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Fundamentals ofMagnetohydrodynamics(MHD)
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Tony Arber
University of Warwick
STFC Advanced School, MSSL September 2013.
Fundamentals of Magnetohydrodynamics
(MHD)
Aim
Derivation of MHD equations from conservation lawsQuasi-neutralityValidity of MHDMHD equations in dierent forms MHD wavesAlfvens Frozen Flux TheoremLine Conservation TheoremCharacteristicsShocks
Applications of MHD, i.e. all the interesting stu!, 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 uid 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
8 equations with 8 unknowns
Maxwell equations
Mass conservation
F = ma for uids
Low frequency Maxwell
Adiabatic equation for uids
Ideal Ohms Law for uids
Mass Conservation - Continuity Equation
x x+x
m =
Z x+xx
dx
F (x+x)F (x)
Mass m in cell of width changes due to rate of mass leaving/entering the cell
xF (x)
@
@t
Z x+xx
dx
!= F (x) F (x+x)
@
@t= limx!0
F (x) F (x+x)x
@
@t+
@F (x)
@x= 0
Mass ux - conservation laws@
@t+
@F (x)
@x= 0
Mass ux per second through cell boundary
In 3D this generalizes to
This is true for any conserved quantity so if conserved
@
@t+r.(v) = 0
F (x, t) = (x, t) vx(x, t)
ZU dx
@U
@t+r.F = 0
Hence applies to mass density, momentum density and energy density for example.
Convective Derivative
In uid dynamics the relation between total and partial derivatives is
Rate of change of quantity at a xed point in space
Convective derivative:
Rate of change of quantity at a point moving with the uid.
Often, and frankly for no good reason at all, write
D
Dtinstead of
d
dt
Adiabatic energy equation
If there is no heating/conduction/transport then changes in uid elements pressure and volume (moving with the uid) is adiabatic
PV = constant
d
dt(PV ) = 0
Where is ration of specic heats
Moving with a packet of uid the mass is conserved so V / 1
d
dt
P
= 0
Momentum equation - Euler uid
x x+x
Total momentum in cell changes due to pressure gradient
P (x) P (x+x)uxx
Now F is momentum ux per second F = uxux
@
@t(ux) +
@F
@x= rP
@
@t(uxx) = F (x) F (x+x) + P (x) P (x+x)
Momentum equation - Euler uid
@ux@t
+ ux@ux@x
= rP
Use mass conservation equation to rearrange as
@ux@t
+ ux@ux@x
= rP
duxdt
= rP
dux(x, t)
dt=
@ux@t
+@x
@t
@ux@x
Since by chain rule
Momentum equation - MHD
du
dt= rPFor Euler uid how does this change for MHD?
Force on charged particle in an EM eld is
F = q(E+ v B)
Hence total EM force per unit volume on electrons is
nee(E+ v B)and for ions (single ionized) is
nie(E+ v B)Where are the electron and ion number densitiesne and ni
Momentum equation - MHD
Hence total EM force per unit volume
If the plasma is quasi-neutral (see later) then this is just
en(ui ue)B = jBWhere j is the current density. Hence
du
dt= rP + jB
Note jxB is the only change to uid equations in MHD. Now need an equation for the magnetic eld and current density to close the system
e(ni ne)E+ (eniui eneue)B
Maxwell equations
Not allowed in MHD!
Initial condition only
Used to update B
Low frequency version used to nd current density j
Low-frequency Maxwell equations
So for low velocities/frequencies we can ignore the displacement current
Displacement current
Quasi-neutrality
For a pure hydrogen plasma we have
Multiply each by their charge and add to get
where is the charge density and j is the current density
From Ampere's law if we look only at low frequencies
Hence for low frequency processes this is quasi-nuetrality
MHD
8 equations with 11 unknowns! Need an equation for E
Maxwell equations
Mass conservation
Momentum conservation
Low frequency Maxwell
Energy conservation
Ohm's Law
Assume quasi-neutrality, subtract electron equation
Note that Ohm's law for a current in a wire (V=IR) when written in terms of current density becomes
This is called the generalized Ohm's law
When uid is moving this becomes
Equations of motion for ion uid is
Magnetohydrodynamics (MHD)
Valid for:
Low frequency
Large scales
If =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
Eulerian form of MHD equations
@
@t= r.(v)
@P
@t= Pr.v
@v
@t= v.r.(v) 1
r.P + 1
jB
@B
@t= r (v B)
j =1
0rB
Final equation can be used to eliminate current density so 8 equations in 8 unknowns
Lagrangian form of MHD equations
j =1
0rB
D
Dt= r.v
DP
Dt= Pr.v
Dv
Dt= 1
r.P + 1
jB
DB
Dt= (B.r)v B(r.v)
D
Dt
B
=B
.rv
=P
( 1)
Alternatives
Specic internal energy density
D
Dt= P
r.v
Conservative form
@
@t= r.(v)
@v
@t= r.
vv + I(P +
B2
2)BB
@E
@t= r.
E + P +
B2
20
v B(v.B)
@B
@t= r(vBBv)
E =P
1 +v2
2+B2
20The 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
=20P
B2
Low beta means dynamics dominated by magnetic eld, high beta means standard Euler dynamics more important
/ c2s
v2A
Assume initially in stationary equilibrium
MHD Waves
r.P0 = j0 B0
Apply perturbation, e.g. P = P0 + P1
Simplify to easiest case with constant and no equilibrium current or velocity
0, P0,B0 = B0z
Univorm B eld
Constant density, pressureZero initial velocityApply small perturbation to system
MHD Waves
Ignore quadratic terms, e.g. P1r.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 A.u = u
Where u = (P1, 1, v1, B1)
Solution requires det| A I| = 0
(Fast and slow magnetoacoustic waves)
(Alfvn waves)
Alfvn speed
Sound speed
B k
Dispersion relation
Incompressible no change to density or pressure
Group speed is along B does not transfer energy (information) across B elds
Alfven Waves
For zero plasma-beta no pressure
Compresses the plasma c.f. a sound wave
Propagates energy in all directions
Fast magneto-acoustic waves
But
So
Magnetic pressureMagnetic tension
Magnetic pressure and tension
Pressure perturbations
Phase speeds:
: Group speeds
Phase and group speeds
Throwing a pebble into a plasma lake...For low plasma beta
Three types of MHD waves
Alfvn waves magnetic tension (=VAk)
Fast magnetoacoustic waves magnetic with plasma pressure (VAk)
Slow magnetoacoustic waves magnetic against plasma pressure (CSk)
transv. velocity density
MHD Waves Movie
vA >> cs
1. The perturbations are waves
2. Waves are dispersionless
3. and k are always real
4. Waves are highly anisotropic
5. There are incompressible - Alfvn waves - and compressible - magnetoacoustic modes
However, natural plasma systems are usually highly structured and often unstable
Linear MHD for uniform media
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
@B
@t= r (v B) +
0r2B
If we assume the resistivity is constant then
Ratio of advective to diusive terms is the magnetic Reynolds number
Rm =0L0v0
Usually in space physics (106-1012). This is based on global scale lengths . If is over a small scale with rapidly changing magnetic eld, i.e. a current sheet, then
Rm >> 1
Rm ' 1L0 L0
Alfvens theorem
d
dt
ZSn.B dS =
ZSn@B
@tdS
Ilv B.dl
= ZSr (E+ v B).n dS
Rate of change of ux through a surface moving with uid
Magnetic ux through a surface moving with the uid is conserved if ideal MHD Ohms law, i.e. no resistivity
Often stated as- the ux is frozen in to the uid
Line Conservation
x(X, t)
x(X+ X, t)
X
Consider two points which move with the uid
x
xi =@xi@Xj
Xj
x(X+ X, 0)
x(X, 0) = X
=@ui@xk
@xk@xj
Xj
= (x.r)u
D
Dtx =
@ui@Xj
Xj
Line Conservation -2
D
Dtx = (x.r)u
Equation for evolution of the vector between two points moving with the uid is
D
Dt
B
=B
.rv
Also for ideal MHD
Hence if we choose to be along the magnetic eld at t = 0 then it will remain aligned with the magnetic eld.
Two points moving with the uid which are initially on the same eld-line remain on the same eld line in ideal MHD
Reconnection not possible in ideal MHD
x
Cauchy Solution
Shown that satisfy the same equation hence B
and x
xi =@xi@Xj
Xj
ImpliesBi=
@xi@Xj
B0j0
Where superscript zero refers to initial values
Bi =@xi@Xj
B0j
0
0
= =
@(x1, x2, x3)
@(X1, X2, X3)
Bi =@xi@Xj
B0j
Cauchy solution
MHD based on Cauchy
Bi =@xi@Xj
B0j
0
= =
@(x1, x2, x3)
@(X1, X2, X3)
P = const
Dv
Dt= 1
r.P + 1
0(rB)B
Only need to know position of uid elements and initial conditions for full MHD solution
dx
dt= v
Resistivity
Coriolis GravityOther
Thermal
Conduction
Radiation Ohmic heating
Other
Non-ideal MHD
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
MHD Characteristics
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
But
w is called the characteristic eld
A = RR1 so R1A = R1
@
@t
R1U
+.
@
@x
R1U
= 0
1. 0 with t x
+ = = w w w R U
This example is for linear equations with constant A
is diagonal so all equations decouple
i.e. characteristics wi propagate with speed
Solution in terms of original variables UThis analysis forms the basis of Riemann decomposition used for treating shocks, e.g. Riemann codes in numerical analysis
Riemann problems
In MHD the characteristic speeds are i.e. the fast, Alfven and slow speeds
vx, vx cf , vx vA, vx cs
Basic ShocksTemperature
x
c2s = P/
T = T (x cst)
Without dissipation any 1D traveling pulse will eventually, i.e. in nite time, form a singular gradient. These are shocks and the dierentially form of MHD is not valid.
Also formed by sudden release of energy, e.g. are, or supersonic ows.
Rankine-Hugoniot relations
U
xS(t)xrxl
ULUR
Integrate equations from xl to xr across moving discontinuity S(t)
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
Jump Conditions
The End