Hepeng Zhang Ben King Bruce Rodenborn

Transcription

1 International School of Physics "Enrico Fermi" - Course CLXXVI Complex materials in physics and biology, 9 June - 9 July 010 Nonlinear dynamics of waves and transport in the atmosphere and oceans: Internal Gravity Waves Harry L. Swinney University of Texas at Austin Hepeng Zhang Ben King Bruce Rodenborn

2 Buoyancy Frequency N(z) ocean depth (km) buoyancy z gravity Sea water density (kg/m 3 )

3 Internal waves 0 z u z y x N u t Linear inviscid governing equation: Assuming plane wave solution: Yields the dispersion relation: sin N (Kundu 004) N k k k z x x is INDEPENDENT OF k k x k z k x z ) ( 0 ),, ( t k z z k x x i z z e u t z x u

4 Internal wave: phase and group velocities Substituting the wave solution into the continuity equation gives k. u = 0 z k phase velocity direction u x group velocity; (particle motion)

5 Internal Waves (observed with vertical laser sheet) N = 0.16Hz = 0.11Hz arcsin N = 43.5 o increasing density

6 Tidal flow over topography produces internal gravity waves Garrett, Science (003) sin N 1. where N Dominant wave period = 4 hours g d dz (semi-diurnal lunar tide)

7 Gulf Stream sea surface temperature Temp. ( o C) New York

8 Internal waves affect ocean currents, which affect climate Rahmstorf, Nature (003)

9 Deep water formation zone Ocean Circulation heating cooling 6 km MIXING by internal wave breaking 60 S Equator 60 N 16,000 km Wunsch and Ferrari (004)

10 Surface air temperature deviation from zonal mean Rahmstorph, Climate Change (000); Nature (005) 80 o +10 o C 60 o 40 o +5 0 o 0 o -0 o -40 o -60 o

11 Internal wave research Most previous: inviscid linear (or inviscid weakly nonlinear, but not viscous weakly nonlinear) -dimensional bouyancy frequency N = constant Present: viscous nonlinear --- boundary layers --- harmonics 3-dimensional --- wave breaking --- turbulent mixing N(z) from ocean data: varies by 100X

12 s Tidal flow on model ocean slope V Asin( t) f a z x N = 1.55 rad/s, a = 36 o

13 Tidal flow on a laboratory model slope FILLING TUBES OSCILLATING MECHANISM 1.00 g/cm 3 Laser sheet 60 cm LASER 1.14 g/cm 3 90 cm Velocity field from Particle Image Velocimetry

14 9 Measured velocity field ( V / A ) z (cm) 6 3 Near-critical 0 a x (cm) x (cm) = 0.38 rad/s, N=1.55 rad/s, A=0.1 cm

15 Resonance: beam angle = slope angle intense internal wave beam ( V / A) 10 8 x (cm) =0.91 rad/s, N=1.55 rad/s, A=0.1 cm x (cm)

16 Resonant boundary current ( V / A ) Internal wave angle = slope angle

17 s Boundary current scaling L Apply boundary layer theory of Dauxios & Young J. Fluid Mech. (1999) At resonance: V V max tide 3 4N N 1/3 L 4/3 a N L 1/3 viscosity

18 Compare boundary layer solution with experiment ( V boundarycurrent ( V tidal flow ) ) max max Zhang, King, Swinney, Phys. Rev. Lett. (008) THEORY EXPERIMENT Perpendicular distance from boundary (cm)

19 Strong shear at resonance leads to instability Kelvin-Helmholtz billows 5 mm

20 Wave breaking: numerical simulation Smith, Moum, Caldwell, J. Physical Oceanography (001) 565 s 1414 s red: low density u m blue: high density u 44 s 6 s

21 FIRST OBSERVATION OF Kelvin-Helmholtz billows in the ocean van Haren & Gostiaux, Geophysical Res. Lett. (010) z 30.0 N, 8.3 W time 50 s 10 m on slope of Great Meteor Seamount (south of Azores), which rises 4500 m from floor to 70 m below surface

22 Continental Slope Angle Selection Land mass Sea level 0. km ~ 5 km z a x Measurements show a ~ 3 o, much smaller than the angle of repose (~0 o ) WHY? Resonant internal waves Cacchione et al. Science (004); Zhang et al., Phys. Rev. Lett. (008)

23 Suspended sediment detaching from near-critical region Contours of suspended sediment concentration z Puig et al., J. Geophysical Research (004) x Measured at continental slope south of the Guardiaro Canyon

24 i Test inviscid weakly nonlinear theory: examine nd harmonic produced in reflection second harmonic, sin h = /N incident internal wave beam, frequency h fundamental, sin r = /N r i i a Determine a for maximum nd harmonic intensity

25 Maximum nd harmonic intensity: inviscid weakly nonlinear theory a Tabaei, Akylas, and Lamb, J. Fluid Mech. (005) Ratio of nd harmonic to incoming beam energy diverges at a max = i Thorpe, The Turbulent Ocean (005) From resonant triad concept and boundary conditions: max tan 30 a max 1 3 4cos 1 i Tabaei et al. Thorpe 15 i incident beam angle i 30 Max angle for fundamental: sin i =1. Max i for nd harmonic: sin i =1/, i.e. i =30 o

26 Internal wave beam reflection measurements N = 1.5 rad/s Wavemaker design: Gostiaux et al., Expts. Fluids (007) Laser 60 cm Laser sheet and seed particles 45cm 90 cm Particle image velocimetry Velocity field

27 Internal wave beam reflection measurements Laser

28 -dimensional pseudo-spectral code (Marcus and Jiang, Berkeley). Solves full nonlinear viscous Navier-Stokes equations in the Boussinesq limit Create wave beam by adding forcing term to Navier-Stokes: Numerical simulations r F ( ˆ z )cos(t ) where 0 0 ) ( ) ( exp ), ( z z x x A z x

29 Computed wave field 60 z (cm) 40 g Rodenborn, Kiefer, Zhang, Swinney (010) second harmonic wavemaker 0.3 Vorticity (rad/sec) 0 0 incident beam a fundamental x (cm) Bouyancy frequency N(z)=constant (linear density gradient)

30 Compare experiments and numerical simulations with theory Plate angle at nd harmonic maximum intensity 30 a max 15 Experiment Simulation Thorpe However, for very small amplitude and low viscosity, numerical simulation agrees with Tabaei et al. 0 i 15 (angle of incident beam) 30

31 Geometric analysis prediction agrees with simulation and experiment Plate angle a max 30 Tabaei et al. Conclude: a little nonlinearity is significant 15 Rodenborn et al. Thorpe 0 15 Incident beam angle i 30

32 Search for 3-dimensional effects: tidal flow over a Gaussian mountain z y x tidal flow mountain height = 7 cm, 1/e half-width =.85 cm

33 3-dimensional numerical simulation Grid: Code: CDP finite volume flow solver (Ham, Stanford) TETRAHEDRA ~ 5 million control volumes

34 Compare simulation and experiment TIDAL FLOW PAST A HEMI-SPHERE oscillating tide velocity (cm/s) simulation experiment vorticity (rad/s) cm King, Zhang, Swinney, Phys. Fluids 1, (009) cm

35 Internal wave generation visualization hemi-sphere FUNDAMENTAL (frequency ) velocity amplitude (cm/s) perpendicular to forcing plane nd HARMONIC (frequency ) velocity amplitude (cm/s) King, Zhang, Swinney Geophysical Res. Lett. (010)

36 Reflection of internal wave from ocean bottom Pingree & New, J. Phys. Oceanography (1991) depth (km) Computed path of internal wave propagation with ( z) where N ( z) arcsin, N( z) g d dz Bay of Biscay

37 Ocean zones SUNLIGHT ZONE TWILIGHT ZONE m: BP Deepwater Horizon well in Gulf of Mexico MIDNIGHT ZONE 54% > 4000 m ABYSS 1.0% > 6000 m ocean topography is well known TRENCHES Google: ocean zones

38 WORLD OCEAN CIRCULATION EXPERIMENT WOCE group: 30 countries Planning: 15 years ( ) Data acquisition: 10 years ( ) Cost: > $10 9 Measure: pressure temperature salinity in depth increments of m at points separated laterally by km

39 WOCE data Each blue dot corresponds to one of the 18,000 ship casts longitude

40 Compute N(z) from WOCE data Depth (km) AVERAGE use sw-bfrq routine ( N, 91.8 W February 1993 This N(z) value is more than 100<N> 4-5 km N < M M M N < M N(z)/(cycles/day)

41 Navier-Stokes Direct Numerical Simulations CDP parallel finite volume flow solver (by Frank Ham, Stanford) Create wave beam: add forcing to momentum equation: where Use measured N(z) r F ( ˆ z )cos(t ) ( x, z) Aexp ( x x 0 ) ( z (13.0 N, W, Middle America Trench) z 0 )

42 Snapshot of simulated internal wave beam depth (km) WAVEMAKER TURNING DEPTH EVANESCENT WAVES x (km) Domain: 6 km x 4 km. Resolution: 15 m 000 time steps/period for 0 periods. Viscosity 1.x10-3 m /s. VORTICITY (10-3 rad/s)

43 Internal wave dynamics Resonant boundary currents ( = a) (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 WOCE data sets: TURNING POINTS are ubiquitous in deep ocean. Open questions: wave breaking correlations mixing Coriolis effect