Jan 04, 2016
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