Solar Corona in EUV
STFC Advanced School U Warwick 2012 – p.1/45
STFC Advanced School U Warwick 2012 – p.2/45
Applications:
Magnetospheres, Sun
and stars, accretion
disks, jets etc, laboratory
plasmas (e.g. fusion
experiments)
Want to understand
physical processes in
plasmas (ionised
conducting fluids)
Motivation
School of Mathematics and Statistics
University of St. Andrews
Thomas Neukirch
Fundamentals of
Magnetohydrodynamics (MHD)
STFC Advanced School U Warwick 2012 – p.3/45
∇×E=−
∂B
,
∂t
=
∇×B
vi (t)
=
i=1
1 ∂E
c2 ∂t
STFC Advanced School U Warwick 2012 – p.4/45
qi vi (t)δ[x − xi (t)] +
∇·B=0
µ0
N
!
i=1
N
1 !
qi δ[x − xi (t)]
!0
qi [E(xi , t) + vi × B(xi , t)]
=
=
dvi
dt
dxi
dt
∇·E
mi
Plasma at most fundamental level: N particle problem
N particle equations plus Maxwell equations (N ! 1)
"Derivation" of MHD in a Nutshell I
...
Magnetic field generation (dynamo processes; lecture
by E Kersalé)
MHD turbulence
Magnetic reconnection (lecture by P Browning)
Instabilities (lecture by P Browning)
MHD shocks and discontinuities
MHD waves (lecture by V Nakariakov)
MHD equilibria (e.g. current sheets, flux tubes, loops,
etc)
Phenomena
STFC Advanced School U Warwick 2012 – p.5/45
STFC Advanced School U Warwick 2012 – p.6/45
Cs = 0: Vlasov equation for collisionless plasmas
Cs [f1 , . . . , fn ] = "collision term"
∂fs
qs
+v·∇x fs + [E(x, t)+v×B(x, t)]·∇v fs = C[f1 , . . . , fn ]
∂t
ms
Leads to equation for one-particle distribution function
fs (x, v, t) equation (for species s of n in total)
BBGKY hierarchy: Reduce problem to one-particle
problem by integrating over N − 1 particle variables xi ,
vi (I am glossing over a lot of maths here)
"Derivation" of MHD in a Nutshell III
i=1
#
N "
∂Γ !
qi
+
(E + vi × B) · ∇vi Γ = 0
vi · ∇xi Γ +
∂t
mi
Liouville equation for Γ, still too nasty
Introduce N particle distribution function
Γ(x1 , v1 ; . . . ; xN , vN ; t)
N particle problem: untractable!
"Derivation" of MHD in a Nutshell II
− qs ns [E(x, t) + us × B(x, t)] = F
#
∂us
+ (us · ∇)us + ∇ · P
∂t
STFC Advanced School U Warwick 2012 – p.8/45
Usually closure condition is some assumption
regarding third or fourth order moments
Need closure condition to truncate moment hierarchy
ms n s
"
∂ns
+ ∇ · (ns us ) = 0
∂t
See first two resulting equations
"Derivation" of MHD in a Nutshell V
STFC Advanced School U Warwick 2012 – p.7/45
Results in an infinite hierarchy of equations: nth
moment equation depends on terms with (n + 1)th
moment
Examples:
$
particle density ns = fs d3 v
$
average velocity us = vfs d3 v/ns
etc
$
Take velocity moments vxk vym vzn fs d3 v of equation for
fs to derive multifluid equations (k, m, n integers)
"Derivation" of MHD in a Nutshell IV
mp np vp +me ne ve
mp np +me ne
=
mp vp +me ve
mp +me
(≈ vp )
STFC Advanced School U Warwick 2012 – p.10/45
MHD is a theory describing large-scale and slow
phenomena compared to kinetic theory
Velocity much smaller than speed of light
Typical time scales much slower than kinetic time
scales, e.g. gyro frequencies
Typical length scales much larger than kinetic length
scales, e.g. gyro radii, skin depth etc
Pressure scalar (see above)
Plasma quasi-neutral (see above)
Assumptions
STFC Advanced School U Warwick 2012 – p.9/45
current density: j = e(np vp − ne ve ) = en(vp − ve )
(total) pressure: p = pp + pe
velocity: v =
charge density: ρc = e(np − ne ) ≈ 0, so ne ≈ np = n
(quasi-neutrality)
mass density: ρ = mp np + me ne = (mp + me )n
(≈ mp n )
Define:
From now assume only two fluids: electrons and
protons (Remark: mp ≈ 1836 me )
"Derivation" of MHD in a Nutshell VI
E+v×B=R
∂B
∂t
∇·B=0
∇×E=−
STFC Advanced School U Warwick 2012 – p.12/45
Poisson equation for E: "solved" by quasi-neutrality
assumption
Solenoidal condition
Faraday’s law
∇ × B = µ0 j
Ampère’s law (displacement current neglected)
MHD Equations: Maxwell Equations
STFC Advanced School U Warwick 2012 – p.11/45
Also needed: Energy equation and Equation of State (will
be discussed later)
Ohm’s law
Equation of Motion (Momentum equation)
&
%
∂v
+ v · ∇v = j × B − ∇p + F
ρ
∂t
∂ρ
+ ∇ · (ρv) = 0
∂t
Mass Continuity equation
MHD Equations: Fluid Equations
V
'
∂ρ
dV = −
∂t
V
'
∇ · (ρv)dV = −
S
'
(ρv) · ndS
(
B2
2µ0
)
I−
S
BB
µ0 .
V
STFC Advanced School U Warwick 2012 – p.14/45
Total momentum P inside volume V changes due to
stresses on boundary and external forces.
where T = ρvv + p +
V
Integrate momentum equation over a volume V :
'
'
'
dP
∂(ρv)
=
dV = − T · ndS + FdV
dt
∂t
Rewrite momentum equation in conservation form:
"
%
&
#
∂(ρv)
B2
BB
+ ∇ · ρvv + p +
I−
=F
∂t
2µ0
µ0
Momentum Conservation
STFC Advanced School U Warwick 2012 – p.13/45
Without flow through boundaries, M in V is conserved.
Mass M inside volume V changes if there is net mass
in- or outflow through the boundary S
dM
=
dt
Integrate continuity equation over a volume V :
Mass Conservation
E+v×B=R
Ideal form
∂B
= ∇ × (v × B)
∂t
STFC Advanced School U Warwick 2012 – p.16/45
∂B
= −∇ × E = ∇ × (v × B − R)
∂t
Combine Faraday’s law and Ohm’s law to obtain the
induction equation
The electric field can be completely eliminated from
the MHD equations
The Induction Equation
STFC Advanced School U Warwick 2012 – p.15/45
ideal: R = 0
resistive: R = ηj (η = resistivity)
more general forms could include: Hall term
j × B/en, (electron) pressure term, inertial terms
etc
R represents different forms of Ohm’s law:
can be regarded as the leading order terms of the
electron fluid equation of motion.
Ohm’s Law
Ohm’s Law
STFC Advanced School U Warwick 2012 – p.17/45
η
∂B
= ∇ × (v × B) − ∇ × [∇ × B]
∂t
µ0
η
= ∇ × (v × B) + ∆B
µ0
Rm =
µ0 L 0 v0
,
η
magnetic Reynolds number
STFC Advanced School U Warwick 2012 – p.18/45
Non-ideal term only important if second derivatives of
B large =⇒ strong current density!
Usually Rm ! 1 for the applications we consider (order
106 − 1012 )
with
1
∂ B̃
∆B̃
= ∇ × (ṽ × B̃) +
Rm
∂ t̃
Non-dimensionalise equation (B = B0 B̃ etc)
Magnetic Reynolds Number
Then
Assume η = constant for simplicity
Resistive MHD: R = ηj
Resistive Induction Equation
S
'
S
*
∂B
n · BdS =
n·
dS − V × B · dl
∂t
S
l
'
= − ∇ × (E + V × B)ndS
'
STFC Advanced School U Warwick 2012 – p.20/45
Important: Current
sheets, magnetic null
points, separators etc
Localized non-ideal
regions can have global
effects!
Violated in localized
regions of strong current
density (large derivatives
of B -field)
Usually Rm ! 1 in solar
applications, i.e. solar
plasma ideal
Resistive MHD: A few remarks
STFC Advanced School U Warwick 2012 – p.19/45
Line conservation (without proof):
for ideal MHD plasma elements
stay on the same field line!
(for detailed discussion, see e.g.
Schindler, 2007)
so magnetic flux conserved if ideal
Ohm’s law applies (V = v)
d
dt
Magnetic Flux and Line Conservation
STFC Advanced School U Warwick 2012 – p.21/45
radiative losses everything else
-.+,
-.+,
L = ∇ · q + Lr − ηj2 − H.
+ ,- .
+,-.
heat flux
Ohmic heating
∂e
+ ρ(v · ∇)e + (γ − 1)ρe∇ · v = −L
∂t
Term on right hand side: Ohmic heating
STFC Advanced School U Warwick 2012 – p.22/45
∂p
+ v · ∇p + γp∇ · v = (γ − 1)η|j|2
∂t
or for resistive MHD (η '= 0)
∂p
+ v · ∇p + γp∇ · v = 0
∂t
Using pressure p, we get for ideal MHD (η = 0, no heat flux
etc)
Energy Equation: Another Form
where
ρ
E.g. using the equation of state for an ideal gas and
internal energy e = p/(γ − 1)ρ
Can be written in different forms depending on
thermodynamic variables used
Energy Equation
STFC Advanced School U Warwick 2012 – p.24/45
Remark: H is only one of infinitely
many "invariants" of ideal MHD
H is a measure of how much a
magnetic field are interlinked,
twisted etc.
V
Magnetic Helicity:
'
H = A · B dV
B=∇×A
Vector potential A:
Magnetic Helicity
STFC Advanced School U Warwick 2012 – p.23/45
More terms necessary if e.g. external forces are
present in the momentum equation
for ideal and resistive MHD!
Have to use momentum equation, multiply by v and
combine with energy equation to get
%
&
∂ 1 2
B2
ρv + ρe +
∂t 2
2µ0
" 2
#
ρv
1
+∇ ·
v + (ρe + p)v + E × B = 0
2
µ0
Energy equations presented above are not in
conservative form!
Energy Conservation
S
STFC Advanced School U Warwick 2012 – p.25/45
because
E = −v × B.
dH
= 0,
dt
H is conserved in ideal MHD, i.e.
STFC Advanced School U Warwick 2012 – p.26/45
(see e.g. Biskamp, 1993, or Biskamp, 2000)
V
In general one finds that (without proof):
'
dH
= −2 E · B dV
dt
Magnetic Helicity Conservation I
V
In many practical situations gauge invariant forms of
magnetic helicity have to be used, e.g.
'
Hrel = (A + A0 ) · (B − B0 ) dV
The surface integral only vanishes if Bn = 0, i.e no field
lines cross boundary
V
H is not gauge invariant in general:
Let A! = A + ∇ψ (same B obviously)
'
'
H ! = H + B · ∇ψ dV = H + ψB · dS
Gauge Invariance
βp =
2µ0 p
B2
STFC Advanced School U Warwick 2012 – p.28/45
Plasma beta: ratio of plasma pressure and magnetic
pressure:
j×B =
1
(∇ × B) × B
µ0
% 2&
B
1
∇
=
(B · ∇)B −
.
µ0
2µ0
,.
+
+ ,- .
magnetic tension magnetic pressure
Lorentz force
Important for MHD equilibria, waves etc
Magnetic pressure and tension
STFC Advanced School U Warwick 2012 – p.27/45
Analogy: Conservation of total mass, but mass density
changes in space and time
However, within the volume helicity density (A · B or
equivalent) will generally be redistributed!
A general remark: Helicity conservation means the
value of the total helicity in a volume does not change!
"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).
Even in non-ideal cases the integral on right hand side
is small, so magnetic helicity is at least approximately
conserved
Magnetic Helicity Conservation II
STFC Advanced School U Warwick 2012 – p.30/45
Often used: Harris Sheet (E. Harris, 1962)
Originally a kinetic equilibrium, but is also an MHD
equilibrium
B 2 (z)
+ p(z) = pT = constant
2µ0
Equilibrium structures: Total pressure across sheet is
constant
Here: non-singular current sheets in 1D (justified by
ratio of length scales)
Current sheets: can be singular MHD structures
(discontinuities) or finite (e.g. neutral sheets)
Current Sheets
STFC Advanced School U Warwick 2012 – p.29/45
Important for defining the connectivity and topology of
magnetic field configurations
Points in space where B = 0
Magnetic Null Points
B02 /(2µ0 ) = p0
Field Lines
p(z) = p0 / cosh2 (z/L) + pb
B = B0 tanh(z/L)x̂
Harris Sheet
STFC Advanced School U Warwick 2012 – p.32/45
STFC Advanced School U Warwick 2012 – p.31/45
µ0 I02
a2
p(r) =
8π 2 (r2 + a2 )2
B0 ar
,
Bφ (r) = 2
r + a2
STFC Advanced School U Warwick 2012 – p.34/45
B0 a2
Bz (r) = 2
r + a2
Gold-Hoyle tube (Gold and Hoyle, 1960) – 1D force-fee
flux tube with Bφ (r) and Bz (r) non-zero
µ0 I0
r
Bφ (r) =
,
2π r2 + a2
Bennett pinch (Bennett 1934) – only Bφ (r) and p(r):
Flux tubes: Examples
STFC Advanced School U Warwick 2012 – p.33/45
Equilibrium (B = (0, Bφ (r), Bz (r))):
0
/
2 (r) + B 2 (r)
Bφ2
B
d
z
φ
+ p(r) +
=0
dr
2µ0
µ0 r
Can be used as models for coronal loops, also for
magnetic structures in solar interior
Simplest case: 1D equilibria in cylindrical geometry
(use r, φ, z as cylindrical coordinates)
Flux tubes
∂
∂y
= 0 =⇒ Invariance in y -direction
Then
STFC Advanced School U Warwick 2012 – p.36/45
A is constant along magnetic field lines!
B · ∇A = (∇A × ey ) · ∇A +By ey · ∇A = 0.
,.
+ ,- .
+
=0
=0 since ∂A
=0
∂y
Satisfy ∇ · B = 0 by B = ∇A × ey + By ey
Assume
Translational Invariance 1
STFC Advanced School U Warwick 2012 – p.35/45
a http://www-solar.mcs.st-and.ac.uk/˜thomas/teaching/mhdlect.pdf
For more details (also on the other cases and with external
forces) : see lecture notes on my web page a
j × B − ∇p = 0
Here just a quick reminder how to do that for translational
invariance without external forces
MHD can be reduced to a single nonlinear elliptic
second-order PDE
Translational, rotational or helical symmetry:
MHD equilibria: Symmetric Systems
and
So
1
j × B − ∇p =
µ0
dg
∇A
dA
STFC Advanced School U Warwick 2012 – p.38/45
%
&
df
dg
−∆A − µ0
− g(A)
∇A = 0
dA
dA
∇By =
By is constant along field lines =⇒ can take By = g(A)
− (∇By × ey ) · ∇A = (∇A × ey ) · ∇By = 0
Translational Invariance 3
STFC Advanced School U Warwick 2012 – p.37/45
Translational Invariance 2
∂x2
∂2
+
∂z 2
∂2
&
A = µ0
d
dA
/
p(A) +
2µ0
By2
0
= F (A)
STFC Advanced School U Warwick 2012 – p.39/45
B=∇×A
STFC Advanced School U Warwick 2012 – p.40/45
Use B directly, ensure B solenoidal by numerical
means
Which gauge for A? Boundary conditions for A?
Vector potential
intrinsically nonlinear
existence of global α and β not guaranteed
(could use four potentials instead).
B = ∇α × ∇β
Euler Potentials (Clebsch representation):
Representation of B to guarantee ∇ · B = 0 much more
difficult
3D MHS
Some analytical solutions known
(for special choices of F (A))
Single nonlinear 2nd order elliptic partial differential
equation:
boundary conditions for A needed (e.g. Dirichlet or von
Neumann)
Grad-Shafranov(-Schlüter) equation for translational
invariance
−
%
Translational Invariance 4
µ0 j = α(r) B
STFC Advanced School U Warwick 2012 – p.42/45
Current density field-aligned/parallel to B everywhere, i.e.
j×B=0
For the corona the plasma beta βp = 2µ0 p/B 2 ( 1 is usually
much smaller than unity, so
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
Force-free Fields 1
STFC Advanced School U Warwick 2012 – p.41/45
What are the appropriate boundary conditions for solving
them?
Further difficulty: these equations are of mixed type!
∇β · ∇ × (∇α × ∇β) = µ0
∂p
∂α
∂p
∇α · ∇ × (∇β × ∇α) = µ0
∂β
Euler Potential Equations
j = α(r)B, B · ∇α = 0
j = αB, α = constant '= 0
STFC Advanced School U Warwick 2012 – p.44/45
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 !)
Nonlinear force-free fields
Linear force-free fields:
Potential fields :
j = 0, α = 0
STFC Advanced School U Warwick 2012 – p.43/45
Force-free Fields 3
∇ × B = α(r) B
B · ∇α = 0
∇·B = 0
Basic equations to solve:
i.e. α is constant along magnetic field lines.
B · ∇α = 0
Since ∇ · j = 0 and ∇ · B = 0 we get
Force-free Fields 2
STFC Advanced School U Warwick 2012 – p.45/45
Schindler, Physics of Space Plasma Activity, Cambridge
UP, 2007
Priest, Solar Magnetohydrodynamics, Reidel, 1982
Goedbloed and Poedts, Principles of
Magnetohydrodynamics, Cambridge UP, 2004
Freidberg, Ideal Magnetohydrodynamics, Plenum Press,
1987
Boyd and Sanderson, The Physics of Plasmas,
Cambridge UP, 2003
Biskamp, Magnetic Reconnection in Plasmas, Cambridge
UP, 2000
Biskamp, Nonlinear Magnetohydrodynamics, Cambridge
UP, 1993
Further Reading