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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