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Applications of MHD Turbulence to Modeling Solar (and Stellar) Coronal Heating and Winds

Applications of MHD Turbulence to Modeling Solar (and Stellar) Coronal Heating and Winds. Steven R. Cranmer Harvard-Smithsonian Center for Astrophysics. Outline: Turbulence as a driver of coronal heating & wind acceleration Kinetic partitioning to protons, electrons, and ions.

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Applications of MHD Turbulence to Modeling Solar (and Stellar) Coronal Heating and Winds

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  1. Applications of MHD Turbulence to Modeling Solar (and Stellar) Coronal Heating and Winds Steven R. CranmerHarvard-SmithsonianCenter for Astrophysics

  2. Outline: • Turbulence as a driver of coronal heating & wind acceleration • Kinetic partitioning to protons, electrons, and ions Applications of MHD Turbulence to Modeling Solar (and Stellar) Coronal Heating and Winds • Results of recent modeling • Open issues (“successes”) (“failures”) • Ion cyclotron resonance? • A laundry list of other possibilities . . . Steven R. CranmerHarvard-SmithsonianCenter for Astrophysics

  3. My own history . . . 1992–1996: hot star theory 1996–1999: UVCS data analysis 1999–2004: Why are heavy ions preferentially heated & accelerated? 2004–now: Why is the whole plasma heated & accelerated?

  4. The Debate in ’08 • Two broad classes of models have evolved that attempt to self-consistently answer the question: How are fast and slow wind streams accelerated? Wave/Turbulence-Driven (WTD) models Reconnection/Loop-Opening (RLO) models Opinionated “position paper:” arXiv: 0804.3058

  5. Open flux tubes: global model • Photospheric flux tubes are shaken by an observed spectrum of horizontal motions. • Alfvén waves propagate along the field, and partly reflect back down (non-WKB). • Nonlinear couplings allow a (mainly perpendicular) cascade, terminated by damping. (Heinemann & Olbert 1980; Hollweg 1981, 1986; Velli 1993; Matthaeus et al. 1999; Dmitruk et al. 2001, 2002; Cranmer & van Ballegooijen 2003, 2005; Verdini et al. 2005; Oughton et al. 2006; many others!)

  6. Alfvén wave reflection • At photosphere: empirically-determined frequency spectrum of incompressible transverse motions (from statistics of tracking G-band bright points) • At all larger heights: self-consistent distribution of outward (z–) and inward (z+) Alfvenic wave power, determined by linear non-WKB transport equation: refl. coeff = |z+|2/|z–|2 TR

  7. Wave / Turbulence-Driven models • Cranmer & van Ballegooijen (2005) solved the transport equations for a grid of “monochromatic” periods (3 sec to 3 days), then renormalized using photospheric power spectrum. • One free parameter: base “jump amplitude” (0 to 5 km/s allowed; ~3 km/s is best)

  8. Anisotropic cascade and dissipation • Traditional (RMHD-like) nonlinear terms have a cascade energy flux that gives phenomenologically simple heating: • We used a generalization based on unequal wave fluxes along the field . . . Z– Z+ • n = 1: usual “golden rule;” we also tried n=2. Z– (e.g., Pouquet et al. 1976; Dobrowolny et al. 1980; Zhou & Matthaeus 1990; Hossain et al. 1995; Dmitruk et al. 2002)

  9. Self-consistent 1D models • Cranmer, van Ballegooijen, & Edgar (2007) computed solutions for the waves & background one-fluid plasma state along various flux tubes... going from the photosphere to the heliosphere. • The only free parameters: radial magnetic field & photospheric wave properties. • Ingredients: • Alfvén waves: non-WKB reflection with full spectrum, turbulent damping, wave-pressure acceleration • Acoustic waves: shock steepening, TdS & conductive damping, full spectrum, wave-pressure acceleration • Radiative losses: transition from optically thick (LTE) to optically thin (CHIANTI + PANDORA) • Heat conduction: transition from collisional (electron & neutral H) to a collisionless “streaming” approximation

  10. Results: turbulent heating & acceleration T (K) Ulysses SWOOPS Goldstein et al. (1996) reflection coefficient

  11. Results: other fast/slow diagnostics • The wind speed is anticorrelated with flux-tube expansion . . . “active region” fields Cascade efficiency: n=1 n=2

  12. Results: in situ turbulence • To compare modeled wave amplitudes with in-situ fluctuations, knowledge about the spectrum is needed . . . • “e+”: (in km2 s–2 Hz–1 ) defined as the Z– energy density at 0.4 AU, between 10–4 and 2 x 10–4 Hz, using measured spectra to compute fraction in this band. Helios (0.3–0.5 AU) Tu et al. (1992) Cranmer et al. (2007)

  13. Results: heavy ion properties • Frozen-in charge states • FIP effect (using Laming’s 2004 theory) Ulysses SWICS Cranmer et al. (2007)

  14. Aside: application to T Tauri winds • Recent work has extended these models to accretion-driven winds of young, solar-type stars (Cranmer 2008, arXiv:0808.2250) • Accretion proceeds by free-falling inhomogeneous clumps impacting the star, and generating MHD waves on the surface (like solar Moreton/EIT waves?). • These “extra” waves give input orders of magnitude more energy into an MHD cascade, and can give rise to stellar winds with dM/dt up to ~10–8M/yr !

  15. Turbulent mass loss solar parameter study “proper” solar models T Tauri models

  16. Turbulent mass loss solar parameter study “proper” solar models T Tauri models

  17. New result: solar wind “entropy” • Pagel et al. (2004) found ln(T/nγ–1) (at 1 AU) to be strongly correlated with both wind speed and the O7+/O6+ charge state ratio. (empirical γ = 1.5) • The Cranmer et al. (2007) models do a good job of reproducing ACE/SWEPAM entropy data (blue region) & Ulysses charge state trends (brown regions).

  18. Smith et al. (2001): Voyager & Omnitape Helios & other 1 AU… Cranmer et al. (2007) L(pole, equator) Joe Borovsky’s “walls” (ACE) The correlation length: apples to oranges?

  19. Problem: too hot at 1 AU ? standard (n=1) model rapid-quenching (n=2) model Ulysses Tp

  20. Electron heat conduction • At ~1 AU, the modeled T(r) is a balance between adiabatic cooling & collisionless conduction. • We’ve used Hollweg (1974):

  21. Progress towards a robust “recipe” Not too bad, but . . . • The turbulent dissipation rate scaling is approximate; needs refining? • Because of the need to determine non-WKB (nonlocal!) reflection coefficients, it may not be easy to insert into global/3D MHD models. • Doesn’t specify proton vs. electron heating (they conduct differently!) • Does turbulence generate enough ion-cyclotron waves to heat the minor ions? • Are there additional (non-photospheric) sources of waves / turbulence / heating for open-field regions? (e.g., flux cancellation events) (B. Welsch et al. 2004)

  22. Multi-fluid collisionless effects? O+5 O+6 protons electrons Polar coronal hole

  23. Alfven wave’s oscillating E and B fields ion’s Larmor motion around radial B-field Preferential ion heating & acceleration • UVCS observations have rekindled theoretical efforts to understand heating and acceleration of the plasma in the (collisionless?) acceleration region of the wind. • Ion cyclotron waves (10–10,000 Hz) suggested as a “natural” energy source that can be tapped to preferentially heat & accelerate heavy ions. lower Z/A faster diffusion

  24. Evidence for ion cyclotron resonance Indirect: • UVCS (and SUMER) remote-sensing data • Helios (0.3–1 AU) proton velocity distributions (Tu & Marsch 2002) • Wind (1 AU): more-than-mass-proportional heating (Collier et al. 1996) (more) Direct: • Leamon et al. (1998): at ω≈Ωp, magnetic helicity shows deficit of proton-resonant waves in “diffusion range,” indicative of cyclotron absorption. • Jian, Russell, et al. (2008) (COSPAR poster): STEREO shows isolated bursts of ~monochromatic waves with ω≈ 0.1–1 Ωp

  25. Where do cyclotron waves come from? (1) Base generation by, e.g., “microflare” reconnection in the lanes that border convection cells (e.g., Axford & McKenzie 1997). • Problems: • Incompatible with radio IPS power spectra (Hollweg 1999) • Minor ions would damp waves before they could resonate with O5+ or protons (Cranmer 2000, 2001) (2) Secondary generation: low-frequency Alfven waves may be converted into cyclotron waves gradually in the corona.

  26. Charge/mass dependence • Assuming there is enough “replenishment” (via, e.g., turbulent cascade?) to counteract local damping, the degree of ion heating depends on the assumed distribution of wave power vs. frequency (or parallel wavenumber). • A simple assumption of a power-law vs. parallel wavenumber shows that the charge-to-mass dependence of the heating may be increasing or decreasing... UVCS O VI (O+5) measurement was used to normalize the heating rate. Mg X (Mg+9) showed a much narrower line profile (despite being so close to O+5 in its charge-to-mass ratio)!

  27. Anisotropic MHD cascade • Can MHD turbulence generate ion cyclotron waves? Many models say no! • Simulations & analytic models predict cascade from small to large k ,leaving k ~unchanged.“Kinetic Alfven waves” with large k do not necessarily have high frequencies.

  28. Anisotropic MHD cascade • Can MHD turbulence generate ion cyclotron waves? Many models say no! • Simulations & analytic models predict cascade from small to large k ,leaving k ~unchanged.“Kinetic Alfven waves” with large k do not necessarily have high frequencies. • In a low-beta plasma, KAWs are Landau-damped, heating electrons preferentially! • Cranmer & van Ballegooijen (2003) modeled the anisotropic cascade with advection & diffusion in k-space and found somek “leakage” . . .

  29. An advection-diffusion cascade model • The Cranmer & van Ballegooijen (2003) advection-diffusion equation: • “Critical balance” (Higdon/Goldreich/Sridhar/others) was built into the eqns . . . • Rapid decay to higher k║ is contained in f(x). Cho et al. (2002) examined various functional forms as fits to numerical simulations (not enough dynamic range?). • CvB2003 solved an approximate version of the advection-diffusion eqn to get: • Key parameter: (β/γ). van Ballegooijen (1986) argued for β/γ ≈ 1 (random walk)

  30. Missed KAW opportunities...? • Hidden in the CvB 2003 paper were a few results that could have been highlighted better... e.g., a prediction for the KAW k┴–7/3 inertial range slope, and: Bale et al. (2005)

  31. Advection-diffusion cascade results • Taking the anisotropic spectrum and using linear Maxwell-Vlasov dissipation rates, the ratio of proton vs. electron heating can be derived as a function of position in the fast solar wind (using the Cranmer & van Ballegooijen 2005 model):

  32. New SUMER constraints • Landi & Cranmer (2009, arXiv:0810.0018) analyzed a set of SUMER line widths that suggest preferential ion heating at r≈ 1.05 to 1.2 Rs in coronal holes. • We produced and compared two independent models: Te • Solve a semi-empirical ion heating equation with an arbitrary normalization for the ion cyclotron wave power. (Each ion is modeled independently of the others.) Normalization varied till agrees w/ data. • Use the Cranmer & van Ballegooijen (2003, 2005) models to predict the ion cyclotron wave power spectrum at a given height.

  33. Example heating model for O VI • How well do we really know the proton temperature? Vary as free parameter... UVCS constraints SUMER constraints • The yellow/green curves seem to do the best... they imply strong Coulomb collisional coupling at the SUMER heights!

  34. Compare all ions at r = 1.069 Rs • Colors: different choices for proton temperature. Black curves: theoretical resonant spectra from Cranmer & van Ballegooijen (2003) advection-diffusion model. y-axis: wave power needed to produce ion heating

  35. Power increase at large Z/A ? • This is not predicted by simple turbulent cascade models. • If it is real, it might be: • Increased wave power from plasma instabilities that are centered around either the alpha (Z/A = 0.5) or proton (Z/A = 1) resonances (Markovskii 2001; Zhang 2003; Laming 2004; Markovskii et al. 2006) ? • Predicted “spectral flattening” due to oblique propagation and/or compressibility effects in dispersion relation? Harmon & Coles (2005) invoked these effects to model the observed IPS density fluctuation spectra. • A kind of “bottleneck effect” wherein the power piles up near the dissipation scale, due to nonlocal interactions between disparate scales in k-space (Falkovich 1994; Biskamp et al. 1998) ???

  36. So does turbulence generate cyclotron waves? Directly from the linear waves? Probably not! How then are the ions heated and accelerated? • When MHD turbulence cascades to small perpendicular scales, the small-scale shearing motions may be able to generate ion cyclotron waves (Markovskii et al. 2006). • Dissipation-scale current sheets may preferentially spin up ions (Dmitruk et al. 2004)? • If MHD turbulence exists for both Alfvén and fast-mode waves, the two types of waves can nonlinearly couple with one another to produce high-frequency ion cyclotron waves (Chandran 2006). • If nanoflare-like reconnection events in the low corona are frequent enough, they may fill the extended corona with electron beams that would become unstable and produce ion cyclotron waves (Markovskii 2007). • If kinetic Alfvén waves reach large enough amplitudes, they can damp via wave-particle interactions and heat ions (Voitenko & Goossens 2006; Wu & Yang 2007). • Kinetic Alfvén wave damping in the extended corona could lead to electron beams, Langmuir turbulence, and Debye-scale electron phase space holes which could heat ions perpendicularly (Matthaeus et al. 2003; Cranmer & van Ballegooijen 2003).

  37. What to do next? • Many of the proposed mechanisms haven’t been tested with realistic coronal plasma conditions! (i.e., plasma beta, driving wave amplitudes & frequencies, etc.) • The mechanisms of “parallel cascade” in low-beta plasmas need to be more fully worked out! (the tail that wags the dog?) The CvB (2003) “advection-diffusion” model is a crass local approx. to a truly nonlocal effect. • Explore relationships between turbulence and reconnection theory... • Better measurements are needed: both remote and in situ!

  38. Future missions • Solar Probe Plus (in to ~20 Rs) is finally moving forward. • CPEX (Coronal Physics Explorer) currently in Phase A concept study: next-generation UVCS & LASCO, capable of probing dozens of ions in coronal holes at UVCS heights! • More traditional “solar physics” missions (SDO) will put new constraints on physics of reconnection & turbulent heating!

  39. Conclusions • UV coronagraph spectroscopy has led to fundamentally new views of the collisionless acceleration regions of the solar wind. • Theoretical advances in MHD turbulence continue to feed back into global models of coronal heating and the solar wind. • The extreme plasma conditions in coronal holes (Tion>> Tp > Te ) have guided us to discard some candidate processes, further investigate others, and have cross-fertilized other areas of plasma physics & astrophysics. • Next-generation observational programs are needed for conclusive “constraints.” For more information: http://www.cfa.harvard.edu/~scranmer/

  40. Extra slides . . .

  41. The solar wind: discovery • 1860–1950: Evidence slowly builds for outflowing magnetized plasma in the solar system: • 1958: Eugene Parker proposed that the hot corona provides enough gas pressure to counteract gravity and accelerate a “solar wind.” • 1962: Mariner 2 provided direct confirmation. • solar flares  aurora, telegraph snafus, geomagnetic storms • comet ion tails point anti-sunward (no matter comet’s motion)

  42. fast slow speed (km/s) Tp(105 K) Te(105 K) Tion / Tp O7+/O6+, Mg/O 600–800 2.4 1.0 > mion/mp low 300–500 0.4 1.3 < mion/mp high In situ solar wind: properties • Mariner 2 detected two phases of solar wind: slow (mostly) + fast streams • Uncertainties about which type is “ambient” persisted because measurements were limited to the ecliptic plane . . . • Ulysses left the ecliptic; provided 3D view of the wind’s source regions. • Helios saw strong departures from Maxwellians. By ~1990, it was clear the fast wind needs something besides gas pressure to accelerate so fast!

  43. Wang et al. (2000) Solar wind: connectivity to the corona • High-speed wind: strong connections to the largest coronal holes hole/streamer boundary (streamer edge) streamer plasma sheet (“cusp/stalk”) small coronal holes active regions • Low-speed wind: still no agreement on the full range of coronal sources:

  44. The coronal heating problem • We still don’t understand the physical processes responsible for heating up the coronal plasma. A lot of the heating occurs in a narrow “shell.” • Most suggested ideas involve 3 general steps: 1. Churning convective motions that tangle up magnetic fields on the surface. 2. Energy is stored in tiny twisted & braided magnetic flux tubes. 3. Collisions (particle-particle? wave-particle?) release energy as heat. Heating Solar wind acceleration!

  45. Coronal heating mechanisms • So many ideas, taxonomy is needed! (Mandrini et al. 2000; Aschwanden et al. 2001) • Where does the mechanical energy come from? • How rapidly is this energy coupled to the coronal plasma? • How is the energy dissipated and converted to heat? vs. waves shocks eddies (“AC”) twisting braiding shear (“DC”) vs. interact with inhomog./nonlin. turbulence reconnection collisions (visc, cond, resist, friction) or collisionless

  46. Reconnection / Loop-Opening models • There is a natural appeal to the RLO idea, since only a small fraction of the Sun’s magnetic flux is open. Open flux tubes are always near closed loops! • The “magnetic carpet” is continuously churning . . . • Open-field regions show coronal jets (powered by reconnection?) that contribute to the wind mass flux. Fisk (2005) Hinode/XRT (X-ray) http://xrt.cfa.harvard.edu STEREO/EUVI (195 Å) courtesy S. Patsourakos

  47. Reconnection / Loop-Opening models • Emerging loops inject both mass and Poynting flux into open-field regions. • Feldman et al. (1999) found correlation between loop-size & coronal temperature. • Fisk et al. (1999), Fisk (2003), Gloeckler et al. (2003), Schwadron & McComas (2003), Schwadron et al. (2005) worked out the solar wind implications . . . Ulysses SWICS Fisk (2003) theory

  48. Alfven wave’s oscillating E and B fields ion’s Larmor motion around radial B-field something else? Departures from thermal equilibrium • UVCS/SOHO observations rekindled theoretical efforts to understand collisionless heating and acceleration effects in the extended corona. • Ion cyclotron waves (10–10,000 Hz) suggested as a “natural” energy source that can be tapped to preferentially heat & accelerate heavy ions. cyclotron resonance-like phenomena MHD turbulence

  49. The extended solar atmosphere . . . Heating is everywhere . . . . . . and everything is in motion

  50. Waves? Start in the photosphere . . . • Photosphere displays convective motion on a broad range of time/space scales: β << 1 β ~ 1 β > 1

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