Transcript Modulation of Solar and Stellar Activity Cycles

Solar Interior Magnetic Fields
and Dynamos
Steve Tobias (Leeds)
5th Potsdam Thinkshop, 2007
Observations
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
SST
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
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
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
Be : stored in ice
cores after 2 years
in atmosphere
14 : stored in tree
C
rings after ~30 yrs
in atmosphere
10
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
1.
2.
3.
4.
Core
Radiative Interior
(Tachocline)
Convection Zone
Visible Sun
1.
2.
3.
4.
5.
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
• 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

RHK
  Ro 1 (Noyes et al 1994)
RHK
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)
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
Nonlinear Including
Rotation,
in B
Gravity etc
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
B
   ( u  B)  2B
(.B  0),
t
 u

   u.u   p  j  B  g  Fviscous  Fother ,
 t


 .( u)  0,
t
D ( p  )
 loss terms,
Dt
p  RT .
Basics for the Sun
4
g


d
Ra 
BASE OF
CZ
 H P
Re  UL

Rm  UL

Pr  

  2 0 p
Pm  
M U
Ro  U
B
2

cs
2WL
PHOTOSPHERE
1020
1016
1013
1012
1010
106
10-7
10-7
105
1
10-3
10-4
0.1-1
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 (~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
B 0
2
   (aB 0  U 0  B 0 )  (   ) B 0 .
t
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
B 0  B0t e    ( A0 Pe )
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  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 largescale field (catastrophic aquenching)
– 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 aeffect 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 aeffect) 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 smallscale 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  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 largescale field (catastrophic aquenching)
– 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 aeffect 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 aeffect) 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 smallscale 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 amplitude 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
Dikpati et al (2006)
• 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?
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) )
Brun, Miesch & Toomre (2004)
• 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 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…