Notes on different aspects of internal waves - including references

<strong>Notes</strong> 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.
Preprint submitted to Elsevier Science 13 March 2008

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], Bourgault
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],
2

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 Stigebrandt
[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], Cacchione
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]. Dörnbrack [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], Umlauf
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 particular
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
3

[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 topography
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], Michallet
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. References
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
4

Garrett [1993], Thorpe [1997, 1999, 2000], Carnevale et al. [2001], Staquet and
Sommeria [2002], Nilsen [2004], Apel et al. [2006]. Armi and Farmer [2002], Manders
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], Nakamura
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],
Dörnbrack [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], Katsumata
[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 particularly
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], Dörnbrack [1998],
Legg and Adcroft [2003], Nash et al. [2004], Kunze and Llewellyn Smith [2004],
5

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]
6

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 averaged
dissipation rates and vertical eddy diffusivity computed from measurements.
Refers to mixing eff. due to internal shear η = Rf(1+Rf) from Gargett and Holloway
[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 measurements.
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.
Thousands of overturning episodes detected. Vertical
∂z
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 is large.
∂z
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
7

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 stratification
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 reported
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 computed
energy fluxes, and studied mixing, and kinetic dissipation rates.
Polzin et al. [1997] is a Science paper with title: Spatial variability of turbulent
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, energy,
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 measurements.
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
8

Kelvin-Helmholtz billows. DX approx. 0.005m. 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 parameterizations.
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, propagation,
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 intrusions
of boundary-layer fluid into interior water with possible implications for
9

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 models.’
Dörnbrack [1998] studied turbulent mixing by breaking gravity waves. In the abstract
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 breakdown
in the second stage. The developing three-dimensional mixed region is organized
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 turbulence’.
MacKinnon and Gregg [2003] describe measurement outside New-England. They
focus on solibores, shear, dissipation rates, diffusivity, and turbulence parameterizations.
Enhanced mixing during Hurricane Edouard described. They point at
episodic events on scales less than 5 m.
10

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 interactions
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 dimensional
nonhydrostatic model experiment’. There is a strong tidal flow through
Kuril Strait, that is shallower than 600m. They do 3D studies with DX = 700m
and DZ = 30m. 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 throughflows.
Intraseasonal and interannual time scales, so internal waves are not ’captured’,
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 5m and DZ in the range 0.25 to
1m, 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 isolated
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 dimensional
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
11

performed with DX = 4km.
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 = 50m 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 approximately
10m to 1m, to the inertial range (less than 1m). On large scales,
from kilometers to say 10m, internal waves dominate variability. They describe
fall off of spectra in this range. They apply a spectral model over a cube (sides
are 20m) and 64 and 128 modes are used, and investigates spectra. Long discussions
of sub-grid closures for such models, including Smagorinsky. Turbulence
anisotropic in buoyancy range and isotropic in inertial range. Interesting statements:’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’.
12

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 measurements
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
13

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
m 2 s −1 found in the ocean interior and 1-5×10 −4 m 2 s −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 compared
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 nearinertial
motions. Global studies based on NCEP-NCAR data of surface winds.
Convergence/divergence in the mixed layer motions pump energy into near inertial
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 dissipation
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
14

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 slope s = 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 between Umax and c is 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. Analysis
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 resolved
scales.
Armi and Farmer [2002] investigate stratified flow over the sill in Knight Inlet and
15

focus on bifurcation fronts and the transition to the unconrolled state. They describe
”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 smallscale
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 mountains.
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 discussion
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 applies
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
16

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 corresponding
model results for Knight Inlet. They claim that the mechanisms that create
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 approximately
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 subsequent
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
17

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 KdVtype
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.
18

References
Y.D. Afanasyev and W.R. Peltier. On breaking internal waves over the sill in
Knight Inlet. Proceedings of the Royal Society of London A, 457:2799–2825,
2001a.
Y.D. Afanasyev and W.R. Peltier. Reply to comment on the paper ’On breaking
internal waves over the sill in Knight Inlet’. Proceedings of the Royal Society
of London A, 457:2831–2834, 2001b.
M.H. Alford. Internal Swell Generation: The Spatial Distribution of Energy Flux
from the Wind to Mixed Layer Near-Inertial Motions. Journal of Physical
Oceanography, 31:2359–2368, 2001.
M.H. Alford and R. Pinkel. Observations of Overturning in the Thermocline:
The Context of Ocean Mixing. Journal of Physical Oceanography, 30:805–832,
2000.
J.R. Apel, L.A. Ostrovsky, Y.A. Stepanyants, and Lynch.J.F. Internal Solitons
in the Ocean. Technical Report WHOI-2006-04, Woods Hole Oceanographic
Institution, 2006.
J. Appt, J. Imberger, and H. Kobus. Basin-scale motion in stratified Upper Lake
Constance. Limnology and Oceanography, 49:919–933, 2004.
L. Armi and D.M. Farmer. Stratified flow over topography: bifurcation fronts
and transition to the uncontrolled state. Proceedings of the Royal Society of
London A, 458:513–538, 2002.
J. Berntsen, J. Xing, and G. Alendal. Assessment of non-hydrostatic ocean models
using laboratory scale problems. Continental Shelf Research, 26:1433–1447,
2006.
J. Berntsen, J. Xing, and A.M. Davies. Numerical studies of internal waves at a
sill: sensitivity to horizontal size and subgrid scale closure, 2008. Submitted to
Continental Shelf Research.
L. Boegman, G.N. Ivey, and J. Imberger. An Internal Solitary Wave parameterization
for Hydrodynamic Lake Models, 2004. Proceeding from the 15th
Australian Fluid Mechanics Conference, 13-17 December 2004.
L. Boegman, G.N. Ivey, and J. Imberger. The degeneration of internal waves
in lakes with sloping topography. Limnology and Oceanography, 50:1620–1637,
2005.
D. Bogucki and C. Garrett. A Simple Model for the Shear-induced Decay of an
Internal Solitary Wave. Journal of Physical Oceanography, 23:1767–1776, 1993.
D. Bourgault and D.E. Kelley. Wave-induced boundary mixing in a partially
mixed estuary. Journal of Marine Research, 61:553–576, 2003.
D. Bourgault and D.E. Kelley. On the Reflectance of Uniform Slopes for Normally
Incident Interfacial Solitary Waves. Journal of Physical Oceanography, 37:1156–
1162, 2007.
19

D. Bourgault, M.D. Blokhina, R. Mirshak, and D.E. Kelley. Evolution of
a shoaling internal solitary wavetrain. Geophysical Research Letters, 34:
doi:10.1029/2006GL028462, 2007.
D.L. Boyer and X. Zhang. Motion of Oscillatory Currents Past Isolated Topography.
Journal of Physical Oceanography, 20:1425–1448, 1990.
D. Cacchione and C. Wunsch. Experimental study of internal waves over a slope.
J.Fluid.Mech., 66:223–239, 1974.
G.F. Carnevale, M. Briscolini, and P. Orlandi. Buoyancy- to inertial-range transition
in forced stratified turbulence. Journal of Fluid Mechanics, 427:205–239,
2001.
P. Chang. A Note on Solitary Waves along Potential Vorticity Fronts on an
f-plane. Journal of Oceanography, 49:477–489, 1993.
P.F. Cummins. Stratified flow over topography: time-dependent comparisons between
model solutions and observations. Dynamics of Atmospheres and Oceans,
33:43–72, 2000.
P.F. Cummins, S. Vagle, L. Armi, and D. Farmer. Stratified flow over topography:
upstream influence and generation of nonlinear waves. Proceedings of the Royal
Society of London A, 459:1467–1487, 2003.
A.M. Davies and J. Xing. The Effect of a Bottom Shelf Front upon the Generation
and Propagation of Near-Inertial Internal Waves in the Coastal Ocean. Journal
of Physical Oceanography, 35:976–990, 2005.
A.M. Davies and J. Xing. On the influence of stratification and tidal forcing upon
mixing in sill regions. Ocean Dynamics, 57:431–451, 2007. doi: 10.1007/s10236-
007-0114-5.
R. Dewey, D. Richmond, and C. Garrett. Stratified Tidal Flow over a Bump.
Journal of Physical Oceanography, 35:1911–1927, 2005.
E. Di Lorenzo, W.R. Young, and S.L. Smith. Numerical and Analytical Estimates
of M2 Tidal Conversion at Steep Oceanic Ridges. Journal of Physical
Oceanography, 36:1072–1084, 2006.
A. Dörnbrack. Turbulent mixing by breaking gravity waves. Journal of Fluid
Mechanics, 375:113–141, 1998.
D.M. Farmer and L. Armi. Stratified flow over topography: models versus observations
. Proceedings of the Royal Society of London A, 457:2827–2830, 2001.
D.M. Farmer and L. Armi. Stratified flow over topography: The role of smallscale
entrainment and mixing in flow establishment. Proceedings of the Royal
Society of London, A455:3221–3258, 1999.
I. Fer. Scaling turbulent dissipation in an Artic fjord. Deep-Sea Research II, 53:
77–95, 2006.
I. Fer and K. Widell. Early spring turbulent mixing in an ice-covered Arctic fjord
during transition to melting . Continental Shelf Research, 27:1980–1999, 2007.
N.P. Fofonoff. Thermal Stability of the World Ocean Thermoclines. Journal of
20

Physical Oceanography, 31:2169–2177, 2001.
O.B. Fringer and R.L. Street. The dynamics of breaking progressive interfacial
waves. Journal of Fluid Mechanics, 494:319–353, 2003.
A.E. Gargett and G. Holloway. Dissipation and diffusion by internal wave breaking.
Journal of Marine Research, 42:15–27, 1984.
Y.N. Goryachkin, V.A. Ivanov, and A.D. Lisichenok. Short-period internal waves
in the frontal zones of non-tidal seas (the Black Sea and the Agean Sea). Physical
Oceanography, 1:535–541, 1990.
J.A.M. Green and A. Stigebrandt. Statistical models and distributions of current
velocities with application to the prediction of extreme events. Estuarine,
Coastal and Shelf Science, 58:601–609, 2003.
Y. Guo and P.A. Davies. Laboratory modelling experiments on the flow generated
by the tidal motion of a stratified ocean over a continental shelf. Continental
Shelf Research, 23:193–212, 2003.
Y. Guo, J.K. Sveen, P.A. Davies, J. Grue, and P. Dong. Modelling the Motion
of an Internal Solitary Wave over a Bottom Ridge in a Stratified Fluid.
Environmental Fluid Mechanics, 4:415–441, 2005.
D.B. Haidvogel. Cross-Shelf Exchange Driven by Oscillatory Barotropic Currents
at an Idealized Coastal Canyon. Journal of Physical Oceanography, 35:1054–
1067, 2005.
K.R. Helfrich. Internal solitary wave breaking and run-up on a uniform slope.
Journal of Fluid Mechanics, 243:133–154, 1992.
K.R. Helfrich and W.K. Melville. Long Nonlinear Internal Waves. Annual Review
of Fluid Mechanics, 38:395–425, 2006.
G. Holloway. Representing topographic stress for large-scale ocean models. Journal
of Physical Oceanography, 22:1033–1046, 1992.
D.a. Horn, J. Imberger, and G.N. Ivey. Internal Solitary Waves in Lakes - a
Closure Problem for Hydrostatic Models, 2000a. A manuscript I found on the
web. Very interesting. Related to other work by this Australians.
D.a. Horn, L.G. Redekopp, J. Imberger, and G.N. Ivey. Internal wave evolution
in a space-time varying field . Journal of Fluid Mechanics, 424:279–301, 2000b.
D.a. Horn, J. Imberger, G.N. Ivey, and L.G. Redekopp. A weakly nonlinear
model of long internal waves in closed basins. Journal of Fluid Mechanics, 467:
269–287, 2002.
P. Hosegood and H. van Haren. Sub-inertial modulation of semi-diurnal currents
over the continental slope in the Faeroe-Shetland Channel. Deep-Sea Research
I, 53:627–655, 2006.
R.X. Huang, W. Wang, and L.L. Liu. Decadal variability of wind-energy input
to the world ocean. Deep-Sea Research II, 53:31–41, 2006.
J.M. Huthnance. Extensive slope currents and the ocean-shelf boundary. Prog.
Oceanog., 29:161–196, 1992.
21

R.N. Ibragimov. Generation of Internal Tides by an Oscillating Flow Along a
Corrugated Slope , 2007. In press.
M. Inall, F. Cottier, C. Griffiths, and T. Rippeth. Sill dynamics and energy
transformation in a jet fjord. Ocean Dynamics, 54:307–314, 2004.
M. Inall, T. Rippeth, C. Griffiths, and P. Wiles. Evolution and distribution of
TKE production and dissipation within stratified flow over topography. Geophysical
Research Letters, 32:L08607,doi:10.1029/2004GL022289, 2005.
M.E. Inall, T.P. Rippeth, and T.J. Sherwin. Impact of nonlinear waves on the
dissipation of internal tidal energy at a shelf break. J. Geophys. Res., 105:
8687–8705, 2000.
C.D. Janzen, J.H. Simpson, M.E. Inall, and F. Cottier. Across-sill circulation
near a tidal mixing front in a broad fjord. Continental Shelf Research, 25:
1805–1824, 2005.
S.R. Jayne and L.C. St.Laurent. Parameterizing Tidal Dissipation over Rough
Topography. Geophysical Research Letters, 28:811–814, 2001.
T.W. Kao, F.-S. Pan, and D. Renouard. Internal solitons on the pycnocline:
generation, propagation, and shoaling and breaking over a slope. Journal of
Fluid Mechanics, 159:19–53, 1985.
K. Katsumata. Two- and three-dimensional numerical models of internal tide
generation at a continental slope. Ocean Modelling, 12:32–45, 2006.
J.M. Klymak and M.C. Gregg. Three-dimensional nature of flow near a sill .
Journal of Geophysical Research, 106:22295–22311, 2001.
J.M. Klymak and M.C. Gregg. The Role of Upstream Waves and a Downstream
Density Pool in the Growth of Lee Waves:Stratified Flow over the Knight Inlet
Sill. Journal of Physical Oceanography, 33:1446–1461, 2003.
J.M. Klymak and M.C. Gregg. Tidally Generated Turbulence over the Knight
Inlet Sill. Journal of Physical Oceanography, 34:1135–1151, 2004.
P.K. Kundu and I.M. Cohen. Fluid Mechanics. Elsevier Academic Press, 2004.
E. Kunze and S.G. Llewellyn Smith. The Role of Small-Scale Topography in
Turbulent Mixing of the Global Ocean. Oceanography, 17:55–64, 2004.
K.G. Lamb. A numerical investigation of solitary internal waves with trapped
cores fomes via shoaling. Journal of Fluid Mechanics, 451:109–144, 2002.
K.G. Lamb. Shoaling solitary internal waves: on a criterion for the formation of
waves with trapped cores. Journal of Fluid Mechanics, 478:81–100, 2003.
K.G. Lamb. On boundary-layer separation and internal wave generation at the
Knight Inlet sill. Proceedings of the Royal Society of London A, 460:2305–2337,
2004.
S. Legg and A. Adcroft. Internal wave breaking at concave and convex continental
slopes. Journal of Physical Oceanography, 33:2224–2246, 2003.
A.K. Liu, J.R. Holbrook, and J.R. Apel. Nonlinear Internal Wave Evolution in
the Sulu Sea. Journal of Physical Oceanography, 15:1613–1624, 1985.
22

J.A. MacKinnon and M.C. Gregg. Mixing on the late-summer new england shelfsolibores,
shear, and stratification. Journal of Physical Oceanography, 33:1476–
1492, 2003.
A.M.M. Manders and L.R.M. Maas. Observations of internal waves in a rectangular
basin with one sloping boundary. Lournal of Fluid Mechanics, 493:59–88,
2003.
B. Marzeion and H. Drange. Diapycnal mixin in a conceptual model of the
Atlantic Meridional Overturning Virculation. Deep-Sea Research II, 53:226–
238, 2006.
R.M. McCabe, P. MacCready, and G. Pawlak. Form Drag due to Flow Separation
at a Headland. Journal of Physical Oceanography, 36:2136–2152, 2006.
E. McPhee-Shaw. Boundary-interior exchange: Reviewing the idea that inernalwave
mixing enhances lateral dispersal near continental margins. Deep-Sea
Research II, 53:42–59, 2006.
H. Michallet and E. Barthelemy. Experimental study of interfacial solitary waves.
Journal of Fluid Mechanics, 366:159–177, 1998.
H. Michallet and G.N. Ivey. Experiments on mixing due to internal solitary
waves breaking on uniform slopes. Journal of Geophysical Research, 104(C6):
13467–13477, 1999.
J.N. Moum and W.D. Smyth. The pressure disturbance of a nonlinear internal
wave train . Journal of Fluid Mechanics, 558:153–177, 2006.
J.N. Moum, D.M. Farmer, W.D. Smyth, K. Armi, and S. Vagle. Structure and
Generation of Turbulence at Interfaces by Internal Solitary Waves Propagating
Shoreward over the Continental Shelf. Journal of Physical Oceanography, 33:
2093–2112, 2003.
J.N. Moum, J.M. Klymak, J.D. Nash, A. Perlin, and W.D. Smyth. Energy Transport
by Nonlinear Internal Waves . Journal of Physical Oceanography, 37:
1968–1988, 2007.
R. Muench. Introduction to the special issue on Ocean Mixing. Deep-Sea Research
II, 53:2–4, 2006.
R. Muench, T. McDougall, and H. Burchard. Preface to special edition on Ocean
Mixing. Deep-Sea Research II, 53:1–1, 2006.
W.H. Munk and C. Wunsch. Abyssal recipes II: energetics of tidal and wind
mixing. Deep-Sea Research I, 45:1978–2010, 1998.
J.R. Munroe and K.G. Lamb. Topographic amplitude dependence of internal
wave generation by tidal forcing over idealized three-dimensional topography.
Journal of Geophysical Research, 110:doi:10.1029/2004JC002537, 2005.
T. Nakamura and T. Awaji. Tidally induced diapycnal mixing in the Kuril
Straits and its role in water transformation and transport: A three-dimensional
nonhydrostatic model experiment. Journal of Geophysical Research, 109:
doi:10.1029/2003JC001850, 2004.
23

T. Nakamura, T. Awaji, T. Hatayama, K. Akitomo, T. Takizawa, T. Kono,
Y. Kawasaki, and M. Fukasawa. The Generation of Large-Amplitude Unsteady
Lee Waves by Subinertial K1 Tidal Flow: A Possible Vertical Mixing Mechanism
in the Kuril Straits . Journal of Physical Oceanography, 30:1601–1621,
2000.
J.D. Nash and J.N. Moum. River Plumes as a Source of Large-Amplitude
Internal Waves in the Coastal Ocean. Nature, 437:400–403, 2005.
doi:10.1038/nature03936.
J.D. Nash, E. Kunze, J.M. Toole, and R.W. Schmitt. Internal Tide Reflection and
Turbulent Mixing on the Continental Slope. Journal of Physical Oceanography,
34:1117–1134, 2004.
A.L. New and R.D. Pingree. An intercomparison of internal solitary waves in the
Bay of biscay and resulting from Korteweg-de Vries- Type theory. Progress in
Oceanography, 45:1–38, 2000.
F. Nilsen. Forcing of a Two-Layered Water Column over a Sloping Seafloor.
Journal of Physical Oceanography, 34:2659–2676, 2004.
J. Nycander. Generation of internal waves in the deep ocean by tides. Journal of
Geophysical Research, 110:doi:10.1029/2004JC002487, 2005.
L.A. Ostrovsky and Y.A. Stepanyants. Do internal solitons exist in the ocean?
Reviews of Geophysics, 27:293–310, 1989.
R. Parsmar and A. Stigebrandt. Observed Damping of Barotropic Seiches through
Baroclinic Wave Drag in the Gullmar Fjord. Journal of Physical Oceanography,
27:849–857, 1997.
G. Pawlak and L. Armi. Mixing and entrainment in developing stratified currents.
Journal of Fluid Mechanics, 424:45–73, 2000.
W.R. Peltier and C.P. Caulfield. Mixing Efficiency in Stratified Shear Flows .
Annual Review of Fluid Mechanics, 35:135–167, 2003.
R.T. Pierrehumbert and B. Wyman. Upstream Effects of Mesoscale Mountains.
Journal of The Atmospheric Sciences, 42:977–1003, 1985.
A.J. Plueddemann and J.T. Farrar. Observations and models of the energy flux
from the wind to mixed-layer inertial currents. Deep-Sea Research II, 53:5–30,
2006.
I. Polyakov. An Eddy Parameterization Based on Maximum Entropy Production
with Application to Modeling of the Arctic Ocean Circulation. Journal of
Physical Oceanography, 31:2255–2270, 2001.
K.L. Polzin, J.M. Toole, J.R. Ledwell, and R.W. Schmitt. Spatial variability of
turbulent mixing in the abyssal ocean. Science, 276:93–96, 1997.
T.P. Rippeth, N.R. Fisher, and J.H. Simpson. The Cycle of Turbulent Dissipation
in the Presebce of Tidal Straining. Journal of Physical Oceanography, 31:2458–
2471, 2001.
O.A. Saenko and W.J. Merryfield. On the Effect of Topographically Enhanced
24

Mixing on the Global Ocean Circulation. Journal of Physical Oceanography,
35:826–835, 2005.
R.M. Samelson. Large scale circulation with locally enhanced vertical mixing.
Journal of Physical Oceanography, 28:712–726, 1998.
H. Segur and J.L. Hammack. Soliton models of long internal waves. Journal of
Fluid Mechanics, 118:285–304, 1982.
T.J. Sherwin, M.E. Inall, and R. Torres. The seasonal and spatial variability of
small-scale turbulence at the Iberian margin. Journal of Marine Research, 60:
73–100, 2002.
W.A.M.N. Smith, J. Katz, and T.R. Osborn. The effect of waves on subgridscale
stresses, dissipation and model coefficients in the coastal ocean bottom
boundary layer. Journal of Fluid Mechanics, 583:133–160, 2007.
W.D. Smyth, J.D. Nash, and J.N. Moum. Differential Diffusion in Breaking
Kelvin-Helmholtz Billows. Journal of Physical Oceanography, 35:1004–1022,
2005.
M.A. Spall. Large-scale circulations forced by localized mixing over a sloping
bottom. Journal of Physical Oceanography, 31:2369–2384, 2001.
M.W. Stacey. The Interaction of Tides with the Sill of a Tidally Energetic Inlet.
Journal of Physical Oceanography, 14:1105–1117, 1984.
M.W. Stacey. Some Aspects of the Internal Tide in Knight Inlet, British Columbia
. Journal of Physical Oceanography, 15:1652–1661, 1985.
M.W. Stacey and Y. Gratton. The Energetics and Tidally Induced Reverse Renewal
in a Two-Silled Fjord. Journal of Physical Oceanography, 31:1599–1615,
2001.
C. Staquet and J. Sommeria. Internal Gravity Waves: From Instabilities to Turbulence.
Annual Review of Fluid Mechanics, 34:559–593, 2002.
M. Stastna and K. Rowe. On weakly nonlinear decritions of nonlinear internal
gravity waves in a rotating reference frame. Atlantic Electronic Journal of
Mathematics, 2:30–54, 2007.
J.K. Sveen, Y. Guo, P.A. Davies, and J. Grue. On the breaking of internal solitary
waves at a ridge. Journal of Fluid Mechanics, 469:161–188, 2002.
S.A. Thorpe. The effects of rotation on the nonlinear reflection of internal waves
from a slope. Journal of Physical Oceanography, 30:1901–1909, 2000.
S.A. Thorpe. On the Interactions of Internal Waves Reflecting from Slopes. Journal
of Physical Oceanography, 27:2072–2078, 1997.
S.A. Thorpe. On internal wave groups. Journal of Physical Oceanography, 29:
1085–1095, 1999.
J.M. Toole, R.W. Schmitt, K.L. Polzin, and E. Kunze. Near-boundary mixing
above the flanks of a midlatitude seamount. Journal of Geophysical Research,
102:947–959, 1997.
Y. Tseng and J.H. Ferziger. Mixing and available potential energy in stratified
25

flows. Physics of Fluids, 13:1281–1293, 2001.
L. Umlauf and U. Lemmin. Interbasin exchange and mixing in the hypolimnion
of a large lake: The role of long internal waves. Limnology and Oceanography,
50:1601–1611, 2005.
H. Van Haren. Details of stratification in a sloping bottom boundary
layer of Great Meteor Seamount. Geophysical Research Letters, 32:
doi:10.1029/2004GL022298, 2005.
H. Van Haren, L.St. Laurent, and D. Marshall. Small and mesoscale processes
and their impact on the large scale: an introduction. Deep-Sea Research Part
II, 51:2883–2887, 2004.
S.K. Venayagamoorthy and O.B. Fringer. Numerical simulations of the interavtion
of internal waves with a shelf break. Physics of Fluids, 18:076603, 2006.
J.-Y. Vincont, S. Simoens, M. Ayrault, and J.M. Wallace. Passive scalar dispersion
in a turbulent boundary layer from a line source at the wall and downstream
of an obstacle. Journal of Fluid Mechanics, 424:127–167, 2000.
V.I. Vlasenko and K. Hutter. Generation of second mode solitary waves by the
interaction of a first mode soliton with a sill. Nonlinear Processes in Geophysics,
8:223–239, 2001.
V.I. Vlasenko and K. Hutter. Transformation and disintegration of strongly nonlinear
internal waves by topography in stratified lakes. Annales Geophysicae,
20:2087–2103, 2002a.
V.I. Vlasenko and K. Hutter. Numerical Experiments on the Breaking of Solitary
Internal Waves over a Slope-Shelf Topography. Journal of Physical Oceanography,
32:1779–1793, 2002b.
V.I. Vlasenko, N. Stashchuk, and K. Hutter. Water exchange in fjords induced
by tidally generated internal lee waves. Dynamics of Atmospheres and Oceans,
35:63–89, 2002.
G.W. Wake, E.J. Hopfinger, and G.N. Ivey. Experimental study on resonantly
forced interfacial waves in a stratified circular cylindrical basin. Journal of
Fluid Mechanics, 582:203–222, 2007.
D.-P. Wang. Tidally generated internal waves in partially mixed estuaries. Continental
Shelf Research, 26:1469–1480, 2006.
K. Widell. Ice-ocean interaction and the under-ice boundary layer in an Arctic
fjord. PhD thesis, Geophysical Institute, University of Bergen, Norway, 2006.
S. Wijffels and G. Meyers. An Intersection of Oceanic Waveguides: Variability
in the Indonesian Throughflow Region. Journal of Physical Oceanography, 34:
1232–1253, 2004.
K.B. Winters and E.A. D’Asaro. Direct Simulation of Internal Wave Energy
Transfer. Journal of Physical Oceanography, 27:1937–1945, 1997.
K.B. Winters, P.N. Lombard, J.J. Riley, and E.A. D’Asaro. Available potential
energy and mixing in density-stratified fluids . Journal of Fluid Mechanics,
26

289:115–128, 1995.
K.B. Winters, H.E. Seim, and T.D. Finnigan. Simulation of non-hydrostatic,
density-stratified flow in irregular domains. International Journal for Numerical
Methods in Fluids, 32:263–284, 2000.
C. Wunsch. On oceanic boundary mixing. Deep-Sea Research, 17:293–301, 1970.
C. Wunsch and R. Ferrari. Vertical Mixing, Energy, and the General Circulation
of the Oceans. Annual Review of Fluid Mechanics, 36:281–314, 2004.
J. Xing and A.M. Davies. Processes influencing tidal mixing in the region of sills.
Geophysical Research Letters, 33:L04603 doi:10.1029/2005GL025226, 2006a.
J. Xing and A.M. Davies. Influence of stratification and topography upon internal
wave spectra in the region of sills. Geophysical Research Letters, 33:L23606
doi:10.1029/2005GL025226, 2006b.
J. Xing and A.M. Davies. On the importance of non-hydrostatic processes in
determining tidally induced mixing in sill regions. Continental Shelf Research,
27:2162–2185, 2007.
E.D. Zaron and G.D. Egbert. Verification studies for a z-coordinate primitiveequation
model: Tidal conversion at a mid-ocean ridge. Ocean Modelling, 14:
257–278, 2006.
27