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

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

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

4. E.g., A Prominence B --> Thermal Blanket + Stability Magnetic tube w. cool plasma 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. or in 3D:In 2D: * _ _ _ _ _ _ * Equation:

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: (i) Solve (ii) Sketch one field line (iii) 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 (i)Eqn. of field lines: (ii) 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) B y =x (b) B x =1, B y =x Ex 2.3 Sketch the field lines for (a) B x =y, B y =a 2 x (b) B x =y, B y =-a 2 x

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 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 * _ _ _ _ _ _ _ _ _ _*: are secondary variables Induction eqn + eqn of motion --> basic processes

23. INDUCTION EQUATION B changes due to transport + diffusion III -- * _ _ _ _ _ _ _ _ _ _ _ _ _ * eg, L 0 = 10 5 m, v 0 = 10 3 m/s --> R m = 10 8 I >> II in most of Solar System --> B frozen to plasma -- keeps its energy Except Reconnection -- j & B large

24. (a) If R m << 1  The induction equation reduces to  B is governed by a diffusion equation --> field variations on a scale L 0 diffuse away on time * _ _ _ _ _ *  E.g.: sunspot ( = 1 m 2 /s, L 0 = 10 6 m), t d = 10 12 sec; for whole Sun (L 0 = 7x10 8 m), t d = 5x10 17 sec with speed

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

26. Magnetic Flux Conservation: Magnetic Field Line Conservation:

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

28. Typical Values on Sun PhotosphereCorona N (m -3 )10 23 10 15 T (K)600010 6 B (G)5 - 10 3 10 10 6 - 110 -3 v A (km/s)0.05 - 1010 3 [N (m -3 ) = 10 6 N (cm -3 ),B (G) = 10 4 B (tesla) = 3.5 x 10 -21 N T/B 2, v A = 2 x 10 9 B/N 1/2 ]

29. Magnetic force: Tension B 2 /----> * _ _ _ _ _ _ _ _ _ _* Magnetic field lines have a Pressure B 2 /(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. T = 5000 K, H = * _ _ _ _ _ *; T = 2 x 10 6 K, H = * _ _ _ _ _ * On Earth H = 9 km, so on munro (1 km) p = 0.9 p 0 or on Everest (9 km) p = 0.37 p 0

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