Floquet Theory for Internal Gravity Waves in a Density-Stratified - PowerPoint PPT Presentation

Floquet Theory for Internal Gravity Waves in a Density-Stratified Fluid Yuanxun Bill Bao Senior Supervisor: Professor David J. Muraki August 3, 2012

Density-Stratified Fluid Dynamics Density-Stratified Fluids ⊲ density of the fluid varies with altitude ⊲ stable stratification: heavy fluids below light fluids, internal waves ⊲ unstable stratification: heavy fluids above light fluids, convective dynamics Buoyancy-Gravity Restoring Dynamics ⊲ uniform stable stratification: dρ/dz < 0 constant ⊲ vertical displacements ⇒ oscillatory motions

Internal Gravity Waves ⊲ evidence of internal gravity waves in the atmosphere ⊲ left: lenticular clouds near Mt. Ranier, Washington ⊲ right: uniform flow over a mountain ⇒ oscillatory wave motions ⊲ scientific significance of studying internal gravity waves ⊲ internal waves are known to be unstable ⊲ a major suspect of clear-air-turbulence

Gravity Wave Instability: Three Approaches Triad resonant instability (Davis & Acrivos 1967, Hasselmann 1967) primary wave + 2 infinitesimal disturbances ⇒ exponential growth ⊲ ⊲ perturbation analysis Direct Numerical Simulation (Lin 2000) ⊲ primary wave + weak white-noise modes ⊲ stability diagram ⊲ unstable Fourier modes Linear Stability Analysis & Floquet-Fourier method (Mied 1976, Drazin 1977) ⊲ linearized Boussinesq equations & stability via eigenvalue computation

My Thesis Goal ⊲ Floquet-Fourier computation: over-counting of instability in wavenumber space ⊲ Lin’s DNS: two branches of disturbance Fourier modes ⊲ goal: to identify all physically unstable modes from Floquet-Fourier computation

My Thesis Goal ⊲ Floquet-Fourier computation: over-counting of instability in wavenumber space ⊲ Lin’s DNS: two branches of disturbance Fourier modes ⊲ goal: to identify all physically unstable modes from Floquet-Fourier computation

The Governing Equations Boussinesq Equations in Vorticity-Buoyancy Form Dη Db −N 2 w ∇ · � − b x u = 0 ; = ; = Dt Dt ⊲ incompressible, inviscid Boussinesq Fluid ⊲ Euler equations + weak density variation (the Boussinesq approximation) ⊲ Brunt-Vaisala frequency N : uniform stable stratification, N 2 > 0 ⊲ 2D velocity: � u ( x, z, t ) ; buoyancy: b ( x, z, t ) � u � − ψ z � � = − � ⊲ streamfunction: � u = ∇ × ψ ˆ y = w ψ x ⊲ vorticity: � y = ∇ 2 ψ ˆ ∇ × � u = η ˆ y

Exact Plane Gravity Wave Solutions buoyancy b buoyancy b z y + x − − + + − Dη Db −N 2 w = − b x = Dt Dt ⊲ dynamics of buoyancy & vorticity ⇒ oscillatory wave motions ⊲ exact plane gravity wave solutions     ψ − Ω d /K N 2  2 A sin( Kx + Mz − Ω d t ) b =    N 2 K/ Ω d η ⊲ primary wavenumbers: ( K, M ) N 2 K 2 ⊲ dispersion relation: Ω 2 d ( K, M ) = K 2 + M 2 .

Linear Stability Analysis ⊲ dimensionless exact plane wave + small disturbances ˜       ψ − Ω ψ  2 ǫ sin( x + z − Ω t ) ˜ b 1 = + b      η 1 / Ω η ˜ 1 ⊲ ǫ : dimensionless amplitude & dimensionless frequency: Ω 2 = 1 + δ 2 ⊲ linearized Boussinesq equations δ 2 ˜ ˜ ψ xx + ψ zz = ˜ η ˜ η + ˜ − ψ/ Ω , sin( x + z − Ω t ) ) ˜ η t + b x 2 ǫJ ( Ω˜ = 0 ˜ ˜ 2 ǫJ ( Ω˜ ˜ b t − ψ x − b + ψ , sin( x + z − Ω t ) ) = 0 ⊲ δ = K/M : related to the wave propagation angle (Lin: δ = 1 . 7 ) ⊲ Jacobian determinant � � f x g x � � J ( f, g ) = = f x g z − g x f z � � f z g z � �

Linear Stability Analysis ⊲ dimensionless exact plane wave + small disturbances ˜       ψ − Ω ψ  2 ǫ sin( x + z − Ω t ) ˜ b 1 = + b      η 1 / Ω η ˜ 1 ⊲ ǫ : dimensionless amplitude & dimensionless frequency: Ω 2 = 1 + δ 2 ⊲ linearized Boussinesq equations δ 2 ˜ ˜ ψ xx + ψ zz = ˜ η ˜ η + ˜ − ψ/ Ω , sin( x + z − Ω t ) ) η t ˜ + b x 2 ǫJ ( Ω˜ = 0 ˜ ˜ 2 ǫJ ( Ω˜ ˜ b t − ψ x − b + ψ , sin( x + z − Ω t ) ) = 0 ⊲ system of linear PDEs with non-constant, but periodic coefficients ⊲ analyzed by Floquet theory ⊲ classical textbook example: Mathieu equation (Chapter 3)

Floquet Theory: Mathieu Equation Mathieu Equation: d 2 u k 2 − 2 ǫ cos( t ) � � dt 2 + u = 0 ⊲ second-order linear ODE with periodic coefficients Floquet theory: u = e − iωt · p ( t ) = exponential part × co-periodic part ⊲ ⊲ Floquet exponent ω ( k ; ǫ ) : Im ω > 0 → instability ⊲ goal: to identify all unstable solutions in ( k, ǫ ) -space

Floquet Theory: Mathieu Equation Mathieu Equation: d 2 u k 2 − 2 ǫ cos( t ) � � dt 2 + u = 0 Two perspectives: perturbation analysis ⇒ two branches of Floquet exponent ⊲ ⊲ away from resonances: ω ( k ; ǫ ) ∼ ± k � k = 1 � : ω ( k ; ǫ ) ∼ ± 1 ⊲ resonant instability at primary resonance 2 + i ǫ 2 ⊲ Floquet-Fourier computation of ω ( k ; ǫ ) ⊲ a Riemann surface interpretation of ω ( k ; ǫ ) with k ∈ C

Floquet-Fourier Computation ⊲ Mathieu equation in system form: � u � � u � � � d 0 1 = i k 2 − 2 ǫ cos( t ) v 0 v dt ⊲ Floquet-Fourier representation: � u ∞ � = e − iωt · � c m e − imt � v   ... ... m = −∞    ...    S 0 ǫ M ⊲ ω ( k ; ǫ ) as eigenvalues of Hill’s bi-infinite matrix:     ...    ǫ M S 1    ⊲ 2 × 2 real blocks: S m and M   ... ... truncated Hill’s matrix: − N ≤ m ≤ N ⊲ ⊲ real-coefficient characteristic polynomial ⊲ compute 4 N + 2 eigenvalues: { ω n ( k ; ǫ ) } ǫ = 0 , eigenvalues from S n blocks: ω n ( k ; 0) = − n ± k & all real-valued ⊲ ⊲ ǫ ≪ 1 , complex eigenvalues may arise from ǫ = 0 double eigenvalues

Floquet-Fourier Computation ω n ( k ; ǫ ) : real • ; complex • ⊲ ε = 0.1 2 ⊲ ‘——’: ω n ( k ; 0) = − n ± k 1.5 ‘——’: ω 0 ( k ; 0) = ± k 1 ⊲ ω n ( k ; ǫ ) curves are close to ω n ( k ; 0) 0.5 Re ω 0 ⊲ two continuous curves close to ± k −0.5 ⊲ the rest are shifted due to −1 � u ∞ −1.5 � = e − i ( ω 0 + n ) t · � c m + n e − imt � v −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 k−axis m = −∞ ⊲ For each k , how many Floquet exponents are associated with the unstable solu- tions of Mathieu equation? two or ∞ ? Both! ⊲ two is understood from perturbation analysis ⊲ ∞ will be understood from the Riemann surface of ω ( k ; ǫ ) with k ∈ C

Floquet-Fourier Computation ω n ( k ; ǫ ) : real • ; complex • ⊲ ε = 0.1 2 ⊲ ‘——’: ω n ( k ; 0) = − n ± k ‘——’: ω 0 ( k ; 0) = ± k 1 ⊲ ω n ( k ; ǫ ) curves are close to ω n ( k ; 0) Re ω 0 ⊲ two continuous curves close to ± k ⊲ the rest are shifted due to −1 � u ∞ � = e − i ( ω 0 + n ) t · � c m + n e − imt � v −2 −2 −1 0 1 2 k−axis m = −∞ ⊲ For each k , how many Floquet exponents are associated with the unstable solu- tions of Mathieu equation? two or ∞ ? Both! ⊲ two is understood from perturbation analysis ⊲ ∞ will be understood from the Riemann surface of ω ( k ; ǫ ) with k ∈ C

A Riemann Surface Interpretation of ω ( k ; ǫ ) ε = 0.1 2 1 Re ω 0 −1 −2 −2 −1 0 1 2 k−axis ⊲ Floquet-Fourier computation with k ∈ C → the Riemann surface of ω ( k ; ǫ ) ⊲ surface height: real ω ; surface colour: imag ω ⊲ layers of curves for k ∈ R become layers of sheets for k ∈ C ⊲ the two physical branches belong to two primary Riemann sheets ⊲ How to identify the two primary Riemann sheets? ⊲ more understanding of how sheets are connected

A Riemann Surface Interpretation of ω ( k ; ǫ ) ⊲ zoomed view near Re k = 1 / 2 shows Riemann sheet connection ⊲ branch points: end points of instability intervals ⊲ loop around the branch points ⇒ √ type ⊲ branch cuts coincide with instability intervals (McKean & Trubowitz 1975)

A Riemann Surface Interpretation of ω ( k ; ǫ ) ⊲ zoomed view near Re k = 0 shows Riemann sheet connection ⊲ branch points: two on imaginary axis ⊲ loop around the branch points ⇒ √ type branch cuts to ± i ∞ give V-shaped sheets ⊲

A Riemann Surface Interpretation of ω ( k ; ǫ ) ε = 0.1 i √ 2 ǫ imag k − i √ 2 ǫ − 1 − 0 . 5 0 0 . 5 1 real k branch cuts: instability intervals & two cuts to ± i ∞ ⊲ ⊲ two primary sheets: upward & downward V-shaped sheets ⊲ associated with the two physically-relevant Floquet exponents ⊲ the other sheets are integer-shifts of primary sheets

A Riemann Surface Interpretation of ω ( k ; ǫ ) ε = 0.1 i √ 2 ǫ imag k − i √ 2 ǫ − 1 − 0 . 5 0 0 . 5 1 real k branch cuts: instability intervals & two cuts to ± i ∞ ⊲ ⊲ two primary sheets: upward & downward V-shaped sheets ⊲ associated with the two physically-relevant Floquet exponents ⊲ the other sheets are integer-shifts of primary sheets