Coherent vortices in rotating geophysical flows A.Provenzale, ISAC-CNR and CIMA, Italy Work done with: Annalisa Bracco, Jost von Hardenberg, Claudia Pasquero A.Babiano, E. Chassignet, Z. Garraffo, J. Lacasce, A. Martin, K. Richards J.C. Mc Williams, J.B. Weiss

Rapidly rotating geophysical flows are characterized by the presence of coherent vortices: Mesoscale eddies, Gulf Stream Rings, Meddies Rotating convective plumes Hurricanes, the polar vortex, mid-latitude cyclones Spots on giant gaseous planets

Vortices form spontaneously in rapidly rotating flows: Laboratory experiments Numerical simulations Mechanisms of formation: Barotropic instability Baroclinic instability Self-organization of a random field

Rotating tank at the “Coriolis” laboratory, Grenoble diameter 13 m, min rotation period 50 sec rectangular tank with size 8 x 4 m water depth 0.9 m PIV plus dye Experiment done by A.Longhetto, L. Montabone, A. Provenzale, C. Giraud, A. Didelle, R. Forza, D. Bertoni

Characteristics of large-scale geophysical flows: Thin layer of fluid: H << L Stable stratification Importance of the Earth rotation

Navier-Stokes equations in a rotating frame

Incompressible fluid: D  /Dt = 0

Thin layer, strable stratification: hydrostatic approximation

Homogeneous fluid with no vertical velocity and no vertical dependence of the horizontal velocity

The 2D vorticity equation

In the absence of dissipation and forcing, quasigeostrophic 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

With small dissipation:

Is this all ?

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

Vortex dynamics: Processes of vortex formation Vortex motion and interactions Vortex merging: Evolution of the vortex population

Vortex dynamics: Vortex motion and interactions: The point-vortex model

Vortex dynamics: Vortex merging and scaling theories

Vortex dynamics: Introducing forcing to get a statistically-stationary turbulent flow

Particle motion in a sea of vortices Formally, a non-autonomous Hamiltonian system with one degree of freedom

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

Example: the stratospheric polar vortex

Global effects of the vortex velocity field: Properties of the velocity distribution

Velocity pdf in 2D turbulence (Bracco, Lacasce, Pasquero, AP, Phys Fluids 2001) Low Re High Re

Velocity pdf in 2D turbulence Low Re High Re

Velocity pdf in 2D turbulence Vortices Background

Velocity pdfs in numerical simulations of the North Atlantic (Bracco, Chassignet, Garraffo, AP, JAOT 2003) Surface floats 1500 m floats

Velocity pdfs in numerical simulations of the North Atlantic

A deeper look into the background: Where does non-Gaussianity come from Vorticity is local but velocity is not: Effect of the far field of the vortices

Background-induced Vortex-induced

Vortices play a crucial role on Particle dispersion processes: Particle trapping in individual vortices Far-field effects of the ensemble of vortices Better parameterization of particle dispersion in vortex-dominated flows

How coherent vortices affect primary productivity in the open ocean Martin, Richards, Bracco, AP, Global Biogeochem. Cycles, 2002

Oschlies and Garcon, Nature, 1999

Equivalent barotropic turbulence Numerical simulation with a pseudo-spectral code

Three cases with fixed A (12%) and I=100: “Control”: NO velocity field (u=v=0) (no mixing) Case A: horizontal mixing by turbulence, upwelling in a single region Case B: horizontal mixing by turbulence, upwelling in mesoscale eddies

29% more than in the no-mixing control case

139% more than in the no-mixing control case

The spatial distribution of the nutrient plays a crucial role, due to the presence of mesoscale structures and the associated mixing processes Models that do not resolve mesoscale features can severely underestimate primary production

Single particle dispersion For a smooth flow with finite correlation length For a statistically stationary flow particle dispersion does not depend on t 0

Single particle dispersion Time-dependent dispersion coefficient

Properties of single-particle dispersion in 2D turbulence (Pasquero, AP, Babiano, JFM 2001)

Parameterization of single-particle dispersion: Ornstein-Uhlenbeck (Langevin) process

Properties of single-particle dispersion in 2D turbulence

Parameterization of single-particle dispersion: Langevin equation

Parameterization of single-particle dispersion: Langevin equation

Why the Langevin model is not working: The velocity pdf is not Gaussian

Why the Langevin model is not working: The velocity autocorrelation is not exponential

Parameterization of single-particle dispersion with a non-Gaussian velocity pdf: A nonlinear Langevin equation (Pasquero, AP, Babiano, JFM 2001)

Parameterization of single-particle dispersion with a non-Gaussian velocity pdf: A nonlinear Langevin equation

The velocity autocorrelation of the nonlinear model is still almost exponential

A two-component process: vortices (non-Gaussian velocity pdf) background (Gaussian velocity pdf) T L (vortices) << T L (background)

A two-component process:

Geophysical flows are neither homogeneous nor two-dimensional

A simplified model: The quasigeostrophic approximation  = H/L << 1 neglect of vertical accelerations hydrostatic approximation Ro = U / f L << 1 neglect of fast modes (gravity waves)

A simplified model: The quasigeostrophic approximation

Simulation by Jeff Weiss et al