22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14: Formulation of the Stability Problem
Hierarchy of Formulations of the MHD Stability Problem for Arbitrary 3-D
Systems
1. Linearized equations of motion
2. Normal mode → eigenvalue approach
3. Variational approach
4. Energy principle
5. Extended Energy Principle
Linearized Equations of Motion
Assume we have an equilibrium satisfying
static
J0 × B0 = ∇p0
V0 = E0 = 0
∇ × B0 = u0 J0
ρ0 arb.
∇ × B0 = 0
Q0 = Q0 ( x ) → 3D in general
Linearize the Equation
i ( x, t )
Q ( x, t ) = Q0 ( x ) + Q
1
Energy:
dp
+ rp∇ ⋅ v = 0
dt
amp L:
μ ⋅ J =∇×B
∂ρ 1
=0
+ ∇ ⋅ ρ0 v
1
∂t
∂p 1 =0
+ v1 ⋅ ∇p0 + rp0∇ ⋅ v
1
∂t
i
μ J = ∇ × B
∇ ⋅B:
∇ ⋅B = 0
i =0
∇ ⋅B
1
Faraday:
∂B
= −∇ × E = ∇ × v × B
∂t
i
∂B
1
×B
= ∇× v
1
0
∂t
∂v
i − ∇p
ρ0 1 = J1 × B0 + J0 × B
1
1
∂t
Mass:
Momentum:
∂ρ
+ ∇ ⋅ ρv = 0
∂t
ρ
dv
= J × B − ∇p
dt
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
1
0 1
(
)
Lecture 14
Page 1 of 8
Simplify the PDE’s by introducing the displacement sector ξ and appropriate
initial conditions
a.
= ∂ξ
v
1
∂t
b. ξ is the plasma displacement away from equilibrium
Initial Conditions: Assume the plasma is in its equilibrium position moving away
with a small velocity
i ( x, 0 ) = ρ ( x, 0 ) = p ( x, 0 ) = 0
ξ ( x, 0 ) = B
1
1
1
( x, 0 ) = ∂ξ ( x, 0 ) ≠ 0
v
1
∂t
(
i = 0 not needed, redundant
Simplify the equations ∇ ⋅ B
1
)
Express all quantities in terms of ξ
( )
∂ρ 1
= 0 → ∂ ⎡ρ + ∇ ⋅ ρ ξ ⎤ = 0
+ ∇ ⋅ ρ0 v
1
1
0 ⎦
∂t
∂t ⎣
∂ p1 + ξ ⋅ ∇p0 + rp0∇ ⋅ ξ = 0
∂t
∂ i
B1 − ∇ × ξ × B0 = 0
∂t
i
μ J = ∇ × B
Mass:
Energy:
(
Faraday:
Ampere
Momentum:
∂2 ξ
p 1 = −ξ 0∇p0 − rp0∇ ⋅ ξ
)
(
(
0 1
1
ρ0
ρ 1 = −∇ ⋅ ρ0 ξ
(
))
i = ∇ × ξ × B
B
1
0
(
)
μ0 J1 = ∇ × ∇ × ξ × B0
)
∂v
1
i − ∇ρ
= J1 × B0 + J0 × B
1
1
∂t
()
= F ξ
C ξ ( x, 0 ) = 0,
∂ξ
( x, 0) = given, + B.C.
a.
ρ0
b.
( ∇ × B0 ) × ⎡∇ × ξ × B ⎤ + ∇ ξ ⋅ ∇p + rp ∇ ⋅ ξ
1 ⎡
∇ × ∇ × ξ × B0 ⎤ × B0 +
F ξ =
0 ⎦
a
0
⎦
⎣
μ0 ⎣
μ0
∂t2
()
(
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
)
∂t
(
)
(
)
Lecture 14
Page 2 of 8
Initial Value Approach
Solve the linear equations of motion
Advantages: a. directly gives time evolution of the system
b. fastest growing mode automatically appears
c. good stand for nonlinear calculations
Disadvantages: a. more information contained than required to determine
stability
b. extra work is required analytically and numerically to
determine this information
c. tough to find marginal stability
Normal Mode Approach
A more efficient procedure that treats one mode at a time
1. The initial ξ ( x, 0 ) can be decomposed into normal modes
2. Each mode is then analyzed separately
3. To do this we fourier analyze in time
i ( x, t ) = Q ( x ) e−iωt
Q
ξ ( x, t ) = ξ ( x ) e−iωt
4. Why is this legitimate?
i , J ,p do not have any time durations. Hence,
5. Note: The equations for ρ 1 ,B
1 1 1
no explicit ω ' s appear. We find (drop 0 subscript)
( )
ρ1 = −∇ ⋅ ρξ
p1 = −ξ ⋅ ∇p − rp∇ ⋅ ξ
(
B1 = ∇ × ξ × B
(
)
μ0 J1 = ∇ × ∇ ξ × B
)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14
Page 3 of 8
The momentum equation becomes ( F has no time derivation)
()
−ω2ρξ = F ξ
()
F ξ = J1 × B + J × B1 − ∇p1
We only need B.C. → ω2 eigenvalue
This is normal mode approach
Advantages: a. more amenable to analysis
b. directly addresses stability question (examine from ω )
c. more convenient numerically
Disadvantages: a. cannot be generalized for nonlinear calculation
b. still relatively complicated
G
Properties of F
1. To proceed further (variational approach, energy principle) we need to
understand the properties of the force operator F ( ξ )
2. We show that
a. F ( ξ ) is self adjoint
b. ω2 is purely real
c. the normal modes are orthogonal
3. Self adjointness (2 procedures)
a. subtle but elegant
b. direct but complicated
The basic self adjoint property is associated with the conservation of energy;
there is no dissipation in the system
Self Adjoint Property
a.
∫ η ⋅ F ( ξ) dr = ∫ ξ ⋅F ( η) dr
where ξ and n are any two arbitrary, independent sectors satisfying the
boundary conditions
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14
Page 4 of 8
b. Simple self adjoint equations:
∫
∫
η⋅
c. A non self adjoint equation:
η ⋅ ξ dr ,
∂ξ
∂x
∫
η⋅
∂2 ξ
∂x2
∫
dr = − ξ ⋅
∂η
∂x
dr =
∫
ξ⋅
∂2
∂x2
η dr
dr
Direct Demonstration: Very Tedious Calculation
1. Assume n ⋅ ξ = n ⋅ η = 0 as plasma boundary (can be generalized)
⎧⎪ 1
B ⋅ ∇ξ⊥
2. η ⋅ F ξ dr = − dr ⎨
⎪⎩ μ0
()
(
∫
)(B ⋅ ∇η ) + rp (∇ ⋅ ξ ) (∇ ⋅ η)
⊥
+
B2
∇ ⋅ ξ⊥ + 2ξ⊥ ⋅ κ ∇ ⋅ η⊥ + 2η⊥ ⋅ κ
μ0
−
4B2
ξ⊥ ⋅ κ η⊥ ⋅ κ + η⊥ ξ⊥ : ∇∇
μ0
(
)(
(
)(
) (
⎛
)
⎝
⎞ ⎫⎪
⎟⎟ ⎬
⎪
0 ⎠⎭
2
) ⎜⎜ p + 2Bμ
G
F is self adjoint by inspection: switch ξ and η , get the same result.
Show that ω2 is real
()
>
2. −ω2 ρ ξ dr =
∫ξ
1. −ω2ρ ξ = F ξ
∫
2
∫ dr ξ
*
()
*
⋅ F ξ dr
( )
*
*
3. Similarly −ω2*ρ ξ = F ξ
>
∫ dr ξ
real operator
∫
2
4. −ω2* ρ ξ dr =
∫ξ
*
( ) dr
*
⋅F ξ
5. Subtract the equations
(
− ω2 − ω2*
)∫ρ ξ
2
dr =
∫ dr ⎡⎢⎣ξ
*
()
( )
*
⋅ F ξ − ξ ⋅ F ξ ⎤⎥
⎦
=0 because of the self adjoint property
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14
Page 5 of 8
6. Therefore
• ω2 = ω2*
• ω2 is real
This has important consequences
1. At marginal stability we note that by definition ωi = 0
2. In ideal MHD ωr = 0 also!! This is a big help. There is no need to find ωr
3.
()
−ω2ρξ = F ξ
real
ξ real
real
4. That ξ is real is not initially obvious ξ (r, t ) = ξ (r ) e−iωt
5. We continue to allow complex ξ to simplify fourier analysis in space, later on.
ξ (r ) = ξ (r ) eimθτikz
for example
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14
Page 6 of 8
Show that the Normal Modes are Orthogonal
1. Consider two normal modes (assume real ξ now)
( )
−ω2mρ ξm = F ξm
( )
>
∫ ⋅ξ
n
dr
∫ ⋅ξ
dr
⋅ ξn dr =
∫ ⎡⎣ξ
−ω2mρ ξn = F ξn
>
m
2. Subtract
(ω
2
n
2
− ωm
)∫ρξ
m
n
( )
( )
⋅ F ξm − ξm ⋅ F ξn ⎤ dr
⎦
=0 by the self adjoint property
3. For n ≠ m, ωn2 ≠ ω2m
∫ρξ
n
⋅ ξm dr = 0 orthogonal property
4. For n=m choose
∫ρξ
2
m
dr = 1 orthonormal
Spectrum of E
In general F exhibits both discrete eigenvalues and continua
⎛
F⎞
Spectrum: −ω2ρ ξ = F ( ξ ) → ξ = ⎜ ω2 + ⎟
ρ⎠
⎝
⎛
F⎞
The points where ⎜ ω2 + ⎟
ρ⎠
⎝
−1
⎡⎣initial conditions⎤⎦
−1
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
do not exist define the spectrum of F
Lecture 14
Page 7 of 8
Examples
Continua significantly complicate MHD analysis for general initial value problems.
They require more than just picking up the pole contributions from the displace
transform.
However the continua lie on stable side of the spectrum and thus do not affect
stability.
Accumulation points: these provide a simple necessary condition for stability.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 14
Page 8 of 8