IUTAM Conference on Turbulence in the Atmosphere and Oceans

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


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


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



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.


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).


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.


From J.Mc Williams, JFM, 1989


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


We use a two-layer QG model with external strain and rotation


General situation of the present study


Point vortex model


Results


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




Comparison of numerical (RK4) integration of point vortex trajectory with the solution of the amplitude equation (blue)


What happens when increases ? Vortex trajectories move from inside to outside of the neutral trajectory

When < c when > c


What happens when increases again (Poincare sections, =10^-3, 10^-1) ?


Zoom on Poincare section, =10^-1, resonance islands


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


Finite-area vortex model (h1=h2)

1/ = Rd (with R=1)

No strain, no external rotation

Polvani, JFM 1991


Rd/R=0.5 =+/-2s <0 /4.


Alignment In white the

upper layer PV

In colors, the Okubo-Weiss criterion value

Alignment is fast,but vortex ellipticity and filaments remain longer


Steady states Slight erosion

of the vortex, weak motion on long time


Oscillation near steady states

Vortex moves away along extension axis and closer to origin along compression axis

Progressive erosion of the vortex


Unstable alignment at large initial distances

The vortex reaches the center along the compres-sion axis, but does not align


Corotation around the center of the plane

Vortex erosion


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


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)

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