superdiffusion blocking Lagrangian Coh. Structures ocean currents internal waves and climate Coriolis force leads to barriers to transport (KAM torus)

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1 Oceanic & Atmospheric Dynamics YESTERDAY Coriolis force leads to 2D flow (distances > ~100 km) jets and vortices barriers to transport (KAM torus) TODAY superdiffusion blocking Lagrangian Coh. Structures TOMORROW ocean currents internal waves and climate

2 Vortex dynamics in a flow with a strong Coriolis force camera quasi-2-dimensional 2u ~ 5 ( u ) u laser sheet 3-dimensional u ~ ( u ) u 150 cm dia. rotating table

3 Coherent structures in turbulent flow in a rotating tank Fluid depth 48 cm. Horizontal laser sheet 8 cm below top. Ruppert-Felsot, Praud, Swinney, Phys. Rev. E 72 (2005) 40 cm

4 Velocity and vorticity fields determined by Particle Image Velocimetry (PIV) Largest velocity ~ 7 cm/s 128x128 vectors 0.3 cm spatial resolution z Vorticity ( u) [s -1 ] 12 cm

5 Vortices form, merge, and dissipate z [s -1 ] 40 cm

6 Eastward-westward jets on Jupiter 75º north 75º south from Cassini spacecraft

7 Merger of Jupiter s three White Ovals Hubble Space Telescope WFPC2

8 JUPITER S GREAT RED SPOT First report on Jupiter s Great Red Spot Voyager II photo

9 How can Jupiter s Great Red Spot persist for centuries? Sufficient conditions found in a model: Marcus, Nature 331 (1988) low Rossby number (strong Coriolis effect) turbulent flow strong shear Beta effect: Coriolis effect varies with latitude

10 two vortices Merger of two laboratory vortices t = 0 sec t = 2 sec Spot merger-2 t = 4 sec t = 6 sec one vortex Sommeria, Meyers, Swinney Nature (1988)

11 How do coherent jets and vortices affect transport in the turbulent oceans and atmosphere?

12 Determine transport: track ocean floats x : stop Garfield et al., J. Physical Oceanography (1998) Pacific Ocean o : start

13 Describe motion as diffusive stop <(Dr) 2 > = 2Dt So diffusion coefficient: D ( D r) 2t 2 start random walk For floats released near the California coast: D ( Dr) 2t cm 2 /s Garfield et al., J. Physical Oceanography (1998) (compare molecular diffusion D = 2 X 10-5 cm 2 /s)

14 Track an ocean float start

15 Lab experiment: track particles in a rotating fluid 1.5 Hz 90 sec trajectories 40 particles Solomon, Weeks, & Swinney, Phys. Rev. Lett. 71 (1993) Reynolds Number = 2000 Rossby number = 0.1 f annulus =1.5 Hz

16 Particle flights and sticking flight sticking sticking long flight Weeks, Urbach, Swinney Physica D 97 (1996)

17 Track particles

18 60 Particle displacements q (rad) time (sec)

19 Mean Square Displacement <(Dq) 2 > t g SUPER DIFFUSION t (sec)

20 Anomalous diffusion <(Dr) 2 > ~ t g with g 1 1 < g < 2 superdiffusion Example: continuous time random walk model Zumofen et al., J. Stat. Phys. (1989) 0 < g < 1 subdiffusion walk stick Shlesinger, Klafter, Wong, J. Stat. Phys. (1982)

21 Probability distribution function for flight of length q P(q) P flight (q) ~ q - m Solomon, Weeks, & Swinney Phys. Rev. Lett. (1993); Weeks & Swinney, Phys. Rev. E (1998)

22 Divergent 2 nd moment of P(q) with P(q) ~ q - m q 2 Then for m < 3, q 0 2 q 2 P( q )dq ~ q 3m hence no Central Limit Theorem Lévy distributions: divergent 2 nd moment Paul Lévy ( s): mathematics Shlesinger, Mandelbrot, Klafter, West (1980s): theoretical physics

23 Why are there sometimes long periods of colder than usual weather? ATMOSPHERIC BLOCKING Usual zonal flow Blocked COLD IN GERMANY POLAND, RUSSIA

24 Experiment with mountain ridges 1.5 Hz Zonal flow mimick Rocky Mountains and Alps mountain ridges Blocked HIGHS LOWS pumping 390 cm 3 /s pumping 260 cm 3 /s Weeks, Tian, Urbach, Ide, Swinney, Ghil, Science (1997)

25 Intermittent transitions between zonal and blocked flow pumping 260 cm 3 /s Tian, Weeks, Ide, Urbach, Baroud, Ghil, Swinney, J. Fluid Mech. (2001)

26 Zonal and blocked regimes Percent time blocked blocked zonal

27 How can coherent structures be identified in turbulent flows? Use velocity field measurements to compute -- vorticity or -- pressure or -- strain or -- energy or -- wavelets or -- Okubo-Weiss criterion or BUT threshold values are ambiguous coherence measures are frame dependent

28 George Haller, Physica D 149 (2001) Identify coherent structures by computing the finite time Lyapunov exponent field to obtain Lagrangian Coherent Structures

29 Aleksandr Lyapunov Lyapunov Exponents: rate of separation of nearby points Consider two points in phase space with infinitesimal separation lim r ( t 0) Then the largest Lyapunov exponent is 1 r ( t) log t r (0 t ) For laboratory systems results depend on: noise level number of data points sampling rate dimension of phase space dimension of attractor Wolf, Swift, Swinney, Vastano: Physica D 16 (1985)

30 Lagrangian Coherent Structures (LCS) in turbulent flows Compute finite time Lyapunov exponent field: the finite-time rates of separation of nearby points throughout the field Extract Lagrangian Coherent Structures: maximizing curves ridges of the finite time Lyapunov exponent field Results are insensitive to integration time George Haller, Physica D (2001)

31 Determination of Finite Time Lyapunov Exponents Use flow map: F t t ( x ) 0 0 t 0 tt0 0 t 0 x x( ;, x ) F ( x ) 0 LCS are locations of extrema for the deformation field: Theorem LCS maximize the largest eigenvalue of the Cauchy-Green strain tensor field to give Direct Lyapunov Exp. field t t T t DLE x ) log F ( x ) F ( x ) t 0 ( max t 0 t

32 Direct Lyapunov Exponent field Direct Lyapunov Exponent (s-1) Black lines are maximizing curves (ridges) of the DLE field

33 Maximizing curves (ridges) are transport barriers Transport across a ridge is negligible Shadden, Lekien, Marsden, Physica D (2005)

34 The Lagrangian Skeleton of Turbulence UNSTABLE (attracting ridge) STABLE (repelling ridge) Mathur, Haller, Peacock, Ruppert-Felsot, Swinney, Phys. Rev. Lett. 98, (2007)

35 Real time velocities in Monterey Bay from surface radar data (mangen.org) Lekien et al., Physica D (2005) SANTA CRUZ RADAR 40 km MONTEREY Longitude (degrees)

36 Pollution control in Monterey, California use real-time Lagrangian Coherent Structures to time the release of sewage Latitude (degrees) SANTA CRUZ Finite time Lyapunov Exponent MONTEREY Longitude (degrees)

37 Applications: Lagrangian Coherent Structures Made possible by velocity field time series data large scale parallel computing Transport in ocean eddies, hurricanes, Structures in flow past cars, planes, trucks, Detection of clear air turbulence J. Marsden, Caltech, G. Haller, McGill University, C. Garth, U California Davis,

38 Transport in oceans and the atmosphere Super transport (non-diffusive) <(Dx) 2 > ~ t g (g > 1); Lévy distributions OZONE HOLE Lagrangian Coherent Structures: ridges of the finite time Lyapunov Exponent field, yield the skeleton of turbulence Applications of Lagrangian Coherent Structures e.g., time release of pollutants MONTEREY BAY