Загрузил Dmitriy Romanenkov

# Dauxois () Internal Waves

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```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
•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
-&gt; ω &lt; N
-&gt; Anisotropic propagation
-&gt; Orthogonal phase and
group velocities
-&gt; 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
• Surfactant to prevent clustering
• Particles = passive tracers
• 2d Motion visualized by illuminating
a laser sheet
Camera
-Fincham &amp; Delerce, Exp. Fluids 29, 13 (2000) Uvmat (Coriolis)
-Meunier &amp; 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, Fl&oacute;r, Chomaz &amp; Bonneton, Exp. in Fluids 11, 255 (1991)
How internal waves are generated in
Oceans?
Numerical Simulation
Maug&eacute; &amp; 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 &amp; 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 &amp; N.J. Balmforth,
J. Phys. Ocean., 38, 235 (2008).
MIT, Boston
A. Paci, J. Flor, Y. Dosman,
F. Auclair, (2008)
M&eacute;t&eacute;o-France, Toulouse
I. Pairaud, C. Staquet, J. Sommeria,
LEGI, Grenoble
How internal waves are generated in
Laboratories?
Internal Waves Generation in a Laboratory
Oscillating cylinder
Gortler (1943), Mowbray &amp; Rarity (1967), Peacock &amp; Weidmann (2005),…
Drawbacks:
-Several Beams
-Beam’s Width ~ Wavelength
Cacchione &amp; Wunsch (1973), Teok et al (1973), Gostiaux et al (2006), ...
Drawbacks:
-Presence of Harmonics
-Beam not well defined
Parametric Instability
Benielli &amp; 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 &lt;Time Period &lt; 60s
u ~1 mm/s
1s &lt;Time Period &lt; 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
-Only one beamAnd
-Wavelength &lt;&lt; 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 &gt;γ
Θ &lt;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 Sandstr&ouml;m Bermuda slope
- 1982 Eriksen
North Pacific
- 1993 Gilbert
Nova Scotia
- 1998 Eriksen
Fiberlying Guyot
The velocity spectrum over tilted
topography (γ=26&deg;) 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 &amp; 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 &amp; Lamb 2005 but away from the critical case
Experimental Test ?
Qualitative results: classical Schlieren
Theory
Experiment
Dauxois, Didier &amp; Falcon, Phys. Fluids (2004)
Dauxois &amp; 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&deg;
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&lt;α) :
–
Fundamental slightly
perturbed
–
Critical
–
Super-critical (θ&gt;α) :
–
Fundamental strongly perturbed
Internal Waves Attractors
Generation of interfacial solitons by
internal waves impinging
on a thermocline
Solitons
Massachusetts Bay &amp;
Cape Cod Bay
Envisat ASAR APP
07-AUG-2003 2:30 UTC
Radar backscattering from the sea surface
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
Maug&eacute; 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 &amp; 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
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