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

Coherent Structures in Geophysical Turbulence

• motivated by atmospheres and oceans
• study high resolution computations
• see organization into coherent structures
• model as dynamical systems of interacting structures

Jeffrey B. Weiss University of Colorado, Boulder 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(108)

• past many bifurcations

complex temporal behavior

• large range of scales

room for complex spatial behavior

• high dimensional

low-dimensional models inadequate but not 1023 - 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

• inviscid invariants
• energy spectrum
• start at k0, go to k1=k0/2, k2=2k0:

E(k0) = E(k1)+E(k2) k0

2 E(k0) = k1 2 E(k1) + k2 2 E(k2)

=> E(k1) = 4 E(k2): inverse energy cascade => Z(k1) = Z(k2)/4: direct enstrophy cascade

E = 1 2 u2dV Z

∫

= 1 2 q2dV

∫

E(t) = E(k,t)dk, Z(t) =

∫

k 2E(k,t)dk

∫

Dissipation

• time dependence
• 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

dE dt ~ −νZ dZ dt ~ −ν ∇Z

2

Hurricane Dennis: August 1999

Gulf Stream and Vortices

Jupiter

from Cassini spacecraft, Nov 2000

• 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

Coherent Structures

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:

vorticity streamfunction

• different vorticity - streamfunction relation

2D QG

qt +ψxqy −ψyqx = D + F q = ∇2D

2 ψ

q = ∇2D

2 ψ + ∂z

1 S(z)∂zψ

S=1

 →   ∇3D

2 ψ

q( r x ,t) ψ( r x ,t)

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 full field background

Bracco, et al, 2000a Siegel and W, 1997

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 four-vortex scattering

• pposite-sign dipole

tripole merger

Point Vortex Dynamics

• N vortices
• position zi=(xi,yi), circulation Γi
• close pair decouples
• isolated pair integrable

H = Γ

i i, j

∑

Γj ln| zi − z j |

Γi dzi dt = J∇ ziH H = Hpair + Hothers +ε(t)Hinteraction

More pv dynamics

• low D integrable subsystem within high D

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 tturnover)
• 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 time avg ensemble avg

2D Turbulence Velocity PDF

• long tails due to vortices

(Bracco, et al, 2000b)

induced by background

induced by vortices

vort tot backgnd Gaussian

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

(Bracco, et al, 2000a) (McWilliams, et al, 1999)

2D 3D QG

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 Δx = 12.5 km

QG Ocean Basin Turbulence

High Re: N = 2048 Δx = 1.6 km

Jet as transport barrier

• effective barrier at low Re
• not a barrier at high Re

′ v ′ q

q

′ v ′ q

q

′ v ′ q = −κ ∂q ∂y

N=256 N=2048

Eddy Diffusion:

κ > 0 κ < 0 κ > 0

Ocean Basin Velocity PDF

• long tails in models and obs
• same cause?

QG Model and GCM Ocean Floats (Bracco, et al, 2000c)

Entire Basin

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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