Intermediate Nonlinear Development of a Linetied gmode


 Liliana Rice
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1 Intermediate Nonlinear Development of a Linetied gmode Ping Zhu University of WisconsinMadison In collaboration with C. C. Hegna and C. R. Sovinec (UWMadison) A. Bhattacharjee and K. Germaschewski (U. New Hampshire) gmode (fusion): a.k.a. magnetic RayleighTaylor 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 APSDPP Meeting, November 2, 2006, Philadelphia, PA p.1/22
2 ELM Onset: Breaching of PeelingBallooning Stability Boundaries PeelingBallooning Boundary for a JETlike Pedestal [Snyder, Wilson, Ferron, Lao, Leonard, Osborne, Turnbull, Mossessian, Murakami, and Xu, Phys. Plasmas (2002)] Nonlinear Linetied gmode p.2/22
3 Ballooning Structure Continues to Dominate in Nonlinear ELMs NIMROD Simulations of ELMs in a DIIIDlike Equilibrium [D.P. Brennan, E.D. Held, S.E. Kruger, A.Y. Pankin, D.D. Schnack and C.R. Sovinec, UWCPTC 058 (2005)] Nonlinear Linetied gmode p.3/22
4 Linetied gmode 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? Linetied gmode: 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. Linetied boundary condition is to provide nonzero k (ballooning). First step towards the understanding of nonlinear ballooning instability. Nonlinear Linetied gmode p.4/22
5 Shearless Slab Configuration of Linetied 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 Linetied gmode 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 ydirection and discontinous structure in xdirection. This was also found in earlier numerical solutions of the CA equation. a BallooningInterchangeCode (Developed at CMRS, UNH) b Sovinec, Glasser, Barnes, Gianakon, Nebel, Kruger, Schnack, Plimpton, Tarditi, Chu, and the NIMROD team, J. Comput.Phys. (2004) Nonlinear Linetied gmode p.13/22
14 BIC Simulation: Nonlinear Growth and Pattern velocity max e4 1e6 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 Linetied gmode p.14/22
15 RayleighTaylor (RT) Finger Pattern in NIMROD Simulation y (0.1) initial spectrum: n y = z x RTfinger width (in y): 0.1 RTfinger 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 Linetied gmode 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 Highn 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 ydirection; ɛ n 1/2 regime: ξ x l z l x 3.6 mode width in xdirection. n Nonlinear Linetied gmode 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 Linetied gmode 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 xdirection mode width in xdirection. 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 Linetied gmode 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 xdirection mode width in xdirection. 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 CowleyArtun regime intermediate nonlinear regime Nonlinear Linetied gmode 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 Linetied gmode p.19/22
21 Governing Eqns for Intermediate Nonlinear Regime Are Derived a Denote the linear local ballooning equations for gmode 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 Linetied gmode 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 xcomponent of velocity; Right: Zoomin 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 nottoosmall u x0 = Nonlinear Linetied gmode p.20/22
23 A Case with Smaller Initial Perturbation: Agreement Stands velocity max e04 1e05 1e06 1e07 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: Zoomin 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 linetied gmode. Nonlinear Linetied gmode p.21/22
24 Summary and Discussion A linetied gmode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). Nonlinear Linetied gmode p.22/22
25 Summary and Discussion A linetied gmode evolves sequentially through ɛ n 1, ɛ n 1, ɛ n 1/2,..., phases (ɛ mode amplitude, n k /k ). ɛ n 1 : CowleyArtun 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 finitetime singularity; Finite time singularity has not been observed in direct MHD simulations. Nonlinear Linetied gmode p.22/22
26 Summary and Discussion A linetied gmode 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 Linetied gmode p.22/22
27 Summary and Discussion A linetied gmode 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 Nonideal, 2fluid effects may play controlling roles Plasmoid/blob formation ( EME : Edge Mass Ejection?) Nonlinear Linetied gmode p.22/22