IMPRSIMPRS IntroIntro CourseCourse Germerode,Germerode, 19.2.200219.2.2002

SSoollaarr CCoonnvveeccttiioonn && MMaaggnneettiissmm

ManfredManfred SchüsslerSchüssler Max-Planck-InstitutMax-Planck-Institut fürfür AeronomieAeronomie Katlenburg-LindauKatlenburg-Lindau ConvectionConvection && magnetism:magnetism: closelyclosely relatedrelated TThhee ssoollaarr ccoonnvveeccttiioonn zzoonnee n 200200 MmMm thickthick layerlayer inin turbulentturbulent motionmotion

n VelocitiesVelocities rangerange fromfrom 100100 m/sm/s (bottom)(bottom) toto 1010 km/skm/s (top)(top) n EnergyEnergy fluxflux nearlynearly completelycompletely transportedtransported byby convectiveconvective motionmotion WWhhaatt iiss ccoonnvveeccttiioonn??

n FlowFlow drivendriven byby thermalthermal buoyancybuoyancy

n ConvectiveConvective instabilityinstability

è Viewgraphs... WHAT IS CONVECTION ?

• Transport of energy (internal, kinetic, latent, ...) by macroscopic motions (flows) driven by dynamical instability of a static equi- librium

• ... driven by an entropy gradient in thermodynamically open system

• Examples : air above a “radiator”, earth’s troposphere, stellar convection zones

• Convective motions normally are overturning (↔ oscillatory)

• treat solar atmospheric flow structures as “convective” in spite of stable stratification of observable layers (“overshoot”)

• “Sun as a Lab”: Concentrate on solar convection zone (→ huge Reynolds number Re = ul/ν, small Prandtl number σ = ν/κ, strong stratification, pressure fluctuations important → labora- tory convection results inapplicable)

• Not discussed here, but important : Effects of/on rotation, os- cillations, heating of upper atmosphere CONVECTION IN ASTROPHYSICS → everybody’s darling/problem

• Basic energy transport mechanism, besides radiation

• Leads to strong structuring of late type star surface regions (→ 1D-models inadequate) which affects the determination of atmospheric structure and abundances of elements

• Generates large-scale magnetic fields (dynamo) and small- scale magnetic structures (flux tubes)

• Provides mechanical energy for heating of chromospheres and coronae, drives stellar winds/mass loss

• Drives global oscillations

• Generates differential rotation

• Causes mixing and disturbs stellar evolution

• Is everywhere: ⇒ envelopes of late-type stars ⇒ cores of early-type stars ⇒ accretion disks ⇒ planetary interiors and atmospheres

• Solar atmosphere is a unique testing ground for understanding stellar convection : Processes can be observed on their natural time scales and length scales

LaboratoryLaboratory resultsresults andand numericalnumerical experimentsexperiments forfor Rayleigh-BénardRayleigh-Bénard convectionconvection withwith highhigh RaRa

17 n lab:lab: RaRa upup toto 1010 (Helium(Helium gasgas atat ~5~5 K)K) 5 1/3 n RaRa >> 2.52.5 1010 :: turbulence,turbulence, NuNu ~~ RaRa 7 2/7 n RaRa >> 44 1010 :: “hard“hard turbulence”,turbulence”, NuNu ~~ RaRa n coherentcoherent threads/plumes/thermalsthreads/plumes/thermals ofof buoyantbuoyant fluidfluid connectconnect toptop && bottombottom n flowsflows drivendriven byby threads,threads, passivepassive non-buoyantnon-buoyant fluidfluid betweenbetween n non-localnon-local energyenergy transport,transport, globalglobal circulationcirculation n withwith rotation:rotation: vortexvortex interactioninteraction ofof plumes,plumes, coalescencecoalescence n SolarSolar convectionconvection stronglystrongly differentdifferent fromfrom Rayleigh-Bénard:Rayleigh-Bénard: boundaryboundary structure,structure, stratification,stratification, compressibility...compressibility... FilamentaryFilamentary flowsflows inin Rayleigh-BénardRayleigh-Bénard convectionconvection

Kerr 1996 TimeTime evolutionevolution ofof filamentaryfilamentary flowsflows inin Rayleigh-BénardRayleigh-Bénard convectionconvection

Kerr 1996 SOLAR CONVECTION : OBSERVATIONAL APPROACHES

Continuum images, filtergrams ⇒ spatial structure : bright granules & interconnected network of dark intergranular lanes ⇒ temporal evolution : formation, dissolution, break-up of gran- ules, “exploding granules”

Spectrograms ⇒ line-of-sight velocities via Doppler effect (“line wiggles”) ⇒ upflow in granules, downflow in intergranular lanes ⇒ velocity distributions & gradients (line asymmetries) ⇒ supergranulation (Dopplergrams)

Proper motions of “tracers” ⇒ horizontal velocities ⇒ mesogranulation by “local correlation tracking” GGrraannuullaattiioonn:: SSoollaarr ssuurrffaaccee ccoonnvveeccttiioonn SSoollaarr ggrraannuullaattiioonn GGrraannuullaattiioonn uunndd llaabboorraattoorryy ccoonnvveeccttiioonn GranulationGranulation asas aa convectiveconvective phenomenonphenomenon SSuuppeerrggrraannuullaattiioonn SSuuppeerrggrraannuullaattiioonn aanndd mmaaggnneettiicc ffiieelldd:: tthhee CCaa++ nneettwwoorrkk GGrraannuullaattiioonn,, ssuunnssppoottss,, && ssmmaallll--ssccaallee mmaaggnneettiicc ffiieelldd ““MMeessooggrraannuullaattiioonn”” ?? 6. Numerical Simulations

Solar convection, in particular near the surface, is unsufficiently de- scribed by concepts like mixing length, linear superposition of eigen- modes, Boussinesq approximation, etc. because of

→ strong density stratification, → nonlinearity of the flows, → non-stationarity, → compressibility effects, shocks.

⇒ Numerical simulations

Equations and problems

Continuity:

∂ρ = −∇ · (ρ u) ∂t ⇒ compressibility, strong stratification

Equation of motion:

∂ u  1 ρ  + u · ∇ u = −∇p + ρ g + (∇ × B) × B + f ∂t 4π visc

⇒ Re = u · l/ν À 1 → small scales → “sub-grid” transport coefficients (viscosity, thermal & magnetic diffu- sion) ⇒ steep gradients, concentrated flows Energy equation:

∂T T γ +u·∇T = − (γ−1)∇·u− ∇·(Frad+Fvisc+Fmag +Fturb) ∂t χp ρcp

⇒ partial ionization effects → thermodynamical quantities must be determined self-consistently ⇒ surface layers → radiation has to be treated accurately, radiative energy loss to space drives the whole flow → solve transfer equation, preferably non-grey (opacity distri- bution functions)

Magnetic induction equation

∂B = ∇ × (u × B) − ∇ × (η ∇ × B) ∂t m

⇒ steep gradients, concentrated fields & currents ⇒ “sub-grid” magnetic diffusivity in addition... → equation of state, → equations for ionization equilibrium, → equations for thermodynamic quantities (χp, γ, cp, ...) → determination of opacities, → integration of radiative transfer equation along many lines of sight for many angles, calculation of mean intensity and radiative flux → determination of diagnostic information (mean quantities, cor- relations, single and average spectral line profiles, continuum intensity maps...) → visualization of flow properties and time evolution → ... • Boundary problems

Real boundaries: → top : τ = 1, radiation, steep T -gradient → bottom : overshoot, thermal boundary layer

Artificial boundaries: → transmitting, non-reflecting, open

Initial conditions: → simulation time > thermal relaxation time

• Approaches

Numerical experiments → simplified physics, consider general properties without attempt- ing to model the Sun → Graham (1975, 1977), Hurlburt et al, (1984, 1986), Chan & Sofia (1986, 1987), Hossain & Mullan (1990), Malagoli et al. (1990), Cattaneo et al. (1991)

Global convection → simulate bulk of solar convection zone, neglect surface regions, trade resolution for total size → Gilman & Glatzmaier (1981,...)

Surface convection → simulate observable solar convection on granular/mesogranular scale → Nordlund (1984,...), Gadun (1986), Stein & Nord- lund (1989), Steffen (1989) • Results : Simulation of convection

→ Strong inhomogeneities, steep gradients (temperature, velocity)

→ Significant pressure fluctuations, “buoyancy braking”

→ Supersonic velocities, shocks

→ Asymmetry of up- and downflows, non-local dynamics ⇒ strong, narrow downflows ↔ broad upflows ⇒ 3D, turbulent: Coherent pattern of downdrafts, disor- ganized upflows ⇒ topology changes with depth → “granulation” is a shallow phenomenon

→ Downdrafts play a prominent role: ⇒ sites of buoyancy driving ⇒ kinetic energy flux ⇒ helical motion due to “bathtub effect”

→ Other transport processes strongly non-local (magnetic fields, angular momentum...)

→ Essential features of observed granulation are reproduced by simulations with radiative transport: ⇒ isolated, hot upflows – network of cool downflows ⇒ timescales, spatial scales, “exploding granules” ⇒ average line profiles

CCoommppuutteerr--ssiimmuullaatteedd ccoonnvveeccttiioonn

•• BoussinesqBoussinesq modelmodel •• RayleighRayleigh number:number: 55 10105 •• 3D,3D, 512512´´512512´´9797 meshmesh •• widewide box,box, aspectaspect ratio:ratio: 1010 •• “(meso)granulation”“(meso)granulation”

Cattaneo & Emonet (2001) CCoommppuutteerr--ssiimmuullaatteedd ccoonnvveeccttiioonn

•• BoussinesqBoussinesq modelmodel •• RayleighRayleigh number:number: 55 10105 •• 3D,3D, 512512´´512512´´9797 meshmesh •• widewide box,box, aspectaspect ratio:ratio: 1010 •• “(meso)granulation”“(meso)granulation”

Cattaneo & Emonet (2001)

Temperature fluctuations SimulatedSimulated long-livedlong-lived convectiveconvective downflowsdownflows

Virtual “corks” are carried by the horizontal flow. They accumulate in downflow regions. ‘‘RReeaalliissttiicc’’ ssoollaarr ssiimmuullaattiioonnss n elaborateelaborate physics:physics: partialpartial ionization,ionization, radiation,radiation, compressible,compressible, openopen box,box, transmittingtransmitting boundaries,boundaries, spectralspectral lineline diagnosticsdiagnostics (Stokes(Stokes profiles)profiles) n ++ :: approximationapproximation toto solarsolar conditionsconditions n ++ :: directdirect comparisoncomparison withwith observationobservationss n -- :: computationalcomputational restrictionsrestrictions (box(box size,size, resolution)resolution) n -- :: ReynoldsReynolds numbersnumbers muchmuch belowbelow solarsolar valuesvalues

è viewgraphsviewgraphs AveragedAveraged energyenergy fluxesfluxes inin aa simulationsimulation ofof solarsolar convectionconvection

Stein & Nordlund, 2000 SSiimmuullaattiioonn aanndd oobbsseerrvvaattiioonn

SimulationSimulation (original)(original)

SimulationSimulation (smoothed)(smoothed)

ObservationObservation CChhaannggee ooff ddoowwnnffllooww ttooppoollooggyy

down

up Stein & Nordlund, 1998 DownflowDownflow structure:structure: plumes/thermalsplumes/thermals

Entropy Vorticity DDiissttrriibbuuttiioonn ooff vvoorrttiicciittyy

Stein & Nordlund, 1998 Penetrating convection/overshoot SimulatedSimulated convectionconvection inin aa solar-likesolar-like sphericalspherical shellshell Miesch 1998 AA “new“new paradigm”paradigm” forfor solar/stellarsolar/stellar convectionconvection ?? (Spruit,(Spruit, 1997)1997)

n OldOld paradigmparadigm (mixing-length(mixing-length model):model): n fullyfully developeddeveloped turbulenceturbulence withwith aa hierarchyhierarchy ofof “eddies”“eddies” n quasi-local,quasi-local, diffusion-likediffusion-like transporttransport n flowsflows drivendriven byby locallocal entropyentropy gradientgradient

n NewNew paradigmparadigm (lab(lab && numericalnumerical experiments):experiments): n turbulentturbulent downdrafts,downdrafts, laminarlaminar isentropicisentropic upflowsupflows n flowsflows drivendriven byby surfacesurface entropyentropy sinksink (radiative(radiative cooling)cooling) n non-diffusivenon-diffusive transporttransport n largerlarger scalesscales (meso/supergranulation)(meso/supergranulation) drivendriven byby compressingcompressing andand mergingmerging downdraftsdowndrafts è viewgraphviewgraph