1. Double cycles and instabilities in EULAG-MHD simulations of solar convection Paul Charbonneau Département de Physique, Université de Montréal
The solar dynamo problem
EULAG-MHD
The « millenium simulation »
Double-cycle dynamo behavior
MHD instabilities in the tachocline
Conclusions
Collaborators: Piotr Smolarkiewicz, Mihai Ghizaru, Dario Passos,
Antoine Strugarek, Jean-François Cossette, Patrice Beaudoin,
Caroline Dubé, Nicolas Lawson, Étienne Racine, Gustavo Guerrero
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2. The solar magnetic cycle
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3. Dynamo problems HARD The kinematic dynamo problem: TURBULENCE
« To find a flow u that can lead to field amplification when
substituted in the MHD equation »
MUCH HARDER
The self-excited dynamo problem:
TURBULENCE
« To find a flow u that can lead to field amplification when
substituted in the MHD equation, while being dynamically
consistent with the fluid equations including the Lorentz force »
HARDEST
The solar/stellar dynamo problem(s):
????????
« To find a flow u that leads to a magnetic field amplification
and evolution in agreement with observational inferences for
the Sun and stars »
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4. The magnetic self-organization conundrum
How can turbulent convection, a flow with a length scale <<R
and coherence time of ~month, generate a magnetic component
with scale ~R varying on a timescale of ~decade ??
Mechanism/Processes favoring organization on large spatial scales: 1. rotation (cyclonicity); 2. differential rotation (scale ~R); and 3. turbulent inverse cascades.
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5. Rotation and differential rotation (1)
No rotation
Rotation at solar rate
Vertical (radial) flow velocity, in Mollweide projection
[ from Guerrero et al. 2013, Astrophys. J., 779, 176 ]
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6. Rotation and differential rotation (2)
Helioseismology
HD simulation
MHD simulation
Angular velocity profiles, in meridional quadrant
Differential rotation in the Sun and solar-type stars is powered
by turbulent Reynolds stresses, arising from rotationally-induced
anisotropy in turbulent transport of momentum and heat
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7. EULAG-MHD simulations
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8. Simulation framework Simulate anelastic convection in thick,
rotating and unstably stratified fluid shell
of electrically conducting fluid, overlaying
a stably stratified fluid shell.
Recent such simulations manage to reach
Re, Rm ~ , at best; a long way from
the solar/stellar parameter regime.
Throughout the bulk of the convecting
layers, convection is influenced by
rotation, leading to alignment of
convective cells parallel to the rotation axis.
Stratification leads to downward pumping
of the magnetic field throughout the
convecting layers.
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9. Selected milestones
Gilman 1983: Boussinesq MHD simulation, producing large-scale magnetic fields with polarity reversals on yearly timescale; but non-solar large-scale organization.
Glatzmaier 1984, 1985: Anelastic model including stratification, large-scale
fields with polarity reversals within a factor 2 of solar period; tendency for
poleward migration of the large-scale magnetic field.
Brun et al. 2004: Strongly turbulent MHD simulation, producing copious
small-scale magnetic field but no large-scale magnetic component.
Browning et al. 2006: Demonstrate the importance of an underlying,
convectively stable fluid layer below the convection zone in producing
a large-scale magnetic component in the turbulent regime.
Brown et al. 2010, 2011: Obtain irregular polarity reversals of thin, intense
toroidal field structure in a turbulent simulation rotating at 5X solar.
Ghizaru et al. 2010: Obtain regular polarity reversals of large-scale magnetic component on decadal timescales, showing many solar-like characteristics.
Nelson et al. 2012, 2013: Autonomous generation of buoyantly rising flux-ropes structures showing sunspot-like emergence patterns.
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10. The MHD equations
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11. EULAG-MHD
EULAG: a robust, general solver for geophysical flows; developed
by Piotr Smolarkiewicz and collaborators at MMM/NCAR
EULAG-MHD: MHD generalization of above; developed mostly
at UdeM in close collaboration with Piotr Smolarkiewicz
Core advection scheme: MPDATA, a minimally dissipative
iterative upwind NFT scheme; equivalent to a dynamical, adaptive
subgrid model.
Thermal forcing of convection via volumetric Newtonian cooling term
in energy equation, pushing reference adiabatic profile towards a
very slightly superadiabatic ambiant profile
Strongly stable stratification in fluid layers underlying convecting layers.
Model can operate as LES or ILES
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12. MHD simulation of global dynamos [ Ghizaru et al
MHD simulation of global dynamos [ Ghizaru et al. 2010, ApJL, 715, L133 ]
Temperature perturbation
Radial flow component
Radial magnetic field component
> Que faisons nous > Simulations MHD
Electromagnetic induction by internal flows is the engine powering the solar
magnetic cycle. The challenge is to produce a magnetic field well-structured
on spatial and temporal scales much larger/longer than those associated
with convection itself. This is the magnetic self-organisation problem.
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13. Kinetic and magnetic energies
(120 s.d.=10 yr)
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14. Simulated magnetic cycles (1)
Large-scale organisation of the magnetic field takes place primarily
at and immediately below the base of the convecting fluid layers
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15. Magnetic cycles (1) Zonally-averaged Bphi at r/R =0.718
Zonally-averaged Bphi at -58o latitude
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16. Successes and problems
KiloGauss-strength large-scale magnetic fields, antisymmetric about
equator and undergoing regular polarity reversals on decadal timescales.
Cycle period four times too long, and strong fields concentrated
at mid-latitudes, rather than low latitudes.
Internal magnetic field dominated by toroidal component and
strongly concentrated immediately beneath core-envelope interface.
Well-defined dipole moment, well-aligned with rotation axis,
but oscillating in phase with internal toroidal component.
Reasonably solar-like internal differential rotation, and solar-like
cyclic torsional oscillations (correct amplitude and phasing).
On long timescales, tendency for hemispheric decoupling, and/or
transitions to non-axisymmetric oscillatory modes.
Cyclic modulation of the convective energy flux, in phase with the
magnetic cycle.
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17. The « millenium simulation » [ Passos & Charbonneau 2014, A&A, in press ]
Define a SSN proxy, measure cycle characteristics (period, amplitude…) and compare to observational record.
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18. Characteristics of simulated cycles (1) [ Passos & Charbonneau 2014, A&A, in press ]
Define a SSN proxy, measure cycle characteristics (period, amplitude…) and compare to observational record.
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19. Characteristics of simulated cycles (2) [ Passos & Charbonneau 2014, A&A, in press ]
0.957/0.947
[ 0.763/0.841 ]
r =
-0.395/-0.147
[ / ]
r =
0.688/0.738
[ 0.322/0.451 ]
r =
-0.465/-0.143
[ 0.185/ ]
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20. Characteristics of simulated cycles (3)
Hemispheric cycle amplitude show a hint of bimodality
Usoskin et al. 2014,
A&A 562, L10;
From 3000yr 14C
time series
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21. Characteristics of simulated cycles (4)
Hemispheric cycle amplitude show a hint of bimodality
Usoskin et al. 2014,
A&A 562, L10;
From 3000yr 14C
time series
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22. « Double cycles » [ with P. Beaudoin, C. Simard, J.-F. Cossette ]
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23. Short quasiperiodic variability (1)
Evidence for short-term (~0.5 – 2 yr) quasiperiodic variability
is found is a great many indicators of solar activity:
Sunspot number and area
Radio flux
Total and spectral irradiance
p-mode acoustic frequencies
Interplanetary magnetic field
Flaring rate
Solar wind speed
( For more, ask Scott McIntosh )
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24. Short quasiperiodic variability (1)
BISON
nHz
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25. Short quasiperiodic variability (1)
GONG
nHz
nHz
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26. Short quasiperiodic variability (2)
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27. Short quasiperiodic variability (3)
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28. Short quasiperiodic variability in millenium simulation (1)
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29. Short quasiperiodic variability (4)
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30. « Double dynamo » action (1)
Peaks at high latitude, but significant
Amplitude down to equatorial regions
m s-1
Peaks at low
latitudes within
convection zone;
tachocline mostly at high latitudes.
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31. « Double dynamo » action (2)
Scenario:
« Long » dynamo mode powered by turbulent emf
(a2W dynamo mode)
« short » dynamo mode powered by rotational shear
in equatorial portion of convection zone (aW dynamo mode)
If so, then Parker-Yoshimura sign rule should apply:
dynamo wave propagates away from equatorial plane
along isocontours of angular velocity
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32. Short quasiperiodic variability (4)
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33. Validation against mean-field model
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34. Tachocline instabilities [ M.Sc. Thesis N. Lawson ]
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35. MHD instabilities in stably stratified stellar interiors
Tayler instabilities (feeds on B)
Magnetoshear instabilities (feeds on B and grad W)
Balbus-Hawley instability (feeds on B and W)
… and many more…
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36. Characteristics of simulated cycles (4)
Hemispheric cycle amplitude show a hint of bimodality
Instability leads to cyclic exchange
of energy between axisymmetric
and non-axisymmetric large-scale
magnetic components, with a
characteristic phase lag.
Nonlinear development leads to
characteristic « clamshell » pattern.
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37. The overshoot layer and tachocline
Due to very low dissipation
and strong stratification,
the overshoot layer is very
thin; rotational shear extends
further below due to magnetic
torques.
Strong magnetic fields accumulate
in stable layer in response to
overshoot and turbulent pumping
in overlying convecting layers.
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38. Magnetic field accumulation in « tachocline »
Poynting flux is downwards
throughout bulk of convecting
and stable layers, at all
phases of the magnetic cycle
t = (Tachocline magnetic energy/Poynting flux = 8 yr
Consistent with « passive » accumulation and dissipation
of tachocline magnetic field, resulting from pumping from above.
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39. A clamshell instability ?
Phase pattern of magnetoshear instability !
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40. A clamshell instability ?
…maybe not…
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41. A Tayler instability ? Tayler stability criterion for purely
toroidal axisymmetric magnetic field:
Tayler-stable throughout most of convectively stable layers
(of course...) …excepts near polarity inversion lines.
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42. A Tayler instability ?
Plausible scenario: high latitudes are Tayler unstable; instability front
moves quiescently equatorward with polarity inversion; when it reaches
mid-latitude toroidal field bands, causes runaway destruction of B…
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43. Mode transitions from an instability ?
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44. Where do we go next ? Understand what sets the cycle period(s)
Understand physical underpinnings of the cyclic
modulation of the convective energy flux
Understand role of tachocline instabilities in long term
stability of simulations, and possible role in triggering
Maunder-Minimum-like period of strongly reduced activity
Comparative benchmark with ASH simulations
Get closer to surface
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45. FIN Collaborators: Piotr Smolarkiewicz (NCAR), Mihai Ghizaru,
Étienne Racine (CSA), Jean-François Cossette, Patrice Beaudoin,
Nicolas Lawson, Amélie Bouchat, Corinne Simard, Caroline Dubé,
Dario Passos
HAO 07/2014

46. Short quasiperiodic variability (4)
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47. Turbulent diffusivity
Turn now to the second term in our EMF development:
In cases where u is isotropic, we have , and thus:
The mathematical form of this expression suggests that can
be interpreted as a turbulent diffusivity of the large-scale field.
for homogeneous, isotropic turbulence with correlation time ,
it can be shown that
This result is expected to hold also in mildly anisotropic, mildly
inhomogeneous turbulence. In general,
I.3.4.3
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48. Turbulent pumping (1)
Turbulent pumping is a basic physical effect arising in inhomogeneous,
anisotropic turbulence; mathematically, it shows up as the antisymmetric
part of the alpha-tensor relating the turbulent EMF to the mean
magnetic field:
Extracting the symmetric part of the tensor yields:
where
acts as a velocity in the mean-field dynamo equations. For mildly
anisotropic, inhomogeneous turbulence:
I.3.4.3
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49. Turbulent pumping (2) m s-1 Poleward transport of surface
magnetic field by turbulent pumping;
speed in range 1-3 m s-1 above
+/- 45o latitudes
Equatorward transport of deep
magnetic field by turbulent pumping
between +/- 15 and 60o latitudes;
speed 1-2 m s-1
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50. Turbulent pumping (3) m s-1 Upward transport of magnetic field
by turbulent pumping in subsurface
layers; speed exceeding 1 m s-1
above +/- 60o latitudes
Downward transport of magnetic field
by turbulent pumping in bulk of deep
convection zone; speed exceeding
1 m s-1 between +/- 15 and 60o latitude
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51. Active region decay (1) III.2.2.1 Peak polar cap flux: ~1014 Wb
Synoptic magnetogram courtesy D. Hathaway, NASA/MSFC
[ ]
Toroidal flux emerging in active regions in one cycle: ~1017 Wb
III.2.2.1
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52. (2b)
Sunspot pairs are the photospheric manifestation of an emerging, formerly toroidal magnetic flux rope generated in the deep interior ;
after surface decay and transport by diffusion, differential rotation, and the surface meridional flow…
…an axisymmetric dipole moment is produced; this Babcock-Leighton mechanism produces a poloidal field from a toroidal component.
III.6.2.2
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53. Active region decay (3)
Zonal means
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54. Active region decay (3) III.2.2.1.4 The Babcock-Leighton
[ Simulation of surface magnetic flux evolution by A. Lemerle ]
The Babcock-Leighton
mechanism is definitely
seen operating at the
solar photosphere! But,
does it really feed back
into the dynamo loop ?
III
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55. Formation of magnetic flux strands (1) [ Nelson et al. 2013, Astrophys
Formation of magnetic flux strands (1) [ Nelson et al. 2013, Astrophys. J., 762: 73 ]
Recent, very high resolution 3D MHD simulations of solar convection
Have achieved the formation of flux-rope-like super-equipartition-strength
« magnetic strands » characterized by a significant density deficit in their
core; ripped from the parent large-scale structure by turbulent entrainement,
subsequent buoyant rise ensues.
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56. Formation of magnetic flux strands (2) [ Nelson et al. 2014, Solar Phys., 289, 441 ]
The strands
« remember »
their origin !
The strands develop a pattern
of East-West tilt similar to that
inferred obervationally for the sun
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57. Stellar cycles (1) B.P. Brown et al. 2011, Astrophys. J., XXX, YYY
ASH LES: at solar rotation
rate and luminosity, no
large-scale field produced;
At 3X solar rotation, steady
axisymmetric large-scale
field is produced;
At 5X solar rotation, irregular
polarity reversals occur.
Convection + Rotation
breed magnetism !!
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58. Solar/stellar magnetism
« If the sun did not have a magnetic
field, it would be as boring a star as
most astronomers believe it to be »
(Attributed to R.B. Leighton)
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59. FIN Collaborators: Piotr Smolarkiewicz (NCAR), Mihai Ghizaru,
Étienne Racine (CSA), Jean-François Cossette, Patrice Beaudoin,
Nicolas Lawson, Amélie Bouchat, Corinne Simard, Caroline Dubé,
Dario Passos
HAO 07/2014