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2. MHD Equations

2. MHD Equations. 2.1 Introduction. Many processes caused by magnetic field (B). Sun is NOT a normal gas . Sun is in 4th state of matter ("PLASMA"). behaves differently from normal gas:. B and plasma -- coupled (intimate, subtle). B exerts force on plasma -- stores energy.

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2. MHD Equations

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  1. 2.MHD Equations 2.1 Introduction Many processes caused by magnetic field (B) Sun is NOT a normal gas Sun is in 4th state of matter ("PLASMA") behaves differently from normal gas: B and plasma -- coupled (intimate, subtle) B exerts force on plasma -- stores energy

  2. MOST of UNIVERSE is PLASMA: Ionosphere --> Sun (8 light mins) Learn basic behaviour of plasma from Sun

  3. Magnetic Field Effects E.g., A Sunspot B exerts a force: • - creates intricate structure • * _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ *

  4. E.g., A Prominence Magnetic tube w. cool plasma B --> Thermal Blanket + Stability What is global equilibrium? / fine structure ? * _ _ _ _ _ _ _ *

  5. E.g., a Coronal Mass Ejection * _ _ _ _ _ _ _ _ _ _ _ _ _ _ *

  6. E.g., A Solar Flare (from TRACE) B stores energy - converted to other forms • _ _ _ _ _ _ _ • _ _ _ _ _ _ _ • _ _ _ _ _ _ _*

  7. 2.2 Flux Tubes & Field Lines Magnetic Field Line -- Curve with tangent in direction of B. Equation: In 2D: * _ _ _ _ _ _ * or in 3D:

  8. Magnetic Flux Tube -- Surface generated by set of field lines intersecting simple closed curve. Strength (F) -- magnetic flux crossing a section i.e., * _ _ _ _ _ _ _ *

  9. But --> No flux is created/destroyed inside flux tube So is constant along tube Ex 2.1 Prove the above result that, if , then is constant along a flux tube.

  10. If cross-section is small, * _ _ _ _ _* B lines closer --> A smaller + B increases Thus, when sketching field lines, ensure they are closer when B is stronger

  11. To sketch magnetic field lines: Solve Sketch one field line Sketch other field lines, remembering that B increases as the field lines become closer (iv) Put arrows on the field lines

  12. EXAMPLE Sketch the field lines for Eqn. of field lines: * _ _ _ _ _ _ _ _ _ _ * Sketch a few field lines: ? arrows, spacing

  13. (iii) Directions of arrows:

  14. (iv) Spacing At origin B = 0.* _ _ _ _ _ _ _ _ _ _ _ _ _ _* Magnetic reconnection & energy conversion

  15. **Examples Ex 2.2 Sketch the field lines for (a) By=x (b) Bx=1, By=x Ex 2.3 Sketch the field lines for (a) Bx=y, By=a2x (b) Bx=y, By=-a2x

  16. 2.3 Plasma Theory • -- the study of the interaction between a magnetic field and a plasma, treated as a continuous medium/set of p’cles • But there are different ways of modelling a plasma: (i) MHD -- fluid eqns + Maxwell (ii) 2-fluid-- electron/ion fluid eqns + Maxwell (iii) Kinetic -- distribution function for each species of particle

  17. Eqns of Magnetohydrodynamics

  18. Magnetohydrodynamics (MHD) Unification of Eqns of: (a) Maxwell

  19. (b) Fluid Mechanics or (D / Dt)

  20. 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

  21. 2.4 Induction Equation _ _ _ _ _ _ Describes: how B moves with plasma / diffuses through it

  22. N.B. • In MHD, v and B are * _ _ _ _ _ _ _ _ _ _*: • Induction eqn • + eqn of motion • --> basic processes • are secondary variables

  23. INDUCTION EQUATION I II • B changes due to transport + diffusion • -- * _ _ _ _ _ _ _ _ _ _ _ _ _* eg, L0 = 105 m, v0 = 103 m/s --> Rm = 108 • I >> II in most of Solar System --> B frozen to plasma -- keeps its energy Except Reconnection -- j & B large

  24. (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 • E.g.: sunspot ( = 1 m2/s, L0 = 106 m), td = 1012 sec; • for whole Sun (L0 = 7x108 m), td = 5x1017 sec

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

  26. Magnetic Flux Conservation: Magnetic Field Line Conservation:

  27. 2.5EQUATION of MOTION (1) (2) (3) (4) * _ _ _ _ _ _ _ * * _ _ _ _ _ _*

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

  29. Magnetic force: TensionB2/ ----> * _ _ _ _ _ _ _ _ _ _* Magnetic field lines have a PressureB2/(2 )----> * _ _ _ _ _ _ _ _ _ _ *

  30. *EXAMPLE

  31. **Examples Find Magnetic Pressure force, Magnetic Tension force and j x B force for Ex 2.4 (a) (b)

  32. Hydrostatic Equilibrium (1) (2) (3) (4) • In most of corona, (3) dominates • Along B, (3) = 0, so (2) + (4) important * _ _ _ _ _ _ _ _ _*

  33. Example MHS Eqm. along B:

  34. On Earth H =9 km, so on munro (1 km) p = 0.9p0 or on Everest (9 km)p = 0.37 p0 T = 5000 K, H = * _ _ _ _ _*; T = 2 x 106 K, H = * _ _ _ _ _ *

  35. We assumed in deriving MHD eqns -- v<<c, = constant, and plasma continuous When is MHD valid ? • Can treat plasma as a continuous medium when Chromosphere Corona • When MHD can still be valid when particles “collide” with B ri = 1 m(corona) • MHD equations can be derived by taking integrals of a kinetic equation for particles (but tricky)

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