Simulating the Rise of Low Twist Flux Ropes in the Convection Zone Mark Linton (Naval Research Lab) James Leake (George Mason University)

Observations of Solar Active Region Twist Measurements of active regions twists: α = ~3x θ deg /m, (z vertical). Average | α| < /m. Pevtsov, Canfield & Metcalf (1995), Longcope, Fisher & Pevtsov (1998). These findings are consistent with what is produced by turbulence acting on rising convection zone flux tubes, with zero initial twist. Longcope, Fisher & Pevtsov (1998).

2D: Untwisted Buoyant Flux Ropes Will Fragment During Rise Longcope, Fisher & Arendt (1996) Sch ü ssler (1979) Active region observations are consistent with the emergence of untwisted flux ropes. 2D MHD simulations show that untwisted buoyant flux ropes quickly fragment and stop rising. Question: What allows flux ropes to rise to surface and emerge with low or zero levels of twist?

Does Viscosity Affect this Fragmentation? 10ν 0 ν 0 /10 ν0 ν0 Lare2D simulation of rise and fragmentation of untwisted flux ropes, at different viscosities (greyscale of B axial ). ν 0 t rise /R 2 = 3x10 -3 as in Longcope et al. (1996). Convection zone viscosity ~10 -9 ν 0. Decreasing the viscosity enhances fragmentation, so this does not help with maintaining coherency.

Maintaining Flux Rope Coherence with Twist Conclusion: twist allows flux ropes to maintain their coherence when they rise through the convection zone. Flux ropes reach a steady-state velocity (v rise ) when their buoyancy balances the drag force. For a twist of ~30x10 -8 /m, the flux tube resists breakup during its rise from -40Mm. Emonet & Moreno-Insertis (1998). Rise of twisted flux rope in 2D, 300x700 grid simulation (L-R symmetric). Velocity of the tube apex (circles) and center (stars) versus time.

Buoyant Rise of Twisted Flux Ropes, Test Run With Lare2D Code Test of Emonet & Moreno-Insertis (1998) run with the Lare2D code at resolution of 600x700, with no left-right asymmetry assumed. Times shown are: [0, 1.4, 2.7, 4.1, 5.4, 6.8] t 0. The results of Emonet & Moreno- Insertis are well reproduced.

Flux Rope Evolution as a Function of Initial Twist Twist decreases in strength from left to right as ~[60, 30, 10, 0]x10 -8 /m (for z 0 =-40Mm). This results in decreasing coherence of flux ropes. At zero twist, flux rope entirely splits into two (Emonet & Moreno-Insertis 1998).

Theory of twist limit for breakup Predict coherence of tube if maximum twist Alfven speed is greater than the flux tube rise speed: v A,θ > v rise. (Emonet & Moreno-Insertis 1998) For constant twist B θ =qrB z, maximum v A,θ ~ B 0 qR/ √ρ. Terminal rise speed: drag balances buoyancy. gδρ/ρ=C D v rise 2 /(2πR)~ v rise 2 /R, v rise ~ √(Rg δρ/ρ) ~ √(Rg/β) (for isothermal buoyancy, δρ/ρ ~ 1/β). This gives q cr = α axis /2 > 1/√(H p R), where pressure scale height is H p =p/(g ρ).

α axis = 27x10 -8 /m Tube core remains intact. Twist threshold for breakup at high η α axis = 22x10 -8 /m Tube breaks up. 2D (Lare2D) simulations for resistivity η/v Aaxis H p =3x10 -4, R=.6Mm, z 0 =-40Mm. Critical twist for flux rope coherence of α axis > 22x10 -8 /m. Agrees with Abbett et al. (2000) 3D simulations for the same flux rope and resistivity.

Lare2D simulations for zero explicit resistivity. Now the critical twist for flux rope coherence is α axis > 16x10 -8 /m. Effect of resistivity: diffuses axial field, decreasing the magnetic pressure of the tube. Plasma pressure compresses tube, increasing the central density. This can even make tube negatively buoyant. Conclusion: Keeping resistivity as low as possible (in line with solar values) is important for coherence. α axis = 16x10 -8 /m Tube splits. Twist threshold for breakup at η=0 11t 0 8t 0 4t 0 Time: 12t 0 8t 0 4t 0 α axis = 18x10 -8 /m Tube holds together.

Effect of Curvature on Buoyant Flux Tubes Buoyant rise of twisted flux ropes in 3D, with arched axes. Left panels: untwisted flux rope. Right panels: twisted flux rope. The untwisted flux rope breaks up, while the twisted flux rope keeps a coherent core, as for 2D results. Abbett, Fisher & Arendt (2000) Left panels: α=0. Right panels: α=12/z 0 at axis. α=30x10 -8 /m for z 0 =-40Mm or α=6x10 -8 /m for z 0 =-200Mm.

Fragmentation of Flux Ropes Versus Curvature of Axis Flux ropes break up for these simulations, where α axis = 15x10 -8 /m is smaller than the critical twist for this setup of α axis > 24x10 -8 /m (if z 0 =-40Mm). Curvature decreases from SS1 to LL1, and L1 has no curvature. The tubes split up increasingly rapidly as the curvature decreases. Conclusion: 3D curvature effects play significant role in keeping flux ropes coherent. Abbett, Fisher & Arendt (2000). Authors conclude that this is due to twisting up of tubes during breakup. Could tension force also be playing a role?

Coherence of Flux Ropes at Zero Twist Fan (2001) Tubes maintain coherence of cross section for rise of at least on density scale height. Concludes that added coherence is due to slow rise from initial neutral buoyancy, plus counter-rotation on each side of apex, which adds twist (similar to mechanism in Abbett et al 2000). Formation of arched, buoyant flux tube due to undular instability in magnetic flux sheet at base of convection zone.

Effect of Solar Rotation on Rising Tubes Coriolis effect acting on rising flux ropes has significant effect on coherence, deflecting vortical flows which otherwise break up flux rope. This allows zero twist field to maintain through rise over tens of Mm. Abbett, Fisher & Fan (2001) Cross sections of rising tubes with (left: LFLL) and without (right: SS0) effects of rotation, with zero twist (B 0 =27 kG). Untwisted flux rope rising under effects of solar rotation (B 0 =100 kG).

Theory of twist limit for breakup Why does curvature in 3D decrease q cr or α cr, as found by Abbett et al (2001)? Hypothesis – downward tension force reduces v rise, therefore reducing the critical twist. Force balance is then: (gδρ-κB 0 2 /2)/ρ=C D v rise 2 /(2πR), where κ is the downward curvature of the tube axis. Twist limit should then scale as: α axis ~ (..)*√(gδρ - κB 0 2 /2)/ρ Instead of ~(…)*√(gδρ/ρ). Coherence of tubes in rotating Sun may be partly due to this, also, as Coriolis effect slows down tube rise speed. Test this hypothesis in 2.5D by simulating tubes with smaller buoyancy δρ/ρ, e.g. p ~ ρ 5/ ξ, so δρ / ρ ~ ξ/(5β). Try the following initial buoyancy states: ξ=5 isothermal, δρ / ρ ~ 1/β. ξ=3 isentropic, δρ / ρ ~ 3/(5β). ξ=1 low buoyancy, δρ / ρ ~ 1/(5β).

4t 0 /√(ξ/5) 8t 0 /√(ξ/5) 11t 0 /√(ξ/5) Time: Low buoyancy: ξ=1 α axis = 11x10 -8 /m Rise at twist limit for different buoyancies Isothermal: ξ=5 α axis = 18x10 -8 /m Isentropic: ξ=3 α axis = 14x10 -8 /m For p~ρ 5/ ξ, buoyancy is: δρ / ρ ~ ξ/(5β). Prediction: v rise ~√(ξ) α crit ~√(ξ). Rise speed scales as predicted, though ξ =1 case rises slightly faster than expected. Twist limits: ξ=3 vs ξ=5 follows predicted scaling. ξ=1 limit is 25% larger than predicted.

Rise of 3D Tubes Through Convection Zone At high twist, most of tube rises, but twist effect generates wrong sign of tilt. At low twist (shown here) 45% of flux makes it to near surface with α~5x Fan (2009) Twisted flux ropes rise coherently. Jouve & Brun (2007) Untwisted flux ropes break up during rise, even with solar rotation. Coriolis effect alone does not keep tubes coherent here.

Large radius tube starting from base of convection zone can rise with lower twist, as α~ 1/√(RH p ). Some breakup, but 40% of original axial flux makes it up to -50Mm, consistent with Fan (2009) 3D result. Cartesian domain, covering most of convection zon: z=-200Mm to -20Mm, x=+/-90Mm. Tube starts at z=-184Mm, with radius=2Mm B 0 =92kG, axial flux=10 22 Mx α axial =6.6x10 -8 /m Tests of Coherence over Long Rise Scales

At α axial =4.4x10 -8 /m, just 12% of original flux rises to -50Mm (not shown). At α axial =3.3x10 -8 /m (above), flux rope splits up entirely. Limit may decrease further when add in low buoyancy, curvature and rotation effects (see Abbett et al, Jouve & Brun, Fan simulations). However, convective flows, not addressed here, may increase minimum amount of twist required (see, eg, Abbett et al 2004 simulations). Lower Twist Limit for Deep Tubes

Twist effects on Emergence into Corona Next question: Even after flux rope makes it to surface, must still emerge into corona to be observed. Murray et al (2006) find that twist must be larger than α axis =120x10 -8 /m for emergence. This is ~40x larger than the average twist of most active regions. How do such low twist regions emerge? Height of top of tube at various twists α axis, in units of 1.2x10 -5 /m. For α axis < 120x10 -8 /m, the tube does not emerge. α axis =360x10 -8 /m α axis =120x10 -8 /m α axis =240x10 -8 /m

Twist Trapped Below Surface in Emergence? Simulation by Fan (2009) shows that significant portion of twist remains below surface when high twist tube emerges. Twist at and above surface is ~10% of twist in in subsurface tube. Twist in average, sub-surface tube could therefore be ~10 -7 /m rather than ~10 -8 /m observed at surface. Archontis & Torok (2008) show emergence of tube with α 0 ~ 60x10 -8 /m. Surface twist could therefore be ~ 6x10 -8 /m. Left: fieldlines from two projections for emerged, twisted flux rope. Below: α versus depth for recently emerged flux rope, in units of α 0 = 667x10 -8 /m. Twist above surface has α ~ 67x10 -8 /m.

Summary / Questions Many simulations of flux ropes rising in convection zone have twists larger than that observed. Is this high level required? The critical twist for coherence is largely determined by the rise speed: decreasing the initial buoyancy can decrease the critical twist required. Does axial curvature slow down the flux rope rise sufficiently to keep rope coherent, or is it the added twisting of the arched legs (Abbett et al 2000)? Problem less severe for large flux ropes, as the critical twist ~ 1/ √(H p R): 2D isothermal 2Mm rope at base of convection zone needs α axial ~ 5x10 -8 /m. Curvature / Coriolis / low buoyancy slow down tube, giving lower necessary twist. But not without limit – convective flows will also destroy tubes, and these do not depend on tube speed. Why are do flux ropes emerging through the photosphere need twists ~40x larger than those observed (Murray et al. 2006)? Can lower twist ropes emerge with less twist, given more time (Archontis & Torok 2008)? Or is most of this twist unobservable, as it is trapped below the photosphere (Fan 2009)?