Thierry Dauxois
Internal Waves
A. Two-layer stratification: Dead Water ExperimentsB. Linear stratification: Internal Wave Beams
-Generation-Propagation-Reflection
C. Realistic stratification: Solitons generation
• I am neither an oceanographer, nor an astrophysicist,
but only a physicist.
Goals of this talk
•Interest for a nonlineartheoretical physicist• New domain of applications• Unusual wave equations, Paradox
• This is why I will focus on the physical mechanisms, studied one after the other, an approach complementaryto the other one (I hope!).
•Interest for oceanographers?• Although difficult questions are already considered• Simple problems have not been addressed• Newexperimental techniques might help
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
Two Layer Stratification
• Tension• Weight of the boat• Depths of layers• Difference of densities
Parameters: - Ekman, 1904- Maas, 2005
Internal Waves at a density interface
Surface Gravity Waves Mass/ Spring
Frequency depends only on restoring force
η=η0 sin(kx-ωt) η=η0 sin (ωt)
ω2=gk tanh (kH) ω2=k0/m
Consider a two-layer system
Large amplitude internal waves
• in the ocean ∆ρ/ρ ~1/1000
• if similar velocities in both layers η1~1000η0
• 100 m internal displacement 10 cm surface expression
Generation of internal waves: 3 layers
• \MATTHIEU\STAGEROMAIN (3 couches avec arret)
• \MATTHIEU\STAGEROMAIN (3 couches avec arret) zoom
For the ocean,
period ~ 30 min
• Slow oscillations
• Wave propagationExample:
Lower density
Higher density
Brunt-Vaisala Frequency
Competition betweengravityand buoyancy
Basic Equations
Navier-Stokes Eq.
Incompressible flows
Mass conservation
Restricting to 2D and introducing the streamfunction
one gets within the Boussinesq approximation
-> ωωωω < N
ωωωω
-> Anisotropic propagation
2D 3D
-> Orthogonal phase and group velocities
-> No wavelength selection
-Streamfunction-Pressure-Density
valid for
Plane wavesolution
Unusual Wave Equation
Surface Waves
• Direction of propagation: Free
• Wavelength controled by the frequency: ω=ck
• Group and phase velocities are parallel
St. Andrew cross
Internal Waves
• Direction of propagation: ω=N sin θ• Wavelength not controled by the frequency: Free• Group and phase velocities are orthogonal
where
Unusual Wave Equation
Nonlinear equation (inviscid case)
Shear Waves, uniform or not, are solutions
But… -Superposition of waves generates nonlinearities-Importance of topography
Tabei, Akylas, Lamb 2005
-Fincham & Delerce, Exp. Fluids 29, 13 (2000) � Uvmat (Coriolis)
-Meunier & Leveque, Exp. Fluids 35, 408 (2003) � DPIV soft (Irphe)
Particle Image Velocimetry (PIV) technique
Camera
Quantitative measurements of the velocity field
• Fluid seeded with 400 microns diameter particle polystyrene beads• Beads of different densities• Surfactant to prevent clustering• Particles = passive tracers• 2d Motion visualized by illuminatinga laser sheet
Dalziel, Hughes, Sutherland, Exp. Fluids 28, 322 (2000).
Synthetic Schlieren Technique
Grid
Camera
Quantitative measurements of the density gradient
Experiment Theory
Isopycnals= lines with the same density
Dye Plane Coloration
Hopfinger, Flór, Chomaz & Bonneton, Exp. in Fluids 11, 255 (1991)
• Internal-tidegeneration 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
Maugé & Gerkema (2006)
Numerical Simulation
L. Gostiaux, T. Dauxois, Laboratory experiments on the generation of internal tidal beams over steep slopes Physics of Fluids 19, 028102 (2007)
Internal tide generation over a continental shelf
Critical angle
R=1.5 cm R=3 cm R=4.5 cm
Synthetic Schlieren laboratory experiments
Emission via oscillating bodies
Analogy for internal tide generation between-Curved static topography of local curvature R in oscillating fluid-Oscillating cylinder of radius R in static fluid
L. Gostiaux, T. Dauxois, Laboratory experiments on the generation of internal tidal beams over steep slopes Physics of Fluids 19, 028102 (2007)
Internal tide generation over a continental shelf
Frequency of Tides define an angle throughthe dispersion relation ω=N sin θ
θ
Topography OCEAN
Generationpointosculatorycylinder
Hearley & Keady have shown (JFM 97) that the longitudinal velocitycomponent of each beam of the St Andrews cross generated by an oscillating cylinder is
Analogy for internal tide generation
-Curved static topography of local curvature R in oscillating fluid-Oscillating cylinder of radius R in static fluid
-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)
T. Peacock, P. Echeverri & N.J. Balmforth, J. Phys. Ocean., 38, 235 (2008).
A. Paci, J. Flor, Y. Dosman,F. Auclair, (2008)
I. Pairaud, C. Staquet, J. Sommeria, M. Maddizadeh (2008)
Recent prolongations
MIT, Boston
Météo-France, Toulouse
LEGI, Grenoble
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
L. Gostiaux, H. Didelle, S. Mercier, T. Dauxois A novel internal waves generator, Exp. Fluids 42, 123 (2007)
150 cm
90cm
14 cm
15cm
Pocket SizeOriginal version
Wavelength = 12 cmu~ 1 cm/s
10s <Time Period < 60s
Wavelength = 4 cmu ~1 mm/s
1s <Time Period < 60s
Principle of the Novel GeneratorL. Gostiaux, H. Didelle, S. Mercier, T. Dauxois A novel internal waves generator, Experiments in Fluids 42, 123 (2007)
Boundary conditions generates internal waves
Camshaft
Plates moved by two camshafts, imposing the relative position of the plates.
1) Generation of plane internal waves
T 2T
3T 4T
Advantages: -Well defined beam -Wavelength << Width-Only one beam -Emission localized in space
vphase
vgroup
And the profile is very flexible
2) Generation of Internal Tide Mode 1
Even without vertical forcing, this is an excellent modeT. Peacock, M. Mercier, T. Dauxois, Internal-tide scattering by 2d topography, in preparation (2009)
ExperimentalResult (PIV)
Principle-Only horizontal forcing-Without vertical forcing
Enveloppe of the cames
3) Generation of an Internal Tide Beam
Experimental Result usingSynthetic Schlieren technique
Internal tide Real Part
The reflected ray keeps the same angle with respect to gravity
Reflection of Internal Waves: an old Paradox
An example of topographical effects where nonlinearities are important
Θ >γ Θ <γ
Up slope Down slope
Critical angle: θ=γ
Energy Focusing
• Energy focalisation• Linear mechanism of transfer of energy to small scales
• Singularity at the critical point
• Trapping of the waves
• Formation of nonlinear structures?
• Role of the dissipation ?
Critical case θ=γ
Old Mystery : Philipps, 1966 !
Reflection: from a Ray to a Beam
- 1966 SandströmBermuda slope- 1982 Eriksen North Pacific- 1993 Gilbert Nova Scotia- 1998 Eriksen Fiberlying Guyot
Observation: in the ocean
The velocity spectrum over tiltedtopography (γ=26°) has an energypeak corresponding to the criticalfrequency
First Theoretical Remark
Vanishing group velocity at the critical angle
infinite time to reach the paradoxal stationary solution !
Generation of a second harmonic propagatingat a different angle
ω2=2ω1=2(N sin θ1)=N sin θ2
θn= arcsin (n sin θ1)
θ2
θ1
Transience and Nonlinearity are important
where
Analytical solution (Dauxois & Young JFM 99)
Creation of an array of vortices along the slope
Nice prolongations for a beam with a finite widthby Tabei, Akylas & Lamb 2005 but away from the critical case
One obtains a final amplitude equationOne obtains a final amplitude equation which is linear !
Dauxois & Young, J. Fluid. Mech. (1999)
Dauxois, Didier & Falcon, Phys. Fluids (2004)
Overturning instability
Qualitative results: classical Schlieren
Theory Experiment
Quantitative measurements
Harmonic 1
Harmonic 3Harmonic 2
Time dependent picture
Qualitative Measurements
–
– Sub-critical (θ<α) :– Fundamental slightly
perturbed
– Critical
– Super-critical (θ>α) :– Fundamental strongly perturbed
Harmonic 1 Harmonic 2 Harmonic 3Differences between sub and supercritical cases
smoothsurface
roughsurface
• Microwaves (radar waves) do not penetrate into water.
• Thus, the radar senses onlythe sea surface roughness.
Radar backscattering from the sea surface
Generation of Internal Solitary Waves in a Laboratory
1. Control the stratification2. Generate the internal tide beam3. Measure the interfacial waves
Acoustic Probes
Thermocline
Emission/Receptionof an acoustic signal
Reflection of the acoustic signal
Perspectives
Reflection on slopes
Diffraction by slits
Reflection on convex slopes
Scattering by a seamount ?
1) Fundamental questions
2) Oceanographic questions
Perspectives
• 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• ???
�Dissipation of Internal Waves: from generation to fate
Munk & Wunsch (1998)
�Interaction between Internal Wave and Vortices
�Effect of the Coriolis force