Nonlinear dynamics of waves and transport in the atmosphere and oceans Harry L. Swinney University of Texas at Austin

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1 International School of Physics "Enrico Fermi" - Course CLXXVI Complex materials in physics and biology, 9 June - 9 July 1 Nonlinear dynamics of waves and transport in the atmosphere and oceans Harry L. Swinney University of Texas at Austin Joel Sommeria Coriolis Lab, CNRS, University it of Grenoble Steven Meyers University of South Florida Tom Solomon Bucknell University (Pennsylvania) Eric Weeks Emory University (Atlanta, Georgia) Brendan Plapp U.S. Department of Homeland Security Charles Baroud Ecole Polytechnique (Paris) Sunny Jung Virginia Polytech Jori Ruppert-Felsot finance (Tokyo) Hepeng Zhang U. Texas & Shanghai Jaiotong University Ben King University of Texas Bruce Rodenborn University of Texas Lecture 1: Jets and vortices Ocean dynamics: relevance to climate Coriolis force >> inertia (on scales greater than ~ 1 km) Hamiltonian Dynamics of particle motion in viscous flows Invariant curves (KAM tori): barriers to transport Lecture : Coherent structures Vortex mergers and dissipation Lévy statistics in transport Lagrangian Coherent Structures and their applications Lecture 3: Internal gravity waves Definition; relevance to global ocean circulation Internal wave generation by tidal flow Nonlinear effects; Turning points Persistent coherent structures: JETS & VORTICES Jupiter: from Voyager I (1978) LOW NORTH POLE isobars at p=1/ atm Feb 17 HIGH Kurioshio Current Japan Aerospace Exploration Agency Benjamin Franklin s Gulf Stream map (1785) From Sundry Maritime Observations JAPAN JAPAN Sea surface temperature 1 o C 3 o C 1

2 Antarctic circumpolar current How do jets and vortices affect transport t and climate? Climate prediction: Global Climate Models RADIATION SOLAR EARTH CO, OZONE, METHANE, CLOUDS horiontal: 1 km vertical: layers CONVECTION HEAT WATER RAIN SNOW MOMENTUM Rise in mean global temperature from 8 of the 3 + global climate models CCSR/Tokyo CCCma/Canada CSIRO/Australia Hadley Ctr/UK GFDL/Princeton MPI/Germany } NCAR/Colorado 5 o C AUSTRALIA --UK ICE MIXING Model rather than simulate lengths 1 cm to 1 km (diffusion: < 1 cm). Double processor speed every 18 months for 5 years 6X resolution increase Navier-Stokes equations (neglecting change of phase, heat transport, radiation, etc.) fields: velocity u(r,t), density (r,t), pressure p(r,t) inertial reference frame u ( u ) u p u t incompressible fluid: u Reynolds number Re inertia dissipation u u u ~ kinematic viscosity U L U L In atmosphere and oceans typically Re ~ UL Effect of earth s rotation u 1 ( u ) u p u ( r ) u t centrifugal Coriolis 1 ( r ) ( ), The centrifugal term, r axis can be incorporated into pressure and doesn t change the dynamics. But the Coriolis term has an enormous effect on the dynamics: Atmosphere and oceans: U~ m/s, =4x1-5 rad/s, L~1 km Our lab experiments: U~.3 m/s, =4 rad/s, L~.3 m So in both cases: Coriolis L 1 inertia U Rossby number ~ 1

3 Water swirls down a sink drain See youtube.com/watch?v=zbvntsa-qoq for a demonstration of the direction of the swirl: Counter-clockwise in northern hemisphere Clockwise in the southern hemisphere u 1 ( u ) u p u u t Then with x ẑˆ u L 5 ~ ~ 1 u u U u ( x, y,, t) flow is -dimensional for slow rapid small change rotation dissipation In a sink U ~.5 m/sec, EARTH = 1 rev/(864 s), L ~. m Coriolis inertia Thus the Coriolis effect is negligible at small length scales. (the youtube demonstration is a hoax) Taylor-Proudman theorem -dimensionality of oceanic and atmopheric flow due to Rapid rotation (Taylor-Proudman) Lab experiments Rotating on annulus rotating flow University of Texas Thinness: ocean & troposphere ~1 km thick, while horiontal scales ~1 km Stratification: suppresses vertical motions leads to internal gravity waves effect: sloping bottom mimics variation of earth s Coriolis force with latitude water SINKS ring of 1 holes SOURCES ring of 1 holes (hole diameter.5 cm) 86 cm cm Weak radial pumping produces strong jet stream STRONG JET: u u r Radial forcing ur u r sinks sources Measure dimensionality of flow Rotating annulus velocity probes = cm Pump outward: anti cyclonic (counter rotating) jet Pump inward: cyclonic (co rotating jet) 3

4 Velocity measured with two probes displaced in (axially) by 16 cm: almost no -dependence quasi--dimensional flow u () t Rossby number =.1 Flow in rotating tank for low pumping Solution of Navier-Stokes for ANY system at low Re: unique symmetry of boundary conditions globally stable Low pumping: axisymmetric jet time exposure of tracer particles Rossby # =. Reynolds # = 37 = 6.8 rad/s Flux = 5.6 cm 3 /s Pump faster: axisymmetric jet becomes unstable 5 fold wave (Rossby wave) Dye visualiation of the Rossby waves m = 3 m = 6 Linear stability analysis of Navier-Stokes eq.: axisymmetric jet Rossby waves Marcus & Lee, Phys. Fluids (1995) particle streak lines in co rotating frame of the Rossby wave m = 7 4

5 PUMPING RATE Onset of instability of circular jet: compare experiment with theory WAVY JET theory 9 OCT. 6: NASA s Casini spacecraft reveals biarre 6-sided feature encircling the north pole of Saturn Each side 13,8 km (cm 3 /sec) CIRCULAR JET Period 1h 39min 4s Latitude 78 o North / (H) Solomon, Holloway, Swinney Phys. Fluids A 5 (1993) -dimensional Hamiltonian dynamics: KAM tori as barriers to transport in planetary flows What does Hamiltonian dynamics have to do with the flow of a viscous fluid? Particles in a D incompressible flow Aref, J Fluid Mech (1984) Fluid velocity u(x,y,t) in terms of stream function (x,y,t): u x u y y x Kolmogorov-Arnold-Moser (KAM) theorem Quasi-periodic motions of Hamiltonian dynamical systems persist under small perturbations 1954: discussed by Kolmogorov Thenthe position (x,y) ofa passive scalar particle is given by integrating dx x dt y dy y dt x HAMILTON S EQUATIONS (1883) with H = 196: proved by Moser for smooth twist t maps 1963: proved by Arnold for analytic Hamiltonian systems KOLMOGOROV MOSER tracer particle motion is described by Hamiltonian ARNOLD KAM tori : tori that are deformed yet persist under weak perturbation H H H 1 5

6 Model Hamiltonian for particle dynamics Zonal flow with 5 waves: the unperturbed (integrable) system Behringer, Meyers, & Swinney, Phys. Fluids (1991) In the co-rotating ti frame of the 5 waves, r,,t) r P (r,,t) ZONAL FLOW WITH 5 WAVES PERTURBATION WITH 6 WAVES time-independent flow in wave co-rotating frame particles initially along these lines remain on invariant curves (tori) 5-fold jet with 1% perturbation by 6-fold wave P with.1) chaotic sea where particles initially along these lines m 1 m1 5-fold jet with 5% perturbation by 6-fold wave CHAOTIC SEA particles initially along these lines chaotic sea persistent KAM TORI Behringer, Meyers, Swinney Phys. Fluids (1991) inject dye Jet: barrier to transport Oone hole: contained by a KAM torus ghost of the KAM invariant torus Sommeria, Meyers, Swinney, Nature (1989) oone concentration: red: high blue: low polar night jet (KAM torus) south pole oone hole broken torus: Cantorus (Percival, 1979) 6

7 Conclusions (Lecture 1) Coriolis force (- xu) makes atmospheric and oceanic turbulence different (long-lived jets and vortices, e.g., jet stream, high and low pressure systems, ) flow is approximately -dimensional Hamilton s equations describe motion of particles in the viscous flow Barriers to transport -- invariant mathematical curves (KAM tori) Lecture Coherent structures Vortex formation, mergers, and persistence Jets and vortices lead to particle transport described by Lévy statistics Lagrangian Coherent Structures (LCS) Eastward-westward jets and vortices on Jupiter Merger of Jupiter s three White Ovals Hubble Space Telescope WFPC 75º north 75º south from Cassini spacecraft JUPITER S GREAT RED SPOT First report on Jupiter s Great Red Spot Voyager II photo 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 Test in a laboratory experiment that satisfies these conditions 7

8 two vortices Merger of two laboratory vortices Sommeria, Meyers, Swinney Nature (1988) t = sec t = 4 sec t = sec Spot merger t = 6 sec one vortex How do coherent jets and vortices affect transport in the turbulent oceans and atmosphere? Determine transport: track ocean floats x : stop Garfield et al., J. Physical Oceanography (1998) Describe motion as diffusive <(r) > = Dt So diffusion coefficient: ( r) D tt stop start random walk Pacific Ocean o : start For floats released near the California coast: ( r) 8 D 1.31 cm /s t Garfield et al., J. Physical Oceanography (1998) (compare molecular diffusion D = X 1-5 cm /s) Track a single ocean float Lab experiment: track particles in a rotating fluid 9 sec trajectories 4 particles 1.5 H start Solomon, Weeks, & Swinney, Phys. Rev. Lett. 71 (1993) Reynolds Number = Rossby number =.1 f annulus =1.5 H 8

9 Particle flights and sticking flight 6 Particle displacements sticking (rad) 4 sticking long flight Weeks, Urbach, Swinney Physica D 97 (1996) time (sec) Mean Square Displacement <( > t SUPER DIFFUSION SO Anomalous diffusion <(r) > ~ t with 1 < < superdiffusion < < 1 subdiffusion walk Example: continuous time random walk model Zumofen et al., J. Stat. Phys. (1989) stick Shlesinger, Klafter, Wong, J. Stat. Phys. (198) t (sec) Probability distribution function for flight of length P() P flight () ~ Solomon, Weeks, & Swinney Phys. Rev. Lett. (1993); Weeks & Swinney, Phys. Rev. E (1998) Divergent nd moment of P() with P() ~ P( )d ~ Then for < 3, 3 Lévy distributions: divergent nd moment hence no Central Limit Theorem Paul Lévy ( s): mathematics Shlesinger, Mandelbrot, Klafter, West (198s): theoretical physics 9

10 Why are there sometimes long periods of colder than usual weather? ATMOSPHERIC BLOCKING Usual onal flow COLD IN GERMANY POLAND, RUSSIA Blocked Experiment with mountain ridges 1.5 H ridges mimick, e.g., Rocky Mountains and Alps Zonal flow mountain ridges Blocked HIGHS LOWS pumping 39 cm 3 /s pumping 6 cm 3 /s Weeks, Tian, Urbach, Ide, Swinney, Ghil, Science (1997) Intermittent transitions between onal and blocked flow pumping 6 cm 3 /s Percent time blocked Zonal and blocked regimes blocked onal Tian, Weeks, Ide, Urbach, Baroud, Ghil, Swinney, J. Fluid Mech. (1) Vortex dynamics in a flow with a strong Coriolis force Coherent structures in turbulent flow in a rotating tank quasi dimensional turbulence u ~ 5 ( u ) u camera laser sheet Fluid depth 48 cm. Horiontal laser sheet 8 cm below top. Ruppert-Felsot, Praud, Swinney, Phys. Rev. E 7 (5) 4 cm 3 dimensional turbulence 1 u ~ ( u ) u 5 noles 15 cm dia. rotating table 1

11 Velocity and vorticity fields determined by Particle Image Velocimetry (PIV) Vortices form, merge, and dissipate Largest velocity ~ 7cm/s 18x18 vectors.3 cm spatial resolution [s -1 ] Vorticity ( u ) [s -1 ] 4 cm 1 cm 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 George Haller, Physica D 149 (1) Identify coherent structures by computing the finite time Lyapunov exponent field to obtain Lagrangian Coherent Structures Aleksandr Lyapunov Lyapunov Exponents: rate of separation of nearby points Consider two points in phase space with infinitesimal separation ( t ) Then the largest Lyapunov exponent is r 1 r ( t ) lim log t t r () 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) 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: maximiing curves ridges of the finite time Lyapunov exponent field Results are insensitive to integration time George Haller, Physica D (1) 11

12 Determination of Finite Time Lyapunov Exponents Use flow map: F t t ( x ) t tt t x x(;, x ) F ( x ) Direct Lyapunov Exponent field determined from laboratory data LCS are locations of extrema for the deformation field: Theorem LCS maximie the largest eigenvalue of the Cauchy Green strain tensor field to give Direct Lyapunov Exp. field t t T t DLE t ( x ) log max Ft ( x) Ft ( x) Direct Lyapunov Exponent (s -1 ) Black lines are maximiing curves (ridges) of the DLE field Maximiing curves (ridges) are transport barriers UNSTABLE (attracting ridge) The Lagrangian Skeleton of Turbulence STABLE (repelling ridge) Mathur, Haller, Peacock, Ruppert-Felsot, Swinney, Phys. Rev. Lett. 98,1445 (7) Transport across a ridge is negligible Shadden, Lekien, Marsden, Physica D (5) Real time velocities in Monterey Bay from surface radar data (mangen.org) Lekien et al., Physica D (5) Pollution control in Monterey, California use real-time Lagrangian Coherent Structures to time the release of sewage SANTA CRUZ RADAR Latitude (degrees) SANTA CRUZ Finite time Lyapunov Exponent km MONTEREY MONTEREY Longitude (degrees) Longitude (degrees) 1

13 Computation of Lagrangian Coherent Structures is now made possible by velocity field time series data large scale parallel computing APPLICATIONS: 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, Lecture : conclusions Super transport (non-diffusive) <(x) > ~ t ( > 1); Lévy distributions 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 OZONE HOLE MONTEREY BAY Lecture 3 Internal Gravity Waves ---waves that arise in fluids whose density varies depth Hepeng Zhang Ben King Bruce Rodenborn Mark Stone Harry L. Swinney Buoyancy Frequency N() ocean depth (km) buoyancy gravity Sea water density (kg/m 3 ) Internal waves Linear inviscid governing equation: (Kundu 4) t Assuming plane wave solution: Yields the dispersion relation: k k x x k N x u N u u ( x,, t) u sin N y x i( kxxkt) e k k x k Internal wave: phase and group velocities Substituting the wave solution into the continuity equation gives k. u = k phase velocity direction u x group velocity; (particle motion) is INDEPENDENT OF k 13

14 Internal Waves (observed with vertical laser sheet) MOVIE N =.16H =.11H Tidal flow over topography produces internal gravity waves arcsin = 43.5 o N increasing density Garrett, Science (3) sin where N Dominant wave period = 1. 4 N hours g d d (semi-diurnal lunar tide) Gulf Stream sea surface temperature Temp. ( o C) Internal waves affect ocean currents, which affect climate New York Rahmstorf, Nature (3) Deep water formation one Ocean Circulation heating cooling Surface air temperature deviation from onal mean Rahmstorph, Climate Change (); Nature (5) 8 o 6 o +1 o C 4 o +5 6km MIXING by internal wave breaking o o - o -4 o 6 S Equator 6 N 16, km Wunsch and Ferrari (4) -6 o 14

15 Internal wave research Most previous: inviscid linear (or inviscid weakly nonlinear, but not viscous weakly nonlinear) Present: viscous nonlinear --- boundary layers --- harmonics Tidal flow on model ocean slope V Asin( t) f -dimensional 3-dimensional --- wave breaking --- turbulent mixing x bouyancy frequency N = constant N() from ocean data: varies by 1X N = 1.55 rad/s, = 36 o Tidal flow on a laboratory model slope FILLING TUBES 1. g/cm Laser sheet 3 OSCILLATING MECHANISM 6 cm (cm) (cm) Measured velocity field ( V / A) near-critical Near critical LASER 1.14 g/cm 3 9 cm 4 6 x (cm) (cm) 8 Velocity field from Particle Image Velocimetry =.38 rad/s, N=1.55 rad/s, A=.1 cm Resonance: beam angle = slope intense internal wave beam ( V angle / A) Resonant boundary current ( V / A) Internal wave angle = slope angle x (cm) (cm) , - near-critical x (cm) =.91 rad/s, N=1.55 rad/s, A=.1 cm 8 15

16 Boundary current scaling Compare boundary layer solution with experiment ( V boundary current max ( V ) ) tidal flow max L Apply boundary layer theory of Dauxios & Young J. Fluid Mech. (1999) Zhang, King, Swinney, Phys. Rev. Lett. (8) THEORY EXPERIMENT At resonance: V V max tide 3 N 4N 1/ 3 L 4/3 a L N 1/ 3 Perpendicular distance from boundary (cm) viscosity Strong shear at resonance leads to instability Kelvin-Helmholt billows Wave breaking: numerical simulation 565 s 1414 s red: low density Smith, Moum, Caldwell, J. Physical Oceanography (1) u m blue: high density u 5 mm 44 s 6 s FIRST OBSERVATION OF Kelvin Helmholt billows in the ocean van Haren & Gostiaux, Geophysical Res. Lett. (1) 1 m time 5 s on slope of Great Meteor Seamount (south of Aores), which rises 45 m from floor to 7 m below surface 3. N, 8.3 W Continental Slope Angle Selection Land mass x. km Sea level ~ 5 km Measurements show ~ 3 o, much smaller than the angle of repose (~ o ) WHY? Resonant internal waves Cacchione et al. Science (4); Zhang et al., Phys. Rev. Lett. (8) 16

17 i Suspended sediment detaching from near-critical region Contours of suspended sediment concentration Puig et al., J. Geophysical Research (4) Test inviscid weakly nonlinear theory: examine nd harmonic produced in reflection incident internal wave beam, frequency second harmonic, sin h = /N h fundamental, sin r = /N r i x Measured at continental slope south of the Guardiaro Canyon i Determine for maximum nd harmonic intensity Maximum nd harmonic intensity: inviscid weakly nonlinear theory Tabaei, Akylas, and Lamb, 3 J. Fluid Mech. (5) Ratio of nd harmonic to incoming beam energy max diverges at max = i Thorpe, The Turbulent Ocean (5) From resonant triad concept and boundary conditions: max tan 1 3 4cos i 1 15 Tabaei et al. Thorpe 15 incident beam angle i i Max angle for fundamental: sin i =1. Max i for nd harmonic: sin i =1/, i.e. i =3 o 3 Internal wave beam reflection measurements Wavemaker design: Gostiaux et al., Expts. Fluids (7) Laser Laser sheet and seed particles Particle image velocimetry N = 1.5 rad/s 9 cm 45cm Velocity field 6 cm Internal wave beam reflection measurements Numerical simulations -dimensional pseudo-spectral code (Marcus and Jiang, Berkeley). Solves full nonlinear viscous Navier-Stokes equations in the Boussinesq limit Laser Create wave beam by adding forcing term to Navier-Stokes: where r F ( ˆ )cos(t ) ( x, ( x x) ( ) ) Aexp 17

18 6 (cm) 4 g incident beam Computed wave field wavemaker Rodenborn, Kiefer, Zhang, Swinney (1) second harmonic fundamental x (cm) Bouyancy frequency N()=constant (linear density gradient).3 Vorticity (rad/sec) -.3 Compare experiments and numerical simulations with theory Plate angle at nd harmonic maximum intensity 3 max 15 i 15 Thorpe (angle of incident beam) 3 Experiment Simulation However, for very small amplitude and low viscosity, numerical simulation agrees with Tabaei et al. Geometric analysis prediction agrees with simulation and experiment Plate angle max 3 15 Tabaei et al. Rodenborn et al. Conclude: a little nonlinearity is significant Search for 3-dimensional effects: tidal flow over a Gaussian mountain x y Thorpe tidal flow mountain height = 7 cm, 1/e half width =.85 cm 15 Incident beam angle i 3 3-dimensional numerical simulation Grid: Code: CDP finite volume flow solver (Ham, Stanford) Compare simulation and experiment TIDAL FLOW PAST A HEMI-SPHERE oscillating tide TETRAHEDRA velocity (cm/s) simulation experiment vorticity (rad/s) ~ 5 million control volumes cm King, Zhang, Swinney, Phys. Fluids 1, (9) cm 18

19 visualiation hemi sphere Internal wave generation perpendicular to forcing plane King, Zhang, Swinney Geophysical Res. Lett. (1) FUNDAMENTAL (frequency ) nd HARMONIC (frequency ) velocity amplitude (cm/s) velocity amplitude (cm/s) Reflection of internal wave from ocean bottom Pingree & New, J. Phys. Oceanography (1991) depth (km) Bay of Biscay Computed path of internal wave propagation with ( ) arcsin, ( ) N where N ( ) g d d Ocean ones SUNLIGHT ZONE TWILIGHT ZONE m: BP Deepwater Horion well in Gulf of Mexico 54% > 4 m MIDNIGHT ZONE ABYSS WORLD OCEAN CIRCULATION EXPERIMENT WOCE group: 3 countries Planning: 15 years ( ) Data acquisition: 1 years ( ) Cost: > $1 9 1.% > 6 m TRENCHES ocean topography is well known Google: ocean ones Measure: pressure temperature salinity (and other quantities) in depth increments, typically m at points separated laterally by 4-1 km WOCE data Each blue dot corresponds to one of the 18, ship casts use sw-bfrq routine Compute N() from WOCE data ( Depth (km) AVERAGE 13. N, 91.8 W February 1993 This N() value is more than 1<N> 4-5 km longitude M N < M M N < M King, Stone, Swinney (1) N()/(cycles/day) 19

20 Navier-Stokes Direct Numerical Simulations CDP parallel finite volume flow solver (by Frank Ham, Stanford) Create wave beam: add forcing to momentum equation: r F (ˆ )cos(t ) where ( x x) ( ) ( x, ) Aexp Use measured N() (13. N, W, Middle America Trench) Snapshot of simulated internal wave beam depth (km) WAVEMAKER TURNING DEPTH EVANESCENT WAVES 4 6 x (km) Domain: 6 km x 4 km. Resolution: 15 m time steps/period for periods. Viscosity 1.x1-3 m /s. VORTICITY (1-3 rad/s) Internal wave dynamics Resonant boundary currents ( = ) (V max ) (slope length) 4/3 overtuning and mixing determine global Continental Slopes (~3 o ) Wave reflection: Weakly nonlinear inviscid theory fails, but experiment, simulation, & geometric analysis agree. Tidal flow over 3D topography: intense harmonics perpendicular to the forcing Analysis of 18 data sets from World Ocean Circulation Experiment reveal TURNING POINTS, below which internal waves can t propagate. Some open questions important for climate modeling How can small length scales in the oceans be modeled? (scales from 1 km down to 1 cm are too small to be resolved in numerical simulations of climate) What is the combined effect of the Coriolis force and the buoyancy frequency N() on internal wave dynamics? [N()varies by more than orders of magnitude] How do nonlinear processes (e.g., wave breaking, wave-wave interaction, wave-boundary interaction) lead to the mixing that provides energy to drive the global ocean circulation?