Solar and Stellar dynamos

1
Solar and Stellar dynamos

• Steve Tobias (Applied Maths, Leeds)
• Stellar Dynamo Meeting Leeds
• Dec 13 17 2004

2
Talk Summary

• Observations
• Some basic theory
• Mean Field Electrodynamics
• The simplest solutions
• a-effect, a-quenching
• Big argument
• Other mechanisms for producing poloidal field
• What physics do we need to understand?
• Cartoon Pictures of solar dynamo a-effect
• Distributed
• Flux conveyor
• Interface Dynamos

• Robust (general) results from mean field models
• Mathematical considerations of the mean field
  equations
• Low order models
• Global Solar Dynamo Models
• Specifically Stellar Models
• See reviews by Ossendrijver (2003), Weiss
  Tobias (2005)

3
Observations Solar
Magnetogram of solar surface shows radial
component of the Suns magnetic field. Active
regions Sunspot pairs and sunspot groups. Strong
magnetic fields seen in an equatorial band
(within 30o of equator). Rotate with sun
differentially. Each individual sunspot lives
1 month. As cycle progresses appear closer to
the equator.
4
Sunspots
Dark spots on Sun (Galileo) cooler than
surroundings 3700K. Last for several days (large
ones for weeks) Sites of strong magnetic
field (3000G) Axes of bipolar spots tilted by
4 deg with respect to equator Arise in pairs
with opposite Polarity Part of the solar
cycle Fine structure in sunspot umbra and
penumbra
5
Observations Solar (a bit of theory)
Sunspot pairs are believed to be formed by the
instability of a magnetic field generated deep
within the Sun. Flux tube rises and breaks
through the solar surface forming active regions.
This instability is known as Magnetic
Buoyancy. It is also important in Galaxies
and Accretion Disks and Other Stars.
Wissink et al (2000)
6
Observations Solar
BUTTERFLY DIAGRAM last 130 years
Migration of dynamo activity from mid-latitudes
to equator
Polarity of sunspots opposite in each hemisphere
(Hales polarity law). Tend to arise in active
longitudes DIPOLAR MAGNETIC FIELD Polarity of
magnetic field reverses every 11 years. 22 year
magnetic cycle.
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Observations Solar

• 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.

8
Observations Solar
SUNSPOT NUMBER last 400 years
Maunder Minimum
Modulation of basic cycle amplitude (some
modulation of frequency) Gleissberg Cycle 80
year modulation MAUNDER MINIMUM Very Few Spots ,
Lasted a few cycles
Coincided with little Ice Age on
Earth
Abraham Hondius (1684)
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Observations Solar
RIBES NESME-RIBES (1994)
BUTTERFLY DIAGRAM as Sun emerged from minimum
Sunspots only seen in Southern Hemisphere
Asymmetry Symmetry soon
re-established. No Longer
Dipolar? Hence (Anti)-Symmetric modulation when
field is STRONG Asymmetric
modulation when field is weak
10
Observations Solar - helicities

• Important observational indices for dynamo theory
  are kinetic helicity , current helicity
  and magnetic helicity
• can be estimated from vector
  magnetograms
• At a given latitude, distribution about a mean
  value that is negative at in Northern Hemisphere
• is not easy to measure
• Suggestive that negative in NH (Berger
  Ruzmaikin 2000)
• Proxy of kinetic helicity can be measured (Duvall
  Gizon 2000) and is
  negative in the NH

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Observations Solar (Proxy)
PROXY DATA OF SOLAR MAGNETIC ACTIVITY AVAILABLE
SOLAR MAGNETIC FIELD MODULATES AMOUNT OF
COSMIC RAYS REACHING EARTH responsible
for production of terrestrial isotopes
stored in ice cores after 2 years in
atmosphere stored in tree rings after 30
yrs in atmosphere
BEER (2000)
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Observations Solar (Proxy)
Cycle persists through Maunder Minimum (Beer et
al 1998)
DATA SHOWS RECURRENT GRAND MINIMA WITH A WELL
DEFINED PERIOD OF 208 YEARS
Wagner et al (2001)
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Solar Structure
Solar Interior

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

Visible Sun

• Photosphere
• Chromosphere
• Transition Region
• Corona
• (Solar Wind)

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The Large-Scale Solar Dynamo

• Helioseismology shows the internal structure of
  the Sun.
• Surface Differential Rotation is maintained
  throughout the Convection zone
• Solid body rotation in the radiative interior
• Thin matching zone of shear known as the
  tachocline at the base of the solar convection
  zone (just in the stable region).

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Torsional Oscillations and Meridional Flows

• In addition to mean differential rotation there
  are other large-scale flows
• Torsional Oscillations
• Pattern of alternating bands of slower and faster
  rotation
• Period of 11 years (driven by Lorentz force)
• Oscillations not confined to the surface
  (Vorontsov et al 2002)
• Vary according to latitude and depth

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Torsional Oscillations and Meridional Flows

• Meridional Flows
• Doppler measurements show typical meridional
  flows at surface polewards velocity 10-20ms-1
  (Hathaway 1996)
• Poleward Flow maintained throughout the top half
  of the convection zone (Braun Fan 1998)
• No evidence of returning flow
• Meridional flow at surface advects flux towards
  the poles and is probably responsible for
  reversing the surface polar flux

17
Observations Stellar (Solar-Type Stars)
Stellar Magnetic Activity can be inferred by
amount of Chromospheric Ca H and K
emission Mount Wilson Survey (see e.g.
Baliunas ) Solar-Type Stars show a
variety of activity.
Cyclic, Aperiodic, Modulated, Grand Minima
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Observations Stellar (Solar-Type Stars)

• Activity is a function of spectral
  type/rotation rate of star
• As rotation increases activity increases
• 
  modulation increases
• Activity measured by the relative Ca II HK
  flux density
• 
  
• 
  
  (Noyes et al 1994)
• But filling factor of magnetic fields also
  changes
• 
  (Montesinos Jordan
  1993)
• Cycle period
• Detected in old slowly-rotating G-K stars.
• 2 branches (I and A) (Brandenburg et al 1998)
• WI 6 WA (including Sun)
• Wcyc/Wrot Ro-0.5
  (Saar Brandenburg 1999)

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Large and Small-scale dynamos

• LARGE SCALE
• Sunspots
• Butterfly Diagram
• 11-yr activity cycle
• Coronal Poloidal Field
• Systematic reversals
• Periodicities
• ------------------------------
• Field generation on scales
• gt LTURB

• SMALL SCALE
• Magnetic Carpet
• Field Associated with granular and supergranular
  convection
• Magnetic network
• ---------------------------------
• Field generation on scales
• LTURB

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Basics for the Sun
Dynamics in the solar interior is governed by
the following equations of MHD
INDUCTION
MOMENTUM
CONTINUITY
ENERGY
GAS LAW
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Basics for the Sun
BASE OF CZ
PHOTOSPHERE
(Ossendrijver 2003)
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Modelling Approaches

• Because of the extreme nature of the parameters
  in the Sun and other stars there is no obvious
  way to proceed.
• Modelling has typically taken one of three forms
• Mean Field Models (85)
• Derive equations for the evolution of the mean
  magnetic field (and perhaps velocity field) by
  parametrising the effects of the small scale
  motions.
• The role of the small-scales can be investigated
  by employing local computational models
• Global Computations (1)
• Solve the relevant equations on a
  massively-parallel machine.
• Either accept that we are at the wrong parameter
  values or claim that parameters invoked are
  representative of their turbulent values.
• Maybe employ some sub-grid scale modelling e.g.
  alpha models
• Low-order models
• Try to understand the basic properties of the
  equations with reference to simpler systems (cf
  Lorenz equations and weather prediction)
• All 3 have strengths and weaknesses

23
Kinematic Mean Field Theory
24
For simplicity, ignore large-scale flow, for the
moment.
Induction equation for mean field
where mean emf is
This equation is exact, but is only useful if we
can relate
to
Consider the induction equation for the
fluctuating field
Where pain in
the neck term
25
Traditional approach is to assume that the
fluctuating field is driven solely by the
large-scale magnetic field.
i.e. in the absence of B0 the fluctuating field
decays. i.e. No small-scale dynamo (not really
appropriate for high Rm turbulent fluids)
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Postulate an expansion of the form
where aij and ßijk are pseudo-tensors, determined
by the statistics of the turbulence.
Simplest case is that of isotropic turbulence,
for which aij adij and ßijk ßeijk. Then mean
induction equation becomes
BUT WHAT ARE a and b ? MORE LATER
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BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS
Add back in the mean flow U0 and the mean field
equation becomes
Now consider simplest case where a a0 cos q and
U0 U0 sin q ef In contrast to the induction
equation, this can be solved for
axisymmetric mean fields of the form
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BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

• Linear growth-rate of B0 depends on dimensionless
  combination of parameters.
• Critical parameter given by
• If D gt Dc then exponentially growing solutions
  are found dynamo action.
• Estimates suggest Da 2, DW 103 for the
  Sun and hence one can make the aW-approximation
  where the a-effect is ignored in generating the
  toroidal field.
• Can also have a2W and a2 dynamos may be of
  relevance for fully convective or more rapidly
  rotating stars.

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BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

• In general B0 takes the form of an exponentially
  growing dynamo wave that propagates.
• Direction of propagation depends on sign of
  dynamo number D.
• If D gt 0 waves propagate towards the poles,
• If D lt 0 waves propagate towards the equator.
• In this linear regime the frequency of the
  magnetic cycle Wcyc is proportional to D1/2
• Solutions can be either
• dipolar or quadrupolar

30

• Crucial questions
• Mean field electrodynamics therefore seems to
  work very well -
• but there are some very obvious questions to ask
• How can we calculate a and b? What will these be
  in the Sun.
• Can we relate them to the properties of the
  flow in the
• kinematic regime?
• 2. Even if we know how a and b behave
  kinematically, what is the
• role of the Lorentz force on the transport
  coefficients a and ß?
• 3. How weak must the large-scale field be in
  order for it to be
• dynamically insignificant? Dependence on
  Rm?

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1. How can we calculate a and b? Can we relate
them to the properties of the flow in the
kinematic regime?

• Of course a and b can only really be calculated
  by determining
• But we can only know b if we solve the
  fluctuating field equation.
• Analytic progress can be made by making one of
  two approximations
• Either Rm or the correlation time of the
  turbulence tcorr is small.
• Then can ignore pain in the neck G term in
  fluctuating field equation.
• Get famous results that a is related to the
  helicity of the flow with a constant of
  proportionality given by the small parameter e.g.

• Note we have parameterised correlations between u
  and b by correlations between u and w

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1. How can we calculate a and b? Can we relate
them to the properties of the flow in the
kinematic regime?

• We could do some numerical experiments and simply
  measure a
• The best way to do this is to impose a known mean
  field B and then calculate numerically
• If the Lorentz force is switched off (the field
  is weak) then this gives a kinematic calculation
  of a.
• Example Choose a flow with Rm not small and not
  at short correlation time and simply evaluate a.
• So we solve the kinematic induction equation
• With an applied mean field to calculate E.
• Here we choose u to be the famous G-P flow

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1. How can we calculate a and b? Can we relate
them to the properties of the flow in the
kinematic regime?

• For this flow the a term is a tensor.
• The a-effect is a very sensitive function of Rm.
• It even changes sign.
• It can in no way be related in a simple manner to
  the helicity of the flow (bit of a strange flow
  as it has infinite correlation time)
• Neither of the approximations work very well at
  high Rm
• Changes in correlation times may change results

a
g
Rm
Rm
Courvoisier et al 2004
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Rotating turbulent convection (Cattaneo
Hughes 2005)
Sometimes it is not even possible to calculate a
turbulent a-effect
Boussinesq convection. Taylor number, Ta
4O2d4/?2 5 x 105, Ro 1/10-1/5 Prandtl
number Pr ?/? 1, Magnetic Prandtl number Pm
?/? 5. Critical Rayleigh number 59
008. Anti-symmetric helicity distribution
anti-symmetric a-effect. Maximum
relative helicity 1/3.
35
Ra 150 000 Weak imposed field in
x-direction. Temperature on a horizontal slice
close to the upper boundary.
Ra 150,000. No dynamo at this Rayleigh
number but still an a-effect. Mean field of
unit magnitude imposed in x-direction
(essentially kinematic) Self-consistent dynamo
action sets in at Ra ? 200,000.
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e.m.f. and time-average of e.m.f. Ra
150,000 Imposed Bx 1. Imposed field
extremely weak kinematic regime.
time
time
time
37
Cumulative time average of the e.m.f. Not
fantastic convergence.
a the ratio of e.m.f. to applied magnetic
field is very small. And even depends on Rm!!!
Note this is not a-suppression (field too
weak) It appears that the a-effect here is not
turbulent (i.e. fast), but diffusive (i.e. slow).
However in models of rotating compressible
convection, Ossendrijver et al (2002) find a
significant a-effect -- fewer cells in box?
-- less turbulent? -- less incoherence
(decoherence)?
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No evidence of significant energy in the large
scales either in the kinematic eigenfunction
or in the subsequent nonlinear evolution. Picture
entirely consistent with an extremely feeble
a-effect.
Note Jones Roberts (2002) find a large mean
field. As you go to bigger boxes and more cells
it is harder to get a mean field or measure an
a-effect
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2. How are a and b modified by the mean field in
the Nonlinear Regime?

• This is a CRUCIAL question.
• Assume kinematic theory is OK (hmm)
• The mean field ltBgt will act back on the
  turbulence so as to switch off the generation
  mechanism via the Lorentz Force.
• When does this happen?
• Traditional argument
• This occurs when mean field reaches equipartition
  with the turbulence so

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2. How are a and b modified by the mean field in
the Nonlinear Regime?

• But
• It is the small scale magnetic field that will
  act back on the small-scale turbulence.
• The dynamo will switch off when the small-scale
  magnetic energy becomes comparable with the
  small-scale kinetic energy of the flow.
• There are many different possibilities, but it
  seems clear that due to amplification by the
  turbulence the small scale magnetic field is much
  bigger than the mean magnetic field

From a simple scaling it follows that where p
is a flow and geometry dependent coefficient
(pgt0)
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2. How are a and b modified by the mean field in
the Nonlinear Regime?

• This poses a major problem for mean field theory
  (see Proctor 2003Diamond et al 2004 for an
  erudite discussion)
• If true then this implies that the a-effect (and
  probably the b-effect) is switched off when the
  mean magnetic field is small (i.e. when
• Hence the source term (a) will be
  catastrophically quenched when the mean field is
  very small.
• Is this correct?
• Two ways of checking
• Analytical results based on approximations
• Numerical results at moderate Rm

42
2. How are a and b modified by the mean field in
the Nonlinear Regime? ANALYTICAL RESULTS

• Really again we have to solve the induction
  equation for b and the momentum equation for u to
  calculate a and b via E
• But some analytical progress is made by following
  two strands
• Take some exact results
• e.g. Integrated Ohms Law
• Conservation of magnetic helicity due to small
  scales
• General Conservation of magnetic helicity (e.g.
  Brandenburg Dobler 2001)

43
2. How are a and b modified by the mean field in
the Nonlinear Regime? ANALYTICAL RESULTS

• The second strand is to combine these exact
  results (or modifications thereof) with an
  approximate result for a in the nonlinear regime
• Note This is not an exact result.
• It is derived using the EDQNM approximation
• Only applies (if at all) for short correlation
  times and assumes that the magnetic field does
  not affect the correlation time of the
  turbulence.
• Also have to be careful about the meanings of
  in this formula (Proctor 2003)

Pouquet, Frisch Léorat 1976
44
2. An aside what do Pouquet, Frisch and Léorat
actually do?(Proctor 2003)

• They take a state of pre-existing helical MHD
  turbulence which is happily minding its own
  business with small scale b and u (but no large
  scale field?).
• They add a weak mean field to this and
  linearise the equations to get equations for the
  perturbations to u and b (u and b)
• They make a short correlation time approximation
  to say that

45
2. How are a and b modified by the mean field in
the Nonlinear Regime? ANALYTICAL RESULTS

• By combining these two results (or similar) it is
  possible to get formulae for a (and b)
• Gruzinov and Diamond 1994,1995, 1996
• Blackman Field 2000, Blackman Brandenburg
  2002
• Kleeorin et al 1995
• And many more
• e.g.

46
2. How are a and b modified by the mean field in
the Nonlinear Regime? NUMERICAL RESULTS

• As in the kinematic regime
  can be calculated numerically and related to an
  applied mean field
• This can be done for forced flows or for
  convection for various values of and Rm.
• e.g. for Galloway-Proctor flow solve

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Components of e.m.f. versus time.
48
(No Transcript)
49
Mean Field Hydrodynamics

• Of course, mean field theory can be played on the
  Navier-Stokes equations (see e.g. Rüdiger 1989).
• Solve equations for mean flow and parameterise
  small-scale interactions.
• This is even more dodgy as there is no closure
  that relates the small scale flows to the mean
  velocity.
• Also have to worry about Galilean Invariance of
  equations

Reynolds Stress
Maxwell Stress
Mean Lorentz Force
50
Other Possible Mechanisms for Producing Poloidal
Field

• In addition to the conventional turbulent driven
  a-effect, there have been other mechanisms
  suggested for generating a large scale poloidal
  field
• Most of these are dynamic and rely on the
  presence of a large-scale toroidal field.

51
Other Possible Mechanisms for Producing Poloidal
Field

• Poloidal field generated by magnetic buoyancy
  instability in connection with rotation or shear
• Either the instability of (thin) magnetic flux
  tubes
• Or more likely the instability of a layer of
  magnetic field (cf Nics talk)
• Joint Instability of field and differential
  rotation in the tachocline (Gilman, Dikpati etc)
• Produces a mean flow with a net helicity
• Decay and dispersion of tilted active regions at
  the solar surface (Babcock-Leighton mechanism)

52
TURBULENT CONVECTION
ROTATION
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
53
TURBULENT CONVECTION
ROTATION
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
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TURBULENT CONVECTION
ROTATION
Reynolds Stress
ltui ujgt
L-effect
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
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TURBULENT CONVECTION
ROTATION
Reynolds Stress
ltui ujgt
L-effect
HELICAL/CYCLONIC CONVECTION u
LARGE-SCALE MAG FIELD ltBgt
W-effect
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
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TURBULENT CONVECTION
ROTATION
Reynolds Stress
ltui ujgt
L-effect
Turbulent EMF
E ltu x bgt
Turbulent amplification of ltBgt
a,b,g-effect
LARGE-SCALE MAG FIELD ltBgt
W-effect
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
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TURBULENT CONVECTION
ROTATION
Reynolds Stress
ltui ujgt
L-effect
Maxwell Stresses
L-quenching
Turbulent EMF
E ltu x bgt
Turbulent amplification of ltBgt
a,b,g-effect
Small-scale Lorentz force
a-quenching
LARGE-SCALE MAG FIELD ltBgt
W-effect
Large-scale Lorentz force
STRONG LARGE SCALE SUNSPOT FIELD ltBTgt
Malkus-Proctor effect
58
Recent Mean Field Modelling

• We have seen that basic mean field modelling
  works well at a fundamental level (gives
  migrating wave solutions, oscillatory dipoles
  etc)
• But we do not have a deep understanding of the
  nature of the mean field coefficients
• amplitude
• dependence on field strength
• form in the Sun and other stars
• How can we proceed to gain an understanding of
  the nature of the solar dynamo?

59
Split Infinitives and Mean field Modelling

• Fowler's remark on the split infinitive is
  well-known
• "The English-speaking world may be divided into
  those who neither know nor care what a split
  infinitive is, those who don't know, but care
  very much, those who know and approve, those who
  know and condemn, and those who know and
  distinguish."

• The same can be said about mean field modelling
• The world may be divided into four categories
• Those who neither know nor care about
  catastrophic a-quenching.
• Those that seem to know (theyve been to enough
  meetings!), but dont care.
• Those who know and care, but do some mean field
  modelling
• Those who know and condemn

60
Illustrative vs Imitative Modelling

• Illustrative modelling.
• Investigate the mathematical properties of the
  mean field equations to try to understand how
  these equations behave as the inputs are varied.
• How does L-quenching affect the dynamo?
• How do dipole modes interact with quadrupole
  modes in the nonlinear regime?
• What would be the consequences of catastrophic
  a-quenching?
• How do transport coefficients affect the period,
  amplitude and direction of travel of dynamo
  solutions?
• What processes might lead to modulation and Grand
  Minima?
• What are the effects of poles and boundary
  conditions
• What is the underlying mathematical structure of
  such equations linear and nonlinear behaviour
• Which (if any) of these results are robust?

61
Illustrative vs Imitative Modelling

• Imitative modelling.
• Try to put as many effects as possible into a
  model to reproduce as many of the features of the
  solar cycle as possible
• There are a lot of observations to be matched.
• Cycle length, cycle amplitude, migration of
  magnetic activity, phase relation between
  poloidal and toroidal fields, active latitudes,
  active longitudes, form of individual cycles,
  torsional oscillations.
• There are a lot of free parameter to play around
  with.
• form of quenching catastrophic, regular, dynamic
• Other form of nonlinearity Malkus-Proctor
  effect, L-quenching
• Effects due to magnetic buoyancy

62
Distributed Dynamo Scenario

• Here the poloidal field is generated throughout
  the convection zone by the action of cyclonic
  turbulence.
• Toroidal field is generated by the latitudinal
  distribution of differential rotation.
• No role is envisaged for the tachocline
• Angular momentum transport would presumably be
  most effective by Reynolds and Maxwell stresses

63
Distributed Dynamo Scenario

• PROS
• Scenario is possible wherever convection and
  rotation take place together
• CONS
• Computations show that it is hard to get a
  large-scale field
• Mean-field theory shows that it is hard to get a
  large-scale field (catastrophic a-quenching)
• Buoyancy removes field before it can get too
  large

64
Flux Transport Scenario

• Here the poloidal field is generated at the
  surface of the Sun via the decay of active
  regions with a systematic tilt (Babcock-Leighton
  Scenario) and transported towards the poles by
  the observed meridional flow
• The flux is then transported by a conveyor belt
  meridional flow to the tachocline where it is
  sheared into the sunspot toroidal field
• No role is envisaged for the turbulent convection
  in the bulk of the convection zone.

65
Flux Transport Scenario

• PROS
• Does not rely on turbulent a-effect therefore all
  the problems of a-quenching are not a problem
• Sunspot field is intimately linked to polar field
  immediately before.
• CONS
• Requires strong meridional flow at base of CZ of
  exactly the right form
• Ignores all poloidal flux returned to tachocline
  via the convection
• Effect will probably be swamped by a-effects
  closer to the tachocline
• Relies on existence of sunspots for dynamo to
  work (cf Maunder Minimum)

66
Modified Flux Transport Scenario

• In addition to the poloidal flux generated at the
  surface, poloidal field is also generated in the
  tachocline due to an MHD instability.
• No role is envisaged for the turbulent convection
  in the bulk of the convection zone in generating
  field
• Turbulent diffusion still acts throughout the
  convection zone.

67
Interface Dynamo scenario

• The dynamo is thought to work at the interface of
  the convection zone and the tachocline.
• The mean toroidal (sunspot field) is created by
  the radial diffential rotation and stored in the
  tachocline.
• And the mean poloidal field (coronal field) is
  created by turbulence (or perhaps by a dynamic
  a-effect) in the lower reaches of the convection
  zone

68
Interface Dynamo scenario

• PROS
• The radial shear provides a natural mechanism for
  generating a strong toroidal field
• The stable stratification enables the field to be
  stored and stretched to a large value.
• As the mean magnetic field is stored away from
  the convection zone, the a-effect is not
  suppressed
• Separation of large and small-scale magnetic
  helicity
• CONS
• Relies on transport of flux to and from
  tachocline how is this achieved?
• Delicate balance between turbulent transport and
  fields.
• Painting ourselves into a corner

69
The Tachocline

• In two of these scenarios the tachocline plays a
  key role in the dynamo process.
• Generates toroidal field through shear
• Stores the strong magnetic field in a stably
  stratified layer.
• The dynamics of this layer is therefore of
  fundamental importance for the solar dynamo.
• Strongly magnetised
• Strong differential rotation
• Strong stratification
• Many instabilities
• Joint instability of differential rotation and
  toroidal field (cf MRI)
• Magnetic Buoyancy Instabilities
• Shear Flow Instabilities

70
Robust Results from Mean Field Models

• There have been an infinite number of mean field
  models of the solar dynamo.
• Are there any results that tend to occur no
  matter what assumptions are put into the model?
• These can emerge from considerations of the
  underlying mathematical structure of the
  equations coupled to the physical context
• or from simply doing lots of runs and seeing
  what sorts of things emerge

71
Robust Results from Mean Field Models

• For aW-dynamo equations fields can appear in
  either an oscillatory or stationary bifurcation.
• Steady modes are favoured when the a-effect is
  more prevalent.
• Separation of regions of a and W tends to lead to
  oscillatory modes
• Dynamo lengthscales tend to be of the order of
  the region of generation
• Dynamo waves tend to follow lines of constant
  rotation
• If generated in convection zone either propagate
  radially or are steady
• If generated in tachocline propagate
  latitudinally.
• For oscillatory modes if Dlt0 fields migrate
  towards the equator
• This is not always the case though (even without
  meridional flows)
• Meridional Flows of 1ms-1 near the are able to
  change the direction of propagation of dynamo
  waves.

72
Robust Results from Mean Field Models

• Amplitude of dynamo solutions tends to increase
  as (D-Dc) increases.
• Initially B2 increases linearly with D-Dc
• Saturation of amplitude as move to more
  supercritical dynamo numbers.
• As (D-Dc) increases solution becomes more
  irregular.
• Sequence of bifurcations leading to chaotic
  solutions in general
• Spatio-temporal fragmentation of solutions

73
Robust Results from Mean Field Models

• In the absence of a strong meridional flow the
  a-effect must be localised fairly close to the
  equator in order to get dynamo waves localised
  near the equator (e.g. Markiel Thomas 1999
  Markiel 2000 Bushby 2004)
• In the absence of a meridional flow a-effects
  near to the tachocline are more effective than
  those close to the surface in generating strong
  magnetic fields (e.g. Mason et al 2003)
• The addition of transport effects (e.g. magnetic
  buoyancy or magnetic pumping as well as
  meridional flows) that moves around flux has a
  significant effect on the cycle period and phase
  relation of the dynamo.
• Stochastic effects can be very important when the
  field is weak (e.g. Hoyng 1993, Ossendrijver
  Hoyng 1996)
• e.g. when DDc or when the dynamo has been
  modulated by nonlinear effects.

74
Robust Results from Mean Field Models

• In addition to the equatorial branch of solutions
  a polar branch of poleward propagating modes are
  often also found (e.g. Bushby 2004)
• These are usually weaker.

75
Robust Results from Mean Field Models

• The presence of the poles and the equator means
  that dynamo action is harder to excite than for
  dynamo waves (see e.g. Worledge et al 1997)
• The frequency of dynamo waves in the linear and
  nonlinear regime is a sensitive function of the
  amplitude of the solution, the frequency rapidly
  moves away from that of the kinematic
  eigenfunctions (problem for relating stellar
  dynamo periods to dynamo calculations see later)

76
Robust Results from Mean Field Models

• In the nonlinear regime, linear eigenfunctions
  (dipoles and quadrupoles) can interact to give
  modes that are neither symmetric nor
  antisymmetric about the equator (mixed modes)
  (Jennings Weiss 1991)
• These modes can interact in an interesting way
  for sufficiently large D, yielding doubly
  periodic and chaotic solutions (see e.g.
  Brandenburg et al 1989)

77
Robust Results from Mean Field Models

• In the nonlinear regime, dynamic nonlinearities
  (e.g. Malkus-Proctor effect, L-quenching, dynamic
  a-effect) automatically leads to modulation of
  basic cycle (e.g. Tobias 1996, 1997 Küker et al
  1999)
• Introduces another time-scale to the problem and
  also complex dynamics.
• Continual exchange of energy between mean field
  and mean flows.
• Also leads naturally to the formation of
  torsional oscillations

Pipin 1999
78
Robust Results from Mean Field Models

• These two modulational effects can interact to
  lead to very complicated spatio-temporal dynamics
• Grand Minima
• Flipping of field parity.

Beer et al 1998
79
Robust Results from Mean Field Models

• Torsional oscillations are generated by dynamic
  nonlinearities that modify the mean differential
  rotation (MP-effect or L-quenching)
• These are a sensitive function of where the
  magnetic field is generated and the
  stratification of the medium.
• Can be generated at the base of the convection
  zone, and manifest themselves towards the
  surface. (Bushby 2004)

80
Low-order models Mathematical Aspects

• Mathematically the structure of the dynamo
  equations can be understood using the techniques
  of dynamical systems and symmetries of the
  problem.
• Reduced sets of ODEs can be derived by either
  truncating the PDEs, or by using a Normal Form
  Analysis.
• These low-order models can shed light on the
  bifurcation structure of the full PDEs

KNOBLOCH ET AL (1998)
81
Low-order models Mathematical Aspects

• For example the interaction between dipole and
  quadrupole modes in the nonlinear regime can be
  understood by considering the symmetries of the
  underlying model and the solutions (Knobloch
  Landsberg 1996)
• Furthermore the competition between the two types
  of modulation can also be studied.

PDE
ODE
KNOBLOCH ET AL (1998)
82
Predictability

• It has been claimed recently that we are now at
  the stage where we can start to make predictions
  about future activity from mean field models
  (Dikpati et al , 2004).
• Given the uncertainties of the input parameters
  and the chaotic nature of nonlinear systems, this
  is an interesting claim!

Bushby Tobias (2005)
83
Global Solar Dynamo Calculations

• Why not simply solve the relevant equations on a
  big computer?
• Large range of scales physical processes to
  capture.
• Early calculations could not get into turbulent
  regime dominated by rotation (Gilman Miller
  (1981), Glatzmaier Gilman (1982), Glatmaier
  (1985a,b) )
• Calculations on massively parallel machines are
  now starting to enter the turbulent MHD regime.
• Focus on interaction of rotation with convection
  and magnetic fields.

Brun, Miesch Toomre (2004)
84
Global Solar Dynamo Calculations

• Computations in a spherical shell of
  (magneto)-anelastic equations
• Filter out fast magneto-acoustic modes but
  retains Alfven and slow modes
• Spherical Harmonics/Chebyshev code
• Impenetrable, stress-free, constant entropy
  gradient bcs

85
Global Computations Hydrodynamic State

• Moderately turbulent Re 150
• Low latitudes downflows align with rotation
• High latitudes more isotropic
• Coherent downflows transport angular momentum
• Reynolds stresses important
• Solar like differential rotation profile

86
Global Computations Dynamo Action

• For Rm gt 300 dynamo action is sustained.
• ME 0.07 KE

• Br is aligned with downflows
• Bf is stretched into ribbons

87
Global Computations Saturation

• Magnetic energy is dominated by fluctuating field
• Means are a lot smaller
• ltBTgt 3 ltBPgt

• Dynamo equilibrates by extracting energy from the
  differential rotation
• Small scale field does most of the damage!
• L-quenching

88
Global Computations Structure of Fields

• The mean fields are weak and show little
  systematic behaviour

• The field is concentrated on small scales with
  fields on smaller scales than flows

89
Stellar Dynamos

• Most of the dynamo modelling effort has naturally
  focussed on the Sun.
• Some progress has been made in describing the
  dynamo action in 3 other classes of stars
• Solar-type stars moderate rotators with deep
  convective envelopes
• Rapidly rotating stars
• Fully Convective stars

90
Solar-like stars

• Try to use observations to calibrate solar dynamo
  models e.g. measure magnetic field amplitude and
  cycle period and try to infer the behaviour of a
  as a function of Ro.
• Traditional to use mean field theory with various
  assumptions.
• Problem results sensitive to assumptions
• a Bn 0.3 lt n lt 1.5
• h Bm m 0.75 (Saar
  Brandenburg 1999)

91
Solar-like stars

• Frequency changes due to 2 effects
• (a) Changes in length-scale
• (b) changes in dynamo wavespeed
• sensitive to transport effects
• Get different dependence for different
  nonlinearities.

Tobias (1998)
92
Solar-like stars
Tobias (1998)

• Interestingly for all the nonlinear mechanisms
  considered the change in frequency from its
  linear value has the same dependence as a
  function of amplitude of solution!

93
Rapidly Rotating stars

• It is very dangerous to extrapolate the results
  from solar mean field models to stars that rotate
  much more rapidly
• For rapidly rotating stars DW/W is much smaller
  than for the Sun and so differential rotation is
  likely to play less of a role.
• Rapid rotation means that Reynolds stresses are
  less likely to be able to transport angular
  momentum away from rotation being constant on
  cylinders.

94
Rapidly Rotating stars

• For a mean field model with plausible
  parametrisations of a and W it is possible to
  determine the role of changing the nature of the
  differential rotation profile.
• Dynamo is more likely to be of a2W type.
• Tangent cylinder plays a large role in confining
  magnetic activity to high latitudes.
• Polar branch much more pronounced (cf polar spots)

Bushby (2003)
95
Fully Convective stars

• For fully convective stars (e.g. fully convective
  T-Tauri stars) magnetic fields are still
  observed.
• With the absence of a stable layer and strong
  shear it is difficult to see how a strong mean
  field can be built up (cf fully computational
  models)
• Dynamo action is likely to be small-scale.
• If a large scale field can be generated, then
  mean field theory indicates that this is likely
  to be steady and perhaps non-axisymmetric.

Küker Rüdiger
96
Conclusions/Speculations and Annoying Questions

• Why does mean-field theory work so well?
• Input parameters need to be constrained
• Requires a full understanding of MHD turbulence
• Turbulent a-effect
• Turbulent diffusion
• Measurement of mean flows.
• What can serious(?) computations teach us
• Small scale (parts of the jigsaw)
• Large scale (global dynamics)
• We can learn a lot from the mathematical
  structure of the equations.