Random Walk on the Surface of the Sun Chung-Sang Ng Geophysical Institute, University of Alaska Fairbanks UAF Physics Journal Club September 10, 2010
Collaborators/Acknowledgements Amitava Bhattacharjee, UNH Liwei Lin, UNH Work supported by NSF, NASA
Outline Introduction to the heating problem of the solar corona. Parker s model for the heating of solar corona based on formation of current sheets. Some results from numerical simulations. Theoretical understandings based on random walk. Conclusions.
Coronium? Unusual emission lines (first observed in 1869 eclipse), e.g., a 530.3 nm green line --- new element (Coronium)? In fact, those are from highly ionized ions (Fe XIV for the green line) [Grotrian (1939) and Edlen (1942)] Corona must be very hot! http://www.fourmilab.ch/images/eclipse_2001/figures/l09.html
X-ray emission from the sun is observed since 1950 s. Nowaday X-ray/EUV images are important tools in observations of the corona. X-ray emission Hotter object emits radiation in shorter wavelength.
b ~ 2.9 x 10-3 mk (Wien s displacement constant) --- for black body radiation. Wien's displacement law White hot is hotter red hot. λ max = b/t Light from the sun has a spectrum peaks ~ 500 nm ==> solar surface ~ 6000 K A spectrum peaks ~ 3 nm (X-ray) ==> T ~ 10 6 K http://en.wikipedia.org/wiki/image:wiens_law.svg
Solar corona: heating problem [http://science.nasa.gov/ssl/pad/solar/quests.htm]
Solar corona: heating problem T steam /T ice ~ 1.4 T corona /T photosphere ~ 300
Solar corona: basic parameters photosphere corona Temperature ~ 6 10 3 K ~ 10 6 K Density ~ 10 23 m 3 ~10 12 m 3 Time scale ~ 10 4 s ~ 20s Magnetic fields (10~100 G) --- role in heating? Two main theories: Alfvén wave (AC) current sheets (DC)
Magnetic field in the solar corona Extreme UV image taken by the Transition Region and Coronal Explorer (TRACE) [http://trace.lmsal.com/] [Knight, 1st Ed. 2004]
Quasi equilibrium in coronal field Extreme UV movie taken by Solar TErrestrial RElations Observatory (STEREO) In 36 hours Most of the time in quasi equilibrium [http:// stereo.gsfc.nasa.gov/]
Parker's model (1972) of coronal heating
Parker's model (1972) of coronal heating
Heating by currents -- Ampère s law plasma can carry electric current magnetic field is related to electric current current density J from J = 1 B D Ampère s law: 1 B µ 0 t µ 0
Heating by currents -- Ohm s law Electric field η is resistivity) E = ηj (in the rest frame of the plasma, For T ~ 10 6 K (k B T ~ 100eV), η ~ 5x10-7 ohm-m (η ~ 7x10-7 ohm-m for stainless steel, 2x10-8 ohm-m for copper) E L = ηl A J A ( ) ΔV = IR Faraday s B law: t = E = η B µ 0 = η µ 0 2 B [Knight]
Heating by current sheets Magnetic diffusion equation: B t = η µ 0 2 B Magnetic diffusion time τ d = µ 0 L 2 /η For resistivity η ~ 5 10 7 ohm - m, L ~ 10 7 m (~ 1% R S ), τ d ~ 8 10 6 years! Requires τ d ~ 10 4 s, then L ~ 10 2 m, i.e., needs to develop small spatial scales: current sheets
Frozen-in field line condition In ideal MHD (η = 0), magnetic field lines are frozen in the fluid. B (t + dt) B (t)
Parker's Model (1972): use η = 0 Straighten a curved magnetic loop Photosphere
Tangential discontinuities as current sheets J = B /µ 0 B back B front L L Total current I = B d a /µ 0 = B d l /µ 0 0 even for. L 0
Tangential discontinuities in Parker s model Photospheric motion has time scale much longer than Alfvén time. τ ~ 10 4 s Corona in quasi-equilibrium --- most of the time. Parker s model [Astrophys. J. 1972]: No smooth force-free equilibrium exists (assume η = 0), due to complex photospheric motions. This works only when the equilibrium becomes unstable [Ng & Bhattacharjee, Phys. Plasmas 1998]
Simulations of Parker's model Start with a uniform B field. Apply random footpoint motion that twists field. Current layers appear/disrupt. Quasi-equilibrium ( B J = 0) most of the time, but becomes unstable when J getting large. Recent results from a 3D pseudo-spectral parallelized Reduced MHD (RMHD) code show that heating rate independent of η (inverse of Lundquist number, which is very large in the solar corona). B
B from random footpoint motion z = L l B L B z B w z = 0 B ~ B z l/l~ B z v p t/l
Constant footpoint motion --- exact solution B B z ~ v Lτ r L l r L = v L w 2 Lη >>1 where l r is the distance a photospheric footpoint move in a resistive time τ r ~ w 2 /η. Unphysically large H ~ W d ~ B l 2 z v r L. L η 1 A reference case for the theory and simulation. w 2
Step length: l Random walk Location after N steps: (x,y) Distance after N steps: L Average location: <x> = <y> = 0 Expected distance: <L> = N 1/2 l l N = 2, L = 2 1/2 l l 2 1/2 l l N = 3, L = 3 1/2 l, etc http://pages.physics.cornell.edu/ sethna/statmech/computerexercises/ RandomWalk/RandomWalk.html Random walk in a random velocity field v: If l = vτ c, L(in time t) = (t/τ c ) 1/2 l = v (tτ c ) 1/2, if t > τ c
B y B z B from random footpoint motion If dissipation is due to Ohmic heating with resistivity η ~ l c L ~ v 0 L τ cohτ r ~ v 0 L τ coh w 2 η >>1 where l c = v 0 τ r τ coh is the statistically expected distance moved by a footpoint with velocity v 0 in a random walk motion in a resistive time τ r ~ w 2 /η. Heating rate W d ~ η J 2 d 3 x ~ ηb 2 y (Lw 2 ) /w 2 ~ v 0 H ~ W L B 2 z τ coh w 2 d is independent of η. w 2 If w v 0 τ coh, H ~ B 2 z v 0 w /L, which is roughly of the same order of magnitude required for coronal heating. However, B y is unphysically large for a small η. 2
Simulation of heating in tectonics model The tectonics model is simulated numerically in 2D [Ng & Bhattacharjee, ApJ, 2008]. τ coh = 20 ~ 0.002τ r Average heating rate almost independent of η. Same heating rate even with instabilities or reconnection.
Random drive --- transverse B/small τ coh B y (t) 1 t t 0 B 2 y (x, t )d 2 xd t 1/ 2 τ coh = 20 ~ 0.002τ r B y has almost a η -1/2 dependence. not physical η -1/2 is still very small
3D Simulation of Parker s model Magnetic energy limited by disruptions. η = ν = 0.000625 (64x64x16) η = 0.0003125, ν = 0.000625 (256x256x32)
3D Simulation of Parker s model Average magnetic field strength saturated in time. η = ν = 0.000625 (64x64x16) η = 0.0003125, ν = 0.000625 (256x256x32)
3D Simulation of Parker s model Energy dissipation rate saturated in time. η = ν = 0.000625 (64x64x16) η = 0.0003125, ν = 0.000625 (256x256x32)
3D Simulation of Parker s model J max larger for smaller η. η = ν = 0.000625 (64x64x16) η = 0.0003125, ν = 0.000625 (256x256x32)
Formation of thin current layers. η = ν = 0.00125 (128x128x32)
Formation of thin current layers. η = 0.0003125, ν = 0.000625 (256x256x32)
Random drive in 3D RMHD Average energy dissipation rate saturated in small η. Longcope & Sudan (1994): P F ~ v F B η 1/ 3
Random drive in 3D RMHD Average magnetic field strength saturated in small η. Note that B z = 1. Longcope & Sudan (1994):
Slow Sweet-Parker reconnection From [Gurnett and Bhattacharjee, 2005] Slow reconnection rate: U in = 2V A /S 1/ 2 ( S = µ 0 V A L /η) Thin and long current sheet: Δ = 2L /S 1/ 2
Sweet-Parker reconnection: Heating rate: Scaling analysis in 3D If τ E < τ c, no random walk: If τ E > τ c, random walk: Substituting numerical parameters shows that transition at around η = 10-3
B from random footpoint motion If dissipation is due instability/reconnection when B y ~ f B z B y ~ f ~ v 0 B z L τ cohτ E τ E ~ ( fl /v 0 ) 2 /τ coh Heating rate W d ~ B 2 y (Lw 2 ) /τ E ~ v 2 0 H ~ W L B 2 z τ coh w 2 d is independent of f (and dissipation mechanism). w 2 If w v 0 τ coh, H ~ B 2 z v 0 w /L, which is roughly of the same order of magnitude required for coronal heating. Now there is no unphysically large, if instability and reconnection dissipates energy fast enough when f ~ O(1). B y
Conclusions Parker's model of coronal heating is studied using 2D and 3D RMHD simulations. Scaling laws with resistivity found in simulations can be understood by using the concept of random walk of photospheric footpoint motion. The saturation of the heating rate in the small η limit seems to be robust regardless of the dissipation mechanism and how the magnetic field production is limited. This heating rate found in simulations and theory is at the level needed for coronal heating.