Boundary induced amplification and nonlinearinstability of interchange modes
Jupiter Bagaipo, A. B. Hassam
Institute for Research in Electronics and Applied PhysicsUniversity of Maryland, College Park
Stability, Energetics, and Turbulent Transportin Astrophysical, Fusion, and Solar Plasmas
08-12 April 2013
Outline
I IntroductionI Ideal MHD interchange modeI MotivationI Two-dimensional nonlinear instability result
I Boundary induced instabilityI AmplificationI Nonlinear instability
I Conclusion and future work
Interchange mode
B
g∆p
Growth rate:
γ2g ≡ gρ′0ρ0
Toroidal device
r
v||
F ~ v|| /r
p1
p2
p3
2
Unstable when:p1 > p2 > p3
Transverse field stabilization
B
g∆p
Dispersion relation:ω2 = k2V 2
Ay − γ2g
Motivation
Behaviour at marginal stability for ideal interchange modes isimportant in:
I Tokamaks,
I Reversed Field Pinches,
I Stellarators,
I etc...
Understanding the tradeoff between the deviation from marginalityand residual convection is useful for B-field coil design instellarators.
Previous result
ÑÓ
Ρ
BÓ
0
gÓ
a
-a
Ρ�
x
y
B0 = Bc + b, ρ ∝ A ∼ ∂tγg∼∣∣∣∣ bBc
∣∣∣∣1/2 � 1
1
k2A = −2bA+
1
4
k2 − 3
k2 + 1A3
Amplitude dependent stability
E0 =1
2k2A2 + U(A), U(A) = bA2 − 1
16
k2 − 3
k2 + 1A4
E0
U HAL
Ac = 2√
2
√k2 + 1
k2 − 3× b1/2
Numerical marginal conditions amplitude test
Linearly stable
∆B/Bc ≈ 0.04%
Small amplitude
a0 = 10−4
Linearly stable
∆B/Bc ≈ 0.04%
Large amplitude
a0 = 10−2
Numerical simulation result1
Stable runs are marked with a blue circle and unstable runs with ared ’x’. The theoretical boundary is shown as the line a0 ∝
√b.
1Phys. Plasmas 18 122103 (2011).
Rippled boundary problem
ÑÓ
Ρ
BÓ
0
gÓ
a
-a
∆
x
y
ρ′0 > 0 and B0 constant
δ
a� 1
Reduced equations in 2D
Complete, nonlinear set:
∂tψ = {ψ,ϕ}z·~∇⊥×ρ(∂t~u + {ϕ, ~u}) = {ψ,∇2
⊥ψ}+ g∂yρ
~u = z×∇⊥ϕ~B⊥ = z×∇⊥ψ
Where, in general, ρ = ρ(ψ) and
{f, h} ≡ ∂xf∂yh− ∂yf∂xh.
“Simple-minded” linear calculation
Boundary condition:
At x = ±a− δ cos(ky)
∂yψ = −δ k sin(ky)∂xψ
Quasistatic equilibrium:
ψ = B0x+ ψ
(B20∇2⊥ + gρ′0)∂yψ = −B0{ψ,∇2
⊥ψ}
To lowest order in δ/a:
ψ =δB0
cos(kxa)cos(kxx) cos(ky)
k2x =gρ′0B2
0
− k2
Boundary induced amplification
Critical condition:
B0 → Bc when kx → kc ≡ π/2aB2
c (k2c + k2) ≡ B2ck
2⊥ = gρ′0
Amplification scaling:
B0 = Bc + b and kx = kc −∆kx
b/Bc ∼ ∆kx/kc � 1
To lowest order in b/Bc:
ψ ≈ δ/a
b/Bc
kck2⊥Bc cos(kxx) cos(ky)
Core penetration
Dirac delta gradient:
ρ′0(x) = ρ′0aδ(x)
Solution:
ψ = B0δka cosh(kx)− 1
2κ2a2 sinh(k|x|)
ka cosh(ka)− 12κ
2a2 sinh(ka)cos(ky), κ2 ≡ gρ′0
B20
∆
b
∆
a0
Ρ0¢HxL
Ψ�
k®¥
Ψ�
k®0
Limits:
ψk→0 ∝ B0δ1− 1
2κ2a|x|
1− 12κ
2a2
ψk→∞ ∝ 2B0δe−ka cosh(kx)
Global amplification
∆
b
∆
a0
Ρ0¢HxL
Ψ�
0
Ψ�
Nonlinear scaling and expansion
Previous scaling:
|ψ/ψ0| ∼ ε ≡ (b/Bc)1/2
⇒ δ
a∼ ε3
Expand ψ in ε:
ψ = ψ0 + ψ1 + ψ2 + ψ3 + · · ·ψ0 = (Bc + b)x
ψi+1
ψi∼ ε
First order
Equation:
B2c (∇2
⊥ + k2⊥)∂yψ1 = 0
∂yψ1|x=±a = 0
Solution:
ψ1 =
[A+
δ/a
b/Bc
kck2⊥Bc
]cos(kcx) cos(ky)
Where A is the plasma response.
A
Bc/kc∼ ε
Second order
Equation:
B2c (∇2
⊥ + k2⊥)∂yψ2 = −Bc{ψ1,∇2⊥ψ1}
= 0
∂yψ2|x=±a = 0
Solution:
ψ2 = −1
4
kcBc
[A+
δ/a
b/Bc
kck2⊥Bc
]2sin(2kcx)
Third order
Equation:
B2c (∇2
⊥ + k2⊥)∂yψ3 + 2Bcb∇2⊥∂yψ1 = −Bc
({ψ1,∇2
⊥ψ2}+{ψ2,∇2
⊥ψ1})
∂yψ3|x=±a = −δBck sin(ky)
Secular term:
ψ3 = Bcδ
ax sin(kcx) cos(ky)
−2BcbA+k2c4
k2 − 3k2ck2⊥
[A+
δ/a
b/Bc
kck2⊥Bc
]3= 0
Time evolution
Time dependence:
A = A(t) where τA∂t ∼ εkeep δ “small”
Result:
1
k2A = −2bA+
1
4
k2 − 3
k2 + 1
[A+
2
π
δ
b
1
k2 + 1
]3Normalized with kc, Bc, and VAc equal to 1.
Boundary induced nonlinear instability2
U(A; δ) = bA2 − 1
16
k2 − 3
k2 + 1
[A+
2
π
δ
b
1
k2 + 1
]4U HA;∆=0L
U HA;∆<∆cL
U HA;∆>∆cL
δc =π
2
√32
27
(k2 + 1)3
k2 − 3× b3/2
2Phys. Plasmas (Letters) 20 020704 (2013).
Summary and conclusionsIn marginally stable interchange mode systems boundaryperturbations...
1. ...are amplified:ψ ∝ δ/b
2. ...induce nonlinear instability.
Therefore, marginal system is highly sensitive to boundaryperturbations:
δc ∝ b3/2
Future workEnergy principle generalization:
I δW with nonlinear effects
I δW with boundary perturbations
Realistic geometry (shear, curvature, etc)
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