The barotropic vorticity equation (with free surface)

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Presentation transcript:

The barotropic vorticity equation (with free surface)

Barotropic Rossby waves (rigid lid)

Rossby waves

The 2D vorticity equation ( f plane, no free-surface effects )

In the absence of dissipation and forcing, 2D barotropic flows conserve two quadratic invariants: energy and enstrophy As a result, one has a direct enstrophy cascade and an inverse energy cascade

Two-dimensional turbulence: the transfer mechanism As a result, one has a direct enstrophy cascade and an inverse energy cascade

Two-dimensional turbulence: inertial ranges As a result, one has a direct enstrophy cascade and an inverse energy cascade

Two-dimensional turbulence: inertial ranges As a result, one has a direct enstrophy cascade and an inverse energy cascade

Two-dimensional turbulence: inertial ranges As a result, one has a direct enstrophy cascade and an inverse energy cascade log k log E(k) k -3 k -5/3 EZ

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  (u 2 +v 2 )/2 Q=(s 2 -  2 )/2

The Okubo-Weiss parameter  u 2 +v 2 Q=s 2 -  2

The Okubo-Weiss field in 2D turbulence  u 2 +v 2 Q=s 2 -  2

The Okubo-Weiss field in 2D turbulence  u 2 +v 2 Q=s 2 -  2

Coherent vortices trap fluid particles for long times (contrary to what happens with linear waves)

Motion of Lagrangian particles in 2D turbulence 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=s 2 -  2

Instability of vorticity filaments  Q=s 2 -  2

Existing vortices stabilize vorticity filaments: Effects of strain and adverse shear  Q=s 2 -  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

 Q=s 2 -  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

Vortex merging and filamentation in 2D turbulence

Vortex merging and filamentation in QG turbulence: role of the Green function