iutam conference on turbulence in the atmosphere and oceans
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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 - PowerPoint PPT PresentationTRANSCRIPT
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Vertical alignment of geostrophic vorticeswith external strain and rotation
Xavier Carton, Xavier Perrot, Alan Guillou* Universite de Bretagne Occidentale, Brest
Isaac Newton Institute for Mathematical Sciences, Cambridge, UKDecember 8-12, 2008
(*) On leave of absence from Universite d'Orsay
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
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
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.
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).
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.
IUTAM Conference on Turbulence in the Atmosphere and Oceans
From J.Mc Williams, JFM, 1989
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
We use a two-layer QG model with external strain and rotation
IUTAM Conference on Turbulence in the Atmosphere and Oceans
General situation of the present study
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Point vortex model
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Results
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
IUTAM Conference on Turbulence in the Atmosphere and Oceans
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)
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
What happens when increases again (Poincare sections, =10^-3, 10^-1) ?
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Zoom on Poincare section, =10^-1, resonance islands
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
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Rd/R=0.5 =+/-2s <0 /4.
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Steady states Slight erosion
of the vortex, weak motion on long time
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
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
IUTAM Conference on Turbulence in the Atmosphere and Oceans
Corotation around the center of the plane
Vortex erosion
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
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)