Intermediate Nonlinear Development of a Line-tied g-mode

Transcription

1 Intermediate Nonlinear Development of a Line-tied g-mode Ping Zhu University of Wisconsin-Madison In collaboration with C. C. Hegna and C. R. Sovinec (UW-Madison) A. Bhattacharjee and K. Germaschewski (U. New Hampshire) g-mode (fusion): a.k.a. magnetic Rayleigh-Taylor instability Driven by density gradient and gravity g (g ρ < 0) A prototype model for ballooning instabilities in magnetized plasmas Examples: ELMs in tokamaks, substorms in magnetotail 48th APS-DPP Meeting, November 2, 2006, Philadelphia, PA p.1/22

2 ELM Onset: Breaching of Peeling-Ballooning Stability Boundaries Peeling-Ballooning Boundary for a JET-like Pedestal [Snyder, Wilson, Ferron, Lao, Leonard, Osborne, Turnbull, Mossessian, Murakami, and Xu, Phys. Plasmas (2002)] Nonlinear Line-tied g-mode p.2/22

3 Ballooning Structure Continues to Dominate in Nonlinear ELMs NIMROD Simulations of ELMs in a DIII-D-like Equilibrium [D.P. Brennan, E.D. Held, S.E. Kruger, A.Y. Pankin, D.D. Schnack and C.R. Sovinec, UW-CPTC 05-8 (2005)] Nonlinear Line-tied g-mode p.3/22

4 Line-tied g-mode is a Prototype Model for Ballooning Instability Questions on nonlinear ballooning instability in context of ELMs What is the major and most relevant nonlinear regime? Nonlinear growth rate and (terminal/saturation) magnitude? Line-tied g-mode: similar physics, simpler geometry Linear ballooning equation of a torus ω 2 F 2 Φ = η (F 2 η Φ) + β [κ v + q (η θ k )κ g ]Φ, F k Interchange drive κ p > 0 RT drive g ρ < 0. Line-tied boundary condition is to provide nonzero k (ballooning). First step towards the understanding of nonlinear ballooning instability. Nonlinear Line-tied g-mode p.4/22

5 Shearless Slab Configuration of Line-tied Flux Tubes z x y ρ 0 ρ 0 g 0 ( κ) B x ( ) d p dx 0 + B = ρ 0 g ˆx, g = gˆx, B 0 = B 0 ẑ ρ 0 (x 0 ) = ρ c + ρ h tanh (x 0 + L c )/L ρ, p 0 (x 0 ) = ρ 0 (x 0 ) L ρ pedestal width; 2ρ h pedestal height Nonlinear Line-tied g-mode p.5/22

6 We Have 2 Small Parameters ɛ ξ L z 1, n 1 k k 1 Nonlinear ballooning expansion ξ( nx 0, ny 0, z 0, t) = ɛ i n j 2 i=1 j=0 ( ˆxξ x{i,j} + ŷ ) ξ y{i,j} + ẑξ z{i,j}. n We expand ideal MHD equation (in Lagrangian form) accordingly [Pfirsch and Sudan, Phys. Fluids B (1993)] ρ 0 J 0R 2 ξ t 2 = [ ] p 0 0 J γ + (B 0 0 R) 2 [ ( )] B0 2J R J B0 0 J 0R + ρ 0 J 0R g where R(r 0, t) = r 0 + ξ(r 0, t), 0 = r 0, J(r 0, t) = 0 R. Nonlinear Line-tied g-mode p.6/22

7 Evolution Through Linear and Nonlinear Regimes Outline of the rest of this talk: Linear regime: ɛ n 1 (definition: ɛ ξ L z 1, 1st nonlinear regime: ɛ n 1 detonation theory comparison with simulation 2nd nonlinear regime: ɛ n 1/2 intermediate nonlinear theory comparison with simulation n 1 k k 1) Nonlinear Line-tied g-mode p.7/22

8 In Regime ɛ n 1 We Recover Linear Mode Equations To leading order, linear mode equations have local ballooning structure: 2 t ξ x{1,0} = Γ 2 A 2 z ξ x{1,0} + 2 t ξ z{1,0} = where [ g L ρ + Γ 2 A g 2 ] ξ x{1,0} + (1 + γβ) c 2 s 1 + γβ 2 z ξ z{1,0} g 1 + γβ zξ x{1,0}, Γ 2 A = B2 ρ, L ρ = ρ ρ, c2 s = γp ρ, and β = p B 2. g 1 + γβ zξ z{1,0}, Near marginal stability, above eqns reduce to one local eigenmode eqn [Zweibel and Bruhwiler, Astrophys. J. (1992); Cowley and Artun, Phys. Rep. (1997)]: ( g zξ 2 x{1,0} + L ρ Γ 2 A ) + g2 c 2 sγ 2 ξ x{1,0} g2 A c 2 sγ 2 A ξx{1,0} 1 + γβ = Γ2 Γ 2 ξ x{1,0}, A where ξx{1,0} L 1 z Lz 0 dzξ x{1,0}. Nonlinear Line-tied g-mode p.8/22

9 Next 2 Higher Orders of n 1/2 Recover Global Envelope Equation Local eigenmode equation LH(z) = (Γ 2 /Γ 2 A )H(z) only determines profile along field lines ξ x{1,0} = ξ x(1) (x, y, τ)h(z), (τ = n 1 2 t) envelope equation for ξ x(1) (x, y, τ) determines global ballooning structure across field lines [Connor, Hastie, & Taylor, Proc. R. Soc. Lond. A. (1979); Dewar & Glasser, Phys. Fluids (1983)] C 0 Γ 2 A where C 0 = H 2 + g2 c 4 s τ 2 yξ 2 nγ 2 x(1) = C 1 Γ 2 yξ 2 x(1) + C 2 xξ 2 x(1), A ( z ) 2 dz (H H ), C 1 = H 2, C 2 = (H ) 2, and H dh/dz. 0 Nonlinear Line-tied g-mode p.9/22

10 Linear Mode Pattern Has 3 Spatial Scales (MHD Simulation) Contour of flow component in the direction of g and ρ Mode width in 3 directions: l y = L y = 0.1, l z = L z = 128, l x L ρ = 5 Nonlinear Line-tied g-mode p.10/22

11 When ɛ n 1 Nonlinearity Modifies Envelope Equation When ɛ n 1, plasma is incompressible at lowest order: ξ = J {1, 1} = x ξ x{1,0} + y ξ y{1,0} = 0 Local eigenmode equation for ξ x{1,0} remains same as linear, so that ξ x{1,0} = ξ x(1) (x, y, τ)h(z), (τ = n 1 2 t) We call regime ɛ n 1 CA regime; it is governed by the nonlinear envelope equation ( CA equation ) [Cowley and Artun, Phys. Rep. 1997]. C 0 Γ 2 τ 2 2 nγ 2 yξ x(1) = C 1 A Γ 2 yξ 2 x(1) + C 2 xξ 2 x(1) + C 3 xξ 2 x(1) 2 2 yξ x(1) + A }{{} linear envelope equation explosive {}}{ C 4 2 yξ 2 x(1) }{{} ɛ n 1 regime nonlinearity Nonlinear Line-tied g-mode p.11/22

12 Nonlinear Envelope (CA) Equation Predicts Finite Time Singularity [Cowley and Artun, Phys. Rep. (1997)] Left: Growth of ξ max x Φ max 150 (t s t) 2.15, where t s = Right: Two dimensional picture of ξ x Φ(x, y, t) at t = 10.6 in the asymptotic stage approaching the singularity. Nonlinear Line-tied g-mode p.12/22

13 Direct MHD Simulations Have Not Found Finite Time Singularity Direct MHD simulations with BIC a and NIMROD b have found that the mode remains bounded in magnitude throughout the linear and nonlinear phase, with a slightly lower growth than the linear phase. [Zhu, Bhattacharjee, and Germaschewski, Phys. Rev. Lett. (2006); Zhu, Hegna, and Sovinec, Phys. Plasmas (2006)] Direct simulations have shown that the nonlinear stage is characterized with the formation of finger pattern in y-direction and discontinous structure in x-direction. This was also found in earlier numerical solutions of the CA equation. a Ballooning-Interchange-Code (Developed at CMRS, UNH) b Sovinec, Glasser, Barnes, Gianakon, Nebel, Kruger, Schnack, Plimpton, Tarditi, Chu, and the NIMROD team, J. Comput.Phys. (2004) Nonlinear Line-tied g-mode p.13/22

14 BIC Simulation: Nonlinear Growth and Pattern velocity max e-4 1e-6 u x u z u y t1 t2 t3 1e time (τ A ) n 1 = k z /k y = Γ 2 /Γ 2 A = (Γ2 A B2 /ρ) ξ x0 /L z n 1 L x = 64, L y = 0.1, L z = 128 grid size: Nonlinear Line-tied g-mode p.14/22

15 Rayleigh-Taylor (RT) Finger Pattern in NIMROD Simulation y (0.1) initial spectrum: n y = z x RT-finger width (in y): 0.1 RT-finger length (in x): 5 Flux tube length (in z): 128 pressure contour -25 t=70 P finger width ~ 0.1 finger length ~ 5 t=70 pressure P Mesh: 64x16 2D elements in x z plane, with biquadratic polynomials; in y direction, 5 Fourier modes included: 0 n y 5. RT finger width: L y = 0.1; finger length: L ρ = 5 Nonlinear Line-tied g-mode p.15/22

16 Mode Scales and Regimes in Terms of n in Simulations Characteristic scale parameter: n k /k = k y /k z. Equilibrium spatial scales: L z = 128 flux tube length, L ρ ( ) 1 d ln ρ = 5 pedestal width dx High-n mode spatial scales: l i mode width in i = x, y, z directions l y : l x : l z = 1 n : 1 n : 1 = k z : k x : k y for n = 1280, l z = L z = 128, l y = 0.1 l x = l z n = l y l z = 3.6 intermediate scale ɛ n 1 (CA) regime: ξ x l z n l y 0.1 mode width in y-direction; ɛ n 1/2 regime: ξ x l z l x 3.6 mode width in x-direction. n Nonlinear Line-tied g-mode p.16/22

17 Transition from (CA) Regime ɛ n 1 to Regime ɛ n 1/2 1 ε=n -1/2 (Int) 0.1 (u x ) max 0.01 ε=n -1 (CA) t1 t2 t3 case a_rtp032105x time (τ A ) t t 1 46: ɛ n 1, u x 0.008, ξ x 0.1 mode width in y; t t 2 80: ɛ n 1/2, u x 0.3, ξ x 3.6 mode width in x. t1 < t < t2: finger formation initiates during the transition; t2 < t < t3: finger pattern becomes prominant as the mode proceeds through the regime ɛ n 1/2. Nonlinear Line-tied g-mode p.17/22

18 When ɛ n 1/2 Mode Transitions to Intermediate Nonlinear Regime Definition of Intermediate Nonlinear Regime: ξ x l x convection in x-direction mode width in x-direction. In this nonlinear regime, coupling to sound wave physics becomes important. CA equation, and its detonation solution, valid for CA regime when ɛ n 1, does not apply in this intermediate nonlinear regime. Nonlinear Line-tied g-mode p.18/22

19 When ɛ n 1/2 Mode Transitions to Intermediate Nonlinear Regime Definition of Intermediate Nonlinear Regime: ξ x l x convection in x-direction mode width in x-direction. In this nonlinear regime, coupling to sound wave physics becomes important. CA equation, and its detonation solution, valid for CA regime when ɛ n 1, does not apply in this intermediate nonlinear regime. ɛ n 1 ɛ n 1 ɛ n 1 2. Γ 2 /Γ 2 A O(n 1 ) Γ 2 /Γ 2 A O(n 1 2 ) linear regime Cowley-Artun regime intermediate nonlinear regime Nonlinear Line-tied g-mode p.18/22

20 Governing Eqns for Intermediate Nonlinear Regime Are Derived a At lowest order of n 1/2, Lagrangian compression is no longer zero due to enhanced nonlinearity x ξ x(0) + y ξ y(0) = [ξ x(0), ξ y(0) ] x ξ x(0) y ξ y(0) y ξ x(0) x ξ y(0) Next order equations similar to linear mode in form (γp + B 2 )J 1 2 B2 z ξ z(0) ρgξ x(0) = F 1 (z, t), ρ 2 t ξ z(0) = γp z J 1 2 ρg zξ x(0), At the third order, nonlinearity appears explicitly ρ y 2 t ξ x(0) + ρ[ξ x(0), 2 t ξ x(0) ] ) = y (B 2 2 zξ x(0) + ρ gξ x(0) + ρgj 1 2 +ρg[ξ x(0), J 1 2 ] + B2 [ξ x(0), 2 zξ x(0) ] a Zhu, Hegna, and Sovinec, Phys. Plasmas (2006) Nonlinear Line-tied g-mode p.19/22

21 Governing Eqns for Intermediate Nonlinear Regime Are Derived a Denote the linear local ballooning equations for g-mode as 2 t ξ x(0) = L x (ξ x(0), ξ z(0) ), 2 t ξ z(0) = L z (ξ x(0), ξ z(0) ). The governing equations for ɛ n 1/2 regime can be written as y ( 2 t ξ x(0) L x (ξ x(0), ξ z(0) ) ) + [ξ x(0), 2 t ξ x(0) ] = gb2 γp + B 2 [ξ x(0), z ξ z(0) ] + B2 ρ [ξ x(0), 2 zξ x(0) ], 2 t ξ z(0) = L z (ξ x(0), ξ z(0) ). Nonlinearity directly modifies local ballooning mode equations. The governing equations are solved numerically and compared with direct MHD simulations in next two slides. a Zhu, Hegna, and Sovinec, Phys. Plasmas (2006) Nonlinear Line-tied g-mode p.19/22

22 Theory Agrees with Simulation in Linear and Nonlinear Regimes 1 ε=n -1/2 (Int) 1 ε=n -1/2 (Int) (u x ) max ε=n -1 (CA) theory simulation time (τ A ) (u x ) max 0.01 ε=n -1 (CA) theory simulation time (τ A ) Left: Growth of x-component of velocity; Right: Zoom-in of nonlinear phase The numerical solution and the direct MHD simulation are set up with the same equilibrium, initial and boundary conditions. This case has a short linear phase due to a not-too-small u x0 = Nonlinear Line-tied g-mode p.20/22

23 A Case with Smaller Initial Perturbation: Agreement Stands velocity max e-04 1e-05 1e-06 1e-07 ε=n -1 (CA) ε=n -1/2 (Int) u x (theory) u x (simulation) u z (theory) u z (simulation) 1e time (τ A ) velocity max u x (theory) u x (simulation) u z (theory) u z (simulation) ε=n -1 (CA) ε=n -1/2 (Int) time (τ A ) Left: Growth of u x and u z ; Right: Zoom-in of nonlinear phase This case has a substantial linear phase (u x0 = 10 5 ). Theory and simulation have different initial transient phases. Good agreement for both x and z components of flow velocity. Intermediate regime: major nonlinear regime of line-tied g-mode. Nonlinear Line-tied g-mode p.21/22

24 Summary and Discussion A line-tied g-mode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). Nonlinear Line-tied g-mode p.22/22

25 Summary and Discussion A line-tied g-mode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). ɛ n 1 : Cowley-Artun regime Convection in x mode width in y (ξ x l y ); Local eigenmodes along each field lines remain same as linear; Nonlinearity modifies the envelope equation CA equation; CA equation predicts finite-time singularity; Finite time singularity has not been observed in direct MHD simulations. Nonlinear Line-tied g-mode p.22/22

26 Summary and Discussion A line-tied g-mode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). ɛ n 1/2 : Intermediate Nonlinear Regime Convection in x mode width in x (ξ x l x ); Nonlinearity directly modifies the growth of local eigenmodes; sound wave physics becomes present; The linear and nonlinear growth obtained from the intermediate nonlinear equations agree with direct MHD simulations; Intermediate regime is further developed into late nonlinear phase: thin finger pattern and discontinuous structures. Nonlinear Line-tied g-mode p.22/22

27 Summary and Discussion A line-tied g-mode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). ɛ n 1/2 : Intermediate Nonlinear Regime Convection in x mode width in x (ξ x l x ); Nonlinearity directly modifies the growth of local eigenmodes; sound wave physics becomes present; The linear and nonlinear growth obtained from the intermediate nonlinear equations agree with direct MHD simulations; Intermediate regime is further developed into late nonlinear phase: thin finger pattern and discontinuous structures. Late nonlinear phase: MHD model may not apply Non-ideal, 2-fluid effects may play controlling roles Plasmoid/blob formation ( EME : Edge Mass Ejection?) Nonlinear Line-tied g-mode p.22/22