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

?

?

?

Rossby scenario


Rossby scenario (Rossby, 1938 J. Mar. Res v.1; Gill, 1982)

€

DQDt

= ∂tQ +V.∇Q = 0

Looking for an adjusted steady statecorresponding to an initial state usinglagrangian conservation of PV

€

∇V.V + 2Ω× V = −1ρo

∇p

(we assume here no dissipation in the flow)

€

Qi =f +ωi

hiFor rotating shallow-water models the PV is equal to

and we should also satisfy the angular momentum (or mass) conservation :

€

L = rfv(rf )+rf2

2=rc2

2€

p = ρog*hand

geostrophic or gradient-wind balance


120 x 90 cm tank

Experimental setup

on rotating turntable

Unité de mécanique UME, ENSTA Palaiseau, France


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

€

Rd =g*ho2Ωo

€

g* =ρ1 − ρ2

ρ1

g

δ =ho/H<<1


Outcropping lensePV distribution

€

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


Outcropping lenseExperimental adjustment

Bu =(Rd/Rc)2= 0.11 α =h/Rd= 0.76 δ =h/H= 0.08


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


Outcropping lenseDissipatif mechanism

Small scale (3D) and transient instability at the initial stage of adjustment

Kinetic energy dissipation


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


PV patchPV distribution

Q

Rc

Q

RcRf

€

Qo=foho(r≥Rc )

PV discontinuity

€

Qin =fo

ho+Δh(r ≤ Rc )

Qo

anticyclonic cyclonic

Rf

Front propagation

Qo

Front propagation

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Qo=foho(r≥Rc )

€

Qin =fo

ho−Δh(r ≤ Rc )


PV patchExperimental adjustment

anticyclone Bu=0.2 λ=0.5

cyclone Bu=0.2 λ=0.5


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


PV patchExperiment vs Rossby adjustment

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Ro =Vmax

2ΩoRmax

Ro > 0 = cyclones

Ro < 0 = anticyclones

0.26

- 0.08

Ro << 1


Uniform PV frontExperimental configuration

ρ1 ρ2

ρ3

h

H

RcL

Three layers experiment !

€

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


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

€

Q = f /h0 = (ωi(x)+ f) /hi


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


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


Uniform PV frontFormation of small scale cyclones during the adjustment of large-scale front

Rc/Rd=3.3

Top view


Conclusions

€

DQDt

= 0+ ...

€

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.


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.

Geostrophic adjustment : the experimental reality · Geostrophic adjustment: the experimental...

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