Coherent vorticesin rotating geophysical flows

A. Provenzale, ISAC-CNR and CIMA, Italy

Work done with:Annalisa Bracco,

Jost von Hardenberg, Claudia Pasquero

A. Babiano, E. Chassignet, Z. Garraffo,J. Lacasce, A. Martin, K. Richards

J.C. Mc Williams, J.B. Weiss

Rapidly rotating geophysical flowsare characterized by the presence of

coherent vortices:

Mesoscale eddies, Gulf Stream Rings, Meddies

Rotating convective plumes

Hurricanes, the polar vortex, mid-latitude cyclones

Spots on giant gaseous planets

Vortices form spontaneouslyin rapidly rotating flows:

Laboratory experiments

Numerical simulations

Mechanisms of formation:Barotropic instabilityBaroclinic instability

Self-organization of a random field

Rotating tank at the “Coriolis” laboratory, Grenoble

diameter 13 m, min rotation period 50 sec

rectangular tank with size 8 x 4 mwater depth 0.9 m

PIV plus dye

Experiment done by A. Longhetto, L. Montabone, A. Provenzale,C. Giraud, A. Didelle, R. Forza, D. Bertoni

Characteristics of large-scale geophysical flows:

Thin layer of fluid: H << L

Stable stratification

Importance of the Earth rotation

Navier-Stokes equations in a rotating frame

sin2

),(,),(

0),,(

0

1

ˆ1

2

22

2

22

f

vuuwuV

spF

SinksSourcesDt

Dsz

wu

Dt

D

z

wwg

z

p

Dt

Dw

z

uuuzfp

z

uwuu

t

u

Dt

uD

Incompressible fluid: D/Dt = 0

),(,),(

0),,(

0

1

ˆ1

2

22

2

22

vuuwuV

spF

SinksSourcesDt

Dsz

wu

z

wwg

z

p

Dt

Dw

z

uuuzfp

z

uwuu

t

u

Dt

uD

Thin layer, strable stratification:hydrostatic approximation

uz

w

gz

p

z

ww

Dt

Dw

0

0

2

22

Homogeneous fluid with no vertical velocityand no vertical dependence of the horizontal velocity

xyvuu

u

uuzfpuut

uz

uw

,),(

0

ˆ1

,0,0

2

0

0

The 2D vorticity equation

2

2

2

0ˆ

ˆ

,),0,0(

utDt

D

uz

uzfut

u

The 2D vorticity equation

2222

2

2

,

t

utDt

D

In the absence of dissipation and forcing,quasigeostrophic flows conserve

two quadratic invariants:energy and enstrophy

dxdyV

Z

dxdyV

E

V

V

22

2

1

2

11

As a result, one has a direct enstrophy cascadeand an inverse energy cascade

Two-dimensional turbulence:the transfer mechanism

2221

21

2

2

21

21

EkEkEk

EkZ

ZZZ

EEE

As a result, one has a direct enstrophy cascadeand an inverse energy cascade

Two-dimensional turbulence:inertial ranges

3/5

3/22

3/13

)(

/1

)(

constant

kkE

lk

ludkkE

lul

u

As a result, one has a direct enstrophy cascadeand an inverse energy cascade

Two-dimensional turbulence:inertial ranges

3

22

2

2

)(

/1

)(

constant

kkE

lk

ludkkE

lul

uZ

As a result, one has a direct enstrophy cascadeand an inverse energy cascade

With small dissipation:

22

2

2

1

constant

tEZ

tE

Is this all ?

Vortices form,and dominate the dynamics

Vortices are localized, long-lived concentrations

of energy and enstrophy:Coherent structures

Vortex dynamics:

Processes of vortex formation

Vortex motion and interactions

Vortex merging: Evolution of the vortex population

Vortex dynamics:Vortex motion and interactions:

The point-vortex model

222 )()(

log4

1

jiji

ijjji

i

j

jj

j

jj

yyxxR

RH

x

H

dt

dy

y

H

dt

dx

ij

Vortex dynamics:Vortex merging and scaling theories

72.0

,,,

constant

constant

2/2/4/

2

22

42

tZttatN

a

aNZ

aNE

Max

Max

Max

Max

Vortex dynamics:

Introducing forcing to get a statistically-stationary turbulent flow

Ft

2222

,

Particle motion in a sea of vortices

xtYXv

dt

dY

ytYXu

dt

dX

tjtYtX

jjj

jjj

jj

),,(

),,(

timeatparticleththeofpositiontheis))(),((

Formally, a non-autonomous Hamiltonian systemwith one degree of freedom

Effect of individual vortices:Strong impermeability of the vortex edgesto inward and outward particle exchanges

Example: the stratospheric polar vortex

Global effects of the vortex velocity field:

Properties of the velocity distribution

Velocity pdf in 2D turbulence(Bracco, Lacasce, Pasquero, AP, Phys Fluids 2001)

Low Re High Re

Velocity pdf in 2D turbulence

Low Re High Re

Velocity pdf in 2D turbulence

Vortices Background

Velocity pdfs in numerical simulationsof the North Atlantic

(Bracco, Chassignet, Garraffo, AP, JAOT 2003)

Surface floats 1500 m floats

Velocity pdfs in numerical simulationsof the North Atlantic

A deeper look into the background:Where does non-Gaussianity come from

Vorticity is local but velocity is not:

xyvu

,),(

2

Effect of the far field of the vortices

Effect of the far field of the vortices

Background-induced Vortex-induced

Vortices play a crucial role onParticle dispersion processes:

Particle trapping in individual vortices

Far-field effects of theensemble of vortices

Better parameterization of particle dispersionin vortex-dominated flows

How coherent vortices affect primary productivity in the open ocean

Martin, Richards, Bracco, AP, Global Biogeochem. Cycles, 2002

yv

xu

tdt

d

HDwDZPZPg

Pg

dt

dD

ZZZPg

Pg

dt

dZ

PZPg

PgP

Nk

N

dt

dP

ZDPNk

NNNs

dt

dN

sDZP

ZZ

P

ZD

/)1(

)(

22

2

22

2

2

2

0

Oschlies and Garcon, Nature, 1999

Equivalent barotropic turbulence

Numerical simulation with a pseudo-spectral code

xv

yu

fR

q

DFqtq

,

],[

22

Three cases with fixed A (12%) and I=100:

“Control”: NO velocity field (u=v=0) (no mixing)

Case A: horizontal mixing by turbulence, upwelling in a single region

Case B: horizontal mixing by turbulence, upwelling in mesoscale eddies

29% more than in the no-mixing control case

139% more than in the no-mixing control case

The spatial distribution of the nutrient plays a crucial role, due to the presence of mesoscale structures

and the associated mixing processes

Models that do not resolve mesoscale features can severely underestimate primary production

Single particle dispersion

N

jjjjj tYtYtXtX

NttA

1

20

200

2 )]()([)]()([1

),(

For a smooth flow with finite correlation length

For a statistically stationary flow particle dispersion does not depend on t0

02

02 where)(),( ttAttA

regime)(brownianlargeat)(

regime)(ballisticsmallat2)(2

22

KA

EA

Single particle dispersion

N

jjjjj tYtYtXtX

NttA

1

20

200

2 )]()([)]()([1

),(

Time-dependent dispersion coefficient

regime)(brownianlargeat2)(

regime)(ballisticsmallat)(

2

)()(

20

2

2

LTKK

K

AK

Properties of single-particle dispersionin 2D turbulence

(Pasquero, AP, Babiano, JFM 2001)

Parameterization of single-particle dispersion:Ornstein-Uhlenbeck (Langevin) process

)/exp(1(12)(

2exp

2

1)(

)/exp()()()(

)(2)'()(

0

)(

2

2

2

0

2/1

LLL

L

LL

TTTK

uup

TtutuR

dttttdWtdW

dW

dWT

dtT

udu

dtuUdX

Properties of single-particle dispersionin 2D turbulence

Parameterization of single-particle dispersion:Langevin equation

Parameterization of single-particle dispersion:Langevin equation

Why the Langevin model is not working:The velocity pdf is not Gaussian

Why the Langevin model is not working:The velocity autocorrelation is not exponential

Parameterization of single-particle dispersionwith a non-Gaussian velocity pdf:

A nonlinear Langevin equation(Pasquero, AP, Babiano, JFM 2001)

dttttdWtdW

dW

dWT

dtu

u

Tdu

LL

)(2)'()(

0

)/1(2

/2

0

2/122

2

Parameterization of single-particle dispersionwith a non-Gaussian velocity pdf:A nonlinear Langevin equation

The velocity autocorrelation of the nonlinear model

is still almost exponential

A two-component process:vortices (non-Gaussian velocity pdf)background (Gaussian velocity pdf)

TL (vortices) << TL (background)

'

)/1(2

/2

2/1

2/122

2

dWT

dtT

udu

dWT

dtu

u

Tdu

uuu

B

B

B

BB

V

V

VV

VV

VV

BV

V

V

A two-component process:

Geophysical flows are neither homogeneousnor two-dimensional

A simplified model:The quasigeostrophic approximation

= H/L << 1 neglect of vertical accelerations hydrostatic approximation

Ro = U / f L << 1 neglect of fast modes (gravity waves)

A simplified model:The quasigeostrophic approximation

z

gzN

zzN

f

zq

xv

yu

Dissqt

q

y

qv

x

qu

t

q

Dt

Dq

)(

)(

,

,

2

2

22

Simulation by Jeff Weiss et al