Notes on different aspects of internal waves - including references
Jarle Berntsen
Department of Mathematics, University of Bergen, Johs. Bruns gt. 12, N-5008 Bergen
Abstract
Short summaries of papers and references.
1 Classification of papers
There is a large and growing literature on internal waves. In the papers different aspects of the waves may be in focus, and also different methods for studying them may be used. Here I try to make a simple classification system for these papers according to the aspects in focus and methods used. Usually one paper falls into several classes.
A. Generation of internal waves.
B. Propagation of internal waves.
C. Breaking and mixing of internal waves.
D. Fjord/Loch/Sill processes.
E. Shelf Slope/Shelf Edge processes.
F. Measurements.
G. Mathematical modelling and theory.
H. Numerical studies.
I. Laboratory experiments.
J. Energetics of internal waves.
K. Wind generation.
L.Fresh water/river water influences.
M.Other aspects of internal waves.
N. Most of the topics above.
A.The generation of internal waves is discussed briefly in many papers, but often in rather vague terms. I have found explanations and discussions of the generation in Parsmar and Stigebrandt [1997], Bourgault et al. [2007], Green and Stigebrandt [2003], Jayne and St.Laurent [2001], Stacey [1984, 1985], Stacey and Gratton [2001], Dewey et al. [2005], Klymak and Gregg [2001, 2003, 2004], Van Haren et al. [2004], Vlasenko and Hutter [2001], Nycander [2005], and Guo and Davies [2003], Farmer and Armi [1999], Nakamura et al. [2000], Di Lorenzo et al. [2006], Vlasenko et al. [2002], Cummins et al. [2003], Haidvogel [2005]. See Alford [2001], Appt et al. [2004], Boegman et al. [2005], Umlauf and Lemmin [2005], Nakamura and Awaji [2004], Inall et al. [2004], Lamb [2002, 2003], Wunsch and Ferrari [2004], Munroe and Lamb [2005], Apel et al. [2006], Helfrich and Melville [2006], Wang [2006], Wake et al. [2007]. Chang [1993], Horn et al. [2000a].
B.The propagation of internal waves is discussed in Bourgault et al. [2007], Bour- gault and Kelley [2007], Stacey [1984, 1985], Stacey and Gratton [2001], Klymak and Gregg [2001, 2003, 2004], Van Haren et al. [2004], Vlasenko and Hutter [2001], Bourgault and Kelley [2003], Moum et al. [2003, 2007], Moum and Smyth [2006],
Nakamura et al. [2000], Vlasenko and Hutter [2002a,b], Vlasenko et al. [2002], and Cacchione and Wunsch [1974], Helfrich [1992], Helfrich and Melville [2006], Michallet and Barthelemy [1998], Michallet and Ivey [1999], and Fringer and Street [2003], Gargett and Holloway [1984], Farmer and Armi [1999], Cummins et al. [2003], Haidvogel [2005], Munroe and Lamb [2005]. Bogucki and Garrett [1993], MacKinnon and Gregg [2003], Nakamura and Awaji [2004], Katsumata [2006], Venayagamoorthy and Fringer [2006]. Thorpe [1999, 2000], Lamb [2002, 2003], Manders and Maas [2003], Appt et al. [2004], Boegman et al. [2005], Umlauf and Lemmin [2005], Apel et al. [2006], Wang [2006], Wake et al. [2007]. Chang [1993], Horn et al. [2000a].
C. Breaking and mixing of internal waves is discussed in Parsmar and Stige- brandt [1997], Bourgault et al. [2007], Bourgault and Kelley [2007], Fer and Widell [2007], Widell [2006], Jayne and St.Laurent [2001], Stacey [1984, 1985], Stacey and Gratton [2001], Dewey et al. [2005], Klymak and Gregg [2001, 2003, 2004], Van Haren et al. [2004], Vlasenko and Hutter [2001], Bourgault and Kelley [2003], Inall et al. [2000, 2004, 2005], Moum et al. [2007], Moum and Smyth [2006], Cac- chione and Wunsch [1974], Nakamura et al. [2000], Saenko and Merryfield [2005], Winters et al. [1995], Winters and D’Asaro [1997], Zaron and Egbert [2006], Wunsch [1970], Vlasenko and Hutter [2002a,b], Vlasenko et al. [2002], Moum et al. [2003], and Helfrich [1992], Helfrich and Melville [2006], Michallet and Ivey [1999], Sveen et al. [2002], Guo and Davies [2003], Guo et al. [2005], Fringer and Street [2003], Gargett and Holloway [1984], Farmer and Armi [1999], Cummins et al. [2003], Haidvogel [2005]. Further studies in Polzin et al. [1997], Nash et al.
[2004], Wunsch and Ferrari [2004], Smyth et al. [2005], and Lamb [2004], Xing and Davies [2006a,b, 2007], Davies and Xing [2007]. D¨ornbrack [1998], Tseng and Ferziger [2001], Legg and Adcroft [2003], Peltier and Caulfield [2003], Kunze and Llewellyn Smith [2004], Munroe and Lamb [2005], McPhee-Shaw [2006]. Bogucki and Garrett [1993], MacKinnon and Gregg [2003], Nakamura and Awaji [2004], Venayagamoorthy and Fringer [2006]. Kao et al. [1985], Carnevale et al. [2001], Staquet and Sommeria [2002], Manders and Maas [2003], Appt et al. [2004], Um- lauf and Lemmin [2005], Wang [2006]. Toole et al. [1997], Thorpe [1999, 2000], Samelson [1998], Alford [2001], Spall [2001], Boegman et al. [2005]. Cummins [2000], Fofonoff [2001], Afanasyev and Peltier [2001a,b], Polyakov [2001], Armi and Farmer [2002], Lamb [2002, 2003]. Horn et al. [2000a].
D. In many paper the focus is on internal waves in fjords or lochs, and in par- ticular their interaction, generation and breaking, with sills. See Parsmar and Stigebrandt [1997], Fer and Widell [2007], Widell [2006], Green and Stigebrandt [2003], Jayne and St.Laurent [2001], Stacey [1984, 1985], Stacey and Gratton [2001], Dewey et al. [2005], Klymak and Gregg [2001, 2003, 2004], Vlasenko and Hutter [2001], McCabe et al. [2006], Nakamura et al. [2000], Nakamura and Awaji
[2004], Zaron and Egbert [2006], Vlasenko and Hutter [2002a,b], Vlasenko et al.
[2002], ,Di Lorenzo et al. [2006], Farmer and Armi [1999], Cummins et al. [2003], Xing and Davies [2006a,b, 2007], Davies and Xing [2007], Cummins [2000], Farmer and Armi [2001], Afanasyev and Peltier [2001a,b], Armi and Farmer [2002], Lamb [2004], Inall et al. [2004, 2005], Wijffels and Meyers [2004], Janzen et al. [2005], Boegman et al. [2005], Hosegood and van Haren [2006], Wang [2006].
E.Along shelf slopes and shelf edges there are strong interactions between topog- raphy and internal waves. See Bourgault et al. [2007], Bourgault and Kelley [2007], Jayne and St.Laurent [2001], Bourgault and Kelley [2003], Inall et al. [2000], Moum et al. [2003, 2007], Moum and Smyth [2006], and Helfrich [1992], Michal- let and Ivey [1999], Guo and Davies [2003], Legg and Adcroft [2003], Nash et al.
[2004], Haidvogel [2005], Munroe and Lamb [2005]. Toole et al. [1997], Thorpe [1997, 1999, 2000], MacKinnon and Gregg [2003], Fofonoff [2001], Lamb [2002, 2003], Hosegood and van Haren [2006], Katsumata [2006], Venayagamoorthy and Fringer [2006], Apel et al. [2006]. Chang [1993].
F. Many measurement programs have focused on studying these waves. Refer- ences include Parsmar and Stigebrandt [1997], Bourgault et al. [2007], Fer and Widell [2007], Widell [2006], Green and Stigebrandt [2003], Stacey [1984], Stacey and Gratton [2001], Dewey et al. [2005], Klymak and Gregg [2001, 2003, 2004], Van Haren et al. [2004], Bourgault and Kelley [2003], Inall et al. [2000, 2004, 2005], Moum et al. [2003, 2007], Moum and Smyth [2006], Polzin et al. [1997], Nakamura et al. [2000], Nash and Moum [2005], Gargett and Holloway [1984], Farmer and Armi [1999], Cummins et al. [2003], Nash et al. [2004], McPhee-Shaw [2006], Toole et al. [1997], Thorpe [1999], Boyer and Zhang [1990], Bogucki and Garrett [1993], Helfrich and Melville [2006], Apel et al. [2006]. Ostrovsky and Stepanyants [1989], MacKinnon and Gregg [2003], Wijffels and Meyers [2004], Appt et al. [2004], Boegman et al. [2005], Umlauf and Lemmin [2005], Hosegood and van Haren [2006]. Fofonoff [2001], Armi and Farmer [2002], Janzen et al.
[2005], Smith et al. [2007]
G. Many mathematical models have been developed to get more insight in these waves. A classic text book is Kundu and Cohen [2004] and references therein.
See also Ibragimov [2007], Green and Stigebrandt [2003], Jayne and St.Laurent [2001], Stacey [1984], Bourgault and Kelley [2003], Nycander [2005], McCabe et al. [2006], and Segur and Hammack [1982], Ostrovsky and Stepanyants [1989], Helfrich [1992], Helfrich and Melville [2006], Guo et al. [2005], Michallet and Barthelemy [1998], Nash and Moum [2005], Winters et al. [1995], Winters and D’Asaro [1997], Zaron and Egbert [2006], Wunsch [1970], Fringer and Street [2003], Gargett and Holloway [1984]. See also energy budgets in Wunsch and Ferrari [2004]. See Cacchione and Wunsch [1974], Kao et al. [1985], Bogucki and
Garrett [1993], Thorpe [1997, 1999, 2000], Carnevale et al. [2001], Staquet and Sommeria [2002], Nilsen [2004], Apel et al. [2006]. Armi and Farmer [2002], Man- ders and Maas [2003], Boegman et al. [2005] Chang [1993], Horn et al. [2000a].
H. Numerical models have also been used in many studies of internal waves. See Bourgault et al. [2007], Bourgault and Kelley [2007], Jayne and St.Laurent [2001], Klymak and Gregg [2003], Vlasenko and Hutter [2001, 2002a,b], Vlasenko et al.
[2002], Bourgault and Kelley [2003], Nycander [2005], McCabe et al. [2006], Naka- mura et al. [2000], Nakamura and Awaji [2004], Saenko and Merryfield [2005], Winters et al. [1995], Winters and D’Asaro [1997], Zaron and Egbert [2006], Stacey [1985], Bogucki and Garrett [1993], Stacey and Gratton [2001], Michallet and Barthelemy [1998], Di Lorenzo et al. [2006], Fringer and Street [2003], Smyth et al. [2005], Farmer and Armi [1999], Cummins et al. [2003], Haidvogel [2005], Lamb [2002, 2003], Xing and Davies [2006a,b, 2007], Davies and Xing [2007], D¨ornbrack [1998], Tseng and Ferziger [2001], Legg and Adcroft [2003], Peltier and Caulfield [2003], Munroe and Lamb [2005], Helfrich and Melville [2006].
Carnevale et al. [2001], Nilsen [2004], Venayagamoorthy and Fringer [2006], Kat- sumata [2006], Wang [2006]. Samelson [1998], Polyakov [2001], Spall [2001], Appt et al. [2004], Umlauf and Lemmin [2005], Smith et al. [2007]. Cummins [2000], Afanasyev and Peltier [2001a,b], Lamb [2004]. Chang [1993], Horn et al. [2000a].
I. Internal waves are studied in laboratory experiments. Here this note is partic- ularly weak. A few references: Helfrich [1992], Michallet and Barthelemy [1998], Michallet and Ivey [1999], Sveen et al. [2002], Guo et al. [2005], Klymak and Gregg [2003], Vlasenko and Hutter [2001], Haidvogel [2005]. See also Cacchione and Wunsch [1974], Segur and Hammack [1982], Kao et al. [1985], Boyer and Zhang [1990], Guo and Davies [2003], McPhee-Shaw [2006], Helfrich and Melville [2006]. Manders and Maas [2003], Boegman et al. [2005], Bourgault and Kelley [2007], Wake et al. [2007].
J.Internal waves are believed to play an important role in the energy transfers in the ocean. Energy fluxes and the transfer of energy from the waves to irreversible mixing is the topic in many studies. See Parsmar and Stigebrandt [1997], Stacey [1984, 1985], Stacey and Gratton [2001], Klymak and Gregg [2003, 2001, 2004], Alford [2001], Vlasenko and Hutter [2001], Nycander [2005], Inall et al. [2000, 2004, 2005], Moum et al. [2007], Moum and Smyth [2006], McCabe et al. [2006], Nakamura et al. [2000], Nakamura and Awaji [2004], Nash and Moum [2005], Winters et al. [1995], Winters and D’Asaro [1997], Zaron and Egbert [2006], Di Lorenzo et al. [2006], Fringer and Street [2003], Gargett and Holloway [1984].
Also Helfrich [1992], Helfrich and Melville [2006], Michallet and Ivey [1999], Tseng and Ferziger [2001], Guo and Davies [2003], Kao et al. [1985], D¨ornbrack [1998], Legg and Adcroft [2003], Nash et al. [2004], Kunze and Llewellyn Smith [2004],
Wunsch and Ferrari [2004], Munroe and Lamb [2005]. Fofonoff [2001], Carnevale et al. [2001], Lamb [2002, 2003], Boegman et al. [2005], Venayagamoorthy and Fringer [2006], Wang [2006], Smith et al. [2007], Bourgault and Kelley [2007].
Horn et al. [2000a].
K. Nilsen [2004], Davies and Xing [2005]. Alford [2001], MacKinnon and Gregg [2003], Appt et al. [2004], Umlauf and Lemmin [2005].
L.Fresh water/river influence. Rippeth et al. [2001].
M.Pierrehumbert and Wyman [1985], Huthnance [1992], Apel et al. [2006], Smith et al. [2007].
N. Liu et al. [1985], Horn et al. [2000b, 2002], Boegman et al. [2004]
2 Some short keyword notes on specific papers
Gargett and Holloway [1984] is referenced by many. Considers, from theory and observations, kinetic energy dissipation rate, dissipation of APE as functions of N, and suggest in the end mixing efficiency approx 0.26. Many assumptions. No numerical exp.
Inall et al. [2000]: Impact of nonlinear waves on the dissipation of internal tidal energy at a shelf break. The shelf considered is the Malin Shelf. Tidally aver- aged dissipation rates and vertical eddy diffusivity computed from measurements.
Refers to mixing eff. due to internal shearη=Rf(1 +Rf) from Gargett and Hol- loway [1984] and value 0.20. in the discussion. Observations and theory, but no numerical experiments.
Inall et al. [2004] describes measurements at the sill in Loch Etive. Tidal jet fjord during spring tide, and wave basin during neap tide. They state:’ the stagnant patch is a result of small-scale instabilities entraining recirculating fluid from the lower layer’. (consistent with Farmer and Armi (1999).) Wave drag, Form drag, and energy losses also investigated.
Inall et al. [2005] focused on energy losses based on the Loch Etive sill measure- ments. Main losses due to internal wave radiation and horizontal eddy shedding.
Alford and Pinkel [2000]: observations of overturning in the thermocline: The context of ocean mixing. Theory and observations with FLIP outside California with 2-6 m vertical resolution of shear, strain, and Ri. They also investigate effective strain rates, ∂w∂z. Thousands of overturning episodes detected. Vertical eddy diffusivity estimated to approx. 0.89×10−4 m2s−1 or 0.70× 10−4 m2s−1 depending on how the data is analysed. PDFs are computed for many measures (PDF = Probability Density Functions). Overturning occurs when ∂w∂z is large.
Observations in the ocean interior, 100-400m where the full depth is 1500m, so away from bottom. No numerical studies. Gregg (1989) should be checked.
In the preface and editorial to a special issue on ocean mixing, see Muench et al.
[2006], Muench [2006], the state of the art and areas where more focus is needed are described in general terms. Very good overviews and discussions.
Sveen et al. [2002] studied breaking over solitary waves at a ridge in a tank for a two layer stratification. The ridge does not extend over the bottom layer. An interesting breaking criterion is that if U becomes larger than 0.7 times the linear wave speed, then breaking occurs.
Guo and Davies [2003] studied tidally driven waves interacting with a slope with
an edge. Sensitivity to non-dimensional parameters discussed. Also sensitivity to slope angle. Relate findings to numerical results. Mixing and over-turning as functions of the parameters involved.
Guo et al. [2005] extends studies in Sveen et al. [2002] and allows linear stratifica- tion of the upper layer. Comparisons theory versus measurements also discussed.
Overturning and mixing in focus.
Haidvogel [2005] compares numerical results for waves created at a coastal canyon with measurements from a rotating tank. He has applied hydrostatic SEOM.
Good agreement.
Helfrich [1992] studied wave breaking and run-up on a uniform slope. He compares with existing theory, and study mixing and sensitivity to slope angle.
Michallet and Barthelemy [1998] performed experimental studies of interfacial solitary waves. Measurements related to solutions of KdV-type equations and to solutions of Euler equations, to investigate match/mis-match between theory and real waves.
Michallet and Ivey [1999] performed experimental studies of interfacial solitary waves breaking at a slope. This paper basis also for numerical experiments re- ported in Berntsen et al. [2006]. Mixing efficiency important topic. How is this efficiency as a function of slope steepness?
Nash et al. [2004] measured internal waves at the slope off Virginia. They com- puted energy fluxes, and studied mixing, and kinetic dissipation rates.
Polzin et al. [1997] is a Science paper with title: Spatial variability of turbu- lent mixing in the abyssal ocean. Points at the Mid-Atlantic ridge as an area of intensified mixing.
Wunsch and Ferrari [2004] give an excellent overview over: ”Vertical mixing, en- ergy, and the general circulation of the oceans.”
Stacey [1984] discuss some aspects of the internal tide in Knight Inlet. Energy transfers, and mathematical and numerical modelling.
Stacey and Gratton [2001] studies internal waves in Saguenay Fjord in Canada.
Mixing and energy budgets in focus. Numerical results are related to measure- ments.
Davies and Xing [2005] studied with a 2D cross coast model near inertial internal waves that are wind generated. Effects near fronts investigated.
Smyth et al. [2005] apply DNS to investigate differential diffusion in breaking
Kelvin-Helmholtz billows. DX approx. 0.005 m. Very interesting.
In Xing and Davies [2006a,b, 2007], Davies and Xing [2007] the MITgcm is used to study the flow at the sill in a loch similar to Loch Etive. It is shown that non-hydrostatic effects are important, and sensitivity to the parameters involved is investigated. Also influence on power spectra. Strong waves in the lee of the sill on inflow.
Tseng and Ferziger [2001] discuss mixing and APE in stratified flows. Introductory discussion of Ozimodov scale and Thorpe scale for over-turning. The computation of APE in focus.
Winters and D’Asaro [1997] describe direct simulation of energy wave transfer.
This is not DNS, DX = 312,5m. The link between large scale internal waves and dissipation in large scale models discussed at the end. They point at parameteri- zations.
Klymak and Gregg [2004] studies turbulence and energy budgets at Knight Inlet.
Relates dissipation rate, N, and Thorpe scale. Separate energy transfers into Internal waves, dissipation, bottom friction, and 3D vortices. Here IW losses seem very important. Dissipation rates reaching 10−4 Wkg−1.
Munroe and Lamb [2005] applies POM to study internal wave generation, propa- gation, and linear energy fluxes over idealized topography (Gaussian seamount).
Horizontal grid sizes in the range from 3 km to 1 km. Only baroclinic energy flux.
(U-prime P-prime term)
Legg and Adcroft [2003] applies the MITgcm to study the interactions of internal waves with slopes. Studies with concave and convex slopes. Discussion of the critical slope angle. Importance of non-hydrostatic pressure. Energy budgets and mixing efficiency are computed.
Kunze and Llewellyn Smith [2004] discuss the role of small-scale topography in turbulent mixing of the global ocean. This is a review type paper. Some major statements: ’Mixing is localized.’ ’... resolving topographic wavelengths of 50-100 km may be sufficient to quantify this process’. As far as I understand it they do not believe very small scale topographic features play an important role in the ’global’ energy picture. Thus they are optimistic in the sense that large scale coarse resolution models can capture the ’important’ physics in global climate type studies. Is this true?
McPhee-Shaw [2006] reviews boundary layer-interior ocean exchanges. The focus is on ’gravitational collapse’ after ’episodic mixing as a means of generating in- trusions of boundary-layer fluid into interior water with possible implications for
dispersal near ocean margins.’
Helfrich and Melville [2006] is a review paper with many references to good work.
From the abstract:’... an overview of the properties of steady internal solitary waves and the transient processes of wave propagation and evolution, primarily from the point of view of weakly nonlinear theory, of which the Korteweg-de Vries equation is the most frequently used example. However, the oceanographic important processes of wave instability and breaking, generally inaccessible with these models, are also discussed. Furthermore, observations often show strongly nonlinear waves whose properties can only be explained with fully nonlinear mod- els.’
D¨ornbrack [1998] studied turbulent mixing by breaking gravity waves. In the ab- stract he describe the evolution:’ In the first one the flow is two-dimensional:
internal waves propagate vertically upwards and create a convectively unstable region beneath the critical level. Convective instability leads to turbulent break- down in the second stage. The developing three-dimensional mixed region is or- ganized into shear-driven overturning rolls in the plane of wave propagation and into counter-rotating streamwise vortices in the spanwise plane.’ Good numerical studies, with interesting figures. Also energy considerations.
Peltier and Caulfield [2003] studied mixing efficiency in stratified shear flows.
They point at the value of 0.2 of the mixing efficiency. Describes very well how this parameter may be computed, and Kristine Selvikvaag used their approach in her Masterdegree. They include numerical studies of wave breaking and computation of mixing efficiency. It is wave breaking in the interior of the fluid, and not up a slope.
Moum et al. [2003] studied ’Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf’. Detailed observations show many small scale features including a very nice picture of Kelvin-Helmholtz instabilities. Linear stability analysis suggest fastest growing modes of lengt scales between 3 and 4.5 m, the KH-billows seen have vertical scale of 10 m and horizontal scale of approximately 50 m. Also even smaller scale turbulence seen. They point at shear instabilities with ’wavelengths of meters or less’, and ’apparently spontaneous generation of turbulence at the interfaces in cases in which we cannot observe the instability leading to turbu- lence’.
MacKinnon and Gregg [2003] describe measurement outside New-England. They focus on solibores, shear, dissipation rates, diffusivity, and turbulence parameter- izations. Enhanced mixing during Hurricane Edouard described. They point at episodic events on scales less than 5 m.
Venayagamoorthy and Fringer [2006] compute non-hydrostatic and non-linear contributions to energy fluxes up an incline. They apply cross sectional numerical models. The energy flux splitting and budgets very interesting. Laboratory scale numerical experiments.
Nilsen [2004] used a two-layered model forced by wind to investigate the interac- tions between barotropic and baroclinic modes over a topography similar to the one outside Norway. Good explanations of the interior forced motions.
Nakamura and Awaji [2004] studies ’Tidally induced diapycnal mixing in the Kuril Strait and its role in water transformation and transport: A three dimen- sional nonhydrostatic model experiment’. There is a strong tidal flow through Kuril Strait, that is shallower than 600 m. They do 3D studies with DX = 700 m and DZ = 30 m. They describe the generation of unsteady lee waves, and length scales of 4-6 km and periods of about 4 hours are suggested. This is far larger waves than is seen in Xing and Davies [2006a], Berntsen et al. [2008]. They also describe eddy induced transport of mixed water. 3D very important here.
Segur and Hammack [1982] discuss soliton models of long internal waved. The investigate the KdV-model, a finite-depth eq. due to Joseph (1977) and Kubota, Ko and Dobbs (1978). Comparisons with laboratory experiments. This two layer, constant depth models.
Wijffels and Meyers [2004] analyse measurements from the Indonesian through- flows. Intraseasonal and interannual time scales, so internal waves are not ’cap- tured’, but variation in density profiles documented.
Vlasenko and Hutter [2002b] studied breaking of solitary internal waves over a slope with a numerical model. DX in the range 1 to 5 m and DZ in the range 0.25 to 1 m, and Pacanowski and Philander for sub-grid parameterization. They focus on the breaking part, and a nice breaking criterion is established. For gentle slopes, more of the energy can go into a dispersive wave trail and less into breaking.
Boyer and Zhang [1990] performed laboratory experiments of flow past an iso- lated seamount similar to Fieberling Guyot and related results to measurements to observations around Fieberling Guyot. The motion depends on the Rossby number, and for strong enough forcing they see eddy shedding. The flow regimes discussed in terms of non-dimensional numbers. Constant N is assumed.
Katsumata [2006] used POM to study the internal tide generation and energy fluxes at a continental slope (outside Australia). He discuss two and three dimen- sional models of internal tide, and state that by using 2D models the energy fluxes may be underestimated. An along shelf scale of the model domain of at least 5 internal tide wave lengths is necessary to capture the energy fluxes. Experiments
performed with DX = 4 km.
Ostrovsky and Stepanyants [1989] give a very good review of observations of internal waves up til then and mathematical models, KdV and more.
Bogucki and Garrett [1993] studied shear-induced decay of an internal solitary wave. The thickening of the the interface between the two layers is in focus, and they derive formulas for the damping rate. Overview of theoretical models is given. Criterion for when thickening occur is given.
Wang [2006] investigated internal waves in a channel generated by tide flowing over a hill or a valley. He used the POM with DX = 50 m and looked at the transfer of energy to the internal wave for different topgraphies. For this case, non-hydrostatic processes may be important, and he also point at that.
Thorpe [1997] studied interactions of internal waves at a slope. It is interactions between the incident wave and the reflected wave that interact to second order.
Interactions depend on the slope angle. A theoretical model for the interactions is developed.
Staquet and Sommeria [2002] is a review paper on internal gravity waves with subtitle: From instabilities to turbulence. Mechanisms of steepening and breaking and the final process from breaking into small-scale turbulence is discussed. Both theory, laboratory and numerical experimets are reviewed. The energy cascade is also reviewed, and possible parameterizations are discussed.
Hosegood and van Haren [2006] describe observations made in the Faeroe-Shetland Channel near the interface between water masses. Internal tidal motions in focus, and kinetic energy spectra are given and discussed. Modulation periods of 3.3 to 4.3 days are related to changes in background stratification and low-frequency vorticity. The source for this may be continental shelf waves.
Carnevale et al. [2001] discuss the transition from the buoyancy range, from ap- proximately 10 m to 1 m, to the inertial range (less than 1 m). On large scales, from kilometers to say 10 m, internal waves dominate variability. They describe fall off of spectra in this range. They apply a spectral model over a cube (sides are 20 m) and 64 and 128 modes are used, and investigates spectra. Long dis- cussions of sub-grid closures for such models, including Smagorinsky. Turbulence anisotropic in buoyancy range and isotropic in inertial range. Interesting state- ments:’Between these extremes, the dynamics is a competition between waves and turbulence. The nature of this intermediate range, called the buoyancy or the saturation range, is highly controversial.’ ’The inertial range terminates in the dissipation range for scales of a few centimetres or below’.
Kao et al. [1985] did measurements of solitary waves in a tank, both generation, propagation, and shoaling and breaking over a slope are discribed. Results related to KdV type theory. They state:’... the onset of wave breaking was governed by shear instability, which was initiated when the local gradient Richardson number became less than 1/4.’
Cacchione and Wunsch [1974] did an experimental study of internal waves op an incline. They considered constant N, and investigated run up along the incline depending on the ratio between wave angle and the slope angle. If the angle of the slope is critical (ratio = 1), ’ a striking instability is observed and the waves are heavily damped’. Also review of theory on the problem.
Appt et al. [2004] reported on basin scale motion in the stratified Lake Constance based on observations and numerical modelling. The oscillations are wind driven.
Surges may be generated. ’The reflection of the surge from the northwestern boundary induced a vertical mode-two reponse leading to an intrusion in the metalimnion that caused a three-layer velocity structure in the smaller subbasin.’
Umlauf and Lemmin [2005] studied waves in Lake Geneva using both measure- ments and the model of Burchard and Bolding. They looked at mixing in the hypolimnion and the role of long internal waves. The forcing is wind. Internal kelvin waves important also here.
Huthnance [1992] give a review of slope currents and ocean-shelf interaction. The focus is not on internal waves. However, internal waves will be strongly influenced by the slope currents.
Manders and Maas [2003] investigate internal wave rays in a rotating recangular tank with one slope. Theory and measurements, wave-wave ineractions of many kinds.
Boegman et al. [2004] discuss mixing and parameterization of mixing in a lake.
Interesting study of the mixing efficiency as a function of the Iribarren number
= Slope/sqrt(amp/lambda).
Boegman et al. [2005] studied internal waves in a tank similar to a lake with sloping topography. Breaking and mixing in focus. Mixing efficiency as a function of the Iribarren number discussed. Results related to findings from Lake Pusiano in north Italy. Can Vmax also be related to Iribarren number? See description of
’spilling breakers’ and ’breaker height’. Very relevant. Also: From a wind event:
generation of a high-frequent solitary wave packet with substantial energy. Also relevant for Jons study.
Thorpe [1999] investigated breaking of tidally driven internal waves and wave
groups for constant N. Breaking criterions in terms of dimensonless parameters are discussed, and the size of the regions with breaking is in focus. In which situations do we get large volumes of fluid with enhanced mixing, and not only small local events?
Thorpe [2000] investigated the effects of rotations on the nonlinear reflection of internal waves from a slope. Constant N assumed. The Lagrangian alongslope drift may increase considerably, and the level with strongest drift is no longer at z = 0.
The direction of the drift near sea bed may be reversed. Eulerian upslope currents associatd with reflection may become stronger. Effects of mixing and other effects not easily studied analytically ignored, and here he points at possible numerical studies.
Toole et al. [1997] measured near boundary mixing above the flanks of Fieberling Guyot, a seamount. Elevated leves of shear and strain found in a 500 m thick layer above the bottom. Turbulent diffusivitiy estimates of approximately 0.1 ×10−4 m2s−1 found in the ocean interior and 1-5 ×10−4 m2s−1 in the boundary layer.
The global significance of this estimated.
Spall [2001] investigated large scaled circulation forced by localized mixing over a sloping bottom. Idealised studies. Internal waves are not explicitly resolved.
Feedback to large scale from local mixing, and investigations of the parameters governing this, in focus.
Samelson [1998] also studied large scaled circulation forced by localized mixing.
He compared the circulation for the case of constant vertical diffusivity as com- pared to the circulation when using a vertical diffusivity that varied in space (Hot spots). Also idealised studies to investigate basic mechanisms. Internal waves are not explicitly resolved.
Alford [2001] studied the spatial distribution of energy flux from the wind to near- inertial motions. Global studies based on NCEP-NCAR data of surface winds.
Convergence/divergence in the mixed layer motions pump energy into near in- ertial waves. Idealised studies. The role of ’events’ discussed. The role of these inertial motions in the total energy transfers necessary to maintain the global circulation discussed. The small scale internal waves are not resolved.
Rippeth et al. [2001] studied, based on measurements in Liverpool Bay,a ROFI system (Region of Freshwater Influence) and investigated how the rate of dissi- pation of turbulent kinetic energy depended on horizontal density gradients and the tidal cycle.
Bourgault and Kelley [2007] study the reflectance of internal waves on a slope and relate 2D slice model results to lab. experiments described in Helfrich [1992] and
Michallet and Ivey [1999]. They argue that in lab. experiments side wall effects are important, and suggest a simple way to parameterize this drag that is similar to how bottom drag is parameterized in 2D (x,y) shallow water models. They investigate Reflectance as a function of the Iribarren number (ξ =s/sqrta0/Lw).
The reflectance increases with Iribarren number(or slopes = slope angle.) In the real ocean these side wall effects will not be present, so they argue for wider tank exp. and/or three dimensional numerical studies.
Polyakov [2001] applied statistical mechanics of potential vorticity to derive an eddy parameterization based on the maximum entropy production (MEP). The eddy parameterization includes the ’Neptun effect’ (Holloway [1992]). This is parameterisation of mesoscale eddies, rather than internal waves. However, the technique for deriving the parameterization is very interesting. He applied the parameterization to flow in the Arctic, and achieves a resonable flow pattern.
Fofonoff [2001] ”describe how the nonlinear effect of contraction on mixing of seawater, referred to as ”cabbeling”, may determine major features of the ocean’s temperature and salinity structures.” Analysis based on measurments of vertical profiles in different oceans. The paper is not on internal waves, but internal waves may play a role. ”A mechanism for creating slopes at the thermocline, such as internal waves, may enhance the process.”
Apel et al. [2006] is a very good review paper on internal solitons in the oceans with many good references, a review of mathematical models, and measurements.
Also effects on sound propagation is discussed.
Lamb [2002] studied solitary waves near the surface generated at a shelf edge and propagating on to the shelf for different stratification. The focus is on possible trapped cores that are fairly shallow. Criteria for the formation of these cores are discussed. The reltionship betweenUmax andcis important. Limit approximately 1. The work was followed up in Lamb [2003] where also additional effects of background currents were considered.
Wake et al. [2007] investigated resonantly forced interfacial waves in a circular tank with focus on stratification effects. References to work by among others Faltinsen.
Smith et al. [2007] applies PIV to measure in the BBL of the coastal ocean. Analy- sis of the data are used to calculate subgrid-scale stresses (SGS), and to evaluate common SGS models like the Smagorinsky model. Under certain conditions, also negative energy fluxes are found, which means feedback from unresolved to re- solved scales.
Armi and Farmer [2002] investigate stratified flow over the sill in Knight Inlet and
focus on bifurcation fronts and the transition to the unconrolled state. They de- scribe ”a wedge of partly mixed fluid downstream of a bifurcation point or plunge point”. They also point at small-scale shear instabilities that are responsible for the initial phase of the mixing. They refer to this as ” a striking example of small- scale processes contributing to the larger-scale response”. They also point at the importance of the boundary layer separation. They have a nice analysis leading to a criterion for the transition to bifurcation. Also nice schematics illustrating the processes.
Pierrehumbert and Wyman [1985] studied upstream effects of mesoscale moun- tains. Atmospheric paper. For large Froude number significant low level blocking effects.
Cummins [2000] addressed the flow at Knight Inlet with a 2D version of the POM.
Focus on the hydraulically controlled high drag state over the sill similar to the observed one. In the model: mixing due to internal wave overturning whereas observations have no apparent overturning. The importance of flow separation in the lee of the sill crest emphasized. Grid resolution near sill 10 m. Results reported to be robust to grid sizes in the range from 5 to 30 m. Smagorinsky diff and vis with Smagorinsky and CM = 0.1 and Mellor-Yamada vertically. 101 sigma-layers used vertically. High resolution near sea bed.
Lamb [2004] studied the flow at Knight Inlet. Focus on boundary-layer separation and internal wave generation. Only numerical results and and some results from a potential flow model shown. No measurements are given, but findings related to measurements. No-slip/Slip condition sensitivity investigated. Sensitivity to subgrid scale closure and sensitivity to density profile both vertically and across sill. Strong internal lee waves in all cases except one. Topographic drag discussed.
Farmer and Armi (1999) point at small scale mixing and a shear instability that create a wedge of mixed fluid. Lamb did not find this in his studies, and a dis- cussion of possible reasons is given. His domain from -3000m to +3000m and DX from 1 to 10 m. He points at some simulations with DX = 0.50 m and DZ
= 0.20 m over the sill, and says that ’at this time no connection between these instabilities and the formation of a wedge ...’.
Afanasyev and Peltier [2001a] applied their numerical model to study breaking internal waves over the sill in Knight Inlet. The model is non-hydrostatic, but ap- plies a slip bottom boundary condition, so it does not allow for bottom boundary separation. As far as I can see: also rigid lid. DX = 5 m and DZ = 0.5 m and they denote this as DNS. Viscosities and diffusivities are set to zero, so I would expect some numerical dissipation. Their main conclusion is that it is the breaking of a forced stationary internal wave, resulting in irreversible mixing, that creates the body of well-mixed fluid in the lee of the sill. They do not find Kelvin-Helmholtz
type instabilities, and argues against the role of such instabilities in the creation of the well mixed wedge, arguing against the conclusion in Farmer and Armi [1999]. They point at the lacking boundary layer separation as a reason for the lack of these instabilities. They define three different relevant Froude numbers.
Also they make simple estimates of the horizontal and vertical scales of the waves as functions of N and topography.
Farmer and Armi [2001] discuss discrepancies between measurements and corre- sponding model results for Knight Inlet. They claim that the mechanisms that cre- ate the wedge of mixed fluid in the numerical observations are wrong. Numerical models produce internal waves rather than shear instabilities, and therefore the timing, build up, and processes leading to the wedge are wrong. In particular they
’attack’ results given in Afanasyev and Peltier (2001). The key questions agreed on is: ”Where does the intermediate layer of mixed fluid come from?” Boundary layer separation also plays a crucial role in the formation of the wedge/layer of mixed fluid.
Afanasyev and Peltier [2001b] responded to the remarks given in Farmer and Armi [2001] to Afanasyev and Peltier [2001a]. They acknowledge that the role of KH instabilities has to be further investigated. The surface reverse flow and boundary layer separation may also play a role here, and at least BL separation was absent in the studies in Afanasyev and Peltier [2001a]. There was also a controversy over the hydraulic analytical approaches in this is further clarified in this response.
Janzen et al. [2005] measured across-sill circulation near a tidal mixing front in a Scottish open fjord, the Clyde Sea. Partially unexplained stong across sill exchanges. Affected by wind.
Chang [1993] address solitary waves along potential vorticity fronts on an f-plane.
Dispersion introduced by short waves may balance non-linear steepening. One Gaussian type perturbation may develop into a train of solitary waves.
Horn et al. [2000a] describe in a note that I found on the web from approxi- mately 2000 solitary waves in lakes and discuss how they may be parameterized in hydrostatic models.
Horn et al. [2000b] describe an extended KdV equation to study evolution and propagation of internal solitary waves. First they describe, with good references, how such waves are generated from intial larger scale waves. They point at sub- sequent breaking, but do not model this phase. Very good paper.
Horn et al. [2002] show how any initial displacement of the interface in a closed basin may develop into: a packet of solitary waves, a dispersive long wave or a
train of dispersive oscillatory waves. They use two independent KdV equations to satisfy closed BCs. Very interesting.
Liu et al. [1985] describe observations of solitary wave trains/packets in the Sulu Sea. They derive a KdV type equation that include the effects of topography and spreading on this sea, and numerical results based on this equation reproduce very well the measured wave packets.
Goryachkin et al. [1990] describe measurements and analysis of internal wave events in the Black sea and the Aegean Sea (non-tidal).
Pawlak and Armi [2000] discuss mixing and entrainment in gravity currents down an incline. They classify different regions down the slope. Sensitivity to slope angle investigated.
Vincont et al. [2000] descibe a tank experiment of scalar dispersion in a turbulent boundary layer behind an obstacle. Relevant for Coral reefs?
New and Pingree [2000] describes both measurements and results from a KdV- type model for the Bay of Biscay. The model is apparently able to capure major features of the internal solitary wave packets. An interesting remark is:” The magnitude of the vertical velicities, however, is signficantly underestimated by the theory ...” The link to mixing also addressed.
Stastna and Rowe [2007] derive a KdV type equation that include rotational effects. Compares solutions to the solutions from the full euler eqns. + Ostrovsky eqn.
Other papers in the pile: Munk and Wunsch [1998], Winters et al. [2000], Sherwin et al. [2002], Van Haren [2005], Plueddemann and Farrar [2006], Huang et al.
[2006] and Muench et al. [2006], Muench [2006], Marzeion and Drange [2006], Fer [2006].
+ some more in other piles, that I have to check up.
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