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Fundamentals ofMagnetohydrodynamics (MHD)
Alan Hood (thanks to Thomas Neukirch)
School of Mathematics and Statistics
University of St Andrews
STFC Advanced School, University of Leeds, 2015 – p. 1/46
Motivation
Solar Corona in EUV
• Want to understandphysical processes inplasmas (ionisedconducting fluids)
• Applications:Magnetospheres, Sunand stars, accretiondisks, jets etc,laboratory plasmas(e.g. fusionexperiments)
STFC Advanced School, University of Leeds, 2015 – p. 2/46
Phenomena
• MHD equilibria (e.g. current sheets, flux tubes,loops, etc)
• MHD waves (lecture by Valery Nakariakov)
• MHD shocks and discontinuities
• Instabilities
• Magnetic reconnection (lecture by Chris Owen)
• MHD turbulence (lecture by Sandra Chapman)
• Magnetic field generation (dynamo processes;lecture by Paul Bushby)
• . . .
STFC Advanced School, University of Leeds, 2015 – p. 3/46
"Derivation" of MHD in a Nutshell I
Plasma at most fundamental level: N particle problemN particle equations plus Maxwell equations (N ≫ 1)
dRi
dt= Ri(t), mi
d2Ri
dt2= qi[E(Ri, t) + Ri ×B(Ri, t)]
∇ · E =1
ǫ0
N∑
i=1
qiδ[R−Ri(t)]
∇×B = µ0
N∑
i=1
qiRi(t)δ[R−Ri(t)] +1
c2∂E
∂t
∇×E = −∂B
∂t, ∇ ·B = 0
STFC Advanced School, University of Leeds, 2015 – p. 4/46
"Derivation" of MHD in a Nutshell II
• N particle problem: untractable!
• Introduce N particle distribution function
Γ(x1,v1; . . . ;xN ,vN ; t) =N∏
i=1
δ(xi −Ri)δ(vi − Ri)
• Liouville equation for Γ, still too nasty
∂Γ
∂t+
N∑
i=1
[
vi · ∇xiΓ +
qimi
(E+ vi ×B) · ∇viΓ
]
= 0
STFC Advanced School, University of Leeds, 2015 – p. 5/46
"Derivation" of MHD in a Nutshell III
• BBGKY hierarchy: Reduce problem toone-particle problem by integrating over N − 1particle variables xi, vi (I am glossing over a lot ofmaths here)
• Leads to equation for one-particle distributionfunction fs(x,v, t) equation (for species s of n intotal)
∂fs∂t
+v·∇xfs+qsms
[E(x, t)+v×B(x, t)]·∇vfs = Cs[f1, . . . , fn]
• Cs[f1, . . . , fn] = "collision term"
• Cs = 0: Vlasov equation for collisionless plasmas
STFC Advanced School, University of Leeds, 2015 – p. 6/46
"Derivation" of MHD in a Nutshell IV
• Take velocity moments∫vkxv
my v
nz fs d
3v of equation
for fs to derive multifluid equations (k,m, nintegers)
• Examples:
• particle density ns =∫fsd
3v
• average velocity us =∫vfsd
3v/ns
• pressure tensor P si,j = ms
∫(vi−ui)(vj −uj)fsd
3v
• etc
• Results in an infinite hierarchy of equations: nth
moment equation depends on terms with (n+ 1)th
momentSTFC Advanced School, University of Leeds, 2015 – p. 7/46
"Derivation" of MHD in a Nutshell V
• See first two resulting equations
∂ns∂t
+∇ · (nsus) = 0
msns
[∂us
∂t+ (us · ∇)us
]
+∇ · Ps
− qsns[E(x, t) + us ×B(x, t)] = F
• Need closure condition to truncate momenthierarchy
• Usually closure condition is some assumptionregarding third or fourth order moments
STFC Advanced School, University of Leeds, 2015 – p. 8/46
"Derivation" of MHD in a Nutshell VI
• From now assume only two fluids: electrons andprotons (Remark: mp ≈ 1836me)
• Define:
• charge density: ρc = e(np − ne) ≈ 0, sone ≈ np = n (quasi-neutrality)
• mass density: ρ = mpnp +mene = (mp +me)n
(≈ mpn )
• velocity: v = mpnpup+meneue
mpnp+mene= mpup+meue
mp+me(≈ up)
• current density: j = e(npup − neue) = en(up − ue)
• (total) scalar pressure: p = pp + pe
STFC Advanced School, University of Leeds, 2015 – p. 9/46
Assumptions
• Plasma quasi-neutral (see above)
• Pressure scalar (see above)
• Typical length scales much larger than kineticlength scales, e.g. gyro radii, skin depth etc
• Typical time scales much slower than kinetic timescales, e.g. gyro frequencies
• Velocity much smaller than speed of light
MHD is a theory describing large-scale and slowphenomena compared to kinetic theory
STFC Advanced School, University of Leeds, 2015 – p. 10/46
MHD Equations: Fluid Equations
Mass Continuity equation
∂ρ
∂t+∇ · (ρv) = 0
Equation of Motion (Momentum equation)
ρ
(∂v
∂t+ v · ∇v
)
= j×B−∇p+ F
Ohm’s lawE+ v ×B = R
Also needed: Energy equation and Equation of State(will be discussed later)
STFC Advanced School, University of Leeds, 2015 – p. 11/46
MHD Equations: Maxwell’s Equations
Ampère’s law (displacement current neglected)
∇×B = µ0j
Faraday’s law
∇× E = −∂B
∂t
Solenoidal condition
∇ ·B = 0
Poisson equation for E: "solved" by quasi-neutralityassumption
STFC Advanced School, University of Leeds, 2015 – p. 12/46
Mass Conservation
• Integrate continuity equation over a volume V :
dM
dt=
∫
V
∂ρ
∂tdV = −
∫
V
∇ · (ρv)dV = −
∫
S
(ρv) · ndS
• Mass M inside volume V changes if there is netmass in- or outflow through the boundary S
• Without flow through boundaries, M in V isconserved.
STFC Advanced School, University of Leeds, 2015 – p. 13/46
Momentum Conservation
• Rewrite momentum equation in conservationform:
∂(ρv)
∂t+∇ ·
T︷ ︸︸ ︷
ρvv +
(
p+B2
2µ0
)
I −BB
µ0
= F
• Integrate momentum equation over a volume V :
dP
dt=
∫
V
∂(ρv)
∂tdV = −
∫
S
T · ndS +
∫
V
FdV
• Total momentum P inside volume V changes dueto stresses on boundary and external forces.
STFC Advanced School, University of Leeds, 2015 – p. 14/46
Ohm’s Law
• Ohm’s LawE+ v ×B = R
can be regarded as the leading order terms of theelectron 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, inertialterms etc
STFC Advanced School, University of Leeds, 2015 – p. 15/46
The Induction Equation
• The electric field can be completely eliminatedfrom the MHD equations
• Combine Faraday’s law and Ohm’s law to obtainthe induction equation
∂B
∂t= −∇×E = ∇× (v ×B−R)
• Ideal form∂B
∂t= ∇× (v ×B)
STFC Advanced School, University of Leeds, 2015 – p. 16/46
Resistive Induction Equation
• Resistive MHD: R = ηj
• Assume η = constant for simplicity
• Then
∂B
∂t= ∇× (v ×B)−
η
µ0∇× [∇×B]
= ∇× (v ×B) +η
µ0∆B
STFC Advanced School, University of Leeds, 2015 – p. 17/46
Magnetic Reynolds Number
• Non-dimensionalise equation (B = B0B etc)
∂B
∂t= ∇× (v × B) +
1
Rm∆B
with
Rm =µ0L0v0η
, magnetic Reynolds number
• Usually Rm ≫ 1 for the applications we consider(order 106 − 1012)
• Non-ideal term only important if secondderivatives of B large =⇒ strong current density!
STFC Advanced School, University of Leeds, 2015 – p. 18/46
Magnetic Flux and Line Conservation
d
dt
∫
S
n ·BdS =
∫
S
n ·∂B
∂tdS −
∮
l
V ×B · dl
= −
∫
S
[∇× (E+V ×B)] · ndS
so magnetic flux conserved ifideal Ohm’s law applies (V = v)
Line conservation (withoutproof): for ideal MHD plasmaelements stay on the same fieldline! (for detailed discussion,see e.g. Schindler, 2007)
STFC Advanced School, University of Leeds, 2015 – p. 19/46
Resistive MHD: A few remarks
• Usually Rm ≫ 1 insolar applications, i.e.solar plasma ideal
• Violated in localizedregions of strongcurrent density (largederivatives of B-field)
• Localized non-idealregions can haveglobal effects!
• Important: Currentsheets, magnetic nullpoints, separators etc
STFC Advanced School, University of Leeds, 2015 – p. 20/46
Energy Equation
• Can be written in different forms depending onthermodynamic variables used
• E.g. using the equation of state for an ideal gasand internal energy e = p/(γ − 1)ρ
ρ∂e
∂t+ ρ(v · ∇)e+ (γ − 1)ρe∇ · v = −L
where
L = ∇ · q︸︷︷︸
heat flux
+
radiative losses︷︸︸︷
Lr − ηj2︸︷︷︸
Ohmic heating
−
everything else︷︸︸︷
H.
STFC Advanced School, University of Leeds, 2015 – p. 21/46
Energy Equation: Another Form
Using pressure p, we get for ideal MHD (η = 0, no heatflux etc)
∂p
∂t+ v · ∇p+ γp∇ · v = 0
or for resistive MHD (η 6= 0)
∂p
∂t+ v · ∇p+ γp∇ · v = (γ − 1)η|j|2
Term on right hand side: Ohmic heating
STFC Advanced School, University of Leeds, 2015 – p. 22/46
Energy Conservation
• Energy equations presented above are not inconservative form!
• Have to use momentum equation, multiply by v
and combine with energy equation to get
∂
∂t
(1
2ρv2 + ρe +
B2
2µ0
)
+∇ ·
[ρv2
2v + (ρe+ p)v +
1
µ0E×B
]
= 0
for ideal and resistive MHD!
• More terms necessary if e.g. external forces arepresent in the momentum equation
STFC Advanced School, University of Leeds, 2015 – p. 23/46
Magnetic Helicity
• Vector potential A:
B = ∇×A
• Magnetic Helicity: H =∫
V
A ·B dV
• H is a measure of how much magnetic fieldsare interlinked, twisted etc.
• Remark: H is only one of infinitely many"invariants" of ideal MHD
STFC Advanced School, University of Leeds, 2015 – p. 24/46
Gauge Invariance
• H is not gauge invariant in general:Let A′ = A+∇ψ (same B obviously)
H ′ = H +
∫
V
B · ∇ψ dV = H +
∫
S
ψB · dS
• The surface integral only vanishes if Bn = 0, i.e nofield lines cross boundary
• In many practical situations gauge invariant formsof magnetic helicity have to be used, e.g.
Hrel =
∫
V
(A+A0) · (B−B0) dV
STFC Advanced School, University of Leeds, 2015 – p. 25/46
Magnetic Helicity Conservation I
• In general one finds that (without proof):
dH
dt= −2
∫
V
E ·B dV
(see e.g. Biskamp, 1993, or Biskamp, 2000)
• H is conserved in ideal MHD, i.e.
dH
dt= 0,
becauseE = −v ×B.
STFC Advanced School, University of Leeds, 2015 – p. 26/46
Magnetic Helicity Conservation II
• Even in non-ideal cases the integral on right handside is small, so magnetic helicity is at leastapproximately conserved
• "Small" here means that other quantities (e.g.magnetic energy) change much more rapidly thanH (see e.g. Schindler, 2007, for details).
• A general remark: Helicity conservation meansthe value of the total helicity in a volume does notchange!
• However, within the volume helicity density (A ·Bor equivalent) will generally be redistributed!
• Analogy: Conservation of total mass, but massdensity changes in space and time
STFC Advanced School, University of Leeds, 2015 – p. 27/46
Magnetic pressure and tension
• Important for MHD equilibria, waves etc
• Lorentz force
j×B =1
µ0(∇×B)×B
=1
µ0(B ·∇)B
︸ ︷︷ ︸
magnetic tension
− ∇
(B2
2µ0
)
︸ ︷︷ ︸
magnetic pressure
.
• Plasma beta: ratio of plasma pressure andmagnetic pressure:
βp =2µ0p
B2
STFC Advanced School, University of Leeds, 2015 – p. 28/46
Magnetic Null Points
• Points in space where B = 0
• Important for defining the connectivity andtopology of magnetic field configurations
STFC Advanced School, University of Leeds, 2015 – p. 29/46
Current Sheets
• Current sheets: can be singular MHD structures(discontinuities) or finite (e.g. neutral sheets)
• Here: non-singular current sheets in 1D (justifiedby ratio of length scales)
• Equilibrium structures: Total pressure acrosssheet is constant
B2(z)
2µ0+ p(z) = pT = constant
• Often used: Harris Sheet (E. Harris, 1962)Originally a kinetic equilibrium, but is also anMHD equilibrium
STFC Advanced School, University of Leeds, 2015 – p. 30/46
Harris Sheet
• B = B0 tanh(z/L)x
• p(z) = p0/ cosh2(z/L) + pb
• B20/(2µ0) = p0
STFC Advanced School, University of Leeds, 2015 – p. 31/46
Field Lines
STFC Advanced School, University of Leeds, 2015 – p. 32/46
Flux tubes
• Simplest case: 1D equilibria in cylindricalgeometry (use r, φ, z as cylindrical coordinates)
• Can be used as models for coronal loops, also formagnetic structures in solar interior
• Equilibrium (B = (0, Bφ(r), Bz(r))):
d
dr
(
B2φ(r) + B2
z (r)
2µ0+ p(r)
)
+B2φ
µ0r= 0
STFC Advanced School, University of Leeds, 2015 – p. 33/46
Flux tubes: Examples
• Bennett pinch (Bennett 1934) – only Bφ(r) andp(r):
Bφ(r) =µ0I02π
r
r2 + a2, p(r) =
µ0I20
8π2a2
(r2 + a2)2
• Gold-Hoyle tube (Gold and Hoyle, 1960) – 1Dforce-fee flux tube with Bφ(r) and Bz(r) non-zero
Bφ(r) =B0ar
r2 + a2, Bz(r) =
B0a2
r2 + a2
STFC Advanced School, University of Leeds, 2015 – p. 34/46
MHD equilibria: Symmetric Systems
Translational, rotational or helical symmetry:
MHD can be reduced to a single nonlinear ellipticsecond-order PDE
Here just a quick reminder how to do that fortranslational invariance without external forces
j×B−∇p = 0
For more details (also on the other cases and withexternal forces) : see lecture notes.a
ahttp://www-solar.mcs.st-and.ac.uk/ thomas/teaching/mhdlect.pdf
STFC Advanced School, University of Leeds, 2015 – p. 35/46
Translational Invariance 1
Assume ∂∂y = 0 =⇒ Invariance in y-direction
Satisfy ∇ ·B = 0 by B = ∇A× ey + Byey
Then
B · ∇A = (∇A× ey) · ∇A︸ ︷︷ ︸
=0
+By ey · ∇A︸ ︷︷ ︸
=0 since ∂A∂y
=0
= 0.
A is constant along magnetic field lines!
STFC Advanced School, University of Leeds, 2015 – p. 36/46
Translational Invariance 2
Take
B · (j×B−∇p) = B · ∇p = (∇A× ey) · ∇p = 0
p is constant along field lines =⇒ can take p = f(A)
So
∇p =df
dA∇A
Also
j×B =1
µ0{−∆A∇A− [(∇By × ey) · ∇A] ey − By∇By} .
STFC Advanced School, University of Leeds, 2015 – p. 37/46
Translational Invariance 2
Take
B · (j×B−∇p) = B · ∇p = (∇A× ey) · ∇p = 0
p is constant along field lines =⇒ can take p = f(A)
So
∇p =df
dA∇A
Also
j×B =1
µ0{−∆A∇A− [(∇By × ey) · ∇A] ey − By∇By} .
STFC Advanced School, University of Leeds, 2015 – p. 37/46
Translational Invariance 3
− (∇By × ey) · ∇A = (∇A× ey) · ∇By = 0
By is constant along field lines =⇒ can take By = g(A)
So
∇By =dg
dA∇A
and
j×B−∇p =1
µ0
(
−∆A− µ0df
dA− g(A)
dg
dA
)
∇A = 0
STFC Advanced School, University of Leeds, 2015 – p. 38/46
Translational Invariance 4
−
(∂2
∂x2+
∂2
∂z2
)
A = µ0d
dA
(
p(A) +B2y
2µ0
)
= F (A)
Grad-Shafranov(-Schlüter) equation for translationalinvariance
Single nonlinear 2nd order elliptic partial differentialequation:boundary conditions for A needed (e.g. Dirichlet orvon Neumann)
Some analytical solutions known(for special choices of F (A))
STFC Advanced School, University of Leeds, 2015 – p. 39/46
3D MHS
• Representation of B to guarantee ∇ ·B = 0 muchmore difficult
STFC Advanced School, University of Leeds, 2015 – p. 40/46
3D MHS
• Representation of B to guarantee ∇ ·B = 0 muchmore difficult
• Euler Potentials (Clebsch representation):
B = ∇α×∇β
intrinsically nonlinearexistence of global α and β not guaranteed(could use four potentials instead).
STFC Advanced School, University of Leeds, 2015 – p. 40/46
3D MHS
• Representation of B to guarantee ∇ ·B = 0 muchmore difficult
• Euler Potentials (Clebsch representation):
B = ∇α×∇β
intrinsically nonlinearexistence 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, University of Leeds, 2015 – p. 40/46
3D MHS
• Representation of B to guarantee ∇ ·B = 0 muchmore difficult
• Euler Potentials (Clebsch representation):
B = ∇α×∇β
intrinsically nonlinearexistence 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 numericalmeans
STFC Advanced School, University of Leeds, 2015 – p. 40/46
Euler Potential Equations
∇β · ∇ × (∇α×∇β) = µ0∂p
∂α
∇α · ∇ × (∇β ×∇α) = µ0∂p
∂β
Further difficulty: these equations are of mixed type!
What are the appropriate boundary conditions forsolving them?
STFC Advanced School, University of Leeds, 2015 – p. 41/46
Force-free Fields 1
For the rest of this lecture I shall focus on force-freefields, because they are most relevant for the solarcorona, e.g. for extrapolation of the coronal magneticfield from photospheric measurements
For the corona the plasma beta βp = 2µ0p/B2 ≪ 1 is
usually much smaller than unity, so
j×B = 0
Current density field-aligned/parallel to B everywhere,i.e.
µ0j = α(r)B
STFC Advanced School, University of Leeds, 2015 – p. 42/46
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, University of Leeds, 2015 – p. 43/46
Force-free Fields 3
• Potential fields : j = 0, α = 0
• Linear force-free fields: j = αB, α = constant 6= 0
• Nonlinear force-free fields j = α(r)B, B · ∇α = 0
All three classes are used for extrapolation of coronalmagnetic fields, but the last one is the most importantclass (but also most difficult to calculate !)
STFC Advanced School, University of Leeds, 2015 – p. 44/46
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, PlenumPress, 1987
• Goedbloed and Poedts, Principles of
Magnetohydrodynamics, Cambridge UP, 2004
STFC Advanced School, University of Leeds, 2015 – p. 45/46
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, University of Leeds, 2015 – p. 46/46