Stability, cyclone-anticyclone asymmetry and frequency selection in rotating shallow-water wakes

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1 Stability, cyclone-anticyclone asymmetry and frequency selection in rotating shallow-water wakes T. Dubos 1, G. Perret 1,2, A. Stegner 1, J.-M. Chomaz 3 and M. Farge 1 1 IPSL/Laboratoire de Meteorologie Dynamique, Ecole Normale Superieure, Paris 2 Laboratoire de Mécanique, Physique et Géosciences, Universite du Havre 3 LadHyX, Ecole Polytechnique, Palaiseau APS Division of Fluid Dynamics, 2007

2 Is This a Von Karman Wake?

3 Outline 1 Scales, Parameters, Regimes Linear Selection Nonlinear Evolution 2 Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter?

4 Outline Scales, Parameters, Regimes Linear Selection Nonlinear Evolution 1 Scales, Parameters, Regimes Linear Selection Nonlinear Evolution 2 Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter?

5 Model, Scales, Parameters One-layer, rotating shallow-water dynamics Scales, Parameters, Regimes Linear Selection Nonlinear Evolution Scale with radius R of the obstacle, upstream velocity U. Define layer thickness H(1 + λψ) with ψ = O(1) Viscosity Reynolds number Gravity wave speed c 2 = g H Coriolis Re = 2UR/ν Fr = U/c Ro = U/fR Rossby deformation radius R d = c/f Bu = (R d /R) 2 = (Ro/Fr) 2 As Ro 0, g δh/l fu λ = Ro/Bu

6 Cyclone-Anticyclone Asymmetry Scales, Parameters, Regimes Linear Selection Nonlinear Evolution Rotating tank - salt water - PIV Ro = 0.1, Bu = 1, λ = 0.1 Ro = 0.1, Bu = 0.1, λ = 1

7 Cyclone-Anticyclone Asymmetry Scales, Parameters, Regimes Linear Selection Nonlinear Evolution Potential vorticity q = f + ω H(1 + λψ) = f 1 Ro ψ Bu 1 ψ + λψ ψ +... H λ = Ro/Bu controls cyclone-anticyclone asymmetry quasi-geostrophic : Bu = O(1), λ = O(Ro) 1 frontal-geostrophic : λ = O(1), Bu = O(Ro) 1 incompressible 2D : λ 1 and Bu 1

8 Scales, Parameters, Regimes Linear Selection Nonlinear Evolution Linear Selection Linear stability analysis of symmetric parallel flow at geostrophic balance velocity profile measured just downstream the obstacle Ro = 0.1, Bu = 1, λ = 0.1

9 Nonlinear Evolution Scales, Parameters, Regimes Linear Selection Nonlinear Evolution DNS started from symmetric parallel flow Experiments

10 Outline Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter? 1 Scales, Parameters, Regimes Linear Selection Nonlinear Evolution 2 Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter?

11 Convective vs Absolute Instability Is a localized wave packet able to propagate upstream? Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter? The classical Von Karman vortex street appears after the velocity profile behind the cylinder becomes absolutely unstable. Convective Absolute

12 Spatio-Temporal Stability Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter? Parallel basic flow measured in experiments just behind the cylindrical obstacle. Feed with initial localized perturbatiom at x = x 0. Growth rate σ(v g) along ray x = x 0 + v gt. Ro = 0.1, Bu = 1, λ = 0.1 Ro = 0.1, Bu = 0.1, λ = 1 Absolute instability? Convective instability

13 Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter? Self-Sustained Oscillator or Noise Amplifier? Ro = 0.35 Bu = 0.9 Ro = 0.1 Bu = 0.16 weak upstream noise stronger upstream noise

14 Control parameter Linear Spatio-Temporal Stability Self-Sustained Oscillator or Noise Amplifier? Control parameter? 10 0 Global stability diagram Unstable Strouhal number St(Ro, Bu, Re = 200) 0.55 Ro 10 1 Bu=(Ro/Fr) 2 =cst St Von Karman Stable Fr Bu

15 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

16 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

17 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

18 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

19 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

20 Cyclone-Anticyclone Asymmetry Controlled by λ = Ro/Bu = O(1) Linear instability of parallel wake flow favors anticyclones Strong anticyclones prevent further growth of cyclones Self-Sustained Oscillator / Noise Amplifier Bu > Bu c 0.3 : Wake globally unstable, vortex street is a self-sustained oscillator Bu < Bu c 0.3 : Wake globally stable, amplifies upstream noise into vortex street, shedding frequency most convectively unstable mode Strouhal number dominantly controlled by Bu

Inertial instability of von Kármán street in a rotating shallow-water layer

Inertial instability of von Kármán street in a rotating shallow-water layer Samuel Teinturier, Alexandre Stegner, Michael Ghil, Samuel Viboud & Henri Didelle Laboratoire de Météorologie Dynamique, CNRS,