High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)

The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Researchunder sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.

Dynamics of the solar convection zone

Matthias Rempel (HAO/NCAR)

Outline

Observations– Large scale magnetic field– Solar cycle– Large scale flows: differential rotation, meridional flow

Differential rotation– Structure of convection– Origin of differential rotation

Solar dynamo– Basic ingredients of a dynamo– Formation of sunspots

Measurement of magnetic field Zeeman effect

– Splitting of spectral lines

– Linear+Circular polarization

Thermal and turbulent broadening of spectral lines

– Splitting not observable except for strongest field (sunspots)

Most field diagnostics are based on polarization signal

– Gives strength and orientation of field

Sunspots on solar disc

Regions of strong magnetic field (3000 Gauss) About 20000km diameter Lifetime of a few weeks

PSPT (blue)PSPT (CaK)

Changing X-ray activity over 11 years

Yohkoh X-ray images

Butterfly diagram + sunspot area over time

Hale’s law

Joy’s law

Solar cycle propertiesButterfly diagram

– Equatorward propagation of activity starting from 35 degrees latitude over 11 years (individual lifetimes of sunspots ~ a few weeks)

Hale’s polarity law– Opposite polarity of bipolar groups in north and south

hemisphere– Polarity in individual hemisphere changes every 11 years

Joy’s law– Bipolar groups are tilted to east-west direction– Leading polarity closer to equator– Tilt angle increases with latitude

Evolution of radial surface field

Everything together

D. Hathaway NASA (MSFC)

Longterm variations

Variability over the past 10000 years Cosmogenic isotopes

– 14C and 10Be produced by energetic cosmic rays

– Cosmic rays modulated by magnetic field in heliosphere

– Longterm record in ice cores (14C and 10Be ) and treerings (14C)

Normal activity interrupted by grant minima ~100 years duration

Persistent 11 year cycleUsokin et al. (2007)

Large scale flows

R. Howe (NSO)

Differential rotation in convection zone, uniform rotation in radiation zone (shear layer in between: Tachocline)

Cycle variation of DR (torsional oscillations, 1% amplitdude)

Differential rotation and meridional flow changes through solar cycle

Surface Doppler measurement R. Ulrich (2005)

Changes is DR

Meridional flow

Butterfly diagram

Radial field

Internal dynamics of convection zone

What drives large scale mean flows (differential rotation + meridional flow)?– Answer: small scale flows:

Reynolds stresses (correlations of turbulent motions) can drive large scale flows

Relevant for angular momentum transport:

How to model the solar convection zone

3D numerical simulations– Solve the full set of equations (including small and

large scale flows) on a big enough computer– Problem: Computers not big enough– Only possible to simulate ingredients

Meanfield models– Solve equations for mean flows only– Problem: need good model for correlations of small

scale flows (not always available)– Can address the full problem, but not from first

principles

Correlations caused by Coriolis force

North-South motions:

negative (poleward)

East-West motions:

positive (equatorward)

Latitudinal transport:

Average: zero unless East-West dominates

Structure of convection close to surface

3D simulation (M. Miesch)

Structure of convection in lower convection zone

3D simulation (M. Miesch)

Coriolis-force causes large scale convection rolls in deep convection zone

Balance between pressure and Coriolis force– Cyclonic rolls: lower pressure– Anti-cyclonic rolls: higher

pressure

Angular momentum transport

Positive– Faster rotating equator– -component of momentum

equation What determines radial

profile of DR?– Force balance between

Coriolis, pressure and buoyancy forces

– r--component of momentum equation

Profile of differential rotation

Latitudinal variation of entropy essential for solar like rotation profile

Possible causes– Anisotropic convective

energy transport (influence of rotation on convection

– Tachocline About 10K temperature

difference between pole and equator (T~106 K at base of CZ)

Results from 3D simulations

3D simulation (M. Miesch)

Summary: differential rotation

Turbulent angular momentum transport– Correlations between meridional (north south) and

longitudinal (east west) motions caused by Coriolis force– Anisotropic convection (“banana cells”)

Radial profile of differential rotation– Determined through force balance in meridional plane – Thermal effects important (about 10K latitudinal

temperature variation needed) Boundary layer (tachocline) important

The MHD induction equation

Basic laws (Ohm’s law, non-relativistic field transformation, Ampere’s law:

Combination of the three:

Differential rotation

Axisymmetry + differential rotation

Induction equation in spherical coordinates:

Properties of solution

Poloidal field always decayingToroidal field can grow significantly in the beginning

– Stretching of field linesToroidal field is also decaying in the long run

– The source of toroidal field decays with the poloidal field

What is missing?– Regeneration of poloidal field

Who can do it?– Again: small scale field and flows

Meanfield induction equation

– Decomposition of velocity and magnetic field:

– Averaging of induction equation:

– Turbulent induction effects:

Induction effect of helical convection

Negative kinetic helicity in northern hemisphere

Induces a poloidal field from toroidal field parallel to the current

of the toroidal field

Turbulent induction effects

-effect induces field parallel to electric currentt increases the effective diffusivity for meanfield

(turbulent diffusivity)

Meanfield Dynamos

The -effect closes the dynamo loop: regeneration of poloidal field from toroidal field

Some more general properties

2-dynamo

– Stationary field

– Poloidal, toroidal field

similar strength

-dynamo

– Periodic solutions, travelling waves

– Toroidal field much stronger than

poloidal field

So — what is the sun doing?Strong differential rotation (observed), periodic

behaviour -dynamoPropagation of activity belt

– Dynamo wave (requires radial shear)

– Advection effect (meridional flow) Location of -effect

– Bulk of convection zone (helical convection positive )

– Base of convection zone (helical convection negative , tachocline instabilities of both signs )

– Rising flux tubes (positive )

Dynamo waveSurface shear layer

– Positive – Very short time scales– Significant flux loss

Tachocline shear layer– Negative (in low latitudes)– Longer time scales, stable stratification allows for flux

storage

Role of tachocline

Stable stratification, long time scales– Formation of large scale field, likely origin of field forming sunspots

Problems of a pure tachocline dynamo– Much stronger shear of opposite sign in high latitudes (strong

poleward propagating activity belt)– Very short wavelength of dynamo wave (strongly overlapping cycles)

Browning et al. (2006)

Advection Meridional flow

– Poleward at surface (observed)

– Return flow not observable through helioseismology (so far)

– Equatorward at base of CZ• Mass conservation• Theory: meanfield models +

3D simulations

– Additional also turbulent advection effects (latitudinal

pumping)

Rising magnetic flux tubes Flux tubes ‘bundle of fieldlines’

form in tachocline Rising field due to buoyancy Fluid draining from apex Coriolis force causes tilt of the

top part of tube– Tilt increases with latitude

as observed

Net effect: positive

3D simulation of rising flux-tube

Flux tube looses a lot of flux during rise (tube has to be twisted in the beginning)

Twist reduces tilt angle

(Y. Fan)

Observations of ‘Surface’ -Effect and Flux Transport

.

Schematic of flux-transport dynamo

Latitudinal shear producing toroidal field

-effect from decay of active regions

Transport of field by meridional flow

Flux-transport dynamo with Lorentz-force feedback on DR and meridional flow

Feedback of Lorentz-force on DR and MC included

Moderate variations of DR and MC

– No significant change of dynamo

High latitude variations of DR– Poleward propagation,

amplitude similar to observed

Summary: The essential ingredients of the solar dynamo I

The sun is a -dynamo– Differential rotation profile (helioseismology)– Dominance of toroidal field (sunspots)– Cyclic behavior

Tachocline important for large scale organization of toroidal field (boundary layer)– Bulk of convection has too short time scales– Flux loss in convection zone due to magnetic

buoyancy and pumping– Stable stratification allows for storage

Summary: The essential ingredients of the solar dynamo II

Advection by meridional flow– Certainly important at surface (observed)– Equatorward meridional flow in lower convection zone

(theory, mass conservation)– How important compared to turbulent effects?

• Magnetic diffusivity• Turbulent pumping (in radius and latitude)

Flux-transport dynamos are very successful models (consistent with observational constraints), but more research required

Summary: The essential ingredients of the solar dynamo III

Sunspot formation– Origin of field: stable stratification at base of convection zone– Strong magnetic flux tubes rising through convection zone

(magnetic buoyancy)– Coriolis force leads to systematic tilt

Open questions– How to keep flux tube coherent in turbulent convection zone?

• Initial twist of tube required, but that also influences tilt angle– Rising tubes prefer long wave numbers (m=1,2)

• Sunspots are of much shorter wave number– Decoupling between emerged sunspot and its magnetic root at

base of convection zone?

The Future

Much more computing power– Better understanding of essential ingredients in the

short run– 3D dynamo model in the long run

Observational constraints– Helioseismology

• Meridional flow• Magnetic field in convection zone?

– Solar-stellar connection• How do cycle properties depend on rotation rate and

depth of convection zone?