the barotropic vorticity equation (with free surface)
DESCRIPTION
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 PresentationTRANSCRIPT
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
