Internal Waves. Thierry Dauxois

Transcription

1 Internal Waves A. Two-layer stratification: Dead Water Experiments B. Linear stratification: Internal Wave Beams -Generation -Propagation -Reflection C. Realistic stratification: Solitons generation Thierry Dauxois

2 Stratified Fluids Atmosphere Density Stability Ocean

3 Goals of this talk I am neither an oceanographer, nor an astrophysicist, but only a physicist. This is why I will focus on the physical mechanisms, studied one after the other, an approach complementary to the other one (I hope!). Interest for a nonlinear theoretical physicist New domain of applications Unusual wave equations, Paradox Interest for oceanographers? Although difficult questions are already considered Simple problems have not been addressed New experimental techniques might help

4 Two Layer Stratification Light and fresh water Dense and salted water When caught in dead water, the boat appeared to be held back by some mysterious force. In calm weather, the boat was capable of 6 to 7 knots. When in dead water, he was unable to make 1.5 knots. Fritjof Nansen, a Norvegian explorer in his epic attempt to reach the North Pole

5 Internal Waves at a density interface Parameters: Tension Weight of the boat Depths of layers Difference of densities - Ekman, Maas, 2005

6 The boat Before fishing After fishing

7 Generation of internal waves: 2 layers fresh salted water

8 Surface Gravity Waves Mass/ Spring η=η 0 sin(kx-ωt) η=η 0 sin (ωt) ω 2 =gk tanh (kh) ω 2 =k 0 /m Frequency depends only on restoring force

9 Consider a two-layer system in the ocean ρ/ρ ~1/1000 if similar velocities in both layers η 1 ~1000η m internal displacement 10 cm surface expression Large amplitude internal waves

10 Generation of internal waves: 3 layers \MATTHIEU\STAGEROMAIN (3 couches avec arret) \MATTHIEU\STAGEROMAIN (3 couches avec arret) zoom

11 B) Linear Stratification Tree-layer system Linear Stratification

12 Brunt-Vaisala Frequency Lower density Competition between gravity and buoyancy Higher density Example: For the ocean, period ~ 30 min Slow oscillations Wave propagation

13 Basic Equations Navier-Stokes Eq. Incompressible flows Mass conservation Restricting to 2D and introducing the streamfunction one gets within the Boussinesq approximation

14 Unusual Wave Equation valid for -Streamfunction -Pressure -Density Plane wave solution -> ω < N -> Anisotropic propagation -> Orthogonal phase and group velocities -> No wavelength selection ω 2D 3D

15 Surface Waves Direction of propagation: Free Wavelength controled by the frequency: ω=ck Group and phase velocities are parallel

16 Internal Waves St. Andrew cross Direction of propagation: ω=n sin θ Wavelength not controled by the frequency: Free Group and phase velocities are orthogonal

17 Internal Waves Propagation Constant N Non Constant N Linear Propagation Nonlinear Propagation

18 Typical Density Profile

19 Unusual Wave Equation Nonlinear equation (inviscid case) Shear Waves, uniform or not, are solutions where Tabei, Akylas, Lamb 2005 But -Superposition of waves generates nonlinearities -Importance of topography

20 How internal waves are visualized in Laboratories?

21 Particle Image Velocimetry (PIV) technique Fluid seeded with 400 microns diameter particle polystyrene beads Beads of different densities Surfactant to prevent clustering Particles = passive tracers 2d Motion visualized by illuminating a laser sheet Camera -Fincham & Delerce, Exp. Fluids 29, 13 (2000) Uvmat (Coriolis) -Meunier & Leveque, Exp. Fluids 35, 408 (2003) DPIV soft (Irphe) Quantitative measurements of the velocity field

22 Synthetic Schlieren Technique Camera Grid Dalziel, Hughes, Sutherland, Exp. Fluids 28, 322 (2000). Quantitative measurements of the density gradient

23 Dye Plane Coloration Isopycnals= lines with the same density Experiment Theory Hopfinger, Flór, Chomaz & Bonneton, Exp. in Fluids 11, 255 (1991)

24 How internal waves are generated in Oceans?

25 Numerical Simulation Maugé & Gerkema (2006) Internal-tide generation close to the critical slope region Propagation of the internal-tide energy along beams to the deep ocean Series of reflections between the sea bed and the surface

26 Internal tide generation over a continental shelf L. Gostiaux, T. Dauxois, Laboratory experiments on the generation of internal tidal beams over steep slopes Physics of Fluids 19, (2007)

27 Emission via oscillating bodies Critical angle Synthetic Schlieren laboratory experiments R=1.5 cm R=3 cm R=4.5 cm Analogy for internal tide generation between -Curved static topography of local curvature R in oscillating fluid -Oscillating cylinder of radius R in static fluid

28 Internal tide generation over a continental shelf Frequency of Tides define an angle through the dispersion relation ω=n sin θ θ Topography Generation point osculatory cylinder OCEAN L. Gostiaux, T. Dauxois, Laboratory experiments on the generation of internal tidal beams over steep slopes Physics of Fluids 19, (2007)

29 Analogy for internal tide generation -Curved static topography of local curvature R in oscillating fluid -Oscillating cylinder of radius R in static fluid Hearley & Keady have shown (JFM 97) that the longitudinal velocity component of each beam of the St Andrews cross generated by an oscillating cylinder is -with the non-dimensional parameter -s longitudinal coordinate along the beam -σ transversal distance across the beam

30 Comparison Theory vs Experiment Experiment Theory L. Gostiaux, T. Dauxois, Laboratory experiments on the generation of internal tidal beams over steep slopes Physics of Fluids 19, (2007)

31 Recent prolongations T. Peacock, P. Echeverri & N.J. Balmforth, J. Phys. Ocean., 38, 235 (2008). MIT, Boston A. Paci, J. Flor, Y. Dosman, F. Auclair, (2008) Météo-France, Toulouse I. Pairaud, C. Staquet, J. Sommeria, M. Maddizadeh (2008) LEGI, Grenoble

32 How internal waves are generated in Laboratories?

33 Internal Waves Generation in a Laboratory Oscillating cylinder Gortler (1943), Mowbray & Rarity (1967), Peacock & Weidmann (2005), Drawbacks: -Several Beams -Beam s Width ~ Wavelength Excitation with a Paddle Cacchione & Wunsch (1973), Teok et al (1973), Gostiaux et al (2006),... Drawbacks: -Presence of Harmonics -Beam not well defined Parametric Instability Benielli & Sommeria (1998) Drawbacks: -Generation in the whole domain

34 A Novel Internal Wave Generator Original version 150 cm Pocket Size 14 cm 90 cm 15 cm Wavelength = 12 cm 10s <Time Period < 60s u~ 1 cm/s Wavelength = 4 cm 1s <Time Period < 60s u ~1 mm/s L. Gostiaux, H. Didelle, S. Mercier, T. Dauxois A novel internal waves generator, Exp. Fluids 42, 123 (2007)

35 Principle of the Novel Generator L. Gostiaux, H. Didelle, S. Mercier, T. Dauxois A novel internal waves generator, Experiments in Fluids 42, 123 (2007) Plates moved by two camshafts, imposing the relative position of the plates. Camshaft Boundary conditions generates internal waves

36 1) Generation of plane internal waves T 2T v phase 3T 4T v group Advantages: -Well defined beam -Only one beam -Wavelength << Width -Emission localized in space And the profile is very flexible

37 2) Generation of Internal Tide Mode 1 Enveloppe of the cames Principle -Only horizontal forcing -Without vertical forcing Experimental Result (PIV) Even without vertical forcing, this is an excellent mode T. Peacock, M. Mercier, T. Dauxois, Internal-tide scattering by 2d topography, in preparation (2009)

38 3) Generation of an Internal Tide Beam Internal tide Real Part Experimental Result using Synthetic Schlieren technique

39 Reflection of internal waves: The mystery of the critical angle

40 Reflection of Internal Waves: an old Paradox An example of topographical effects where nonlinearities are important Up slope Down slope The reflected ray keeps the same angle with respect to gravity Θ >γ Θ <γ Critical angle: θ=γ

41 Reflection: from a Ray to a Beam Energy Focusing Critical case θ=γ Singularity at the critical point Trapping of the waves Energyfocalisation Linear mechanism of transfer of energy to small scales Formation of nonlinear structures? Role of the dissipation? Old Mystery : Philipps, 1966!

42 Observation: in the ocean Sandström Bermuda slope Eriksen North Pacific Gilbert Nova Scotia Eriksen Fiberlying Guyot The velocity spectrum over tilted topography (γ=26 ) has an energy peak corresponding to the critical frequency

43 First Theoretical Remark Vanishing group velocity at the critical angle infinite time to reach the paradoxal stationary solution! Generation of a second harmonic propagating at a different angle ω 2 =2ω 1 =2(N sin θ 1 ) =N sin θ 2 θ 2 θ 1 θ n = arcsin (n sin θ 1 ) Transience and Nonlinearity are important

44 Analytical solution (Dauxois & Young JFM 99) One obtains a final amplitude equation which is linear! Creation of an array of vortices along the slope where Nice prolongations for a beam with a finite width by Tabei, Akylas & Lamb 2005 but away from the critical case

45 Experimental Test?

46 Qualitative results: classical Schlieren Theory Experiment Dauxois, Didier & Falcon, Phys. Fluids (2004) Dauxois & Young, J. Fluid. Mech. (1999) Overturning instability

47 Quantitative results: synthetic Schlieren Critical case Re=1! Clear energy focalization

48 Large scale experiments at higher Reynolds number Coriolis turntable located in Grenoble

49 Experiments without rotation Experiments without rotation α=10

50 Quantitative measurements Qualitative Measurements Time dependent picture Harmonic 1 Harmonic 2 Harmonic 3

51 Differences between sub and supercritical cases Harmonic 1 Harmonic 2 Harmonic 3 Sub-critical (θ<α) : Fundamental slightly perturbed Critical Super-critical (θ>α) : Fundamental strongly perturbed

52 Internal Waves Attractors

53 Generation of interfacial solitons by internal waves impinging on a thermocline

54 Solitons Massachusetts Bay & Cape Cod Bay Envisat ASAR APP 07-AUG :30 UTC

55 Radar backscattering from the sea surface Microwaves (radar waves) do not penetrate into water. Thus, the radar senses only the sea surface roughness. smooth surface rough surface

56 A realistic example: The Bay of Biscay Maugé et Gerkema, NL Processes in Geophysics 15, 233 (2008)

57 Generation of Internal Solitary Waves in a Laboratory 1. Control the stratification 2. Generate the internal tide beam 3. Measure the interfacial waves

58 Generation of Internal Solitary Waves in a Laboratory Top View Side View

59 Acoustic Probes Emission/Reception of an acoustic signal Thermocline Reflection of the acoustic signal

60

61 Deformation of the interface Solitons Generation temps 1 st probe 2 nd probe 3 rd probe

62 Solitons Generation Deformation of the interface temps 1 st probe 2 nd probe 3 rd probe

63 Perspectives 1) Fundamental questions Diffraction by slits Reflection on slopes convex slopes 2) Oceanographic questions Scattering by a seamount?

64 Perspectives Dissipation of Internal Waves: from generation to fate Localized mixing at internal tide generation sites Wave-wave interactions such at the Parametric Subharmonic Instability Interaction of internal waves with mesoscale structures. Scattering by finite-amplitude bathymetry??? Interaction between Internal Wave and Vortices Effect of the Coriolis force Munk & Wunsch (1998)

65 Thanks Matthieu Mercier (Lyon) Romain Vasseur (Lyon) Louis Gostiaux (Grenoble) Denis Martinand (Marseille) Tom Peacock (MIT, USA) Manikandan Mathur (MIT, USA) Theo Gerkema (Texel, Netherlands) Funds PATOM IDAO LEFE -Topogi 3D (2005) -PIWO (2008) 2008