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The barotropic vorticity equation (with free surface)

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,

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The barotropic vorticity equation (with free surface)

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  1. The barotropic vorticity equation (with free surface)

  2. Barotropic Rossby waves (rigid lid)

  3. Barotropic Rossby waves (rigid lid)

  4. Barotropic Rossby waves (rigid lid)

  5. Rossby waves

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

  7. 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

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

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

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

  11. 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

  12. Is this all ?

  13. Vortices form, interact, and dominate the dynamics Vortices are localized, long-lived concentrations of energy and enstrophy: Coherent structures

  14. 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

  15. Coherent vortices in 2D turbulence

  16. Qualitative structure of a coherent vortex |z| (u2+v2)/2 Q=(s2-z2)/2

  17. The Okubo-Weiss parameter z u2+v2 Q=s2-z2

  18. The Okubo-Weiss field in 2D turbulence z u2+v2 Q=s2-z2

  19. The Okubo-Weiss field in 2D turbulence z u2+v2 Q=s2-z2

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

  21. Motion of Lagrangian particles in 2D turbulence Formally, a non-autonomous Hamiltonian system with one degree of freedom

  22. The Lagrangian view

  23. Effect of individual vortices: Strong impermeability of the vortex edges to inward and outward particle exchanges

  24. Example: the stratospheric polar vortex

  25. Vortex formation: Instability of vorticity filaments Dressing of vorticity peaks But: why are vortices coherent ? Q=s2-z2

  26. Instability of vorticity filaments z Q=s2-z2

  27. Existing vortices stabilize vorticity filaments: Effects of strain and adverse shear z Q=s2-z2

  28. Processes of vortex formation and evolution in freely-decaying turbulence: Vortex formation period Inhibition of vortex formation by existing vortices

  29. 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, ….

  30. A model for vortex dynamics: The (punctuated) point-vortex model

  31. Beyond 2D: Free-surface effects Dynamics on the b-plane Role of stratification z Q=s2-z2

  32. The discarded effects: free surface

  33. The discarded effects: dynamics on the b-plane

  34. Filtering fast modes: The quasigeostrophic approximation in stratified fluids

  35. The stratified QG potential vorticity equation

  36. Vortex merging and filamentation in 2D turbulence

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

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