Linear Instability of a Wave in a Density-Stratified Fluid Yuanxun Bill Bao, David J. Muraki Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada Introduction • A fluid with depth-dependent density is said to be density-stratified . (ocean & atmosphere) • Buoyancy & gravity-driven oscillatory waves (internal gravity waves ) can be generated by a displacement of a fluid element at the interface of stratified fluids. • Physical realization: a strong wind flowing over a mountain range. • One possible configuration is a steady laminar flow. Figure 1: Streamlines of a laminar flow (A. Nenes) and a lenticular cloud formed over Mt.Fuji [4] . • Laminar flow, however, may become unstable because small disturbances can grow in time to make the flow more complicated or even turbulent. • We are interested in characterizing instabilities of these waves in terms of wavenumber (k,m). Equations for a Density-Stratified Fluid Equations of Motion ∇· u = 0 (1) Dρ Dt = ρ 0 g N 2 w (2) D u Dt = - 1 ρ 0 ∇p - g ρ 0 ρ ˆ z (3) • (1) Zero-divergence, (2) Conservation of mass, (3) Conservation of Momentum • Velocity u =(u,v,w), density ρ( x,t), pressure p( x,t) • Boussinesq approximation and Brunt- V¨ais¨al¨a frequency N 2D Streamfunction Formulation (dimensionless) η t + b x + J (η,ψ ) = 0 b t - ψ x + J (b,ψ ) = 0 • Streamfunction ψ (x,z,t): u = ψ z , w = -ψ x ; Buoyancy b(x,z,t) • Vorticity: η = ψ zz + δ 2 ψ xx ; Hydrostatic limit: δ → 0; Laplacian: δ → 1 • Advection from Jacobian: J (f,ψ )= f x ψ x f z ψ z = f x ψ z - ψ x f z = uf x + wf z Simple Nonlinear Solutions ψ b = -ω 1 2 sin(x + z - ωt) • Buoyancy-gravity as restoring forces ⇒ oscillatory wave e i(k x x+k z z -ωt) • Linear dispersion relation: ω 2 (k x ,k z )= k 2 x k 2 z +δ 2 k 2 x • All (k x ,k z )-pairs satisfying linear dispersion relation give exact nonlinear solutions ! • A simple sinusoidal one: k x = k z = 1, ω< 0. Linearized Equations ˜ η t + ˜ b x - J ( ω ˜ η + ω (1 + δ 2 ) ˜ ψ, 2 sin(x + z - ωt) ) = 0 ˜ b t - ˜ ψ x - J ( ω ˜ b + ˜ ψ, 2 sin(x + z - ωt) ) = 0 • Goal: to characterize the linear instability of a simple sinusoidal wave • Linearize w.r.t the nonlinear wave ψ b = -ω 1 2 sin(x + z - ωt)+ ˜ ψ ˜ b • Linear PDEs with periodic, non-constant coefficients • A problem for Floquet Theory Instability via Floquet Theory Textbook ODE example: Mathieu Equation ¨ u +(α + β cos t)u =0 ⇒ ˙ u ˙ v = 0 1 -α - β cos t 0 u v Figure 2: Mathieu stability spectrum • Floquet solution: u(t)= e ρt +∞ -∞ c n e int = exponential part × co-periodic part Floquet Analysis for PDEs • Product of exponential & co-periodic Fourier series ˜ ψ ˜ b = e i(kx+mz -Ωt) +∞ -∞ v n e in(x+z -ωt) • Floquet exponent Im Ω(k,m; ) > 0 ⇒ instability • Hill’s infinite matrix & generalized eigenvalue problem . . . . . . . . . S 0 M 1 M 0 S 1 . . . . . . . . . - Ω . . . Λ 0 Λ 1 . . . • 2 × 2 real blocks: M n (k,m); S n (k,m) symmetric ; Λ n (k,m) diagonal • Truncated matrix -N ≤ n ≤ N & compute eigenvalues {Ω(k,m; )} PDE Unstable Spectrum • Maximum Growth Rate ( =0.1, δ = 0) • Artificial periodicity due to index shift ⇒ multiple counting (D.J. Muraki) ˜ ψ ˜ b = e i((k +q )x+(m+q )z -(Ω+ω q )t) +∞ -∞ v n+q e in(x+z -ωt) • Goal: Rules for determining Ω Figure 3: maximum growth rate vs “center-of-mass” uniqueness by D. J. Muraki Instability via Perturbation Methods • Simple analogy from real polynomial pertur- bation. • Complex roots only come from multiple root perturbation. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1 0 1 x-axis y-axis Polynomial Perturbation (distinct roots vs double root) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 1 2 3 x-axis y-axis Eigenvalue Degeneracy & Triad Resonances • 0 < 1, instabilities via complex conjugate Ω from multiple eigenvalues at =0 • Double root appearing in adjacent (n =0, 1) Fourier modes: ω (k,m)+ ω (1, 1) = ω (k +1,m +1) • Unstable (k,m)-pair by PDE perturbation -3 -2 -1 0 1 2 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 k-axis m-axis Triad resonance trace Figure 4: Triad resonant trace and unstable spectrum • Internal gravity waves are unstable, a small perturbation can result in more complicated flows or even turbulences. 0 2 4 6 8 10 12 0 2 4 6 sinusoidal internal gravity wave at t = 0 x-axis z-axis 0 2 4 6 8 10 12 0 2 4 6 internal gravity wave at t = 4 x-axis z-axis Figure 5: Small disturbances grow to make a more complicated flow pattern References [1] D. J. Muraki, Unravelling the Resonant Instabilities of a Wave in a Stratified Fluid, 2007 [2] P. G. Drazin, On the Instability of an Internal Gravity Wave, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 356, No. 1686 (1977), 411-432 [3] D. W. Jordan and P. Smith (1987), Nonlinear Ordinary Differential Equations (Second Edition), Oxford Uni- versity Press, New York. pp. 245-257 [4] A. Nenes, laminar flow grid plot, [Image] Available: http://nenes.eas.gatech.edu/CFD/Graphics/d2grd.jpg A lenticular cloud over Mt. Fuji, [Image] Available: http://ecotoursjapan.com/blog/?p=123, November 30, 2009