ARCHiVES
Interference and Resonance of
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Internal Gravity Waves
by
Sasan John Ghaemsaidi
LIBRARIES
B.S., University of Texas at Austin (2009)
S.M., Massachusetts Institute of Technology (2011)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
Sasan John Ghaemsaidi, 2015. All rights reserved.
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August 17, 2015
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Thomas Peacock
Associate Professor
Thesis Supervisor
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David Hardt
Chairman, Department Committee on Graduate Studies
Accepted by ...........
2
Interference and Resonance of
Internal Gravity Waves
by
Sasan John Ghaemsaidi
Submitted to the Department of Mechanical Engineering
on August 17, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Internal waves are propagating disturbances within stratified fluids, arising from a
balance of gravity, buoyancy, and rotation. As well as being of fundamental scientific
interest, they are ubiquitous in a variety of forms in the Earth's oceans, where they are
responsible for driving vertical mixing. And it is the rule, rather than the exception,
that internal waves propagate through a varying background density stratification.
We begin by theoretically studying internal waves that are harmonically forced at
a horizontal level above a semi-infinite, non-uniform density stratification. Starting
with a two-layer model, we identify the existence of resonance peaks and diminution
troughs in the wave transmission spectra, and provide physical insight through the
application of ray theory. Thereafter, we proceed to consider smoothly varying stratifications, demonstrating that these resonance and diminution features persist beyond
simple models. We conclude by considering the relevance of the results to geophysical
settings. As an example, we demonstrate that an ocean stratification is inherently
tuned to transmit internal wave energy to the deep ocean at specific combinations of
wavelength and frequency.
Subsequently, we perform a laboratory experimental study of an internal wave field
generated by harmonic, spatially-periodic surface forcing of a strongly-stratified, thin
upper layer sitting atop a weakly-stratified, deep lower layer. In linear regimes, the
energy flux associated with relatively high frequency internal waves is prevented from
entering the lower layer by virtue of evanescent decay. In the experiments, however,
we find that the development of parametric subharmonic instability (PSI) in the
upper layer transfers energy from the forced primary wave into a pair of subharmonic
daughter waves, each capable of penetrating the weakly-stratified lower layer. We
find that around 10% of the primary wave energy penetrates into the lower layer via
this nonlinear wave-wave interaction for the regime we study.
With an emphasis on assessing the role of interference in tuning wave transmission, we perform a series of laboratory experiments in order to measure resonance
and diminution in the aforementioned non-uniform stratification. We find that the
occurrence of destructive interference in the upper stratification layer naturally yields
diminution of the transmitted wave. Conversely, constructive interference results in
a notable amplification of the wave field over time scales on the order of the forcing
period; the development of nonlinear wave-wave interactions due to wave amplifica-
3
tion is observed over longer time scales. Good agreement is obtained between the
experimental results and a weakly viscous, long wave model of our system within the
linear regime.
Given the ubiquity of layering in environmental stratifications, an interesting example being double-diffusive staircase structures in the Arctic water column, we furthermore present the results of a joint theoretical and laboratory experimental study
investigating the impact of multiple layering on internal wave propagation. We first
present results for a simplified model that demonstrates the nontrivial impact of multiple layering. Incident waves of particular length and time scales can experience
constructive interference taking place within the alternating stratified and mixed layers, which in turn appreciably enhances wave transmission. Thereafter, utilizing a
weakly viscous, linear model that can handle arbitrary vertical stratifications, we
perform a comparison of theory with experiments finding excellent qualitative and
quantitative agreement. We conclude by applying this model to a case study of a
staircase stratification profile obtained from the Arctic Ocean, finding a rich landscape of transmission behavior.
Thesis Supervisor: Thomas Peacock
Title: Associate Professor
4
To -my -mother,
Shahnaz Ghaemaghamian,
an ambassador of love, vision,
sacrifice, and strength.
5
6
Acknowledgments
A sincere thank you to Prof. Torn Peacock for introducing me to the beauty of
internal gravity waves, and for his support and willingness to discuss any aspect of my
research; to my thesis committee members, Profs. John Bush, Thierry Dauxois, Pierre
Lermusiaux, Manikandan Mathur, and Gareth McKinley, for their time, enriching
discussions, and support; to Profs. Abeyaratne, Gossard, Parks and Turitsyn for
making my dynamics and solid mechanics teaching experiences so enjoyable.
I have had the good fortune of being a member of a remarkable research group,
whose members to date are Michael Allshouse, Severine Atis, Brian Doyle, Paula
Echeverri, Margaux Filippi, Maha Haji, Ray Hardin, Nils Holzenberger, Hussain
Karimi (honorary homie), Matthieu Leclair, Manikandan Mathur, Matthieu Mercier,
Zaim Ouazzani, and Spencer Wilson. I have also had the pleasure of collaborating
with Thierry Dauxois, Sylvain Joubaud Philippe Odier of ENS Lyon, as well as Hayley
Dosser and Luc Rainville of the University of Washington. I am grateful for the help
and support of Andy Gallant and Andy Ryan of the central machine shop, who are
the true engineers.
A very special thank you to Paula Echeverri for taking the time to introduce me
to the basics of internal waves during my first few months at MIT; to Manikandan
Mathur for enthusiastically supporting my work and for helping me become an experimentalist; Matthieu Mercier for countless enriching discussions and for teaching
me new experimental techniques; Michael Allshouse for his invaluable support and
guidance. A special thank you to Louis Gostiaux for making my time in Lyon so
memorable and enjoyable; only with (for) Louis could the harvesting of potatoes be
so fun. I sincerely appreciate the support, guidance and advice of Leslie Regan, Joan
Kravit and Una Sheehan.
Maha Haji, thank you for making the past three years so memorable and uplifting.
You have been there through the lows and the highs, and its been a righteous journey
by virtue of your presence.
Any acknowledgement in this thesis of my friends and the profound role they have
played in my life would fail to do true justice. Please know that I stand as a truly
blessed man for having you all in my life.
Last, but most certainly not least, a recognition of my family: my mother, Shahnaz
Ghaemaghamian, my father, Bahman Saidi, and my brother, Mike Saidi. You three
have made my life extraordinarily beautiful.
7
8
Contents
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Forced internal waves in a non-uniform stratification
2.1 System description and governing equations . . . . .
2.2 Sharp interface model . . . . . . . . . . . . . . . . . .
2.2.1 Analytical approach . . . . . . . . . . . . . .
2.2.2 Physical interpretation of transmission spectra
2.3 Finite-width interface model . . . . . . . . . . . . . .
2.4 Geophysical application . . . . . . . . . . . . . . . .
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
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Stratified fluids . . . . . . . . . . . . . . . .
Internal waves in a uniform stratification . .
Internal waves in a non-uniform stratification
Geophysical role . . . . . . . . . . . . . . . .
Thesis overview . . . . . . . . . . . . . . . .
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1.1
1.2
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1.5
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Introduction
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Nonlinear internal wave penetration via parametric subliarmonic instability
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3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3
3.4
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5
Discussion and conclusion
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Experimental investigations of internal wave
tion
4.1 Theoretical formulation . . . . . . . . . . . .
4.2 Experimental set-up . . . . . . . . . . . . .
4.3 Results & discussion . . . . . . . . . . . . .
4.3.1
Qualitative observations . . . . . . .
4.3.2
Linear transmission . . . . . . . . . .
4.3.3
Nonlinear consequences of resonance .
4.4 Conclusion . . . . . . . . . . . . . . . . . . .
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resonance and diminu. . . . . . . . . . . . . .
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5
The
5.1
5.2
5.3
5.4
impact of multiple layering
M otivation . . . . . . . . . . .
Mathematical formulation . .
Impact of layering . . . . . . .
Experimental studies . . . . .
5.4.1 Apparatus & Methods
6
transmission
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R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Geophysical application . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
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5.4.2
5.5
5.6
on internal wave
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Conclusion
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A WKBJ condition for plane wave approximation of boundary conditions
89
10
List of Figures
1-1
1-2
1-3
1-4
Stratification profiles of the equatorial Pacific Ocean (140'W, 00 N)
obtained from the World Ocean Database 2013 (Boyer et al. 2013).
(a) Salinity S and temperature T profiles are shown in black and red,
respectively. Panels (b) and (c) present the corresponding density p
and buoyancy frequency N profiles. . . . . . . . . . . . . . . . . . . .
18
Fluid parcel in (a) static equilibrium and (b) slightly perturbed. The
red and black arrows respectively denote the gravitational and buoyancy forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Snapshot of an internal wave field generated by a vertically oscillating
cylinder (width into the page). The color plot denotes the total velocity magnitude ult and the arrows denote local velocity directions; the
vertical arrow on the right denotes the orientation of gravity. The solid
black line positioned along each wave beam denotes the group velocity vector Cg and the orthogonal dashed line denotes the wave number
vector k (which is coincident with the phase velocity cp). . . . . . . .
22
Snapshots of the total velocity lul field for increasing excitation frequencies: (a) w/N = 1/2 (0 = 30'), (b) w/N = 1/v2 (0 = 450), and
(c) w/N = V/2 (0 = 60'). The velocity scale is (mm/s). . . . . . . .
22
1-5
Internal wavefields constructed by different mathematical formulations.
Panel (a) demonstrates a downward propagating plane wave; panel
(b) presents a vertical mode-1 propagating left to right. The principal difference between the two approaches is that vertical modes have
their vertical length scales pre-determined by the horizontal boundaries, whereas plane waves are free to embody any length scale. The
experimental wavefield in panel (b) was obtained by Manikandan Mathur. 23
1-6
An internal wave beam with spectral profile if(k) = k exp (-4k2 ) propagating through a (a) uniform and a (b) non-uniform stratification; the
corresponding buoyancy frequency profiles N(z) are shown in the left
panels. The incident beam amplitude has been normalized to unity,
and the spatial scales have been normalized by the dominant wavelength A 0 = 27r/ko, where 'I(k) is maximum at k = ko. Arrows are
placed to indicate the direction of energy transport in each scenario. .
11
25
1-7
Downward propagating internal wave beam through a non-uniform
stratification (left panel) with the amplitude spectrum 4l (k) = exp [- (k - ko
where ko = 10. Two mathematical approaches to computing the wavefield ') are presented: the (a) WKBJ approximation, and the (b) nu27
merical solution to (1.15). . . . . . . . . . . . . . . . . . . . . . . . .
2-1
Sketch depicting the system configuration. A traveling wave disturbance imposed at z = 0, which can correspond to the base of a mixedlayer, generates a downward propagating incident (I) wave and an
upward propagating reflected (R) wave in the upper layer of buoyancy
frequency N1 , while a transmitted (T) wave propagates downward in
the semi-infinite layer of buoyancy frequency N2 ; the sinusoidal black
and straight gray arrows denote the group velocity and wave vector
orientations, respectively. The stratification, shown on the right, comprises a transition layer of width A between the upper and deep ocean
-L. . . . . . . . . . . . . . . . .
stratification that is centered on z
35
Logarithmic magnitude of F4 , and F1,' for scenario (i) in (a, b), and
scenario (ii) in (c, d) as a function of ImilL for 0.01 < lm2/m1.i < 100
(denoted by the numbers located beside the black to gray lines), with
the dashed line corresponding to 1m2 /m1l = 1. The sketches above
the figure illustrate the corresponding propagating/propagating (i.e.
scenario (i)) and propagating/evanescent scenario (i.e. scenario (ii)) of
each column. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .
38
(a) Schematic of the ray paths within a finite-depth Ni layer for general
The gray arrows indicate the incident
and inilL.
values of 1m2 /m1,
and reflected rays (within the Ni layer) and the transmitted rays (entering the N2 layer); the black arrows at the upper-boundary represent
the forcing for points at which internal wave rays encounter the upper boundary. Streamfunction amplitudes T( associated with then In
transmitted ray are presented as a function of InmilL, the plane-wave
reflection coefficient r, and the initial transmitted wave amplitude TM.
(b) Ray paths corresponding to a resonant case for the vertical velocity
component, where nmi1L = (p - 0.5) 7r and Im21 < Imi1. The black
arrows in the upper (N1 ) layer indicate the instantaneous orientation
of the velocity component associated with internal waves that are cither directly excited by the upper boundary forcing or result from
reflections; the black arrows in the lower (N2 ) layer indicate the instantaneous velocity component of the resulting transmitted wave. . .
-40
2-2
2-3
12
2-4
2-5
2-6
The transmission parameter 0." (solid black line) as a function of the
number of reflections n for [milL = 0.57r and (a) lm2 /m I=
0.01
(resonance) and (b) 1m 2/mi = 100 (diminution). The analytical expression for r., which corresponds to F(,
is plotted as a dashed, gray
line for each case. As the number of reflections n increases, the series
expression (2.11) converges to the analytical expression (2.8). In (a)
the convergence is monotonic, whereas in (b) the converging solution
oscillates rapidly around the final value, resulting in a seemingly solid
black area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Logarithmic magnitude of FV, for (a) scenario (i), and (b) scenario (ii)
as a function of m 1 L for [r 2 /nJ = 0.1 (black lines) and rM2 /MII =
10 (gray lines). The transmission spectra were computed for IrnilA
values of 0, 1, 2, and 10, which are labelled for each curve, as well
as being denoted by increasing line thicknesses. The sketches to the
right illustrate the corresponding scenario of each plot. Note that the
abscissa begins at a point that is half of max(ImiIA) in order for A/2 <
L. The spectra corresponding to I'M 2 /'m1j = 10, Im11 A = 10 in figure
(b) has been omitted for clarity. . . . . . . . . . . . . . . . . . . . . .
44
The phase adjustment 6# introduced by a finite-width transition layer
as a function of Im 2 /nm1, demonstrating that 6# < 0 (> 0) for Jim2/"m1l <
1 (> 1).
2-7
3-1
3-2
3-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
(a) A stratification profile of the equatorial Pacific Ocean (140'W,
0 0 N), obtained from the World Ocean Database 2013. The measured
profile is denoted by the solid line, and a slightly-smoothed stratification profile incorporating a constant, mean deep ocean stratification is
denoted by the dashed line. The surface mixed-layer spans the upper
47 m, and the pycnocline is located between the gray dashed lines, at
which the boundary conditions are applied. The weakly stratified deep
ocean has mean buoyancy frequency N 2 ~ 0.003 s-'. (b) Magnitude
of the normalized energy flux E* transmission spectra for the stratification shown in (a) as a function of kL for two different values of
w /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Schematic of the experimental configuration. A sample particle image
is overlaid to show the field of view captured by the CCD camera. The
measured buoyancy frequency profile N(z) is overlaid to the right. . .
51
Snapshots of the vertical velocity wave field at successive times illustrating the evolution of the wavefield due to the development of PSI
in the upper Ni layer. The velocity fields have been normalized by the
characteristic velocity scale Awo, and the rectangular boundary denotes the region used to perform the time-frequency analysis presented
in figure 3-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Time-averaged spectrograms, S (w, t), for the u (red) and zv (blue)
velocity components normalized by the maximum quantity . . . . . .
53
13
3-4
3-5
4-1
4-2
4-3
4-4
4-5
4-6
Wave components comprising the wave field in the rightmost panel of
figure 3-2. Panels in (a) and (b) correspond to the wave fields associated with the incident and reflected primary wave in the N1 layer.
Panels in the left column are the primary wave components oscillating
at wo; the middle and right columns correspond to the subharmonic
daughter waves oscillating at w, and w 2 , respectively. The orientation
of the group velocity vectors are denoted by the black arrows in each
plot. For clarity, we plot the a velocity component normalized by the
characteristic velocity scale Awo ... . . . . . . . . . . . . . . . . . . .
54
(Left) power transmission profiles calculated by horizontally-integrating
the vertical energy flux field for the downward wave components shown
in figure 3-4. (Right) buoyancy frequency profile (solid line) and corresponding temporal frequencies (dashed lines). . . . . . . . . . . . .
56
Buoyancy frequency profiles from two separate experimental stratifications measured by a conductivity-temperature probe (black and grey
lines) and the corresponding fitted tanh profile (red line). . . . . . . .
60
(a) panoramic view of the experimental set-up during the filling process. (b) sketch detailing the system dimensions. A sample particle
image is overlapped to depict the location and size of the measurement
field of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Snapshots of the experimental wave fields corresponding to the four
frequencies indicated in the transmission spectrum; (a, c) wave fields
excited at frequencies corresponding to resonance, (b, d) wave fields
excited at frequencies corresponding to diminution. The vertical velocities have been normalized by the characterized velocity scale Aw.
63
Spectrum of the transmission parameter It.W (black line) as a function
of excitation frequency w. The experimentally measured values are
given by the red data points. We note that the standard deviation of
the measurements are at most ofl the order of the marker size. ....
64
Transmission time histories collected at (x, z) = (161, -19.7) cm for
(a) resonance (w = 0.875 s- 1 ), and (b) diminution (w = 0.74 s-). The
shaded intervals represent the time it takes for a wave ray initiated at
the surface to travel downwards, reflect upwards, and to make contact
with the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
(a) Spatially averaged spectrogram S, (w, t) of a resonant wave field
excited at w = 0.875 s- normalized by So (the mean time-frequency
spectrum for the excitation frequency w). (b) Corresponding transmitted wave signal at (x, z)
(161, -19.7) cm normalized by Aw. (c)
Frequency spectrum at t = 40T. . . . . . . . . . . . . . . . . . . . . .
67
-
14
(a) Daily (grey) and 30-day running-mean (black) wind speed neasureinents at the mooring site in the Northern Chukchi Sea; red arrows
indicate storms. (b) Magnitude of the inertial currents as a function
of depth and time, in in s-.
Image is reproduced from Rainville
&
5-1
Woodgate (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2
71
(a) A density stratification profile collected from the Canada Basin in
the Arctic Ocean (Rainville & Winsor 2008). (b) Magnified view of
5-3
5-4
5-5
5-6
a section of the profile in (a) revealing the double-diffusive layering
structure; the red line indicates a 5 m bin-averaged smoothed density
profile. (c) The corresponding buoyancy frequency profiles N(z). .
71
Sketch of the model system. A harmonic, plane internal wave of
amplitude I and wavenumber k, = (k, Imi 1) incident at angle 01 =
tan- (k/m I) to the horizontal, encountering a transition region with
multiple layers between z = zi and z = z 2 , and emerging with a transmitted amplitude T at angle 02= tan-1 (k/m 21). . . . . . . . . . . .
72
Normalized transmission coefficients for T for sharp interface systems
comprising (a) a single mixed layer of thickness L and (b) two mixed
layers of thickness L/2 separated by a stratification layer of thickness A = L/2. The black circle in panels (a) and (b) corresponds to
mOL ~ 3.5 and 01 = 45'. Panels (c) and (d) present the corresponding theoretical results for the experimental profiles considered in 5.4,
incorporating viscosity; panels (e) and (f) present the corresponding
theoretical transmission for total velocity lul =
Iu
W 2 . The
sketches to the right of plots (a, b) and (c) - (f) illustrate the corresponding stratifications. The dashed regions in plots (e) and (f) denote
the experimental parameter space studied in 5.4. . . . . . . . . . . .
74
Schematic of the experimental arrangement, and the two measured
density stratifications, Ap = p- 1000, for the (a) single, and (b) double
mixed layer studies. A cylinder, positioned 2 cm below the free surface,
was used to generate a downward, left to right propagating internal
wave beam that was visualized by means of particle image velocimetry
(PIV); an actual raw PIV image is overlaid to show the location and
size of the field of view . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Results for (a) - (c) single and (d) - (f) double mixed layers; snapshots
of the experimental (middle column) and theoretical (right column)
vertical velocity wavefields are presented at an arbitrary phase. The
wavefields have been normalized by the characteristic vertical velocity
am plitude Aw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
15
5-7
5-8
Experimental total velocity Fourier spectra of the (a) incident wavefield
and transmitted wavefields for the (b) single and (c) double mixed layer
stratifications. The total velocity Fourier component has been normalized by the maximum incident value. Panels (d) - (f) present the
corresponding theoretical wavefields. The transmission spectra of the
wavefields for a complete field of view are presented in panels (h) and
(i) for the single and double mixed layer stratifications, respectively.
Panel (a) is reproduced in panels (d) and (g), as the experimental incident spectrum is the basis for all three sets of results. The dashed
boundary corresponds to that in figures 5-4(e, f). . . . . . . . . . . .
Transmitted energy flux (period averaged in t) normalized by the incident energy flux, ISqrI/IEx, as a function of excitation frequency w,
for the (a) staircase and (b) 5 m bin-averaged stratifications. Incident
vertical wave lengths are A, = 10 m (-), 20 m (- -), 30 m (- - ),
and 50 mr ( ... ). The inset shown in panel (a) presents the near-inertial
wave (w = 1.05f) energy transmission against vertical wavelength A.
16
81
82
Chapter 1
Introduction
Waves subtly embrace our lives. Our first cries after birth penetrate the tense air
by means of mechanical pressure waves. As we grow, we receive nourishment via
peristaltic wave motion. Once we are old enough to play, our hair streams and
oscillates as we run freely through the wind. Instruments that produce the music we
love do so through vibrational wave motion in the strings, and amplifying acoustic
waves through their bodies.
We find a more familiar example by looking over bodies of water, where we see the
atmosphere pinching the free surface, subsequently giving rise to both smooth and
crisp undulations that form surface gravity waves. These waves survive on the balance
of gravity and fluid inertia. This thesis concerns wave motions found within the body
of the ocean: internal gravity waves. We begin this chapter in 1.1 by providing
a general overview of stratified fluids, and their ability to sustain oscillatory motion
within their body. Next, in 1.2 and 1.3 we respectively introduce the general theory
behind internal waves in uniform and non-uniform stratifications. We conclude in 1.4
by detailing the geophysical significance of internal wave dynamics, and in 1.5 with
an overview of the thesis.
1.1
Stratified fluids
The ocean and atmosphere are prime examples of stratified environments, which are
characterized by measurable variations in salinity, temperature and, more generally,
density with respect to one or more spatial coordinates. For most applications, density
stratifications exhibit non-uniformity in the direction of gravity, and may take the
form of somewhat discrete layers or continuous distributions. For this thesis, we
focus on scenarios primarily relevant to the ocean, but also the atmosphere, and thus
density stratifications that are naturally continuous.
The ocean can reach depths of thousands of meters. From the unwelcoming abyss
to the exposed surface, the typical oceanic water column can comprise an intricate
array of features. For example, figure 1-1 presents salinity, temperature, and density
profiles measured by conductivity-temperature-depth (CTD) sensors deployed in the
equatorial Pacific Ocean (Boyer et al. 2013). In the upper 50 - 100 of the ocean lies
17
1
0
20
7O
30
(b)
(a)
(c)
-250
-500-
-750-
1000
34.5
34.6
34.7
S (psu)
34.8
1020
1025
1030
p (kg in-')
1035
0
0.04
0.02
N (s-')
0.06
Figure 1-1: Stratification profiles of the equatorial Pacific Ocean (140'W, 00 N) obtained from the World Ocean Database 2013 (Boyer et al. 2013). (a) Salinity S and
temperature T profiles are shown in black and red, respectively. Panels (b) and (c)
present the corresponding density p and buoyancy frequency N profiles.
what is commonly referred to as the surface mixed layer, identified by the uniformity
of the temperature T and salinity S profiles shown in figure 1-1(a). Wind action at
the ocean surface, coupled with thermal convection in the upper portion of the water
column, yields a well-mixed region of water with effectively no stratification.
As shown in figure 1-1(a), the T and S profiles change substantially with depth,
which together with pressure p induce variations in density p(S, T, p). The corresponding density distribution is presented in figure 1-1(b), which comprises regions
characterized by distinct features: a uniformly stratified surface layer separated from
the weakly stratified depths by a relatively thin transition layer. In the roughly 200 m
below the surface mixed layer there is a dramatic change in temperature separating
the cold, dense water of the deep from the warmer, lighter water of the surface; this
notable transition in temperature, salinity, and density is respectively referred to as
the thermocline, halocline, and pycnocline. Below the pycnocline lies the weakly
stratified depths of the ocean, which span the remainder of the water column.
For a gravitationally stable stratification, the density must increase monotonically
with depth. With this basic stability condition satisfied, we apply our focus to a
parcel of fluid at some arbitrary position in the water column. By virtue of the
aforementioned gravitational stability of the system, an infinitesimal perturbation to
the position of this fluid parcel will not cause any sort of unbounded growth. Upon
'A similar scenario is observed in the atmosphere, with temperature variations and ambient
pressure playing the dominant role in influencing gas density (Sutherland 2010).
18
(a)
I
(b)
i9
6
t
pXzo)
z
p(ZO)
pAzo)
'(-zo)
I> p~zo)
1
6z(t
1
t
Figure 1-2: Fluid parcel in (a) static equilibrium and (b) slightly pcrturbed. The red
and black arrows respectively denote the gravitational and buoyancy forces.
displacement, the fluid parcel will experience two body forces: gravity and buoyancy,
both of which act to counter the other, as shown in figure 1-2. This counterbalance
of restoring forces induces an oscillatory motion that is described by:
p (zo) 6Z + p (zo +
z)g - p(zo) g = 0,
gravity
buoyancy
inertia
(1.1)
where 6z is the vertical perturbation, zO is the initial vertical position, and g is gravity.
After taking the Taylor expansion of the buoyancy term in (1.1) about zo, we obtain
the linearized equation:
(1.2)
5z + N26z = 0,
where
dp
N2 = g
p dz
(1.3)
From (1.2) and (1.3) it is evident that the natural oscillatory frequency of a displaced fluid element in a stable density stratification is N, which is commonly referred to as the buoyancy, or Brunt-Vsiissld, frequency (Gill 1982). The buoyancy
frequency is effectively used as a metric to characterize a particular stratification,
as shown in figure 1-1(c). A typical oceanic buoyancy frequency profile contains
a mixed surface layer, in which N = 0, lying atop a pycnocline that has a peak
(which translates to oscillatory periods of
buoyancy frequency of N ~ 10-2 S
Th = 5 cph). Below the pyenocline lie the weakly stratified depths of the ocean
where N = 10-- s-' (TN a 0.5 cph).
19
1.2
Internal waves in a uniform stratification
The occan surface sustains gravity waves that are confined to the interface separating
two fluids of dramatically different densities, water and air. On the other hand, density stratified fluids support internal gravity waves that are free to propagate within
the body of the fluid. As a result, stratified fluids can respond to the imposition of
external forcing by generating internal waves that transport energy through coherent
wave motions.
Two-dimensional, inviscid internal waves are governed by the following equation
(LeBlond & Mysak 1978, Gill 1982, Sutherland 2010):
Ott
(V 2 ,O) +
N 2 09, 4 + f 2
22
= 0,
(1.4)
where 0 is the streamfunction describing the velocity field via (u, w)
(-&V', 0.4),
N is the buoyancy frequency of the linear density stratification, and f is the Coriolis
parameter (i.e. background rotational frequency). Equation (1.4) admits harmonic
solutions in space and time, thus yielding plane wave solutions of the form:
V7(x, zt) = T exp [i (k - x - wt)] ,
(1.5)
where 'I is a complex amplitude characterizing the magnitude and phase of the
streamfunction fluctuations, the wavenumber vector k = (k, rn) characterizes the
spatial frequency of the waves in the (x, z) plane, and w is the temporal wave frequency.
Substitution of (1.5) into (1.4) yields the dispersion relation,
N k + fIm 2
2
,p-12
Wk
(1.6)
which links the spatial and temporal arrangement of the wave solutions, meaning that
a wave with a given temporal frequency w has four unique wavenumbers, (k,
n) and
(-k, t7n), that satisfy the governing equation. For a more intuitive picture, we define
the angle separating the wave number vector k and the z-axis to be 6 = tan- (k/rm),
which then allows (1.6) to be recast as:
W
N
2
sin 2 6 +
f 2 cos 2 0.7)
In essence, an increase in the excitation frequency results in a shallower k, that in turn
produces steeper lines of constant phase (which are orthogonal to k). Additionally,
from (1.6) we see that N and f respectively set the tipper and lower limits of the
internal wave pass band (provided f < N): propagating internal waves exist for
f < w < N, whereas evanescent decay takes place for w < f or w > N.
Insight into kinematics and energy transport is also provided by the dispersion
relation in (1.6). The phase velocity vector c = w/k specifies the flux of wave
crests (i.e. lines of constant phase), which is naturally aligned with the wave number
vector. The group velocity vector denotes the direction of energy propagation, and is
20
formally defined 2 Cg = (kDw, 0op). An intriguing aspect of internal wave behavior is
the orthogonality of the phase and group velocity vectors c% - eg = 0, where energy
propagates along lines of constant phase.
Gortler (1943), and subsequently Mowbray & Rarity (1967), were first to provide
experimental evidence of internal wavefields by means of the Schlieren flow visualization technique (this technique highlights changes in density by utilizing a light source
to image alterations to the refractive index of the fluid). Figure 1-3 presents a more
modern-day snapshot of an internal wave field generated by a vertically oscillating
cylinder in a- linear density gradient and in the absence of background rotation (i.e.
f = 0); particle image velocimetry (PIV) was employed in visualizing the velocity
field, which is the technique used to gather experimental data for this thesis. Four
wave beams emanate away from the cylinder, transporting energy away from the
generation source. The orthogonal phase configuration with respect to the group
velocity is denoted by the dashed and solid arrows, respectively. We further note
that the length scale of the waves is set by the characteristic scale of the localized
disturbance, which for this example is the diameter of the cylinder. A demonstration
of the manner in which an internal wave field evolves with excitation frequency is
given in figure 1-4. The increasing steepness of the wave beams with increasing w
exemplify the dispersion relation (1.7).
The theoretical construction of uni-directional wave beams, such as those presented in figures 1-3 and 1-4, requires the linear superposition of a continuous spectrum of plane waves propagating at the same temporal frequency (Thomas & Stevenson 1972, Kistovich & Chashechkin 1998); for example, the expression for a downward,
left to right propagating wave of arbitrary profile will be of the form:
j
T (k) exp [i (kx + k cot Oz - wt)] dk,
(1.8)
where T (k) describes the amplitude distribution. The particular form of T (k) determines the resultant wave form (e.g. 'I oc exp (-k2 ) yields a finite-width wave
2
The appropriate context for the consideration of group velocity is the propagation of narrowband wave pulses. Waves characterized by a narrow-band about a dominant wave number ko written:
'(x, t) =
(k) exp [i (k - x - w (k) t)] dk,
where
w (k)
w (ko) + (k - ko) - VkW (ko).
After expanding w (k) to first order and substituting into the integral expression we find:
J~
' /'(k) exp
[ik (x - VkW (ko) t)] dk,
which represents a carrier envelope of the wave pulse that travels with the group velocity:
.
C9 = VkW (ko)
21
0
0.2
0.4
|u| (mm/s)
C9
10
0
Q
9
10
-20
0
20
x (cm)
Figure 1-3: Snapshot of an internal wave field generated by a vertically oscillating
cylinder (width into the page). The color plot denotes the total velocity magnitude
Jul and the arrows denote local velocity directions;
denotes the orientation of gravity. The solid black
beam denotes the group velocity vector cg and the
the wave number vector k (which is coincident with
(a)
the vertical arrow on the right
line positioned along each wave
orthogonal dashed line denotes
the phase velocity ca).
(C)
(b)
10
0.6
0
0.3
10
-10
0
x (cm)
10
-10
0
x (cm)
10
-10
0
x (ci)
10
Figure 1-4: Snapshots of the total velocity Jul field for increasing excitation frequencies: (a) w/N = 1/2 (6 = 300), (b) w/N = 1/V2 (0 = 450), and (c) w/N =
v/3/2 (0 = 600). The velocity scale is (mm/s).
22
(b)
(a)
4-T-_-
-1
TT-X
//
/////
Figure 1-5: Internal wavefields constructed by different mathematical formulations.
Panel (a) demonstrates a downward propagating plane wave; panel (b) presents a
vertical mode-i propagating left to right. The principal difference between the two
approaches is that vertical modes have their vertical length scales pre-determined by
the horizontal boundaries, whereas plane waves are free to embody any length scale.
The experimental wavefield in panel (b) was obtained by Manikandan Mathur.
&
beam with a Gaussian envelope). As such, plane waves are essentially the building
blocks of freely propagating waveforms, and can be used to characterize a variety of
geophysical scenarios (Booker & Bretherton 1967, Lindzen & Barker 1985, Tabaci
Akylas 2003).
An experimental snapshot of a developing, downward propagating plane wave is
presented in figure 1-5(a), where the well-established plane wave in the upper half of
the field of view is shown to be the manifestation of (1.5). Typically, the application of
plane waves is most appropriate for scenarios in which the vertical extent of the ocean
does not come into consideration. For instance, the ocean is effectively semi-infinite in
the vertical direction for waves that experience appreciable decay/dissipation as they
propagate downwards, with no reflections off the sea floor taking place. Additionally,
complicated forcing profiles applied along a boundary (i.e. storms and translating
pressure fronts) can be decomposed into a distribution of plane waves, which can be
individually used to solve the governing equation and then combined to produce the
resultant wavefield via Fourier superposition (Mathur & Peacock 2009, 2010).
The presence of vertical boundaries introduces no-vertical flux boundary conditions at the ocean bottom and surface, which effectively renders a waveguide. As such,
the finite vertical extent of the system poses constraints that must be accounted for;
to do so, we modify the ansatz presented in (1.5) to read:
(x,(,xt) ,0x T (z; k) exp [i (kx - wt)I .
(1.9)
Substituting (1.14) into (1.4) yields an ordinary differential equation governing the
23
vertical structure of the wave field:
X2, + k 2
(N 2
W2
W2)
-
f
= 0.
(1.10)
2
The no-vertical flux boundary conditions are explicitly stated as T (0) = I(-H) = 0,
oc I. The governing equation (1.10) and boundary
which utilize the fact that w =
conditions together comprise a Sturm-Liouville problem that has particular eigenvalues k,, satisfying the constraint m.,H = n7r, which correspond to the eigenfunctions
(i.e. vertical modes) T, = sin (mz). As a result, the solution for an arbitrary
wavefield can now be written as a summation of vertical modes:
'(x, z, t)
Re
(z
, sin ( 7hz exp [i (kx - wt)]
,
(1.11)
\n=1
where
k~' n7r
H
(
o2
__ f2 )1/2
(.2
N2_-W2
and 41,, are complex amplitudes that define the amplitude and phase of the n-th
mode. From (1.11) one can observe that as the mode number increases, so does the
number of nodes spanning the vertical, while the vertical length scale characterizing
the vertical structure of the wavefield decreases. An experimental snapshot of a
propagating mode-1 internal wave is presented in figure 1-5(b), from which we can
see that (1.11) yields a traveling vortex for n = 1.
Modal constructs are traditionally used to model the complicated internal wavefields associated to topographic generation(Eckart 1961, Llewellyn Smith & Young
2002, 2003, Echeverri et al. 2009). In such scenarios, the time-periodic tidal motion
of the ocean induces flow past underwater topography which perturbs the stable background density field and concurrently generates internal waves that radiate energy
away. The conversion from tidal energy to internal waves is strongly dependent on
the specific features of the topographic source, which influence the modal amplitude
distribution of the resultant wavefield; an indication of the energy flux associated to
each constitutive mode is derived by coupling the corresponding modal amplitudes
and length scales (P6tr6lis et al. 2006, Echeverri & Peacock 2010).
1.3
Internal waves in a non-uniform stratification
For stratifications with nonlinear density gradients, the vertical structure of internal
waves is no longer sinusoidal, but dependent on the vertical structure of the local
buoyancy frequency profile N(z). In such environments, we begin with a slightly
modified form of the governing equation:
Ott (V 2 V) + N(z) 2 91 '.' + f 2&040= 0,
24
(1.13)
(a)
o
(b)
0
.)
0.
1
0
1
0
4
3
2
5
x/A
N
k exp (-4k2 ) propaFigure 1-6: An internal wave beam with spectral profile 4(k)
gating through a (a) uniform and a (b) non-uniform stratification; the corresponding
buoyancy frequency profiles N(z) are shown in the left panels. The incident beam
amplitude has been normalized to unity, and the spatial scales have been normalized
by the dominant wavelength Ao = 27/ko, where 41(k) is maximum at k = ko. Arrows
are placed to indicate the direction of energy transport in each scenario.
which admits harmonic solutions in x and t, thus yielding separable solutions of the
form:
(1.14)
t) oc 4 (z; k) exp [i (kx - wt)] .
z(x,.-,
Substituting (1.14) into (1.13) yields a variable coefficient ordinary differential equation governing the vertical wave structure:
2
N( 1,) 2 _2)
41z + k2 (NK2
j)
41
0.
(1.15)
Generally, solutions to (1.15) are obtained numerically, and depending on the boundary conditions, (1.15) may be treated either as an initial- or boundary-value problem.
The occurrence of refraction and scattering is what differentiates wave propagation through a non-uniform medium from a uniform medium. Figure 1-6 presents
snapshots of a model wave beam traveling through a uniform and non-uniform strat25
ification. In the constant N scenario i.e. figure 1-6(a), the wave beam propagates
through the uniform medium without experiencing any adjustment to its structure.
In a non-uniform medium i.e. figure 1-6(b), however, the incident wave beam notably
undergoes refraction (changes in the orientation of its group velocity, and thus its
travel path) and scattering (redistribution of its energy into an array of reflected and
transmitted waves). Thus, the consequences of internal wave propagation through a
non-uniform stratification has the potential to be quite sophisticated and involved.
It is possible to theoretically calculate wave transmission through non-uniform
stratifications provided that the incident waves have sufficiently small length scales
relative to the stratification. In such scenarios, one can employ the Wentzel-Kramers-
Brillouin-Jeffreys (WKBJ) approximation (Brekhovskikh 1980, Cheng 2007) to obtain
a solution of the form:
T ~ exp [ikO(z)] ,
+ k-101 + k-
# ~o
2
+. . .,
(1.16)
where # represents some unknown phase function, and k sets the length scale of the
wave (which is assumed to be small i.e. k >> 1). Substituting (1.16) into (1.15) yields:
ik#22 - k 2
2
+ k2 ,92 = 0,
(1.17)
where V2 = (N(z) 2 _ W 2 ) / (W1 _ f 2 ); separating the terms of (1.17) into various
orders of k produces a set of linear differential equations whose solutions comprise
the series terms of the ansatz in (1.16):
0
(1) :0#2Z
0
(k-1) :
o (k-
2
) :
- 02 = 0
ioo,,2 - 20,201,.z = 0
igi,22 - #2
-200,-02,2
= 0
(1.18)
After carefully solving for each term, the second-order WKBJ approximation of the
wave solution is constructed:
T ~
1
[
f
/,d exp t~ik JZ i (z') d'2z' .
(1.19)
Figure 1-7 presents a comparison of a wavefield obtained by both the WKBJ approach and by numerically solving (1.15). We find that for this particular wavefield, in
which the length scales are significantly smaller than the length scales characterizing
the variation of the background stratification, the second-order WKBJ approximation
yields results that are effectively identical to the those that are numerically obtained.
This example illustrates an essential feature of WKBJ approximations: the wave energy fully transmits through the nonuniform media with no reflections taking place.
The physical idea behind this being that the incident wave is able to gradually adjust
to variations in the background stratification. Unfortunately, only a limited number
26
(b) numerical
(a) WKBJ
0
-10
I
-
-20
-30
1
0
N
0
20
40
60
80
0
20
40
60
80
X/A
x/A
Figure 1-7: Downward propagating internal wave beam through a non-uniform strat2
ification (left panel) with the amplitude spectrum kI (k) = exp [- (k - ko) ], where
ko = 10. Two mathematical approaches to computing the wavefield ' are presented:
the (a) WKBJ approximation, and the (b) numerical solution to (1.15).
of realistic scenarios are appropriate for the application of WKBJ techniques; in this
thesis, we go beyond the WKBJ limit by numerically solving the variable coefficient
differential equations.
A variety of studies have pursued the propagation of internal waves through complex, non-uniform stratifications beyond the small wavelength limit. Certain variable
buoyancy frequency profiles can be idealized as discrete regions of constant N, which
allows one to adopt a piece-wise approach to computing wave transmission by judiciously applying continuity conditions at the relevant interfaces (Tolstoy 1973). In
such a context, Sutherland & Yewchuk (2004) investigated the ability of internal
waves to tunnel through weakly stratified layers which, interestingly, were shown to
be analogous to the tunneling of quantum particles through potential barriers. The
piece-wise linear theory of Sutherland &:. Yewchuk (2004) was further extended by
Nault & Sutherland (2007) to compute the energy transmission through arbitrary
stratification and shear profiles. With an emphasis on geophysical relevance, Mathur
& Peacock (2009) made use of the methods of Nault S Sutherland (2007) and Fourier
superposition to construct internal wave beams and to realistically model the phenomena of ducting which arises when a wave beam impinges upon regions of variable
stratification. Later, Mathur & Peacock (2010) studied the selective transmission of
internal waves propagating through a non-uniform stratification, which was found to
have an interesting mathematical analogy to the classical Fabry-Perot interferometer.
Altogether, these studies laid the theoretical groundwork for the modeling of internal
wave propagation through complex stratifications that we pursue in this thesis.
Laboratory experiments can act as both validation of theory, and a means of
accessing dynamical regimes that are otherwise out of reach of mathematical models. Inspired by the turbulent consequences of overturning, Delisi & Orlanski (1975)
27
performed a series of experiments exploring the reflection and overturning of internal
waves incident upon a constant density layer. 3 The reflection process, which mimicked
a wave incident to the pycnocline, produced a complex pattern, which, for critical
amplitudes, eventually lead to gravitational instability and breaking at the transition
point. Similarly, Sutherland & Yewchuk (2004), and Mathur & Peacock (2009, 2010)
incorporated rigorous laboratory studies to confirm their theoretical formulations.
All of the aforementioned studies paint a picture of the complex wave pattern
taking place in regions of non-uniform stratification. In such patterns comprising the
overlap of various waves, the interaction of the constitutive waves can play a pivotal
role in the dynamics. The phenomena of constructive (destructive) interference, which
is the growth (reduction) in energy by virtue of interactions that are in (out of) phase
with each other, will play a natural role in our studies of internal wave dynamics in
non-uniform stratifications.
1.4
Geophysical role
Oceanic manifestations of internal waves arise from a variety of sources. One outstanding example is the interaction of the tides with prominent sea-bottom topographies. The displacement of the barotropic tidal flow introduces baroclinic 4 perturbations in the form of internal tides (i.e. internal waves that propagate at the tidal
frequency) (Garrett & Munk 1979, Garrett & Kunze 2007). Surface wind action
and storms offer another primary avenue through which external forcing mechanisms
generate internal waves oscillations, their importance to the oceanic energy budget
being comparable in magnitude to the contribution of internal tides (Munk & Wunsch
1998). As first outlined in 1.2, internal tides are generally modeled as a superposition of vertical modes whose amplitude spectra are determined by the topography
and flow conditions, whereas surface excited internal waves are typically evaluated as
vertical propagating free waves (although sonic studies have utilized vertical modes
in their formulations e.g. Kundu (1993)). In this thesis, we utilize free waves in our
theoretical formulations.
Field measurements find the stratified state of the ocean to persist even in the face
of large scale circulation. An energy source for vertical mixing is therefore needed in
order for the ocean to counteract the advection introduced by this global circulation,
and to maintain its stratified state (Ledwell et al. 2000). Munk (1966) reconciled
measurements and models where the dominant processes were turbulent mixing and
vertical upwelling, arid attempted to identify physical processes that gave rise to the
necessary advection and diffusion. Subsequently, Munk & Wunsch (1998) determined
that internal waves potentially possessed the mechanical power necessary to maintain
the ocean density distribution. Of the ~ 3.7 TW of power introduced by the orbit of
3
Stratifications containing a dramatic jump in density characterize, to first order, the upper ocean
pycnocline as well as atmospheric inversion layers.
4Barotropic signifies
the alignment of pressure and density lines (i.e. pressure is constant along
constant density lines), whereas baroclinic refers to the misalignment of pressure and density lines
(i.e. pressure is variable along constant density lines) (Gill 1982).
28
the Moon around the Earth, ~ 1 TW is attributed to the tidal generation of internal
waves in the ocean. An additional ~ 1.2 TW is introduced by atmospheric forcing
and wind action at the ocean surface. Together, these convective motions in the ocean
provide an important source of mixing (Garrett 2003).
In addition to the contribution of internal waves to maintaining the state of the
ocean, they also play a role in global and regional environmental dynamics. Surface
signatures of internal tides, visualized from satellite imagery, indicate length scales
that are 0(10-100) km (Apel et al. 1975). Additionally, satellite altimetry has been
used by Ray & Mitchum (1997) to detect internal waves propagating 0(100-1000) km
distances away from their source of generation. The scattering that takes place once
these waves come across sea bottom topography, such as seamounts and ridges, can
redistribute energy flux into smaller length scales waves, therefore increasing the
likelihood of mixing taking place (Johnston & Merrifield 2003). On a regional scale,
the associated flux of temperature and nutrients has notable biological implications for
ocean systems. Coral reefs, for instance, experience nutrient deposition and cooling
from the shoaling and breaking of large amplitude internal waves (Leichtcr et al. 1996).
Internal waves have also been observed to promote photosynthesis in the upper ocean
by inducing vertical displacements of phytoplankton (McGowan & Hayward 1978).
Atmospheric forcing is a dominant source of internal waves generated near the
surface region of the water column. The energy injected into the surface mixed layer
via storms and pressure fronts give rise to internal waves that primarily propagate
near the Coriolis frequency i.e. near-inertial waves (Pollard 1980, Kunze & Sanford
1984, Alford et al. 2013, Soares & Richards 2013).
D'Asaro (1985) examined the
space-time distribution of mixed layer energy flux in North America, where the highly
intermittent data indicated that peak events correspond to translating pressure fronts
of
0(100)
0
km scale. A more global perspective is offered by Alford (2001), who
assessed the energy transfer from winds to motions in the surfaced mixed layer and
found their global mean energy flux to be of the same order as estimates of tidal
input. On smaller length and time scales, spatially periodic vortical fluid motion
within the surface mixed layer due to Langmuir circulation (Polton et al. 2008) and
periodic Kelvin-Helmholtz billows due to shear (Pham et al. 2009) are also potentially
significant sources of internal waves but are relatively unstudied.
The outlook of internal waves in the Arctic Ocean is of particular interest. Field
measurements have shown that the internal wave energy levels in the Arctic have historically been less than their lower latitude counterparts (Levine et al. 1987, D'Asaro
& Morehead 1991), presumably due to the weak tidal currents (Levine et al. 1985)
and the presence of protective sea ice cover (Padman 1995). Satellite data, however,
reveals a marked decline in Arctic sea ice coverage (Comiso et al. 2008) as well as increases in the duration of ice-free time periods and melt seasons (Markus et al. 2009).
This reality is augmented by recent observations of energetic internal wave activity
during ice-free seasons by Rainville & Woodgate (2009); alternatively, the presence
of ice cover yields weak measurements, even during strong storm events. Similar
conclusions were drawn from the field studies of Dosser et al. (2014), who found a
positive correlation between measurable wave energy and sea ice degradation. The
prospect of a serious reduction in Arctic ice cover exposes this environmentally sen-
29
sitive body of water to a variety of processes that will have unknown implications on
Arctic dynamics (Rainville et al. 2011).
1.5
Thesis overview
Having provided an introduction to internal wave dynamics in stratified fluids, and
a general discussion of their role in the environment, we now detail the focus of this
thesis. This thesis investigates the geophysically relevant problem of internal wave
generation by processes in the upper ocean. Specifically, we study the wave dynamics
that take place in complex stratifications and how transmission is influenced. We
detail sonic of the rich consequences that stem from wave interference, and discuss
possible environmental implications.
We begin, in chapter 2, by putting forth a linear framework by which to assess
the transmission of boundary forced internal waves in a two-layer stratification 5 . We
find such a stratification, subject to plane wave boundary forcing, has transmission
spectra spanning diminution troughs and resonance peaks; using well-established ray
arguments we find that transmission is keenly dependent on the coherences of waves
spanning the ocean pycnocline. We subsequently present a framework for quantifying
transmission in continuously varying stratifications, which is then applied to a stratification from the equatorial Pacific in order to assess energy transmission to the deep
ocean.
A study of nonlinear wave-wave interactions, termed parametric subharmonic instability (PSI), is provided in chapter 3. Results of a laboratory study modeling the
boundary forcing of internal waves in a stratification inspired by the ocean are presented. We observe the gradual destabilization of the primary forced mother wave
and the consequent generation of subharmonic daughter waves. The birthed daughter
waves prove to be effective carriers of energy to lower depths, most importantly in
regimes where the mother wave is evanescent in the lower stratification layer. vVe
quantify the power transmission associated to the generation of subharmonics and
assess the relevance of PSI to ocean environments.
Chapter 4 presents a detailed experimental study of the model described in chapter
2. In focusing on a geophysically relevant parameter space, we perform a series of
experiments that correspond to the peaks and troughs of the transmission spectra
associated to our linear model. The experimental wavefields effectively demonstrate
the phenomena of constructive and destructive interference, and present excellent
agreement with linear theory. Under conditions where resonance is expected (at the
peaks of the transmission spectra), however, we observe the dramatic destabilization
of the wavefield over long time scales due to PSI induced by constructive interference,
which was the focus of our studies presented in chapter 3.
Looking to extend our model to a problem of direct geophysical interest, in chapter 5 we investigate the effect of double-diffusive layering, which is prevalent in the
5
A two-layer stratification is a conventional, first-order approximation to an oceanic stratification
comprising both the lpycnocline and weakly stratified lower layer.
30
Arctic Ocean, on internal wave transmission. We take a joint theoretical and experimental approach to modeling wave transmission through multiple mixed layers. Essentially, we find that the interaction between the evanescent and propagating waves
for incident waves of particular length and time scales is able to yield constructive
interference, thus yielding enhanced transmission. Having obtained excellent agreement between our idealized model and experiments, we apply our model to a realistic
staircase stratification profile from the Arctic Ocean.
A summary of the contributions extended by this thesis is given in chapter 6. We
outline directions for future research and identify outstanding questions.
31
32
Chapter 2
Forced internal waves in a
non-uniform stratification
It is the rule, rather than the exception, that internal waves propagate through a
varying background density stratification. To understand internal wave propagation
in these environments many studies have utilized ray tracing techniques, which operate under the slowly varying assumption (though there are techniques to correct
for such limitations, such as matching methods and Airy functions) (Broutman et al.
2004). In this chapter we utilize linear theory to consider a scenario in which waves
are forced harmonically (i.e. at a particular frequency and horizontal wave number)
at a horizontal level in nonuniform stratifications, taking into account background rotation, but for simplicity not considering vertical shear. Whilst being of fundamental
interest, there is a geophysical motivation behind the study. Both the ocean and atmosphere are non-uniformly stratified systems with effectively semi-infinite domains,
the ocean and atmosphere being bounded from above and below, respectively. Internal waves can also be excited at or near these boundaries. In the ocean, for example,
Langmuir cells (Thorpe 2004) and Kelvin-Helmholtz billows (Pham et al. 2009) drive
motions of the base of the mixed layer with a definite spatial and temporal frequency,
and a previous fundamental study has investigated the vertical cellular structure of
an internal wave field induced by spatially periodic wind forcing at the ocean surface
(Wunsch 1977).
Related classical studies have considered some aspects of the scenarios we investigate here, such as using reflection and transmission coefficients to investigate
waveguides (Tolstoy 1973) and the possibility of true resonance induced by a forced
boundary (Nappo 2013). Here, we advance these works by first providing a more complete analysis, and clearer physical description of conditions for resonance. Thereafter,
we study the detuning of transmission by continuous stratifications without the restriction of the WKBJ approximation, enabling the results to be reasonably applied
to real world scenarios. Finally, we note that in scenarios involving more spatiotemporally complicated boundary conditions, our approach may be used to model
the Fourier components (in space and time) of the forcing and resultant wave field.
The organization of the chapter is as follows. The physical system and the internal
waves sustained therein are described in 2.1. In order to elucidate the key phenomena
33
and underlying physics, in 2.2 we investigate a two-layer system with sharp interface,
before progressing to smoothly varying stratifications with finite-width interfaces in
2.3. Application of the results to an example oceanographic scenario is presented in
2.4. Finally, our conclusions and ideas for future research directions and applications
are given in 2.5.
2.1
System description and governing equations
We begin by considering a semi-infinite fluid system with the /zz-coordinatesystem
oriented as presented in figure 2-1; z = 0 corresponds to the horizontal level at
which a disturbance is imposed on the system and from which the internal wave
field originates (note that this level can be located at any vertical position within the
stratification, not just at the surface). For our initial studies, the density stratification
of the fluid is characterized by an upper layer of constant stratification N separated
from a semi-infinite bottom layer of constant stratification N2 by a transition region
of characteristic width A centered on z
-L. A model profile that nicely exemplifies
this is the function:
NZ)N(
N1 +
Ni)
=) N2q+N
2
(r.
- N2>
tanhi N1 (2.1)
tah
2
+ L
(A/6
Note that the A/6 factor in (2.1) ensures that the profile approaches 99.5% of the
constant N value at the z = -L
A/2 boundaries of the region of varying stratification.
Using (1.15) as our governing equation, we find that stratification regions of constant N yield solutions of the form T oc exp (imz), where:
i =
k
N
/2
(W2 - f2)
(2.2)
Throughout the thesis, for simplicity, we assume f to be smaller than N and w. For
f < w < N the vertical wavenumber m is real and so Real(m) > 0 and Real(m) < 0
denote downward and upward propagating internal waves, respectively; for W > N,
the vertical wavenumuber m is imaginary, in which case Imag(m) > 0 and Imag(m) < 0
correspond to upward and downward evanescent decay, respectively.
The information in the streamufunction field 4' can be manipulated to yield additional field variables of interest; namely, the velocity field components u = -&z', t
defined such that 0 1v = -fit and Y= '4, as well as the isopycual displacement field
5, the pressure field p, and the density perturbation field p. The variables that are
linearly related to T are given by:
f
w
'1'
P
ik
-k/w
-pok4 2/(Wg)
34
exp [i (kr - WOt).
(2.3)
Az
x
N2
N1
N(z)
k1
i
------
---
-L
1J9
A/2
Figure 2-1: Sketch depicting the system configuration. A traveling wave disturbance
a
imposed at z = 0, which can correspond to the base of a mixed-layer, generates
(R)
downward propagating incident (I) wave and an upward propagating reflected
wave in the upper layer of buoyancy frequency N1 , while a transmitted ('T) wave propagates downward in the semi-infinite layer of buoyancy frequency N2 ; the sinusoidal
black and straight gray arrows denote the group velocity and wave vector orientations,
of
respectively. The stratification, shown on the right, comprises a transition layer
-L.
=
z
on
centered
width A between the upper and deep ocean stratification that is
35
The remaining class of variables proportional to ' are:
II-1
{p}
=
f'if
/w
exp [i (kx - wt)],
(2.4)
po (f2 _ W2) /(kuj)
PI
where the prime indicates differentiation with respect to z.
We consider an imposed traveling wave disturbance at z = 0 of the form:
(X, t) = To exp [i (kx - wt)] ,
(2.5)
where To is the complex amplitude of the disturbance, though To may be replaced
by an amplitude relevant to variables proportional to T' i.e.
'.
The objective
is to then determine the underlying wave field that is consistent with this imposed
disturbance. In so doing, we recognize that quantifying the downward transmission
of internal waves will depend on the particular physical variable under consideration,
with relations (2.3) and (2.4) revealing there to be two options. To that end, the
streamfunction 4, as well as the imposed condition at z = 0, can readily be related
to other physical variables of interest by utilizing the relations provided in (2.3) and
(2.4).
2.2
2.2.1
Sharp interface model
Analytical approach
In order to identify key phenomena and clarify the underlying physics, we begin by
considering internal waves in a two-layer fluid separated by a sharp interface, i.e. the
limit A -+ 0 of (2.1):
N (Z)
N1,
-L < Z < 0(26
7Ni2)<=
< - L.
X,
(2.6)
For this scenario, the vertical structure of the wave fields in the two constant stratification regions can be written:
{
Iexp (imIz) + R exp (-imiz),
-L < Z <0
Texp (imz),
z < --L,
,
where mi and m2 denote vertical wave numbers in regions of constant N1 and N 2
respectively, the coefficients I and R represent the amplitudes of the downward and
upward propagating components of the internal wave field in the upper layer, respectively, and T is the amplitude of the transmitted, downward propagating wave field
in the lower layer (see figure 2-1). We note that I, R and T are the net result of the
forcing and multiple reflections from the forcing level z
0 and the sharp interface
at z = -L.
At z = 0, the boundary condition (2.5) yields a single equation, I + R = 1. Fur-
36
thermore imposing kinematic matching conditions on w and u at the sharp interface
Z = -L, corresponding to continuity of vertical velocity and pressure, respectively, enables computation of the unknown wave amplitudes, from which parameters relating
the properties of the transmitted wave to the imposed upper-boundary disturbance
can be determined. For the perturbation variables 4', w, 6 and p, which we recall are
all linearly related, the transmission parameter is:
qj
0)
[cos (m1L) + i
--
mi7
1
sin
(RIL)
.
The transmission parameter for a, v and p, which are both linearly related to
similarly derived:
=-
",t,,v
=
'(Z
(
/
L) =
0)
(2.8)
)
TJizL)
cos (m1L) + i
in,
(M2
sin (m 1 L)
.
4',
are
(2.9)
The transmission parameters (2.8) and (2.9) depend only on the non-dimensional
quantities m2 /'mi and 'm1 L, the former being the ratio of the vertical wavelengths in
the two layers and the latter being the product of the thickness of the upper layer
to the vertical wavenumber in the upper layer. We recall that in each layer the
eigenvalues m are set by the parameters w, f, N, and k via equation (2.2).
Recalling that the vertical wavenumbers in and M 2 can either be real or imaginary, depending on whether or not w falls within the pass-band [f, N], we consider
the following two scenarios:
(i) m., mn
(ii) M2 < 0
>
0
< M2
:
:
Propagating waves in both layers (i.e. f < w < N1, N2 );
Propagating/evanescent waves in the upper/lower layer
(i.e. f < N2 <w < N1 ).
jI'm 2 I implies N1 5 N2
In light of our simplifying assumption that f < N1 , N2, mi
for scenario (i), since the vertical wave numbers are both real; this is not so for
scenario (ii), in which case one m is real and the other is imaginary (see (2.2)).
Figure 2-2 presents the magnitude of the transmission parameters F., and F4
for scenarios (i) and (ii) as a function of minifL and 1m2 /m1 I (the modulus of these
quantities are presented, given that M 2 is complex for scenario (ii)). For scenario (i),
addressed in figures 2-2(a) and (b), the system exhibits resonance peaks or diminution
troughs for Im1iL = (p - 0.5)7r, where p is an integer. Whether such a value of ImilL
corresponds to a peak or trough depends on two factors: the ratio Im 2 /r1ni and
whether the physical variable under consideration is linearly related to ' or V". When
I'm 2 /'m1j < 1, for example, a peak for a variable related to 4' (i.e. w) corresponds
to a trough for a variable related to 0' (i.e. u), and vice versa. From either relation
= lm2 /m1i
at a transmission peak
(2.8) or (2.9), it is readily found that IFJr
1mi
=
IT21 by virtue of there
and there is unity transmission for the trivial case of
l
being no change in stratification between the two layers. Curiously, there is also unity
transmission for all variables provided Im 1 IL = p-r. We note that qualitatively similar
transmission spectra have been reported for freely propagating waves in nonuniform
37
0
0
-L
z
zo
0.01 0.1 [1
(C)
0.01
2
itI
101
0
10
-
0
-
1
-2-
(d)
(b)
100
101 100
I1
10
5
0.1
S0
0I
j
01
01
01.01
-2
0
0.57r
27r
ImilL
0
0.57r
7r
1.57r
27r
ImilL
Figure 2-2: Logarithmic magnitude of F4 and F, for scenario (i) in (a, b), and seenario (ii) in (c, d) as a function of ImilL for 0.01 < 1m2 /niI < 100 (denoted by the
numbers located beside the black to gray lines), with the dashed line corresponding
to lm2/Mil= 1. The sketches above the figure illustrate the corresponding propagating/propagating (i.e. scenario (i)) and propagating/evanescent scenario (i.e. scenario
(ii)) of each column.
38
stratifications (Nault & Sutherland 2007, Mathur & Peacock 2009), although the
maximum transmission factor for such studies has been unity as there was no total
forcing of the wave field via a boundary. However, in our studies of boundary forced
waves substantial amplification is possible.
For scenario (ii), propagating waves exist only in the upper layer while in the
lower layer the wave field is evanescent. The spectra in figures 2-2(c) and 2-2(d)
have infinite resonant peaks between troughs, and the values of |m.1|L at which these
peaks occur are a smooth function of 1rm 2 /m 1j. For lm 2 /mil -+ 0, corresponding
to very weak vertical decay in the lower layer compared to the vertical propagation
wavelength in the upper layer (i.e. waves are almost propagating in the lower layer),
the resonance peaks and diminution troughs expectedly align with their counterparts
shown in scenario (i). As Im 2 /miI -+ oo, corresponding to an increasingly rapid
vertical decay in the lower layer compared to the vertical propagation wavelength in
the upper layer, the resonance peaks and troughs shift to larger Inn1 L values, up to a
maximum shift of ir/2. Depending on whether the physical variable being considered
is related to 0 or ,', the values of the troughs systematically decrease or increase as
a function of jm2 /m1 1, respectively.
2.2.2
Physical interpretation of transmission spectra
The features of the transmission spectra contained in figure 2-2 can be physically
interpreted by considering the reflection properties of internal waves at the sharp
interface and upper boundary (Tolstoy 1973, Nappo 2013).
Scenario (i)
As illustrated in figure 2-3(a), we begin by considering the fate of an internal wave
emitted by a local disturbance of strength To, which is linearly related to vertical
velocity, at some arbitrary location along the upper boundary, taken to be the origin.
The resulting downward propagating wave is incident on the sharp interface, resulting
1 =
in reflected and transmitted waves of amplitude RM' = rI)
and TM
I) + RM,
respectively, satisfying continuity of velocity across the interface; the parameter
in
-
M2
=
rM(2.10)
i1 + 'i 2
is the sharp-interface, plane-wave reflection coefficient for i) and the superscripts
(e.g. R(1 )) denote the number of reflection events experienced by the wave. For
'miI < I'm 2 1 (i.e. N1 < N2 ), r is a negative quantity, implying that in addition to
any magnitude adjustment upon reflection there is a ir change of phase; conversely,
the phase remains unchanged when mil > 1rm 21 (i.e. Ni > N 2 ). Subsequently,
the reflected wave propagates upwards until it encounters the upper boundary at a
horizontal location d = 21m1 IL/k away from the point of origin. At this point, the
resultant downward propagating wave is a combination of the contribution from local
upper-boundary forcing, 'Io exp (ikd) = To exp (2ilmn IL), and the total reflection of
RM. Within the context of this linear model the reflection is taking place on what
39
(a)
Toe 4i Im I Ib
L
_ _d
I
0
9
<I
~>
//
/7
24~\
L
t
(b)
I
To
T(
im L
1
.pj4.})
)
2
1,(2.)
T(1) = (I + r) To
-
I (r4i1 I, 2i I mrrlI+ 2) T1
Iq
Iq
kt
K;
K
>4
X 4
x
>4
>4
>4
/
''5
(1
(I +I Irl) TIo
< +Iri)
- (1+2
Irl +
1r-
2
) Ijo
(1 +21rl +21r1
2
+ 1r.)
0m
Figure 2-3: (a) Schematic of the ray paths within a finite-depth N, layer for general
values of 11.2/1i1 and i 1 |L. The gray arrows indicate the incident and reflected
rays (within the N1 layer) and the transmitted rays (entering the N2 layer); the
black arrows at the upper-boundary represent the forcing for points at which internal
wave rays encounter the upper boundary. Streamfunction amplitudes T(,) associated
with the n"' transmitted ray are presented as a function of jm 1 IL, the plane-wave
(b) Ray
reflection coefficient r, and the initial transmitted wave amplitude T(.
paths corresponding to a resonant case for the vertical velocity component, where
ImIL = (p - 0.5) r and ImI < 1mI1. The black arrows in the upper (N 1 ) layer
indicate the instantaneous orientation of the velocity component associated with internal waves that are either directly excited by the upper boundary forcing or result
from reflections; the black arrows in the lower (N2 ) layer indicate the instantaneous
velocity component of the resulting transmitted wave.
40
is effectively a rigid lid at z = 0; it therefore experiences another 7r phase-change
for 'w (and therefore for 4'), while no phase-change for u (and thus 4'). Thereafter,
the resulting wave propagates down to the sharp interface where the processes just
described are repeated over many consecutive reflections. As the number of reflections
within the N, layer increases, successive waves are characterized by a superposition
of amplitudes from increasingly many reflections.
The defining factor in determining the amplitude of a variable associated with
a transmitted wave is the degree of coherence between the interacting waves within
the N1 layer. Depending on the particular values of Irn.1 L and jm 2 /mi1, the incident
and reflected waves within the N1 layer can experience constructive or destructive
interference, or any degree of coherence therein. As an example, the configuration of
constructive interference for the vertical velocity component (or indeed any variable
linearly related to 4'), resulting in finite peaks of amplitude Im 2 /'m 1- in figure 2-2(a),
is illustrated schematically in figure 2-3(b): this requires |m121 < ImlI, in which case
reflection at the sharp interface conserves the sign of 4', while reflection at z = 0 always
inverts it. Therefore, if a disturbance imposed at the upper boundary is out of phase
with the disturbance initiated to the left of it, which is the case for ImlIL = (p -. 5)7r,
then subsequently generated waves are in phase with the previous waves reflected up
from the sharp interface and back down from the upper boundary. Consequently,
after a number of reflections, all the waves reaching a given point present the same
sign for their vertical velocity component, resulting in a constructive interference.
The converse is true for the horizontal velocity component, for which incident and
reflected waves have opposite signs, resulting in a very weak transmitted horizontal
velocity component, corresponding to the troughs of amplitude rm2 /m1 Iobserved in
figure 2-2(b) at 'm L = (p - 0.5)7r.
Conveniently, the ray approach also enables a series representation of the transmission parameters as a function of the number of prior reflections:
T("i
S(1+
(-r)(_-) exp [2(j - 1)ilmIL] .
r)
(2.11)
j=l
As n -+ 00, the series converges to the analytical expression for F, as defined in
(2.8); the convergence of the series and analytical solution is validated for both resonant and diminution cases in figures 2-4(a) and 2-4(b), respectively. We note that
0(100) number of reflections is necessary before convergence between p) and F4 , is
observed to within ~ 10%, but after only 0(10) reflections the vertical velocity component of the transmitted wave field is magnified by a factor of around 25 in figure
2-4(a). This latter observation is insightful as it reveals that this ray approach makes
it possible to estimate transmission parameters for forcing comprising only a finite
number of wavelengths or occurring only for a finite time interval, in which case only
a finite number of reflection events can occur. This may prove useful for applying the
model to real-world scenarios for which forcing may have a dominant frequency and
wavenunuber but is inevitably spatiotemporally restricted.
41
100
501
0
(b)
0.02
0
-
-
50
100
-
200
U
-
0.1
300
Figure 2-4: The transmission parameter r(' (solid black line) as a function of the
number of reflections n for lmilL = 0.57r and (a) 1m2/m1= 0.01 (resonance) and (b)
Im2 /m 1 = 100 (diminution). The analytical expression for F.,', which corresponds
to I:, , is plotted as a dashed, gray line for each case. As the number of reflections
n increases, the series expression (2.11) converges to the analytical expression (2.8).
In (a) the convergence is monotonic, whereas in (b) the converging solution oscillates
rapidly around the final value, resulting in a seemingly solid black area.
42
Scenario (ii)
For this scenario, the N2 layer sustains evanescent waves. The fundamental difference
with scenario (i) is that because M 2 is complex the plane wave reflection coefficient
r, defined in (2.10), can be rewritten as:
r = exp (i4>),
<P = 2 arctan
,
(2.12)
which is of unity magnitude and has a phase <D that is dependent on Im2 /m1i. We
note that <D can have any value in the range from zero (i.e. Im2 /n1I -+ 0) to 7r (i.e.
Im2 /mlI -+ oc). The phase difference between a wave generated at x = d = 21niIL/k
0 (see figure 2-3(a)) is
and the upper-boundary reflection of a wave generated at x
therefore:
(2.13)
,
AO = 2Jm1IL from which it is evident that the phase 4P is crucial in determining the degree of
coherence within the N, layer. As Im2 /rnmI increases so does <b, and this is responsible
for the resonant peaks shifting to higher values of milmL for larger values of IM 2 /m1l.
Finally, a consequence of Irl being unity is that the amplitude of the resonance peaks
is infinite rather than finite; the underlying physical reason is the evanescent nature
of the N2 layer that prevents any downward leakage of energy.
2.3
Finite-width interface model
Moving towards more realistic scenarios, we now consider the impact of a finitewidth transition region between the upper and lower stratification layers (i.e. A # 0
in (2.1)), which requires a numerical solution. Assuming a constant stratification in
the deep ocean, the upper and deep-ocean boundary conditions are:
T = 'Do,
Z= 0,(21.
00.(2.14)
In order to make the numerical computation practical, we apply the lower boundary
condition at z = -L -A/2 rather than at the unattainable z = -oc. As demonstrated
in Appendix A, for a wavenumber k this is valid provided the WKB condition:
m12 >>
)
(>2
1/3
(2.15)
-L; if it is not, the lower boundary condition can simply be applied
is satisfied at z
at a deeper location such that the criterion is satisfied for all k values considered.
Equation (1.15) is then solved for stratification (2.1) and boundary conditions (2.14)
using the Matlab function bVp4c, with the definition of the transmission parameter
43
Imil A
ImIJA 0. 1,2
=10
+
oZ
-1
I-mIJA0
Im IA =0,12
2-
__0
-2L
-Z+
-4-61
5
27r
3.57r
37r
2.5?r
47r
4.5-x
m1IL
Figure 2-5: Logarithmic magnitude of >, for (a) scenario (i), and (b) scenario (ii) as
a function of m 1 L for Ir1 2 /T1 = 0.1 (black lines) and IM2/mi = 10 (gray lines).
The transmission spectra were computed for mi A values of 0, 1, 2, and 10, which
are labelled for each curve, as well as being denoted by increasing line thicknesses.
The sketches to the right illustrate the corresponding scenario of each plot. Note that
the abscissa begins at a point that is half of max(IinlIA) in order for A/2 < L. The
spectra corresponding to
1m 2 /rnl =
10,
ImilA
= 10 in figure
(b) has been omitted
for clarity.
modified to account for the presence of the transition layer:
V,
=
+ (z = -L - A/2)
~
r
T (ZI = 0)
(2.16)
The inclusion of the transition layer introduces an additional non-dimensional parameter, mIA, relating the thickness of the transition layer to the vertical wavenumber
within the N layer, which augments the number of parameters that characterize the
, m1A).
sharp interface system (i.e. mL, 11/m9
Figure 2-5 presents the effect of the parameter ImilIA on F , for scenarios (i) and
(ii); the values used are in1 A = 0, 1, 2, and 10. For both scenarios, the numerically
obtained spectra in the limit ImIA = 0 are consistent with the sharp interface results
presented in figure 2-2. As Irm I1A increases there is a systematic detuning of the
44
increasing A
0=0
102
100
10-2
m2/mI
Figure 2-6: The phase adjustment 6# introduced by a finite-width transition layer as
a function of 1rn2 /rM1n, demonstrating that 6# < 0 (> 0) for I.m 2 /mil < 1 (> 1).
system. For IM 2 /rniI < 1 and I'm2 /M1I > 1 the spectral peaks shift towards smaller
and larger ImiIL values, respectively. For scenario (i), the peaks and troughs decrease
in amplitude and become less distinct with increasing ImiA, whereas for scenario (ii)
the peaks remain sharp as ImilA increases. We also note that for Im 2/mi| > 1 the
diminution troughs in scenario (ii) experience a dramatic decline in magnitude as
Im 1 IA grows.
The horizontal shifts of the peaks of the spectra presented in figure 2-5(a) are
explained by a phase shift of reflected waves introduced by the finite-with transition
layer. While a plane wave reflected from a sharp interface experiences a phase shift
> 1, respectively, the presence of a finiteof 0 or 7 for lm 2 /7n1i < 1 or 1"1 2 /m
width transition layer introduces an additional phase shift 6#, which we calculated
numerically using an established approach(Mathur & Peacock 2009). It is plotted as
a function of Ir1. 2 /m1 in figure 2-6. This induced phase adjustment is negative for
I'm 2 /mi I < 1 and positive for 1m 2 /m I > 1, with the magnitude of 60 increasing with
A. Rewriting the phase difference expressed in (2.13) to make explicit the effect of
the finite-width transition layer, one obtains:
(21,iL + 1601 - 7r,
=
1"m2/m1|I
+A
(2.17)
2|m1|jL - 1601,
1"12/71-1>1
1.
From (2.17), we see that for a given phase difference A# between surface forcing
and a ray reflected back up to the surface, depending on whether 1m2 /m 1 I < 1 or
> 1, the quantity im IL must be smaller or larger, respectively, for the finite-1I
ln2/m
width interface compared to the sharp interface (for which 60 = 0). Thus, compared
to sharp interface results, the transition spectra for a finite-width interface in figure
2-5(a) shift to the left for |m2/m1I < 1 and to the right for In.2 /m1| > 1.
45
2.4
Geophysical application
The results of this fundamental study may have some relevance to geophysical scenarios. For example, in many locations the ocean stratification can be roughly approximated as a mixed layer sitting atop a two layer stratification, with a strongly
stratified upper layer (the pycnocline) and a weakly stratified deep ocean (Pollard
1970, Fructus & Grue 2004). Various processes occur in the mixed layer to excite internal waves in the ocean below by virtue of driving vertical motion of the base of the
mixed layer. Relevant to our model are reports of relatively high-frequency internal
waves with a distinct horizontal wavelength and forcing frequency excited by travel-
ing Langmuir cells (Thorpe 2004) and Kelvin-Helmholtz billows (Pham et al. 2009).
The scenario of low-frequency near-inertial wave excitation by mixed-layer motions
is a very important one, but in this case the forcing is due to localized storms and
thus more impulsive in nature (Gill 1984); our analysis could be used to investigate
the response of the ocean to a Fourier component of such forcing.
As a proof-of-concept, we consider a stratification from the World Ocean Database
2013; this single cast profile was obtained on 24th April 2012 at location 140'W, 0 0 N.
At this latitude, background rotation is effectively negligible, so we thus set f = 0.
Figure 2-7(a) presents the stratification, wherein a 47 in-deep surface mixed-layer
sits atop an L =101 m thick transition region containing a pycnocline with peak
buoyancy frequency N,,, ~ 0.041 s-, below which lies the weakly stratified deep
ocean of buoyancy frequency N2
0.003 s1 In this region of the equatorial Pacific,
internal waves have been observed, associated with boundary layer eddies traveling
within the surface mixed-layer (Wijesekera & Dillon 1991). More specifically, the
velocity difference between the moving eddies and the stratification beneath generates
shear, and undulations at the base of the surface mixed-layer arising from this shear
consequently excite internal waves with observed horizontal wavelengths spanning
0(102) - 0(10,) m.
A pertinent piece of information to which we have direct access via our theory
is the energy flux transmission parameter, which quantifies the susceptibility of the
deep ocean to internal wave energy injected via mixed-layer forcing. To demonstrate
that the approach can be used in this regard, we begin by expressing the energy flux
(E, S.) transmitted to the deep ocean as:
E
=
(pWt 2 _f2) W
' +v'2),
4kw
Po (w 2 _ .f2
(pw)t,
=
_
)
EPO
(2.18)
(e"4',- TI
),
(2.19)
where K) reflects spatio-temporal averaging over the wavelength and period, as indicated by the subscripts, and I(z = -148 in) = Tl.+il' (note that for two-dimensional
forcing in the z-plane the spatio-temporally averaged energy flux E, is zero, since
p and v are out of phase). To present the results, we normalize the energy flux by a
46
(a)
0
-50
(b)
100
150
--
/N2-
=
0.5
-
/N., =
0.9
101
300
300-
105000
10
10 1
10
kL
0.02
0.04
N (s-')
Figure 2-7: (a) A stratification profile of the equatorial Pacific Ocean (140'W, 0 0 N),
obtained from the World Ocean Database 2013. The measured profile is denoted by
the solid line, and a slightly-smoothed stratification profile incorporating a constant,
mean deep ocean stratification is denoted by the dashed line. The surface mixed-layer
spans the upper 47 in, and the pycnocline is located between the gray dashed lines,
at which the boundary conditions are applied. The weakly stratified deep ocean has
mean buoyancy frequency N2 a 0.003 s-1. (b) Magnitude of the normalized energy
flux E* transmission spectra for the stratification shown in (a) as a function of kL
for two different values of w/N.
characteristic scale based on the forcing amplitude, i.e.
E* =
pOJN-2kT
,S
(2.20)
where To is the imposed stream function amplitude at z = -47 m.
The results in figure 2-7(b) show the normalized energy flux spectrum E*(k, w)j
at z = - 148 in for w/N 2 =0.5 and 0.9. While the studies in the preceding sections
presented results in terms of m.* 1 L and m 2 /m.i, it is now practical to consider the results
in terms of kL and w/N given the direct application to the ocean. For w/N2=0.5 and
0.9 we find that transmitted energy flux to the deep ocean exhibits a clear peak value
0.4, respectively, corresponding to horizontal wavelengths
for kL ~ 0.2 and kL
on the order of 1000 n. There is also an array of smaller peaks for kL ~ 1 - 5,
corresponding to horizontal wavelengths of around 100 in. In regards to energy flux
transmission to the deep ocean, therefore, our approach reveals that this stratification
is selective to forcing at the discerned combinations of wavelength and frequency.
4i7
2.5
Conclusion
We have studied the internal wave response of semi-infinite, non-uniform stratifications to harmonic forcing input at a horizontal level. To provide insight and facilitate understanding, the transmission characteristics of an idealized, two-layer, sharpinterface system were first investigated and rationalized. The key factor was found to
be the degree of constructive or destructive interference between multiple reflections
in the upper stratification layer. This study also helps understanding of qualitatively
similar transmission and reflection peaks in related previous studies (Nault & Sutherland 2007, Mathur & Peacock 2009). In cases of constructive interference, even weak
forcing can excite a strong system response, although whether or not resonance occurs is also dependent on which physical variable is being considered: for example,
a resonance peak for the vertical velocity component corresponds to a diminution
trough for the horizontal velocity component.
The introduction of a finite-width transition region between the upper and lower
stratification layer preserves the transmission features of the sharp-interface system
provided that the waves have long vertical wavelengths compared to the width of
the transition region. When the width of the transition region is similar in scale to
the vertical wavelength of the waves, however, the system becomes detuned, with
the resonance peaks shifting and diminishing in amplitude. As an example of a
potential geophysical application, this model was used to study the response of a
sample ocean stratification representative of the equatorial Pacific to forcing at the
base of the mixed layer. It was established that in the absence of shear resonant
conditions exist for the physically important quantity of energy flux, revealing that
a given ocean stratification is tuned to selectively transmit internal wave energy at
particular combinations of horizontal wavelength and frequency of forcing.
48
-------------------------------------,,,
"I'll,
I'll1__.-.,- -11',
1
-1 -
-
- -
I -.
kiWWW"WWW" _ -1-1
Chapter 3
Nonlinear internal wave
penetration via parametric
subharmonic instability
&
Of substantial interest is how internal waves excited in the upper ocean may penetrate
into the deep ocean, where they may potentially drive deep-ocean mixing (Munk
Wunsch 1998). A variety of upper ocean processes contribute to the internal wave
energy budget (Langmuir cells (Thorpe 1975, Polton et al. 2008) are a prominent
example), and it has yet to be well understood how the associated energy is distributed
throughout the water column. In this chapter, we describe the results of a laboratory
experiment in which internal waves are excited in a relatively shallow, upper layer of
high buoyancy frequency, N1 , sitting atop a deeper layer of low buoyancy frequency,
N2 . The internal wave field is excited at a relatively high frequency wo (i.e. N2 < w0 <
N1 ) via boundary forcing imposed at the top of the upper layer; as such, the wave field
excited at the forcing frequency is confined to the upper layer because it is evanescent
in the lower layer. By observing the evolving wave dynamics over sufficiently long
time scales, we study the development and consequences of instabilities in this setting.
We begin in 3.1 by providing an overview of the problem and instability mechanism; in 4.2 we detail the experimental set-up used to create and measure the
internal wavefield and ensuing instability. The experimental results are presented in
3.3, and the radiation of energy is analyzed in 3.4. A summary of the chapter is
provided in 3.5.
3.1
Background
There have been many studies regarding the propagation of linear internal wave fields
through non-uniform density stratifications, as characterized by the vertical variation
of the local buoyancy frequency N(z). A standard procedure that provides substantial
insight is to calculate the transmission properties through piecewise linear density gradients by matching boundary conditions at interfaces (Sutherland & Yewchuk 2004).
A notable advance on such methods was the development of an analytical approach
49
-
-
'__-
_
(I wouldn't consider it 'analytical.' What do you think?) to tackle propagation
through stratifications without requiring any assumptions regarding the relative scale
of the waves and variations of the background stratification (Nault & Sutherland
2007). This approach was subsequently extended to address the propagation of internal wave beams through nonuniform stratifications by Mathur & Peacock (2009), the
predictions being validated by close agreement with laboratory experiments. All the
aforementioned examples being linear models, however, the propagating wave field
does not experience wave-wave interactions.
It is well established that nonlinearities can give rise to wave-wave interactions
and resonant growth via the so called parametric subharmonic instability (PSI) (Mied
1976, Drazin 1977, Koudella & Staquet 2006). This process takes internal wave energy
from a primary, 'mother' wave of frequency wo and wavenumber ko, and distributes it
among a pair of 'daughter' waves of frequencies w, and w 2 , and wavenumbers k, and
k 2 ; the emergency of PSI requires the satisfaction of a temporal resonance condition,
wO,
(3.1)
k 1 + k) = ko.
(3.2)
W1 + W2
as well as a spatial resonance condition,
&
The subharmonic daughter waves birthed by PSI are characterized by smaller vertical
length scales, and thus higher vertical shear, and slower group velocities. In light
of this, field studies (MacKinnon et al. 2013) and numerical simulations (Gayen
Sarkar 2013) have sought to investigate PSI driven processes to explore its potential
to drive mixing processes in the occan. Complementary laboratory studies in uniform
stratifications have demonstrated conditions under which internal wave fields become
unstable to minute perturbations via PSI (Joubaud et al. 2012, Bourget et al. 2013).
3.2
Experimental set-up
A schematic of the experimental configuration, incorporating a plot of the measured experimental stratification, is presented in figure 5-5. A 1.6 m x 0.4 in x
0.17 m acrylic tank was filled with salt-stratified water of depth 0.35 in using the
double bucket method. The density stratification, measured using a conductivitytemperature probe, comprised a roughly 8 cm thick upper layer of peak buoyancy
frequency N1
1.25 s- 1 and an approximately 25 cm deep lower layer of buoyancy
frequency N2
0.93 s-'. The non-uniform stratification was established by first filling the top and subsequently the bottom stratified layer, adjusting the bucket density
ratios in between these two filling processes.
A horizontal wave generator was mounted atop the tank in such a manner that
the vertical motion of the array of plates within the generator forced the upper few
millimetres of the stratification, thereby generating downward, left-to-right propagating internal waves. The wave generator spanned 32 cm and contained four horizontal
wavelengths of A = 7.7 cm in addition to several buffer plates that smoothly tapered
50
0.32 m
00.5
1
1.5
LOM
1.6 m
Figure 3-1: Schematic of the experimental configuration. A sample particle image
is overlaid to show the field of view captured by the CCD camera. The measured
buoyancy frequency profile N(z) is overlaid to the right.
from the amplitude of forcing to zero at either end of the generator. A traveling wave
was excited by the wave generator, imposing a vertical velocity on the stratification
of the form w(x, z = 0, t) = Awo cos (kox - wot), where x is the horizontal coordinate,
t is time, the amplitude A = 2.5 mm, the frequency wo ~~1.03 s-' and the horizontal
wavenumber ko = 2YT/A - 81.6 m- 1 . By choosing the forcing frequency of the wave
generator such that N2 < wo < N 1, the upper layer acted as a waveguide supporting
propagating internal waves, with evanescent decay taking place in the lower N2 layer.
O(10),
The characteristic Reynolds number for the experiment was Re = Awo/vko
to be
viscosity
expect
thus
we
s-1;
m2
x
10-6
1
v
is
where the kinematic viscosity
present and have a reasonable, but not dominant, role in the experiments.
Particle image velocimetry (PIV) was used to obtain velocity field data. Prior
to filling, the water was seeded with hollow glass spheres of mean diameter 10 pim,
which were subsequently illuminated by an approximately 2 mm thick laser sheet. The
evolving motion of the internal wave field was recorded by a CCD camera positioned
normal to, and 1.5 n away from, the front face of the tank. The measurement field
of view covered a 60 cm x 32 cm (2400 pixels x 1300 pixels) area below the wave
generator (as illustrated in figure 5-5). The nominally two-dimensional velocity fields
were visualized in the mid-plane of the tank, so as to minimize sidewall effects. Images
were recorded at a frame rate corresponding to 32 images per oscillatory period. These
images were analyzed using the LaVision DaVis 7.2 PIV processing software package
to obtain the experimental velocity fields.
t
10 To
20
40
x (cm)
t= 30T
t= 20T
t =15 T)
-10
-30'
0
60 0
20
40
x (cm)
20
60 0
40
(cm)
60 0
40
20
X (cm)
60
Figure 3-2: Snapshots of the vertical velocity wave field at successive times illustrating
the evolution of the wavefield due to the development of PSI in the upper Ni layer.
The velocity fields have been normalized by the characteristic velocity scale AwO,
and the rectangular boundary denotes the region used to perform the time-frequency
analysis presented in figure 3-3.
3.3
Results
Snapshots illustrating the temporal evolution of the experimental wave field are presented in figure 3-2. As demonstrated by the snapshot at time t = lOTO, where
TL = 27/wo, waves are initially restricted to the upper Ni layer and experience
evanescent decay below, consistent with linear expectations. At t = 15TO, the einergence of PSI becomes evident in the N, layer in the form of a modest wrinkling of
the primary wave field, these perturbations being the subharmonics superposed onto
the primary wave field. PSI is clearly established by t = 20TO, by which time the
downward propagating daughter waves penetrate the lower N2 layer at two different
propagation angles, one to the right and one to the left, thus revealing the presence of
two daughter waves oscillating at a frequency less than N2 . By t = 30TO, the daughter waves are well established throughout the lower N2 layer, with PSI continuing to
operate in the upper N1 layer.
To shed light on the frequency content of the wave field, we perform a timefrequency analysis (Flandrin 1999) by means of the following transform:
S., (w, t)
K
v (Tr)h (t -
T)
exp (iwT) dT 2)
(3.3)
where v represents an arbitrary velocity component, and h is a smoothing Hamming
window of unity energy. For a representative picture, we spatially average the spectrograms over the rectangular domain indicated in the rightmost panel of figure 3-2;
this region is chosen because it contains all wave components that arise throughout
the duration of the experiment. More specifically, spectrograms are computed for
both the u and w velocity components by time-averaging the last five periods of the
data set, by which time PSI had clearly been established.
Figure 3-3 presents the results of the analysis, with the spectra having prominent
0.73 si and w
1.03 s-1, as well as at w,1
peaks at the forcing frequency, w()
52
.
..
I
I
100
I
I
I
WO
00
~~W2
LUO
-2
+ W2
10-
10 41
0
1
1.5
11
0.5
1
1
2.5
2
W (s-')
Figure 3-3: Time-averaged spectrograms, S (w, t), for the
components normalized by the maximum quantity.
n (red) and w (blue) velocity
.
0.31 s-1, corresponding to the two daughter waves. These frequency peaks satisfy
the resonance condition wo 1 + W2 to within 1%, thus confirming the formation
of a resonant wave triad; the spatial resonance condition, which is a well established
experimental feature (Joubaud et al. 2012, Bourget et al. 2013), is also checked and
confirmed. The relatively high forcing frequency results in steeply oriented constant
phase lines for the forced wave field, which in turn yields strong and weak vertical
and horizontal velocities, respectively. Thus, in figure 3-3 the frequency results for
the w velocity field more prominently feature the steeper primary wave, whereas
the u velocity field more readily highlights the shallower-propagating, subharmonic
daughter waves. Notably, the u velocity spectra also has a strong signal at W = 0,
which corresponds to a mean flow likely generated via entrainment by the horizontal
phase velocity of the boundary forcing (Bourget et al. 2013). One can also identify
additional nonlinear interactions, albeit at significantly weaker amplitudes, between
the primary and daughter waves at frequencies w0 + w 1 and Wo + c 2
Using the Hilbert transform filtering technique (Mercier et al. 2008), we decompose
the experimental wave field into its different directionalities. Figure 3-4 presents the
six wave components that comprise the wave field once PSI has been established
(rightmost panel in figure 3-2, t = 30TO); for clarity, the u velocity has been used
to plot the constitutive waves. We see that the upper N1 layer contains the incident
primary wave and also a primary wave reflected from the relatively sharp N-to-N2
transition in the stratification, as demonstrated by the results in the left-most panels
of figure 3-4(a) and 3-4(b), respectively. Each of these wave fields at the primary
frequency o spawn a pair of subharmonic daughter waves via PSI. Generally, a
consequence of the spatial resonant triad condition is that the wr-daughter wave will
have a group velocity oriented in the same quadrant as that of the primary wave
(i.e. sgn (kj) = sgn (ko) and sgn (mi 1 ) = sgn (mno)) while the wc-daughter wave has
53
WI
W(
(a)
10
-20~
-30
(b)
7WOR .W
10.
-20
-30
0
4)
20
X (cm)
60 0
40
20
.r (cm)
60 0
40
20
60
X (Cm)
Figure 3-4: Wave components comprising the wave field in the rightmost panel of
figure 3-2. Panels in (a) and (b) correspond to the wave fields associated with the
incident and reflected primary wave in the NI layer. Panels in the left column are the
primary wave components oscillating at wo; the middle and right columns correspond.
to the subharinonic daughter waves oscillating at wi and w 2 , respectively. The orientation of the group velocity vectors are denoted by the black arrows in each plot. For
clarity, we plot the v velocity component normalized by the characteristic velocity
scale Awo.
54
-. 4--
-
its group velocity oriented in the opposite direction (i.e. sgn (k2 ) = -sgn (ko) and
sgn (in 2 ) = -sgn (mo)); as mentioned previously, we checked, and confirmed, that this
spatial resonance condition was satisfied via now-standard procedures (Joubaud et al.
2012). Commensurate with these requirements, the incident and reflected primary
waves yield downward propagating w, and w 2 subharmonics, respectively, and since
both subharmonic frequencies are smaller than N2 , in contrast to the primary wave,
they are capable of penetrating into the lower stratification layer.
3.4
Energetics
From our PIV velocity field measurements, we calculate the energy flux field for the
different frequency components using a recently established method by Lee et al.
(2014). First, the stream function field ')(x, z) is obtained from the two-dimensional
velocity field data via the relation (u, a)) =(-&2, Oxo) and path integration. The
stream function field, as well as the background density and wave frequency, arc
subsequently incorporated into the expression for energy flux:
2
w)w +W
u.
(3.4)
iwp4' ((N 2
+
With the energy flux field in hand, the vertical power radiation profiles (per unit
width) are calculated via:
P=
(J) -d,
(3.5)
where L C [0,60] cm is the horizontal domain of the experimental field of view,
and the brackets (-) denote time period averaging. We note that the primary wave
field is weakly nonlinear by virtue of PSI taking place, and that the aforementioned
definitions neglect viscosity. Given the small amplitudes of our wave fields and the
modest impact of viscosity in our experiments, however, it is reasonable to expect
that this approach gives suitable order-of-magnitude estimates.
Figure 3-5 presents the power transmission profiles computed by separately applying the energy flux calculations to each of the downward propagating wave components presented in figure 3-4. As expected, there is no power transmission of
relatively high frequency waves (i.e. w > N2 ) into the weakly stratified lower layer.
There is, however, notable power transmission for daughter waves at frequency w, and
some transmission at frequency w 2 . More quantitatively, we find that approximately
0(10%) of the power of the primary wave is transmitted in the form of the daughter
wave of frequency wi, made possible by PSI, and around 0(1%) is transmitted at frequency w 2 . The subsequent decay of this power with depth is attributable to viscous
effects, which affect the daughter waves somewhat more than the primary wave, due
to the smaller length scales of the former compared to the latter.
55
__ -A
iW
__--
I
I
I
I
I
I
I
I
I
I
I
-10I
S-15
-
-20
10~3
102
0
10
10-
0
N (s-
P (nW cmrn)
1
0.5
1
)
-251
10-4
Figure 3-5: (Left) power transmission profiles calculated by horizontally-integrating
the vertical energy flux field for the downward wave components shown in figure 3-4.
(Right) buoyancy frequency profile (solid line) and corresponding temporal frequencies (dashed lines).
3.5
Discussion and conclusion
In summary, our experiments provide the first demonstration of a novel process by
which energy injected at relatively high frequencies into the strongly-stratified upperlayer of a water column can penetrate into an otherwise forbidden, weakly-stratified
lower-layer lying beneath. The underlying mechanism is PSI, which extracts energy
from the primary wave field and distributes some of it into a pair of daughter wave
fields, both of which are not evanescent, but propagating, in the lower layer. For the
experimental configuration we used to demonstrate this scenario, we found this to
be an 0(10%) effect in regards to power transmission, although more studies would
be required to determine how this varies with the strength of the wave field and
other system parameters such as wo, ko, N1 and N2 . Building oi this discovery,
in future studies it would be of interest to investigate whether this scenario might
have relevance to the occan, which is also characterized by a strongly stratified upper
layer and a. weakly stratified abyss. Physical processes such as Langmuir circulation
(Thorpe 2004, Polton et al. 2008) and shear-driven instabilities (Pham et al. 2009),
for example, are known to excite relatively high-frequency internal waves in the upper
ocean, but relatively little is known about the magnitude and fate of the energy flux
associated with them.
56
Chapter 4
Experimental investigations of
internal wave resonance and
diminution
Harmonic surface forcing of a non-uniform stratification comprising a relatively thin,
highly stratified upper layer sitting atop a deep, weakly stratified lower layer was
shown in chapter 2 to produce a rich spectra of transmission features. The transmitted wave response is predicted to range from diminution to resonant growth depending
on the degree of coherence between the constitutive waves in the upper stratification
layer. As shown through ray tracing in an idealized sharp interface model, destructive
interference in the upper layer naturally yields diminution of the transmitted wave.
Conversely, constructive interference produces gradual amplification for successive
reflections. In this chapter, we perform a series of laboratory experiments to investigate the role of interference in tuning wave transmission. The goal is to validate the
transmission spectrum derived in chapter 2, as well as obtain experimental evidence
of internal wave resonant behavior.
We start by developing a weakly viscous, long wave model in 4.1 in order to
compute a theoretical transmission spectrum incorporating the limitations of the
experimental set-up. In 4.2 we detail the experimental set-up and measurement
techniques used to obtain internal wave fields. The resultant internal wave fields are
described and the transmission results are compared to theory in 4.3. A summary
of the work and concluding thoughts are offered in 4.4.
4.1
Theoretical formulation
We consider the two-dimensional scenario of internal waves propagating through a
nonuniform background density stratification. The mathematical formulation has
been previously detailed in chapter 2, though we restate them for simplicity. Small
amplitude internal waves in a Boussinesq fluid are known to satisfy:
Ott (V 2,4') + N(z)
2
57
".Xo
=
v= V 44',
(1.1)
where 1 is the streamfunction, N(z) is the local buoyancy frequency, and v is the
kinematic viscosity. Assuming harmonic, horizontally periodic solutions of the form
0 oc
) exp [i (kx - wt)] yields an equation governing the vertical structure of the
wave field:
io
T2Z22 +
im
- 2k2
N_)2
#22 + k 2 k + iW (.N
-
)
= 0.
(4.2)
The system model is identical to that presented in figure 2-1, in which a transition
region lies between two regions of constant stratification. In this chapter, however, we
incorporate the effects of viscosity in order to compare with the subsequent laboratory
experiments.
Within the upper and lower constant stratification regions the wave field has
solutions of the form T oc exp (iMiz), where:
Mi~ mi + cm' + 0(E2).
(4.3)
Here, E =vk2 /w is the viscous perturbation parameter (<1 in the long wavelength
limit), the index i = 1, 2 indicates the upper and lower constant stratification regions,
mi is the inviscid vertical wavenumber and mT' is the weakly viscous correction, given
by:
mi
k
(N/w)2 _1
(4.4)
m
(
ik (Ni1/w) 4
(4.51)
_=TF
2
(N /w) 2 - 1
For disturbances within the internal wave frequency pass band' (i.e. 0 < w < Ni),
positive (negative) values for mi denote downward (upward) energy propagation; positive (negative) values of n' denote upward (downward) decay. We note that for real
systems in which a constant stratification is never truly achieved, the assumption of
plane waves in any given region more formally requires the following WKB J condition
be satisfied (see appendix A for details):
m >>
k2NN'/
3
(4.6)
where m represents a local vertical wavenumber.
To find the vertical wave structure arising from periodic surface forcing of amplitude To, we first write the solution at the relevant boundaries as:
T = Iexp [i (mi
- mn-j) Z] + R exp [i (-n
'I =Texp [i (11 2 - m) z
1
+ m') z],
z = 0,
(4.7)
z= -L - A/2,
,
where mi anid m"' are obtained by respectively solving (4.4) and (4.5) for N(zi), and
'Here we neglect background rotation
f.
58
WAWNAAW
"---
""' """"
I, R and T are the incident, reflected the transmitted wave amplitudes, respectively. The treatment of the variable coefficient equation in (4.2) as a boundaryvalue problem requires the application of two boundary condition at both z = 0 and
z = -L - A/2. Using (4.7), we write the unknown coefficients I, R, and T as
functions of T and T, by taking the necessary number of derivatives. Thereafter, the
boundary conditions applied at each boundary are given by:
}
T,
=
0,
(4.8)
n) T
= i (m2-
m) 2
-2
z = -L - A/2.
(4.9)
The subsequent set of equations are numerically solved using the MATLAB bvp4c
function to obtain AP(z).
In chapter 2 we studied the transmission of an infinite plane wave forcing at the
upper boundary z = 0. In this chapter, however, we modify the formulation to account for periodic forcing of finite span in order to connect with practical limitations.
The appropriate expression for the boundary forcing at z = 0 is written:
40 (x, t) = To [H(x) -
H(x - L)] exp [i (kx - wt)] ,
(4.10)
where H(x) is the Heaviside step function, and the disturbance is taken to lie over
x E [0, C]. The corresponding Fourier transform of (4.10) is:
(K
1
exp [i (k - K) L]
1 (k -Kn)
(.1
where K is the wavenumber parameter. With T (z) already in hand, the final form of
wavefield is derived:
(4.12)
4.2
27r
S(X,
j
Experimental set-up
The experiments were performed in a glass tank of dimensions 5.46 x 0.55 x 0.6 in 3
which housed a salt-stratified fluid of depth 0.6 in. A non-uniform stratification of
the form presented in (2.1) is established by means of the standard double-bucket
technique (Oster & Yamamoto 1963). Figure 4-1 presents stratification profiles of
two separate experiments measured by a conductivity-temperature (CT) probe which
clearly depicts the sigmoid distribution. Two sets of experiments were performed,
with the second set aiming for more detail after the first set. The two corresponding
stratification profiles were effectively identical to each other. A tanh function is
fitted to the two experimental measurements, from which we find the thickness of
,
)=(K) 1(z) exp [i (Kx - wt)] dK.
59
0
-
to-
-20
-30
0.6
0.8
N (s-
1
1.2
)
0.4
Figure 4-1: Buoyancy frequency profiles from two separate experimental stratifications measured by a conductivity-temperature probe (black and grey lines) and the
corresponding fitted tanh profile (red line).
14 em, buoyancy frequency values of the upper and
the transition region to be Alower layers to be N, ~ 1.08 s- and N2 ~ 0.66 s-i respectively, and an upper layer
thickness of L ~ 14.3 cm.
A 1.63 n long wave generator (Gostiaux et al. 2007, Mercier et al. 2010) was positioned horizontally above the wave tank, as shown in figure 4-2. The wave generator
utilized for these experiments contained 84 individual plates, which were configured
such that their vertical motion simulated a harmonic, traveling plane wave described
by the function w (x, t) = Aw cos (kx - wt) (where w is the vertical velocity along a
horizontal plane, A = 3 nim is the wave amplitude, W is the forcing frequency, and
k is the horizontal wave number). Additionally, lighter material plates were used in
order to decrease the load and to maintain the rigidity of the structure.
By virtue of periodically forcing a vertical boundary, a downward, right to left
invariant into the
propagating incident wave was generated; the boundary forcing is
page, and thus the resultant wavefield is expectedly two-dimensional. Given the forc0 (10 - 100),
ing parameters, the corresponiding Reynolds number is Re = Aw/vk
2
where v ~ 1 x 10-6 m s-1 is the kinematic viscosity; as a result, we expect viscous
effects to be weak. In order to recreate the conditions necessary for resonance
and diminution, it was essential to minimize the damping effects of viscosity while
maximizing the number of reflections that take place over the finite span of the wave
generator. Given that m" ~ k' a large enough wave length was needed to overcome
viscous dissipation, whereas a smaller length scale increased the number of wave
lengths comprising the wave generator profile, and thus the number of reflections
that take place in the upper N, layer. As a result, the optimal wave length for the
60
p
traverse
probe
wave generator
(a)
pumps
z
stratification
1.63 m
ST
5.46 m
Figure 4-2: (a) panoraminc view of the experimental set-up during the filling process.
(b) sketch detailing the systemn dinensions. A sample particle image is overlapped to
depict the location and size of theimeasurement field of view.
61
wave generator was chosen to be A = 27/k a 22.6 cm.
Prior to filling, hollow glass spheres of mean diameter 8-12 ptm were dispersed
in the fluid. An approximately 2 mm thin light sheet created by a pulsed Nd:YAG
laser unit illuminated the particles in a vertical plane parallel to the tank front wall.
In order to minimize wall effects, the laser sheet bisected the width of the working
section. An Imager Pro X 4M LaVision CCD camera with a resolution of 2, 042 x 2, 042
pixels was placed 1 m in front of the tank to visualize the particles at a frame rate of
24 images per forcing period. The images were calibrated and cross-correlated using
the DaVis particle image velocimetry (PIV) software package to obtain velocity field
data. For each experimental run, 100 forcing periods worth of data was collected in
order to capture both the short and long time scale dynamics.
4.3
Results & discussion
4.3.1
Qualitative observations
Figure 4-3 presents snapshots of the wave field excited at frequencies that correspond
to resonant peaks and diminution troughs in the transmission spectra. As the forcing frequency increases wave transmission crosses two resonant peaks separated by
a diminution trough before approaching the evanescent regime. Figure 4-3(a) corresponds to the resonant peak occurring at the lowest frequency, which yields propagating waves throughout the vertical domain. For relatively small times scales, the
wave field is seemingly stable; for sufficiently long time scales (i.e. t = 50T) the wave
field experiences dramatic wave-wave interactions which introduce daughter waves
with smaller vertical length scales. Moreover, the development of PSI prevents the
unbounded amplification of the wave field as it experiences constructive interference
in the upper N, layer. PSI effectively extracts the growing energy of the forced wave
field and distributes it to the birthed daughter waves.
Excitation frequencies that yield destructive interference in the N1 layer naturally correspond to diminution in the transmission spectrum. Figure 4-3(b) presents
snapshots of the wave field at short and long time scales. Compared to the resonant
wave field (figure 4-3(a)), this particular wave field is notably weakened, and remains
weakened with time. Details of the diminution process will be provided in 4.3.2.
The wave field displayed in figure 4-3(c) coincides with the resonant peak occurring
at the higher forcing frequency. Similar to 4-3(a), the wave field clearly experiences
amplification while maintaining stability over short time scales. As time progresses,
however, the growing wave velocities induce PSI and the subsequent redistribution of
energy into additional wave frequencies.
As the forcing frequency approaches the N1 buoyancy frequency, transmission predictably drops. This is reflected in figure 4-3(d), where the wave amplitude decreases
with time, without the emergency of any instability mechanisms. In all, we observe
a dramatic landscape of transmission behavior for the buoyancy frequency profiles in
figure 4-1. Resonant behavior is clearly observed at the resonant frequencies of the
transmission spectra, which ultimately open the door to nonlinear wave-wave inter62
St=50T
t=--10 T
(a)
-40
(b)
-20
-40
K'
*
S. -20
(c)
(d)
-40
-20-
x (cm)
x (cm)
150
130
I
6
-
150
130
-
.5I0A
4
2
0. 4
II
0.8
0.6
Figure 4-3: Snapshots of the experimental wave
quencies indicated in the transmission spectrum;
cies corresponding to resonance, (b, d) wave fields
to diminution. The vertical velocities have been
locity scale Aw.
63
I
fields corresponding to the four fre(a, c) wave fields excited at frequenexcited at frequencies corresponding
normalized by the characterized ve-
7
6-
5-4-
-3
2-
1
0
0.4
0
00
0.5
0.6
0.7
0.8
0.9
1
W (s-)
Figure 4-4: Spectrum of the transmission parameter 1F14 , (black line) as a function
of excitation frequency w. The experimentally measured values are given by the red
data points. We note that the standard deviation of the measurements are at most
on the order of the marker size.
actions. Conversely, at the troughs of the transmission spectra we observe a notable
weakening of the forced wave field and long term stability.
4.3.2
Linear transmission
Figure 4-4 presents the transmission spectrum corresponding to the fitted buoyancy
frequency profile shown in figure 4-1. Two resonant peaks, one located at w ~ 0.65 s-'
and a notably thinner one at w ~ 0.89 s-,
sandwich a, diminution trough spanning
0.7 < w < 0.85 s-.
We perform a series of experiments spanning 0.55 < w < 0.93 s-1 in order to
validate the transmission parameter >, ; the red data points in figure 4-4 indicate
the experimental results. Depending on whether a particular frequency reflects resonance or diminution, two different schemes are applied to quantify transmission.
For resonant frequencies we select the peak amplitude within the linear regime (before the development of nonlinearitics) to be taken as the transmitted amplitude;
for diminution frequencies we present the steady-state amplitude as the transmitted
amplitude. Normalizing the measured transmitted velocities by the characteristic
forcing amplitude Aw yields the transmission parameter F,4. The data points presented in figure 4-4 are the mean and standard deviation of the data points spanning
(165 - 0.25A, 165) cm. There is generally very low variability in the experimental
measurements by virtue of the standard deviation being smaller than the marker size
of the plotted data point.
There is excellent agreement between the experimental measurements and the
theoretical transmission spectrum. The wave fields excited at the resonant frequencies
are naturally constrained with respect to how much amplification may occur. Two
64
WIT
'P ..... "
_ , , "
1 11
-Vm-
linear
(a)
nonlinear
4
-
-2
-41
(b)
0
10
20
40
30
50
60
70
t/T
Figure 4-5: Transmission time histories collected at (x, z) = (161, -19.7) cm for (a)
1
resonance (w = 0.875 s ), and (b) diminution (w = 0.74 s-1). The shaded intervals
represent the time it takes for a wave ray initiated at the surface to travel downwards,
reflect upwards, and to make contact with the surface.
primary factors account for this: the finite span of the applied boundary forcing and
viscous damping. Both of these factors impose limitations on the energy of the system,
to
which in turn translates to limitations in wave growth. As a result, it is impossible
the
observe true resonance in reality. Nevertheless, our measurements clearly indicate
existence of resonance peaks in the transmission spectrum by virtue of the dramatic
wave amplification in the vicinity of these resonance peaks. Lastly, an additional
noteworthy deviation occurs around the vicinity of the second resonant peak located
at w ~~0.89 s-1. Interestingly, the experiments indicate a wider transmission peak
compared to the narrow theoretical peak.
Further insight into the role of interference is gained by looking at the time history of the transmitted wave signals. Figure 4-5 presents examples of time series
a gradual
undergoing resonance and diminution. In figure 4-5(a) we clearly observe
enlargement of the wave amplitude over the first 20 forcing periods; this particular
65
span of time reflects the linear regime. Once the critical amplitude is reached at
t ~ 22T, however, PSI starts to take form and one clearly observes period tripling
arising from the nonlinearities. Furthermore, the gradual amplification observed in
the time series validates the idealized sharp interface model outlined in 2.2.2 in
which we demonstrate how constructive interference yields amplitude growth with
increasing number of wave reflections (see figure 2-4(a)).
Figure 4-5(b) presents a time series reflecting diminution. The steady-state amplitude of the wave signal is a fraction of the surface forced amplitude, thus confirming
the occurrence of destructive interference. What is particularly noteworthy, however,
is the manner in which steady-state is achieved. Just as demonstrated in figure 24(b), we experimentally observe how alternating wave reflections within the upper N,
layer oscillate in wave amplitude towards a final diminutive value. The finite span of
the wave generator used in our experimental set-up naturally limits the number of
reflections that can possibly take place. For the particular forcing frequency of this
time series (w = 0.74 s-'), the maximum number of wave reflections that can occur
over the seven horizontal wavelengths of the wave generator is seven. In figure 4-5(b)
we observe that the transmitted wave amplitude oscillates over eight reflections; had
the wave generator extended beyond seven horizontal wave lengths, the number of
oscillations in the transmitted wave amplitude would presumably increase as well,
thus leading to an even smaller steady-state amplitude.
4.3.3
Nonlinear consequences of resonance
The emergence of resonance in forced systems naturally breaks the linear assumptions
used to model their physics. In our study, we observe the emergency of nonlinearities
for wave fields excited at resonant, or near-resonant frequencies, as demonstrated in
figure 4-5(a). Qualitatively, we are able to identify this nonlinear mechanism as PSI by
virtue of the presence of subharmonics in the wave fields generated at the resonant
frequencies in figures 4-3(a, c). For confirmation, we compute the spectrogram of
a resonant wave field forced at w = 0.875 s-1 by applying the transform in (3.3)
and spatially averaging over the upper N1 layer. The representative time-frequency
spectrum is shown in figure 4-6(a); in computing the spectrogram, we choose to work
with the horizontal velocity field in order to augment the subharmonic signals. From
the spectrograim, we clearly identify the emergence of two subharmonic daughter
waves around t ~ 22T, as well as two higher harmonics. The temporal resonant
condition w
w 1+ w 2 is satisfied to within 2%, thus confirming the development of
PSI. By performing this analysis, we find that each experiment in which resonance
takes place invariably experiences the development of PSI.
4.4
Conclusion
In this chapter we experimentally validate the theoretically findings of chapter 2.
A series of experiments were performed spanning a large portion of a transmission
spectrum containing two resonant peaks and a diminution trough. We find there to
66
0
-1
-2
-3
log (S. (w, 0) /So)
(a)
(c)
2
W+ W
1 .5
LJ + W02
1
I
W2
0.5
0
1
0
tO
(b)
F+,
4
10
100
S, (w,t =40T)/So
0
-4
0
10
20
30
50
4()
60
70
80
90
t/T
Figure 4-6: (a) Spatially averaged spectrogram S, (w, t) of a resonant wave field
excited at w = 0.875 s-I normalized by So (the mean time-frequency spectrum for
the excitation frequency w). (b) Corresponding transmitted wave signal at (x, z)
(161, -19.7) cm normalized by Aw. (c) Frequency spectrum at t = 40T.
67
be excellent agreement between linear theory and experimental measurements. Both
resonance and diminution were observed to take place in the experiments. In the case
of resonance, the amplification of the transmitted wave eventually gave way to PSI,
which redistributed energy into two subharmonic daughter waves thereby preventing
further growth of the primary wave. As expected, wave fields that correspond to
diminution troughs experienced notable wave suppression and remained stable over
relatively long time scales. In all, we confirm that the interference taking place in an
internal wave guide can dramatically alter the transmission of wave energy into the
deep lower layer. Depending on the particular length and time scales of the primary
forced wave, as well as the number of wave reflections taking place in the upper
layer, the deep ocean response to periodic boundary forcing can potentially be quite
dramatic and consequential.
68
Chapter 5
The impact of multiple layering on
internal wave transmission
Wave propagation through layered media is a classical problem of particular relevance
to numerous fields of study, from geophysics to manufacturing to electromagnetism.
A standard starting point for addressing such problems is the simplification of both
the incident wave as well as the media in which it propagates. As such, the simplest
waveform is the plane harmonic wave, and the layered media may be idealized as
an assemblage of discrete layers each of uniform property. From here, solutions are
obtained by carefully applying matching and continuity conditions at each interface
and solving the system of algebraic equations for the desired unknown wave amplitudes. Subsequently, more complex problems may be approached by means of Fourier
integrals, WKBJ techniques, ray tracing, and normal modes (for more details, see the
canonical text by Brekhovskikh (1980)).
The objective of this chapter is to consider the propagation of internal waves
through continuously varying stratifications with multiple layers, a scenario relevant
to the Arctic Ocean. We start in 5.1 by highlighting the geophysical relevance of
this work. Then, in 5.2 we outline our analytical approach, which extends beyond
the WKBJ approximation and accounts for viscosity so as to enable comparison with
laboratory experiments. In 5.3 we use a simple model to demonstrate that the
presence of multiple layers has a nontrivial impact. To validate both the model and
this fundamental insight, we perform a comparison between theory and laboratory
experiments in 5.4. Thereafter, we apply our model to an Arctic Ocean stratification
and assess the impact of a double-diffusive staircase in 5.5. Concluding remarks and
future directions are discussed in 5.6.
5.1
Motivation
An evolving scenario of interest is the increasingly influential role that internal waves
may play in the dynamics of the Arctic Ocean (Rainville et al. 2011). Historically,
the existence of protective ice cover has shielded the Arctic Ocean from atmospheric
storms, and as a result internal wave energy levels in the region have been much
69
weaker than in lower latitude oceans (Levine et al. 1987). The rapid loss of protective
ice cover during summer months, however, suggests that change may be afoot. Figure
5-1 presents field measurements from the Northern Chukchi Sea in the Arctic Ocean
(Rainville & Woodgate 2009). Mooring data corresponding to time periods in which
sea-ice cover is present do not indicate any notable velocity readings. During ice-free
time periods, however, strong currents are observed in the water column, especially
during strong storm events. Consequently, the presence of internal waves in this body
of water is linked to both surface disturbances and the absence of protective ice cover.
In considering the associated evolution of internal wave activity, a striking feature
of the region that must be accounted for is the existence of complex stratifications,
such as multiple pycnoclines and intricate double-diffusive staircase structures (Padman & Dillon 1988), these being a vertical series of mixed layers (i.e. effectively zero
stratification) separated by thin stratified layers; an example is presented in figure
5-2. Such layers are known to horizontally extend hundreds of kilometers (Timmermans et al. 2008). In light of the prominence of double-diffusive layering in the Arctic
water column, we seek to assess the dynamics of an internal wave incident upon a
stratification interposed by an array of mixed layers.
5.2
Mathematical formulation
We consider linear, two-dimensional internal waves propagating through a nonuniform
background density stratification, satisfying the equation (Sutherland 2010):
(N(z) 2 + t)
&,x
+
(f2 +
Ot)
4'2
vDtV4',
(5.1)
where v is the kinematic viscosity. Assuming harmonic, horizontally periodic solutions
of the form 4' Dc T1(z) exp i (kx - wt) yields the equation:
7
+ (i
'
-
zz + k2
2k2
k" + i
(N(=) 2
= 0.
w
(5.2)
We note that our model does not account for horizontal variations in the background
stratification by virtue of the ansatz comprising a periodic solution in x.
Figure 5-3 presents a sketch of the system used in our mathematical formulation,
in which a layered transition region lies between two regions of constant stratification.
Within the upper and lower constant stratification regions the wave field has solutions
of the form T rc exp (Mz), which upon substitution into (5.2) yields:
M
-
i (AT2 + B
exp (i tan 1 (B /AT) /2),
(5.3)
where
A:
B+ =
k 2 -F
,
2wv
2
2w
-I
(1 +
(tan 1 (,y) /2)
,
(5.4)
(I + 7214cos (tan-' (,y) /2))
(5.5)
70
-T2)t/4 sin
a)
..-4
..
*4
5
A S 0 N
D
J F M A M
J J
A S
O
N
D
J F M A M J J A
S
02 m
s'
:3s
-C
60
Figure 5-1: (a) Daily (grey) and 30-day running-mean (black) wind speed measurements at the mooring site in the Northern Chukchi Sea; red arrows indicate storms.
(b) Magnitude of the inertial currents as a function of depth and time, in in s-.
Image is reproduced from Rainville &. Woodgate (2009).
0
(C)
(b)
(a)
-175
-100
-200
-185
-300
-400
-500
-600
-195
1024 1026 1028
p (kg m-)
1027.62
1027.52
0
0.01
0.02
N (rad s-')
Figure 5-2: (a) A density stratification profile collected from the Canada Basin in the
Arctic Ocean (Rainville & Winsor 2008). (b) Magnified view of a section of the profile
m
in (a) revealing the double-diffusive layering structure; the red line indicates a 5
frequency
buoyancy
bin-averaged smoothed density profile. (c) The corresponding
profiles N(z).
'71
11
N(z)
Az
k1
01
1
3;Z
-
-
-
T
Figure 5-3: Sketch of the model system. A harmonic, plane internal wave of amplitude I and wavenumber ki = (k, muI) incident at angle 01 = tan-'(k/Imui) to the
horizontal, encountering a transition region with multiple layers between z = 1 and
z 2, and emerging with a transmitted amplitude T at angle 02= tan 1 (k/m2
I).
and
4wvk (
2
-
(5.6)
f2)
(P2 - f2)2'
Of the four eigenvalue solutions, two correspond to weakly-damped internal waves
and two correspond to highly-damped, oscillatory boundary layers. We neglect the
boundary layer solutions, and retain only the weakly damped solutions, M (A-, B+)I
for which k < oW/v. As a result, the vertical wavenumber has the expected inviscid
component:
(11
2
1/2
(5.7)
(W-2 - f2)
where the subscript i = 1, 2 indicates the upper arid lower constant stratification
regions, and viscous decay is captured by m'
R e (Mi (A-, B+)). Note that for
propagating disturbances (i.e. f < w < Ni), positive (negative) values for m' and m'
denote downward (upward) energy propagation and decay, respectively. For systems
in which a constant stratification is never truly achieved, the assumption of plane
mi = i Ilm (Mi (A-, B+)) =
k
waves in any given region more formally requires the following WKBJ condition be
satisfied:
rnl>>
(W2
_ f2
.
(5.8)
We look to compute internal wave transmission through a nonuniform stratification N(z) with upper and lower boundaries at z = Z1 and 22, respectively, containing
multiple layers in between. To do so, we impose a downward propagating wave at 71
and solve for the transmitted wave amplitude at z2. The general solutions at these
72
I-
71
1-
TWWMMMWqW"'-."
,
-1-
1,
,
".1-
.1-
11-1.11111111"-
two boundaries are written:
S= Iexp [(in + m') z + Kcxp [- (mi
4'
+ n) z],
Texp [(n 2 + m) z],Z
z =5.9,
= Z2,
where I is a prescribed incident wave amplitude, and R and T are respectively the
reflected and transmitted wave amplitudes. The boundary conditions in (5.9) can be
applied at vertical locations at which (5.8) is satisfied, and there is no requirement
that the wave field be WKBJ compliant in between these two boundaries. In order to
evaluate (5.2) as a boundary value problem, we write the unknown coefficients R and
T as functions of T so that two conditions are applied at each boundary, namely T2
and 422. The subsequent set of equations are numerically solved using the MATLAB
bvp4c function to obtain R and T.
Having obtained a solution 4(z), this can be used to determine the values of all
key field variables via the standard linear relations:
w
{
p
U
v }=
P
ik
T}
exp [i (kx -wt)],
-k/w
(5.10)
-pOkN2/(Wg)
z
po (f 2
-1
if /w
_ W 2 ) / (kw)
exp [i (kx - wt)],
(5.11)
where (u, v, w) is the velocity field, 6 is the isopycnal displacement, p is the density
perturbation field, and p is the pressure field.
5.3
Impact of layering
In the inviscid, sharp-interface limit one can derive analytical expressions for T and
R (Sutherland & Yewchuk 2004). To demonstrate the nontrivial impact of multiple
layers, we briefly consider two straightforward scenarios, illustrated in figure 5-4: a
single mixed layer of depth L (identical to Sutherland & Yewchuk (2004)) and a
pair of mixed layers of depth L/2 separated by a stratified layer of width A = L/2.
The two systems have the same vertical extent lacking stratification, in which waves
are evanescent, but the second system has the additional feature of an intervening
stratified layer.
In figures 5-4(a) and 5-4(b) we present the transmission coefficient (ITI/|I|) for
the two scenarios as functions of the incident propagation angle 01 = tan- (k/Iml 1)
and the evanescent decay exponent mOL, where
ml0
(5.12)
i
is the evanescent decay scale calculated using equation (5.7) with N = 0. As shown
73
0.5
0
5
0.5
0
1
0.6
0
1
1.2
N
N
(a)
4
3L
2
N
KD
f)N
.(d)
3
L/2
z2
00
4z L 12
60
30
01
()
90
0
60
30
01
(0)
90 0
60
30
90
01(0)
Figure 5-4: Normalized transmission coefficients for I for sharp interface systems
comprising (a) a single mixed layer of thickness L and (b) two mixed layers of thickL/2. The black circle
ness L/2 separated by a stratification layer of thickness A
6
45'. Panels (c) and (d)
in panels (a) and (b) corresponds to moL ~ 3.5 and 1
present the corresponding theoretical results for the experimental profiles considered
in 5.4, incorporating viscosity; panels (e) and (f) present the corresponding theo1,12 + Iw| 2 . The sketches to the right
retical transmission for total velocity lul =
of plots (a, b) and (c) - (f) illustrate the corresponding stratifications. The dashed
5.4.
regions in plots (e) and (f) denote the experimental parameter space studied in
74
in figure 5-4(a), an internal wave experiences strong transmission for mixed layers
that are small compared to the evanescent decay length (i.e. moL < 1). Because of
interference patterns within the mixed layer, however, there is a clear enhancement
in tunneling for internal waves incident around 45'. The splitting of the mixed layer
into two equal parts by a stratified layer adds significant new features, as shown in
figure 5-4(b). There is an intricate sequence of transmission ridges, corresponding to
transmission for much larger values of moL than for the single layer case. For example,
the scenario rnOL ~ 3.5 and 01 = 45', for which the single layer case has effectively
zero transmission, has almost complete transmission for the two layer system.
Varying the value of A alters the specific details of the results but the qualitative
appearance of a sequence of transmission ridges is persistent, and in the limit of A -* 0
the results in figure 5-4(b) collapse to those in figure 5-4(a). Extending the study to a
stratification comprising three layers and/or layers of greater or lesser strength than
the upper and lower bounding stratifications produces qualitatively similar results,
with transmission ridges. The existence of these transmission ridges can be attributed
to the interplay between the interference patterns of evanescent waves in the mixed
layers (Sutherland & Yewchuk 2004) and propagating waves within the stratified
layers (Ghaemsaidi et al. 2015).
There is huge scope for variation of parameters, as one could also consider the
impact of mixed layers and intermediate stratification layers of varying thickness, of
varying intermediate stratification, and of having any number of layers; it would be
easy to become overwhelmed. As such, there is one key message to take away from
this motivating example: the impact that multiple layering has on the transmission
properties of a stratification can be complex and depend sensitively on the details of
the layering structure.
5.4
5.4.1
Experimental studies
Apparatus & Methods
As illustrated in figure 5-5, experiments were performed in 0.54 m deep salt-stratified
water housed in a glass wave tank 5.46 in long and 0.55 m wide. A partition running
along the length of the tank created a working section of width 0.2 m, and parabolic
reflection barriers positioned at both ends of the tank reflected waves generated in the
working section to the rear 0.35 in-wide dissipation section (Echeverri et al. 2009).
Filling of the wave tank was via the traditional double bucket method and two different stratifications were investigated: a single mixed layer spanning -7 cm, and
two mixed layers each of depth -3.5 cm separated by a ~7 cm deep stratified layer.
To create the mixed layers we temporarily paused the double bucket filling in order
to introduce constant density fluid into the wave tank. The density stratifications
were measured using a calibrated PME probe attached to a Parker linear traverse
for position control; the buoyancy frequency and density profiles are shown in figures
5-4(e, f) and 5-5(a, b), respectively.
To create an internal wave field with a dominant characteristic wavenumber, a
75
A
(b)
(a)
k
7.5cm
0
5.46 m
50 0
(kg n--3
50
)
Ap
Figure 5-5: Schematic of the experimental arrangement, and the two measured density
stratifications, Ap = p - 1000, for the (a) single, and (b) double mixed layer studies.
A cylinder, positioned 2 cm below the free surface, was used to generate a downward,
left to right propagating internal wave beam that was visualized by means of particle
image velocinetry (PIV); an actual raw PIV image is overlaid to show the location
and size of the field of view.
76
...
__
-
Ml "' II-
Ii " I "'116LI1161 "'I -
Ii I W
cylinder of diameter 7.5 cm that spanned the 0.2 m width of the working section
of the tank was used to generate a downward propagating internal wave beam of
characteristic wave number ~ 45 m 1 . The cylinder was oscillated vertically with an
amplitude of A = 7+0.07 mm over a range of excitation frequencies using a steppermotor operated Parker linear traverse. Each excitation frequency was separated by
a sufficient amount of time to allow the background stratification to return to its
base state, which was persistent throughout the experiment. The frequency range
was limited since both the incident and transmitted waves needed to be contained
within the ~ 50 x 50 cm 2 experimental field of view; lower frequency wave beams
with relatively shallow propagation angles would have required larger fields of view,
while higher frequency wave beams are too steep to reliably differentiate between the
incident and reflected wave components. As a result of the parabolic end-walls, no
unwanted reflections from the tank boundaries were observed.
Hollow glass spheres of 8-12 pin diameter and 0.1-1.5 specific gravity were used to
seed the stratified fluid. A light sheet created by a pulsed Nd:YAG laser illuminated
the particles in the mid-plane of the working section parallel to the front wall of the
tank. An Imager Pro X 4M LaVision CCD camera with a resolution of 2,042 x 2,042
pixels, positioned 1.2 m in front of the tank was used to visualize the particles at a
frame rate of 16 images per forcing period. The images were calibrated and crosscorrelated using the DaVis particle image velocimetry (PIV) software package to
obtain velocity field data. Since no obvious nonlinear features were observed, the
velocity fields were Fourier filtered at the forcing frequency in order to suppress noise.
For comparison with theory, incident across-beam profiles were collected as close
to the generation source as possible, and the Fourier spectrum was calculated via
d6(k)
J
w(qj) exp
(.k)
d7-,
(5.13)
where n = x sin 0+z cos 0 is the across-beam coordinate. This provided the input data
needed for our theoretical model to predict the transmitted and reflected wave fields.
For convenience, we make use of another form of the internal wave dispersion relation
presented in (5.7), w/N = sin 0, where 0 is the angle between the group velocity vector
and the horizontal. In order to truncate the integral limits in (5.13), we confirmed
that the across-beam profile of the incident wave beam was reasonably contained
within the field of view. With 61(k) in hand, we computed the vertical structure of
the wavefield i(z; k) for each value of k, and constructed the final solution using
linear superposition (Mathur & Peacock 2009):
{f
w(x, z, t) =& t(z; k) exp(ikx - iwt) dk
For these experiments,
f =
.
(5.14)
0 and v = 1 x 106 m 2 s-.
77
5.4.2
Results
Figures 5-6(a) and 5-6(d) present the experimental stratification profiles whose transmission properties are presented as a function of 01 and moL in figures 5-4(c, e) and
5-4(d, f), respectively. The stratification for the first set of experiments comprised a
single mixed layer of thickness 7.5
0.1 cm (figure 5-6(a)). We note that the lower
1
stratification (N 2 = 1.24 rad s ) is slightly stronger than the upper stratification
(N = 0.95 rad s- 1 ) and this is readily accounted for in our model. In the second
set of experiments, the upper and lower stratifications were the same as in figure
5-6(a), and the transition region comprised upper and lower mixed layers of thickness
3.1 0.1 cm and 3.6 0.1 cm, separated by a 7.3
0.1 cm stratified layer. Notably,
our calculations for the experimental stratifications (figures 5-4(c - f)) retain the key
features of the idealized results (figures 5-4(a, b)).
The comparisons of the experimental and theoretical vertical velocity fields ill figure 5-6 show very good agreement; indeed, the differences are almost indiscernible.
For the single layer experiment (figures 5-6(a - c)), we observe an incident wave beam
largely reflecting from the mixed layer; a coherent transmitted wave beam is observed
below the mixed layer, albeit with a greatly diminished amplitude. For the experiment with two mixed layers, although the total vertical extent of evanescent behavior
(~6.7 cm) matches that of the individual mixed layer experiment within 10%, we see
a notable increase in the amplitude of the transmitted wave beam (figures 5-6(d- f)).
When comparing the experimental and theoretical wave fields for this scenario, sonic
additional features appear in the upper right hand corner of the experimental results
(figure 5-6(e)) compared to the theoretical results (figure 5-6(f)). These are due
to reflections of the experimental wave field by the free surface; a process that, for
simplicity, was not incorporated into the model.
For a more quantitative comparison of the experimental and theoretical results, in
figure 5-7 we present experimental and theoretical Fourier spectra of the total velocity
.
U
xL = /u12 +lw! for the incident and transmitted wave beams as functions of mr1
Fourier spectra of the transmitted across-beam profiles for single and double mixed
layers are shown in figures 5-7(b) and 5-7(c) for experiments, and the corresponding
theoretical predictions are presented in figures 5-7(c) and 5-7(f). Notably different
transmission spectra are observed for the two stratifications, with the transmitted
amplitudes in the two layer case being substantially greater than the single layer
case.
To connect the experiments to the discussion in 5.3, the total velocity transmission spectra presented in figures 5-4(e, f) are reimapped from mo to rn1 , and then
multiplied by the experimentally measured total velocity incident spectra (figure 57(a)) to obtain the theoretical transmission spectra. The results, presented in figures
5-7(h, i), compare very well with those in figures 5-7(b, c), and 5-7(e, f); the two-layer
system has stronger transmission and the results for the single layer stratification similarly show a transmission ridge leaning to the left, in contrast to rightward-leaning
transmission ridge for the double-layer stratification. Notably, the ridge in figure 57(i) is sharper and stronger than those in figures 5-7(c) and 5-7(f ); this is because the
experimental field of view was limited in horizontal extent, preventing us from resolv-
78
--
,
-
-
I-
"IMPPM lo
,
1.'r11"M1"P1qr""RM"T1""
014
I
(a)
(b)
(c)
-01
-0.2
U?,,/ (Aw)
.1-0.3
I
0.4
-0.4
I
-0.5
d)
I
I
I
(M
(e)T
-0.1
-0.2
-0.4
n-0.3
-0.4
-0.5
0 0.5 1 1.5
N (s')
0
0.1
0.2
r (i)
0.3
0.4
0.5 0
0.1
0.3
0.2
0.4
0.5
S(m1)
Figure 5-6: Results for (a) - (c) single and (d) - (f) double mixed layers; snapshots
of the experimental (middle column) and theoretical (right column) vertical velocity
wavefields are presented at an arbitrary phase. The wavefields have been normalized
by the characteristic vertical velocity amplitude Aw.
79
ing the Fourier transform of the complete transmitted wavefield. The results in figure
5-7(i) nevertheless confirm that the effect we observe is indeed due to a transmission
ridge that arises from multiple layering, as delineated by the dashed boundaries in
figures 5-4(e, f), and not due to effects such as the sensitivity of transmission to small
changes in the stratification (Gregory & Sutherland 2010).
5.5
Geophysical application
We now look to apply the model to study the impact of multiple layering for a
geophysical scenario. As an example, we consider the transmission characteristics
of the density stratification presented in figure 5-2, which was collected from the
Arctic Ocean during the 2005 Beringia expedition (Rainville & Winsor 2008). This
continuous density profile is characterized by thin stratified layers separated by mixed
layers produced by double-diffusive instabilities (figure 5-2(b)). The corresponding
buoyancy frequency profile (figure 5-2(c)) consists of fine peaks of large N separated
by 0(1) in layers of N ~ 0. For this illustrative example, we assume that the
buoyancy frequency is constant above -170 m and below -200 m, and matches the N
values at the respective boundaries, thus satisfying (5.8). This staircase stratification
differs from the idealized profiles considered in 5.3 and 5.4 in that the stratified
layers are thinner than the mixed layers and are of notably higher N than the mean
background stratification. As we shall see, however, the existence of transmission
peaks and troughs is again a standout feature of the results.
To assess the impact of double-diffusive layering, we compare the transmission
properties of the staircase structure to those of a bin-averaged profile, indicated in
figures 5-2(b, c) by the black and red lines, respectively. Using the inviscid version
of the model developed in 5.2, we compute T (w; A.) as a function of frequency for
prescribed vertical wavelengths. Over the Canada Basin, the inertial period ranges
from Tf = 12.15 to 12.45 h, yielding f ~~1.42 x 10-4 rad s-, which sets the lower
bound on wave frequency. Characteristic vertical wavelengths for near-inertial waves
in this region are in the range 10-50 in (Cole et al. 2014). Via the linear relations
(5.10) and (5.11) we are free to calculate the transmission for a wide range of physical
variables, from which we choose to work with the time-averaged total energy flux,
E = (pu).
We first consider the staircase stratification profile, for which the results of the
energy flux transmission ratio, 1ST1/IEl, are presented in figure 5-8(a). For an incident vertical wavelength of A. = 10 in, we see ~ 60% suppression of energy flux for
near-inertial waves (i.e. low frequency disturbances near f). There are two transmission peaks, at w ~ 0.004 rad s- and w ~ 0.0055 rad s-i; for w > 0.006 rad s-1 there
is no energy flux transmission as the internal waves are evanescent below 200 m. The
results are quite similar for internal waves of vertical wave length 20 m, with ~ 40%
suppression of near-inertial wave energy flux, but only a single transmission peak in
the mnid-frequency range. Detuning of the transinission spectra is observed for larger
A., which is characterized by a monotonic decline in transmission between low and
high frequency waves.
80
0.8
0.4
04
0. 5
0.2
0.2
0
0
0.7
(f)
0.6
0
0.8
0.4
0.7
0.6
0. 5
0.2
0.2
0
04
0
0.4
1
0.2
0.5
01
0
0.8
~'
0.7
(h)
I
I
>1
0.5
I,,'
0.6
I
0
I
1
50
100
iI ( I 'I)
150
1
100
50
IM1I (.11-1)
150
1
50
100
150
ImiI (in')
Figure 5-7: Experimental total velocity Fourier spectra of the (a) incident wavefield
and transmitted wavefields for the (b) single and (c) double mixed layer stratifications. The total velocity Fourier component has been normalized by the maximum
incident value. Panels (d) - (f) present the corresponding theoretical wavefields. The
transmission spectra of the wavefields for a complete field of view are presented in
panels (h) and (i) for the single and double mixed layer stratifications, respectively.
Panel (a) is reproduced in panels (d) and (g), as the experimental incident spectrum
is the basis for all three sets of results. The dashed boundary corresponds to that in
figures 5-4(e, f).
81
0.5
(a)
1
=1.05f
W
(b)
/- 11EII vs.A
0.8
10
0.6
-
......
.\ 0.4
1
50
100
0.8
0.6
..
0.4
-
0.2
-0.2
00
f
1
2
3
W
5
4
(s )
6
7
f
8
X103
1
2
3
4
W (s I)
5
6
7
8
x10
3
Figure 5-8: Transmitted energy flux (period averaged in t) normalized by the incident
energy flux, ISrI/IErI, as a function of excitation frequency W, for the (a) staircase
and (b) 5 in bin-averaged stratifications. Incident vertical wave lengths are A, = 10 in
(-), 20 m (- -), 30 in (- - ), and 50 in (- - - ). The inset shown in panel (a) presents
the near-inertial wave (w = 1.05f) energy transmission against vertical wavelength
AZ*
Near-inertial wave transmission is strongly dependent on the degree of wave interference in the layered stratification, with particular vertical wave lengths experiencing
strong transmission (as denoted by the inset in figure 5-8(a)). The effect of wave interference diminishes once A' substantially exceeds the length scale of the layers in
the stratification, with transmission ultimately approaching unity for sufficiently large
values of A_. The dynamic regime of the Arctic Ocean system, however, is such that
the interference phenomena demonstrated in the previous sections will prominently
impact near-inertial wave energy transmission. For comparison, the transmission
properties of the bin-averaged stratification (shown in figure 5-2(c)) are presented in
figure 5-8(b). For 10 < A- < 50 i, all temporal frequencies with f < w < N have
complete transmission; the energy flux sharply dropping to zero upon transition into
the evanescent regime, as expected.
We note that our analysis does not consider interfacial waves, which are known to
significantly influence transmission properties of internal waves across a single mixed
layer (Sutherland & Yewchuk 2004). For this study, however, if one considers the
relative importance of wave and hydrostatic pressures (Drazin & Reid 1981), the role
of interfacial waves is found to be insignificant provided:
gk (A/p/ p
gk(App) 2<i.
(5.15)
0 (10-') and k 0
0
0 (102 - 101) m1 for
For the Arctic layering scenario, Ap/p
high frequency internal waves (and two orders of magnitude smaller for near-inertial
waves), arid so this condition is clearly satisfied and the role of interfacial waves can
be ignored.
82
MIR NP
1-1
The details of the transmission spectra are sensitive to the particulars of the density stratification; for example reducing or increasing the flanking N values or altering
the strength, size, spacing and number of intermediate layers can reduce or increase
the transmission to the deep ocean. But the key message remains unchanged, namely:
we find two sets of strikingly different results for the actual and smoothed versions of
an Arctic Ocean density stratification. The presence of a staircase structure within
the water column yields a rich landscape of energy transmission that is dependent
on the time and length scale of the incident wave. Smoothing such stratifications by
means of bin-averaging eliminates the distinct features of the Arctic water column and
in turn provides an erroneous prediction of energy transmission to the deep ocean.
5.6
Conclusion
We have performed a fundamental study of the impact of multiple layers on the
propagation of internal waves in a density stratification. Starting with a simple stratification, it was demonstrated that multiple layers can have counter-intuitive effects
due to the interplay between evanescent and propagating waves in neighboring layers. A set of laboratory experiments was performed and the experimental results
were compared with the predictions of a weakly viscous model, the excellent agreement between the two served to validate the model and its interesting predictions.
As a demonstration of the broad applicability of the model, which could be used
for ocean or atmosphere inspired studies, we investigated a scenario relevant to the
Arctic Ocean, finding that the presence of a double-diffusive staircase structure has
a non-trivial effect on wave transmission. The degree of wave interference within
the multiple layers, which is dependent on the time and length scale of the internal
waves, is a prime determinant. of transmission. These results apply to any layered
system, and it is not necessary that the origin of the layering be a double-diffusion
mechanism. Future research directions of interest would be to ascertain whether the
presence of layering can promote or suppress the onset of instability and associated
internal wave driven mixing, and to incorporate the impact of vertical shear.
83
84
Chapter 6
Conclusion
This thesis investigates the effect of interference and resonance on the the transmission of internal gravity waves through non-uniform density stratifications. A joint
theoretical and experimental approach is adopted in order to model wave transmission through two geophysically relevant stratifications: one being an idealization of
the upper ocean pycnocline, and the other being double-diffusive layers present in the
Arctic water column.
In chapter 2 we presented a theoretical model of boundary forced internal waves
in a two layer stratification encompassing a thin, highly stratified upper layer sitting
atop a deep, weakly stratified lower layer. We began by computing the wave transmission for an idealized two layer stratification separated by a sharp interface, from
which we found the transmission spectrum to span resonant peaks and diminution
troughs. The application of ray theory lead to a physical interpretation of the analytical results, which showed wave coherence within the upper layer to be the defining
factor in tuning the degree of transmission. Essentially, constructive interference of
the constitutive waves in the upper layer led to resonance, whereas destructive interference produced diminution. We subsequently extended the model to account
for finite-width transition regions separating the upper and lower layers, where we
discovered a general detuning of the transmission features as the transition width increased. Nonetheless, we found the key features of the transmission spectrum, namely
resonance and diminution, to persist beyond simple models.
Having detailed the linear transmission features of a two-layer stratification in
chapter 2, we performed an experimental study of boundary forced internal waves in
chapter 3, where we found the development of nonlinearitics to play a profound role
in the radiation of energy into the deep ocean. Specifically, we studied the dynamics
of high frequency internal waves, which render the lower stratified layer evanescent
according to linear theory. On short time scales, the internal wave energy is solely
contained within the upper layer waveguide. However, the rise of resonant wave-wave
interactions (i.e. parametric subharmnonic instability) over long time scales invariably
gave rise to two subharmonic daughter waves capable of penetrating into the lower
evanescent layer. By quantifying the energy flux of the wave field, we found that
approximately 10% of the primary wave energy was transmitted into the lower layer
through this nonlinear mechanism. We identified parametric subharmonic instability
85
as an effective mechanism in the redistribution of internal wave energy in geophysically
relevant stratifications.
Chapter 4 details the laboratory experiments that served as confirmation of the
theoretical transmission predictions of chapter 2. Both resonance and diminution of
transmitted waves were observed through experiments using an internal wave generator positioned above a two-layer buoyancy frequency profile. By scanning a broad
range of excitation frequencies we were able to cover the relevant portion of the transmission spectrum. Time histories of the transmitted waves clearly showed resonant
growth as well as the fascinating process of diminution, with both confirming their
respective physical processes delineated by the application of ray theory in chapter 2.
Notably, we observed the dramatic destabilization of the wave field due to wave-wave
interactions in the upper layer once a critical amplitude was reached during resonant
growth. Overall, excellent agreement was established between the linear experimental
results and the theoretical predictions.
In chapter 5 we studied the transmission of an internal wave beam through an
array of double-diffusive mixed layers spanning the water column. Beginning with an
idealized model, we confirmed that internal wave transmission through a stratification containing a single mixed layer undergoes significant decay. The introduction of
a stratified layer into the mixed layer, however, dramatically altered the transmission
spectra by giving rise to unity transmission tongues, thus making total transmission
possible by virtue of constructive interference taking place in the intervening stratified layer. Experiments confirmed the enhancement of transmission through a double
mixed layer compared to a single mixed layer by means of constructive interference;
excellent quantitative was obtained between experiment and a viscous theoretical
model that was developed. After the validation of our theoretical model, we subsequently studied an Arctic density stratification characterized by a double-diffusive
staircase containing an array of mixed layers. In comparing the Arctic stratification to a bin-averaged version, we found the Arctic profile to be characterized by
a rich transmission landscape, whereas the bin-averaged profile predicted a simple,
erroneous transmission spectrum.
The essential idea of this thesis is the significant role interference can play in tuning wave transmission through non-uniform media. Constructive interference leads
to resonant growth in the case of boundary forcing, and total transmission through
layered media in the case of an incident wave bean. In the former scenario, resonance
naturally leads to a breakdown of linearity and gives rise to nonlinear wave-wave interactions which in turn introduce subharmonic waves characterized by smaller vertical
length scales, higher shear, and slower group velocities. In the latter scenario, enhanced transmission allows incident wave energy to penetrate deeper into the ocean,
thereby introducing a potential source of deep ocean mixing. The aforementioned
features are ripe for the onset of energy dissipation and mixing, with significant geophysical implications as it involves the non-trivial distribution of energy in the ocean
introduced by surface forcing mechanisms. Identification of the potential energetic
consequences of internal waves generated in the upper ocean by near-surface processes
can contribute to improved climate models, global energy budget calculations, and
the planning of ocean field studies.
86
This thesis provides a mathematical framework for calculating the transmission of
boundary generated, linear internal gravity waves. A series of laboratory experiments
serve as validation for our theoretical model, and further highlight the importance
of interference in tuning the transmission of internal waves through geophysically
relevant density stratifications. Moreover, we identify the effectiveness of nonlinear
resonant wave-wave interactions (i.e. PSI) in distributing energy into weakly stratified
regions of the water column. Through theoretical analysis and laboratory experiments
we show that destructive interference of internal waves leads to diminution, whereas
constructive interference induces resonance over short time scales and destabilization
through PSI over long time scales. With regards to non-uniformities introduced by
double-diffusive layering within Arctic density stratifications, we perform a fundamental study demonstrating the profound role played by interference on enhancing
wave transmission through layered media. A joint theoretical and experimental analysis shows how the introduction of a stratified layer within a layer of uniform density
increases transmission by ~ 100%, which has dramatic implications for internal waves
in the Arctic Ocean.
87
88
Appendix A
WKBJ condition for plane wave
approximation of boundary
conditions
We begin by assuming a wave-like solution with an unknown phase function of the
form:
(A.1)
T ~ exp (i(z)).
Upon substituting this function into (1.15) one obtains the phase evolution equation:
"-
(0')2 + k 2
N
0.
- 2
)) W
(A.2)
A weakly varying stratification region implies weak variation in the local vertical wave
number m(z) = #'(z):
A.3)
(0, )2 .
4"f<
In this context, the dominant balance in (A.2) over Z E (-oo, -L - A/2) becomes:
(N
2(
1/2
!2
2 _
( W2
/2)
_
which approximates the wave number M 2 as in (2.2). The condition given by (A.3)
is then recast as:
m2>
k 2NN'
2_
f
1/3
,
(A.5)
which must be satisfied in order to truncate the boundary condition at z = -oc to
Z = -L - A/2.
89
90
"llop MIN lips',
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