Coherent vortices in rotating geophysical flows A Provenzale
- Slides: 76
Coherent vortices in rotating geophysical flows A. Provenzale, ISAC-CNR and CIMA, Italy B. Work done with: C. Annalisa Bracco, D. Jost von Hardenberg, E. Claudia Pasquero F. Babiano, E. Chassignet, Z. Garraffo, G. J. Lacasce, A. Martin, K. Richards H. 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, B. 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: Dr/Dt = 0
Thin layer, strable stratification: hydrostatic approximation
Homogeneous fluid with no vertical velocity and no vertical dependence of the horizontal velocity
The 2 D vorticity equation
The 2 D 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 2 D turbulence (Bracco, Lacasce, Pasquero, AP, Phys Fluids 2001) Low Re High Re
Velocity pdf in 2 D turbulence Low Re High Re
Velocity pdf in 2 D 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
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 A. 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 statistically stationary flow particle dispersion does not depend on t 0 For a smooth flow with finite correlation length
Single particle dispersion Time-dependent dispersion coefficient
Properties of single-particle dispersion in 2 D turbulence (Pasquero, AP, Babiano, JFM 2001)
Parameterization of single-particle dispersion: Ornstein-Uhlenbeck (Langevin) process
Properties of single-particle dispersion in 2 D 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) TL (vortices) << TL (background)
A two-component process:
Geophysical flows are neither homogeneous nor two-dimensional
A simplified model: The quasigeostrophic approximation d = 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
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