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Magnetic Flux Tube -- Surface Generated by Set of Field Lines Intersecting Simple Closed Curve

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Magnetic Flux Tube -- Surface Generated by Set of Field Lines Intersecting Simple Closed Curve 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 ? Central to Solar System source energy, wind Touchstone for astronomy Even more true today ! Fundamental cosmic processes particle acceln, turbulence, instabilities, winds, coronae Magnetic fields crucial Revolution in understanding Yohkoh, Ulysses, SOHO, TRACE, RHESSI Major progress - basic qns unanswered Role of Theory ? Not -- reproduce images Nor explain every observation *Understand Basic Processes -- step-by-step -- simple -> sophisticated model *Listen to Observers -- clues The Sun Amazingly rich variety of MHD phenomena The Corona (from Yohkoh) A Magnetic World Magnetic Field 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 Magnetohydrodynamics MHD - the study of the interaction between a magnetic field and a plasma, treated as a continuous medium 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 Chromosphere (,Tn==10 10 ) Lcm >> 3 Corona Tn10616, 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 Flux Tube -- 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 Interaction of B and Plasma Unification of Eqns of: (i) Maxwell —¥B/m = j + ∂∂ D/t , —.B = 0, —¥E= -∂∂ B/,t —.D = rc, where B =me H, D = E, E = j / s . (ii) Fluid Mechanics dv Motion p, r dt =-— d Continuity r .v=0, dt +—r Perfect gas p= Rr T, Energy eqn. ............. where d/=/+ dt∂∂ t v.— In MHD 1. Assume v << c --> Neglect ∂D/∂t —¥B/m = j -()1 2. Extra E on plasma moving E+vB=j/¥-s ()2 3. Add magnetic force dv p jB r dt =-—+ ¥ Eliminate E and j: take curl (2), use (1) for j 4. INDUCTION EQUATION ∂ B =-—¥E=—¥( vB ¥ - j/)s ∂ t =—¥¥()vB -h —¥—¥ ( B) 2 =—¥¥()vB +h — B, 1 whereh = is magnetic diffusivity ms Induction Equation ∂ B 2 =—¥¥()vB +h — B ∂ t N.B.: (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 +h — B ∂ t AB (iv) B changes due to transport + diffusion (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 The induction equation reduces to ∂ B 2 =—h B ∂ t 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 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 = h 2 ∂ t ∂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.
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