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 ? ² Transport of energy (internal, kinetic, latent, ...) by macroscopic motions (flows) driven by dynamical instability of a static equi- librium ² ... driven by an entropy gradient in thermodynamically open system ² Examples : air above a “radiator”, earth’s troposphere, stellar convection zones ² Convective motions normally are overturning ($ oscillatory) ² treat solar atmospheric flow structures as “convective” in spite of stable stratification of observable layers (“overshoot”) ² “Sun as a Lab”: Concentrate on solar convection zone (! huge Reynolds number Re = ul=º, small Prandtl number σ = º=·, strong stratification, pressure fluctuations important ! labora- tory convection results inapplicable) ² Not discussed here, but important : Effects of/on rotation, os- cillations, heating of upper atmosphere CONVECTION IN ASTROPHYSICS ! everybody’s darling/problem ² Basic energy transport mechanism, besides radiation ² Leads to strong structuring of late type star surface regions (! 1D-models inadequate) which affects the determination of atmospheric structure and abundances of elements ² Generates large-scale magnetic fields (dynamo) and small- scale magnetic structures (flux tubes) ² Provides mechanical energy for heating of chromospheres and coronae, drives stellar winds/mass loss ² Drives global oscillations ² Generates differential rotation ² Causes mixing and disturbs stellar evolution ² Is everywhere: ) envelopes of late-type stars ) cores of early-type stars ) accretion disks ) planetary interiors and atmospheres ² 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: @½ = ¡r ¢ (½ u) @t ) compressibility, strong stratification Equation of motion: 0@ u 1 1 ½ @ + u ¢ r uA = ¡rp + ½ g + (r £ B) £ B + f @t 4¼ visc ) Re = u ¢ l=º À 1 ! small scales ! “sub-grid” transport coefficients (viscosity, thermal & magnetic diffu- sion) ) steep gradients, concentrated flows Energy equation: @T T γ +u¢rT = ¡ (γ¡1)r¢u¡ r¢(Frad+Fvisc+Fmag +Fturb) @t Âp ½cp ) 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 = r £ (u £ B) ¡ r £ (´ r £ 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 (Âp; 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 ! ... ² Boundary problems Real boundaries: ! top : ¿ = 1, radiation, steep T -gradient ! bottom : overshoot, thermal boundary layer Artificial boundaries: ! transmitting, non-reflecting, open Initial conditions: ! simulation time > thermal relaxation time ² 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) ² 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