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