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Chapter 6: Spectral Theory Overview Spectral Theory: Overview 6-1 Chapter 6: Spectral Theory Overview Intuitive approach to stability: two viewpoints for study of stability, linearization • and Lagrangian reduction; [ book: Sec. 6.1 ] Force operator formalism: equation of motion, Hilbert space, self-adjointness of • the force operator; [ book: Sec. 6.2 ] Quadratic forms and variational principles: expressions for the potential energy, • different variational principles, the energy principle; [ book: Sec. 6.4 ] Further spectral issues: returning to the two viewpoints; [ book: Sec. 6.5 ] • Extension to interface plasmas: boundary conditions, extended variational princi- • ples, Rayleigh–Taylor instability. [ book: Sec. 6.6 ] Spectral Theory: Intuitive approach to stability (1) 6-2 Two viewpoints How does one know whether a dynamical system is stable or not? • W0 a b W1 Energy W1 W0 ξ ξ Force F F stable unstable Method: split the non-linear problem in static equilibrium (no flow) and small (linear) • time-dependent perturbations. Two approaches: exploiting variational principles involving quadratic forms (energy), • or solving the partial differential equations themselves (forces). Spectral Theory: Intuitive approach to stability (2) 6-3 Aside: nonlinear stability Distinct from linear stability, involves finite amplitude displacements: • (a) system can be linearly stable, nonlinearly unstable; (b) system can be linearly unstable, nonlinearly stable (e.g. evolving towards the equilibrium states 1 or 2). a b 1 2 Quite relevant for topic of magnetic confinement, but too complicated at this stage. • Spectral Theory: Intuitive approach to stability (3) 6-4 Linearization Start from ideal MHD equations: • ∂v ρ( + v v) = p + j B ρ Φ , j = B , (1) ∂t · ∇ −∇ × − ∇ ∇× ∂p = v p γp v , (2) ∂t − · ∇ − ∇· ∂B = (v B) , B =0 , (3) ∂t ∇× × ∇· ∂ρ = (ρv) . (4) ∂t −∇ · assuming model I (plasma–wall) BCs: n v =0 , n B =0 (at the wall) . (5) · · Linearize about static equilibrium with time-independent ρ , p , B , and v =0 : • 0 0 0 0 j B = p + ρ Φ , j = B , B =0 , (6) 0 × 0 ∇ 0 0∇ 0 ∇× 0 ∇· 0 n B =0 (at the wall) . (7) · 0 Spectral Theory: Intuitive approach to stability (4) 6-5 Time dependence enters through linear perturbations of the equilibrium: • v(r,t) = v1(r,t) , p(r,t) = p0(r) + p1(r,t) , all, except v1: f1(r,t) f0(r) . (8) B(r,t) = B0(r)+ B1(r,t) , | |≪| | ρ(r,t) = ρ0(r) + ρ1(r,t) , Inserting in Eqs. (1)–(4) yields linear equations for v , p , B , ρ (note strange order!): • 1 1 1 1 ∂v ρ 1 = p + j B + j B ρ Φ , j = B , (9) 0 ∂t −∇ 1 1 × 0 0 × 1 − 1∇ 1 ∇× 1 ∂p 1 = v p γp v , (10) ∂t − 1 · ∇ 0 − 0∇· 1 ∂B 1 = (v B ) , B =0 , (11) ∂t ∇× 1 × 0 ∇· 1 ∂ρ 1 = (ρ v ) . (12) ∂t −∇ · 0 1 Since wall fixed, so is n, hence BCs (5) already linear: n v =0 , n B =0 (at the wall). (13) · 1 · 1 Spectral Theory: Intuitive approach to stability (5) 6-6 Lagrangian reduction Introduce Lagrangian displacement vector field ξ(r,t): • plasma element is moved over ξ(r,t) away from the equilibrium position. ξξ (r,t) B0 r Velocity is time variation of ξ(r,t) in the comoving frame, ⇒ Dξ ∂ξ v = + v ξ , (14) Dt ≡ ∂t · ∇ D involving the Lagrangian time derivative (co-moving with the plasma). Dt Spectral Theory: Intuitive approach to stability (6) 6-7 Linear (first order) part relation yields • ∂ξ v v = , (15) ≈ 1 ∂t only involving the Eulerian time derivative (fixed in space). Inserting in linearized equations, can directly integrate (12): • ∂ρ 1 = (ρ v ) ρ = (ρ ξ) . (16) ∂t −∇ · 0 1 ⇒ 1 −∇ · 0 Similarly linearized energy (10) and induction equation (11) integrate to p = ξ p γp ξ , (17) 1 − · ∇ 0 − 0∇· B = (ξ B ) (automatically satisfies B =0) . (18) 1 ∇× × 0 ∇· 1 Inserting these expressions into linearized momentum equation yields • ∂2ξ ρ = F p (ξ), B (ξ),ρ (ξ) . (19) 0 ∂t2 1 1 1 Equation of motion with force operator F. ⇒ Spectral Theory: Force operator formalism (1) 6-8 Force Operator formalism Insert explicit expression for F Newton’s law for plasma element: • ⇒ ∂2ξ F(ξ) π B ( Q)+( B) Q +( Φ) (ρξ)= ρ , (20) ≡ −∇ − × ∇× ∇× × ∇ ∇· ∂t2 with change of notation (so that we can drop subscripts 0 and 1): π p = γp ξ ξ p, (21) ≡ 1 − ∇· − · ∇ Q B = (ξ B) . (22) ≡ 1 ∇× × Geometry (plane slab, cylinder, torus, etc.) defined by shape wall, through BC: • n ξ =0 (at the wall) . (23) · Now count: three 2nd order PDEs for vector ξ sixth order Lagrangian system; • ⇒ originally: eight 1st order PDEs for ρ , v , p , B eight order Eulerian system. 1 1 1 1 ⇒ Third component of B1 is redundant ( B1 = 0 ), and equation for ρ1 produces • trivial Eulerian entropy mode ω =0 (with∇· ρ =0 , but v =0 , p =0 , B =0 ). E 1 6 1 1 1 Neglecting this mode, Lagrangian and Eulerian representation equivalent. ⇒ Spectral Theory: Force operator formalism (2) 6-9 Ideal MHD spectrum Consider normal modes: • iωt ξ(r,t)= ξˆ(r)e− . (24) Equation of motion becomes eigenvalue problem: ⇒ F(ξˆ)= ρω2ξˆ . (25) − For given equilibrium, collection of eigenvalues ω2 is spectrum of ideal MHD. • { } Generally both discrete and continuous (‘improper’) eigenvalues. ⇒ 1 The operator ρ− F is self-adjoint (for fixed boundary). • The eigenvalues ω2 are real. ⇒ Same mathematical structure as for quantum mechanics! ⇒ Spectral Theory: Force operator formalism (3) 6-10 Since ω2 real, ω itself either real or purely imaginary • In ideal MHD, only stable waves ( ω2 > 0 ) `or exponential instabilities ( ω2 < 0 ): ⇒ (a) ξ ω t - σ σ (b) ξ ω iν t - iν Crudely, F(ξˆ) ξˆ for ω2 > 0 and ξˆ for ω2 < 0 (cf. intuitive picture). ⇒ ∼− ∼ Spectral Theory: Force operator formalism (4) 6-11 Dissipative MHD In resistive MHD, operators no longer self-adjoint, eigenvalues ω2 complex. • Stable, damped waves and ‘overstable’ modes ( instabilities): ⇒ ≡ (a) ω ξ t - σ - iν σ - iν (b) ω ξ - σ + iν σ + iν t Spectral Theory: Force operator formalism (5) 6-12 Stability in ideal MHD For ideal MHD, transition from stable to unstable through ω2 =0 : marginal stability. • Study marginal equation of motion ⇒ F(ξˆ)=0 . (26) In general, this equation has no solution since ω2 =0 is not an eigenvalue. ⇒ Can vary equilibrium parameters until zero eigenvalue is reached, e.g. in tokamak • stability analysis, the parameters β 2µ p/B2 and ‘safety factor’ q 1/I . ≡ 0 1 ∼ p Find critical curve along which ω2 =0 is an eigenvalue: ⇒ β unstable stable q 1 this curve separates stable from unstable parameter states. ⇒ Spectral Theory: Force operator formalism (6) 6-13 Physical meaning of the terms of F Rearrange terms: • F(ξ)= (γp ξ) B ( Q)+ (ξ p)+ j Q + Φ (ρξ) . (27) ∇ ∇· − × ∇× ∇ · ∇ × ∇ ∇· First two terms (with γp and B) present in homogeneous equilibria, last three terms only in inhomogeneous equilibria (when p, j, Φ =0 ). ∇ ∇ 6 Hogeneous equilibria • isotropic force (γp ξ) : compressible sound waves; ⇒ ∇ ∇· anisotropic force B ( Q) : field line bending Alfv´en waves; ⇒ × ∇× waves always stable (see below). ⇒ Inhomogeneous equilibria have pressure gradients, currents, gravity • potential sources for instability: will require extensive study! ⇒ Spectral Theory: Force operator formalism (7) 6-14 Homogeneous case Sound speed c γp/ρ and Alfv´en speed b B/√ρ constant, so that • ≡ ≡ 1 2 2 ρ− F(ξˆp)= c ξˆ + b ( ( (b ξˆ))) = ω ξˆ . (28) ∇∇ · × ∇× ∇× × − Plane wave solutions ξˆ exp(ik r) give ∼ · 1 2 2 2 2 ρ− F(ξˆ)= (k b) I (b + c ) kk + k b (kb + bk) ξˆ = ω ξˆ (29) − · − · · − recover the stable waves of Chapter 5. ⇒ Recall: slow, Alfv´en, fast eigenvectors ξˆ , ξˆ , ξˆ form orthogonal triad • s A f can decompose any vector in combination of these 3 eigenvectors of F; ⇒ eigenvectors span whole space: Hilbert space of plasma displacements. ⇒ Extract Alfv´en wave (transverse incompressible k ξ =0 , B and k along z ): • · 2 ˆ 2 ˆ 1 2 ∂ ξy 2 2 ∂ ξy 2 ρ− Fˆ = b = k b ξˆ = = ω ξˆ , (30) y ∂z2 − z y ∂t2 − y Alfv´en waves, ω2 = ω2 k2b2, dynamical centerpiece of MHD spectral theory. ⇒ A ≡ z Spectral Theory: Force operator formalism (8) 6-15 Hilbert space Consider plasma volume V enclosed by wall W , with two displacement vector fields • (satisfying the BCs): ξ = ξ(r,t) (on V ) , where n ξ =0 (at W ) , · (31) η = η(r,t) (on V ) , where n η =0 (at W ) . · Define inner product (weighted by the density): 1 ξ, η ρ ξ∗ η dV , (32) h i≡ 2 · Z and associated norm ξ ξ, ξ 1/2 . (33) k k≡h i All functions with finite norm ξ < form linear function space, a Hilbert space. • k k ∞ Force operator F is linear operator in Hilbert space of vector displacements. ⇒ Spectral Theory: Force operator formalism (9) 6-16 Analogy with quantum mechanics Recall Schr¨odinger equation for wave function ψ : • Hψ = Eψ. (34) Eigenvalue equation for Hamiltonian H with eigenvalues E (energy levels). ⇒ continuous E = 0 Spectrum of eigenvalues E consists of discrete ⇒ spectrum for bound states{ (E} < 0 ) and continuous discrete spectrum for free particle states (E > 0 ). Norm ψ ψ,ψ 1/2 gives probability to find particle in the volume. ⇒ k k≡h i Central property in quantum mechanics: Hamiltonian H is self-adjoint linear operator • in Hilbert space of wave functions, ψ ,Hψ = Hψ ,ψ . (35) h 1 2i h 1 2i Spectral Theory: Force operator formalism (10) 6-17 Back to MHD How about the force operator F? Is it self-adjoint and, if so, what does it mean? • Self-adjointness is related to energy conservation.
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