Thierry Dauxois

Internal Waves

A. Two-layer stratification: Dead Water ExperimentsB. Linear stratification: Internal Wave Beams

-Generation-Propagation-Reflection

C. Realistic stratification: Solitons generation

Stratified FluidsAtmosphere

Ocean

Density

Stability

• 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

Before fishing After fishing

The boat

fresh

salted

water

Generation of internal waves: 2 layers

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

Tree-layer system

Linear Stratification

B) Linear Stratification

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

Internal Waves Propagation

Constant N Non Constant N

Linear Propagation Nonlinear Propagation

Typical Density Profile

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

How internal waves are visualized in Laboratories?

-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)

How internal waves are generated in Oceans?

• 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

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

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

Reflection of internal waves:The mystery of the critical angle

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 !

Experimental Test ?

Dauxois & Young, J. Fluid. Mech. (1999)

Dauxois, Didier & Falcon, Phys. Fluids (2004)

Overturning instability

Qualitative results: classical Schlieren

Theory Experiment

Clear energy focalization

Re=1 !

Critical case

Quantitative results: synthetic Schlieren

Coriolis turntablelocated in Grenoble

Large scale experiments at higher Reynolds number

α=10°

Experiments without rotationExperiments without rotation

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

Internal Waves Attractors

Generation of interfacial solitons by internal waves impinging

on a thermocline

Massachusetts Bay & Cape Cod Bay

Envisat ASAR APP

07-AUG-2003 2:30 UTC

Solitons

smoothsurface

roughsurface

• Microwaves (radar waves) do not penetrate into water.

• Thus, the radar senses onlythe sea surface roughness.

Radar backscattering from the sea surface

Maugé et Gerkema, NL Processes in Geophysics 15, 233 (2008)

A realistic example: The Bay of Biscay

Generation of Internal Solitary Waves in a Laboratory

1. Control the stratification2. Generate the internal tide beam3. Measure the interfacial waves

Generation of Internal Solitary Waves in a Laboratory

Top View

Side View

Acoustic Probes

Thermocline

Emission/Receptionof an acoustic signal

Reflection of the acoustic signal

Solitons Generation

temps

Deformation ofthe interface

1st probe

2nd probe

3rd probe

Solitons Generation

temps

Deformation ofthe interface

1st probe

2nd probe

3rd probe

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

Thanks

Tom Peacock(MIT, USA)

Romain Vasseur(Lyon)

Theo Gerkema(Texel, Netherlands)

Manikandan Mathur (MIT, USA)

Matthieu Mercier(Lyon)

-Topogi 3D (2005)-PIWO (2008)

-2005 PATOM-2006 IDAO-2007 LEFE 2008

Denis Martinand(Marseille)

Louis Gostiaux(Grenoble)

Funds