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 Stratified Fluids Atmosphere Density Stability Ocean 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 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 Internal Waves at a density interface Parameters: • Tension • Weight of the boat • Depths of layers • Difference of densities - Ekman, 1904 - Maas, 2005 The boat Before fishing After fishing Generation of internal waves: 2 layers fresh salted water Surface Gravity Waves Mass/ Spring η=η0 sin(kx-ωt) η=η0 sin (ωt) ω2=gk tanh (kH) ω2=k0/m Frequency depends only on restoring force Consider a two-layer system • in the ocean ∆ρ/ρ ~1/1000 • if similar velocities in both layers • 100 m internal displacement η1~1000η0 10 cm surface expression Large amplitude internal waves Generation of internal waves: 3 layers • \MATTHIEU\STAGEROMAIN (3 couches avec arret) • \MATTHIEU\STAGEROMAIN (3 couches avec arret) zoom B) Linear Stratification Tree-layer system Linear Stratification Brunt-Vaisala Frequency Lower density Competition between gravity and buoyancy Higher density Example: For the ocean, • Slow oscillations period ~ 30 min • Wave propagation Basic Equations Navier-Stokes Eq. Incompressible flows Mass conservation Restricting to 2D and introducing the streamfunction one gets within the Boussinesq approximation Unusual Wave Equation -Streamfunction valid for -Pressure -Density Plane wave solution -> ω < N -> Anisotropic propagation -> Orthogonal phase and group velocities -> No wavelength selection 2D ω 3D Surface Waves • Direction of propagation: Free • Wavelength controled by the frequency: ω=ck • Group and phase velocities are parallel 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 Internal Waves Propagation Constant N Non Constant N Linear Propagation Nonlinear Propagation Typical Density Profile 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 How internal waves are visualized in Laboratories? 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 Synthetic Schlieren Technique Camera Grid Dalziel, Hughes, Sutherland, Exp. Fluids 28, 322 (2000). Quantitative measurements of the density gradient Dye Plane Coloration Isopycnals= lines with the same density Experiment Theory Hopfinger, Flór, Chomaz & Bonneton, Exp. in Fluids 11, 255 (1991) How internal waves are generated in Oceans? 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 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, 028102 (2007) 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 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, 028102 (2007) 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 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, 028102 (2007) 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 How internal waves are generated in Laboratories? 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 A Novel Internal Wave Generator Original version Pocket Size 150 cm 14 cm 90 cm 15 cm Wavelength = 12 cm Wavelength = 4 cm u~ 1 cm/s 10s <Time Period < 60s u ~1 mm/s 1s <Time Period < 60s L. Gostiaux, H. Didelle, S. Mercier, T. Dauxois A novel internal waves generator, Exp. Fluids 42, 123 (2007) 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 1) Generation of plane internal waves T 2T vphase 3T 4T vgroup Advantages: -Well defined beam -Only one beamAnd -Wavelength << Width the -Emission profile is very inflexible localized space 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) 3) Generation of an Internal Tide Beam Internal tide Real Part Experimental Result using Synthetic Schlieren technique Reflection of internal waves: The mystery of the critical angle 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 Reflection: from a Ray to a Beam Energy Focusing Critical case θ=γ • Singularity at the critical point • Trapping of the waves • Energy focalisation • Formation of nonlinear structures? • Linear mechanism of transfer of energy to small scales • Role of the dissipation ? Old Mystery : Philipps, 1966 ! Observation: in the ocean - 1966 Sandström Bermuda slope - 1982 Eriksen North Pacific - 1993 Gilbert Nova Scotia - 1998 Eriksen Fiberlying Guyot The velocity spectrum over tilted topography (γ=26°) has an energy peak corresponding to the critical frequency 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 ω2=2ω1=2(N sin θ1) =N sin θ2 θn= arcsin (n sin θ1) θ2 θ1 Transience and Nonlinearity are important Analytical solution (Dauxois & Young JFM 99) One obtains a final amplitude equation which is linear ! where Creation of an array of vortices along the slope Nice prolongations for a beam with a finite width by Tabei, Akylas & Lamb 2005 but away from the critical case Experimental Test ? Qualitative results: classical Schlieren Theory Experiment Dauxois, Didier & Falcon, Phys. Fluids (2004) Dauxois & Young, J. Fluid. Mech. (1999) Overturning instability Quantitative results: synthetic Schlieren Critical case Re=1 ! Clear energy focalization Large scale experiments at higher Reynolds number Coriolis turntable located in Grenoble Experiments without rotation Experiments without rotation α=10° Quantitative measurements Qualitative Measurements Time dependent picture Harmonic 2 Harmonic 1 Harmonic 3 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 Internal Waves Attractors Generation of interfacial solitons by internal waves impinging on a thermocline Solitons Massachusetts Bay & Cape Cod Bay Envisat ASAR APP 07-AUG-2003 2:30 UTC 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 A realistic example: The Bay of Biscay Maugé et Gerkema, NL Processes in Geophysics 15, 233 (2008) Generation of Internal Solitary Waves in a Laboratory 1. Control the stratification 2. Generate the internal tide beam 3. Measure the interfacial waves Generation of Internal Solitary Waves in a Laboratory Top View Side View Acoustic Probes Emission/Reception of an acoustic signal Thermocline Reflection of the acoustic signal Solitons Generation Deformation of the interface 1st probe temps 2nd probe 3rd probe Solitons Generation Deformation of the interface 1st probe temps 2nd probe 3rd probe Perspectives 1) Fundamental questions Diffraction by slits Reflection on convex slopes slopes 2) Oceanographic questions Scattering by a seamount ? 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) Thanks Matthieu Mercier (Lyon) Romain Vasseur (Lyon) Tom Peacock (MIT, USA) Funds -2005 PATOM -2006 IDAO -2007 LEFE Louis Gostiaux (Grenoble) Manikandan Mathur (MIT, USA) Denis Martinand (Marseille) Theo Gerkema (Texel, Netherlands) -Topogi 3D (2005) -PIWO (2008) 2008