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II. MAGNETOHYDRODYNAMICS

II. MAGNETOHYDRODYNAMICS. (Space Climate School, Lapland, March, 2009) Eric Priest (St Andrews). CONTENTS. 1. Introduction 2. Flux Tubes 3. MHD Equations 4. Induction Equation 5. Equation of Motion 6. Solar MHD 7. 2D magnetic reconnection 8. 3D reconnection

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II. MAGNETOHYDRODYNAMICS

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  1. II. MAGNETOHYDRODYNAMICS (Space Climate School, Lapland, March, 2009) Eric Priest (St Andrews)

  2. CONTENTS 1. Introduction 2. Flux Tubes 3. MHD Equations 4. Induction Equation 5. Equation of Motion 6. Solar MHD 7. 2D magnetic reconnection 8. 3D reconnection Conclusions

  3. 1. INTRODUCTION Magnetic Field Effects: -- exerts a force (creates structure) -- provides insulation -- stores energy (released in CME or flare)

  4. Magnetohydrodynamics • MHD - the study of the interaction between a magnetic field and a plasma, treated as a continuous medium • This assumption of a continuous medium is valid for • length-scales • Chromosphere • Corona

  5. 2. FLUX TUBES Magnetic Field Line -- Curve w. tangent in direction of B. Equation: In 2D: * _ _ _ _ _ _* or in 3D:

  6. Magnetic Flux Tube Surface generated by set of field lines intersecting simple closed curve. Strength (F) -- magnetic flux crossing a section i.e., *_ _ _ _ __ _ _ _ * (ii) But ---> Fis constant along tube (iii) If cross-section is small, *_ _ _ _ _ _ _ *

  7. Eqns of Magnetohydrodynamics Model interaction of B and plasma (conts medium)

  8. 3. FUNDAMENTAL EQUATIONS of MHD • Unification of Eqns of: • (i) Maxwell

  9. (ii) Fluid Mechanics or (D / Dt)

  10. In MHD • 1. Assume v << c --> Neglect*_ _ _ * • 2. Extra E on plasma moving *_ _ _ _* • 3. Add magnetic force * _ _ _ _* • Eliminate E and j: take curl (2), use (1) for j

  11. 4. INDUCTION EQUATION * *

  12. Induction Equation N.B.: (i) --> B if v is known primary variables (ii) In MHD, v and B are * *: induction eqn + eqn of motion --> basic physics (iii) are secondary variables (iv) B changes due to transport + diffusion

  13. Induction Equation A B magnetic • (v) -- * * Reynolds number eg, L0 = 105 m, v0 = 103 m/s --> Rm = 108 (vi) A >> B in most of Universe --> B moves with plasma -- keeps its energy Except SINGULARITIES -- j & B large Form at NULL POINTS, B = 0 --> reconnection

  14. (a) If Rm << 1 • The induction equation reduces to • B is governed by a diffusion equation • --> field variations on a scale L0 • diffuse away on time * * with speed

  15. (b) If Rm >> 1 The induction equation reduces to and Ohm's law --> Magnetic field is “* *” frozen to the plasma

  16. 5. EQUATION of MOTION (1) (2) (3) (4) • In most of corona, (3) dominates • Along B, (3) = 0, so (2) + (4) important

  17. Magnetic force: TensionB2/ ----> force when lines curved Magnetic field lines have a PressureB2/(2 )----> force from high to low B2

  18. Ex Expect physically: (check mathematically)

  19. Ex Expect physically: (check mathematically)

  20. Equation of Motion (1) (2) (3) (4) * * Plasma beta * * Alfvén speed

  21. Typical Values on Sun N (m-3) = 106N (cm-3), B (G) = 104 B (tesla) = 3.5 x 10 -21N T/B2, vA = 2 x 109B/N1/2

  22. 6. In Solar MHD Waves, Instabilities, We study Equilibria, Magnetic reconnection in dynamos, magnetoconvection, sunspots, prominences, coronal loops, solar wind, coronal mass ejections, solar flares

  23. Example Shapes - caused by magnetic field (force-free) Fineness - small scale of heating process + small Structure along loops - hydrostatics/hydrodynamics (--H)

  24. 7. MAGNETIC RECONNECITON • Reconnection is a fundamental process in a plasma: • Changes the topology • Converts magnetic energy to heat/K.E • Accelerates fast particles • In Sun ---> Solar flares, CME’s / heats Corona

  25. In 2D takes place only at an X-Point -- Current very large --> ohmic heating -- Strong diffusion allows field-lines to break / change connectivity and diffuse through plasma

  26. Reconnection canoccur when X-point collapses v d Small current sheet width --> magnetic field diffuses outwards at speed * * = _ _ _

  27. If magnetic field is brought in by a flow (vx = - Ux/a vy = Uy/a) then a steady balance can be set up

  28. Sweet-Parker (1958) Simple current sheet - uniform inflow

  29. Petschek (1964) • Sheet bifurcates - Slow shocks - most of energy • Reconnection speedve-- any rate up to maximum

  30. 8. 3D RECONNECTION Many New Features (i) Structure of Null Point Simplest B = (x, y, -2z) 2 families of field lines through null point: Spine Field Line FanSurface

  31. (ii) Global Topology of Complex Fields In 2D -- Separatrix curves In 3D -- Separatrix surfaces

  32. In 2D, reconnection at X transfers flux from one 2D region to another. In 3D, reconnection at separator transfers flux from one 3D region to another. In complex fields we form the SKELETON -- set of nulls, separatrices -- from fans

  33. (iii) 3D Reconnection Can occur at a null point or in absence of null At Null -- 3 Types of Reconnection: Spine reconnection Fan reconnection Separator reconnection

  34. Numerical Expt (Linton & Priest) [3D pseudo-spectral code, 2563 modes.] Impose initial stagn-pt flow v = vA/30 Rm = 5600 Isosurfaces of B2:

  35. B-Lines for 1 Tube Colour shows locations of strong Ep stronger Ep Final twist

  36. 9. CONCLUSIONS • Reconnection fundamental process - - 2D theory well-developed - 3D new voyage of discovery: topology reconnection regimes (+ or - null) • Coronal heating • Solar flares

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