Fundamentals of
Magnetohydrodynamics
(MHD)
Tony Arber
University of Warwick
STFC Advanced School, MSSL September 2013.
Aim
Derivation of MHD equations from conservation laws
Quasi-neutrality
Validity of MHD
MHD equations in different forms
MHD waves
Alfven’s Frozen Flux Theorem
Line Conservation Theorem
Characteristics
Shocks
Applications of MHD, i.e. all the interesting stuff!, will
be in later lectures covering Waves, Reconnection and
Dynamos etc.
Derivation of MHD
Possible to derive MHD from
• N-body problem to Klimotovich equation, then take
moments and simplify to MHD
• Louiville theorem to BBGKY hierarchy, then take
moments and simplify to MHD
• Simple fluid dynamics and control volumes
First two are useful if you want to study kinetic theory along the
way but all kinetics removed by the end
Final method followed here so all physics is clear
Ideal MHD
Maxwell equations
Mass conservation
F = ma for fluids
Low frequency Maxwell
Adiabatic equation for fluids
Ideal Ohm’s Law for fluids
8 equations with 8 unknowns
Mass Conservation - Continuity Equation
F (x)
m=
Z
F (x +
x+ x
x)
⇢ dx
x
x+ x
x
Mass m in cell of width x changes due to rate of mass
leaving/entering the cell F (x)
!
Z x+ x
@
⇢ dx = F (x) F (x + x)
@t
x
F (x)
@⇢
= lim
x!0
@t
F (x +
x
@⇢ @F (x)
+
=0
@t
@x
x)
Mass flux - conservation laws
@⇢ @F (x)
+
=0
@t
@x
Mass flux per second through cell boundary
F (x, t) = ⇢(x, t) vx (x, t)
In 3D this generalizes to
@⇢
+ r.(⇢v) = 0
@t
This is true for any conserved quantity so if
@U
+ r.F = 0
@t
Z
U dx conserved
Hence applies to mass density, momentum density and energy
density for example.
Convective Derivative
In fluid dynamics the relation between total and partial
derivatives is
Convective derivative:
Rate of change of quantity at a point
moving with the fluid.
Rate of change of quantity at a fixed
point in space
Often, and frankly for no good reason at all, write
D
d
instead of
Dt
dt
Adiabatic energy equation
If there is no heating/conduction/transport then changes in fluid
element’s pressure and volume (moving with the fluid) is adiabatic
PV
Where
= constant
is ration of specific heats
d
(P V ) = 0
dt
Moving with a packet of fluid the mass is conserved so V / ⇢
✓ ◆
d P
=0
dt ⇢
1
Momentum equation - Euler fluid
P (x)
P (x +
⇢ux x
x+
x
x)
x
Total momentum in cell changes due to pressure gradient
@
(⇢ux x) = F (x)
@t
F (x +
x) + P (x)
P (x +
Now F is momentum flux per second F = ⇢ux ux
@
@F
(⇢ux ) +
=
@t
@x
rP
x)
Momentum equation - Euler fluid
Use mass conservation equation to rearrange as
@ux
@ux
⇢
+ ⇢ux
= rP
@t
@x
✓
◆
@ux
@ux
⇢
+ ux
= rP
@t
@x
dux
⇢
=
dt
rP
Since by chain rule
dux (x, t)
@ux
@x @ux
=
+
dt
@t
@t @x
Momentum equation - MHD
du
=
For Euler fluid ⇢
dt
rP how does this change for MHD?
Force on charged particle in an EM field is
F = q(E + v ⇥ B)
Hence total EM force per unit volume on electrons is
ne e(E + v ⇥ B)
and for ions (single ionized) is
ni e(E + v ⇥ B)
Where ne and ni are the electron and ion number densities
Momentum equation - MHD
Hence total EM force per unit volume
e(ni
ne )E + (eni ui
ene ue ) ⇥ B
If the plasma is quasi-neutral (see later) then this is just
en(ui
ue ) ⇥ B = j ⇥ B
Where j is the current density. Hence
du
⇢
=
dt
rP + j ⇥ B
Note jxB is the only change to fluid equations in MHD. Now
need an equation for the magnetic field and current density to
close the system
Maxwell equations
Not allowed in MHD!
Initial condition only
Used to update B
‘Low’ frequency version
used to find current
density j
Low-frequency Maxwell equations
Displacement current
So for low velocities/frequencies we can ignore the
displacement current
Quasi-neutrality
For a pure hydrogen plasma we have
Multiply each by their charge and add to get
where σ is the charge density and j is the current density
From Ampere's law
Hence for low frequency processes
if we look only at low frequencies
this is quasi-nuetrality
MHD
Maxwell equations
Mass conservation
Momentum conservation
Low frequency Maxwell
Energy conservation
8 equations with 11 unknowns! Need an equation for E
Ohm's Law
Equations of motion for ion fluid is
Assume quasi-neutrality, subtract electron equation
This is called the generalized Ohm's law
Note that Ohm's law for a current in a wire (V=IR) when
written in terms of current density becomes
When fluid is moving this becomes
Magnetohydrodynamics (MHD)
Valid for:
• Low frequency
• Large scales
If η=0 called ideal MHD
Missing viscosity, heating,
conduction, radiation, gravity,
rotation, ionisation etc.
Validity of MHD
Assumed quasi-neutrality therefore must be low frequency
and speeds << speed of light
Assumed scalar pressure therefore collisions must be sufficient
to ensure the pressure is isotropic. In practice this means:
• mean-free-path << scale-lengths of interest
• collision time << time-scales of interest
• Larmor radii << scale-lengths of interest
However as MHD is just conservation laws plus low-frequency
MHD it tends to be a good first approximation to much of the
physics even when all these conditions are not met.
Eulerian form of MHD equations
@⇢
=
@t
r.(⇢v)
@P
=
@t
P r.v
@v
=
@t
v.r.(v)
1
1
r.P + j ⇥ B
⇢
⇢
@B
= r ⇥ (v ⇥ B)
@t
1
j=
r⇥B
µ0
Final equation can be used to eliminate current density so 8
equations in 8 unknowns
Lagrangian form of MHD equations
D⇢
=
Dt
DP
=
Dt
Dv
=
Dt
Alternatives
⇢r.v
D✏
=
Dt
P r.v
1
1
r.P + j ⇥ B
⇢
⇢
DB
= (B.r)v B(r.v)
Dt
1
j=
r⇥B
µ0
P
r.v
⇢
Specific internal energy density
P
✏=
⇢(
1)
D
Dt
✓
B
⇢
◆
B
= .rv
⇢
Conservative form
@⇢
=
@t
@⇢v
=
@t
r.(⇢v)
✓
B
r. ⇢vv + I(P +
)
2
✓✓
@E
=
@t
r.
@B
=
@t
r(vB
E=
P
2
2
B
E+P +
2µ0
◆
v
BB
◆
B(v.B)
◆
Bv)
⇢v 2
B2
+
+
1
2
2µ0
The total energy density
Plasma beta
A key dimensionless parameter for ideal MHD is the
plasma-beta
It is the ratio of thermal to magnetic pressure
2µ0 P
=
B2
Low beta means dynamics dominated by magnetic field, high
beta means standard Euler dynamics more important
c2s
/ 2
vA
MHD Waves
Univorm B field
Constant density, pressure
Zero initial velocity
Apply small perturbation to system
Assume initially in stationary equilibrium
r.P0 = j0 ⇥ B0
Simplify to easiest case with ⇢0 , P0 , B0 = B0 ẑ constant and
no equilibrium current or velocity
Apply perturbation, e.g. P = P0 + P1
MHD Waves
Ignore quadratic terms, e.g. P1 r.v1
Linear equations so Fourier decompose, e.g.
P1 (r, t) = P1 exp i(k.r
!t)
Gives linear set of equations of the form Ā¯.ū = ū
Where ū = (P1 , ⇢1 , v1 , B1 )
Solution requires det|Ā¯
¯ =0
I|
Dispersion relation
(Fast and slow magnetoacoustic waves)
B
k
α
(Alfvén waves)
Alfvén speed
Sound speed
Alfven Waves
• Incompressible – no change to density or pressure
• Group speed is along B – does not transfer energy
(information) across B fields
Fast magneto-acoustic waves
For zero plasma-beta – no pressure
• Compresses the plasma – c.f. a sound wave
• Propagates energy in all directions
Magnetic pressure and tension
But…
So…
Magnetic pressure
Magnetic tension
Pressure perturbations
Phase and group speeds
Phase speeds:
: Group speeds
MHD Waves Movie
Throwing a ‘pebble’ into a ‘plasma lake’...
For low plasma beta vA >> cs
transv. velocity
density
Three types of MHD waves
• Alfvén waves
magnetic tension (ω=VAk‫)װ‬
• Fast magnetoacoustic waves
magnetic with plasma pressure (ω≈VAk)
• Slow magnetoacoustic waves
magnetic against plasma pressure (ω≈CSk‫)װ‬
Linear MHD for uniform media
1. The perturbations are waves
2. Waves are dispersionless
3. ω and k are always real
4. Waves are highly anisotropic
5. There are incompressible - Alfvén waves - and compressible magnetoacoustic – modes
However, natural plasma systems are usually
highly structured and often unstable
Non-ideal terms in MHD
Ideal MHD is a set of conservation laws
Non-ideal terms are dissipative and entropy producing
•
•
•
•
Resistivity
Viscosity
Radiation transport
Thermal conduction
Resistivity
Electron-ion collisions dissipate current
If we assume the resistivity is constant then
@B
⌘ 2
= r ⇥ (v ⇥ B) +
r B
@t
µ0
Ratio of advective to diffusive terms is the magnetic Reynolds
number
µ0 L0 v 0
Rm =
⌘
Usually in space physics Rm >> 1 (106-1012). This is based on
global scale lengths L0 . If L0 is over a small scale with rapidly
changing magnetic field, i.e. a current sheet, then Rm ' 1
Alfven’s theorem
Rate of change of flux through a surface moving with fluid
d
dt
Z
Z
I
@B
n.B dS =
n
dS
v ⇥ B.dl
@t
S
S
l
Z
=
r ⇥ (E + v ⇥ B).n dS
S
Magnetic flux through a surface moving with
the fluid is conserved if ideal MHD Ohm’s law,
i.e. no resistivity
Often stated as- the flux is frozen in to the fluid
Line Conservation
x(X + X, t)
x(X + X, 0)
x
x(X, t)
X
x(X, 0) = X
Consider two points which move with the fluid
@xi
xi =
Xj
@Xj
D
@ui
x=
Xj
Dt
@Xj
@ui @xk
=
Xj
@xk @xj
= ( x.r)u
Line Conservation -2
Equation for evolution of the vector between two points moving
with the fluid is
D
x = ( x.r)u
Dt
Also for ideal MHD
D
Dt
✓
B
⇢
◆
B
= .rv
⇢
Hence if we choose x to be along the magnetic field at t = 0
then it will remain aligned with the magnetic field.
Two points moving with the fluid which are initially on the same
field-line remain on the same field line in ideal MHD
Reconnection not possible in ideal MHD
Cauchy Solution
B
Shown that
and x satisfy the same equation hence
⇢
@xi
xi =
Xj
@Xj
Implies
Bi
@xi Bj0
=
⇢
@Xj ⇢0
Where superscript zero refers to initial values
@xi 0 ⇢
Bi =
Bj 0
@Xj
⇢
⇢0
=
⇢
@(x1 , x2 , x3 )
=
@(X1 , X2 , X3 )
Cauchy solution
@xi Bj0
Bi =
@Xj
MHD based on Cauchy
@xi Bj0
Bi =
@Xj
⇢0
=
⇢
@(x1 , x2 , x3 )
=
@(X1 , X2 , X3 )
P = const ⇢
Dv
=
Dt
1
1
r.P +
(r ⇥ B) ⇥ B
⇢
µ0 ⇢
dx
=v
dt
Only need to know position of fluid elements and initial
conditions for full MHD solution
Non-ideal MHD
Resistivity
Coriolis
Thermal
Conduction
Gravity
“Other”
Radiation
Ohmic heating
“Other”
MHD Characteristics
Sets of ideal MHD equations can be written as
All equations sets of this types share the
same properties
• they express conservation laws
• can be decomposed into waves
• non-linear solutions can form shocks
• satisfy L1 contraction, TVD constraints
Characteristics
is called the Jabobian matrix
For linear systems can show that Jacobian matrix is a
function of equilibria only, e.g. function of p0 but not p1
Properties of the Jacobian
Left and right eigenvectors/eigenvalues are real
Diagonalisable:
Characteristic waves
This example is for linear equations with constant A
But A = R⇤R
@
R
@t
1
so
1
R
1
A = ⇤R
@
U + ⇤.
R
@x
1
1
U =0
∂w
∂w
+ Λ.
= 0 with w = R −1U
∂t
∂x
w is called the characteristic field
Riemann problems
⇤ is diagonal so all equations decouple
i.e. characteristics wi propagate with speed
In MHD the characteristic speeds are vx , vx ± cf , vx ± vA , vx ± cs
i.e. the fast, Alfven and slow speeds
Solution in terms of original variables U
This analysis forms the
basis of Riemann
decomposition used
for treating shocks, e.g.
Riemann codes in
numerical analysis
Basic Shocks
Temperature
c2s = P/⇢
T = T (x
cs t)
x
Without dissipation any 1D traveling pulse will eventually, i.e. in
finite time, form a singular gradient. These are shocks and the
differentially form of MHD is not valid.
Also formed by sudden release of energy, e.g. flare, or supersonic
flows.
Rankine-Hugoniot relations
U
UL
UR
x
S(t)
xl
xr
Integrate equations from xl to xr across moving discontinuity S(t)
Jump Conditions
Use
Let xl and xr tend to S(t) and use conservative form to get
•Rankine-Hugoniot conditions for a discontinuity moving at speed vs
•All equations must satisfy these relations with the same vs
The End