k 2 E ( k , t ) dk E ( t ) = E ( k , t ) dk , Z ( t ) = start at - PowerPoint PPT Presentation

Coherent Structures in Geophysical Turbulence Jeffrey B. Weiss University of Colorado, Boulder • motivated by atmospheres and oceans • study high resolution computations • see organization into coherent structures • model as dynamical systems of interacting structures Collaborators: Bracco, di Frischia, von Hardenberg, McWilliams, Provenzale, Siegel, Yavneh

What is Geophysical Turbulence? • large scale motion of atmospheres and oceans • extremely turbulent large Reynolds number: Re =UL/ ν ~ O(10 8 ) • past many bifurcations complex temporal behavior • large range of scales room for complex spatial behavior • high dimensional low-dimensional models inadequate but not 10 23 - statistical mechanics inappropriate • needs fast computers with lots of memory

Classes of Turbulence • 3D homogeneous isotropic turbulence • turbulent convection • geophysical turbulence • stable • rotating weak vertical motions • anisotropic thin fluid • similarities to 2D

Environmental Influences • fast rotation small Rossby number: Ro = U/ Lf <<1 • strong stable stratification small Froude number: F = U/ NH <<1 • rotation and stratification => anisotropy • small vertical velocity w << u, v • inverse turbulent cascade => coherent structures

Inverse Cascade E = 1 = 1 • inviscid invariants ∫ u 2 dV ∫ q 2 dV Z 2 2 • energy spectrum ∫ ∫ k 2 E ( k , t ) dk E ( t ) = E ( k , t ) dk , Z ( t ) = • start at k 0 , go to k 1 =k 0 /2, k 2 =2k 0: E(k 0 ) = E(k 1 )+E(k 2 ) 2 E(k 0 ) = k 1 2 E(k 1 ) + k 2 2 E(k 2 ) k 0 => E(k 1 ) = 4 E(k 2 ): inverse energy cascade => Z(k 1 ) = Z(k 2 )/4: direct enstrophy cascade

Dissipation • time dependence dE dZ 2 dt ~ −ν Z dt ~ −ν ∇ Z • since dZ/dt ≤ 0 , Z(t) ≤ Z(0), and as ν -> 0 , dE/dt ->0 • energy is conserved • enstrophy gradients grow so enstrophy dissipation remains finite • enstrophy decays • connection to cascade: • dissipation acts at small scales

Hurricane Dennis: August 1999

Gulf Stream and Vortices

Jupiter from Cassini spacecraft, Nov 2000

Coherent Structures • what we really want to predict • spatially localized • partly unknown how to derive them from PDE • provide reduced description • degrees of freedom: • N ~ O(10-100) structures • location • internal degrees of freedom • plus random component? • conjecture: population of structures define attractor

Structures vs. Wavenumbers • traditional theories of turbulence: • wavenumber space • random phase approx. common • structures: • local in physical space • random phase approximation fails

Basic Approach • seek generic properties of planetary fluids • study simple planetary fluids • extrapolate to atmospheres and oceans • simple systems provide roadmap

2D and QG Equations • same vorticity equation: q ( r x , t ) vorticity ψ ( r x , t ) streamfunction q t + ψ x q y − ψ y q x = D + F • different vorticity - streamfunction relation 2 ψ q = ∇ 2 D 2D 1 2 ψ + ∂ z q = ∇ 2 D S ( z ) ∂ z ψ QG 2 ψ  →   ∇ 3 D S = 1

2D Turbulence vorticity colored blue (-) to red (+)

2-Fluid View of 2D Turbulence 1. coherent vortices 2. classical turbulent background structures arrest cascade, cause cascade theories to fail Bracco, et al, 2000a Siegel and W, 1997 full field background

Vortex Dynamical Systems • structured component: • system of interacting vortices • statistics of vortex population • reduced degrees of freedom • conservative motion: • valid when structures well separated • Hamiltonian dynamics • hierarchy of models – point vortices – elliptical vortices – etc.

Dissipation • dissipation: • high Re => dissipation small, but not zero • character of dissipation: – occurs when vortices close => rare – rapidly, and strongly transforms vortices – intermittency • punctuated dynamical system • Hamiltonian evolution • dissipative transformation at isolated points in time jumps to new Hamiltonian ∞ • Re -> dissipation becomes more intermittent

Simple Vortex Behavior in 2D Turbulence same-sign merger opposite-sign dipole four-vortex scattering tripole merger

Point Vortex Dynamics • N vortices • position z i =(x i ,y i ), circulation Γ i ∑ H = Γ j ln| z i − z j | Γ i i , j dz i dt = J ∇ z i H Γ i • close pair decouples H = H pair + H others + ε ( t ) H interaction • isolated pair integrable

• low D integrable subsystem within high D More pv dynamics chaotic system • close pairs have long lifetimes • Cantori? • close pairs have high speeds • same-sign pairs rotate • opposite-sign pairs translate

Point Vortex Velocity PDF • N=100, long integration, O(100 t turnover ) • episodes of fast velocity due to close pairs (Weiss, et al, 1998)

More vel pdf • close pairs => flights • pdfs with long tails • non/slow ergodicity Gaussian ensemble avg time avg

2D Turbulence Velocity PDF • long tails due to vortices (Bracco, et al, 2000b) vort backgnd tot Gaussian induced by vortices induced by background

3D QG Turbulence potential vorticity colored blue (-) to red (+)

Structure Based Scaling Theory • mean vortex theory • avg size, amplitude, … • global quantities due to vortex component • assumes • algebraic evolution t α • self-similar temporal evolution • a few exponents, predicts others ( Bracco, et al, 2000a )

Scaling theory graphs • works well for 2D simulations • some controversy in experiments • OK in 3D QG simulations 2D 3D QG ( McWilliams, et al, 1999 ) ( Bracco, et al, 2000a )

Ocean Basin Turbulence • ocean more QG than atmosphere • basins boundaries • inhomogeneous • variation of Coriolis force with latitude • β -plane • horizontal anisotropy • allows jets • stationary turbulence

Ocean Circulation

QG Ocean Basin Model ( Siegel, et al, 2001) • QG PV eqn on β -plane • rectangular basin • 3200 x 3200 x 5 km • solid, no slip sides and bottom • vary horizontal resolution • six vertical levels • forcing • steady, zonal surface wind • bottom Ekman drag • massively parallel state-of-the-art code

QG Ocean Basin Turbulence increasing Reynolds number

QG Ocean Basin Turbulence Low Re: N = 256 Δ x = 12.5 km

QG Ocean Basin Turbulence High Re: N = 2048 Δ x = 1.6 km

Jet as transport barrier  effective barrier at low Re  not a barrier at high Re

q = − κ ∂ q v ′ ′ Eddy Diffusion: ∂ y q N=256 v ′ ′ q q N=2048 v ′ q ′ κ > 0 κ < 0 κ > 0

Ocean Basin Velocity PDF • long tails in models and obs • same cause? Entire Basin QG Model and GCM Ocean Floats (Bracco, et al, 2000c)

Conclusions • coherent structures ubiquitous • seen empirically (obs and models) • no theory for existence • represent reduced degrees of freedom • dynamical system of interacting structures • punctuated dynamics • intermediate D = low D + high D? • coherent vortices responsible for non-Gaussian pdf? • eddies in ocean model cause • cross-jet transport • negative eddy diffusivity

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