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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 adisplacement 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 tomake 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=
ρ0
gN2w (2)
D~u
Dt= −
1
ρ0
~∇p−g
ρ0ρz (3)
• (1) Zero-divergence, (2) Conservation ofmass, (3) Conservation of Momentum
• Velocity ~u = (u, v, w), density ρ(~x, t),pressure p(~x, t)
•Boussinesq approximation and Brunt-Vaisala frequency N
2D Streamfunction Formulation (dimensionless)
ηt + bx + J(η, ψ) = 0
bt − ψx + J(b, ψ) = 0
• Streamfunction ψ(x, z, t): u = ψz, w = −ψx; Buoyancy b(x, z, t)
• Vorticity: η = ψzz + δ2ψxx; Hydrostatic limit: δ → 0 ; Laplacian: δ → 1
• Advection from Jacobian: J(f, ψ) =fx ψxfz ψz
= fxψz − ψxfz = ufx + wfz
Simple Nonlinear Solutions
(
ψ
b
)
=
(
−ω1
)
2ε sin(x + z − ωt)
• Buoyancy-gravity as restoring forces ⇒ oscillatory wave ei(kxx+kzz−ωt)
• Linear dispersion relation: ω2(kx, kz) =k2
x
k2z+δ
2k2x
• All (kx, kz)-pairs satisfying linear dispersion relation give exact nonlinear solutions!
• A simple sinusoidal one: kx = kz = 1, ω < 0.
Linearized Equations
ηt + bx − εJ(
ωη + ω(1 + δ2)ψ , 2 sin(x + z − ωt))
= 0
bt − ψx − εJ(
ωb + 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
)
=
(
−ω1
)
2ε 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) = eρt
+∞∑
−∞
~cneint
= exponential part × co-periodic part
Floquet Analysis for PDEs
• Product of exponential & co-periodic Fourier series
(
ψ
b
)
= ei(kx+mz−Ωt)
+∞∑
−∞
~vnein(x+z−ωt)
• Floquet exponent Im Ω(k,m; ε) > 0 ⇒ instability
• Hill’s infinite matrix & generalized eigenvalue problem
. . . . . .
. . . S0 εM1εM0 S1
. . .. . . . . .
− Ω
. . .Λ0
Λ1. . .
• 2 × 2 real blocks: Mn(k,m); Sn(k,m) symmetric ; Λn(k,m) diagonal
• Truncated matrix −N ≤ n ≤ N & compute eigenvalues Ω(k,m; ε)
PDE Unstable Spectrum
• Maximum Growth Rate (ε = 0.1, δ = 0)
• Artificial periodicity due to index shift ⇒ multiple counting (D.J. Muraki)
(
ψ
b
)
= ei((k+q)x+(m+q)z−(Ω+ωq)t)
+∞∑
−∞
~vn+qein(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 rootperturbation.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
0
1
x−axis
y−a
xis
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−a
xis
Eigenvalue Degeneracy & Triad Resonances
• 0 < ε 1, instabilities via complex conjugate Ω from multiple eigenvalues at ε = 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−a
xis
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 oreven turbulences.
0 2 4 6 8 10 120
2
4
6 sinusoidal internal gravity wave at t = 0
x−axis
z−a
xis
0 2 4 6 8 10 120
2
4
6 internal gravity wave at t = 4
x−axis
z−a
xis
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. SeriesA, 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.jpgA lenticular cloud over Mt. Fuji, [Image] Available: http://ecotoursjapan.com/blog/?p=123, November 30, 2009