the barotropic vorticity equation (with free surface)

The barotropic vorticity equation (with free surface)

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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 )

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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

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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

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=∂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

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(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