Exponential Growth and Filamentary Structure of Nonlinear Ballooning - PowerPoint PPT Presentation

Exponential Growth and Filamentary Structure of Nonlinear Ballooning Instability 1 Ping Zhu in collaboration with C. C. Hegna and C. R. Sovinec University of Wisconsin-Madison Sherwood Conference Denver, CO May 5, 2009 1Research supported by U.S. Department of Energy.

Close correlation found between onset of type-I ELMs and peeling-ballooning instabilities ◮ Breaching of P-B stability boundary correlated to ELM onset (DIII-D) [Snyder et al. , 2002] .

Close correlation found between onset of type-I ELMs and peeling-ballooning instabilities ◮ P-B modes were identified in ELM precursors (JET) [Perez et al. , 2004] .

Close correlation found between onset of type-I ELMs and peeling-ballooning instabilities ◮ Filament structures persist after ELM onset (MAST) [Kirk et al. , 2006] .

Causal relation between ballooning instability and ELM dynamics remains unclear ◮ Questions on ELM dynamics: ◮ Is ELM onset triggered by ELM precursors? How? ◮ Is ballooning instability responsible for ELM precursor or ELM itself?

Causal relation between ballooning instability and ELM dynamics remains unclear ◮ Questions on ELM dynamics: ◮ Is ELM onset triggered by ELM precursors? How? ◮ Is ballooning instability responsible for ELM precursor or ELM itself? ◮ Questions on ballooning instability (this talk): ◮ How fast does it grow in nonlinear stage? ◮ How does ballooning mode structure evolve nonlinearly?

Different nonlinear regimes of ballooning instability can be characterized by the relative strength of nonlinearity ε in terms of n − 1 ◮ Nonlinearity and ballooning parameters n − 1 ∼ k � ε ∼ | ξ | ∼ λ ⊥ ≪ 1 , ≪ 1 L eq k ⊥ λ � ◮ For ε ≪ n − 1 , linear ballooning mode theory [Coppi, 1977; Connor, Hastie, and Taylor, 1979; Dewar and Glasser, 1983] ◮ For ε ∼ n − 1 , early nonlinear regime [Cowley and Artun, 1997; Hurricane, Fong, and Cowley, 1997; Wilson and Cowley, 2004; Cowley, Sherwood 2008] ◮ 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; ] ◮ For ε ≫ n − 1 / 2 , late nonlinear regime; analytic theory under development.

Outline 1. Nonlinear ballooning equations ◮ Formulation ◮ Analytic solution 2. Comparison with NIMROD simulations ◮ Simulation setup ◮ Comparison method ◮ Comparison results 3. Summary and Discussion

A Lagrangian form of ideal MHD is used to develop the theory of nonlinear ballooning instability � � J γ + ( B 0 · ∇ 0 r ) 2 J ∇ 0 r · ∂ 2 ξ ρ 0 p 0 ∂ t 2 = −∇ 0 2 J 2 � B 0 � B 0 �� + ∇ 0 r · J · ∇ 0 J · ∇ 0 r (1) ∂ r ( r 0 , t ) = r 0 + ξ ( r 0 , t ) , ∇ 0 = , J ( r 0 , t ) = |∇ 0 r | (2) where ∂ r 0 The full MHD equation can be further reduced for nonlinear ballooning instability using expansion in terms of ε and n − 1 ∞ ∞ � � ε i n − j 2 ξ ( i , j ) ( r 0 , t ) ξ ( r 0 , t ) = (3) i = 1 j = 0

Nonlinear ballooning expansion is carried out for general magnetic configurations with flux surfaces ◮ Clebsch coordinate system (Ψ 0 , α 0 , l 0 ) B = ∇ 0 Ψ 0 × ∇ 0 α 0 (4) ◮ Expansions are based on intermediate nonlinear ballooning ordering ε ≡ | ξ | / L eq ∼ n − 1 / 2 ≪ 1 ∞ � � √ 2 + e ∧ � n − j 2 + B ξ � e ⊥ ξ Ψ √ n ξ α ξ ( n Ψ 0 , n α 0 , l 0 , t ) = (5) 2 j j + 1 j 2 j = 1 ∞ √ � n − j 2 J j J ( n Ψ 0 , n α 0 , l 0 , t ) = 1 + (6) 2 j = 0 where e ⊥ = ( ∇ 0 α 0 × B ) / B 2 , e ∧ = ( B × ∇ 0 Ψ 0 ) / B 2 . ◮ The spatial structure of ξ (Ψ , α, l ) and J (Ψ , α, l ) is ordered to be consistent with linear ideal ballooning theory: Ψ = √ n Ψ 0 , α = n α 0 , l = l 0 .

The linear local ballooning operator will continue to play a fundamental role in the nonlinear dynamics Linear ideal MHD ballooning mode equation ρ∂ 2 t ξ = L ( ξ ) (7) where local ballooning operator L ( ξ ) ≡ B · ∇ 0 ( B · ∇ 0 ξ ) − ∇ 0 ( B · ∇ 0 B ) · ξ � �� 1 � B · ∇ 0 ξ � − 2 κ · ξ ⊥ − BB · ∇ 0 1 + γβ − 2 B · ∇ 0 B � � B · ∇ 0 ξ � − 2 κ · ξ ⊥ (8) 1 + γβ is an ODE operator along field line, with ξ = ξ � B + ξ ⊥ .

A set of nonlinear ballooning equations for ξ are described using the linear operator [Zhu and Hegna, 2008] 2 , ξ � ρ ( | e ⊥ | 2 ∂ α ∂ 2 t ξ Ψ 2 , ∂ 2 2 ]) = ∂ α L ⊥ ( ξ Ψ 2 + [ ξ 1 t ξ 1 2 ) + [ ξ 1 2 , L ( ξ 1 2 )] , (9) 1 1 1 t ξ � 2 , ξ � ρ B 2 ∂ 2 2 = L � ( ξ Ψ 2 ) (10) 1 1 1 L ⊥ ( ξ Ψ , ξ � ) ≡ e ⊥ · L ( ξ ) (11) B ∂ l ( | e ⊥ | 2 B ∂ l ξ Ψ ) + 2 e ⊥ · κ e ⊥ · ∇ 0 p ξ Ψ = + 2 γ p e ⊥ · κ � B ∂ l ξ � − 2 e ⊥ · κ ξ Ψ � , (12) 1 + γβ L � ( ξ Ψ , ξ � ) ≡ B · L ( ξ ) (13) � B ∂ l ξ � − 2 e ⊥ · κ ξ Ψ �� γ p � = B ∂ l , (14) 1 + γβ [ A , B ] ≡ ∂ Ψ A · ∂ α B − ∂ α A · ∂ Ψ B . (15)

The local linear ballooning mode structure continues to satisfy the nonlinear ballooning equations ◮ The nonlinear ballooning equations can be rearranged in the compact form nonlinear � �� � t ξ Ψ − L ⊥ ( ξ Ψ , ξ � ) [Ψ + ξ Ψ , ρ | e ⊥ | 2 ∂ 2 ] = 0 , (16) � �� � linear t ξ � − L � ( ξ Ψ , ξ � ) = 0 . ρ B 2 ∂ 2 (17) ◮ The general solution satisfies t ξ Ψ = L ⊥ ( ξ Ψ , ξ � ) + N (Ψ + ξ Ψ , l , t ) ρ | e ⊥ | 2 ∂ 2 (18) ◮ A special solution is the solution of the linear ballooning equations ( N = 0), for which the nonlinear terms in (16) all vanish.

Implications of the “linear” analytic solution ◮ The solution is linear in Lagrangian coordinates, but nonlinear in Eulerian coordinates ξ = ξ lin ( r 0 ) = ξ lin ( r − ξ ) = ξ non ( r ) .

Implications of the “linear” analytic solution ◮ The solution is linear in Lagrangian coordinates, but nonlinear in Eulerian coordinates ξ = ξ lin ( r 0 ) = ξ lin ( r − ξ ) = ξ non ( r ) . ◮ Perturbation developed from linear ballooning instability should continue to ◮ grow exponentially ◮ maintain filamentary spatial structure

Outline 1. Nonlinear ballooning equations ◮ Formulation ◮ Analytic solution 2. Comparison with NIMROD simulations ◮ Simulation setup ◮ Comparison method ◮ Comparison results 3. Summary and Discussion

Simulations of ballooning instability are performed in a tokamak equilibrium with circular boundary and pedestal-like pressure 0.08 0.06 p 0.04 0.02 3 0 2.5 2 1.5 q 1 0.5 ◮ Equilibrium from 0 2 1.5 ESC solver [Zakharov and 2 > <J || B/B 1 Pletzer,1999] . 0.5 ◮ Finite element mesh 0 0 0.2 0.4 0.6 0.8 1 in NIMROD [Sovinec et 1/2 ( Ψ p /2 π ) al. ,2004] simulation.

Linear ballooning dispersion is characteristic of interchange type of instabilities 0.25 0.2 0.15 γτ A 0.1 0.05 0 0 10 20 30 40 50 toroidal mode number n f_050808x01x02 ◮ Extensive benchmarks between NIMROD and ELITE show good agreement [B. Squires et al., Poster K1.00054, Sherwood 2009] .

Simulation starts with a single n = 15 linear ballooning mode 1000 total 1 n=0 n=15 0.001 kinetic energy n=30 1e-06 n=0 n=15 n=30 1e-09 total 1e-12 f_es081908x01a 1e-15 0 5 10 15 20 25 30 35 40 45 time ( µ s)

Isosurfaces of perturbed pressure δ p show filamentary structure ( t = 30 µ s , δ p = 168 Pa )

For theory comparison, we need to know plasma displacement ξ associated with nonlinear ballooning instability ◮ ξ connects the Lagrangian and Eulerian frames, r ( r 0 , t ) = r 0 + ξ ( r 0 , t ) (19) ◮ In the Lagrangian frame d ξ ( r 0 , t ) = u ( r 0 , t ) (20) dt ◮ In the Eulerian frame ∂ t ξ ( r , t ) + u ( r , t ) · ∇ ξ ( r , t ) = u ( r , t ) (21) ◮ ξ is advanced as an extra field in NIMROD simulations.

Lagrangian compression ∇ 0 · ξ can be more conveniently used to identify nonlinear regimes ◮ Nonlinearity is defined by ε = | ξ | / L eq , but L eq is not specific. ◮ Linear regime ( ε ≪ n − 1 ) ∇ 0 · ξ = ∇ · ξ ≪ 1 (22) ◮ Early nonlinear regime ( ε ∼ n − 1 ) Ψ ξ Ψ + λ − 1 α ξ α + λ − 1 λ − 1 � ξ � ∇ 0 · ξ ∼ (23) n 1 / 2 n − 1 + n 1 n − 3 / 2 + n 0 n − 1 ∼ n − 1 / 2 ≪ 1 . ∼ ◮ Intermediate nonlinear regime ( ε ∼ n − 1 / 2 ) Ψ ξ Ψ + λ − 1 α ξ α + λ − 1 λ − 1 � ξ � ∇ 0 · ξ ∼ (24) n 1 / 2 n − 1 / 2 + n 1 n − 1 + n 0 n − 1 / 2 ∼ 1 . ∼

The Lagrangian compression ∇ 0 · ξ is calculated from the Eulerian tensor ∇ ξ in simulations Transforming from Lagrangian to Eulerian frames, one finds ξ ( r 0 , t ) = ξ [ r − ξ ( r , t ) , t ] (25) ∂ ξ ∇ ξ = ∂ r � ∂ r � ∂ r − ∂ ξ · ∂ ξ = ∂ r ∂ r 0 = ( I − ∇ ξ ) · ∇ 0 ξ (26) The Lagrangian compression ∇ 0 · ξ is calculated from the Eulerian tensor ∇ ξ at each time step using ∇ 0 · ξ = Tr ( ∇ 0 ξ ) = Tr [( I − ∇ ξ ) − 1 · ∇ ξ ] . (27)

Exponential linear growth persists in the intermediate nonlinear regime of tokamak ballooning instability [Zhu, Hegna, and Sovinec, 2008] 100 10 (div 0 ξ ) max ~1 1 dimensionless unit 0.1 (div 0 ξ ) max <<1 0.01 exponential growth of ξ 0.001 0.0001 | ξ | max /a (div 0 ξ ) max 1e-05 1e-06 0 5 10 15 20 25 30 35 time ( τ A = µ s) Dotted lines indicate the transition to the intermediate nonlinear regime when ∇ 0 · ξ ∼ O ( 1 ) .