2d o hydrodynamic instability r n o t 2d? - summer school … · 2006-11-10 · t 2d? jean-marc...

Aoste 2006

2D o

r no

t 2D?

Jean-Marc C

homaz,

Pantxika O

thguy, Paul B

illant

& F

rançois Gallaire

Ladhyx,

CN

RS

/École polytechnique

Palaiseau, F

rance

Hydrodynam

ic instability of quasi tw

o-dimensional flow

s

Rough S

ea at Naruto in A

wa P

rovinceAndo Hiroshige

Aoste 2006

2

Surface vortices on the ocean

Plankton

Tasm

ania

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3

Meddies (M

editerranean eddy)

Cam

paigns AR

CA

NE

-CA

MB

IOS

IFR

EM

ER

1998

Arm

i et al nature 1988

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4

Atm

osphere, Atm

ospheres

Jupiter

© N

asaNeptune

Terre

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5

Saturne, N

AS

A, C

assini spacecraft imaging, betw

een 02/22 and 03/22/2004

Atm

osphere, Atm

ospheres ©

Nasa

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6

Mars A

ttacks ©

Nasa

Long colomnar vortex: dust D

evil

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Existence of 2D

vortices

yS

o many reasons to stay 2D

xT

aylor Proudm

an column

xS

tratification

xS

hallow flow

yS

o many instabilities to go 3D

xS

hear instability

xC

entrifugal instability

xE

lliptic instability

xH

yperbolic instability

xZ

igzag instability

xB

oundary layer instability

1

1

3DQ

uasi2D

Taylor P

roudman

column

Quasi

Geostrophic

Ro=U/ 2W L

h

F=U/ N L

h

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Instabilities of a infinite vortex

Axisym

metric

2D3D

m=

0

2p/k

Perturbations e

i(kx+m

q-wt), s

=-iw

Basic state

2p/m

Uq

Ux

k=0

m≠0, k≠0

Aoste 2006

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Stability of isolated vortices

zInertial gravity w

aves (dimensional analysis on the board)

Inertial wave: (q, angle of k w

ith the vertical)

w 2 = (2W

) 2cos 2q outside the vortexw

2 = (z + 2W ) 2cos 2q inside the vortex

Frequency between 0 and 2W

Trapped w

ave if z>0

Inertial gravity wave:

w 2 = N

2sin2q + (2W

) 2cos 2q outsidew

2 = N 2sin

2q + (z + 2W ) 2cos 2q inside

Trapped w

ave if z >0 and N

< 2W

orz <0 and N > 2W

Aoste 2006

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Stability of isolated vortices

dz /dr =0

Kloosterziel &

van Heijst (1991)

z the vorticity

z =

1/r d(ruq )/dr

yInflexion point (R

ayleigh)

zS

hear instability 2D

z

rr

uq

Aoste 2006

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Stability of isolated vortices

zS

hear instability 2D

Rayleigh E

quation

z

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12

Stability of isolated vortices

zS

hear instability 2DR

ayleigh Equations

z

z

z

Aoste 2006

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Stability of isolated vortices

yS

hear instability 2D physical m

echanism

yS

quire theorem2D

mode m

ost unstable fi k =

0

k // to u and ^ to the shear

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That’s allFor today