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

Fields, flows and activity Observations Fields, flows and activity

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

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.

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

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 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.

Three solar cycles of sunspots Courtesy David Hathaway

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 modulation MAUNDER 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 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

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)

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)

Solar Structure Solar Interior Core Radiative Interior (Tachocline) Convection Zone Visible Sun Photosphere Chromosphere Transition Region Corona (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 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

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)

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 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)

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 Dynamics in the solar interior is governed by the following equations of MHD INDUCTION MOMENTUM CONTINUITY ENERGY GAS LAW

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 10-3-0.4 (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 W-effect – poloidal  toroidal

Mean-field electrodynamics A basic physical picture a-effect – toroidal  poloidal poloidal  toroidal

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

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

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!

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

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 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)

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 a-effect) in the lower reaches of the convection zone

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”

Mean-field electrodynamics A basic physical picture W-effect – poloidal  toroidal

Mean-field electrodynamics A basic physical picture a-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 a and b 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 a-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 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)

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 a-effect) in the lower reaches of the convection zone

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”

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

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)

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)

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

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

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

Global solar dynamo models 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 Bf 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! L-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

Global solar dynamo models 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)

Global Computations: Dynamo Action CZ 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

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…