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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
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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
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Vortices form spontaneouslyin rapidly rotating flows:
Laboratory experiments
Numerical simulations
Mechanisms of formation:Barotropic instabilityBaroclinic instability
Self-organization of a random field
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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
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Characteristics of large-scale geophysical flows:
Thin layer of fluid: H << L
Stable stratification
Importance of the Earth rotation
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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
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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
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Thin layer, strable stratification:hydrostatic approximation
uz
w
gz
p
z
ww
Dt
Dw
0
0
2
22
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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
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The 2D vorticity equation
2
2
2
0ˆ
ˆ
,),0,0(
utDt
D
uz
uzfut
u
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The 2D vorticity equation
2222
2
2
,
t
utDt
D
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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
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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
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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
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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
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With small dissipation:
22
2
2
1
constant
tEZ
tE
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Is this all ?
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Vortices form,and dominate the dynamics
Vortices are localized, long-lived concentrations
of energy and enstrophy:Coherent structures
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Vortex dynamics:
Processes of vortex formation
Vortex motion and interactions
Vortex merging: Evolution of the vortex population
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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
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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
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Vortex dynamics:
Introducing forcing to get a statistically-stationary turbulent flow
Ft
2222
,
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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
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Effect of individual vortices:Strong impermeability of the vortex edgesto inward and outward particle exchanges
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Example: the stratospheric polar vortex
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Global effects of the vortex velocity field:
Properties of the velocity distribution
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Velocity pdf in 2D turbulence(Bracco, Lacasce, Pasquero, AP, Phys Fluids 2001)
Low Re High Re
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Velocity pdf in 2D turbulence
Low Re High Re
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Velocity pdf in 2D turbulence
Vortices Background
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Velocity pdfs in numerical simulationsof the North Atlantic
(Bracco, Chassignet, Garraffo, AP, JAOT 2003)
Surface floats 1500 m floats
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Velocity pdfs in numerical simulationsof the North Atlantic
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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
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Effect of the far field of the vortices
Background-induced Vortex-induced
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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
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How coherent vortices affect primary productivity in the open ocean
Martin, Richards, Bracco, AP, Global Biogeochem. Cycles, 2002
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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
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Equivalent barotropic turbulence
Numerical simulation with a pseudo-spectral code
xv
yu
fR
q
DFqtq
,
],[
22
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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
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29% more than in the no-mixing control case
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139% more than in the no-mixing control case
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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
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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
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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
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Properties of single-particle dispersionin 2D turbulence
(Pasquero, AP, Babiano, JFM 2001)
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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
![Page 60: Coherent vortices in rotating geophysical flows A.Provenzale, ISAC-CNR and CIMA, Italy Work done with: Annalisa Bracco, Jost von Hardenberg, Claudia Pasquero](https://reader035.vdocument.in/reader035/viewer/2022062308/56649f0d5503460f94c207be/html5/thumbnails/60.jpg)
Properties of single-particle dispersionin 2D turbulence
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Parameterization of single-particle dispersion:Langevin equation
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Parameterization of single-particle dispersion:Langevin equation
![Page 63: Coherent vortices in rotating geophysical flows A.Provenzale, ISAC-CNR and CIMA, Italy Work done with: Annalisa Bracco, Jost von Hardenberg, Claudia Pasquero](https://reader035.vdocument.in/reader035/viewer/2022062308/56649f0d5503460f94c207be/html5/thumbnails/63.jpg)
Why the Langevin model is not working:The velocity pdf is not Gaussian
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Why the Langevin model is not working:The velocity autocorrelation is not exponential
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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
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Parameterization of single-particle dispersionwith a non-Gaussian velocity pdf:A nonlinear Langevin equation
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The velocity autocorrelation of the nonlinear model
is still almost exponential
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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
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A two-component process:
![Page 70: Coherent vortices in rotating geophysical flows A.Provenzale, ISAC-CNR and CIMA, Italy Work done with: Annalisa Bracco, Jost von Hardenberg, Claudia Pasquero](https://reader035.vdocument.in/reader035/viewer/2022062308/56649f0d5503460f94c207be/html5/thumbnails/70.jpg)
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Geophysical flows are neither homogeneousnor two-dimensional
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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)
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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
![Page 74: Coherent vortices in rotating geophysical flows A.Provenzale, ISAC-CNR and CIMA, Italy Work done with: Annalisa Bracco, Jost von Hardenberg, Claudia Pasquero](https://reader035.vdocument.in/reader035/viewer/2022062308/56649f0d5503460f94c207be/html5/thumbnails/74.jpg)
Simulation by Jeff Weiss et al
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