Dynamics of the solar convection
zone
Matthias Rempel
(HAO/NCAR)
High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research
under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.
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
PSPT (CaK)
PSPT (blue)
 Regions of strong magnetic field (3000 Gauss)
 About 20000km diameter
 Lifetime of a few weeks
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 properties
 Butterfly 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)
Usokin et al. (2007)
 Normal activity interrupted
by grant minima ~100 years
duration
 Persistent 11 year cycle
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
Changes is DR
Meridional flow
Butterfly diagram
Radial field
Surface Doppler measurement R. Ulrich (2005)
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
Latitudinal transport:
North-South motions:
negative (poleward)
East-West motions:
positive (equatorward)
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 decaying
 Toroidal field can grow significantly in the beginning
– Stretching of field lines
 Toroidal 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 current
 t 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
 -dynamo
– Stationary field
– Periodic solutions, travelling waves
– Poloidal, toroidal field
– Toroidal field much stronger than
similar strength
poloidal field
So — what is the sun doing?
 Strong differential rotation (observed), periodic
behaviour  -dynamo
 Propagation 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 wave
 Surface 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
Browning et al. (2006)
 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)
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
(Y. Fan)
 Flux tube looses a lot of flux during rise (tube has to
be twisted in the beginning)
 Twist reduces tilt angle
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?