Negative energy modes in some models for plasma physics E. Tassi1, in collaboration with P.J. Morrison2 1 Centre de Physique Th´eorique,CNRS, Marseille, France 2 Department of Physics and Institute for Fusion Studies, University of Texas, Austin, USA 1. Introduction • Negative energy modes (NEMs) refer to spectrally stable modes of oscillations possessing negative energy (more precise definition later) • NEMs important because can be destabilized by dissipation • Intuitively, dissipation makes total energy decay ! result of amplitude of NEMs increasing • Also nonlinearity can destabilize equilibria with NEMs 2. NEMs in plasma physics NEMs in low-dimensional dynamical systems but also in models for continuous media In astrophysical and laboratory plasmas NEMs occur in several cases, e.g. • Streaming instabilities (Sturrock, 1958) • Resistive instabilities in magnetically confined plasmas (Greene and Coppi, 1965) • Vlasov-Maxwell (Morrison and Pfirsch, 1989/90/92 Correa-Restrepo and Pfirsch 1992/93/97) • Drift-kinetic equations (Throumoulopoulos and Pfirsch, 1996) • Two-stream instabilities (Kueny and Morrison, 1995 (a,b), Lashmore-Davies, 2007) • Ideal magnetohydrodynamics (Hirota and Fukumoto, 2008 (a,b)) • Magnetorotational instability (Ilgisonis et al., 2007, 2009, Khalzov et al., 2008) • Magnetosonic waves in the solar atmosphere (Joarder et al., 1997) 3. Hamiltonian approach to NEMs • Hamiltonian framework for NEMs : general and unambiguous definition of energy (Morrison and Kotschenreuther, 1989) • Based on Hamiltonian normal form N degree-of-freedom, linear, real Hamiltonian system z_ = JcAz; with z = (q1; ··· ; qN ; p1; ··· ; pN ) A constant 2N × 2N matrix 1 H = A zizj; quadratic Hamiltonian L 2 ij 0N IN Jc = canonical symplectic matrix −IN 0N 4. Normal form for linear Hamiltonian systems • Consider z =z ~ei!t +z ~∗e−i!t (and then drop the tilde) • i!αzα = JcAzα; α = 1; ··· ;N assume N distinct real eigenvalues !α ∗ •− !α are also eigenvalues, associated with zα T T −1 • Define h(α; β) := i!αzβ Ωzα = zβ Azα with Ω = Jc • One can show that h(α; β) = 0; if β 6= −α ∗T ∗T ∗ • h(−α; α) = zα Azα = i!αzα Ωzα is the energy of the mode (zα;!α; zα; −!α) 5. Mode signature • Can choose normalization constant for the eigenvectors so that ∗T zα Ωzα = ±2i: • Consider zα eigenvector associated with !α> 0 ∗T ∗ • If zα Ωzα = −2i then (zα;!α; zα; −!α) is a positive energy mode (PEM) ∗T ∗ • If zα Ωzα = 2i then (zα;!α; zα; −!α) is a negative energy mode ∗T • Indeed, for a PEM the energy is h(−α; α) = i!αzα Ωzα = 2!α> 0 6. Mode signature - II • For stable modes canonical transformation T :(Q1; ··· ;QN ;P1; ··· ;PN ) ! (q1; ··· qN ; p1; ··· pN ) leading to normal form of the Hamiltonian: N 1 X H = σ ! (P 2 + Q2 ) ; L 2 α α α α α=1 with σi 2 {−1; 1g and !α positive eigenvalues of the system • In the normal form stable modes of HL ! sum of harmonic oscillators with different frequencies • Modes with σ = −1 give negative contribution: these are NEMs • If eigenvalues !α and eigenvectors zα are known, the procedure for determining the transformation T is algorithmic 7. Model for electron temperature gradient driven turbulence • Turbulence and formation of structures ("streamers") observed in tokamak fusion devices can be due to instabilities driven by gradients in electron temperature (ETG) @ p p (1 − r2)φ = [φ, r2φ + x] + p ; rx ; (1) @t r @ p p p p = p ; φ + [ rx; φ] ; (2) @t r r • Slab model (G¨urcanand Diamond, 2004) for evolution of pressure fluctua- tions p(x; y) and electrostatic potential φ(x; y) • Coupling of advection equation for p and Charney-Hasegawa-Mima type equation for φ • r / rTe0 provides instability @f @g @f @g • [f; g] := @x @y − @y @x 8. Hamiltonian structure for the ETG model • ETG model possesses Hamiltonian structure (G¨urcanand Diamond, 2005) • Thus it can be cast in the form @χi = fχi;Hg; i = 1; ··· ; n @t with n = 2 field variables and H[χ1; ··· ; χn] Hamiltonian functional •f ; g Poisson bracket: antisymmetric bilinear operator satisfying Leibniz and Jacobi identity p 1 2 2 pp • Field variables χ = Λ := φ − r φ and χ = P := r + rx • Hamiltonian functional and Poisson bracket: 1 Z p H(Λ; P) = d2x ΛL−1Λ − P2 + 2 rPx ; 2 Z 2 fF; Gg = d x(x − Λ)[FΛ;GΛ] − P([FΛ;GP ] + [FP ;GΛ]): with Lf = f − r2f 9. Linearized ETG model • Poisson bracket for the ETG model possesses Casimirs Z Z 2 2 C1 = d xH(P);C2 = d x(Λ − x)F(P) with arbitrary H and F • Casimir C: fC; F g = 0; 8F ) Casimirs are invariants for the dynamics • Linearize the model around equilibria (no flow - linear pressure gradient) −1 Λeq = L Λeq = 0; Peq = αP x (3) with constant αP , yields @ p @ Λ~_ = − L−1Λ~ − r P~; @y @y @ P~_ = α L−1Λ~ : P @y • Equilibria (3) are critical points of free energy functional F := H + C1 + C2 p 2 for F(P) = 0 and H(P) = (1 − r/αP )P =2 10. Hamiltonian structure for the linearized ETG system • Linearized system still Hamiltonian ~ P+1 ~ −ik·x ~ P+1 ~ −ik·x • Λ = k=−∞ Λk(t)e ; P = k=−∞ Pk(t)e yields Hamiltonian system for Fourier amplitudes: p ~_ ky ~ ~ 2 2 2 Λk = i 2 Λk + i rkyPk; where k = ky and k? = kx + k 1 + k? ~_ ky ~ Pk = −iαP 2 Λk ; 1 + k? p +1 +1 ~ 2 ! X k X jΛkj r ~ 2 with HL = HL = 2π 2 − jPkj ; 1 + k αP k=1 k=1 ? +1 X ik @F @G @F @G and bracket fF; Gg = − 2π ~ ~ ~ ~ k=1 @Λk @Λ−k @Λ−k @Λk 2 @F @G @F @G @F @G @F @G −αP + − − : @Λ~ k @P~−k @P~k @Λ~ −k @P~−k @Λ~ k @Λ~ −k @P~k 11. Canonical form • Change of variables: r 1 π ~ ~ ~ ~ qk = 2 (Pk + αP Λk + P−k + αP Λ−k) ; kαP r 1 π ~ ~ ~ ~ pk = −i 2 (Pk + αP Λk − P−k − αP Λ−k) ; kαP rπ rπ q2 = (Λ~ + Λ~ ); p2 = i (Λ~ − Λ~ ) ; k k k −k k k k −k k k k k k • In the real variables z = (q1 ; q2 ; p1; p2) the system becomes canonical: _k k k z = JcA z : 0 a c 0 0 1 k B c b 0 0 C p k p p A = B C ; a = − rαP k; b = 2 −k rαP ; c = rjαP jk: @ 0 0 a −c A 1 + k? 0 0 −c b •) Framework of the general theory previously described 12. Mode signature for ETG model q p k k 2 !s = 2 1 − 1 − 4(1 + k?)αP r ; slow mode 2(1 + k?) q p k k 2 !f = 2 1 + 1 − 4(1 + k?)αP r ; fast mode 2(1 + k?) k k • The system possesses also the eigenvalues !−s;−f = −!s;f 1 p • Equilibria spectrally stable iff αP < 2 4(1+k?) r • If r ! 0 or αP ! 0 then stable drift wave • Eigenvectors 0 1 1 0 1 1 k k B −B∓ C k k ∗ B −B∓ C z = q B C ; z = q B C ; s;f 1 s;f @ −i A −s;−f 1 s;f @ i A −iB∓ iB∓ p b+a± (b+a)2−4c2 where B± = 2c ; 13. Mode signature for ETG model (Tassi and Morrison, 2011) • Consider slow modes: k T k 2 k ∗ k z−s Ωzs = 2i(1 − B−)q1s q1s: 2 • For stable modes one finds 1 − B− > 0 k k ∗ p 1 • Choose normalization constant q1s = q1s = 2 1−B− k k k T k k • Energy of the kth slow mode: h (−s; s) = i!s z−s Ωzs = −2!s k • If !s > 0 the slow mode is a stable but negative energy mode 1 p • This occurs for 0 < αP < 2 4(1+k?) r • If pressure gradient is negative (αP < 0) ) spectral stability without NEM k k k T k k • Fast modes : h (−f; f) = i!f z−f Ωzf = 2!f > 0 8k?; k; r; αP ) Fast modes always PEMs 14. Normal form for linearized ETG model k k k k k k k k k • T :(q1 ; q2 ; p1; p2) ! (Q1;Q2;P1 ;P2 ) with 0 1 1 0 0 1 D− D+ B − B− − B+ 0 0 C q T k = B D− D+ C ;D = B2 − 1 B 0 0 1 − 1 C ± ± @ D− D+ A 0 0 B− − B+ D− D+ puts the Hamiltonian (for stable modes) into its normal form 1X0 2 2 2 2 H0 = !k Qk + P k − !k Qk + P k ; L 2 f 2 2 s 1 1 k k • Slow modes give negative contribution to the energy when !s > 0 15. Dispersion relation for ETG model p p Figure 1: αP = −0:3, r = 0:2 Figure 2: αP = 0:5, r = 0:2 • Instability occurs at k? = 1:22 (Fig. 2) • Collision of eigenvalues of a PEM with a NEM (Kre˘ınbifurcation) 2 • Presence of NEMs reflects in undefiniteness of δ F (Λeq; Peq), where F = H + C1 + C2 • Energy-Casimir method predicts formal stability for F(Peq) = 0, 00 H (Peq) > 1 ) αP < 0 i.e. no NEMs 16. Magnetic reconnection in collisionless plasmas • Magnetic reconnection: modification of the way infinitesimal plasma vol- umes are connected by means of magnetic field lines • Involved in e.g., solar flares, magnetic substorms, sawtooth oscillations in tokamaks Figure 4: Image taken by TRACE. Figure 3: From M. Lockwood, Nature, 409, 677 (2001). • Electron inertia can cause reconnection in high-temperature tokamak plasmas 17.