Eulerian/Lagrangian relationships in 2D turbulence plus a
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Transcript Eulerian/Lagrangian relationships in 2D turbulence plus a
The barotropic vorticity equation (with free
surface)
u
y
, v
x
, 2
D 2
L2
2 y 0
Dt
LD
2
L2 2
L2
0
2 , 2
t
LD
LD
x
Barotropic Rossby waves (rigid lid)
u
y
, v
x
, 2
D 2
y 0
Dt
2
2
,
0
t
x
u U u'
v v'
(y) ' U y '
Barotropic Rossby waves (rigid lid)
'
'
u U u'
, v v'
, ' 2 '
y
y y
x
(y) ' U y '
2
'
2
'U '
0
t
x
x
Barotropic Rossby waves (rigid lid)
2
'
2
'U '
0
t
x
x
' expik x il y i t
k
U
k 2 l2
Rossby waves
The 2D vorticity equation
( f plane, no free-surface effects )
u
y
, v
x
, 2
2
, 2 D F
t
In the absence of dissipation and forcing,
2D barotropic flows conserve
two quadratic invariants:
energy and enstrophy
1
E
A
A
1
Z
A
1 2
1
2
u v dxdy
2
A
A
2
1
dxdy
2
A
A
A
1
2
dxdy
2
1 2 2
dxdy
2
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
the transfer mechanism
E E1 E 2
Z Z1 Z 2
Z k2E
k 2 E k12 E1 k22 E 2
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
inertial ranges
u3
const ant u l1/ 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
u3
3 const ant u l
l
E(k)dk u 2 l 2
Z
k 1/l
E(k) k 3
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Two-dimensional turbulence:
inertial ranges
k-5/3
log E(k)
k-3
E
Z
log k
As a result, one has a direct enstrophy cascade
and an inverse energy cascade
Is this all ?
Vortices form, interact,
and dominate the dynamics
Vortices are
localized, long-lived concentrations
of energy and enstrophy:
Coherent structures
Vortex studies:
Properties of individual vortices
(and their effect on tracer transport)
Processes of vortex formation
Vortex motion and interactions,
evolution of the vortex population
Transport in vortex-dominated flows
Coherent vortices in 2D turbulence
Qualitative structure of a coherent vortex
||
(u2+v2)/2
Q=(s2-2)/2
The Okubo-Weiss parameter
v u
u v
, sn
x y
x y
2
Q sn ss2 2
Q 4 p
2
u
x
Q 4 det
v
x
v u
, ss
x y
u2+v2
u
y
4 2
v
Q=s2-2
y
The Okubo-Weiss field in 2D turbulence
u2+v2
Q=s2-2
The Okubo-Weiss field in 2D turbulence
u2+v2
Q=s2-2
Coherent vortices
trap fluid particles
for long times
(contrary to what happens with linear waves)
Motion of Lagrangian particles
in 2D turbulence
(X j (t),Y j (t)) is the posit ionof the j thparticleat timet
dX j
u(X j ,Y j ,t)
dt
y
dYj
v(X j ,Y j ,t)
dt
x
Formally, a non-autonomous Hamiltonian system
with one degree of freedom
The Lagrangian view
Effect of individual vortices:
Strong impermeability of the vortex edges
to inward and outward particle exchanges
Example: the stratospheric polar vortex
Vortex formation:
Instability of vorticity filaments
Dressing of vorticity peaks
But: why are vortices coherent ?
Q=s2-2
Instability of vorticity filaments
Q=s2-2
Existing vortices stabilize vorticity filaments:
Effects of strain and adverse shear
Q=s2-2
Processes of vortex formation and evolution
in freely-decaying turbulence:
Vortex formation period
Inhibition of vortex formation by existing vortices
Vortex interactions:
Mutual advection (elastic interactions)
Opposite-sign dipole formation (mostly elastic)
Same-sign vortex merging, stripping, etc
(strongly inelastic)
2 to 1, 2 to 1 plus another, ….
A model for vortex dynamics:
The (punctuated) 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
Beyond 2D:
effects
Free-surface
Dynamics on the -plane
Role of stratification
Q=s2-2
The discarded effects: free surface
The discarded effects: dynamics on the -plane
Filtering fast modes:
The quasigeostrophic approximation
in stratified fluids
The stratified QG potential vorticity equation
, vg
y
x
v g ug
2
x y
ug
2
f
2
0
q 0 y 2
z N (z) z
N 2 (z)
g d
dz
q
,q D F
t
Vortex merging and filamentation
in 2D turbulence
Vortex merging and filamentation
in QG turbulence: role of the Green function