Transcript Eulerian/Lagrangian relationships in 2D turbulence plus a

Coherent vortices
in 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 flows
are 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 spontaneously
in rapidly rotating flows:
Laboratory experiments
Numerical simulations
Mechanisms of formation:
Barotropic instability
Baroclinic 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 m
water 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



2
 2  u 

Du u  
u
1

 u  u  w
  p  f zˆ  u     u  2 
Dt t
z

z 

 2
Dw
1 p
2w 

 g     w  2 
Dt
 z
z 

D
w
   u  
0
Dt
z
Ds
 Sources Sinks
Dt
F (  , p, s )  0



V  (u , w) , u  (u, v)
f  2 sin 
Incompressible fluid: D/Dt = 0



2
 2  u 

Du u  
u
1

 u  u  w
  p  f zˆ  u     u  2 
Dt t
z

z 

 2
Dw
1 p
2w 

 g     w  2 
Dt
 z
z 

w
 u 
0
z
Ds
 Sources Sinks
Dt
F (  , p, s )  0



V  (u , w) , u  (u , v)
Thin layer, strable stratification:
hydrostatic approximation
Dw
0
Dt
 2
2w 
   w  2   0
z 

p
  g
z

w
   u
z
Homogeneous fluid with no vertical velocity
and no vertical dependence of the horizontal velocity

u
w0 ,
 0 ,   0
z


u  
1
2
 u  u   p  f zˆ  u    u
t
0
 u  0
   


u  (u, v)   
,
 y x 
The 2D vorticity equation

  u  (0,0,  ) ,    2

 
 u    f   zˆ  u     2
t

  zˆ  u   0
D  

 u      2
Dt
t
The 2D vorticity equation
   2
D  

 u      2
Dt
t
 2
  ,  2    2 2
t


In the absence of dissipation and forcing,
quasigeostrophic flows conserve
two quadratic invariants:
energy and enstrophy
1
1
2
E 
 dxdy
VV 2
1
Z
V
   
2
2
dxdy
V
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
the transfer mechanism
E  E1  E2
Z  Z1  Z 2
Z  k 2E
k 2 E  k12 E1  k 22 E2
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
inertial ranges
 u3

 const ant  u  l 1/ 3

l
E (k )dk  u 2  l 2 / 3
k  1/ l
E ( k )  k 5 / 3
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
inertial ranges
u2
 2  const ant  u  l
 l
E (k )dk  u 2  l 2
k  1/ l
Z
E ( k )  k 3
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
With small dissipation:
E
2
t
2
 const ant
1
Z  E  2
t
2
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
H
j

dt
y j
dx j
H
j

dt x j
dy j
1
H
4

i j
i
j
log Rij
R ij2  ( xi  x j ) 2  ( yi  y j ) 2
Vortex dynamics:
Vortex merging and scaling theories
2
E  N  Max
a 4  constant
2
Z  N  Max
a2
   Max a 2
 Max  constant
N  t 
, a  t / 4
,   t / 2
  0.72
, Z  t  / 2
Vortex dynamics:
Introducing forcing
to get a statistically-stationary turbulent flow
 2
  ,  2    2 2  F
t


Particle motion
in a sea of vortices
( X j (t ),Y j (t )) is the positionof the j  th particleat timet

 u( X j , Y j , t )  
dt
y
dYj

 v( X j , Y j , t ) 
dt
x
dX j
Formally, a non-autonomous Hamiltonian system
with one degree of freedom
Effect of individual vortices:
Strong impermeability of the vortex edges
to 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 simulations
of the North Atlantic
(Bracco, Chassignet, Garraffo, AP, JAOT 2003)
Surface floats
1500 m floats
Velocity pdfs in numerical simulations
of the North Atlantic
A deeper look into the background:
Where does non-Gaussianity come from
Vorticity is local but velocity is not:
   2
   

(u , v)   
,
 y x 
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 on
Particle dispersion processes:
Particle trapping in individual vortices
Far-field effects of the
ensemble of vortices
Better parameterization of particle dispersion
in vortex-dominated flows
How coherent vortices affect
primary productivity in the open ocean
Martin, Richards, Bracco, AP, Global Biogeochem. Cycles, 2002
dN
N
 s (N0  N )  
P  DD   Z Z
dt
kN
dP
N
g P 2

P
Z  P P
2
dt
kN
g P
dZ
g P 2
2

Z   Z Z  Z Z
2
dt
g P
dD
g P
2
 (1   )
Z   P P   Z Z   D D  ws D / H
2
dt
g P
d



 u v
dt t
x
y
2
Oschlies and Garcon, Nature, 1999
Equivalent barotropic turbulence
q
 [ , q]  F  D
t
q 
2

u
y

R
2
f

, v
x
Numerical simulation
with a pseudo-spectral code
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
1
A2 (t , t0 )   [ X j (t )  X j (t0 )]2  [Y j (t )  Y j (t0 )]2
N j 1

For a statistically stationary flow
particle dispersion does not depend on t0
A2 (t, t0 )  A2 ( ) where   t  t0
For a smooth flow with finite correlation length
A2 ( )  2 E 2
at small
(ballistic regime)
A2 ( )  K
at large 
(brownian regime)
Single particle dispersion

N
1
A2 (t , t0 )   [ X j (t )  X j (t0 )]2  [Y j (t )  Y j (t0 )]2
N j 1

Time-dependent dispersion coefficient
A2 ( )
K ( ) 
2
K ( )   2
at small
(ballistic regime)
K ( )  K 0  2 2TL
at large 
(brownian regime)
Properties of single-particle dispersion
in 2D turbulence
(Pasquero, AP, Babiano, JFM 2001)
Parameterization of single-particle dispersion:
Ornstein-Uhlenbeck (Langevin) process
dX  (U  u )dt
u

du   dt  1/ 2 dW
TL
TL
dW  0
dW (t )dW (t ' )  2 (t  t0 )dt
R( )  u (t ) u (t   )  exp( / TL )
 u2 
1

p (u ) 
exp 
2 
2 
 2 
 TL (1  exp( / TL ) 
2
K ( )  2 TL 1 




Properties of single-particle dispersion
in 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 dispersion
with a non-Gaussian velocity pdf:
A nonlinear Langevin equation
(Pasquero, AP, Babiano, JFM 2001)
du  
2
2  u /
TL 2 (1  u /  )
2
2
dt 

1/ 2
L
T
dW  0
dW (t )dW (t ' )  2 (t  t0 )dt
dW
Parameterization of single-particle dispersion
with 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)
u  uV  u B
duV  

2
V
2  uV /  V
TV 2 V (1  uV /  V )
2
2
dt 
uB
B
duB   dt  1/ 2 dW '
TB
TB
V
1/ 2
V
T
dW
A two-component process:
Geophysical flows are
neither homogeneous
nor 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
Dq q
q
q q

u v

  , q   Diss
Dt t
x
y t


u
, v
y
x
2

f
 

2
q 
z  N 2 ( z ) z
g 
2
N ( z)  
 z




Simulation by Jeff Weiss et al