Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Stability, cyclone-anticyclone asymmetryand frequency selection

in rotating shallow-water wakes

T. Dubos1, G. Perret1,2, A. Stegner1,J.-M. Chomaz3 and M. Farge1

1IPSL/Laboratoire de Meteorologie Dynamique, Ecole Normale Superieure, Paris2Laboratoire de Mécanique, Physique et Géosciences, Universite du Havre

3LadHyX, Ecole Polytechnique, Palaiseau

APS Division of Fluid Dynamics, 2007

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Is This a Von Karman Wake ?

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Outline

1 Cyclone - anticyclone asymmetryScales, Parameters, RegimesLinear SelectionNonlinear Evolution

2 Global Stability and Shedding Frequency SelectionLinear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Outline

1 Cyclone - anticyclone asymmetryScales, Parameters, RegimesLinear SelectionNonlinear Evolution

2 Global Stability and Shedding Frequency SelectionLinear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Model, Scales, ParametersOne-layer, rotating shallow-water dynamics

Scale with radius R of the obstacle, upstream velocity U.Define layer thickness H(1 + λψ) with ψ = O(1)

Viscosity⇒ Reynolds number Re = 2UR/ν

Gravity wave speed c2 = g′H Fr = U/c

Coriolis Ro = U/fR

Rossby deformation radius Rd = c/f Bu = (Rd/R)2 = (Ro/Fr)2

As Ro → 0, g′δh/L ∼ fU λ = Ro/Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Cyclone-Anticyclone Asymmetry

Rotating tank - salt water - PIVRo = 0.1, Bu = 1, λ = 0.1

Ro = 0.1, Bu = 0.1, λ = 1

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Cyclone-Anticyclone Asymmetry

Potential vorticity q =f + ω

H(1 + λψ)=

fH

“1− Ro

“∆ψ − Bu−1ψ + λψ∆ψ + . . .

””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

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Linear SelectionLinear stability analysis of symmetric parallel flow at geostrophic balance

velocity profile measured just

downstream the obstacle

Ro = 0.1, Bu = 1, λ = 0.1

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Scales, Parameters, RegimesLinear SelectionNonlinear Evolution

Nonlinear Evolution

DNS started fromsymmetric parallel flow

Experiments

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Linear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

Outline

1 Cyclone - anticyclone asymmetryScales, Parameters, RegimesLinear SelectionNonlinear Evolution

2 Global Stability and Shedding Frequency SelectionLinear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Linear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

Convective vs Absolute InstabilityIs a localized wave packet able to propagate upstream ?

The classical Von Karman vortex street appears after the velocity profilebehind the cylinder becomes absolutely unstable.

Convective Absolute

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Linear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

Spatio-Temporal Stability

Parallel basic flow measured in experiments just behind the cylindrical obstacle.

Feed with initial localized perturbatiom at x = x0 .

Growth rate σ(vg) along ray x = x0 + vg t .

Ro = 0.1, Bu = 1, λ = 0.1

Absolute instability ?

Ro = 0.1, Bu = 0.1, λ = 1

Convective instability

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Linear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

Self-Sustained Oscillator or Noise Amplifier ?

Ro = 0.35Bu = 0.9

Ro = 0.1Bu = 0.16

weakupstream

noise

strongerupstream

noise

−202

−202

−202

0 5 10 15 20 25 30−2

02

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Linear Spatio-Temporal StabilitySelf-Sustained Oscillator or Noise Amplifier ?Control parameter ?

Control parameter

Global stability diagram

10−2 10−1 10010−2

10−1

100

Fr

Ro

Bu=(R

o/Fr)

2 =cst

Unstable

Stable

Strouhal numberSt(Ro, Bu, Re = 200)

0.01 1 100100.10.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Bu

St

Von Karman

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes

Cyclone - anticyclone asymmetryGlobal Stability and Shedding Frequency Selection

Summary

Summary

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 > Buc ' 0.3 : Wake globally unstable, vortex street is aself-sustained oscillator

Bu < Buc ' 0.3 : Wake globally stable, amplifies upstream noise intovortex street, shedding frequency ' most convectively unstable mode

Strouhal number dominantly controlled by Bu

T. Dubos, G. Perret, A. Stegner, J.-M. Chomaz and M. Farge Rotating shallow-water wakes