2D vortices in fluids:2D vortices in fluids:Stability of vortical ensemblesStability of vortical ensembles

Ziv KiznerZiv Kizner

Tripole eddy observed in the Bay of Biscay Pingree & Le Cann, 1992

Tripole: A core vortex & two (n = 2) satellites

Lab experiment Semi-analytical solution

Flow pattern

Vorticity cross sections ,

core

satellite

Quadrupole, n = 3

core

satellite

Vorticity cross section

Pentapole, n = 4

r

Point-vortex concept

DD

L

v Stock’s theorem:

Point vortex:

DDDL

dxdydxdyy

u

x

vdxdy zvvds ˆ)(curl

L

r = R

0L

vds

cR

cR

Rr

22

vds

r

cU

Point-vortex ensemble: Autonomic dynamical system

x

y B

C

A rB,C

rA,C

rA,B

a

b

c

22CA

AC

BA

ABA

r

yyc

r

yybx

,,

22CA

AC

BA

ABA

r

xxc

r

xxby

,,

22CB

BC

BA

BAB

r

yyc

r

yyax

,,

22CB

BC

BA

BAB

r

xxc

r

xxay

,,

22CB

CB

CA

CAC

r

yyb

r

yyax

,,

22CB

CB

CA

CAC

r

xxb

r

xxay

,,

-1

2

-1

Tripole:

1. Invariants of a troika

Impulse:

Hamiltonian:

Stability of a point-vortex tripole

CBAx xxxP 2

CBAy yyyP 2

022 222222 BCACAByx rrrPPP

2. TripoleImpulse:

Hamiltonian:

02Pyx PP troika = tripole

2lnln3 llH )(Hmonotonic function of l

3. Tripole stabilityOn the iso-Hamiltonian sheet

BCACABBCACAB rrrrrr lnln2ln2 ),,(HH

)()( lHrrr BCACAB ,,Hthe tripole supplies a minimum to

P 2

tripole troika

-1

-1

+2

C

B

A

B

C

A

l

l

x

y

Tripole oscillator

y y

y

y

y

yy y

The end

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