Searching for the solar dynamo in a computer

Dhrubaditya Mitra NORDITA

16th January 2011 Bangalore

Collaborators

● Reza Tavakol, QMUL.● Axel Brandenburg, NORDITA.● Petri Kapyla, Helsinki Observatory.● Maarit Mantere, Helsinki Observatory

Solar magnetic field

Solar dynamo: important features.

● Oscillations and polarity reversal, 22 year solar cycle.

● Equatorward migration of sunspots. ● At the solar surface the azimuthally

averaged radial field is rather weak (about 1G) compared to the peak magnetic field in sunspots (about 2 kG).

Turbulence in the sun.

● Observation from MDI (Schou et al 1998)

Convection zone

Radiative core

● Helioseismology

● Large scale shear or differential rotation.

● Convection and rotation.

Tachocline

Dream of dynamo simulations

● A model which incorporates the essential ingredients (MHD, rotation, convection, differential rotation).

● Shows large-scale magnetic field. ● The large-scale magnetic field shows

spatio-temporal behaviour similar to the Sun.

● And the dynamics persists even for very high magnetic Reynolds number.

Compressible Magnetohydrodynamics (MHD)

D UDt =−∇ p

J X B Fvisc f

∂ ∂ t =−∇⋅

U

J = ∇ X B∂ B∂ t = ∇ X

U X B− J

Advective derivative Lorentz force

viscous force

Magnetic diffusivity

PencilPencilCodeCode

● Started in Sept. 2001 by Axel Brandenburg and Wolfgang Dobler

● High order (6th order in space, 3rd order in time)

● Cache & memory efficient● MPI, can run PacxMPI (across countries!)● Maintained/developed by ~40 people (SVN)● Automatic validation (over night or any

time)● Max resolution so far 10243 , 4096 procs

• Isotropic turbulence– MHD, passive scl, CR

• Stratified layers– Convection, radiation

• Shearing box– MRI, dust, interstellar– Self-gravity

• Sphere embedded in box

– Fully convective stars– geodynamo

• Other applications– Homochirality– Spherical coordinates

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Pencil formulationPencil formulation

● In CRAY days: worked with full chunks f(nx,ny,nz,nvar)– Now, on SGI, nearly 100% cache misses

● Instead work with f(nx,nvar), i.e. one nx-pencil● No cache misses, negligible work space, just 2N

– Can keep all components of derivative tensors● Communication before sub-timestep● Then evaluate all derivatives, e.g. call curl(f,iA,B)

– Vector potential A=f(:,:,:,iAx:iAz), B=B(nx,3)

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Switch modulesSwitch modules● magnetic or nomagnetic (e.g. just hydro)● hydro or nohydro (e.g. kinematic dynamo)● density or nodensity (burgulence)● entropy or noentropy (e.g. isothermal)● radiation or noradiation (solar convection, discs)● dustvelocity or nodustvelocity (planetesimals)● Coagulation, reaction equations● Chemistry (reaction-diffusion-advection equations)

Other physics modules: MHD, radiation, partial ionization, chemical reactions, selfgravity

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High-order schemesHigh-order schemes

● Alternative to spectral or compact schemes– Efficiently parallelized, no transpose necessary– No restriction on boundary conditions– Curvilinear coordinates possible (except for

singularities)● 6th order central differences in space● Non-conservative scheme

– Allows use of logarithmic density and entropy– Copes well with strong stratification and temperature

contrasts

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Wavenumber characteristicsWavenumber characteristics

( )kx

dxkxdkeff sin/cos

−=

( ) xkkx

dxkxdk Nyeff δπ / ,cos/cos 222 =

−=

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Evolution of code sizeEvolution of code size

User meetings:User meetings:2005 Copenhagen2006 Copenhagen2007 Stockholm2008 Leiden2009 Heidelberg2010 New York

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Increase in # of auto testsIncrease in # of auto tests

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Faster and bigger machinesFaster and bigger machines

From cartesian to spherical● The code is modular such that it uses only the

vector operator. ● To convert a code from cartesian to spherical

polar we need to recode the vector operators. ● We do this by co-variant derivatives.

Simulations of 3-d spherical dynamos● Inelastic Ash Code (Gilman, Glatzmaier, …,

Brun, Meisch, Browning)● Ash code: Finite-difference in radial direction,

spectral in other two. Simulations done in a spherical shell.

● Weakly compressible pencil code: Finite-difference in all the three directions. Simulations done in spherical wedge

● Ghizaru, Charbonneau and Smolarkiewicz, ApJ 2010.

Kapyla et al 2010

Kapyla et al 2010

Migration:

Results from other groups:

Kinetic helicity and differential rotation:

Beasts: ● Banana cells in convection (Meisch et al 2000).● Sea serpents (Kosovichev et al)

Summary ● Numerical simulation results are consistent with

the theory but neither the theory nor simulations describes the sun.

● Simulations are of course not in the right parameter range.

● Need for sub-grid scale modelling.● Resolving near-surface shear layer, in other

words MASSIVE RESOLUTION. ● Cartesian simulations of forced turbulence

shows large scale magnetic field.

Simulations with two signs of kinetic helicity.

● Consider simulations with two hemispheres with an external force which injects negative (positive) helicity in northern (southern) hemisphere.

● Rotation and stratification in the sun creates the neagtive (positive) kinetic helicity in the northern (southern) hemisphere.

● No differential rotation/shear● We observe large scale magnetic field

which shows fascinating dynamical behaviour.

Space-time diagram

DNSPFC

Mean-Field,PFC

DNS of north withtwo openboundaries

Frequencies of oscillations

Diamonds: DNS results, Asterices : Mean field Results. The frequency of oscillations essentially does not dependon magnetic Reynolds number.

Magnetic helicity in open domains

Caveats

● Although the frequency of oscillations are not resistively limited the initial growth phase is.

● The magnetic field in the mean-field simulations is catastrophically quenched, which may be alleviated by diffusive flux of magnetic helicity across the equator.

● Presumably simulations with stratification+rigid rotation will reproduce this results but at much higher resolutions.

Open questions

● In the absence of any other mechanism, magnetic helicity is transported across equator by diffusion. Can we define a diffusion coefficient and how does that compare with turbulent diffusivity ?

● Include convection+rigid rotation. ● How will differential-rotation change all these ?● How shall the picture change (if at all ) as we go

close to the pole ? (Mean field simulations suggest no change at all.)