IUTAM Conference on Turbulence in the Atmosphere and Oceans

1. Vertical alignment of geostrophic vortices with external strain and rotation Xavier Carton, Xavier Perrot, Alan Guillou* Universite de Bretagne Occidentale, Brest Isaac Newton Institute for Mathematical Sciences, Cambridge, UK December 8-12, 2008 (*) On leave of absence from Universite d'Orsay IUTAM Conference on Turbulence in the Atmosphere and Oceans

2. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Vertical alignment occurs when two like signed vortices at different depths/altitudes andinitially separated join their central axes ; alignment can be monotonic or oscillatory. • This process depends on the initial 3D structure of PV, on stratification, and on external flow. • We will address this latter influence here. • Firstly, we recall observations and previous results

3. IUTAM Conference on Turbulence in the Atmosphere and Oceans • In the atmosphere • Observations and models of (re)-alignment of weak tornado-like vortices (with broad vorticity distributions) Willoughby, JAS47, 1990 – Reasor and Montgomery, JAS58, 2001 ---> • For R/Rd<1, a trapped quasi-mode with azimuthal wavenumber 1 propagates around the vortex column and prevents its realignment • For R/Rd>1, this mode disappears in the continuous spectrum of Rossby waves and alignment via redistribution of PV via sheared RW

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5. IUTAM Conference on Turbulence in the Atmosphere and Oceans • In the ocean • Observations and model of deep anticyclonic vortices (meddies) drifting meridionally under and across a zonal jet (the Azores Current). Anticyclonic meanders form on the jet and align with the meddies – Tychensky and Carton, JGR, 1998 ; Vandermeirsch, Carton, Morel, DAO 2003. • Observation and models of two vortex alignment near the East Australian Current, Creswell and Legeckis, DSR34, 1986 ; Nof and Dewar, DSR41, 1994. Alignment is a relatively slow process (several turnover times) for nonlinear vortices (lens eddies) involving the formation of ”arms” circling the vortices (inertial mechanism) and final oscillations of the aligned vortex.

6. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Initially two distinct lens eddies (Leo and Maria) drift towards each other near the East Australian Current • Finally the two eddies are aligned (see below).

7. IUTAM Conference on Turbulence in the Atmosphere and Oceans • In geophysical turbulence • Vertical alignment is related to the barotropization of vortices in geostrophic turbulence (Rhines, 79; Salmon, 80; Mc Williams, 89-90). • Process study by Polvani (JFM225, 1989) in a two-layer QG model : the alignment of two constant-PV vortices occurs for Rd < R and d < 3.3 R (equal layer thicknesses). • Sutyrin et al (JFM357, 1998) study the alignment of thin-core vortices in a continuously stratified QG model : there is also a critical vertical distance between the two vortices that separates alignment from co-rotation. • But vortex alignment often occurs in the presence of other vort.

8. IUTAM Conference on Turbulence in the Atmosphere and Oceans • From J.Mc Williams, JFM, 1989

9. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Outline : • Analysis of vortex trajectories in a point vortex model : equilibria, stability, resonance, chaos • Alignment of initially circular vortices with piecewise-constant PV (steady external strain) – numerical results • Influence of unsteady strain • Conclusions

10. IUTAM Conference on Turbulence in the Atmosphere and Oceans • We use a two-layer QG model with external strain and rotation

11. IUTAM Conference on Turbulence in the Atmosphere and Oceans • General situation of the present study

12. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Point vortex model

13. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Results

14. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Resonance with oscillatory strain and rotation • Multiple time scale expansion of the point-vortex equations with time-periodic strain and rotation • First-order response varies with the neutral oscillation period and with the forcing period --> primary resonance when they match (harmonic frequency)‏ • Second-order response leads to secondary resonance which is subharmonic • We present here the resonance in the vicinity of the harmonic frequency

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17. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Comparison of numerical (RK4) integration of point vortex trajectory with the solution of the amplitude equation (blue)‏

18. IUTAM Conference on Turbulence in the Atmosphere and Oceans • What happens when  increases ? Vortex trajectories move from inside to outside of the neutral trajectory • When  < c when  > c

19. IUTAM Conference on Turbulence in the Atmosphere and Oceans • What happens when  increases again (Poincare sections, =10^-3, 10^-1) ?

20. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Zoom on Poincare section, =10^-1, resonance islands

21. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Growth of chaos : destabilization of the heteroclinic trajectories (near the hyperbolic points) – confirmed by Melnikov theory • The chaotic domain grows from these regions by successive destabilization of KAM tori (with cantori and chaos)‏ • Vortex trajectories around the neutral points have increasing radii with growing  ; finally, they reach the chaotic domain

22. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Finite-area vortex model (h1=h2)‏ • 1/ = Rd (with R=1)‏ • No strain, no external rotation • Polvani, JFM 1991

23. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Rd/R=0.5 • =+/-2s <0 • /4.

24. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Alignment • In white the upper layer PV • In colors, the Okubo-Weiss criterion value • Alignment is fast,but vortex ellipticity and filaments remain longer

25. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Steady states • Slight erosion of the vortex, weak motion on long time

26. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Oscillation near steady states • Vortex moves away along extension axis and closer to origin along compression axis • Progressive erosion of the vortex

27. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Unstable alignment at large initial distances • The vortex reaches the center along the compres-sion axis, but does not align

28. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Corotation around the center of the plane • Vortex erosion

29. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Influence of time-varying strain and rotation ? • =0.25 : limited influence at small initial distances, but more straining out of the vortices at larger distances. • Steady states were not observed but very slow oscillation regimes. • Observation of corotating figure 8 equilibria (small distances, weak strain)‏ • Regime of « unstable alignment » obtained at smaller distances • =0.5 : vortex erosion is again increased

30. IUTAM Conference on Turbulence in the Atmosphere and Oceans • Conclusions • Point vortex study useful to determine steady states, equilibria and resonances. Oscillation with slowly varying amplitude around neutral points. Chaos grows from heteroclinic trajectories and fills the plane • Phenomenology of alignment with strain and rotation is richer than in isolation. Steady states are recovered as well as oscillations. The « unstable alignment » from large distances does not succeed contrary to 2D merger with strain. • Influence of unsteady strain : fewer equilibria, more oscillations and erosion • Extend to 3DQG (and to coupled SQG-3DQG)‏