Solar Interior Magnetic Fields and Dynamos

1
Solar Interior Magnetic Fields and Dynamos

• Steve Tobias (Leeds)

5th Potsdam Thinkshop, 2007
2
Observations

• Fields, flows and activity

3
Large-scale activity

• Fields, flows and activity

4
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.
5
Sunspots
Dark spots on Sun (Galileo) cooler than
surroundings 3700K. Last for several days (large
ones for weeks) Sites of strong magnetic
field (3000G) Joys Law Axes of bipolar spots
tilted by 4 deg with respect to equator Hales
Law Arise in pairs with opposite polarity Part
of the solar cycle Fine structure in
sunspot umbra and penumbra
SST
6
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-
we are just beginning to understand how strong
coherent tubes may form from weaker layers of
field.
Kersalé et al (2007)
7
Observations Solar (a bit of theory)
Once structures are formed they rise and break
through the solar surface to form active regions
this process is not well understood e.g. why
are sunspots so small?
8
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.
9
Three solar cycles of sunspots
Courtesy David Hathaway
10
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 (see next slide)
• Global Magnetic field reversal.

11
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.

12
Observations Solar
SUNSPOT NUMBER last 400 years
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)
13
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
14
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)
15
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 Distribution of
maxima in activity is consistent with a Gamma
distribution. we have a current maximum
life expectancy for this is short (Abreu et
al 2007)
Wagner et al (2001)
16
Solar Structure
Solar Interior

1. Core
2. Radiative Interior
3. (Tachocline)
4. Convection Zone

Visible Sun

1. Photosphere
2. Chromosphere
3. Transition Region
4. Corona
5. (Solar Wind)

17
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).

18
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

19
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)
• Large fluctuations about this mean with often
  evidence of multiple cells and strong temporal
  variation with the solar cycle (Roth 2007)
• No evidence of returning flow
• Meridional flow at surface advects flux towards
  the poles and is probably responsible for
  reversing the surface polar flux

20
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
21
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)

22
I (i) Small-scale activity

• Fields and flows and activity

23
Small-Scale dynamo action the magnetic carpet
24
Basic Dynamo Theory
Dynamo theory is the study of the generation
of magnetic field by the inductive motions
of an electrically conducting plasma.
Non-relativistic Maxwell equations Ohms Law
Navier-Stokes equations
25
Basic Dynamo Theory
Dynamo theory is the study of the generation
of magnetic field by the inductive motions
of an electrically conducting plasma.
Induction Eqn
Momentum Eqn
Including Rotation, Gravity etc
Nonlinear in B
A dynamo is a solution of the above system for
which B does not decay for large times. Hard to
find simple solutions (antidynamo theorems)
26
Cowlings Theorem (1934)

• Why is dynamo Theory so hard?
• Why are there no nice analytical solutions?
• Why dont we just solve the equations on a
  computer?
• Dynamos are sneaky and parameter values are
  extreme

• It can be shown that a flow or magnetic field
  that is too simple (i.e. has too much symmetry)
  cannot lead to or be generated by dynamo action.
• The most famous example is Cowlings Theorem.
• No Axisymmetric magnetic field can be maintained
  by a dynamo

27
Basics for the Sun
Dynamics in the solar interior is governed by
the following equations of MHD
INDUCTION
MOMENTUM
CONTINUITY
ENERGY
GAS LAW
28
Basics for the Sun
BASE OF CZ
PHOTOSPHERE
(Ossendrijver 2003)
29
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 (5)
• 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

30
The Geodynamo

• The Earths magnetic field is also generated by a
  dynamo located in its outer fluid core.
• The Earths magnetic field reverses every 106
  years on average.
• Conditions in the Earths core much less
  turbulent and are approaching conditions that can
  be simulated on a computer (although rotation
  rate causes a problem).

31
Mean-field electrodynamics

• A basic physical picture

W-effect poloidal ? toroidal
32
Mean-field electrodynamics

• A basic physical picture

a-effect toroidal ? poloidal
poloidal ? toroidal
33
BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS
This can be formalised by separating out the
magnetic field into a mean (B0) and fluctuating
part (b) and parameterising the small-scale
interactions In their simplest form 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
34
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

35
Some solar dynamo scenarios

• Distributed, Deep-seated, Flux Transport,
  Interface, Near-Surface.
• This is simply a matter of choosing plausible
  profiles for a and b depending on your prejudices
  or how many of the objections to mean field
  theory you take seriously!

36
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

37
Near-surface Dynamo Scenario

• This is essentially a distributed dynamo
  scenario.
• The near-surface radial shear plays a key role.
• Magnetic features tend to move with rotation rate
  at the bottom of the near surface shear layer.
• Same pros and cons as before.
• Brandenburg (2006)

38
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.

39
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)

40
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.

41
Interface/Deep-Seated Dynamo

• 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

42
Interface/Deep-Seated Dynamo

• 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

43
Mean-field electrodynamics

• A basic physical picture

W-effect poloidal ? toroidal
44
Mean-field electrodynamics

• A basic physical picture

a-effect toroidal ? poloidal
poloidal ? toroidal
45
Some solar dynamo scenarios

• Distributed, Deep-seated, Flux Transport,
  Interface, Near-Surface.
• This is simply a matter of choosing plausible
  profiles for a and b depending on your prejudices
  or how many of the objections to mean field
  theory you take seriously!

46
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

47
Near-surface Dynamo Scenario

• This is essentially a distributed dynamo
  scenario.
• The near-surface radial shear plays a key role.
• Magnetic features tend to move with rotation rate
  at the bottom of the near surface shear layer.
• Same pros and cons as before.
• Brandenburg (2006)

48
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.

49
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)

50
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.

51
Interface/Deep-Seated Dynamo

• 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

52
Interface/Deep-Seated Dynamo

• 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

53
Predictions of Future activity
Dikpati, de Toma Gilman (2006) have fed sunspot
areas and positions into their numerical model
for the Suns dynamo and reproduced the
amplitudes of the last eight cycles with
unprecedented accuracy (RMS error lt 10). Recent
results for each hemisphere shows similar
accuracy.
Cycle 24 Prediction 160 15
54
Precursor Predictions
Precursor techniques use aspects of the Sun and
solar activity prior to the start of a cycle to
predict the size of the next cycle. The two
leading contenders are 1) geomagnetic activity
from high-speed solar wind streams prior to cycle
minimum and 2) polar field strength near cycle
minimum.
Geomagnetic Prediction 160 25 (Hathaway
Wilson 2006)
Polar Field Prediction 75 8 (Svalgaard,
Cliver, Kamide 2005)
55
Other Amplitude Indicators
Hathaways Law Big cycles start early and leave
behind a short period cycle with a high minimum
(courtesy David Hathaway).
Amplitude-Period Effect Large amp-litude cycles
are preceded by short period cycles (currently at
130 months ? average amplitude)
Amplitude-Minimum Effect Large amplitude cycles
are preceded by high minimum values (currently at
12.6 ? average amplitude)
56
Dynamo Predictions of solar activity

• No (in-depth) understanding of the solar dynamo
• Drive to make predictions
• Drive to tie dynamo theory in with observations
• Tempting to say
• Dynamo driven by what we see at the surface and
  we can use this to predict future activity
• Is this a useful thing to do?

Dikpati et al (2006)
57
Irregularity/Modulation

• Clearly if the cycle were periodic there would be
  no trouble predicting
• Difficulties in predicting arise owing to
  modulation of the basic cycle
• Only 2 possible sources for modulation
• Stochastic
• Deterministic
• (or a combination of the two)

58
Stochastic/Deterministic

• Stochastic modulation (see e.g. Hoyng 1992)
• can still arise even if the underlying physics is
  linear (good)
• Small random fluctuations cause modulation and
  have large effects (bad)
• Best of luck predicting using a physics based
  model.
• Deterministic Modulation (see e.g JWC85)
• Underlying physics nonlinear (bad)
• In best case scenario stochastic fluctuations
  have small effects (shadowing)

59
Prediction from mean-field models

• Stochastic modulation
• Choose a linear flux transport dynamo
• perturb stochastically
• All predictability goes out of the window

Bushby Tobias ApJ 2007
60
Prediction from mean-field models
Bushby Tobias ApJ 2007

• Deterministic modulation
• Long-term predictability is impossible owing to
  sensitive dependence on initial conditions (even
  with exactly the right model)
• Short-term prediction relies on having the model
  exactly correct (sensitivity to model parameters)
• Even if fitted over a large number of cycles

61
Global solar dynamo models

• Large-scale computational dynamos, with and
  without tachoclines

62
Numerics

• Most dynamo models of the future will be solved
  numerically.
• There is a need for
• An understanding of the basic physics via simple
  models
• Careful numerics that does not claim to do what
  it can not.
• The dynamo problem is notoriously difficult to
  get right even the kinematic induction
  equation.
• The history of dynamo computing is littered with
  examples of incorrect results (even famously
  Bullard Gellman).

63
Numerics a list of rules

• Any code that relies on numerical dissipation
  (e.g. ZEUS) will not get dynamo calculations
  correct
• It is vital to treat the dissipation correctly
  (be very careful with hyperdiffusion)
• Unfortunately, if a calculation is under-resolved
  then it may lead to dynamo action when there is
  no dynamo.
• Non-normality of dynamo equations means that
  equations have to be integrated for a long time
  to ensure dynamo action (ohmic diffusion times)
• As a rule of thumb can tell the maximum
  possible Rm by simply knowing the resolution they
  use and the form of the flow.
• Be sceptical of all claims of super-high Rm
  (Rm256 requires at least 963 fourier modes or
  more finite difference points)
• Doubling the resolution buys you a fourfold
  increase in Rm but costs 16 times as much for a
  3d calculation.

64
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)
65
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

66
Global solar dynamo models

• Distributed dynamo computations

67
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
• Meridional flow profiles multiple cells,
  time-dependent

68
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

69
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

70
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

71
Global solar dynamo models

• Addition of a forced tachocline

72
Global Computations Hydrodynamic State

• Tachocline is forced using drag force.
• Convection is allowed to evolve.
• Again get latitudinal differential rotation
• Bit now have radial differential rotation in the
  tachocline as well.
• 13 differential rotation (reduced from non-pen)

73
Global Computations Dynamo Action
CZ
Stable

• Pr0.25, Pm 8
• Strong fluctuating fields 3000G in CZ
• Time averaged ? 300G
• In stable layer field is organised
• Opposite polarity in northern/southern hemisphere

74
Global Computations Dynamo Action

• Time averaged 3000G in stable layer (i.e. 10
  times that in CZ)
• How do you get such an organised systematic field
• Geometry? Rotation? Compressibility (buoyancy?)
• See later