Instabilities of a Wave in a Density-Stratified FluidFloquet Instabilities & Triad Resonance

Yuanxun Bill BaoSupervisor: Professor David Muraki

Simon Fraser University

July 9, 2010

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Laminar flow

I Wind flow across an obstacle (mountain range)

I Flowing air rises up the windward side, sinks down the lee side.

I One possible configuration: steady laminar flow across the mountain

Figure 1: Streamlines of a laminar flow (A. Nenes) and a lenticular cloud over Mt.Fuji

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Mountain Wave & Turbulence

I Breakdown of laminar flow configuration

I Produces time-dependent waves

I Upward deflected air influences layers tens of thousands of feet above(⇒ in-flight turbulence)

Figure 2: Mountain wave cartoon

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Mountain Wave & Turbulence

I Numerical simulation: Courtesy of Professor Craig Epifanio, Texas A&M

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Hazard for Powered Aircrafts, Paradise for Gliders

I The U.S. Aeronautical Information Manual states,”Many pilots go all their lives without understanding what a mountain wave is. Quite a few

have lost their lives because of this lack of understanding.”

I Sign of hazard — Lenticular Clouds

I Enables Gliders to gain remarkable altitudes (50,721 ft).

Figure 3: Mt.Rainier by Tim Thompson Figure 4: Typical glider path

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Equations for a Density-Stratified Fluid

Nonlinear Equations

ηt + bx + J(η, ψ) = 0bt − ψx + J(b, ψ) = 0

I A system of nonlinear PDEs

I Streamfunction: ψ(x, z, t) & Buoyancy: b(x, z, t)

I Vorticity: η = ψzz + δ2ψxx

I Hydrostatic limit: δ → 0 ; Laplacian: δ → 1

I Nonlinearity from Jacobian determinant

J(f, ψ) =fx ψxfz ψz

= fxψz − ψxfz

I Existence of exact solutions?

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A Simple Nonlinear Solution

Nonlinear Equations

(ψzz + δ2ψxx)t + bx + J(ψzz + δ2ψxx, ψ) = 0bt − ψx + J( b, ψ) = 0

Sinusoidal Wave

I Buoyancy-gravity as restoring forces ⇒ oscillatory wave ei(kxx+kzz−ωt)

I Linear dispersion relation: ω2(kx, kz) =k2x

k2z+δ2k2x

I All (kx, kz)-pairs satisfying linear dispersion relation giveexact nonlinear solutions! (Zero-Jacobian)

I A simple one: kx = kz = 1, ω = ± 1√1+δ2(

ψb

)=

(−ω

1

)2ε sin(x+ z − ωt)

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Linearized Equations

Nonlinear Equations

(ψzz + δ2ψxx)t + bx + J(ψzz + δ2ψxx, ψ) = 0bt − ψx + J( b, ψ) = 0

I One exact solution:(ψb

)=

(−ω

1

)2ε sin(x+ z − ωt)

I Goal: to characterize the linearized stability of this simple nonlinear waveLinearize w.r.t the nonlinear wave(

ψb

)=

(−ω

1

)2ε sin(x+ z − ωt) +

(ψ

b

)

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

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Linearized Stability

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

I Linear PDEs with periodic coefficients

I A problem for Floquet Theory

ODE Floquet Theory

The system with T -periodic coefficients u(t) = P(t) · u(t) has solutions of theform

u(t) = eρt · p(t)

where p(t) is a T -periodic function.

I Fundamental matrix Φ(t) and Propagation matrix Φ(T )

I Eigenvalue µ of Φ(T ) and Floquet exponent µ = eρT

I Re ρ > 0⇒ instability

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Floquet Instabilities of the Mathieu Equation

Mathieu Equation

u+ (α+ β cos t)u = 0

⇒(uv

)=

[0 1

−α− β cos t 0

]︸ ︷︷ ︸

P(t)

(uv

)

Floquet Instabilities

I Floquet solution & Fourier periodic parts

u(t) = eρt

+∞∑−∞

~cneint

= exponential part × co-periodic part

I ρ(α, β), Floquet exponent Re(ρ) > 0⇒ instability

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Computation of Mathieu Unstable Spectrum

Naive Approach

I ODE solver & ”sufficiently” large t

I Accuracy and Efficiency?

Fourier Spectral Approach

I Numerically solve for Φ(T )

I Eigenvalue & Floquet exponents

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Floquet Instabilities of PDEs

ηt + bx + iεJ(ωη + ω(1 + δ2)ψ , ei(x+z−ωt) − e−i(x+z−ωt)) = 0

bt − ψx + iεJ(

ωb+ ψ , ei(x+z−ωt) − e−i(x+z−ωt)) = 0

Floquet, Fourier & Linear Algebra

I Product of exponential & co-periodic Fourier series(ψ

b

)= ei(kx+mz−Ωt)

+∞∑−∞

~vnein(x+z−ωt)

I Floquet exponent Im Ω(k,m; ε) > 0⇒ instability

I L.A. formulation: Hill’s infinite matrix & generalized eigenvalue problem

. . .. . .

. . . S0 εM1

εM0 S1

. . .

. . .. . .

− Ω

. . .. . .

. . . Λ0 0

0 Λ1

. . .

. . .. . .

I Block-diagonal with 2× 2 real blocks: Sn(k,m) symmetric ; Λn(k,m) diagonal

I Truncated matrix −N ≤ n ≤ N & compute eigenvalues Ω(k,m; ε)12 / 19

Floquet PDE Unstable Spectrum

I 1st attempt: Maximum Growth Rate (ε = 0.1, δ = 0)

I Artificial periodicity due to index shift ⇒ multiple counting(ψ

b

)= ei((k+q)x+(m+q)z−(Ω+ωq)t)

+∞∑−∞

~vn+qein(x+z−ωt)

I Goal: Rules for determining Ω

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Floquet Spectrum Unravelled

Work by Professor David Muraki

I ”center-of-mass” uniqueness: preserves notion of central wavevector in(k,m)-space

I How to understand unstable spectrum analytically?

I What is the significance of these blue curves?

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Multiple Roots

I Recall Im Ω > 0⇒ instability

I Simple analogy from real polynomial perturbation

I Complex roots can only come from multiple root perturbation

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Eigenvalue Degeneracy

. . .. . .

. . . S0 εM1

εM0 S1

. . .

. . .. . .

− Ω

. . .. . .

. . . Λ0 0

0 Λ1

. . .

. . .. . .

Instability of Small Amplitude Waves (0 < ε 1)

I ε = 0, recovers linear dispersion relation ⇒ real eigenvalues

I ε 6= 0, coefficients of characteristic polynomial are real

I For what (k,m)-pairs are the waves unstable when ε is small?

Double root in adjacent (n = 0, 1) blocks

I Recall linear dispersion relation ω2(k,m) = k2

m2 for δ = 0

I For what (k,m)-pairs do Ω have a double root in n = 0, 1 blocks at ε = 0?

I Ω double root at ε = 0⇒ ω(k,m) + ω(1, 1) = ω(k + 1,m+ 1)

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Triad Resonance

I resonance condition: ω(k,m) + ω(1, 1) = ω(k + 1,m+ 1)

I unstable (k,m)-pair by matrix perturbation theory

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What I have learned...

I Some background knowledge about atmospheric scienceI density stratified flowI mountain wave & turbulence

I Floquet theoryI application to mathieu equationI stability analysis via computationI application to PDEs

I Some perturbation theoryI application to mathieu equation, triad resonance

I Linear AlgebraI (generalized) eigenvalue problemI eigenvalue degeneracy

I Different perspectives on a problem

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Many thanks to ...

I My supervisor Professor David Muraki

I NSERC for project funding

I Professor JF Williams for his suggestions

I Department of Mathematics, Simon Fraser University

Acknowledgments

D. J. Muraki, Unravelling the Resonant Instabilities of a Wave in a Stratified Fluid, 2007

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

D. W. Jordan and P. Smith (1987), Nonlinear Ordinary Differential Equations (Second Edition), Oxford

University Press, New York. pp. 245-257

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, Nov 30, 2009

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