# 2. MHD Equations - University of St Andrews

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 MOST of UNIVERSE is PLASMA: Ionosphere --> Sun (8 light mins)

Learn basic behaviour of plasma from Sun Magnetic Field Effects E.g., A Sunspot B exerts a force: -- creates intricate structure * ____________________* E.g., A Prominence Magnetic tube w. cool plasma B --> Thermal Blanket + Stability

What is global equilibrium? / fine structure ? *_______* E.g., a Coronal Mass Ejection QuickTime and a decompressor are needed to see this picture. *_________ _____ * E.g., A Solar Flare (from TRACE)

B stores energy converted to other forms QuickTime and a decompressor are needed to see this picture. _ _ _ _ _ _ _ _______ _ _ _ _ _ _ _* 2.2 Flux Tubes & Field Lines Magnetic Field Line -- Curve with tangent in direction of B. Equation: dx dy dz

In 2D: * _ _ _ _ _ _ * or in 3D: B = B = B x y z Magnetic Flux Tube -- Surface generated by set of field lines intersecting simple closed curve. Strength (F) -- magnetic flux crossing a section i.e., * _ _ _ _ _ _ _ * But .B = 0 --> No flux is created/destroyed inside flux tube

So F = B.dS is constant along tube Ex 2.1 Prove the above result that, if .B = 0 , then F = B.dS is constant along a flux tube. F = B.dS 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

To sketch magnetic field lines: (i) Solve dy By = dx Bx (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

EXAMPLE Sketch the field lines for Bx =y, By =x (i) Eqn. of field lines: *__________* (ii) Sketch a few field lines: ? arrows, spacing dy By = dx Bx (iii) Directions of arrows: (Bx =y, By =x)

(iv) Spacing (Bx =y, By =x) At origin B = 0.* _ _ _ _ _ _ _ _ _ _ _ _ _ _ * Magnetic reconnection & energy conversion **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 2.3 Plasma Theory

-- the study of the interaction between a magnetic field and a plasma, treated as a continuous medium/set of pcles 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 Eqns of Magnetohydrodynamics Magnetohydrodynamics (MHD) Unification of Eqns of: (a) Maxwell

B / = j + D / t, .B = 0, E = B / t, .D = c , where B = H, D = E, E = j / . (b) Fluid Mechanics dv Motion = p, dt d Continuity + .v = 0, dt Perfect gas p = R T,

Energy eqn. ............. where d / dt = / t + v. or (D / Dt) In MHD 1. Assume v << c --> Neglect * _ _ _ * B/ = j 2. Extra E on plasma moving E +

(1) * _ _ _ _* = j/ (2) 3. Add magnetic force dv dt = p + * _ _ _ _ * Eliminate E and j: take curl (2), use (1) for j

2.4 Induction Equation B = E = (v B j / ) t = (v B) ( B) 2 = (v B) + B, where _ _ _ _ _ _ is magnetic diffusivity Describes: how B moves with plasma / diffuses through it N.B. In MHD, v and B are * _ _ _ _ _ _ _ _ _ _*: Induction eqn + eqn of motion B

t dv dt = (v B) = p + 2 + B jB

--> basic processes j = B / and E = v B + j/ are secondary variables INDUCTION EQUATION B t = (v B) 2

+ B I II B changes due to transport + diffusion I II L0 v0 = = R -- * _ _ _ _ _ _ _ ______*

2 = 1 m /s, L0 = 105 m, v0 = 103 m/s --> Rm = 108 eg, I >> II in most of Solar System --> B frozen to plasma -- keeps its energy Except Reconnection -- j & B large (a) If Rm << 1 The induction equation reduces to

B 2 t = B B is governed by a diffusion equation --> field variations on a scale L0 diffuse away on time * _ _ _ _ _* with speed v d = L0 /t d = L0 E.g.: sunspot ( = 1 m2/s, L0 = 106 m), td = 1012 sec;

8 17 for whole Sun (L = 7x10 m), t = 5x10 sec 0 d (b) If Rm >> 1 The induction equation reduces to

B t = (v B) and Ohm's law --> E + vB = 0 Magnetic field is * _ _ _ _ _ _ _ _ _ _ _ _ _* Magnetic Flux Conservation: Magnetic Field Line Conservation: 2.5 EQUATION of MOTION dv

= p + dt (1) (2) (2) (i) = (3) j B + g (3) (4) p = 2 B / (2 ) * _ _ _ _ _ _ _ *

When <<1, j B do inate B * _ _ _ _ _ _* (ii) (1) (3) vvA = Typical Values on Sun Photosphere Corona N (m-3) 1023

1015 T (K) 6000 106 B (G) 5 - 103 10 106 - 1

10-3 vA (km/s) 0.05 - 10 103 [N (m-3) = 106 N (cm-3), B (G) = 104 B (tesla) = 3.5 x 10 -21 N T/B2, vA = 2 x 109 B/N1/2] Magnetic force: jB

B = ( B) B = (B.) B 2 2 Magnetic field lines have a Tension B / ----> * _ _ _ _ _ _ _ _ _ _* Pressure B2/(2 )----> * _ _ _ _ _ _ _ _ _ _ * 2

*EXAMPLE 2 B B j B = (B.) 2 2 Bx B + By ) (j B)x = (Bx

x y x 2 **Examples Find Magnetic Pressure force, Magnetic Tension force and j x B force for Ex 2.4 (a) B = x y (b) B = y x + x y Hydrostatic Equilibrium dv

= p + dt (1) (2) j B + g (3) (4) In most of corona, (3) dominates Along B, (3) = 0, so (2) + (4) important (2) p0 / L0 = >> 1 for

(4) 0 g L0 << H = p0 0 g * _ _ _ _ _ _ _ _ _* Example Suppose g = - g z MHS Eqm. along B: dp = g,

dz wee = p / (R T ). dp p RT So = , H = . dz H g T =cont p = p0 e z / H

p = p0 e z / H On Earth H = 9 km, so on munro (1 km) p = 0.9 p0 or on Everest (9 km) p = 0.37 p0 T = 5000 K, H = * _ _ _ _ _*; T = 2 x 106 K, H = * _ _ _ _ _ * When is MHD valid ? = constant, We assumed in deriving MHD eqns -- v<

and plasma continuous Can treat plasma as a continuous medium when 2 1 n T L >> pf =300 6 17 3 k 10 K 10 Chromosphere Corona When L < pf collide with B 4

20 (T = 10 , n = 10 ) L >> 3 cm 6 16 (T =10 , n =10 ) L >> 30 k MHD can still be valid when particles L > i =vj / (eB) (ion gyoadiu) ri = 1 m(corona) MHD equations can be derived by taking integrals of a kinetic equation for particles (but tricky)

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