1. Solar Interior Magnetic Fields and Dynamos
Steve Tobias (Leeds)
5th Potsdam Thinkshop, 2007

2. Fields, flows and activity
Observations
Fields, flows and activity

3. Fields, flows and activity
Large-scale activity
Fields, flows and activity

4. Observations: Solar Magnetogram of solar surface
shows radial component of the
Sun’s 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)
Joy’s Law: Axes of bipolar spots tilted by ~4 deg with respect to equator
Hale’s 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 (Hale’s 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 BUTTERFLY DIAGRAM: as Sun emerged from minimum
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 10 Be C 14
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 Core Radiative Interior (Tachocline)
Convection Zone
Visible Sun
Photosphere
Chromosphere
Transition Region
Corona
(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
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 (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 + Ohm’s 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. Cowling’s Theorem (1934) Why is dynamo Theory so hard?
Why are there no nice analytical solutions?
Why don’t 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 Cowling’s 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
1020
1013
1010
10-7
105
10-3
10-4
0.1-1
1016
1012
106
10-7
10-6 1
(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 Earth’s magnetic field is also generated by a dynamo located in its outer fluid core.
The Earth’s magnetic field reverses every 106 years on average.
Conditions in the Earth’s 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
Alpha-effect
Omega-effect
Turbulent diffusivity
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 > 0 waves propagate towards the poles,
If D < 0 waves propagate towards the equator.
In this linear regime the frequency of the magnetic cycle Wcyc is proportional to |D|1/2
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 Sun’s dynamo and reproduced the amplitudes of the last eight cycles with unprecedented accuracy (RMS error < 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
Hathaway’s 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 (Rm~256 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 > 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
<BT> ~ 3 <BP>
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 ActionCZ
Stable
Pr=0.25, 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…