Fundamentals of Magnetohydrodynamics (MHD)

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1 Fundamentals of Magnetohydrodynamics (MHD) Thomas Neukirch School of Mathematics and Statistics University of St. Andrews STFC Advanced School U Dundee 2014 p.1/46

2 Motivation Solar Corona in EUV Want to understand physical processes in plasmas (ionised conducting fluids) Applications: Magnetospheres, Sun and stars, accretion disks, jets etc, laboratory plasmas (e.g. fusion experiments) STFC Advanced School U Dundee 2014 p.2/46

3 Phenomena MHD equilibria (e.g. current sheets, flux tubes, loops, etc) MHD waves (lecture by Ineke De Moortel) MHD shocks and discontinuities Instabilities (lecture by Gunnar Hornig) Magnetic reconnection (lecture by Gunnar Hornig) MHD turbulence Magnetic field generation (dynamo processes; lecture by Paul Bushby)... STFC Advanced School U Dundee 2014 p.3/46

4 "Derivation" of MHD in a Nutshell I Plasma at most fundamental level: N particle problem N particle equations plus Maxwell equations (N 1) dx i dt = v dv i i(t), m i dt E = q i [E(x i,t)+v i B(x i,t)] B = µ 0 N = 1 N q i δ[x x i (t)] ǫ 0 i=1 i=1 E = B t, B = 0 q i v i (t)δ[x x i (t)]+ 1 c 2 E t STFC Advanced School U Dundee 2014 p.4/46

5 "Derivation" of MHD in a Nutshell II N particle problem: untractable! Introduce N particle distribution function Γ(x 1,v 1 ;...;x N,v N ;t) Liouville equation for Γ, still too nasty Γ t + N i=1 [ v i xi Γ+ q i m i (E+v i B) vi Γ ] = 0 STFC Advanced School U Dundee 2014 p.5/46

6 "Derivation" of MHD in a Nutshell III BBGKY hierarchy: Reduce problem to one-particle problem by integrating over N 1 particle variables x i, v i (I am glossing over a lot of maths here) Leads to equation for one-particle distribution function f s (x,v,t) equation (for species s of n in total) f s t +v xf s + q s m s [E(x,t)+v B(x,t)] v f s = C[f 1,...,f n ] C s [f 1,...,f n ] = "collision term" C s = 0: Vlasov equation for collisionless plasmas STFC Advanced School U Dundee 2014 p.6/46

7 "Derivation" of MHD in a Nutshell IV Take velocity moments v k xv m y v n zf s d 3 v of equation for f s to derive multifluid equations (k,m,n integers) Examples: particle density n s = f s d 3 v average velocity u s = vf s d 3 v/n s etc Results in an infinite hierarchy of equations: n th moment equation depends on terms with (n+1) th moment STFC Advanced School U Dundee 2014 p.7/46

8 "Derivation" of MHD in a Nutshell V See first two resulting equations n s t + (n su s ) = 0 [ ] us m s n s t +(u s )u s + P q s n s [E(x,t)+u s B(x,t)] = F Need closure condition to truncate moment hierarchy Usually closure condition is some assumption regarding third or fourth order moments STFC Advanced School U Dundee 2014 p.8/46

9 "Derivation" of MHD in a Nutshell VI From now assume only two fluids: electrons and protons (Remark: m p 1836m e ) Define: charge density: ρ c = e(n p n e ) 0, so n e n p = n (quasi-neutrality) mass density: ρ = m p n p +m e n e = (m p +m e )n ( m p n ) = m pv p +m e v e m p +m e ( v p ) velocity: v = m pn p v p +m e n e v e m p n p +m e n e current density: j = e(n p v p n e v e ) = en(v p v e ) (total) pressure: p = p p +p e STFC Advanced School U Dundee 2014 p.9/46

10 Assumptions Plasma quasi-neutral (see above) Pressure scalar (see above) Typical length scales much larger than kinetic length scales, e.g. gyro radii, skin depth etc Typical time scales much slower than kinetic time scales, e.g. gyro frequencies Velocity much smaller than speed of light MHD is a theory describing large-scale and slow phenomena compared to kinetic theory STFC Advanced School U Dundee 2014 p.10/46

11 MHD Equations: Fluid Equations Mass Continuity equation ρ t + (ρv) = 0 Equation of Motion (Momentum equation) ) Ohm s law ρ ( v t +v v E+v B = R = j B p+f Also needed: Energy equation and Equation of State (will be discussed later) STFC Advanced School U Dundee 2014 p.11/46

12 MHD Equations: Maxwell s Equations Ampère s law (displacement current neglected) Faraday s law B = µ 0 j Solenoidal condition E = B t B = 0 Poisson equation for E: "solved" by quasi-neutrality assumption STFC Advanced School U Dundee 2014 p.12/46

13 Mass Conservation Integrate continuity equation over a volume V : dm dt = V ρ t dv = V (ρv)dv = S (ρv) nds Mass M inside volume V changes if there is net mass in- or outflow through the boundary S Without flow through boundaries, M in V is conserved. STFC Advanced School U Dundee 2014 p.13/46

14 Momentum Conservation Rewrite momentum equation in conservation form: T { ( }} ) { + ρvv+ (ρv) t p+ B2 2µ 0 I BB µ 0 = F Integrate momentum equation over a volume V : dp dt = V (ρv) t dv = S T nds + V FdV Total momentum P inside volume V changes due to stresses on boundary and external forces. STFC Advanced School U Dundee 2014 p.14/46

15 Ohm s Law Ohm s Law E+v B = R can be regarded as the leading order terms of the electron fluid equation of motion. R represents different forms of Ohm s law: ideal: R = 0 resistive: R = ηj (η = resistivity) more general forms could include: Hall term j B/en, (electron) pressure term, inertial terms etc STFC Advanced School U Dundee 2014 p.15/46

16 The Induction Equation The electric field can be completely eliminated from the MHD equations Combine Faraday s law and Ohm s law to obtain the induction equation Ideal form B t = E = (v B R) B t = (v B) STFC Advanced School U Dundee 2014 p.16/46

17 Resistive Induction Equation Resistive MHD: R = ηj Assume η = constant for simplicity Then B t = (v B) η µ 0 [ B] = (v B)+ η µ 0 B STFC Advanced School U Dundee 2014 p.17/46

18 Magnetic Reynolds Number Non-dimensionalise equation (B = B 0 B etc) with R m = µ 0L 0 v 0 η B t = (ṽ B)+ 1 R m B, magnetic Reynolds number Usually R m 1 for the applications we consider (order ) Non-ideal term only important if second derivatives of B large = strong current density! STFC Advanced School U Dundee 2014 p.18/46

19 Magnetic Flux and Line Conservation d dt S n BdS = S = n B t ds l V B dl [ (E+V B)] nds S so magnetic flux conserved if ideal Ohm s law applies (V = v) Line conservation (without proof): for ideal MHD plasma elements stay on the same field line! (for detailed discussion, see e.g. Schindler, 2007) STFC Advanced School U Dundee 2014 p.19/46

20 Resistive MHD: A few remarks Usually R m 1 in solar applications, i.e. solar plasma ideal Violated in localized regions of strong current density (large derivatives of B-field) Localized non-ideal regions can have global effects! Important: Current sheets, magnetic null points, separators etc STFC Advanced School U Dundee 2014 p.20/46

21 Energy Equation Can be written in different forms depending on thermodynamic variables used E.g. using the equation of state for an ideal gas and internal energy e = p/(γ 1)ρ ρ e t +ρ(v )e+(γ 1)ρe v = L where L = q + }{{} heat flux everything else {}}{{}}{ L r ηj }{{} 2 H. Ohmic heating radiative losses STFC Advanced School U Dundee 2014 p.21/46

22 Energy Equation: Another Form Using pressure p, we get for ideal MHD (η = 0, no heat flux etc) or for resistive MHD (η 0) p t +v p+γp v = 0 p t +v p+γp v = (γ 1)η j 2 Term on right hand side: Ohmic heating STFC Advanced School U Dundee 2014 p.22/46

23 Energy Conservation Energy equations presented above are not in conservative form! Have to use momentum equation, multiply by v and combine with energy equation to get ) t ( 1 2 ρv2 +ρe+ B2 2µ 0 [ ρv 2 + for ideal and resistive MHD! 2 v+(ρe+p)v+ 1 µ 0 E B ] = 0 More terms necessary if e.g. external forces are present in the momentum equation STFC Advanced School U Dundee 2014 p.23/46

24 Magnetic Helicity Vector potential A: Magnetic Helicity: H = B = A V A BdV H is a measure of how much a magnetic field are interlinked, twisted etc. Remark: H is only one of infinitely many "invariants" of ideal MHD STFC Advanced School U Dundee 2014 p.24/46

25 Gauge Invariance H is not gauge invariant in general: Let A = A+ ψ (same B obviously) H = H + B ψdv = H + ψb ds V The surface integral only vanishes if B n = 0, i.e no field lines cross boundary In many practical situations gauge invariant forms of magnetic helicity have to be used, e.g. H rel = V (A+A 0 ) (B B 0 )dv S STFC Advanced School U Dundee 2014 p.25/46

26 Magnetic Helicity Conservation I In general one finds that (without proof): dh dt = 2 V E BdV (see e.g. Biskamp, 1993, or Biskamp, 2000) H is conserved in ideal MHD, i.e. dh dt = 0, because E = v B. STFC Advanced School U Dundee 2014 p.26/46

27 Magnetic Helicity Conservation II Even in non-ideal cases the integral on right hand side is small, so magnetic helicity is at least approximately conserved "Small" here means that other quantities (e.g. magnetic energy) change much more rapidly than H (see e.g. Schindler, 2007, for a detailed calculation). A general remark: Helicity conservation means the value of the total helicity in a volume does not change! However, within the volume helicity density (A B or equivalent) will generally be redistributed! Analogy: Conservation of total mass, but mass density changes in space and time STFC Advanced School U Dundee 2014 p.27/46

28 Magnetic pressure and tension Important for MHD equilibria, waves etc Lorentz force j B = 1 µ 0 ( B) B = 1 (B )B µ } 0 {{} magnetic tension ( B 2 2µ 0 ) }{{} magnetic pressure Plasma beta: ratio of plasma pressure and magnetic pressure: β p = 2µ 0p B 2 STFC Advanced School U Dundee 2014 p.28/46.

29 Magnetic Null Points Points in space where B = 0 Important for defining the connectivity and topology of magnetic field configurations STFC Advanced School U Dundee 2014 p.29/46

30 Current Sheets Current sheets: can be singular MHD structures (discontinuities) or finite (e.g. neutral sheets) Here: non-singular current sheets in 1D (justified by ratio of length scales) Equilibrium structures: Total pressure across sheet is constant B 2 (z) 2µ 0 +p(z) = p T = constant Often used: Harris Sheet (E. Harris, 1962) Originally a kinetic equilibrium, but is also an MHD equilibrium STFC Advanced School U Dundee 2014 p.30/46

31 Harris Sheet B = B 0 tanh(z/l)ˆx p(z) = p 0 /cosh 2 (z/l)+p b B0 2/(2µ 0) = p 0 STFC Advanced School U Dundee 2014 p.31/46

32 Field Lines STFC Advanced School U Dundee 2014 p.32/46

33 Flux tubes Simplest case: 1D equilibria in cylindrical geometry (use r, φ, z as cylindrical coordinates) Can be used as models for coronal loops, also for magnetic structures in solar interior Equilibrium (B = (0, B φ (r), B z (r))): d dr ( ) B 2 φ (r)+bz(r) 2 +p(r) 2µ 0 + B2 φ µ 0 r = 0 STFC Advanced School U Dundee 2014 p.33/46

34 Flux tubes: Examples Bennett pinch (Bennett 1934) only B φ (r) and p(r): B φ (r) = µ 0I 0 2π r r 2 +a 2, p(r) = µ 0I 2 0 8π 2 a 2 (r 2 +a 2 ) 2 Gold-Hoyle tube (Gold and Hoyle, 1960) 1D force-fee flux tube with B φ (r) and B z (r) non-zero B φ (r) = B 0ar r 2 +a 2, B z(r) = B 0a 2 r 2 +a 2 STFC Advanced School U Dundee 2014 p.34/46

35 MHD equilibria: Symmetric Systems Translational, rotational or helical symmetry: MHD can be reduced to a single nonlinear elliptic second-order PDE Here just a quick reminder how to do that for translational invariance without external forces j B p = 0 For more details (also on the other cases and with external forces) : see lecture notes on my web page. a a thomas/teaching/mhdlect.pdf STFC Advanced School U Dundee 2014 p.35/46

36 Translational Invariance 1 Assume y = 0 = Invariance in y-direction Satisfy B = 0 by B = A e y +B y e y Then B A = ( A e y ) A+B }{{} y e y A }{{} =0 =0 since A =0 y = 0. A is constant along magnetic field lines! STFC Advanced School U Dundee 2014 p.36/46

37 Translational Invariance 2 Take B (j B p) = B p = ( A e y ) p = 0 p is constant along field lines = can take p = f(a) So Also p = df da A j B = 1 µ 0 { A A [( B y e y ) A]e y B y B y }. STFC Advanced School U Dundee 2014 p.37/46

38 Translational Invariance 2 Take B (j B p) = B p = ( A e y ) p = 0 p is constant along field lines = can take p = f(a) So Also p = df da A j B = 1 µ 0 { A A [( B y e y ) A]e y B y B y }. STFC Advanced School U Dundee 2014 p.37/46

39 Translational Invariance 3 ( B y e y ) A = ( A e y ) B y = 0 B y is constant along field lines = can take B y = g(a) So and j B p = 1 µ 0 ( B y = dg da A A µ 0 df da g(a)dg da ) A = 0 STFC Advanced School U Dundee 2014 p.38/46

40 Translational Invariance 4 ( 2 x z 2 ) A = µ 0 d da ( p(a)+ B2 y 2µ 0 ) = F(A) Grad-Shafranov(-Schlüter) equation for translational invariance Single nonlinear 2nd order elliptic partial differential equation: boundary conditions for A needed (e.g. Dirichlet or von Neumann) Some analytical solutions known (for special choices of F(A)) STFC Advanced School U Dundee 2014 p.39/46

41 3D MHS Representation of B to guarantee B = 0 much more difficult STFC Advanced School U Dundee 2014 p.40/46

42 3D MHS Representation of B to guarantee B = 0 much more difficult Euler Potentials (Clebsch representation): B = α β intrinsically nonlinear existence of global α and β not guaranteed (could use four potentials instead). STFC Advanced School U Dundee 2014 p.40/46

43 3D MHS Representation of B to guarantee B = 0 much more difficult Euler Potentials (Clebsch representation): B = α β intrinsically nonlinear existence of global α and β not guaranteed (could use four potentials instead). Vector potential: B = A Which gauge for A? Boundary conditions for A? STFC Advanced School U Dundee 2014 p.40/46

44 3D MHS Representation of B to guarantee B = 0 much more difficult Euler Potentials (Clebsch representation): B = α β intrinsically nonlinear existence of global α and β not guaranteed (could use four potentials instead). Vector potential: B = A Which gauge for A? Boundary conditions for A? Use B directly, ensure B solenoidal by numerical means STFC Advanced School U Dundee 2014 p.40/46

45 Euler Potential Equations β ( α β) = µ 0 p α α ( β α) = µ 0 p β Further difficulty: these equations are of mixed type! What are the appropriate boundary conditions for solving them? STFC Advanced School U Dundee 2014 p.41/46

46 Force-free Fields 1 For the rest of this lecture I shall focus on force-free fields, because they are most relevant for the solar corona, e.g. for extrapolation of the coronal magnetic field from photospheric measurements For the corona the plasma beta β p = 2µ 0 p/b 2 1 is usually much smaller than unity, so j B = 0 Current density field-aligned/parallel to B everywhere, i.e. µ 0 j = α(r)b STFC Advanced School U Dundee 2014 p.42/46

47 Force-free Fields 2 Since j = 0 and B = 0 we get B α = 0 i.e. α is constant along magnetic field lines. Basic equations to solve: B = α(r)b B α = 0 B = 0 STFC Advanced School U Dundee 2014 p.43/46

48 Force-free Fields 3 Potential fields : j = 0, α = 0 Linear force-free fields: j = αb, α = constant 0 Nonlinear force-free fields j = α(r)b, B α = 0 All three classes are used for extrapolation of coronal magnetic fields, but the last one is the most important class (but also most difficult to calculate!) STFC Advanced School U Dundee 2014 p.44/46

49 Further Reading Biskamp, Nonlinear Magnetohydrodynamics, Cambridge UP, 1993 Biskamp, Magnetic Reconnection in Plasmas, Cambridge UP, 2000 Boyd and Sanderson, The Physics of Plasmas, Cambridge UP, 2003 Freidberg, Ideal Magnetohydrodynamics, Plenum Press, 1987 Goedbloed and Poedts, Principles of Magnetohydrodynamics, Cambridge UP, 2004 STFC Advanced School U Dundee 2014 p.45/46

50 Further Reading (continued) Goedbloed, Keppens, and Poedts, Advanced Magnetohydrodynamics, Cambridge UP, 2010 Priest, Magnetohydrodynamics of the Sun, Cambridge UP, 2014 Schindler, Physics of Space Plasma Activity, Cambridge UP, 2007 STFC Advanced School U Dundee 2014 p.46/46

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