flow
  anomalies should have 22y period. Would also expect a bias in dipole moment
• Dynamo process in and/or just below convection zone. In this case velocity
  anomalies will be driven by Lorentz forces j and so have 11y period.

• Velocity data favours dynamo explanation. If there is a15
  dynamo it must be fast
               Fast and Slow Dynamos
•   At large Rm there are two
    time scales:
    Turnover time Ta ≈ L/U
    Diffusion time Td ≈
•   Growth time of order Ta –
    Fast dynamo
•   Growth time = Ta  fn(Rm)
    >>Ta – Slow dynamo
•   For fast dynamos exponential
    stretching of field lines
    needed: flow must be
•   Growth rate of fast dynamo
    bounded by rate of growth of
    line elements: reduction due
    to cancellation (e.g. 2D flows
    non-dynamos)                       16
The Solar Dynamo II
  DAMTP, University of Cambridge

 Leeds, 7 September 2005
                          The Tachocline
•   Helioseismology has shown that solar rotation is
    almost constant with radius in convection zone:
    thin stably stratified shear layer (tachocline) at
    the base.
•   Tachocline maintained by angular momentum
    transport by meridional circulation and Lorentz
    forces due to strong toroidal fields in radiative
    layer below c.z.
•   Tachocline probably spans overshoot region at
    base of convection zone. Lower Tachocline:
    shellular flows, disconnected from oscillatory
    field above. Upper Tachocline: in adiabatically
    stratified region, penetrated by turbulent plumes.
    Toroidal field probably located here.

Torsional Oscillations

      • Observations of zonal flow
        anomalies shows wavelike behaviour
        with 11y period propagating towards
        equator. Mean field models show
        same effect.
       Modelling the large-scale
                                      Fundamental Physics
                                       (the  dynamo)
• Full numerical simulations only
  just feasible (extreme parameter
  ranges, many scale heights, lots
  of physics)
• Most popular historical
  approach involves assumption
  of two scales (mean field
• Details are becoming
  controversial, but several
  different versions of model still
  widely used

                     Mean field models 1
•   Assumption : fields and flows exist on two scales L and l<< L. Write e.g. magnetic
    field, B=<B>+b where <> is average over small scale, and <b>=0.

•   Reynolds stresses in momentum equation can also be modelled through a „Λ-
    effect‟. Fully three-dimensional models include Coriolis effects.
•   E can in principle be expressed in terms of <B> through equation for b.
    ASSUMING that this relation is local in space and time, can use the ansatz

•   These are the - and -effects; -effect gives e.m.f. parallel to field, while -effect
    gives additional turbulent diffusivity.

                     Mean field models 2
•   Recall mean field equations and - and -effect ansatz.

•   -effect allows one to get round Cowling‟s Theorem and find axisymmetric dynamo

•   -effect in B equation usually ignored cf effect of differential rotation. Resulting model
    an -dynamo. BUT how to calculate  and  ??
           Mean field models 3
• Simplest model of mean-field dynamo action - Parker dynamo waves; x
  gives distance from pole, r gives radius

                          If D>0 then waves move towards poles
                          If D<0 then waves move towards equator

             Linear mean field models
• Can extend model to more realistic spherical
  geometry; solve in spherical shell conductor with
  external insulator.

                                   Linear models
• If , ur even, u odd about equator, and also  odd,
   even then have two parities

•    A even, B odd - dipolar
•    A odd, B even - quadrupolar

•   D now odd about equator; if D <0 in
    N.Hemisphere, solutions show exponentially
    growing waves propagating towards equator.
                           Linear models 2
•   Linearised models with fixed α and
    zonal flow profile easily yield
    oscillatory modes with equatorward
    propagation. But quantitative
    comparison lacking.
•   Cycle frequency ~ (turbulent)
    diffusion time
•   Can produce poleward propagation
    at high latitudes using observed
    differential rotation

       Problems with mean fields 1
• Many problems with actual calculation of  and .
• Have to find b by solving fluctuating field equation

                                                         “Pain in the Neck” term

•Very difficult unless PitN term can be ignored
•Two possibilities :
 (a) Rm small (not appropriate for Sun),

(b) “Short-Sharp” approximation: correlation time c<<turnover time.
Then get (for isotropic turbulence, <u>=0)
                                                            In both cases 
                                                            proportional to
       Problems with mean fields 2
• Do these approximations actually make any sense for real flows?

• Can investigate  by imposing mean field B0 and calculating E= B0

• Recent calculations by Courvoisier et al. show very strange behaviour
with Rm ! Use “GP-flow” (periodic in (x,y), strongly helical)

   u  ( y , x , );       
                                       cos( x  cos(t ))  sin( y  sin( t )) 

                                                               But this flow
                                                                   not
                                Rm                              „turbulent‟
       Problems with mean fields 3
• More realistic flows provided by
turbulent convection in a rotating
layer (Cattaneo & Hughes)

•If Rayleigh no. R= gd3/
sufficiently large, get convection
with helicity (anti-symmetric about
mid plane). Helicity quite large:
<u•> 2/(<u•u> <•>)~0.1

• Can find parameters such that flow
is not a small-scale dynamo, but still
has helicity. Calculate  by adding
uniform field as before. Might
expect that would get significant
effect as Rm is large.                   28
         Problems with mean fields 4
• However it is found that the mean -effect is extremely small, and seems to depend on Rm
even at large Rm.
•To get growthrates/timescales for mean field to be comparable to turnover time and not
diffusion time, need  to be O(|u|). Here there is not even a converged value.
•Same system, but in narrower boxes, yields larger effect - so small values are due to
decoherence between different cyclonic cells.
•Even when (small-scale) dynamo action begins at larger Rayleigh numbers no evidence of
any large-scale field.




                                                                             Bx   29
             Dynamics of mean field
• Dynamical effects appear indynamos 1
                             two ways:
• back reaction of the Lorentz force alters large scale flow (MP mechanism)
• changes to the small scale velocity field change mean field coefficients (-
and -quenching)
•Crucial question: how big does the large scale field have to be before the
coefficients are affected?
•At large Rm expect that |b|>>|<B>| as a result of flux line stretching and
folding: then expect small scale flow to be altered when |b|2~|u|2
(equipartition). Since we have |b|~ Rma|<B>|, a>0, generation will be
affected for |<B>| much less than equipartition values - hard to reconcile with
observed solar field configuration with large zonal fields. Get formula of
form (>0)

•Simple models support this with ~1.
•BUT assumption of usual -effect assumes
This seems unlikely at large Rm                                              30
               Dynamics of mean field
                               dynamos 2
 • When there is a small scale dynamo b exists independently of <B>.
 Suppose we have <B>=0 and MHD turbulence field u,b.
 •Then add small mean field (supposed uniform) <B>; perturbation fields
 u´, b´ satisfy

Approx gives

cf, exact result

 Note only for small <B>; but has been widely used elsewhere!
           What can we learn from
           mean field dynamos 1
• In spite of difficulties with
calculating  etc. it has been
widely used in nonlinear regime
to model aspects of the cycle.
Some results seem robust.
• Can find oscillating wavelike
solutions for a wide range of
flows and  distributions.
• D<0 (in N.Hemisphere) gives
correct sign of propagation, and
this can be justified by
handwaving arguments about
sign of .
• Nonlinear solutions may be
dipolar, quadrupolar or of mixed

         What can we learn from
         mean field dynamos 2
• Nonlinear effects on zonal
flow - either from MP-
mechanism or the mean
effects of small scale Lorentz
force (-quenching) - lead to
torsional oscillations as

• The addition of an equation
for zonal flow introduces a
new timescale: can lead to
various forms of modulation
of the cycle, involving parity
changes, amplitude changes
or both
           What can we learn from
           mean field dynamos 3
• Many many different models of mean field dynamos, but the
symmetries of the system are shared. So even low-order
models with the same symmetries show robust behaviour.

•In particular, there is a clear association between “Grand
Minima” (low amplitude transients) and parity excursions. This
is borne out by sunspot records


                                                                 DIPOLE            QUAD

           Other scenarios
• Distributed -effect models have many
  shortcomings. Other possibilities have been
• Parker interface model
• Deep-seated (buoyancy-driven) model
• Conveyor-belt (flux-transport) models
• Direct numerical simulation

                         Parker model
• Parker recognised importance of
  tachocline in controlling dynamo
• Interface model has -effect small
  below tachocline (no convection)
  but  only significant below
• Different radial scales in two
  regions due to differences in  in
  convective and non-convective        Linear model - but can be
  regions.                             made self-consistent with
• Get travelling wave solutions         models for -quenching
  confined to the interface.
        “Deep-seated” Scenario 1
•   Toroidal field created from
    radial field by radial shear of
    tachocline. Also from
    horizontal poloidal field by
    latitudinal differential rotation.

•   Magnetic buoyancy instability
    (or possibly shear flow type
    instability) produces loops of

•   Loops rise to surface, creating
    bipolar regions and meridional
    field, as a result of rotation due
    to Coriolis forces. Rate of rise
    controlled by balance between
    buoyancy and downward
    pumping effect.

     “Deep-seated” scenario 2
•   Action of Coriolis force
    ensures incomplete
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•   Downward pumping of
    flux elements moves
    flux to overshoot region
    to be stretched by zonal
•   Only weaker fields take
    part in this process:
    stronger elements
    escape to the surface.
•   NB the “-effect” here
    is essentially nonlinear;
    B must exceed

                    Conveyer-belt models
•   Simple Flux Transport model. Poloidal
    field produced by tilting of active
    regions - equivalent to surface -effect,
    but based on B at the base of the
    convection zone.

•   Field transported to poles and then
    down to tachocline by meridional flow.

•   Shear flow at tachocline produces
    toroidal field, which emerges to form
    active regions

•   No role for convection zone: quenching
    not a problem. Can account for phase
    dissferences between sunspot and polar
    poloidal fields.

•   Surface may be untypical of field
    structures deeper down. Needs sunspots
    for dynamo to work. (cf Maunder
    minimum).                                   Modification of this model sees a role for
                                                poloidal field production at the tachocline
                                                            by MHD instability
         Direct simulation models 1
• Anelastic codes have been
  used to produce direct
  numerical simulations of
  dynamo action in a
  convecting spherical shell
  (Miesch, Brun & Toomre)
• Dynamo driven by
  convection and differential
• Differential rotation of
  solar like type produced by
  Reynolds stresses due to
  coherent downflows

           Direct simulation models 2
•   Dynamo is vigorous but no
    evidence of cyclic
•   Magnetic energy ~0.07 KE
•   Large-scale field hard to
    generate due to lack of
    coherence over global
•   Dynamo equilibrates by
    extracting energy from the
    differential rotation,
    magnetic Reynolds stresses

                       TURBULENT                               ROTATION

                                           Reynolds Stress
                                              <u’i u’j>       -effect
   CONVECTION u’              MAG FIELD   b’              Maxwell

                      Turbulent EMF               DIFFERENTIAL              MERIDIONAL
                        E = <u’ x b’>             ROTATION                 CIRCULATION Up
 amplification of        ,,-effect    Small-scale
 <B>                                     Lorentz force
               LARGE-SCALE                -quenching
               MAG FIELD   <B>

                                                                     Lorentz force
                                        STRONG LARGE
(courtesy of S.Tobias)                  SCALE SUNSPOT                Malkus-Proctor
                                                                         effect            42
                                           FIELD <BT>
                    Future Directions
• Mean field models have led to qualitative understanding, but
  detailed calculation of α etc. in dynamic regime is controversial,
  and getting more so!
• Broad elements of basic processes (tachocline, pumping,
  buoyancy…) understood but more detailed calculations needed
  before useful quantitative information obtained.
• Better observations of cyclic behaviour in other stars with
  convective envelopes will help calibrate theories.
• A full-scale numerical model incorporating all relevant physics is a
  long way off; in the medium term any successful model will work
  by „wiring together‟ detailed studies of the different regions –still
  scope for clever theoreticians!


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