2D o Hydrodynamic instability r n o t 2D? - Summer School … · 2006-11-10 · t 2D? Jean-Marc...
Transcript of 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
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Surface vortices on the ocean
Plankton
Tasm
ania
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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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Atm
osphere, Atm
ospheres
Jupiter
© N
asaNeptune
Terre
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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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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
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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
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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
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Stability of isolated vortices
zS
hear instability 2D
Rayleigh E
quation
z
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Stability of isolated vortices
zS
hear instability 2DR
ayleigh Equations
z
z
z
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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