cattaneo@flash.uchicago.edu

The solar dynamo(s)

Fausto Cattaneo

Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas

Chicago 2003

The solar dynamo problemChicago 2003

• Wide range of spatial scales. From global scale to limit of resolution

• Wide range of temporal scales. From centuries to minutes

• Solar activity is extremely well documented

The solar dynamo is invoked to explain the origin magnetic activity

Three important features:

Models are strongly observationally constrained

ObservationsChicago 2003

Hale’s polarity law suggests organization on global scale.

Typical size of active regions approx 200,000Km

Typical size of a sunspot

50,000Km

Small magnetic elements show structure down to limit of resolution (approx 0.3")

Observations: large scaleChicago 2003

• Active regions migrate from mid-latitudes to the equator

• Sunspot polarity opposite in two hemispheres

• Polarity reversal every 11 years

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

10BeC14

Beer (2000)Wagner et al (2001)

Cycle persists through

Maunder Minimum (Beer et al 1998)

Observations: large scaleChicago 2003

Observations: small scaleChicago 2003

Two distinct scales of convection (maybe more)

• Supergranules:– not visible in intensity

– 20,000 km typical size

– 20 hrs lifetime

– weak dependence on latitude

• Granules:– strong contrast

– 1,000km typical size

– 5 mins lifetime

– homogeneous in latitude

Observations: small scaleChicago 2003

Quiet photospheric flux

• Network fields– emerge as ephemeral regions (possibly)

– reprocessing time approx 40hrs

– weak dependence on solar cycle

• Intra-network magnetic elements– possibly unresolved

– typical lifetime few mins

General dynamo principleChicago 2003

Any three-dimensional, turbulent (chaotic) flow with high magnetic Reynolds number is (extremely) likely to be a dynamo.

• Reflectionally symmetric flows:

– Small-scale dynamo action

– Disordered fields; same correlation length/time as turbulence

– Generate but not

• Non-reflectionally symmetric flows:

– Large-scale dynamo; inverse cascade of magnetic helicity

– Organized fields; correlation length/time longer than that of turbulence

– Possibility of

| |2B2

B

| | 22B B

In astrophysics lack of reflectional symmetry associated with

(kinetic) helicity Coriolis force Rotation

Rotational constraints

Introduce Rossby radius Ro (in analogy with geophysical flows)

• Motions or instabilities on scales Ro “feel’’ the rotation.

– Coriolis force important helical motions

– Inverse cascades large-scale dynamo action

• Motions or instabilities on scales < Ro do not “feel” the rotation.

– Coriolis force negligible non helical turbulence

– Small-scale dynamo action

Chicago 2003

Modeling: large-scale generationChicago 2003

Dynamical ingredients

• Helical motions: Drive the α-effect. Regenerate poloidal fields from toroidal

• Differential rotation: (with radius and/or latitude) Regenerate toroidal fields from poloidal. Probably confined to the tachocline

• Magnetic buoyancy: Removes strong toroidal field from region of shear. Responsible for emergence of active regions

• Turbulence: Provides effective transport

Modeling: helical motionsChicago 2003

• Laminar vs turbulent α-effect: – Babcock-Leighton models. α-effect driven by rise and twist of large scale loops and

subsequent decay of active regions. Coriolis-force acting on rising loops is crucial. Helical turbulence is irrelevant. Dynamo works because of magnetic buoyancy.

– Turbulent models. α-effect driven by helical turbulence. Dynamo works in spite of magnetic buoyancy.

• Nonlinear effects: – Turbulent α-effect strongly nonlinearly

suppressed

– Interface dynamos?

– α-effect is not turbulent (see above)

Cattaneo & Hughes

Modeling: differential rotationChicago 2003

• Latitudinal differential rotation: – Surface differential rotation persists throughout the convection zone

– Radiative interior in solid body rotation

• Radial shear: – Concentrated in the tachocline; a thin

layer at the bottom of the convection zone

– Whys is the tachocline so thin? What controls the local dynamics?

No self-consistent model for the solar differential rotation

Schou et al.

Modeling: magnetic buoyancyChicago 2003

Wissink et al.

What is the role of magnetic buoyancy?

• Babcock-Leighton models:– Magnetic buoyancy drives the dynamo

– Twisting of rising loops under the action of the Coriolis force generates poloidal field from toroidal field

– Dynamo is essentially non-linear

• Turbulent models:– Magnetic buoyancy limits the growth of the

magnetic field

– Dynamo can operate in a kinematic regime

Do both dynamos coexist? Recovery from Maunder minima?

Modeling: turbulenceChicago 2003

How efficiently is turbulent transport?• Babcock-Leighton models: Turbulent diffusion causes the dispersal of active regions. Transport of poloidal flux to the poles.

• Interface models: Turbulent diffusion couples the layers of toroidal and poloidal generation

• All models: – Turbulent pumping helps to keep

the flux in the shear region

– Turbulence redistributes angular

momentum

– Etc. etc. etc.

Tobias et al.

Modeling: challengesChicago 2003

No fully self-consistent model exists.

• Self-consistent model must capture all dynamical ingredients (MHD, anelastic)

• Geometry is important (sphericity)

• Operate in nonlinear regime

• Resolution issues. Smallest resolvable scales are– in the inertial range

– rotationally constrained

– stratified

Need sophisticated sub-grid models

temperature

g

hot

cold

time evolution

• Plane parallel layer of fluid

• Boussinesq approximation

• Ra=500,000; P=1; Pm=5

Modeling: small-scale generationChicago 2003

Sim

ula

tion

s b

y L

enz

& C

atta

neo

Modeling: physical parameters Chicago 2003

Re

Rm

Pm

=1

Stars

Liquid metal experiments

simulations

IM

Pm=1

102

103

103 107

• Dynamo must operate in the inertial range of the turbulence

• Driving velocity is rough

• How do we model MHD behaviour with Pm <<1

Pm=1

Pm=0.5

Re=

1100

, Rm

=55

0R

e=55

0, R

m=

550

yes

no

• Does the dynamo still operate? (kinematic issue)

• Dynamo may operate but become extremely inefficient (dynamical issue)

Modeling: kinematic and dynamical issuesChicago 2003

• Relax requirement that magnetic field be self sustaining (i.e. impose a uniform vertical field)

• Construct sequence of simulations with externally imposed field, 8 ≥ Pm ≥ 1/8, and S = = 0.25

• Adjust Ra so that Rm remains “constant”

Pm 8.0 4.0 2.0 1.0 0.5 0.25 0.125

Ra 9.20E+04 1.40E+05 2.00E+05 3.50E+05 7.04E+05 1.40E+06 2.80E+06

Nx, Ny 256 256 256 512 512 512 768

Modeling: magneto-convectionChicago 2003

Simulations by Emonet & Cattaneo

Chicago 2003

B-field (vertical) vorticity (vertical)

Pm = 8.0

Pm = 0.125

Modeling: magneto-convection

Chicago 2003

• Energy ratio flattens out for Pm < 1

• PDF’s possibly accumulate for Pm < 1

• Evidence of regime change in cumulative

PDF across Pm=1

• Possible emergence of Pm independent

regime

Modeling: magneto-convection

SummaryChicago 2003

Two related but distinct dynamo problems.

• Large-scale dynamo– Reproduce cyclic activity

– Reproduce migration pattern

– Reproduce angular momentum distribution (CV and tachocline)

– Needs substantial advances in computational capabilities

• Small scale dynamo– Non helical generation

– Small Pm turbulent dynamo