22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2: Derivation of Ideal MHD Equation
Review of the Derivation of the Moment Equation
1. Starting Point: Boltzmann Equation for electrons, ions and Maxwell Equations
2. Moments of Boltzmann Equation: conservation of mass, momentum and
energy.
⎧
1 ⎫
⎡ dFα ⎛ ∂Fα ⎞ ⎤ ⎪⎪
⎪⎪
∫ ⎢⎢ dt − ⎝⎜ ∂t ⎠⎟ ⎥⎥ ⎨ mα v ⎬ dv
⎪
c⎦ ⎪
⎣
2
⎩⎪mα v 2⎭⎪
mass
momentum
energy
3. Accounting: v = uα ( e, t ) + v, uα = fluid velocity, v = random velocity
nα = ∫ Fα dv
uα =
1
nα
density
∫ vFα dv =
fluid velocity
v
pressure tensor
P α = nαmα v v
pα =
1
mαnα v 2
3
scalar pressure
hα =
nαmα
v2 v
2
heat flux
R α = ∫ mα v Cαβ dv
Qα =
friction due to collisions
mα v 2
∫ 2 Cαβ dv
heat generated due to collisions
General 2 Fluid Equations
∇×E = −
∂B
∂t
∇×B = μ ⋅ J+
∇ ⋅B = 0
1 ∂E
c2 ∂t
∇ ⋅E =
σ
∈0
∂nα
+ ∇ ⋅ (nα uα ) = 0
∂t
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2
Page 1 of 11
mαnα
duα
= qαnα (E + uα × B ) − ∇ ⋅ P α + R α
dt
e, i
dTα
3
+ P α : ∇uα = Qα − ∇ ⋅ hα
nα
2
dt
σ = e (ni − ne )
J = e (ni ui − ne ue )
Physical Assumptions Leading to Ideal MHD
1. Moment equations as they now stand are exact, but not closed.
2. Certain assumptions lead to closure - 1 fluid MHD model
Asymptotic Assumptions
1. MHD is concerned with low frequency - long wavelength macroscopic behavior
2. The first simplification of the 2 fluid equations eliminates short wavelength,
fast time scale phenomena: well satisfied assumptions experimentally
3. Asymptotic assumptions change basic mathematical structure of the time
evolution.
speed of light → ∞
electron inertia → 0
First Asymptotic Assumption c → ∞
1. Maxwell equations → low frequency Maxwell equations
2. Formally let ∈0 → 0
∇ × B = u0 J +
ni − ne =
1 ∂E
≈ μ0 J
c2 ∂t
∈0
∇ ⋅E ≈ 0
e
neglect displacement current
quasineutrality
3. Equations are now Gallilean invariant
4. Conditions for validity:
ω
ωpe
λd ≡
v Te
ωpe
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
a
no plasma oscillations
Lecture 2
Page 2 of 11
ω
∼ v Ti
k
v Te
no high frequency waves
c
5. Note: ne = ni ≡ n does not imply E or ∇ ⋅ E = 0 . Only that
∈0 ∇ ⋅ E en
1
Second Asymptotic Assumption me → 0
1. The electron response time is essentially instantaneous because me
mi
2. We then neglect electron inertia in the momentum equation
0 ≈ −ene (E + ue × B ) − ∇Pe + R e
3. Conditions for validity
ω
ωpe
λd
a
no electron plasma oscillations
ω
ωce
rce
a
no electrons cyclotron oscillations
to B
4. Both c → α, me ∞ 0 assumptions are well satisfied for MHD behavior
Subtle Effect
1. Neglect of electron inertia along B can be tricky
2. For long wavelengths, electrons can still require a finite response time
even though me is small. This is region of the drift wave
3. We shall see that MHD consistently treats motion poorly, but for MHD
behavior, remarkably this does not matter!!
4. To treat such behavior more sophisticated models are required. The
resulting instabilities are much weaker, (and still important) than for MHD.
The two Fluid Equations with Asymptotic Assumptions
∇×E = −
∂B
∂t
∂n
+ ∇ ⋅ nue = 0
∂t
∇ ⋅B = 0
∇ × B = μ0en (ui − ue )
ne = ni = n
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
∂n
+ ∇ ⋅ nui = 0
∂t
Lecture 2
Page 3 of 11
mn
i
due
− en (E + ui × B ) + ∇ ⋅ Pi = R i
dt
3 dTα
+ P α : ∇uα + J ⋅ hα = Qα
n
2 dt
e
en (E + ue × B ) + +∇ ⋅ Pe = R e
Single Fluid Equations
1. Introduce single fluid variable
v = ui
the momentum of fluid is carried by ions since mi = 0
p = pe + pi
total pressure
ρ = mn
i
mass density
J = en (ui − ue ) current density
2. Use all information!!. This is not trivial!! Initially the unknowns are
E , B , J, V , n, p (19 variables). The finally unknowns are
E , B , J, V , n, p (14 variables)
3. Maxwell equations → OK as is in low frequency form
4. Mass conservation
a. Mi × ion
∂ρ
+ ∇ ⋅ ρv = 0
∂t
b. e (ion-electron) → ∇ ⋅ en (ui − ue )
= ∇⋅J= 0
This is automatic from the low frequency Maxwell equations
∇ × B = μ0 J → ∇ ⋅ J = 0
5. Momentum Equation (ion + electron)
a. ρ
(
)
dv
− en (ui − ue ) × B + ∇ ⋅ Pi + Pe = Ri + R e
dt
J×B
∇ ⋅ ⎡(pi + pe ) Ι + Πi + Π e ⎤
⎣
⎦
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
∫ dv ⎡⎣me v cei + mi v cie ⎤⎦ = 0
Lecture 2
Page 4 of 11
b. ρ
(
dv
− J × B + ∇p = −∇ ⋅ Πi + Π e
dt
)
6. Electron Momentum equation
a. E + ue × B =
ue = ui −
R e − ∇ ⋅ Pe
en
J
J
=v−
en
en
b. E + v × B =
1 ⎡
R e − ∇ ⋅ Pe + J × B ⎤
⎦
en ⎣
7. Energy Equation (ions)
Pi : ∇ui
a.
3 d pi
n
+ pi∇ ⋅ ui = Qi − ∇ ⋅ hi − Πi : ∇ui
2 dt n
1
2
b. 1:
3 dpi 3 pi dn
−
2 dt 2 n dt
c. 2:
dn
∂n
∂n
+ ∇ ⋅ nv = 0 =
+ v ⋅ ∇n + n∇⋅v →
= −n∇⋅v
dt
∂t
∂t
pi∇ ⋅ v = −
d. 1+2:
e.
pi dn
n dt
3 dpi 5 pi dn 3 5 3 d pi
−
= n
2 dt 2 n dt 2
dt n5 3
d pi
2 ⎡
=
Qi − ∇ ⋅ hi − Πi : ∇ v ⎤
r
⎦
dt ρ
3ρr ⎣
r=5/3
8. Energy Equation (electrons)
a.
p
d pe
2 ⎡
J
d⎤
= r ⎢Qe − ∇ ⋅ he − Π e : ∇v +
⋅ ∇ er + Π e : ∇
⎥
r
dt ρ
en
en ⎦
3ρ ⎣
ρ
from
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
d
dte
from Π e : ∇ue
Lecture 2
Page 5 of 11
b.
d
∂
=
+ v ⋅ ∇ = ion convective derivation
dt ∂t
Assumptions Leading to Ideal MHD
1. Philosophy: Ideal MHD is concerned with phenomena occurring on certain
length and time scales.
2. Ordering: Using this, we can order all the terms in the one fluid equations.
After ignoring small terms, we obtain ideal MHD.
3. Status: At this point only the assumptions c → ∞, me → 0 have been used
in the equation
Characteristic Length and Time Scales for Ideal MHD
1.
v
∂
∼ ω ∼ Ti
a
∂t
2.
1
∂
∼k ∼
∂x
a
macroscopic MHD phenomena
3. v ∼ v Ti
4. a → macroscopic length
5. v Ti → macroscopic ion velocity
6. a v Ti → corresponding macroscopic time scale
Two Approaches to Ideal MHD
A.
Collision dominated plasma: regions limit to ideal MHD
B.
Collision free limit: also works but for subtle reasons
Collision Dominated Limit
1. The electrons and ions are assumed collision dominated
2. This is the basic requirement to keep the pressure isotropic. Many
collisions keep particle close together. This allows us to divide the plasma
into small fluid element and provides a good physical description.
3. There are 2 conditions for a collision dominated plasma
a. on the time scale of internal there are many collisions, so the plasma
is near maxwellion
•
ions: ion-ion coulomb collisions dominate
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2
Page 6 of 11
•
electrons: electron-ion, electron-electron collisions are
comparable
•
ions: ωτii ∼
•
electrons: ωτee ∼ ωτee ∼
•
Recall: τee ∼ τei ∼ (me mi )
•
The ion condition is most severe
v Ti τii
a
1
12
⎛m ⎞
v Ti
τee ∼ ⎜ e ⎟
a
⎝ mi ⎠
12
v Ti τii
a
v Ti τii
a
1
τii and τEQ ∼ (mi me )
12
τii
1
b. The macroscopic length scale must be much larger than the mean free
path for collisions. λ α = v Tα ταα
λi
v τ
= Ti ii
a
a
•
ions
•
electrons
1 (same as before)
v τ
λe
v τ
∼ Te ee ∼ Ti ii
a
a
a
1 (same as ions)
MHD Limit
1. Use the collision dominated assumption to obtain ideal MHD
2. Several additional assumptions will also be required
3. Various moments in the equations are approximated by classical transport
theory of Braginskii.
4. Transport coefficients can also be derived in the homework problems
Reduction of 1 Fluid Equation
1. Maxwell Equations – OK
2. Mass conservation – OK
3. Momentum Equation
u
2
⎡
⎤
a. ions: Πii ∼ μi ⎢2∇ ⋅ ui − ∇ ⋅ ui ⎥ ∼ μi i
3
a
⎣
⎦
viscosity
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2
Page 7 of 11
b. electrons: Π ee ∼ μe
c. Note: ue = v −
ue
a
J
en
r
J
∇p
T
∼
∼
∼ ii
env Benv aBev Ti
a
1
assume small gyro radius
d. ∴ ui ≈ ue : small difference in the flow velocities generate macroscopic
current density J, but ui − ue
v Ti
e. Ordering:
ue
v
∼ μe Ti
a
a
•
Π ee ∼ μe
•
⎛m ⎞
μe ∼ ⎜ e ⎟
⎝ mi ⎠
•
Πii
μv
∼ i Ti
Pi
api
•
∴
•
Both Π terms are negligible in momentum equation
12
f.
ρ
μi → Π e
Πii
τv
∼ ii Ti
pi
a
Πi
μi ∼ nTi τii viscosity coefficient
1 collision dominated assumption
dv
= J × B − ∇p momentum equation
dt
4. Ohms Law
4
3
2
1
1 ⎡
E + v×B =
R e − ∇ ⋅ Pe + J × B ⎤
⎦
en ⎣
Hall effect
Electron diamagnetism ωre
Resistivity
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2
Page 8 of 11
a. 1 ∼ 2
1 / 4 ∼ J enV ∼
rLi
a
1 small gyro radius assumptions
b. Re ∼ resistivity momentum transfer due to collisions
•
R e = en η d, η =
•
3/4 ∼
12
c.
⎛ me ⎞
⎜
⎟
⎝ mi ⎠
me
me
ne2 τei
12
J
ne2 τei v TiB
∼
me ⎛ rii ⎞ ⎛ me ⎞
⎟
⎜ ⎟∼⎜
eτeiB ⎝ a ⎠ ⎝ mi ⎠
2
⎛ a ⎞ ⎛ rii ⎞
⎜
⎟⎜ ⎟
⎝ VTi τii ⎠ ⎝ a ⎠
2
⎛ a ⎞ ⎛ rii ⎞
⎜
⎟⎜ ⎟
⎝ v Ti τii ⎠ ⎝ a ⎠
1
d. The plasma must be larger enough so that resistive diffusion does not
play an important role.
∂ ⎞
⎛
5. Energy equation ⎜ v ⋅ ∇ ∼
t ⎟⎠
∂
⎝
a. ions: Πi pi
1
1, ( J en ⋅ ∇ ) pe
b. electrons: Π e pe
c.
d pi
2
= r ⎡⎣Qi − ∇ ⋅ hi ⎤⎦
r
dt ρ
3ρ
d.
d pe
2
=
⎣⎡Qe − ∇ ⋅ he ⎦⎤
r
dt ρ
3ρr
∇ ⋅ pe , ( Π e J en)
vpe
e. hi = −κ i∇ Ti − κ ⊥ i∇ ⊥ Ti
f.
dominant contribution is from thermal
conduction
he = −κ e ∇ Te − κ ⊥ e ∇ ⊥ Te
g. In general κ
h. Q1 = −
i.
Qe = −
n ( Ti − Te )
τeq
n ( Te − Ti )
τeq
κ⊥
→ equilibration
+
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
J ⋅ Re
→ equilibration plus ohmic heating
en
Lecture 2
Page 9 of 11
j.
Note: cons. of energy → Qi + Qe − J ⋅ Re en = 0
k. Compare
12
2
•
⎛m ⎞
JRe
ωpe = ⎜ e ⎟
en
⎝ mi ⎠
•
small ohmic heating in MHD time scale
d pi
2
= r
r
dt ρ
3ρ
l.
∴
m.
d pe
2
= r
dt ρr
3ρ
⎛ a ⎞ ⎛ rii ⎞
⎜
⎟⎜ ⎟
⎝ v Ti τii ⎠ ⎝ a ⎠
1
⎡
( Te − Ti ) ⎤
⎢∇n κ i ⋅ ∇ Ti + n
⎥
τEQ ⎥⎦
⎢⎣
(
⎡
⎢∇n κ
⎣⎢
(
)
)
⋅ ∇ Te + n
e
( Ti − Te ) ⎤
⎥
⎦⎥
τEQ
n. But MHD is a single fluid model - 1 pressure, 1 temperature
o. This occurs if τEQ is very small, forcing Te ≈ Ti
p. Small τEQ require
12
⎛m ⎞
q. ⎜ i ⎟
⎝ me ⎠
v Ti τii
a
nT
τEQ
ωp or ωτEQ
1
1
This is more severe than the collision dominated momentum condition
momentum equilibration τ .
energy equilibration τ
r.
If this is true then
•
1st information Te ≈ Ti ≡ T 2
•
2nd information (add equations)
•
d p
1
=
∇
r
dt ρ
3ρr
•
But κ i ≈ (me mi )
•
Thus
(κ
+κ
i
12
∇ ⋅ κ i∇ T
ωp
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
e
)∇ T
κ e, κ
12
⎛m ⎞
∼⎜ i ⎟
⎝ me ⎠
e
≈ nTe τei me
τiiv Ti
a
1
Lecture 2
Page 10 of 11
This gives Ideal MHD Equation
∇ ×E = −
∂B
∂t
∇ × B = u0 J
∇ ⋅B = 0
ni = ne = n
∂p
+ ∇ ⋅ ρv = 0
∂t
ρ
dv
= J × B − ∇p
dt
E + V ×B = 0
d p
=0
dt ρr
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 2
Page 11 of 11