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The barotropic vorticity equation (with free surface). Barotropic Rossby waves (rigid lid). Barotropic Rossby waves (rigid lid). Barotropic Rossby waves (rigid lid). Rossby waves. The 2D vorticity equation ( f plane, no free-surface effects ). In the absence of dissipation and forcing, - PowerPoint PPT Presentation
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The barotropic vorticity equation (with free surface)
€
u = −∂ψ
∂y, v =
∂ψ
∂x, ζ =∇ 2ψ
D
Dt∇ 2ψ −
L2
LD2ψ + β y
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
∂
∂t∇ 2ψ −
L2
LD2ψ
⎛
⎝ ⎜
⎞
⎠ ⎟+ ψ , ∇ 2ψ −
L2
LD2ψ
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥+ β
∂ψ
∂x= 0
Barotropic Rossby waves (rigid lid)
€
u = −∂ψ
∂y, v =
∂ψ
∂x, ζ =∇ 2ψ
D
Dt∇ 2ψ + β y( ) = 0
∂
∂t∇ 2ψ + ψ ,∇ 2ψ[ ] + β
∂ψ
∂x= 0
u =U + u'
v = v '
ψ = Ψ(y) +ψ '= −U y +ψ '
Barotropic Rossby waves (rigid lid)
€
u =U + u'= −∂ψ
∂y= −
∂Ψ
∂y−
∂ψ '
∂y, v = v '=
∂ψ '
∂x, ζ = ζ '=∇ 2ψ '
ψ = Ψ(y) +ψ '= −U y +ψ '
∂
∂t∇ 2ψ '+U
∂
∂x∇ 2ψ '( ) + β
∂ψ '
∂x= 0
Barotropic Rossby waves (rigid lid)
€
∂∂t∇ 2ψ '+U
∂
∂x∇ 2ψ '( ) + β
∂ψ '
∂x= 0
ψ '= exp ik x + i l y − iω t( )
ω
k=U −
β
k 2 + l2
Rossby waves
The 2D vorticity equation ( f plane, no free-surface effects )
€
u = −∂ψ
∂y, v =
∂ψ
∂x, ζ =∇ 2ψ
∂∇2ψ
∂t+ ψ ,∇ 2ψ[ ] = Dζ + F
In the absence of dissipation and forcing,2D barotropic flows conserve
two quadratic invariants:energy and enstrophy
€
E =1
A A
∫ 1
2u2 + v 2
( )dxdy =1
A A
∫ 1
2∇ψ
2dxdy
Z =1
A A
∫ ζ 2
2dxdy
1
A A
∫ 1
2∇ 2ψ( )
2dxdy
As a result, one has a direct enstrophy cascadeand an inverse energy cascade
Two-dimensional turbulence:the transfer mechanism
€
E = E1 + E2
Z = Z1 + Z2
Z = k 2E
k 2E = k12E1 + k2
2E2
As a result, one has a direct enstrophy cascadeand an inverse energy cascade
Two-dimensional turbulence:inertial ranges
€
ετ=u3
l= constant → u ≈ l1/ 3
E(k)dk ≈ u2 ≈ l2 / 3
k ≈1/ l
E(k) ≈ k−5 / 3
As a result, one has a direct enstrophy cascadeand an inverse energy cascade
Two-dimensional turbulence:inertial ranges
€
Z
τ=u3
l3= constant → u ≈ l
E(k)dk ≈ u2 ≈ l2
k ≈1/ l
E(k) ≈ k−3
As a result, one has a direct enstrophy cascadeand an inverse energy cascade
Two-dimensional turbulence:inertial ranges
As a result, one has a direct enstrophy cascadeand an inverse energy cascade
log k
log E(k)
k-3
k-5/3
E Z
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
u2+v2
Q=s2-2
€
=∂v∂x
−∂u
∂y, sn =
∂u
∂x−
∂v
∂y, ss =
∂v
∂x+
∂u
∂y
Q = sn2 + ss
2 −ζ 2
Q = −4∇ 2p
Q = −4 det
∂u
∂x
∂u
∂y∂v
∂x
∂v
∂y
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟= 4 λ2
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 particlesin 2D turbulence
€
(X j (t),Y j (t)) is the position of the j − th particle at time t
dX j
dt= u(X j ,Y j , t) = −
∂ψ
∂y
dY jdt
= v(X j ,Y j , t) =∂ψ
∂x
Formally, a non-autonomous Hamiltonian systemwith one degree of freedom
The Lagrangian view
Effect of individual vortices:Strong impermeability of the vortex edgesto inward and outward particle exchanges
Example: the stratospheric polar vortex
Vortex formation:
Instability of vorticity filamentsDressing 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 evolutionin 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
222 )()(
log4
1
jiji
ijjji
i
j
jj
j
jj
yyxxR
RH
x
H
dt
dy
y
H
dt
dx
ij−+−=
ΓΓ=
∂∂
=Γ
∂∂
−=Γ
∑≠π
Q=s2-2
Beyond 2D:
Free-surface effects
Dynamics on the -plane
Role of stratification
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
€
ug = −∂ψ
∂ y, vg =
∂ψ
∂ x
ζ =∂vg∂ x
−∂ ug∂ y
=∇ 2ψ
q =∇ 2ψ 0 + β y +∂
∂z
f02
N 2(z)
∂ψ
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟
N 2(z) = −g
ρ
dρ
dz
∂q
∂t+ ψ ,q[ ] = Dζ + F
Vortex merging and filamentationin 2D turbulence
Vortex merging and filamentationin QG turbulence: role of the Green function
