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Double cycles and instabilities in EULAG-MHD simulations of solar convection

Paul Charbonneau Département de Physique, Université de Montréal

1. The solar dynamo problem2. EULAG-MHD 3. The « millenium simulation »4. Double-cycle dynamo behavior 5. MHD instabilities in the tachocline6. 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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The solar magnetic cycle

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Dynamo problems

The kinematic dynamo problem:

« To find a flow u that can lead to field amplification whensubstituted in the MHD equation »

« To find a flow u that can lead to field amplification whensubstituted in the MHD equation, while being dynamicallyconsistent with the fluid equations including the Lorentz force »

« To find a flow u that leads to a magnetic field amplificationand evolution in agreement with observational inferences forthe Sun and stars »

The self-excited dynamo problem:

The solar/stellar dynamo problem(s):

HARD

HARDEST

MUCH HARDER

TURBULENCE

TURBULENCE

????????

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The magnetic self-organization conundrum

How can turbulent convection, a flow with a length scale <<Rand coherence time of ~month, generate a magnetic componentwith 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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Rotation and differential rotation (1)

Vertical (radial) flow velocity, in Mollweide projection[ from Guerrero et al. 2013, Astrophys. J., 779, 176 ]

No rotation Rotation at solar rate

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Rotation and differential rotation (2)

Angular velocity profiles, in meridional quadrant

Helioseismology HD simulation MHD simulation

Differential rotation in the Sun and solar-type stars is poweredby turbulent Reynolds stresses, arising from rotationally-induced

anisotropy in turbulent transport of momentum and heat

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EULAG-MHD simulations

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Simulation framework

Simulate anelastic convection in thick,rotating and unstably stratified fluid shellof electrically conducting fluid, overlayinga stably stratified fluid shell.

Recent such simulations manage to reachRe, Rm ~102-103, at best; a long way fromthe solar/stellar parameter regime.

Throughout the bulk of the convectinglayers, convection is influenced byrotation, leading to alignment of convective cells parallel to the rotation axis.

Stratification leads to downward pumpingof the magnetic field throughout the convecting layers.

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Selected milestones

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.

Brun et al. 2004: Strongly turbulent MHD simulation, producing copious

small-scale magnetic field but no large-scale magnetic component.

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.

Gilman 1983: Boussinesq MHD simulation, producing large-scale magnetic fields with polarity

reversals on yearly timescale; but non-solar large-scale organization.

Brown et al. 2010, 2011: Obtain irregular polarity reversals of thin, intense

toroidal field structure in a turbulent simulation rotating at 5X solar.

Nelson et al. 2012, 2013: Autonomous generation of buoyantly rising flux-ropes structures showing sunspot-like emergence patterns.

Ghizaru et al. 2010: Obtain regular polarity reversals of large-scale magnetic component on decadal timescales, showing many solar-like characteristics.

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The MHD equations

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EULAG-MHDEULAG: a robust, general solver for geophysical flows; developed by Piotr Smolarkiewicz and collaborators at MMM/NCAR

EULAG-MHD: MHD generalization of above; developed mostlyat UdeM in close collaboration with Piotr Smolarkiewicz

Core advection scheme: MPDATA, a minimally dissipativeiterative upwind NFT scheme; equivalent to a dynamical, adaptivesubgrid model.

Thermal forcing of convection via volumetric Newtonian cooling termin energy equation, pushing reference adiabatic profile towards avery slightly superadiabatic ambiant profile

Strongly stable stratification in fluid layers underlying convecting layers.

Model can operate as LES or ILES

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MHD simulation of global dynamos [ Ghizaru et al. 2010, ApJL, 715, L133 ]

Electromagnetic induction by internal flows is the engine powering the solarmagnetic cycle. The challenge is to produce a magnetic field well-structuredon spatial and temporal scales much larger/longer than those associatedwith convection itself. This is the magnetic self-organisation problem.

Temperature perturbation Radial flow component Radial magnetic field component

http://www.astro.umontreal.ca/~paulchar/grps > Que faisons nous > Simulations MHD

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Kinetic and magnetic energies

(120 s.d.=10 yr)

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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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Magnetic cycles (1)Zonally-averaged Bphi at r/R =0.718

Zonally-averaged Bphi at -58o latitude

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Successes and problemsKiloGauss-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.

Reasonably solar-like internal differential rotation, and solar-like

cyclic torsional oscillations (correct amplitude and phasing).

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.

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 themagnetic cycle.

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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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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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Characteristics of simulated cycles (2)[ Passos & Charbonneau 2014, A&A, in press ]

r = 0.957/0.947[ 0.763/0.841 ]

r = -0.465/-0.143[ 0.185/-0.117 ]

r = 0.688/0.738[ 0.322/0.451 ]

r = -0.395/-0.147[ -0.552/-0.320 ]

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Characteristics of simulated cycles (3)

Hemispheric cycle amplitude show a hint of bimodality

Usoskin et al. 2014,A&A 562, L10;

From 3000yr 14Ctime series

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Characteristics of simulated cycles (4)Hemispheric cycle amplitude show a hint of bimodality

Usoskin et al. 2014,A&A 562, L10;

From 3000yr 14Ctime series

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« Double cycles » [ with P. Beaudoin, C. Simard, J.-F. Cossette ]

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Short quasiperiodic variability (1)

Evidence for short-term (~0.5 – 2 yr) quasiperiodic variabilityis found is a great many indicators of solar activity:

1. Sunspot number and area2. Radio flux3. Total and spectral irradiance4. p-mode acoustic frequencies5. Interplanetary magnetic field6. Flaring rate7. Solar wind speed8. ( For more, ask Scott McIntosh )

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Short quasiperiodic variability (1)

BISON

2.1-3.5 nHz

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Short quasiperiodic variability (1)

GONG

1.6-2.5 nHz2.5-3.5 nHz

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Short quasiperiodic variability (2)

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Short quasiperiodic variability (3)

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Short quasiperiodic variability in millenium simulation (1)

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Short quasiperiodic variability (4)

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« Double dynamo » action (1)Peaks at high latitude, but significantAmplitude down to equatorial regions

m s-1

Peaks at low latitudes withinconvection zone;tachocline mostly at high latitudes.

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« Double dynamo » action (2)

Scenario:

« Long » dynamo mode powered by turbulent emf( dynamo mode)

« short » dynamo mode powered by rotational shearin equatorial portion of convection zone ( dynamo mode)

If so, then Parker-Yoshimura sign rule should apply:dynamo wave propagates away from equatorial planealong isocontours of angular velocity

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Short quasiperiodic variability (4)

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Validation against mean-field model

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Tachocline instabilities[ M.Sc. Thesis N. Lawson ]

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MHD instabilities in stably stratifiedstellar interiors

1. Tayler instabilities (feeds on B)2. Magnetoshear instabilities (feeds on B and grad 3. Balbus-Hawley instability (feeds on B and 4. … and many more…

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Characteristics of simulated cycles (4)Hemispheric cycle amplitude show a hint of bimodality

Instability leads to cyclic exchangeof energy between axisymmetricand non-axisymmetric large-scalemagnetic components, with a characteristic phase lag.

Nonlinear development leads tocharacteristic « clamshell » pattern.

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The overshoot layer and tachocline

Strong magnetic fields accumulatein stable layer in response to overshoot and turbulent pumpingin overlying convecting layers.

Due to very low dissipationand strong stratification, the overshoot layer is verythin; rotational shear extendsfurther below due to magnetictorques.

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Magnetic field accumulation in « tachocline »

Poynting flux is downwardsthroughout bulk of convectingand stable layers, at allphases of the magnetic cycle

= (Tachocline magnetic energy/Poynting flux = 8 yr

Consistent with « passive » accumulation and dissipationof tachocline magnetic field, resulting from pumping from above.

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A clamshell instability ?

Phase pattern of magnetoshear instability !

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A clamshell instability ?

…maybe not…

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A Tayler instability ?

Tayler-stable throughout most of convectively stable layers (of course...) …excepts near polarity inversion lines.

Tayler stability criterion for purelytoroidal axisymmetric magnetic field:

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A Tayler instability ?

Plausible scenario: high latitudes are Tayler unstable; instability frontmoves quiescently equatorward with polarity inversion; when it reachesmid-latitude toroidal field bands, causes runaway destruction of B…

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Mode transitions from an instability ?

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Where do we go next ?

Understand what sets the cycle period(s)

Understand physical underpinnings of the cyclicmodulation of the convective energy flux

Understand role of tachocline instabilities in long termstability of simulations, and possible role in triggeringMaunder-Minimum-like period of strongly reduced activity

Comparative benchmark with ASH simulations

Get closer to surface

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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

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Short quasiperiodic variability (4)

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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 canbe 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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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 meanmagnetic field:

Extracting the symmetric part of the tensor yields:

where

acts as a velocity in the mean-field dynamo equations. For mildlyanisotropic, inhomogeneous turbulence:

I.3.4.3

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Turbulent pumping (2)Poleward transport of surface magnetic field by turbulent pumping;speed in range 1-3 m s-1 above+/- 45o latitudes

m s-1

Equatorward transport of deep magnetic field by turbulent pumpingbetween +/- 15 and 60o latitudes;speed 1-2 m s-1

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Turbulent pumping (3)Upward transport of magnetic fieldby turbulent pumping in subsurfacelayers; speed exceeding 1 m s-1 above +/- 60o latitudes

m s-1

Downward transport of magnetic fieldby turbulent pumping in bulk of deepconvection zone; speed exceeding1 m s-1 between +/- 15 and 60o latitude

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Active region decay (1)

Synoptic magnetogram courtesy D. Hathaway, NASA/MSFC[ http://solarscience.msfc.nasa.gov/images/magbfly.gif ]Toroidal flux emerging in active regions in one cycle: ~1017 Wb

Peak polar cap flux: ~1014 Wb

III.2.2.1

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(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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Active region decay (3)

Zonal means

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Active region decay (3)[ Simulation of surface magnetic flux evolution by A. Lemerle ]

The Babcock-Leightonmechanism is definitelyseen operating at the solar photosphere! But,does it really feed backinto the dynamo loop ?

III.2.2.1.4

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Formation of magnetic flux strands (1)[ Nelson et al. 2013, Astrophys. J., 762: 73 ]

Recent, very high resolution 3D MHD simulations of solar convectionHave achieved the formation of flux-rope-like super-equipartition-strength « magnetic strands » characterized by a significant density deficit in theircore; ripped from the parent large-scale structure by turbulent entrainement,subsequent buoyant rise ensues.

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Formation of magnetic flux strands (2)[ Nelson et al. 2014, Solar Phys., 289, 441 ]

The strands« remember »their origin !

The strands develop a patternof East-West tilt similar to thatinferred obervationally for the sun

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Stellar cycles (1)B.P. Brown et al. 2011, Astrophys. J., XXX, YYY

ASH LES: at solar rotationrate and luminosity, nolarge-scale field produced;

At 3X solar rotation, steadyaxisymmetric large-scalefield is produced;

At 5X solar rotation, irregularpolarity reversals occur.

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Solar/stellar magnetism

« If the sun did not have a magnetic field, it would be as boring a star asmost astronomers believe it to be »

(Attributed to R.B. Leighton)

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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