Filamentary Structure andExponential Growth of

Nonlinear Ballooning Instability 1

Ping Zhu

in collaboration withC. C. Hegna and C. R. Sovinec

University of Wisconsin-Madison

Plasma Physics SeminarMadison, WI

February 16, 2009

1Research supported by U.S. Department of Energy.

A ballooning instability in a tokamak (NIMROD simulation)

Filamentary structure persists during (type-I) edgelocalized modes (ELMs) in the MAST tokamak

[Kirk et al., 2006]

Both linear and nonlin-ear ELM phases.

ELM filaments resemble linear ballooning modestructure

MAST [Kirk et al. , 2006] ELITE [Snyder et al. , 2005]

NIMROD [Sovinec et al. , 2006]

I Characteristics of ELM filamentsI Toroidal mode number n ∼ 15− 20I Elongated structures aligned with field linesI Persist well into nonlinear phase

I Questions to address in theory:I Why do nonlinear ELM and ballooning filaments resemble

linear ballooning structure?I What is the temporal evolution in the nonlinear regime?

Different nonlinear regimes of ballooninginstability are characterized by the relativestrength of the nonlinearity with powers of n−1

I Nonlinearity and ballooning parameters

ε ∼ |ξ|Leq

� 1, n−1 ∼k‖k⊥

∼Ly

Lz� 1

I For ε � n−1, linear ballooning mode theory [Coppi, 1977; Connor,

Hastie, and Taylor, 1979; Dewar and Glasser, 1983]

I For ε ∼ n−1, early nonlinear regime [Cowley and Artun, 1997; Hurricane,

Fong, and Cowley, 1997; Wilson and Cowley, 2004]

I For ε ∼ n−1/2, intermediate nonlinear regime → this talk[Zhu, Hegna, and Sovinec, 2006; Zhu et al. , 2007; Zhu and Hegna, 2008; Zhu, Hegna, and Sovinec, 2008; ]

I For ε � n−1/2, late nonlinear regime; analytic theory underdevelopment.

Intermediate nonlinear regime was previouslyidentified for line-tied g mode [Plasma Physics Seminar 2006]

z

x

y

g∇ρ0

Left: Equilibrium configuration; Right: ux contour of line-tied g-moded

dx0

(p0 +

B20

2

)= ρ0g · x , g = −gx , B0 = B0z

ρ0(x0) = ρc + ρh tanh (x0 + Lc)/Lρ, p0(x0) = ρ0(x0)Lρ → pedestal width; 2ρh → pedestal height

Transition from early nonlinear regime ε ∼ n−1 tointermediate nonlinear regime ε ∼ n−1/2

[Zhu et al. , 2007]

0 10 20 30 40 50 60 70 80 90time (τ

A)

0.0001

0.001

0.01

0.1

1

(ux) m

ax

case a_rtp032105x01

ε=n-1

(CA)

ε=n-1/2

(Int)

t1 t2 t3

I (ux)max(t = 0) = 10−3

I t1 ∼ 46: ε ∼ n−1, ux ∼ 0.008, ξx ∼ 0.1 → mode width in y ;t2 ∼ 80: ε ∼ n−1/2, ux ∼ 0.3, ξx ∼ 3.6 → mode width in x .

I t1 <∼ t <∼ t2: finger formation initiates during the transition;t2 <∼ t <∼ t3: finger pattern becomes prominent as themode proceeds through the regime ε ∼ n−1/2.

Exponential growth persists in intermediatenonlinear regime of the line-tied g-mode [Zhu et al. , 2007]

0 20 40 60 80 100 120time (τ

A)

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

velo

city

max

ux (theory)

ux (simulation)

uz (theory)

uz (simulation)

ε=n-1

(CA)

ε=n-1/2

(Int)

I (ux)max(t = 0) = 10−5

90 100 110 120 130time (τ

A)

0.001

0.01

0.1

1

velo

city

max

ux (theory)

ux (simulation)

uz (theory)

uz (simulation)

ε=n-1

(CA)

ε=n-1/2

(Int)

I Zoom-in of the left figure

I Simulation results agree with numerical solution of thenonlinear line-tied g mode equations.

I Both simulation and numerical solution of theory indicateexponential-like nonlinear growth. Why?

Outline

1. Nonlinear ballooning equationsI FormulationI Analytic solution

2. Comparison with NIMROD simulationsI Simulation setupI Comparison methodI Comparison results

3. Summary and Discussion

A Lagrangian form of ideal MHD is used to developthe theory of nonlinear ballooning instability

ρ0

J∇0r · ∂2ξ

∂t2 = −∇0

[p0

Jγ+

(B0 · ∇0r)2

2J2

]

+∇0r ·[

B0

J· ∇0

(B0

J· ∇0r

)]+

ρ0

J∇0r · g (1)

where r(r0, t) = r0 + ξ(r0, t),∇0 =∂

∂r0, J(r0, t) = |∇0r| (2)

The full MHD equation can be further reduced for nonlinearballooning instability using expansion in terms of ε and n−1

ξ(√

nx0, ny0, z0, t) =∞∑

i=1

∞∑j=0

εin−j2

(xξx{i,j} +

y√n

ξy{i,j} + zξz{i,j}

).

(3)

Nonlinear ballooning expansion is carried out forgeneral magnetic configurations with flux surfaces

I Clebsch coordinate system (Ψ0, α0, l0)

B = ∇0Ψ0 ×∇0α0 (4)

I Expansions are based on intermediate nonlinearballooning ordering ε ≡ |ξ|/Leq ∼ n−1/2 � 1

ξ(√

nΨ0, nα0, l0, t) =∞∑

j=1

n−j2

(e⊥ξΨ

j2

+e∧√

nξα

j+12

+ Bξ‖j2

)(5)

J(√

nΨ0, nα0, l0, t) = 1 +∞∑

j=0

n−j2 J j

2(6)

where e⊥ = (∇0α0 × B)/B2, e∧ = (B×∇0Ψ0)/B2.I The spatial structure of ξ(Ψ, α, l) and J(Ψ, α, l) is ordered

to be consistent with linear ideal ballooning theory:Ψ =

√nΨ0, α = nα0, l = l0.

The linear local ballooning operator will continueto play a fundamental role in the nonlineardynamics

Linear ideal MHD ballooning and g modes obey

ρ∂2t ξ = L(ξ) (7)

where

L(ξ) ≡ B · ∇0(B · ∇0ξ)− [∇0(B · ∇0B) +∇0(ρg)] · ξ

−BB · ∇0

{1

1 + γβ

[B · ∇0ξ

‖ − ρB · gB2 ξ‖ −

(2κ +

ρgB2

)· ξ⊥

]}−2B · ∇0B + ρg

1 + γβ

[B · ∇0ξ

‖ − ρB · gB2 ξ‖ −

(2κ +

ρgB2

)· ξ⊥

](8)

is an ODE operator, with ξ = ξ‖B + ξ⊥ [Hurricane, Fong, and Cowley, 1997].

A set of nonlinear ballooning equations for ξ aredescribed using the linear operator [Zhu and Hegna, 2008]

ρ(|e⊥|2∂α∂2t ξΨ

12

+ [ξ 12, ∂2

t ξ 12]) = ∂αL⊥(ξΨ

12, ξ‖12) + [ξ 1

2,L(ξ 1

2)], (9)

ρB2∂2t ξ‖12

= L‖(ξΨ12, ξ‖12) (10)

L⊥(ξΨ, ξ‖) ≡ e⊥ · L(ξ) (11)= B∂l(|e⊥|2B∂lξ

Ψ) + 2e⊥ · κe⊥ · ∇0pξΨ

+2γpe⊥ · κ

1 + γβ

(B∂lξ

‖ − 2e⊥ · κξΨ)

, (12)

L‖(ξΨ, ξ‖) ≡ B · L(ξ) (13)

= B∂l

[γp

1 + γβ

(B∂lξ

‖ − 2e⊥ · κξΨ)]

, (14)

[A, B] ≡ ∂ΨA · ∂αB− ∂αA · ∂ΨB. (15)

The local linear ballooning mode structure andgrowth continue to satisfy the nonlinearballooning equations in Lagrangian space

I The nonlinear ballooning equations can be written in thecompact form[

Ψ + ξΨ, ρ|e⊥|2∂2t ξΨ − L⊥(ξΨ, ξ‖)

]= 0, (16)

ρB2∂2t ξ‖ − L‖(ξΨ, ξ‖) = 0. (17)

I The general solution satisfies

ρ|e⊥|2∂2t ξΨ = L⊥(ξΨ, ξ‖) + N(Ψ + ξΨ, l , t), (18)

ρB2∂2t ξ‖ = L‖(ξΨ, ξ‖). (19)

I A special solution to the nonlinear ballooning equations isthe solution of the linear ballooning equations

N(Ψ, l , t) = 0, where Ψ = Ψ + ξΨ. (20)

Implications of the analytic solution

I The solution is linear in Lagrangian coordinates, butnonlinear in Eulerian coordinates

ξ = ξlin(r0) = ξlin(r− ξ) = ξnon(r).

I Linear ballooning mode operator requires solution havingfilamentary structure ξ ∼ A(Ψ, α)H(l)

I Linear ballooning mode structure gives exponential growth

∂2t ξ = Lξ = Γ2A(Ψ, α)H(l) = Γ2ξ.

I Perturbation developed from linear ballooning instabilityshould continue to

I grow exponentiallyI maintain filamentary spatial structure

Outline

1. Nonlinear ballooning equationsI FormulationI Analytic solution

2. Comparison with NIMROD simulationsI Simulation setupI Comparison methodI Comparison results

3. Summary and Discussion

Simulations of ballooning instability are performedin a tokamak equilibrium with circular boundaryand pedestal-like pressure

I Equilibrium fromESC solver [Zakharov and

Pletzer,1999]

I Finite element meshin NIMRODsimulation.

0

0.02

0.04

0.06

0.08

p0

0.51

1.52

2.5

3

q

0 0.2 0.4 0.6 0.8 1

(Ψp/2π)

1/2

0

0.5

1

1.5

2

<J ||B

/B2 >

Linear ballooning dispersion is characteristic ofinterchange type of instabilities

0 10 20 30 40 50toroidal mode number n

0

0.05

0.1

0.15

0.2

0.25γτ

A

f_050808x01x02

I Extensive benchmarks between NIMROD and ELITE showgood agreement [B. Squires et al., 2009].

Simulation starts with a single n = 15 linearballooning mode

0 5 10 15 20 25 30 35 40 45time (µs)

1e-15

1e-12

1e-09

1e-06

0.001

1

1000

kine

tic e

nerg

y

n=0n=15n=30total

elecd=ndiff=0,kvisc=kperp=25,nhypd=1

mx=20,my=32,poly=5,lphi=7

f_es081908x01a

Isosurfaces of perturbed pressure δp showfilamentary structure (t = 30µs, δp = 168Pa)

For theory comparison, we need to know plasmadisplacement ξ associated with nonlinearballooning instability

ξ connects the Lagrangian and Eulerian frames,

r(r0, t) = r0 + ξ(r0, t) (21)

In the Lagrangian frame

dξ(r0, t)dt

= u(r0, t) (22)

In the Eulerian frame

∂tξ(r, t) + u(r, t) · ∇ξ(r, t) = u(r, t) (23)

where u(r, t) is velocity field, ∂t = (∂/∂t)r, and ∇ = ∂/∂r. ξ isadvanced as an extra field in simulations.

Lagrangian compression ∇0 · ξ can be moreconveniently used to identify nonlinear regimes

I Nonlinearity is defined by |ξ|/Leq, but Leq is not specific.I Linear regime

∇0 · ξ = ∇ · ξ (24)

I Early nonlinear regime

∇0 · ξ ∼ λ−1Ψ ξΨ + λ−1

α ξα + λ−1‖ ξ‖ (25)

∼ n1/2n−1 + n1n−3/2 + n0n−1 ∼ n−1/2 � 1.

I Intermediate nonlinear regime

∇0·ξ ∼ λ−1Ψ ξΨ+λ−1

α ξα+λ−1‖ ξ‖ ∼ n1/2n−1/2+n1n−1+n0n−1/2 ∼ 1.

(26)I The Lagrangian compression is sensitive to nonlinearity:

matrix (I−∇ξ)−1 could become singular passing beyondintermediate nonlinear regime.

The Lagrangian compression ∇0 · ξ is calculatedusing the Eulerian tensor ∇ξ

Transforming from Lagrangian to Eulerian frames, one finds

ξ(r0, t) = ξ[r− ξ(r, t), t ] (27)

∇ξ =∂ξ

∂r

=

(∂r∂r− ∂ξ

∂r

)· ∂ξ

∂r0

= (I−∇ξ) · ∇0ξ (28)

The Lagrangian compression ∇0 · ξ is calculated from theEulerian tensor ∇ξ at each time step using

∇0 · ξ = Tr(∇0ξ) = Tr[(I−∇ξ)−1 · ∇ξ]. (29)

Exponential linear growth persists in theintermediate nonlinear regime of tokamakballooning instability [Zhu, Hegna, and Sovinec, 2008]

0 5 10 15 20 25 30 35time (µs)

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

|ξ|max

(m)

(div0ξ)

max

Int

Dotted line indicates the transition to the intermediate nonlinearregime when ∇0 · ξ ∼ O(1)

Lagrangian compression may also be able toidentify other nonlinear ballooning regimes

0 5 10 15 20 25 30 35 40 45time (µs)

1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

|ξ|max

(m)

(div0ξ)

max

Int

20x32,poly=5,lphi=7elecd=ndiff=0,kvisc=kperp=25,nhypd=1f_es081908x01a

I Intermediate nonlinear regime is entered ∼ 28µs.I Large ∇0 · ξ indicates transition to nonlinear regimes.

Perturbation energy grows with the linear growthrate into the intermediate nonlinear regime(vertical line)

0 5 10 15 20 25 30 35 40 45time (µs)

1e-15

1e-12

1e-09

1e-06

0.001

1

1000

kine

tic e

nerg

y

n=0n=15n=30total

elecd=ndiff=0,kvisc=kperp=25,nhypd=1

mx=20,my=32,poly=5,lphi=7

f_es081908x01a

Contours of plasma velocity and displacement ξ att = 5µs, linear phase

Re VR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VPhi

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiP

2.0 2.5 3.0 3.5 4.0-1

.0-0

.50.

00.

51.

0R

Z

Contours of plasma velocity and displacement ξ att = 30µs, intermediate nonlinear phase

Re VR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VPhi

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiP

2.0 2.5 3.0 3.5 4.0-1

.0-0

.50.

00.

51.

0R

Z

Contours of plasma velocity and displacement ξ att = 40µs, start of late nonlinear phase

Re VR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re VPhi

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiR

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiZ

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re XiP

2.0 2.5 3.0 3.5 4.0-1

.0-0

.50.

00.

51.

0R

Z

Distortions of flux surface are consistent withplasma displacement

Re P

2.0 2.5 3.0 3.5 4.0

-1.0

-0.5

0.0

0.5

1.0

R

Z

Re P

2.0 2.5 3.0 3.5 4.0-1

.0-0

.50.

00.

51.

0

R

Z

I Above: Total pressure contours.I Left: t = 30µs; Right: t = 40µs.

Total pressure contours on 4 poloidal slices show3D view of distorted pedestal (t = 40µs)

Total pressure isosurface shows nonlinearfilamentary structure (t = 40µs, p = 3602Pa)

Distribution of plasma displacement vectors (ξ)aligns with pressure isosurface ribbons (t = 40µs)

Zoom-in view of total pressure isosurface anddisplacement (ξ) vectors (t = 40µs)

Distribution of velocity (u) vector is different fromthat of displacement (ξ) vector at t = 40µs

Simulations started from multiple linear ballooningmodes also confirm theory prediction

1e-05

0.0001

0.001

0.01

0.1

1

10

100

|ξ|max

/a

(div0ξ)

max

0 10 20 30 40 50time (τ

A)

1e-8

1e-6

1e-4

1e-2

1

100

kine

tic e

nerg

y

Int

total n=15

n=30 n=0

1e-05

0.0001

0.001

0.01

0.1

1

10

100

|ξ|max

/a

(div0ξ)

max

0 10 20 30 40time (τ

A)

1e-8

1e-6

1e-4

1e-2

1

100

kine

tic e

nerg

y

n=0

n=1

n=6

n=11

n=13

n=15

total

Int

Summary

I There is an exponential growth phase of nonlinearballooning instability

I The growth rate is same as the linear growth rate.I The spatial structure is same as the linear mode (in

Lagrangian space).I This nonlinear phase is defined by ordering ξΨ ∼ λΨ.

I ImplicationsI May explain persistence of filamentary structures in ELM

experiments.I Nonlinear ELM behavior of pedestal may be largely

controlled by its linear properties (which is determined bythe configuration).

Discussion

I What are missing?I Nonlinearity

I What is the “nonlinearity depletion” mechanism?I Marginal unstable configuration (Γ ∼ 0) need nonlinear

filament envelope equation (detonation?)I Late nonlinear regimes: saturation, filament->blob?

I 2-fluid, FLR (finite Larmor radius) effectsI Edge shear flow effectsI RMP (resonant magnetic perturbation) effectsI Geometry (non-circular shape, divertor separatrix/X-point)I Peeling-ballooning couplingI ......