Solar Interior Magnetic Fields and Dynamos

Solar Interior Magnetic Fields and Dynamos

Steve Tobias (Leeds)

5th Potsdam Thinkshop, 2007

Observations

Fields, flows and activity

Large-scale activity

Fields, flows and activity

Observations: SolarMagnetogram of solar surface shows radial component of theSun’s magnetic field.

Active regions: Sunspot pairsand sunspot groups.Strong magnetic fields seen in an equatorial band (within30o of equator).

Rotate with sun differentially.

Each individual sunspot lives~ 1 month.As “cycle progresses” appear closer to the equator.

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 equatorHale’s Law: Arise in pairs with opposite polarityPart of the solar cycleFine structure in sunspotumbra and penumbra

SST

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)

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?

Observations: Solar

BUTTERFLY DIAGRAM: last 130 yearsMigration 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 FIELDPolarity of magnetic field reverses every 11 years.22 year magnetic cycle.

Courtesy David Hathaway

Three solar cycles of sunspots

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.

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.

Observations: Solar

SUNSPOT NUMBER: last 400 years

Modulation of basic cycle amplitude (some modulation of frequency)Gleissberg Cycle: ~80 year modulationMAUNDER MINIMUM: Very Few Spots , Lasted a few cycles Coincided with little Ice Age on Earth

Abraham Hondius (1684)

Observations: Solar

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

RIBES & NESME-RIBES (1994)

Observations: Solar (Proxy)

SOLAR MAGNETIC FIELD MODULATES AMOUNT OF COSMIC RAYS REACHING EARTH responsible for production of terrestrial isotopes

PROXY DATA OF SOLAR MAGNETIC ACTIVITY AVAILABLE

: stored in ice cores after 2 years in atmosphere : stored in tree rings after ~30 yrs in atmosphere

10Be

C14

BEER(2000)

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” isconsistent with a Gamma distribution.

we have a current maximum – lifeexpectancy for this is short (Abreu et al2007)

Wagner et al (2001)

Solar Structure

Solar Interior

1. Core2. Radiative Interior3. (Tachocline)4. Convection Zone

Visible Sun

1. Photosphere2. Chromosphere3. Transition Region4. Corona5. (Solar Wind)

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).

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

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

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

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)

I ~ 6 A (including Sun)cyc/rot ~ Ro-0.5 (Saar & Brandenburg 1999)

HKR1 RoRHK

9.0 RoF

Observations: Stellar (Solar-Type Stars)

I (i) Small-scale activity

Fields and flows and activity

Small-Scale dynamo action – the magnetic carpet

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…

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

Nonlinear in B

A dynamo is a solution of the above system for whichB does not decay for large times.Hard to find simple solutions (antidynamo theorems)

IncludingRotation,Gravity etc

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”

Basics for the Sun

.

terms,loss)(

,0).(

,.

),0.()( 2

TRpDt

pD

t

pt

t

otherviscous

u

FFgBjuuu

BBBuB

Dynamics in the solar interior is governed by the following equations of MHD

INDUCTION

MOMENTUM

CONTINUITY

ENERGY

GAS LAW

Basics for the Sun

PHdgRa

4

ULRe

ULRm

Pr

Pm

LURo 2

scUM

202

Bp

1020

1013

1010

10-7

105

10-3

10-4

0.1-1

1016

1012

106

10-7

10-6

1

110-3-0.4

BASE OF CZ PHOTOSPHERE

(Ossendrijver 2003)

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

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).

Mean-field electrodynamics

A basic physical picture

-effect – poloidal toroidal



-effect – toroidal poloidal poloidal toroidal

.)() 02

0000 BBUB

B

(

t

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 interactionsIn their simplest form the mean field equation becomes

Now consider simplest case where = 0 cos andU0 = U0 sine

In contrast to the induction equation, this can be solved for axisymmetricmean fields of the form

)( 000 eeB Pt AB

Alpha-effect Omega-effect Turbulent diffusivity

• 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 cyc is proportional to |D|1/2

• Solutions can be either

dipolar or quadrupolar

BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

Some solar dynamo scenarios

Distributed, Deep-seated, Flux Transport, Interface, Near-Surface.

This is simply a matter of choosing plausible profiles for and depending on your

prejudices or how many of the objections to mean field theory you take seriously!

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 -quenching)

– Buoyancy removes field before it can get too large

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)

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.

Flux Transport Scenario• PROS

– Does not rely on turbulent-effect therefore all the problems of -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 “-effects” closer to the tachocline

– Relies on existence of sunspots for dynamo to work (cf Maunder Minimum)

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.

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 -effect) in the lower reaches of the convection zone

• 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 -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”

Interface/Deep-Seated Dynamo



-effect – poloidal toroidal



-effect – toroidal poloidal poloidal toroidal

Some solar dynamo scenarios

Distributed, Deep-seated, Flux Transport, Interface, Near-Surface.

This is simply a matter of choosing plausible profiles for and depending on your

prejudices or how many of the objections to mean field theory you take seriously!

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 -quenching)

– Buoyancy removes field before it can get too large

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)

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.

Flux Transport Scenario• PROS

– Does not rely on turbulent-effect therefore all the problems of -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 “-effects” closer to the tachocline

– Relies on existence of sunspots for dynamo to work (cf Maunder Minimum)

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.

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 -effect) in the lower reaches of the convection zone

• 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 -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”

Interface/Deep-Seated Dynamo

Dikpati, de Toma & Gilman (2006) have fed sunspot areas and Dikpati, de Toma & Gilman (2006) have fed sunspot areas and positions into their numerical model for the Sun’s dynamo and positions into their numerical model for the Sun’s dynamo and reproduced the amplitudes of the last eight cycles with unprecedented reproduced the amplitudes of the last eight cycles with unprecedented accuracy (RMS error < 10). Recent results for each hemisphere shows accuracy (RMS error < 10). Recent results for each hemisphere shows similar accuracy.similar accuracy.

Cycle 24 Prediction ~ 160 Cycle 24 Prediction ~ 160 ± 15± 15

Predictions of Future activityPredictions of Future activity

Precursor PredictionsPrecursor 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))

Other Amplitude IndicatorsOther Amplitude IndicatorsHathaway’s Law: Big cycles start early and leave behind a short Hathaway’s Law: Big cycles start early and leave behind a short period cycle with a high minimum (courtesy David Hathaway).period cycle with a high minimum (courtesy David Hathaway).

Amplitude-Period Effect: Large amp-Amplitude-Period Effect: Large amp-litude cycles are preceded by short litude cycles are preceded by short period cycles (currently at 130 period cycles (currently at 130 months months → average amplitude)→ average amplitude)

Amplitude-Minimum Effect: Large Amplitude-Minimum Effect: Large amplitude cycles are preceded by amplitude cycles are preceded by high minimum values (currently at high minimum values (currently at 12.6 12.6 → average amplitude)→ average amplitude)

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)

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)

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)

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

• 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

Prediction from mean-field models

Bushby & Tobias ApJ 2007

Global solar dynamo models

Large-scale computational dynamos, with and without tachoclines

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).

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.

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)

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


Distributed dynamo computations

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

Global Computations: Dynamo Action

• For Rm > 300 dynamo action is sustained.

• ME ~ 0.07 KE

• Br is aligned with downflows

• B is stretched into ribbons

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!

• -quenching

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


Addition of a forced tachocline

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)


• 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

CZ Stable


• 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…

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Solar Interior Magnetic Fields and Dynamos