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)