MAGNETOHYDRODYNAMICS - 2 (Sheffield, Sept 2003) Eric Priest. St Andrews

Transcription

1 MAGNETOHYDRODYNAMICS - 2 (Sheffield, Sept 2003) Eric Priest St Andrews

2 CONTENTS - Lecture 2 1. Introduction 2. Flux Tubes *Examples 3. Fundamental Equations 4. Induction Equation *Examples 5. Equation of Motion *Examples 6. Equilibria 7. Waves 8. Reconnection 9. Coronal Heating 10. Conclusions

3 *EXAMPLE 3. Diffusion of a 1D Field (*hard) Suppose satisfies B = B(x,t) y, ˆ B = t where B(x,t) 2 B h x 2 Find B(x,t) if B(x,0) = { + B 0 for x>0 - B 0 for x<0 Hint: try B=B(x/t 1/2 )

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5 SOLUTION - Ex. 3

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7 *EXAMPLE 4. Advection of a 1D Field (*hard) Consider the effect of a given flow v x = - Ux/a, v y = Uy/a on a magnetic field B = B(x,t) yˆ when R m >> 1: (i) Show that B(x,t) satisfies B t - Ux a B x = UB (ii) If B(x,0) = cos (x/a), solve to find B(x,t) a

8 SOLUTION - Ex. 4

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10 5. EQUATION of MOTION dv r dt = - p + j B + rg (1) (2) (3) (4) In most of corona, (3) dominates Along B, (3) = 0, so (2) + (4) important () 2 p0 / L0 = >> 1 for ( 4) r g L 0 0 << H = p0 r g 0 Scale Height

11 L o < H --- p const L o > H --- p falls Pressure Gradient - p - acts from high to low p isobars - is normal to isobars

12 Magnetic force: j B = ( B) B m Magnetic field lines have a Tension B 2 / B 2 = (B. ) B - Ê ˆ Ë Á m 2m m m ----> force when lines curved Pressure B 2 /(2 )----> force from high to low B 2

13 N.B. dv r dt = - p + j B + rg (1) (2) (3) (4) (i) (2) (3) = b = B /( 2m) When b <<1, j B dominates 2 p Plasma beta (ii) (1) ª (3) Æ v ª v = A B mr Alfvén speed

14 *EXAMPLES 5-7 B 2 j B = (B. ) B - Ê ˆ Ë Á m 2m Find Magnetic Pressure force, Magnetic Tension force and j x B force for : Ex 5. B = yˆ Ex 6. B = xˆ + x yˆ Ex 7. B = yx ˆ + xyˆ x break

15 SOLUTIONS - Ex. 5 B = x yˆ

16 Ex. 6 B = x ˆ + x yˆ

17 Ex. 7 B = y x ˆ + x yˆ

18 Typical Values on Sun Photosphere Chromosphere Corona N (m -3 ) T (K) B (G) b v A (km/s) N(m -3 ) = 10 6 N(cm -3 ), B(G) = 10 4 B(tesla) b = 3.5 x N T/B 2, v A = 2 x 10 9 B/N 1/2

19 6. EQUILIBRIA dv r dt = - p + j B + rg (1) (2) (3) (4) If v << v A, then (1) << (3) and so 0 = - p + j B + r g, where j = B / m,.b = 0, r = p /( R T) (Magnetohydrostatic Equilibrium)

20 0 = - p + j B + r g (2) (3) (4) If L o << H, then (4) << (2) and 0 = - p + j B (Magnetostatic Equilibrium) If L o << 2H/ b, then (4) << (3) and 0 = j B (Force-Free) j = 0 Æ B = 0,.B = 0 Æ 2 B = 0 (Potential)

21 Example Suppose g = - g zˆ MHS Eqm. along B: dp 0 = - - ds or, since ds rgcos q, cos q = dz, dp 0 = - dz - where r = p rg, /( R T). So dp dz p =- = = Æ H H R T,. T const g p = p e - z / H 0 T = 5000 K -- H = 150 km, T = 2 x 10 6 K -- H = 100 Mm

22 Force-Free Fields j B= 0, where j = B / m and. B = 0 Æ j parallel to B Æ B = a B --(*) (i). (*) Æ 0 =.( ab ) = a.b + B. a Æ B. a = 0 a is constant along each field line

23 (ii) If a = a 0 B = a B --(*) uniform, (*) Æ ( B ) = ( a B) = a B = a B Æ ( + a 2 ) B = 0 Constant- a or linear force-free fields

24 7. WAVES (Introduction) Uniform medium of pressure p, density Disturbance v = Æ w = v, p = p + p, r = r + r ( p/ r = c) v, p, r ª exp[ i( k.r - w)] kc 2 2 s 1 1 Æ w / k = Linearise eqns motion, continuity, energy Fourier analyse (i) Sound Waves (B 0 = 0) Waves propagate with speed g c s r Dispersion Relation 0

25 (ii) Magnetic Waves (p 0 = 0) Repeat, but uniform (B 0 ) - include j x B force - assume wave propagates at angle to B 0 Either Alfvén Waves w = kv A cos Incompressible - due to magnetic tension w 2 = kv 2 2 Or A Compressional Alfvén Waves Compressible - due to magnetic pressure - propagate at speed q

26 (iii) MHD Waves (p 0 and B 0 nonzero) Alfvén Wave is unaffected Compressional Alfvén Wave and Sound Wave are coupled: Slow Magnetoacoustic Wave (Slow-Mode) + Fast Magnetoacoustic Wave (Fast-Mode) Propagate slower/faster than Alfvén Wave

27 (iv) Shock Waves Nonlinear sound wave can steepen to a shock wave -- propagates at speed > c s In MHD 3 modes: (1) Slow-mode shock - propagates faster than slow-mode speed - turns B towards normal (2) Fast-mode shock - propagates faster than fast-mode speed - turns B away from normal (3) Finite-amplitude Alfvén Wave - no change in p - reverses tangential magnetic field

28 Slow-mode Alfvén Fast-mode

29 8. MAGNETIC RECONNECTION Reconnection is a fundamental process in a plasma: Changes the topology Converts magnetic energy to heat/k.e Accelerates fast particles In solar system --> dynamic processes:

30 Magnetosphere Reconnection at magnetopause & in tail [C Owen]

31 Solar Corona Magnetic field comes thro' surface --> Solar flares, CME s / heats Corona

32 Reconnection can 2D RECONNECTION Occur when X-point collapses Why?

33 X-Point Field ( B = y x ˆ + x y) ˆ j z B = y - x B x y j B = 0

34 Perturb -? Grow B = y, B = a 2 x x y 2 - a 2 x 2 = const y j z = a 2 m -1 m j B = 2 2 ( a -1)(- a x xˆ + y yˆ)

35 Reconnection In 2D takes place only at an X-Point -- Current very large -- Strong dissipation allows field-lines to break / change connectivity In 2D theory well developed : * (i) Slow Sweet-Parker Reconnection (1958) * (ii) Fast Petschek Reconnection (1964) * (iii) Many other fast regimes -- depend on b.c.'s Almost-Uniform (1986) Nonuniform (1992)

36 Sweet- Parker (1958) Simple current sheet - uniform inflow R mi Mass conservation : Lv Advection / diffusion: v Accelerate along sheet: v Ai i o i = lvo = h / l = v Lv = A, Recon. Rate M v 1 = i = h i v R A 12 / mi

37 *EXAMPLE 8 Energetics of Sweet-Parker model (i) Find ratio (B o /B i ) of outflow to inflow B [Assume E = constant] (ii) Find inflow of e.m. energy in terms of v i, B i, L [Poynting flux is ExH per unit area]

38 (iii) Show inflow of K.E. << inflow of e.m. energy (iv) Compare outflow and inflow of energy Deduce: Sweet-Parker model converts half the inflowing energy into K.E. and half into heat break

39 SOLUTION - Ex. 8 Energetics of Sweet-Parker model (l << L) We have v o =v Ai, v i /v o =l/l=r mi -1/2 <<1. (i)? B o /B i E = const = v o B o = v i B i --> B B o i vi 1 = = << 12 / v R o mi 1

40 (ii) Inflow of e.m. energy? (E x H / area) 2 i E H L = v B i L m (iii) 1 Inflow K E v r i v 2 2 i = = << 2 Inflow E. M. B / m v (iv) Outflow K. E. Inflow E. M. Inflow energy -> 1 2 i 2 1 Ai 1 v r o ( vol) v 2 2 o = 2 2 = 2 = v B L/ m v i i KE.. + Ai Heat Hot fast jets

41 Effect of stagnation-point flow *EXAMPLE 9 v x = - Ux/a v y = Uy/a on magnetic field ( B = Bx ( ) yˆ ),

42 (i) Show E ẑ is constant and Ohm's law becomes E -( Ux / a) B = h db/ dx (ii) Solve for B(x) if B(0)=0. Sketch it. (iii) Show this solution also satisfies steady equations of continuity and motion [if r = constant ] i.e., Exact solution of nonlinear MHD equations! break

43 SOLN. - Ex. 9 Stagnation-point flow v x = - Ux/a v y = Uy/a Mag. field (B y (x)) (i) E = E (x,y) zˆ E = 0 Æ E z E z = = 0 Æ E = const x y Then E + v B = B h Æ E -( Ux / a) B = h db/ dx z

44 Solve E -( Ux / a) B = h db/ dx x small Æ B ª Ex h x large Æ B ª Ea Ux l 2 = h a U Full soln. B( x) = e - x 2 / 2l 2 Ú e + u 2 / 2l 2 du x 0

45 Stagnation-point flow (v x = - Ux/a, v y = Uy/a) Mag. field (B y (x))? An exact solution of MHD eqns E + v B = h B, E = 0 Continuity.( r v) = 0 Motion r( v. ) v = - ( p + B 2 /( 2m)) or r[ - v ( v) + ( v )] = - ( p + B /( 2m)) 2 But v = 0 and r = const So ( p + B 2 /( ) m rv ) = p = p - rv -B s /( 2m)

46 Yohkoh 9. THE CORONA Bright Pts Loops Holes A magnetic world T=few MK How is the Corona Heated?

47 Recent Space Observations Low-freq. waves -- few obsns in plumes and flare-excited loops High-freq. waves --? heating outer corona Most evidence (low corona) --> reconnection elegant explanation for many diverse phenomena

48 How is Reconnection Working in Corona? (i) Drive Simple Recon. at Null by phot c. motions --> X-ray bright point Supported by TRACE (ii) Binary Reconnection -- motion of 2 sources (iii) Separator Reconnection -- complex B (iv) Braiding (v) Coronal Tectonics

49 (ii) Binary Reconnection Many magnetic sources in solar surface Relative motion of 2 sources -- "binary" interaction Suppose unbalanced and connected --> Skeleton Move sources --> "Binary" Reconnection Flux constant - - but individual B-lines reconnect

50 Cartoon Movie (Binary Recon.) Potential B Rotation of one Source about another

51 (iii) Separator Reconnection Relative motion of 2 sources in solar surface Initially unconnected Initial state of numerical expt. (Galsgaard & Parnell)

52 Comput. Expt. (Parnell / Galsgaard Magnetic field lines -- red and yellow Strong current Velocity isosurface

53 (iv) Braiding Parker s Model Initial B uniform / motions braiding

54 Numerical Experiment (Galsgaard) Current sheets grow --> turb. recon.

55 Current Fluctuations Heating localised in space -- Impulsive in time

56 (v) CORONAL TECTONICS? Effect on Coronal Heating of Magnetic Carpet * (I) Magnetic sources in surface are concentrated

57 *(II) Flux Sources Highly Dynamic Magnetogram movie (white +ve, black -ve) Sequence is repeated 4 times Flux emerges... cancels Reprocessed very quickly (14 hrs!!!)? Effect of structure/motion of carpet on Heating

58 Coronal Tectonics Model (Priest, Heyvaerts & Title) Each "Loop" --> surface in many sources Flux from each source topolog y distinct -- Separated by separatrix surfaces As sources move, coronal fields slip ("Tectonics") --> J sheets on separatrices --> Reconnect --> Heat Corona filled w. myriads of separatrix J sheets, heating impulsively

59 Fundamental Flux Units not Network Elements Intense tubes (B G, 100 km, 3 x Mx) Each network element intense tubes Single ephemeral region (XBP) sources Each TRACE Loop finer loops

60 TRACE Loop Reaches to surface in many footpoints. Separatrices form web in corona

61 Corona - Myriads Different Loops Each flux element --> many neighbours But in practice many more connections

62 10. CONCLUSIONS Understanding how B interacts with plasma: Key to many solar system phenomena Two main equations: Induction equation -- advection + diffusion Eqn. motion -- magnetic tension + pressure forces Reconnection - diff t roles in coronal heating - binary, sep r, braiding * XBP - usually driven reconnection * High corona --? high-frequency waves * Coronal loops --? turbulent reconnection

63 Need in future: * Effect magnetic carpet * Effect complex magnetic topology * Test viability of Coronal Tectonics Model Fully understand subtle link photo/corona --> Solar B and Solar Dynamics Observatory In following lectures theoretical MHD effects in: Magnetoconvection (Hughes) Waves & Instabilities (Erdelyi) Solar Wind (Habbal) Magnetosphere (Owen)

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