length scales of stratified turbulence
TRANSCRIPT
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Length scales of stratified turbulence : new insight onThorpe displacements statistics
L. Gostiaux, A. Delache, E. Horne, LMFA, CNRS, EC Lyon, France
H. van Haren, NIOZ, Texel, The Netherlands
A. Cimatoribus, EPFL, Lausanne, Switzerland
A. Venaille, ENS de Lyon - Joel Sommeria, LEGI
Emil Hopfinger Colloquium11-13th May 2016, LEGI, Grenoble, France
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
Plane waves emitted by asinusoidal wave maker atthe Coriolis Platform LEGIT. Dauxois, J. Sommeria, C. Saquet,H. Didelle, S. Viboud
Internal wave breaking over adeep seamount, NIOZH. van Haren, A. Cimatoribus
Numerical stratified turbulence atLMFAC. Cambon, F. Godeferd, A. Delache
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
Plane waves emitted by asinusoidal wave maker atthe Coriolis Platform LEGIT. Dauxois, J. Sommeria, C. Saquet,H. Didelle, S. Viboud
Internal wave breaking over adeep seamount, NIOZH. van Haren, A. Cimatoribus
Numerical stratified turbulence atLMFAC. Cambon, F. Godeferd, A. Delache
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
Plane waves emitted by asinusoidal wave maker atthe Coriolis Platform LEGIT. Dauxois, J. Sommeria, C. Saquet,H. Didelle, S. Viboud
Internal wave breaking over adeep seamount, NIOZH. van Haren, A. Cimatoribus
Numerical stratified turbulence atLMFAC. Cambon, F. Godeferd, A. Delache
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
2009 version ofHopfinger and Toly experiment.A. Venaille, J. Sommeria LEGI
Hopfinger, E. J. & Toly, J.-A. Spatially decaying turbulence and its relation to mixing across density interfacesJournal of Fluid Mechanics, 1976, 78, 155-175
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
2009 version ofHopfinger and Toly experiment.A. Venaille, J. Sommeria LEGI
Dimension : 40× 40× 90cm
Grid : L = 10cm, 5Hz, 8cmamplitude
Lower fluid : water + salt +rhodamine
Upper fluid : water + ethanol
∆ρ =0.1, 0.3, 0.8 & 1.1%
1024× 1024, 10Hz, 25× 25cmHopfinger, E. J. & Toly, J.-A. Spatially decaying turbulence and its relation to mixing across density interfacesJournal of Fluid Mechanics, 1976, 78, 155-175
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
From linear internal waves to stratified turbulence
2009 version ofHopfinger and Toly experiment.A. Venaille, J. Sommeria LEGI
Hopfinger, E. J. & Toly, J.-A. Spatially decaying turbulence and its relation to mixing across density interfacesJournal of Fluid Mechanics, 1976, 78, 155-175
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Thorpe scale mixing estimate
Thorpe 1977:The presence of inversions may be used as a gauge of the verticaldisplacements which occur in the Loch.
Thorpe, S. A. Turbulence and Mixing in a Scottish Loch, Royal Society of London Philosophical Transactions SeriesA, 1977, 286, 125-181
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Thorpe scale mixing estimate
Re-ordering density profiles −→ Thorpe displacement d .
Thorpe scale LT =< d2 >1/2.<> : vertical + successive profiles
In stably stratified regions, d = 0 :χ is the proportion of unstable fluid(also named intermittency index)
Conditional Thorpe scale :L′T =< d2 >1/2 |d 6= 0
(LT =√χL′T )
Thorpe, S. A. Turbulence and Mixing in a Scottish Loch, Royal Society of London Philosophical Transactions SeriesA, 1977, 286, 125-181
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Oceanic Mixing recipies
Relating Thorpe scale to dissipation :Ozmidov Scale Lo = ε1/2N−3/2
Dillon, 1982 : Lo = 0.8LT
Osborn 1980 : kz = Γε/N2
(Γ = 0.2, mixing efficiency)Dillon, 1982
Turbulent dissipation rate ε = 0.64L2TN3
Turbulent diffusivity kz = 0.128L2TN
Ozmidov, R. On turbulent exchange in stable stratified ocean Izvestia Acad. Sci. USSR, Atmosphere and OceanPhysics, 1965, 8Dillon, T. M. Vertical Overturns: A Comparison of Thorpe and Ozmidov Length Scales J. Geophys. Res., AGU,1982, 87, 9601-9613Osborn, T. Estimates of the local rate of vertical diffusion from dissipation measurements J. Phys. Oceanogr.,1980, 10, 83-89
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
On the definition of Thorpe displacement
Thorpe 77 :“The vertical displacements are the minimum distances which fluidparticles need to be moved from the observed profile to producethe synthetic stable profile”
σ(zi )→ σs(zi )
σ(zi ) = σs(zi + di )
The inverted vertical displacements are the distances which fluidparticles need to be moved from the synthetic stable profile toproduce the observed profile
σ(zi + d∗i ) = σs(zi )
d∗i = −dp(i) where p is a permutation operator
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
On the definition of Thorpe displacement
Thorpe 77 :“The vertical displacements are the minimum distances which fluidparticles need to be moved from the observed profile to producethe synthetic stable profile”
σ(zi )→ σs(zi )
σ(zi ) = σs(zi + di )
The inverted vertical displacements are the distances which fluidparticles need to be moved from the synthetic stable profile toproduce the observed profile
σ(zi + d∗i ) = σs(zi )
d∗i = −dp(i) where p is a permutation operator
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
On the definition of Thorpe displacement
Thorpe 77 :“The vertical displacements are the minimum distances which fluidparticles need to be moved from the observed profile to producethe synthetic stable profile”
σ(zi )→ σs(zi )
σ(zi ) = σs(zi + di )
The inverted vertical displacements are the distances which fluidparticles need to be moved from the synthetic stable profile toproduce the observed profile
σ(zi + d∗i ) = σs(zi )
d∗i = −dp(i) where p is a permutation operator
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted Thorpe displacements
Case 1 : solid body rotation
In this case, d∗i = −dp(i) = di
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted Thorpe displacements
Case 2 : Rankine vortex
In this case, d∗(z) is discontinuous
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted displacements in the litterature
LT =< d2 >1/2.Thorpe, 1977
Thorpe, S. A. Turbulence and Mixing in a Scottish Loch, Royal Society of London Philosophical Transactions SeriesA, 1977, 286, 125-181
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted displacements in the litterature
Thorpe scaleLT =< (−d∗)2 >1/2.Dillon, 1982
Dillon, T. M. Vertical Overturns: A Comparison of Thorpe and Ozmidov Length Scales J. Geophys. Res., AGU,1982, 87, 9601-9613
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted displacements in the litterature
Thorpe scaleLT =< (−d∗)2 >1/2.Galbraith & Kelly, 1996
Galbraith, P. S. & Kelley, D. E. Identifying overturns in CTD profiles Journal of Physical Oceanography, 1996, 13,688-702
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Direct and inverted displacements in the litterature
Thorpe scaleLT =< (d∗)2 >1/2.Stansfield et al. 2004
Stansfield, K.; Garrett, C. & Dewey, R. The Probability Distribution of the Thorpe Displacement within Overturnsin Juan de Fuca Strait J. Phys. Oceanogr., Journal of Physical Oceanography, American Meteorological Society,2001, 31, 3421-3434
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
The signature of the overturn in d∗ profiles
Rankine Vortex
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
The signature of the overturn in d∗ profiles
Rankine Vortex
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
The signature of the overturn in d∗ profiles
Mixed patch
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Distinguishing overturns on in situ data
van Haren, H. & Gostiaux, L. Characterizing turbulent overturns in CTD-data Dynamics of Atmospheres andOceans , 2014, 66, 58 - 76
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Oceanic Mixing recipies
Relating Thorpe scale to dissipation :Ozmidov Scale Lo = ε1/2N−3/2
Dillon, 1982 : Lo = 0.8LT
Osborn 1980 : kz = Γε/N2
(Γ = 0.2, mixing efficiency)Dillon, 1982
Turbulent dissipation rate ε = 0.64L2TN3
Turbulent diffusivity kz = 0.128L2TN
Ozmidov, R. On turbulent exchange in stable stratified ocean Izvestia Acad. Sci. USSR, Atmosphere and OceanPhysics, 1965, 8Dillon, T. M. Vertical Overturns: A Comparison of Thorpe and Ozmidov Length Scales J. Geophys. Res., AGU,1982, 87, 9601-9613Osborn, T. Estimates of the local rate of vertical diffusion from dissipation measurements J. Phys. Oceanogr.,1980, 10, 83-89
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Searching for mixing places
Importance of boundaries
Armi, L. Some evidence for boundary mixing in the deep Ocean Journal of Geophysical Research: Oceans, 1978,83, 1971-1979Garrett, C. The role of secondary circulation in boundary mixing Journal of Geophysical Research, 1990, 95,3181-3188
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
The NIOZ High Sampling-Rate Thermistors
NIOZ-HST thermistors
Autonomous sensors (energyand data storage)
Synchronized 1Hztemperature aquisition
1mK relative accuracy
Duration more than one year
Deployment depth 6000mH. van Haren, M. Laan, D.-J. Buijsman, L. Gostiaux,M.G. Smit and E. Keijzer, NIOZ3: independenttemperature sensors sampling yearlong data at a rate of 1Hz, IEEE J. Ocean. Eng., 34, 315-322 (2009).
Hans van Haren,NIOZ, TheNetherlands
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Measurements at Josephine Seamount
Deep Atlantic Ocean seamount (submit at2000m)
Supercritical to subcritical for M2 tide
Mooring line with 100 sensors each, 1mvertical resolution, 3 months at 1Hz
Three different locations (supercritical tosubcritical)
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Measurements at Josephine Seamount
Mooring A, 2210m, N = 1.4 10−3rad/s
H. van Haren, A. Cimatoribus and L. Gostiaux, Where large deep-ocean waves break, Geophysical Research Letters,GL063329, 2015
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Measurements at Josephine Seamount
Mooring B, 2530m, N = 1.0 10−3rad/s
H. van Haren, A. Cimatoribus and L. Gostiaux, Where large deep-ocean waves break, Geophysical Research Letters,GL063329, 2015
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Measurements at Josephine Seamount
Mooring C, 2937m, N = 0.7 10−3rad/s
H. van Haren, A. Cimatoribus and L. Gostiaux, Where large deep-ocean waves break, Geophysical Research Letters,GL063329, 2015
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Analytical overturn
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Analytical overturn
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Deep ocean overturns
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Numerical overturns
High resolution (20483 points, Re=1000 kmaxη ' 3) withpseudospectral method, freely decaying turbulence.A. Delache, E. Horne, LMFA
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Statistics of displacement values
Gobal richardson number Ri = N2H2/u2 with u the rms turbulentvelocity and H the vertical domain size
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Statistics of displacement values
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Statistics of displacement values
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Simple energetic scaling
From the energetic point of view, the potential energy in anoverturn is Ep ' N2d2 (Thorpe 1977)If an overturn is seen as a transfer of kinetic to potential energy,Ec ' Ep leads to d2 ' u2/N2 = H2Ri−1
d ' H/√
Ri
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Conditional Thorpe scale L′T
L′T =< d2 >1/2 |d 6= 0(LT =
√χL′T )
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Conditional Thorpe scale L′T
L′T =< d2 >1/2 |d 6= 0(LT =
√χL′T )
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Effect of intermittency χ
The intermittency influences the Thorpe scale estimate at high Ri
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Energetic approcah
On the other hand, the energetic interpretation Ep ' N2d2 is morevalid with a strong intermittency
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
Perspectives
Comparison with A. Venaille statistical model
DNS in the case of two-layer stratified turbulence
Statistics of Thorpe displacements on oceanic data
L. Gostiaux Length scales of Stratified Turbulence
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The Thorpe scale LT Mixing estimates Thorpe’s displacement statistics Conclusion
The inverted Thorpe displacement
{σ(zi ) = σs(zi + di )σ(zi + d∗i ) = σs(zi )
With z∗i = zi + d∗i , it follows
σ(z∗i ) = σs(z∗i − d∗i )
But [z∗i ] is a permutation of [zi ], which can be written :{z∗i = zp(i)
d∗i = −dp(i)
L. Gostiaux Length scales of Stratified Turbulence