Hydrodynamic stability analysis in terms of action-angle variables

Transcription

1 1/27 Hydrodynamic stability analysis in terms of action-angle variables Makoto Hirota 1 Collaborators: Yasuhide Fukumoto 2, Philip J. Morrison 3, Yuji Hattori 1 1 Institute of Fluid Science, Tohoku University 2 Institute of Mathematics for Industry, Kyushu University 3 Institute for Fusion Studies, The University of Texas at Austin International/Interdisciplinary Seminar on Nonlinear Science March 14, Kashiwa

2 1. Introduction How is Fluid mechanics different from Classical mechanics? 2/27 Infinite degree of freedom Eulerian description for velocity field (vs. Lagrangian description for particle orbit) The partial differential equations (PDEs) for vector and scalar fields Mathematical methods used in classical mechanics are not directly applicable. It is hard to solve the PDEs even by using computers. Linear and weakly nonlinear stability analysis (perturbation analysis) is one of the feasible approaches. Dye Laminar flow Instability Turbulent flow (Reynolds 1883)

3 Outline? Stability theory for Hamiltonian systems Hydrodynamic stability theory Based on the Hamiltonian viewpoint of fluid mechanics, the variational method is shown to be useful for predicting stability (both theoretically and numerically). 3/27 1. Introduction (As a typical and simple hydrodynamic stability problem, we consider) 2. Rayleigh equation stability of inviscid parallel shear flow 3. Action-angle variables (in classical mechanics) 4. Variational stability conditions (in classical mechanics) 5. Wave action (action variable in fluid mechanics) 6. Variational stability conditions for Rayleigh equation 7. Summary

4 4/27 Rayleigh equation Stability of inviscid parallel shear flow Shear flow U = U(x)e y, Disturbance ũ = [ϕ(x)e ik(y ct) + c.c.] e z, (c C, k R) (c U)(ϕ k 2 ϕ) + U ϕ =, ϕ( L) = ϕ(l) = (where is the x-derivative) If there exists an eigenvalue c with Im c >, the flow is unstable. Kelvin-Helmholtz instability: One of the most classical hydrodynamic stability problem But, the stability condition on U(x) is still nontrivial. U (x) everywhere = =\ Stable (Rayleigh 188)

5 History 5/ Rayleigh No inflection point (U ) Stable 195 Fjørtoft One inflection point x I and U (U U I ) > where U I = U(x I ) Stable 1964 Rosenbluth & Simon (Nyquist method) In the limit k, 1 L L U U (U U I ) + dx > Stable U 2 (U U I ) L L 1969 Arnold (variational method) δ 2 E is poitive or negative definite Stable 1991 Barston (variational method) 1999 Balmforth & Morrison (Nyquist method) a necessary and sufficient condition 23, 25 Lin (Tollmien s method) a necessary condition 214 Hirota, Morrison & Hattori (variational method) a necessary and sufficient stability condition These methods requirs a solution of Rayleigh s equation.

6 6/27 General pertubation theory Lagrangian: L[q] = L(q, q) Perturbed orbit q(t) = q (t) + ξ(t) q : known solution, ξ: perturbation Linearized dynamics Weakly nonlinear dynamics at equilibrium "Two-mode" coupling Linear instability Resonant absorption/growth Three-mode coupling Parametric instability Second harmonic resonance etc. Four-mode coupling Modulational instability Nonlinear frequency shift etc. Analysis of ξ is, however, tedious when ξ has a large degree of freedom. Each mode coupling can be reduced to a normal form by the transformation to action-angle variables.

7 Action-angle variables in classical mechanics Periodic motion 7/27 Canonical variables (q, p) Action-angle variables (µ, θ) Angle Normal form of single mode Energy = Frequency Action Action = const. Adiabatic invariance µ const. when a parameter (such as ω) is slowly varying. Averaging If (q 1, p 1 ) is fast oscillation, H H (q, p ) + ω 1 (q, p )µ 1 Averaged equation for (q, p )

8 Instability An instability is caused by a resonance ω 1 = ω 2 between positive and negative energy modes. (Krein 195) H =H + ω 1 µ 1 + ω 2 µ 2 > < Positive energy mode Equilibrium point Energy increase Negative energy mode 8/27 Energy decrease Negative energy mode is also destabilized by energy dissipation effect. Ex. Precession of spinning top Signs of modal energies (called Krein signatures) are important!

9 Variational stability conditions 9/27 Linear Hamiltonian system with N degrees of freedom u = (q 1, q 2,..., q N, p 1, p 2,..., p N ); { ( ) t q = H/ p, I t p = H/ q, t u = J Hu, J =, H = 1 u, Hu I 2 i t u = Lu, L = ij H where H = H and J = J. But, L is non-self-adjoint L L. Lyapunov stability theorem (Oberman and Kruskal 1965, Case 1965, Barston 1977) Q := u, ij P(L)u is a constant of motion, where P is any real polynomial. P and ϵ > s.t. Q ϵ u 2 or Q ϵ u 2 Stable (Hint) If P(L) = L/2, then Q = H. If P(L) = L 3, ij P(L) = H(iJ )H(iJ )H is also self-adjoint. Lots of sufficient stability conditions What choice of P leads to a better condition?

10 1/27 Modal decomposition: u = 2N u α e iω αt = 2N u α e iω αt α=1 α=1 Eigenvalue problem: E(ω α )u α = E(ω α )u α = where E(ω) = ωij H. Accordingly, Q is decomposed into Q = 2N α=1 P(ω α )µ α with µ α = u α, ij u α = q α1 1 E 1,1 (ω α ) D ω (ω α)q α1 where µ α corresponds to the action variable for a neutrally stable mode (ω α R). E 1,1 is the (1, 1) cofactor of the matrix E. D(ω) = det E(ω) is the characteristic polynomial; D(ω α ) =. Even when H = 1 2N 2 α=1 ω αµ α is indefinite, it is possible to make Q positive or negative definite by choosing P.

11 Theorem: If a polynomial P(ω) is chosen such that ω R, then max u P(ω) E 1,1 (ω) D ω Q u 2 > Spectrally unstable; Im ω α > (Necessary and sufficient condition) Q for all neutrally stable modes Q only for growing (Im ω α > ) and damping (Im ω α < ) modes (ω) holds for all A trivial choice is P(ω) = E 1,1 (ω) D ω (ω). But, this choice is not practically useful because D(ω) is needed to construct Q. What is interesting here is that Number of unstable eigenvalues (Im ω α > ) of the non-self-adjoint L one to one relation Number of positive eigenvalues of the self-adjoint Q where Q = u, Qu 11/27

12 Example 2 t 2 ( q1 ) +2 q 2 ( ) ωl ω L t ( q1 q 2 ) = ω 2 ( q1 q 2 ) 12/27 Coriolis force Potential force An instability is caused by resonance between a positive energy mode and a negative energy mode. growing damping D(ω) = ω 4 2(2ω 2 L ω2 )ω 2 + ω 4 E 1,1 (ω) = ω 2 + ω 2 By choosing P(ω) = (ω 2 +ω)ω(ω 2 2 2ωL 2 +ω2 ), we obtain a necessary and sufficient condition as max Q/ u 2 > Unstable.

13 Wave action theory for fluid (mode wave) Vortex tube Uniform background Plane wave: A exp(ik x ωt) Dispersion relation: D(ω, k) = Wave action = D ω (ω, k) A 2 (Auer et al. 1958) 13/27 Weakly non-uniform background (short wavelength limit) Wave packet: A(x) exp(ik x ωt), Local dispersion relation: D(ω, k, x) = Wave action density = D (ω, k, x) A(x) 2 ω Wave-kinetic theory & Weak turbulence theory (Stix 1962, Sagdeev & Galeev 1969) Non-uniform background Eigenvalue problem discrete and continuous spectra (Differential equation) Global eigenmode Singular eigenmode Wave action (= action variable) is nontrivial. r r

14 Continuous spectrum in hydrodynamic disturbance 14/27 Vorticity disturbance stretched by background shear flow U(x) (Case 196) Initial condition e iky e iky iku(x)t : the Doppler shift ku(x) depends on x. Continuous spectrum {ku(x) x R} Continuum mode Integral of infinite number of singular eigenmodes localized on each streamline e.g. delta functions Since the singular function is not square integrable, it has been difficult to calculate wave action (and also wave energy) for continuous spectrum. Van Kampen mode (Morrison & Pfirsch 1992), Rayleigh equation (Balmforth & Morrison 22) General singular mode (Hirota & Fukumoto 28)

15 Action variables for eigenmode and continuum mode 15/27 i t u = Lu, LJ = J L Laplace transform u(t) U(Ω) = iu() Ω L, and define D 1 (Ω) := S = 1 4π 2π u, J 1 u dθ = 1 θ 2πi Eigenvalues {ω n n = 1, 2,... }, µ n = 1 2πi Γ(σ + ) D 1 (Ω)dΩ = n Γ(ω n ) Continuous spectrum ω σ c R, µ(ω) = i 2π D 1 (Ω)dΩ = u(), ij U(Ω) µ n + σ c µ(ω)dω ( ) 1 D (ω n ), (residue) Ω [ D 1 (ω + i) D 1 (ω i) ]. (jump) Deformation = (Hirota & Fukumoto 28)

16 16/27 Rayleigh equation Stability of inviscid parallel shear flow Basic flow U = U(x)e y, Disturbance ũ = [ϕ(x)e iωt+iky + c.c.] e z, (ω C, k R) (c U)(ϕ k 2 ϕ) + U ϕ =, ϕ( L) = ϕ(l) = If there exists an eigenvalue c = ω/k with Im c >, the flow is spectrally unstable. (Case 196) A continuous spectrum exists, c = ω/k {U(x) R x [ L, L]}. Sign of the energy of continuous spectrum = Sign of UU (Balmforth & Morrison 22, Hirota & Fukumoto 28) Kelvin-Helmholtz instability emerges from a contact point between positive- and negative-energy continuous spectra. Analogous to Krein s theory (Hagstrom & Morrison 211) growing damping

17 Variational stability conditions 17/27 In terms of vorticity disturbance w = ϕ := ϕ + k 2 ϕ, i k tw = Lw where L = U U 1 L is non-self-adjoint (L = L ), but has a Hamiltonian property LU = U L. Theorem [Oberman and Kruskal 1965, Case 1965, Barston 1977] Let P(c) be any real polynomial. Then, Q = L L w 1 P(L)wdx = const. U Therefore, If P and ϵ > s.t. Q ϵ u 2 or Q ϵ u 2 (Lyapunov) Stable What is the best choice of P?

18 If U(x) has only one inflection point x = x I, 18/27 the choice of P(c) = c U(x I ) results in (Arnold 1966) The second variation of the energy in the inertial frame moving at the velocity U I = U(x I ) is Q = δ 2 E I = L L ( U UI w U ) 1 wdx. The shear flow U is stable if δ 2 E I is either positive or negative definite. (This includes Rayleigh-Fjørtoft s stability criterion.) growing growing damping damping In this frame, the energy of the continuous spectrum is negative everywhere.

19 If U(x) has multiple inflection points x In, n = 1, 2,..., N I, 19/27 the choice of P(c) = N I n=1 (c U In ), where U In = U(x In ), results in (Barston 1991) The shear flow U is stable if Q = L L w 1 U N I n=1 [ (U UIn ) U 1] wdx, is either positive or negative definite.... still sufficient conditions, but very close to necessary and sufficient one.

20 Theorem: Assume 1) U(x) is an analytic, bounded and strictly monotonic function on [ L, L]. 2) if U (x I ) = at x = x I, then U (x I ). Define the quadratic form Q by choosing P(c) = ν N I n=1 (c U In) where either ν = 1 or ν = 1 is chosen such that 2/27 ν U N I n=1 (U U In ) for all x. Then, max Q w 2 dx > (Spectrally) Unstable (Necessary and sufficient stability condition!) (Hirota, Morrison, Hattori 214) By showing Q > for some w, we can prove instability!

21 Outline of the proof: 21/27 Spectrum: Sp(L) = {c α, c α C ; Im c α, α = 1, 2,..., N} {U(x) R ; x [ L, L]} (discrete) (continuous) Mode decomposition: w = N α=1 ( ) Umax w α e ikcαt + w α e ikc αt + Exponentially growing and damping modes Then, Q is also decomposed into U min ŵ(c)e ikct dc Neutrally stable continuum mode N Q = [P(c α )µ α + P(c α )µ α ] + α=1 Umax U min P(c)ˆµ(c)dc where µ α = L L w αw α /U dx and sgn ˆµ(c) = sgn U (U 1 (c)). (Balmforth & Morrison 22, Hirota & Fukumoto 28) If Q > for some w, there exists at least a growing eigenmode.

22 Numerical tests λ 1 = max Q/ w 2 dx Example 1 U(x) = x + 5x tanh[4(x.5)], x [ 1, 1] 22/ as decreases Corollary: At the marginal stability λ 1, Q is non-singular ( w 1 2 dx < ).

23 Example 2 U(x) = x.2 + sin[8(x.2)]/16, x [ 1, 1] 23/ as decreases Corollary: Number of positive signatures of Q corresponds to number of unstable eigenmodes.

24 5. Summary 24/27 Using Rayleigh s equation, we have demonstrated that the variational method can be improved to give necessary and sufficient stability criteria. Linearized system i t w = klw Variational criterion max Q/ w 2 > non-self-adjoint c 1, c 2, C w(x) C j, Im c j > Unstable self-adjoint λ 1 > λ 2 > R w(x) R λ 1 > Unstable singular as Im c + non-singular around λ = We can determine the stability more efficiently and accurately than directly solving the linearized equation.

25 However, I have a lot of open questions. 25/27 This variational method can calculate neither growth rates nor frequencies of the unstable modes. So far, this method is not useful for systems of finite degree of freedom. Piecewise-linear flows are more complicated. The linear part [x 1, x 2 ] is a set of inflection points, on which the classical Krein collision occurs. min max Q P=c U(x I ) > x I [x 1,x 2 ] w =1 Ustable

26 The variational method is not always applicable to nonmonotonic shear flows, /27 OK Impossible to make all signatures negative We can think of many difficult situations when multiple continuous spectra exist. (e.g., stratified shear flow, MHD etc.) Smoothness of shear flow is mathematically important, but physically not. Is it possible to obtain an unified view of resonance among discrete modes and continuum modes? (like the normal forms)

27 References 27/27 1. Arnold VI Izv. Vyssh. Uchebn. Zaved. Mat. [Sov. Math. J.] 5, Oberman C, Kruskal M J. Math. Phys. 6, Case KM Phys. Fluids 8, Barston EM J. Fluid Mech. 233, Morrison PJ, Pfirsch D Phys. Fluids B 4, Balmforth NJ, Morrison PJ. 22 Hamiltonian description of shear flow. In Large-scale atmosphereocean dynamics II (eds J Norbury, I Roulstone), pp Cambridge, UK: Cambridge University Press. 7. Hirota M, Fukumoto Y. 28 J. Math. Phys. 49, Hirota M 21 J. Plasma Fusion Res. SERIES, 9, Hirota M, Morrison PJ, Hattori Y. 214 Proc. R. Soc. A 47,