Fundamentals of Magnetohydrodynamics (MHD)€¦ · Fundamentals of Magnetohydrodynamics (MHD) Alan...
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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
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Motivation
Solar Corona in EUV
• Want to understandphysical processes inplasmas (ionisedconducting fluids)
• Applications:Magnetospheres, Sunand stars, accretiondisks, jets etc,laboratory plasmas(e.g. fusionexperiments)
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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)
• . . .
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"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
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"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
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"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
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"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
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"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
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"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
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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
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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)
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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
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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.
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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.
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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
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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)
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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
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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!
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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)
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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Magnetic Null Points
• Points in space where B = 0
• Important for defining the connectivity andtopology of magnetic field configurations
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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
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Harris Sheet
• B = B0 tanh(z/L)x
• p(z) = p0/ cosh2(z/L) + pb
• B20/(2µ0) = p0
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Field Lines
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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
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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
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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
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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!
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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} .
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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} .
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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
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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))
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3D MHS
• Representation of B to guarantee ∇ ·B = 0 muchmore difficult
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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).
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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?
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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
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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?
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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
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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
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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 !)
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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
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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
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