Geostrophic Adjustment in an Axisymmetric Vortex - Atmospheric
Geostrophic adjustment : the experimental reality · Geostrophic adjustment: the experimental...
Transcript of Geostrophic adjustment : the experimental reality · Geostrophic adjustment: the experimental...
Alpine Summer School 2006: fronts, waves and vortices
Geostrophic adjustment :the experimental reality
Outcropping lens
Uniform PV front
A.Stegner, V. Mitkin, P. Bouruet-Aubertotcontact: stegner @ lmd.ens.fr
Cyclonic PV patch
Anticyclonic PV patch
Geostrophic adjustment: the experimental reality
Complexity
Time
Quasi-geostrophic+ Slow motions
Primitive equations (RSW)+ Fast motions(waves)
Geostrophic balanceSteady motions
Real world !Rotating and stratified 3D fluid
+ 3D motionsnon-hydrostaticdissipation…
Experimental reality
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Rossby scenario
Geostrophic adjustment: the experimental reality
Rossby scenario (Rossby, 1938 J. Mar. Res v.1; Gill, 1982)
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DQDt
= ∂tQ +V.∇Q = 0
Looking for an adjusted steady statecorresponding to an initial state usinglagrangian conservation of PV
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∇V.V + 2Ω× V = −1ρo
∇p
(we assume here no dissipation in the flow)
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Qi =f +ωi
hiFor rotating shallow-water models the PV is equal to
and we should also satisfy the angular momentum (or mass) conservation :
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L = rfv(rf )+rf2
2=rc2
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p = ρog*hand
geostrophic or gradient-wind balance
Geostrophic adjustment: the experimental reality
120 x 90 cm tank
Experimental setup
on rotating turntable
Unité de mécanique UME, ENSTA Palaiseau, France
Geostrophic adjustment: the experimental reality
ρ1
ρ2
ho
H
RcL
Outcropping lense
bottomless cylinder
Verticallaser sheet 532 nm
Fluorescent dye
Laser induced fluorescence (LIF) to visualize the density interface
Experimental configuration
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Rd =g*ho2Ωo
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g* =ρ1 − ρ2
ρ1
g
δ =ho/H<<1
Geostrophic adjustment: the experimental reality
Outcropping lensePV distribution
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Q =foho
=fo +ω(r)h(r)
; r ≤ Rf
Q
Rc
Q
Rc Rf
initial state (unbalanced) adjusted state
front propagation€
Q =foho
; r ≤ Rc
PV singularity
Geostrophic adjustment: the experimental reality
Outcropping lenseExperimental adjustment
Bu =(Rd/Rc)2= 0.11 α =h/Rd= 0.76 δ =h/H= 0.08
Geostrophic adjustment: the experimental reality
Outcropping lenseExperiment vs Rossby adjustment
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
-10 -5 0 5 10
t=2.5Tft=9.7Tfpredicted
h(cm
)
r(cm)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-8 -6 -4 -2 0 2 4 6 8
predictedt=2.6Tft=9.7Tf
v(cm
.s-1
)
r(cm)
accurate prediction on geopotential discrepency on velocity
Time averaged layer thickness and velocity profile
t=2.5Tf : significant wave activity (filter by averaging)t=10Tf : no more wave activity
Ro ~ 1
Geostrophic adjustment: the experimental reality
Outcropping lenseDissipatif mechanism
Small scale (3D) and transient instability at the initial stage of adjustment
Kinetic energy dissipation
Geostrophic adjustment: the experimental reality
PV patchExperimental configuration
ρ2
RcL
ρ1
ρ2
ρ1ρ1
ρ2 H
hoho
Δh
Δh > 0 anticyclonic PV patch Δh < 0 cyclonic PV patch
λ=Δh/ho= 0.5 δ =h/H= 0.08
shallow-water layers
Geostrophic adjustment: the experimental reality
PV patchPV distribution
Q
Rc
Q
RcRf
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Qo=foho(r≥Rc )
PV discontinuity
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Qin =fo
ho+Δh(r ≤ Rc )
Qo
anticyclonic cyclonic
Rf
Front propagation
Qo
Front propagation
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Qo=foho(r≥Rc )
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Qin =fo
ho−Δh(r ≤ Rc )
Geostrophic adjustment: the experimental reality
PV patchExperimental adjustment
anticyclone Bu=0.2 λ=0.5
cyclone Bu=0.2 λ=0.5
Geostrophic adjustment: the experimental reality
PV patchExperiment vs Rossby adjustment
-0,4
-0,35
-0,3
-0,25
-0,2
-0,15
-0,1
-0,05
0
0 2 4 6 8 10
V/fRd - predictedU/fRd - t=2TU/fRd - t=5TU/fRd - t=10TU/fRd - t=20T
U/fR
d
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0 2 4 6 8 10
V/fRd - predictedU/fRd - t=2TU/fRd - t=5TU/fRd - t=10TU/fRd - t=20T
U/fR
d
r/Rd
Kinetic energy dissipationin anticyclonic PV front
accurate prediction for cyclonic PV front
anticyclone
cyclone
Geostrophic adjustment: the experimental reality
PV patchExperiment vs Rossby adjustment
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Ro =Vmax
2ΩoRmax
Ro > 0 = cyclones
Ro < 0 = anticyclones
0.26
- 0.08
Ro << 1
Geostrophic adjustment: the experimental reality
Uniform PV frontExperimental configuration
ρ1 ρ2
ρ3
h
H
RcL
Three layers experiment !
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ρ2 −ρ1 << ρ3 −ρ2
-two thin upper layers having slightly different densities
-one deep and dense layer whichacts as neutral layer
Horizontal density gradient with uniform h => uniform PV front
ρ1ρ2
Geostrophic adjustment: the experimental reality
Uniform PV frontPV distribution
Initial state Baroclinic adjusted front
rc r1 r2
. .
rc
2 1 12
Q Q
Q = f/h0=1ω2=0ω1=0
ω1= η-1<0ω2= -η<0
η
η(r1)=1 η(r2)=0
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Q = f /h0 = (ωi(x)+ f) /hi
Geostrophic adjustment: the experimental reality
Uniform PV frontExperimental adjustment
t=0
t=0.5Tf
t=1 Tf
t= 1.5Tf
t= 3Tf
Initial state
Gravity current
Mean adjusted state
Baroclinic instability
Bu=(Rd/Rc)2=0.22 δ =h/H= 0.125
Geostrophic adjustment: the experimental reality
Uniform PV frontExperimental versus Rossby adjustment (t=1.5Tf)
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
z/H
r/Rd
Rc/Rd=2.1
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
ModelExperiment
V/fRd
r/Rd
Rc/Rd=3.3
Tilted front position
Upper velocity in the outer layer
Strong cyclonic gradient
ω/f ~ 5Ro ~ 1
Geostrophic adjustment: the experimental reality
Uniform PV frontFormation of small scale cyclones during the adjustment of large-scale front
Rc/Rd=3.3
Top view
Geostrophic adjustment: the experimental reality
Conclusions
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DQDt
= 0+ ...
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Q = (ω + f) /h
Lagrangian conservation of PV is a robust phenomenon even for full 3D rotating, stratified and dissipative flows.
Small-scale and non-hydrostatic instabilities could change the Rossby scenarioand dissipate a significant part of the initial energy of the system.
Such instabilities occurs especially for outcropping fronts (i.e.with PV singularity)where the density interface intersect an horizontal boundary.
Geostrophic adjustment: the experimental reality
Homeworks …
Assuming Rc>>Rd : from cylindrical to cartesian coordinates- x <=> r and y <=> θ- dh/dθ=0 <=> dh/dy=0 and Vθ(r=0)=0 <=> Vy(x=0)=0
Using the rotating shallow-water framework calculate the Rossby adjustedstate for the previous cases:
- an outcropping density anomaly
- a PV patch vortex
- a uniform PV front
Calculate the energy budget from the initial (potential energy only) to theadjusted state as a function of the burger number Bu.