fundamentals of magnetohydrodynamics (mhd)€¦ · fundamentals of magnetohydrodynamics (mhd) alan...

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)


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)

• . . .


"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


"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


"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


"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


"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


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


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)


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


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.


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.


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


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)


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


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!


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)


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


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.


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


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


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


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


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.


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


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


Magnetic Null Points

• Points in space where B = 0

• Important for defining the connectivity andtopology of magnetic field configurations


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


Harris Sheet

• B = B0 tanh(z/L)x

• p(z) = p0/ cosh2(z/L) + pb

• B20/(2µ0) = p0


Field Lines


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


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


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


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!



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} .



− (∇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



−

(∂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))


3D MHS

• Representation of B to guarantee ∇ ·B = 0 muchmore difficult


3D MHS


• Euler Potentials (Clebsch representation):

B = ∇α×∇β

intrinsically nonlinearexistence of global α and β not guaranteed(could use four potentials instead).


3D MHS



B = ∇α×∇β


• Vector potential: B = ∇×A

Which gauge for A? Boundary conditions for A?


3D MHS



B = ∇α×∇β


• Vector potential: B = ∇×A

Which gauge for A? Boundary conditions for A?

• Use B directly, ensure B solenoidal by numericalmeans


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?


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


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


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 !)


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


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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