1. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Self Consistent Solar Wind Models Steven R. Cranmer Harvard-Smithsonian Center for Astrophysics

2. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Self Consistent Solar Wind Models Steven R. Cranmer Harvard-Smithsonian Center for Astrophysics Outline: 1.Five Necessary “Ingredients” 2.Successes of Wave/Turbulence Models (1D) 3.New Approximations for Wave Reflection (3D)

3. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Parker’s isothermal solar wind Gene Parker (1958) considered the steady-state conservation of mass and momentum in a hot (T ≈ 10 6 K) corona. Solutions were independent of the density (everywhere), and they did not require solving the internal energy conservation equation. In the early 1960s, new models with different T(r) profiles, including those consistent with polytropic (P ~ ρ γ ) equations of state. γ < 1.5 ! Sturrock & Hartle (1966) included heat conduction (and T p ≠ T e ), and found that energy addition was needed.

4. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Ingredient #1: “real” coronal heating vs.  Waves & turbulent dissipation?  Reconnection / mass input from loops? What determines how much energy is deposited as heat… ultimately from the “pool” of subphotospheric convection? How much heating is needed to produce the fast & slow solar wind? e.g., Leer et al. (1982) ≈ 8 x 10 5 erg/cm 2 /s (fast wind) ≈ 3 x 10 6 erg/cm 2 /s (slow wind)

5. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland It was realized in the late 1970s that coronal temperatures were probably too low to produce the “fast” solar wind via gas pressure gradients alone. Just as E/M waves carry momentum and exert pressure on matter, acoustic and MHD waves do work on the gas via similar net stress terms. To illustrate the effect, I constructed a grid of Parker-like models, with a ~flat T p (r) and a range of Alfvén wave amplitudes (conserving wave action). Ingredient #2: extra momentum sources Contours: wind speed at 1 AU (km/s) PCHPCH

6. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Ingredient #3: self-regulating mass flux Hammer (1982) & Withbroe (1988) suggested a steady-state energy balance: heat conduction radiation losses — ρvkT 5252 Only a fraction of the deposited heat flux conducts down, but in general, we expect that the mass loss rate should be roughly proportional to F heat. (In practice, the dependence is ~weaker than linear...)

7. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Ingredient #4: in situ conduction & heating Is the internal energy “game” over by the time the solar wind accelerates to its final terminal speed? In situ measurements (0.3–5 AU) say no... T ~ r – 4/3 Proton: Electron: 0.3 AU1 AU5 AU Cranmer et al. (2009)

8. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Ingredient #5: funnel-type field expansion Blah. Peter (2001) Fisk (2005) Empirical models of the open field from the “magnetic carpet” demand superradial expansion in low corona. UV Doppler blue-shifts are consistent with funnel flows (Byhring et al. 2008; Marsch et al. 2008). H I Lyα disk intensity in coronal holes isn’t explainable without funnel flows (Esser et al. 2005). Παντα ρει !

9. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Ingredient #5: funnel-type field expansion Cranmer & van Ballegooijen (2010) produced Monte Carlo models of the magnetic carpet’s connection to the solar wind. Preliminary models suggest the super- granular network is (at least in part) “emergent” from smaller-scale granule motions, diffusion, & rapid bipole emergence (e.g., Rast 2003; Crouch et al. 2007).

10. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland What happens when all of these ingredients are mixed together?

11. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Waves & turbulence in open flux tubes 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)

12. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Dissipation of MHD turbulence Standard nonlinear terms have a cascade energy flux that gives phenomenologically simple heating: Z+Z+ Z–Z– Z–Z– We used a generalization based on unequal wave fluxes along the field... n = 1: usual “golden rule;” we also tried n = 2. Caution: this is an order-of-magnitude scaling! (“cascade efficiency”) (e.g., Pouquet et al. 1976; Dobrowolny et al. 1980; Zhou & Matthaeus 1990; Hossain et al. 1995; Dmitruk et al. 2002; Oughton et al. 2006)

13. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland 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. Some details about the 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

14. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Magnetic flux tubes & expansion factors polar coronal holesf ≈ 4 quiescent equ. streamersf ≈ 9 “active regions”f ≈ 25 A(r) ~ B(r) –1 ~ r 2 f(r) (Banaszkiewicz et al. 1998) Wang & Sheeley (1990) defined the expansion factor between “coronal base” and the source-surface radius ~2.5 R s. TR

15. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Results: turbulent heating & acceleration T (K) reflection coefficient Goldstein et al. (1996) Ulysses SWOOPS

16. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Summary of other results Wind speed is anti-correlated with flux-tube expansion & height of critical point. For more information, see Cranmer (2009, Living Reviews in Solar Phys., 6, 3) Models match in situ data that correlate wind speed with: Integrated heat fluxes |F heat | match empirical req’s: 10 6 to 3 x 10 6 erg/cm 2 /s. Comparison with remote-sensing data (e.g., UVCS) isn’t as far along, because the models are one-fluid… the data showcase multi-fluid collisionless effects. The turbulent heating rate in the corona scales directly with the mean magnetic flux density there, as is inferred from X-rays (e.g., Pevtsov et al. 2003). Temperature (Matthaeus, Elliott, & McComas 2006) Frozen-in charge states [O 7+ /O 6+ ] The FIP effect [Fe/O] Specific entropy [ln(T/n γ–1 )] (Pagel et al. 2004) Turbulent fluctuation energy (Tu et al. 1992) (Zurbuchen et al. 1999)

17. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland TZ Challenge #2: is slow wind composition open/closed? FIP effect modeled with Laming (2004) theory: Alfven waves exert “pressure” on ions, but not on neutrals in upper chromo. Cranmer et al. (2007) Ulysses SWICS Wave pressure is automatically calculated in the model. In chromosphere, | a wp / g | ≈ 0.1, but it acts over ~tens of scale heights. Note that in these models, the “hole/streamer boundary slow wind” has fast-wind-like abundances. Only the “active region slow wind” has enhanced low-FIP abundances.

18. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland TZ Challenge #3: how does heating affect slow/fast wind? How do fast-wind properties in interplanetary space vary from the 1996–1997 minimum to the present minimum? Fractions given as “(new–old)/old” “New” magnetic field model was run with same parameters as old model. –03 % –17 % –14 % –28 % –22 % speed density Temp. P gas P dyn Ulysses polar data B field (input) v, n, T (output) +01 % –22 % –08% –21 % –27 % WTD model output (McComas et al. 2008)(Cranmer et al. 2010, SOHO-23 Proc.)

19. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland What’s stopping us from including this in 3D “global MHD” models of the Sun-heliosphere system? → Non-WKB Alfvén wave reflection!

20. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland How is wave reflection treated? “refl. coef” = |z + |/|z – | 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: TR 3e–5 1e –4 3e –4 0.001 0.003 0.01 0.03 0.1 0.3 0.9

21. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Reflection in simple limiting cases... Many earlier studies solved these equations numerically (e.g., Heinemann & Olbert 1980; Velli et al. 1989, 1991; Barkhudarov 1991; Cranmer & van Ballegooijen 2005). As wave frequency ω → 0, the superposition of inward & outward waves looks like a standing wave pattern: phase shift → 0 phase shift → – π/2 As wave frequency ω → ∞, reflection becomes weak... Cranmer (2010) presented approximate “bridging” relations between these limits to estimate the non-WKB reflection without the need to integrate along flux tubes. See also Chandran & Hollweg (2009); Verdini et al. (2010) for other approaches!

22. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Results: numerical integration vs. approx. “refl. coef” = |z + |/|z – | TR 3e–5 1e –4 3e –4 0.001 0.003 0.01 0.03 0.1 0.3 0.9 3e–5 1e –4 3e –4 0.001 0.003 0.01 0.03 0.1 0.3 0.9 ω0ω0

23. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Results: coronal heating rates Each “row” of the contour plot contributes differently to the total, depending on the power spectrum of Alfven waves... f –5/3 Cranmer & van Ballegooijen (2005) Tomczyk & McIntosh (2009) observational constraints on heating rates

24. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Conclusions For more information: http://www.cfa.harvard.edu/~scranmer/ It is becoming easier to include “real physics” in 1D → 2D → 3D models of the Sun- heliosphere system. Theoretical advances in MHD turbulence continue to help improve our understanding about coronal heating and solar wind acceleration. vs. We still do not have complete enough observational constraints to be able to choose between competing theories.

25. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Extra slides...

26. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland protons electrons O +5 O +6 Multi-fluid collisionless effects! coronal holes / fast wind (effects also present in slow wind)

27. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Low-freq. waves: remote-sensing techniques The following techniques are direct… (UVCS ion heating is more indirect) Intensity modulations... Motion tracking in images... Doppler shifts... Doppler broadening... Radio sounding... Tomczyk et al. (2007)

28. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland 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)

29. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland Results: in situ turbulence To compare modeled wave amplitudes with in-situ fluctuations, knowledge about the spectrum is needed... “e + ”: (in km 2 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. Cranmer et al. (2007) Helios (0.3–0.5 AU) Tu et al. (1992)

30. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland New result: solar wind “entropy” Pagel et al. (2004) found ln(T/n γ–1 ) (at 1 AU) to be strongly correlated with both wind speed and the O 7+ /O 6+ charge state ratio. (empirical γ = 1.5) The Cranmer et al. (2007) models (black points) do a reasonably good job of reproducing ACE/SWEPAM entropy data (blue). Because entropy should be conserved in the absence of significant heating, the quantity measured at 1 AU may be a long-distance “proxy” for the near-Sun locations of strong coronal heating.

31. Self Consistent Solar Wind ModelsS. R. Cranmer, 25 January 2010, ISSI, Bern, Switzerland New result: scaling with magnetic flux density Mean field strength in low corona: If the regions below the merging height can be treated with approximations from “thin flux tube theory,” then: B ~ ρ 1/2 Z ± ~ ρ –1/4 L ┴ ~ B –1/2 B ≈ 1500 G (universal?) f ≈ 0.002–0.1 B ≈ f B,....... and since Q/Q ≈ B/B, the turbulent heating in the low corona scales directly with the mean magnetic flux density there (e.g., Pevtsov et al. 2003; Schwadron et al. 2006; Kojima et al. 2007; Schwadron & McComas 2008)... Thus,