Solar and Stellar dynamos

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

Talk Summary

• Observations • Some basic theory – Mean Field Electrodynamics – The simplest solutions – a -effect, a -quenching – Big argument – Other mechanisms for producing poloidal field • What physics do we need to understand?

• Cartoon Pictures of solar dynamo a -effect – Distributed – Flux conveyor – Interface Dynamos • Robust (general) results from mean field models • Mathematical considerations of the mean field equations • Low order models • Global Solar Dynamo Models • Specifically Stellar Models • See reviews by Ossendrijver (2003), Weiss & Tobias (2005)

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 30 o 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) Axes of bipolar spots tilted by ~4 deg with respect to equator Arise in pairs with opposite Polarity Part of the solar cycle Fine structure in sunspot umbra and penumbra

Observations Solar (a bit of theory)

Sunspot pairs are believed to be formed by the instability of a magnetic field generated deep within the Sun.

Flux tube rises and breaks through the solar surface forming active regions.

This instability is known as Magnetic Buoyancy.

It is also important in Galaxies and Accretion Disks and Other Stars.

Wissink et al (2000)

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.

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 .

Maunder Minimum

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)

RIBES & NESME-RIBES (1994)

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

Observations: Solar - helicities

• • Important observational indices for dynamo theory are kinetic helicity , current 

A.B



J r B r

 can be estimated from vector  magnetograms – At a given latitude, distribution about a mean value that is negative at in Northern Hemisphere • 

A.B

 is not easy to measure – Suggestive that negative in NH (Berger & Ruzmaikin 2000) • Proxy of kinetic helicity can be measured (Duvall & Gizon 2000) (  

u

)

z

(  .

u

) and is negative in the NH

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 10 Be 14 C 10 Be 14 C : stored in ice cores after 2 years in atmosphere : stored in tree rings after ~30 yrs in atmosphere 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 Wagner et al (2001)

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)

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) – 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

R



HK R



HK



Ro

 1 (Noyes et al 1994) But filling factor of magnetic fields also changes

F



Ro

 0 .

9 (Montesinos & Jordan 1993)

Cycle period

–Detected in old slowly-rotating G-K stars.

–2 branches (I and A) (Brandenburg et al 1998) W I ~ 6 W A W cyc / W rot (including Sun) ~ Ro -0.5

(Saar & Brandenburg 1999)

Large and Small-scale dynamos

LARGE SCALE SMALL SCALE Sunspots Butterfly Diagram 11-yr activity cycle Coronal Poloidal Field Systematic reversals Periodicities ----------------------------- Field generation on scales > L TURB Magnetic Carpet Field Associated with granular and supergranular convection Magnetic network -------------------------------- Field generation on scales ~ L TURB

Basics for the Sun

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

INDUCTION MOMENTUM CONTINUITY ENERGY GAS LAW



B



t

  

u



t

  (

u



B

) 

u

.



u

    2

B



p

   

t

  .( 

u

)

D

(

p

   ) 

p



Dt R



T

.

 0 , loss terms,

j



B

(  .

B

 0 ),  

g



F

viscous



F

other

,

Ra



Basics for the Sun

BASE OF PHOTOSPHERE

g

 

d

4 

H P

CZ

10 20 10 16

Re 

UL



Rm

Pr    

UL

  2   0

p B

2

Pm

  

M Ro

 

U U c s

2 W

L 10 13 10 10 10 -7 10 5 10 -3 10 -4 0.1-1 10 12 10 6 10 -7 1 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 (~1%) • Solve the relevant equations on a massively-parallel machine.

• Either accept that we are at the wrong parameter values or claim that parameters invoked are representative of their turbulent values.

• Maybe employ some “sub-grid scale modelling” e.g. alpha models – Low-order models • Try to understand the basic properties of the equations with reference to simpler systems (cf Lorenz equations and weather prediction) • All 3 have strengths and weaknesses

Kinematic Mean Field Theory Starting point is the magnetic induction equation of MHD: 

B



t

   (

u



B

)    2

B

, where

B

is the magnetic field,

u

is the fluid velocity and η is the magnetic diffusivity (assumed constant for simplicity).

Assume scale separation between large- and small-scale field and flow:

B



B

0 

b

,

U



U

0 

u

, where

B

and

U

vary on some large length scale

L

, and

u

and

b

vary on a much smaller scale

l.



B

 

B 0

, 

U

 

U 0

, where averages are taken over some intermediate scale

l « a « L.

For simplicity, ignore large-scale flow, for the moment.

Induction equation for mean field: 

B

0 

t

  

E

   2

B

0 , where mean emf is E  

u



b

 .

This equation is exact, but is only useful if we can relate E to

B

0 .

Consider the induction equation for the fluctuating field: 

b



t

   (

u



B

0 )   

G

   2

b

, Where “pain in the neck term”

Traditional approach is to assume that the fluctuating field is driven solely by the large-scale magnetic field.

i.e. in the absence of

B

0

the fluctuating field decays.

i.e. No small-scale dynamo (not really appropriate for high Rm turbulent fluids)

Under this assumption, the relation between

b

E and

B

0 ) is linear and homogeneous.

and

B

0 (and hence between

Postulate an expansion of the form: E

i

 a

ij B

0

j

 

ijk



B

0 

x k j

  where α

ij

and β

ijk

are

pseudo-

tensors, determined by the statistics of the turbulence.

Simplest case is that of isotropic turbulence, for which α

ij

Then mean induction equation becomes: = αδ

ij

and β

ijk =

βε

ijk.



B

0 

t

  

(

a

B

0 )  (    )  2

B

0 .

α : regenerative term, responsible for large-scale dynamo action. E

B

is an axial vector then α can be non-zero only for turbulence lacking reflexional symmetry (i.e. possessing handedness). β : turbulent diffusivity.

BUT WHAT ARE

a

and



? MORE LATER

BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

Add back in the mean flow U 0 and the mean field equation becomes 

B

0 

t

  

(

a

B

0 

U

0 

B

0 )  (    )  2

B

0 .

Now consider simplest case where a = a 0 cos q and U 0 = U 0 sin q e f In contrast to the induction equation, this can be solved for axisymmetric mean fields of the form

B

0 

B

0

t

e

  (

A

0

P

e

 )

BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

• Linear growth-rate of

B

0 depends on dimensionless combination of parameters.

• Critical parameter given by

D



D

a

D

W  a

L

 W

L

2  • If |D| > D c then exponentially growing solutions are found – dynamo action.

• Estimates suggest |D a | ~ 2, |D W | ~ 10 3 for the Sun and hence one can make the aW -approximation where the a -effect is ignored in generating the toroidal field.

• Can also have a 2 W and a 2 dynamos – may be of relevance for fully convective or more rapidly rotating stars.

BASIC PROPERTIES OF THE MEAN FIELD EQUATIONS

• In general

B

0 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 W cyc is proportional to |D| 1/2 • Solutions can be either dipolar or quadrupolar

Crucial questions Mean field electrodynamics therefore seems to work very well but there are some very obvious questions to ask 1. How can we calculate a and  ? What will these be in the Sun. Can we relate them to the properties of the flow in the kinematic regime?

2. Even if we know how a and  behave kinematically, what is the role of the Lorentz force on the transport coefficients α and β?

3. How weak must the large-scale field be in order for it to be dynamically insignificant? Dependence on

Rm

?

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

• Of course a and  can only really be calculated by determining E • But we can only know

b

 

u



b

 .

if we solve the fluctuating field equation.

• Analytic progress can be made

by making one of two approximations

• Either Rm or the correlation time of the turbulence t corr is small.

• Then can ignore “pain in the neck” G term in fluctuating field equation.

• Get famous results that a is related to the helicity of the flow with a constant of proportionality given by the small parameter e.g.

a   t

corr



u

.

ω

 3 • Note we have parameterised correlations between

u

correlations between

u

and w and

b

by

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

• We could do some numerical experiments and simply measure a • The best way to do this is to impose a

known mean field

B

then calculate numerically

E  

u



b

 .

and

• If the Lorentz force is switched off (the field is weak) then this gives a kinematic calculation of a.

• Example: Choose a flow with Rm not small and not at short correlation time and simply evaluate a.

• So we solve the kinematic induction equation 

B



t

   (

u



B

) 

Rm

 1  2

B

,  .

B

 0 .

With an applied mean field to calculate E.

Here we choose

u

to be the famous G-P flow

u

 ( 

y

,  

x

,  );   3 2  cos(

x

 cos(

t

))  sin(

y

 sin(

t

)) 

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

• For this flow the a • The term is a tensor.

a effect is a very sensitive function of Rm.

• It even changes sign.

• It can in no way be related in a simple manner to the helicity of the flow (bit of a strange flow as it has infinite correlation time) • Neither of the approximations work very well at high Rm • Changes in correlation times may change results… 

Courvoisier et al 2004 Rm Rm

Rotating turbulent convection (Cattaneo & Hughes 2005)

Sometimes it is not even possible to calculate a turbulent

a

-effect

T 0

Ω g

Boussinesq convection.

T 0 + ΔT Taylor number,

Ta

= 4Ω 2 d 4 /ν 2 = 5 x 10 5 , Ro = 1/10-1/5 Prandtl number

Pr = ν/κ

= 1, Magnetic Prandtl number

Pm = ν/η = 5.

Critical Rayleigh number = 59 008.

Anti-symmetric helicity distribution anti-symmetric α-effect.

Maximum relative helicity ~ 1/3.

Ra

= 150 000 Weak imposed field in

x

-direction.

Temperature on a horizontal slice close to the upper boundary.

Ra =

150,000. No dynamo at this Rayleigh number – but still an α-effect.

Mean field of unit magnitude imposed in

x

-direction (essentially kinematic) Self-consistent dynamo action sets in at

Ra

 200,000.

u B

2 2

e.m.f. and time-average of e.m.f.

Ra =

150,000 Imposed

B x =

1.

Imposed field extremely weak – kinematic regime.

E E

y x

E

E

z time time time

Cumulative time average of the e.m.f. Not fantastic convergence.

α – the ratio of e.m.f. to applied magnetic field – is very small. And even depends on Rm!!!

Note this is not

a

-suppression (field too weak) It appears that the α-effect here is not turbulent (i.e. fast), but diffusive (i.e. slow).

However in models of rotating compressible convection, Ossendrijver et al (2002) find a significant

a

-effect -- fewer cells in box?

-- less turbulent?

-- less incoherence (decoherence)?

E E

y

E

x z

B x

No evidence of significant energy in the large scales – either in the kinematic eigenfunction or in the subsequent nonlinear evolution.

Picture entirely consistent with an extremely feeble α-effect.

Note Jones & Roberts (2002) find a large mean field. As you go to bigger boxes and more cells it is harder to get a mean field or measure an a-effect

2. How are a and  modified by the mean field in the Nonlinear Regime? • This is a CRUCIAL question.

• Assume kinematic theory is OK (hmm) • The mean field <

B

> will act back on the turbulence so as to switch off the generation mechanism via the Lorentz Force.

• When does this happen?

• Traditional argument… – This occurs when mean field reaches equipartition with the turbulence so  

u

2   

B

 2 .

2. How are a and  modified by the mean field in the Nonlinear Regime? • But… • It is the small scale magnetic field that will act back on the small-scale turbulence.

• The dynamo will switch off when the small-scale magnetic energy becomes comparable with the small-scale kinetic energy of the flow.

• There are many different possibilities, but it seems clear that due to amplification by the turbulence the small scale magnetic field is much bigger than the mean magnetic field From a simple scaling it follows that: 

B

2  

Rm p



B

 2 .

where p is a flow and geometry dependent coefficient (p>0)

2. How are a and  modified by the mean field in the Nonlinear Regime? • This poses a major problem for mean field theory (see Proctor 2003;Diamond et al 2004 for an erudite discussion) • If true then this implies that the a -effect (and probably the  -effect) is switched off when the mean magnetic field is small (i.e. when 

B

 2   

u

2 

Rm p

• Hence the source term ( a ) will be

catastrophically quenched

when the mean field is very small.

• Is this correct?

• Two ways of checking – Analytical results based on approximations – Numerical results at moderate Rm

2. How are a and  modified by the mean field in the Nonlinear Regime? ANALYTICAL RESULTS • Really again we have to solve the induction equation for

b

and and the momentum equation for

u

 via E to calculate a • But some analytical progress is made by following two strands • Take some exact results – e.g. Integrated Ohm’s Law E

.

B

0   1 

j

.

b

  

e

.

b

 (  a

B

0 2 ).



a.b

 

t

  2 

e.b

   .(

b

 )     (

a



e

)  – General Conservation of magnetic helicity 2001)

dH M dt



Q H



F H

(e.g. Brandenburg & Dobler

2. How are a and  modified by the mean field in the Nonlinear Regime? ANALYTICAL RESULTS The second strand is to combine these exact results (or modifications thereof) with an

approximate

result for a in the nonlinear regime a   t

corr

3 

u

.

ω



b.j

 

Pouquet, Frisch & Léorat 1976

• Note: This is

not

an exact result.

• It is derived using the EDQNM approximation • Only applies (if at all) for short correlation times and assumes that the magnetic field does not affect the correlation time of the turbulence.

formula (Proctor 2003)

2. An aside – what do Pouquet, Frisch and Léorat actually do?

(Proctor 2003) • They take a state of pre-existing helical MHD turbulence which is happily minding its own business with small scale b and u (but no large scale field?).

• They

add

B

equations to get equations for the perturbations to u and b (u’ and b’) • They make a short correlation time approximation to say that

j

,

b

,

u, ω

2. How are a and  modified by the mean field in the Nonlinear Regime? ANALYTICAL RESULTS • By combining these two results (or similar) it is possible to get formulae for a (and  ) – Gruzinov and Diamond 1994,1995, 1996 – Blackman & Field 2000, Blackman & Brandenburg 2002 – Kleeorin et al 1995 – And many more… • e.g.

Ε  a

K



Rm



t

0  0

J 0 .B

0

1 

RmB

0 2 / 2

B eq

/ 2

B eq

B

0  

t

0  0

J 0

2. How are a and  modified by the mean field in the Nonlinear Regime? NUMERICAL RESULTS calculated numerically and related to an applied mean

B

• This can be done for forced flows or for convection for

B

0 • e.g. for Galloway-Proctor flow solve  

t



B



u

t

 

u

.



u

   (

u

 

p

B

)  

j



B



Re Rm

 1  2

B

,

1

 2

u



F

,  .

u

 0 ,  .

B

 0 .

u

 ( 

y

,  

x

,  );   3 2  cos(

x

 cos(

t

))  sin(

y

 sin(

t

)) 

Components of e.m.f.

versus time.

Suggestive of the formula: a  1  a 0

Rm



B

0 2 for γ = O(1).

α versus

Rm

(C, H & Thelen 2002) α versus

B

0 2 (Cattaneo & Hughes 1996)

Mean Field Hydrodynamics

• Of course, mean field theory can be played on the Navier-Stokes equations (see e.g. Rüdiger 1989).

• Solve equations for mean flow and parameterise small scale interactions.

Reynolds Stress Mean Lorentz Force Maxwell Stress

• This is even more dodgy as there is no closure that relates the small scale flows to the mean velocity.

• Also have to worry about Galilean Invariance of equations

Other Possible Mechanisms for Producing Poloidal Field

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

Other Possible Mechanisms for Producing Poloidal Field

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

TURBULENT CONVECTION ROTATION STRONG LARGE SCALE SUNSPOT FIELD

TURBULENT CONVECTION ROTATION DIFFERENTIAL ROTATION

W

MERIDIONAL CIRCULATION U p STRONG LARGE SCALE SUNSPOT FIELD

TURBULENT CONVECTION ROTATION Reynolds Stress

L

-effect DIFFERENTIAL ROTATION

W

MERIDIONAL CIRCULATION U p STRONG LARGE SCALE SUNSPOT FIELD

HELICAL/CYCLONIC CONVECTION u’ TURBULENT CONVECTION ROTATION Reynolds Stress

L

-effect DIFFERENTIAL ROTATION

W

MERIDIONAL CIRCULATION U p LARGE-SCALE MAG FIELD STRONG LARGE SCALE SUNSPOT FIELD

W

-effect

HELICAL/CYCLONIC CONVECTION u’ TURBULENT CONVECTION ROTATION Reynolds Stress

L

-effect SMALL-SCALE MAG FIELD b’ Turbulent amplification of Turbulent EMF

E

=

a,,

-effect LARGE-SCALE MAG FIELD DIFFERENTIAL ROTATION

W

STRONG LARGE SCALE SUNSPOT FIELD

W

-effect MERIDIONAL CIRCULATION U p

HELICAL/CYCLONIC CONVECTION u’ TURBULENT CONVECTION ROTATION Reynolds Stress

L

-effect SMALL-SCALE MAG FIELD b’ Maxwell Stresses

L

-quenching Turbulent amplification of Turbulent EMF

E

=

a,,

-effect LARGE-SCALE MAG FIELD DIFFERENTIAL ROTATION Small-scale Lorentz force

a

-quenching

W

MERIDIONAL CIRCULATION U p

W

-effect STRONG LARGE SCALE SUNSPOT FIELD Large-scale Lorentz force Malkus-Proctor effect

Recent Mean Field Modelling

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

Split Infinitives and Mean field Modelling • Fowler's remark on the split infinitive is well known: • "The English-speaking world may be divided into those who neither know nor care what a split infinitive is, those who don't know, but care very much, those who know and approve, those who know and condemn, and those who know and distinguish." • The same can be said about mean field modelling • “The world may be divided into four categories – Those who neither know nor care about catastrophic a quenching.

– Those that seem to know (they’ve been to enough meetings!), but don’t care. – Those who know and care, but do some mean field modelling – Those who know and condemn”

Illustrative vs Imitative Modelling

• Illustrative modelling.

– Investigate the mathematical properties of the mean field equations to try to understand how these equations behave as the inputs are varied.

• How does L -quenching affect the dynamo?

• How do dipole modes interact with quadrupole modes in the nonlinear regime?

• What would be the consequences of catastrophic a -quenching?

• How do transport coefficients affect the period, amplitude and direction of travel of dynamo solutions?

• What processes might lead to modulation and Grand Minima?

• What are the effects of poles and boundary conditions • What is the underlying mathematical structure of such equations: linear and nonlinear behaviour – Which (if any) of these results are robust?

Illustrative vs Imitative Modelling

• Imitative modelling.

– Try to put as many effects as possible into a model to reproduce as many of the features of the solar cycle as possible • There are a lot of observations to be matched.

– Cycle length, cycle amplitude, migration of magnetic activity, phase relation between poloidal and toroidal fields, active latitudes, active longitudes, form of individual cycles, torsional oscillations. • There are a lot of free parameter to play around with.

a

ij

(

r

,  ,

t

) 

ijk

(

r

,  ,

t

)

u p

(

r

,  ,

t

) – form of quenching: catastrophic, regular, dynamic – Other form of nonlinearity: Malkus-Proctor effect, L -quenching – Effects due to magnetic buoyancy

Distributed Dynamo Scenario

• • • •

Here the poloidal field is generated throughout the convection zone by the action of cyclonic turbulence.

Toroidal field is generated by the latitudinal distribution of differential rotation.

No role is envisaged for the tachocline Angular momentum transport would presumably be most effective by Reynolds and Maxwell stresses

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

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 Dynamo scenario

• • •

The dynamo is thought to work at the interface of the convection zone and the tachocline.

The mean toroidal (sunspot field) is created by the radial diffential rotation and stored in the tachocline.

And the mean poloidal field (coronal field) is created by turbulence (or perhaps by a dynamic

a

effect) in the lower reaches of the convection zone

Interface Dynamo scenario

•

PROS

–

The radial shear provides a natural mechanism for generating a strong toroidal field

–

The stable stratification enables the field to be stored and stretched to a large value.

–

As the mean magnetic field is stored away from the convection zone, the

a

-effect is not suppressed

–

Separation of large and small scale magnetic helicity

•

CONS

–

Relies on transport of flux to and from tachocline – how is this achieved?

– –

Delicate balance between turbulent transport and fields.

“Painting ourselves into a corner”

The Tachocline

• In two of these scenarios the tachocline plays a key role in the dynamo process.

– Generates toroidal field through shear – Stores the strong magnetic field in a stably stratified layer.

• The dynamics of this layer is therefore of fundamental importance for the solar dynamo.

– Strongly magnetised – Strong differential rotation – Strong stratification • Many instabilities – Joint instability of differential rotation and toroidal field (cf MRI) – Magnetic Buoyancy Instabilities – Shear Flow Instabilities

Robust Results from Mean Field Models

• There have been an infinite number of mean field models of the solar dynamo.

• Are there any results that “tend to occur” no matter what assumptions are put into the model?

• These can emerge from considerations of the underlying mathematical structure of the equations coupled to the physical context… • …or from simply doing lots of runs and seeing what sorts of things emerge

Robust Results from Mean Field Models

• For aW -dynamo equations fields can appear in either an oscillatory or stationary bifurcation.

– Steady modes are favoured when the a -effect is more prevalent.

– Separation of regions of a and W tends to lead to oscillatory modes • Dynamo lengthscales tend to be of the order of the region of generation • Dynamo waves tend to follow lines of constant rotation – If generated in convection zone either propagate radially or are steady – If generated in tachocline propagate latitudinally.

• For oscillatory modes if D<0 fields migrate towards the equator – This is not always the case though (even without meridional flows) • Meridional Flows of 1ms -1 near the are able to change the direction of propagation of dynamo waves.

Robust Results from Mean Field Models

• Amplitude of dynamo solutions tends to increase as (D-D c ) increases.

– Initially B 2 increases linearly with D-D c – Saturation of amplitude as move to more supercritical dynamo numbers.

• As (D-D c ) increases solution becomes more irregular.

– Sequence of bifurcations leading to chaotic solutions in general – Spatio-temporal fragmentation of solutions

Robust Results from Mean Field Models

• In the absence of a strong meridional flow the a -effect must be localised fairly close to the equator in order to get dynamo waves localised near the equator (e.g. Markiel & Thomas 1999; Markiel 2000; Bushby 2004) • In the absence of a meridional flow a -effects near to the tachocline are more effective than those close to the surface in generating strong magnetic fields (e.g. Mason et al 2003) • The addition of transport effects (e.g. magnetic buoyancy or magnetic pumping as well as meridional flows) that moves around flux has a significant effect on the cycle period and phase relation of the dynamo.

• Stochastic effects can be very important when the field is weak (e.g. Hoyng 1993, Ossendrijver & Hoyng 1996) – e.g. when D~D c or when the dynamo has been modulated by nonlinear effects.

Robust Results from Mean Field Models

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

Robust Results from Mean Field Models

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

Robust Results from Mean Field Models

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

Robust Results from Mean Field Models

Pipin 1999

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

• Also leads naturally to the formation of torsional oscillations

Robust Results from Mean Field Models

DIPOLE DIPOLE MIN QUAD

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

MIN

Beer et al 1998

Robust Results from Mean Field Models

• Torsional oscillations are generated by dynamic nonlinearities that modify the mean differential rotation (MP-effect or L quenching) • These are a sensitive function of where the magnetic field is generated and the stratification of the medium.

• Can be generated at the base of the convection zone, and manifest themselves towards the surface. (Bushby 2004)

Low-order models: Mathematical

KNOBLOCH ET AL (1998)

Aspects

• Mathematically the structure of the dynamo equations can be understood using the techniques of dynamical systems and symmetries of the problem.

• Reduced sets of ODEs can be derived by either truncating the PDEs, or by using a Normal Form Analysis.

• These low-order models can shed light on the bifurcation structure of the full PDEs

Low-order models: Mathematical

PDE ODE

Aspects

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

KNOBLOCH ET AL (1998)

Bushby & Tobias (2005)

Predictability

• It has been claimed recently that we are now at the stage where we can start to make predictions about future activity from mean field models (Dikpati et al , 2004) .

• Given the uncertainties of the input parameters and the chaotic nature of nonlinear systems, this is an interesting claim!

Global Solar Dynamo Calculations

Brun, Miesch & Toomre (2004)

• 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 Glatmaier (1985a,b) ) (Gilman & Miller (1981), Glatzmaier & Gilman (1982), • Calculations on massively parallel machines are now starting to enter the turbulent MHD regime.

• Focus on interaction of rotation with convection and magnetic fields.

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 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

Global Computations: Dynamo Action • For Rm > 300 dynamo action is sustained.

• ME ~ 0.07 KE • B r is aligned with downflows • B f is stretched into ribbons

Global Computations: Saturation • Magnetic energy is dominated by fluctuating field • Means are a lot smaller • ~ 3 • 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

Stellar Dynamos

• Most of the dynamo modelling effort has naturally focussed on the Sun.

• Some progress has been made in describing the dynamo action in 3 other classes of stars – Solar-type stars: moderate rotators with deep convective envelopes – Rapidly rotating stars – Fully Convective stars

Solar-like stars

• Try to use observations to calibrate solar dynamo models e.g. measure magnetic field amplitude and cycle period and try to infer the behaviour of a as a function of Ro.

• Traditional to use mean field theory with various assumptions.

• Problem: results sensitive to assumptions –  a ~ B ~ B m n 0.3 < n < 1.5

m ~ 0.75 (Saar & Brandenburg 1999)

Solar-like stars

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

Tobias (1998)

Solar-like stars

Tobias (1998)

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

Rapidly Rotating stars

• • •

It is very dangerous to extrapolate the results from solar mean field models to stars that rotate much more rapidly For rapidly rotating stars

W/W

is much smaller than for the Sun and so differential rotation is likely to play less of a role.

Rapid rotation means that Reynolds stresses are less likely to be able to transport angular momentum away from rotation being constant on cylinders.

Rapidly Rotating stars

•

For a mean field model with plausible parametrisations of

a W

it is possible to determine the role of and changing the nature of the differential rotation profile.

• •

Dynamo is more likely to be of

a 2 W

type.

Tangent cylinder plays a large role in confining magnetic activity to high latitudes.

Bushby (2003)

•

Polar branch much more pronounced (cf polar spots)

Fully Convective stars

• •

For fully convective stars (e.g. fully convective T-Tauri stars) magnetic fields are still observed.

With the absence of a stable layer and strong shear it is difficult to see how a strong mean field can be built up (cf fully computational models)

•

Dynamo action is likely to be small-scale.

•

Küker & Rüdiger

If a large scale field can be generated, then mean field theory indicates that this is likely to be steady and perhaps non-axisymmetric.

Conclusions/Speculations and Annoying Questions • Why does mean-field theory work so well?

– Input parameters need to be constrained • Requires a full understanding of MHD turbulence – Turbulent a -effect – Turbulent diffusion • Measurement of mean flows.

• What can serious(?) computations teach us – Small scale (parts of the jigsaw) – Large scale (global dynamics) • We can learn a lot from the mathematical structure of the equations.