1 (Revision 2) Solar MHD Theory MT4510 Prof Alan Hood Dr Clare Parnell 2 Contents 0 Review of Vector Calculus. 5 0.1 OperatorsinVariousCoordinateSystems. ........ 6 0.2 Flux........................................ 8 0.3 VectorIdentities................................ .. 8 0.4 IntegralTheorems................................ 9 1 Maxwell’s Equations and Magnetic Fields 11 1.1 Maxwell’sEquations .............................. 11 1.2 ElectromagneticWavesinaVacuum . .... 12 1.3 MagneticFieldLines .............................. 13 2 MHD Equations 17 2.1 ElectromagneticEquations . .... 17 2.1.1 Maxwell’sEquations . .. .. .. .. .. .. .. 17 2.1.2 Ohm’sLaw................................ 18 2.2 FluidEquations .................................. 18 2.2.1 ContinuityEquation ........................... 19 2.2.2 TheEquationofMotion . .. .. .. .. .. .. 19 2.2.3 TheEnergyEquation........................... 20 2.2.4 Summary of the MHD Equations: Important to know these . ...... 20 2.2.5 GeneralRemarks ............................. 21 3 Magnetic Induction and Magnetic Energy 23 3.1 TheInductionEquation. .. 23 3.2 InductionEquation-TheDiffusionLimit . ....... 25 3.2.1 Diffusioninacurrentsheet. .. 25 3.3 InductionEquation-Frozen-in-FluxTheorem . ......... 28 3.3.1 Frozen-in-Flux Theorem (Alfv´en’s Theorem) . ....... 28 3.4 SteadyState.................................... 33 3.5 MagneticEnergy ................................. 34 4 Magnetic Forces 37 4.1 TheLorentzForce................................. 37 4.1.1 MagneticTensionForce . 38 4.1.2 MagneticPressureForce . 39 4.1.3 MagneticForceBalance . 40 3 4 CONTENTS 5 Magnetohydrostatic Equilibria 41 5.1 HydrostaticPressureBalance . ..... 41 5.2 ThePlasmaBeta - β ............................... 42 5.3 TheForce-FreeApproximation. .... 43 5.4 PotentialFields ................................. 45 5.4.1 UniquenessofPotentialFields . ... 49 5.5 Force-FreeFields ................................ 50 5.5.1 Property of α ............................... 50 5.5.2 Constant α ................................ 50 5.5.3 Non-constant α .............................. 50 5.6 TheVectorMagneticPotential . .... 51 5.6.1 ShearedArcades ............................. 54 5.7 ProminenceEquilibria . .. 55 5.7.1 KippenhahnandSchl¨uterProminenceModel . ..... 56 6 Magnetohydrodynamic Waves 61 6.1 LinearisedMHDEquations. .. 61 6.2 Acoustic (Sound) Waves - Basic Waves Properties . ......... 62 6.2.1 DispersionRelations . 65 6.3 Alfv´enWaves................................... 66 6.3.1 PhaseandGroupSpeed. .. .. .. .. .. .. 69 7 The Solar Wind 73 7.1 Introduction.................................... 73 7.2 Parker’sSolarWindModel . .. 73 Chapter 0 Review of Vector Calculus. Vector Calculus is the fundamental tool used in the formulation and analysis of problems in electromagnetism. It is, therefore, important to review the basics of vectors and vector calculus. The basic definitions of quantities are given below: Scalar function - e.g., p, ρ may be dependent on space and/or time - p(x,y,z,t), with only a magnitude (e.g., pressure or density) and written in light type in books. Vector function - e.g., B, v with a magnitude and direction (e.g. magnetic field, velocity) that may be dependent on space and/or time, written as B or B in books ( B = Bxˆx + By ˆy + Bzˆz). You must identify vectors by underlining them. This is important. Sum of vectors : Assuming that A =(Ax, Ay, Az)= Axˆx + Ay ˆy + Azˆz (and similarly for the vector B), we have A B = C, (C = A B , C = A B , C = A B ). ± x x ± x y y ± y z z ± z Scalar Product : A B = A B + A B + A B = A B cos θ = B A, · x x y y z z | || | · where θ is the acute angle between the two vectors. Note that this works for any orthogonal ˆ ˆ coordinate system. Thus, in cylindrical coordinates, if A =(AR, Aφ, Az)= ARR+Aφφ+Azˆz (and similarly for the vector B), then A B = A B + A B + A B . · R R φ φ z z Vector Product : A B = A B sin θ ˆn = B A where θ is the acute angle between the two vectors. ˆn unit× vector| pointing|| | perpendicular− × to the plane containing A and B. The standard way to calculate A B is to use the determinant method. Note this approach works in all orthogonal coordinate systems.× ˆx ˆy ˆz A B = A A A =(A B A B ) ˆx +(A B A B ) ˆy +(A B A B ) ˆz × x y z y z − z y z x − z x x y − y x Bx By Bz 5 6 CHAPTER 0. REVIEW OF VECTOR CALCULUS. 0.1 Operators in Various Coordinate Systems. Three coordinate systems will be used throughout this course. 1. Cartesian Coordinates (x,y,z). (a) Increment and Volume Element. dl = dxˆx + dyˆy + dzˆz, (0.1) dV = dxdydz. (0.2) (b) Grad. (∇ acts on a scalar function to produce a vector.) This operator gives the rate of change of the scalar function it acts on. The direction is the direction of the maximum increase of the function. Its magnitude gives the slope or rate of increase along this direction. ∂f ∂f ∂f ∇f = ˆx + ˆy + ˆz. (0.3) ∂x ∂y ∂z (c) Div. (∇ acts on a vector function to produce a scalar.) · The divergence of a vector function A = (Ax(x, y, z), Ay(x, y, z), Az(x, y, z)) at a point is a measure of how much the function spreads out or diverges at that point. ∂A ∂A ∂A ∇ A = x + y + z . (0.4) · ∂x ∂y ∂z In1Dsothat A =(Ax(x), 0, 0), if the function Ax(x) is increasing, then ∂Ax(x)/∂x > 0. (d) Curl. (∇ acts on a vector function to produce a vector.) The curl of× a function is a measure of how much the function curls, rotates or twists at the point in question. ∂A ∂A ∂A ∂A ∂A ∂A ∇ A = z y ˆx + x z ˆy + y x ˆz. (0.5) × ∂y − ∂z ∂z − ∂x ∂x − ∂y (e) Laplacian of a Scalar function, f produces a scalar. ∂2f ∂2f ∂2f 2f = ∇ ∇f = + + . (0.6) ∇ · ∂x2 ∂y2 ∂z2 (f) Laplacian of a Vector function A produces a vector. 2A = 2A ˆx + 2A ˆy + 2A ˆz = ∇(∇ A) ∇ (∇ A). (0.7) ∇ ∇ x ∇ y ∇ z · − × × 2. Cylindrical Coordinates (R,φ,z). (a) Increment and Volume Element. dl = dRRˆ + Rdφφˆ + dzˆz, (0.8) dV = RdRdφdz. (0.9) 0.1. OPERATORS IN VARIOUS COORDINATE SYSTEMS. 7 (b) Grad. ∂f 1 ∂f ∂f ∇f = Rˆ + φˆ + ˆz. (0.10) ∂R R ∂φ ∂z (c) Div. 1 ∂ 1 ∂A ∂A ∇ A = (RA )+ φ + z . (0.11) · R ∂R R R ∂φ ∂z (d) Curl. 1 ∂A ∂A ∂A ∂A ∇ A = z φ Rˆ + R z φˆ + × R ∂φ − ∂z ∂z − ∂R 1 ∂ 1 ∂A (RA ) R ˆz. (0.12) R ∂R φ − R ∂φ (e) Laplacian of a Scalar. 1 ∂ ∂f 1 ∂2f ∂2f 2f = R + + . (0.13) ∇ R ∂R ∂R R2 ∂φ2 ∂z2 3. Spherical Coordinates (r,θ,φ) (a) Increment and Volume Element. dl = drˆr + rdθθˆ + r sin θdφφˆ, (0.14) dV = r2 sin θdrdθdφ. (0.15) (b) Grad. ∂f 1 ∂f 1 ∂f ∇f = ˆr + θˆ + φˆ. (0.16) ∂r r ∂θ r sin θ ∂φ (c) Div 1 ∂ 1 ∂ 1 ∂A A = (r2A )+ (A sin θ)+ φ . (0.17) ∇· r2 ∂r r r sin θ ∂θ θ r sin θ ∂φ (d) Curl 1 ∂ ∂A A = (A sin θ) θ ˆr + ∇ × r sin θ ∂θ φ − ∂φ 1 1 ∂A ∂ 1 ∂ ∂A r (rA ) θˆ + (rA ) r φˆ. (0.18) r sin θ ∂φ − ∂r φ r ∂r θ − ∂θ (e) Laplacian of a Scalar. 1 ∂ ∂f 1 ∂ ∂f 1 ∂2f 2f = r2 + sin θ + . (0.19) ∇ r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2 8 CHAPTER 0. REVIEW OF VECTOR CALCULUS. 0.2 Flux The flux F of a vector quantity B through a surface S is given by F = B dS (0.20) · ZS where dS = dSˆn is a vector normal to the surface S, ( B dS = BndS where Bn is the normal component at the surface). By convention dS points outwards· from the surface when the surface is closed. 0.3 Vector Identities You are expected to know and memorize the following vector identities 1. (A B) C = A (B C)= B (C A)=(B C) A = C (A B)=(C A) B × · · × · × × · · × × · 2. A (B C)=(C B) A =(A C)B (A B)C × × × × · − · 3. ∇(fg)= f∇g + g∇f 4. ∇ ∇f = 2f · ∇ 5. ∇ (∇f)=0 × 6. ∇ (∇ A)=0 · × 7. 2A = ∇(∇ A) ∇ ∇ A ∇ · − × × The following are given as a useful reference and will be used in subsequent examples. You would not be expected to memorize them. 1. ∇(a/b)=(1/b)∇a (a/b2)∇b − 2. ∇ (fA)=(∇f) A + f(∇ A) × × × 3. ∇ (fA)=(∇f) A + f(∇ A) · · · 4. ∇ (A B)=(B ∇)A (A ∇)B +(∇ B)A (∇ A)B × × · − · · − · 5. ∇ (A B)= B (∇ A) A (∇ B) · × · × − · × 6. A (∇ B)=(∇B) A (A ∇)B × × · − · 7. ∇(A B)=(B ∇)A +(A ∇)B + B (∇ A)+ A (∇ B) · · · × × × × Example 0.3.1 Simplify k (v B). Using the above vector× identity× (one of the ones you should memorize), we have k (v B)=(k B) v (k v) B. × × · − · This can be used repeatedly to simplify more complicated expressions such as k [k (v B)] B. { × × × } × 0.4. INTEGRAL THEOREMS 9 0.4 Integral Theorems The two theorems that are commonly used are : 1. Divergence Theorem (∇ B)dV = B dS, (0.21) · · Z Z ZV Z ZS where dS = ˆndS and ˆn is the outward pointing normal to the surface. The integral on the right hand side involves the normal component of B integrated over some closed surface (S) and is therefore the flux of B through that surface. The left hand side calculates how much the entire function “diverges” or spreads out within the volume (V) bounded by the closed surface (S). The spreading out of the function within the volume is equal to the flux through the surface. 2. Stokes Theorem (∇ A) dS = A dl (0.22) × · · Z ZS IC The left hand side calculates the flux of the curl of A passing through a surface (S). This just equals the line integral of function around the boundary line (C) of the surface (S).