MAGNETOHYDRODYNAMICS - 1 (Sheffield, Sept 2003) Eric Priest St Andrews CONTENTS - Lecture 1 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. Conclusion 1. INTRODUCTION Why Study the Sun ?

ƒ Central to Solar System source energy, wind ƒ Touchstone for astronomy Even more true today ! ƒ Fundamental cosmic processes particle acceln, turbulence, instabilities, winds, coronae Magnetic fields crucial ƒ Revolution in understanding Yohkoh, Ulysses, SOHO, TRACE, RHESSI Major progress - basic qns unanswered Role of Theory ? ƒ Not -- reproduce images

ƒ Nor explain every observation

ƒ *Understand Basic Processes -- step-by-step -- simple -> sophisticated model

ƒ *Listen to Observers -- clues The Sun Amazingly rich variety of MHD phenomena The Corona (from Yohkoh)

A Magnetic World Effects E.g., A Sunspot

B creates intricate structure E.g., A Prominence

B --> Thermal Blanket + Stability E.g., A Coronal Mass Ejection

B exerts a Force --> Instability E.g., A Solar Flare (from TRACE)

B stores Energy - converted to other forms ƒ MHD - the study of the interaction between a magnetic field and a , treated as a continuous medium

ƒ The assumption of a continuous medium is valid for length-scales

2 -1 Ê T ˆ Ê n ˆ L >> 300Á 6 ˜ Á 17 3 ˜ km Ë10K ¯ Ë 10 m- ¯ 420 ƒ Chromosphere (,Tn==10 10 ) Lcm >> 3 ƒ Corona 616 ()Tn==10, 10 Lkm>> 30 2. FLUX TUBES Magnetic Field Line -- Curve w. tangent in direction of B.

Equation:

dy By dx dy dz In 2D: = , or in 3D: == dx Bx BxyzB B Magnetic -- Surface generated by set of field lines intersecting simple closed curve.

(i) Strength (F) -- magnetic flux crossing a section i.e., F = Ú BdS. (ii) But —=.B 0 --> No flux is created/destroyed inside flux tube So F = Ú BdS. is constant along tube (iii) If cross-section is small,

FBAª

B lines closer --> A smaller + B increases Example

Sketch the field lines for ByBxxy==,

dy By (i) Eqn. of field lines: = dx Bx Æ y2 - x2 = constant

(ii) Sketch a few field lines:

? arrows, spacing (iii) Directions of arrows:

(,ByBxxy== ) (iv) Spacing (,ByBxxy== )

At origin B = 0. A "neutral" or "null" point Magnetic reconnection & energy conversion *EXAMPLES 1 & 2

Sketch field lines for:

Ex 1. By = x ˆ

Ex 2. Bx = ˆˆ + x y

break SOLUTIONS - Ex. 1

B = x yˆ

--> x = const Ex. 2 B = xˆˆ + x y

Æ

1 2 yx=+2 const 3. FUNDAMENTAL EQUATIONS of MHD ƒ Interaction of B and Plasma

ƒ Unification of Eqns of: (i) Maxwell —¥B/m = j + D/t , —.B = 0, ∂∂ —¥E= - B/,t ∂∂ .D = , — c r where B = H, D = E, E = j / . me s (ii) Fluid Mechanics dv Motion p, r dt =-— d Continuity r .v=0, dt +— r Perfect gas p= R T, Energy eqn...... r where d/=/+ dt t v.— ∂∂ In MHD ƒ 1. Assume v << c --> Neglect ∂D/∂t —¥B/m = j -()1 ƒ 2. Extra E on plasma moving E+vB=j/¥-s ()2 ƒ 3. Add magnetic force dv p jB r dt =-—+ ¥ ƒ Eliminate E and j: take curl (2), use (1) for j 4. INDUCTION EQUATION ∂ B =-—¥E=—¥( vB ¥ - j/) ∂t =—¥¥()vB - —¥—¥ ( B) 2 =—¥¥()vB + — B, s h 1 h where= is magnetic diffusivity h ms Induction Equation

∂ B 2 =—¥¥()vB + — B ∂t N.B.: h (i) --> B once v is known

(ii) In MHD, v and B are primary variables: induction eqn + eqn of motion --> basic physics

(iii) j=—¥ B/msand E= -vB+j/ ¥ are secondary variables Induction Equation

∂ B 2 =—¥¥()vB + — B ∂t AB (iv) B changes due to transport + diffusionh (v) -- magnetic A Lv00 ==Rm Reynold number B h

2 5 3 8 e.g., h = 1 m /, s L0 = 10 m, v0 = 10 m/s --> Rm = 10 (vi) A >> B in most of Universe --> B frozen to plasma -- keeps its energy Except SINGULARITIES -- j & B large Form at NULL POINTS, B = 0 (a) If Rm << 1

ƒ The induction equation reduces to ∂ B 2 =—B ∂t h ƒ B is governed by a diffusion equation

--> field variations on a scale L0 2 L0 diffuse away on time td = h h with speed vLtdd= 0 / = L0 2 6 12 ƒ E.g.: sunspot ( h = 1 m /s, L0 = 10 m), td = 10 sec; 8 17 for whole Sun (L0 = 7x10 m), td = 5x10 sec (b) If Rm >> 1 The induction equation reduces to ∂ B =—¥¥()vB ∂t and Ohm's law -->

E+vB=0¥ Magnetic field is "frozen to the plasma" Magnetic Flux Conservation:

Magnetic Field Line Conservation: *EXAMPLE 3. Diffusion of a 1D Field (*hard) Suppose B = B(x,t) y,ˆ where B(x,t) satisfies 2 ∂ B ∂ B = 2 ∂t h∂x Find B(x,t) if

+ B0 for x>0 Hint: try B(x,0) = { B=B(x/t1/2) - B0 for x<0 break *EXAMPLE 4. Advection of a 1D Field (*hard) Consider the effect of a given flow

vx = - Ux/a, vy = Uy/a on a magnetic field By = B(x,t) ˆ when Rm >> 1: (i) Show that B(x,t) satisfies ∂ B Ux∂ B UB - = ∂ t a ∂ x a (ii) If B(x,0) = cos (x/a), solve to find B(x,t) break