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National Library of Canada Acquisitions and Bbliographic Services Bibliothèque nationale du Canada Acquisitions et services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 Ottawa ON K 1 A ON4 Canada Canada Yaur hie Vafre réterence Our fi& Nom reference The author has granted a nonexclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sel1 copies of ths thesis in microform, paper or electronic formats. The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or othenvise reproduced without the author's permission. L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la fome de rnicrofiche/film, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation. Abstract Rotary spectra of horizontal current velocities From moorings in the northeast Pacific are found to have a significant spectral peak at precisely the sum of the semidiunial and local inertial Bequencies. The existence of this spectral peak is, in the absence of any known forcing at the surn frequency, sufficient to assume some form of nonlinear interaction between intemal wave motions ai the inertial and sernidiumal fiequencies. The mooring locations studied in this thesis represent three distinct deep ( 2000 m) oceanic regimes: a topographically rough, a topographically smooth, and a near-coastal region. The amplitude of the spectral peak at the sum fiequency (termed the 7M2 frequrncy) is consistent among regions, implying that the nonlinear interaction is not directly linked to specific characteristics of the regions and may be ubiquitous to the world ocean. Although the measurements are limited, there is no significant variation in intensity of the spectral peak with depth. Weak resonant wave-wave interaction theory is the most readily applicable mode1 for the nonlinear interaction. Although the validity of the weakness assumption used in deriving the coupling eficiency is in question, the physicai mechanism of the coupling of two interna1 gravity waves via a resonant triad is well founded. nie resonance condition prescribes geometric constraints on the propagation and wavelengths of a triad of interacting intemal waves. The propagation and wavelengths of inertial and sernidiumal waves observed in the northeast Pacific are consistent with these constraints and the formation of resonant triads is likely. Anaiysis, based on the likclihood of the global formation of resonant triads between inertial and semidiml waves, indicates thar higher latitudes are favored. An examination of two deep-ocean sites dong Juan de Fuca Ridge shows that elevated energy is not usually coincident in the inertial, semidiuml andjhl? frequency bands. When the energy in a11 three bands is coincident, motions in thejm2 band can be: 1) nonresonantly forced requiring the support of the inertial and semidiurnal motions or, 2) a result of the nonlinear tram fer of energy from the inertial and semidiuml motions. When the energy is not coincident, the energy in thejbl? frequency band can only result fiom nonlinear exchange of energy from the inertial and semidiurnal motions, for which a plausible mechanism is triad resonance. Bispectral analysis indicates that there is nonlinear coupling between the inertial, semidiurnal andm2 frequency bands at the Endeavour segment site but not at the CoAxial segment site. This implies that the Endeavour segment site may be a generation region forjmz waves while the CoAxial site is in the path of propagating fm2 wave energy from a remote source region. Table of Contents.. Abstract... ri.. List of Tables... vri List of Figures... ix Acknowledgements xv Dedication... CHAPTER 1. inrroduction Oceanographic Context Structure of the Thesis... 3 CHAPTER 2. Oceanographic Observations xvii 2.1 Introduction Measurernents Study Regions Juan de Fuca Ridge Quiet Eddy Moonngs... 1 I Near-Coastal Region Power Spectra Motivation... f Rotary Spec tra Regionaiiy Averaged Rotary Spectra Method Application Pervasiveness ofjm2 interactions... 24 Table afc0ntent.s 2.3 Summary and Conclusions CHAPTER 3. Oceanic Nonlinear Interaction Mechanisms Introduction Background Weak Resonant Interaction Mechanisms Elastic Scattering Induced Diffiision Paramrtric Subharmonic Instability Validity Other Models Eikonal Method Direct Numerical Simulation Resonant Interaction Theory... II Discrete Interaction Transport Theory The Transport Equation Sununary and Conclusions CHAPTER 4. Grometrical Considentions and Coupling Efficiency Introduction Geometric Considerat ions Preliminary Local Variability Global Variability Coupling Eficiency Surnmary and Conclusions Introduction $2 Observations Harmonic Tidal Analysis Rotary Spectra Tidal Currents Inertial Currents jm2 and M4 Spectral Peaks Time Series Demodulation Introduction The Signal Mode Method Multiple Filter Technique and Wavelet Analysis Analysis of the Data Discussion CHAPTER 6. Bispectral Analysis Table of Contents 6.1 introduction Higher Order Statistics Moments and Cumulants Cumulant Spectra... IL7 6.3 The Bispectnim Estimation TheModel Method of Estimation Quaciratic Phase Coupling Biphase and Bispectrum Magnitude B icoherence Statistical Issues B icoherence-squared Biphase Data Length Stationarity and Transients Analysis of Data Goals Plots of Bispectra Magnitude B icoherence-squared P Io ts Biphase Discussion CHAPTER 7. Conclusions Nomenclature and Definitions..., Bibliography Appendix A: M2-fInteractions Appendix B: Future Work vii List of Tables Table 2.1. Current rneter deployments from Endeavour Ridge and CoAxial Ridge Table 2.2. Current meter deployments from station QEP Table 2.3. Current meter deployments from the Coastal Oceanic Dynamics Experiment Table 2.4. Spectral density and rotary coefficient at tidal and inertial frequencies Table 2.5. Variance and percentage of total variance ai tidal and inertial frequencies Table 4.1. Representative values from ERAZ and CX- 1. Observed frequenc ies are the peak frequrncies from rotary spectn with 50 degrees of Freedom Table 4.2. Propagating angle, cp, for inertial, semidiunial andjm2 frequency interna1 waves at three depths at stations CX-I and ERAZ Table 5.1. Selected tidal constituents used for the harmonic analysis of tidal currents of the time series from stations ERM and CX Table 5.2. Baroclinic tidal parameters fiom vertical mode decomposition for a m deep basin with a characteristic September stratification from a reg ion near station ERA2. Rossby radius is calculated by clj; where c is phase spced of the mode and f is the Coriolis parameter Table 5.3. Tibl current velocities and rotary components for the four principal semidiurnal tidal constituents, derived using the global inverse tidal model, TPX0.2, of Egbert et al. [1994] List of Tables viii Table 6.1. Data length and resolution as a function of DFT length and sampling fiequrncy Table 6.2. Biphase of bispectral estimates nearest B( f, M,) of horizontal currents frorn ERA Table 9.1. Center frequencies of observed peaks and predicted values of inertial, tidal and derived bands...,...., List of Figures Figure 2.1. Locations of cunent meter moorings used in this thesis... 6 Figure 2.2. Current meter mooring stations ai Endeavour Ridge. The red rectangular area represents the vent field. The map boundaries correspond to the rectangular region in Figure Figure 2.3. Regionally averaged rotary spectra from three distinct oceanographic regions from the northeast Pacific Figure 2.4. Spectral drnsity at the jmz band normalized to the spectral density at the inertial and semidiurnal bands versus depth (denoted by color) and current meter for the Endeavour Ridge region and the Quiet Eddy moo~g region (QEP). Values are displayed as percentages Figure 2.5. Spectral density of & band normalized to the spectral density of the semidiumal (M2) band versus depth (denoted by color) and current meter for the Endeavour Ridge region and the Quiet Eddy mooring region (QEP). Values are displayed as percentages. List of Figures Figure 3.1. The three limiting classes of resonant triads in two dimensional wavenumber space. kh is the horizontal wavenumber and m is the vertical wavenumber Fi y re 4.1. The conic surfaces showing the possible propagation directions prescribed by two muencies of wavevectors onginating fiom the sarne point Figure 4.2. Dispersion relations at station ERA2 for three values of Brunt-Vaisala frequency intersec ting internai waves atj; Mz, andjm2 frequencies Figure 4.3. The four limiting triads in phase space for sum interactions for the situalion O, O: a Figure 4.4. Range of wavelengths and group velocities of inertial and semidimal waves relative to the wavelength and group velocity of the jm2 wave at station CX- 1. The center panel gives the range of viable wavelengths ratios betwsen inertial and semidiurnal waves Figure 4.5. Wavenumbers, relative to the wavenumber of thejmz wave, of the inertial (ki, f~st column of panels) and the semidiurnal (k?, second column of panels) waves as a Function of latitude for the three limiting viads A, B, C, and D as described in Figure Figure 4.6. The variation of primary wavelength ntio,hcf)/k(~~), with latitude for a resonant triad. The two cases are shown,jss.jm2, where the inertial wave,f; propagates in the same vertical direction as the secondary wave,jm2, and, Mt.ss.jM2, where the M2 wave propagates in the same vertical direction as theml wave. The top two panels show the range of ratios of hv)/ h(~?), while the middle two panels explicitly show this range normalized to the mean of the two ratios delineated in the e st two panels. The bottom two panels show the wavelenth relationships between the primary (i, M2) and secondary w2) waves. In the ha1 panel, an inset is used to expand the plot h m 20' upward List qîfigures Figure 4.7. Group velocity, relative to the group velocity of thejm2 wave, of the inertial (ki, frst column of panels) and the semidiurnal (kz, column row of panels) waves as a function of latitude for the three limiting triads A, B, C, and D as described in Figure Figure 4.8. The coupling coefficient, T, as a function of horizontal wavevector angle between kl and k3 as defmed by Figure 4.3. The vertical propagation angles of ki and k3 are set by the fiequencies off andjm2 at station ERA2 thus kl and k3 are equivalent to kjs and kjmz at that station. Triads A and B correspond to JssjM2 and triads D and C to Mz.~~.JM Figure 4.9. The magnitude of the coupling coefficient, T, for the four limiting triads as a function of latitude for N= 10 cpd and N= 19.8 cpd. Rads A and B correspond to the case,jssjm?, and triads C and D to the case, Mz.ss.jM2. The red dotted line signifies the cut-off latitude for triad interactions brtweenjand Ml Figure The magnitude of the coupiing coefficient, T, for the four lirniting triads as a function of latitude for N=48 cpd and N=192 cpd. Triads A and B correspond to the casejssjm2, and triads C and D to the case, M2.ss.N2. The red dotted linc signifies the cut-off latitude for viad interactions betweenjand M Figure 5.1. Rotary spectra of currents. Blue (red) curves are clockwise (counterclockwise) sprcm with 50 degrees of fieedom. (a) 286-day record from CX-1; (b) 292-day record from E W ; (c) and (d) as in (a) and (b), respectively, for the residual time series Figure 5.2. Clockwise rotary velocity of horizontal currents from ER42 versus log fiequency I and time; colorbars indicate velocities in cm s Figure 5.3. Counterclockwise rotary velocity of horizontal currents fkom ERA2 versus log fiequency and tirne; colorbars indicate velocities in cm s List of Figures Figure 5.4. Clockwise rotary velocity of horizontal currents from CX- I versus log fiequency l and time; colorbars indicate velocities in cm s* Figure 5 S. Counterclockwise rotary velocity of horizontal cunents from CX- 1 versus log frequency and time; colorbars hdicate velocities in cm s Figure 5.6. Fifih order elliptic band pass filtering of the magnitude of velocity squared from the time series fiom station CX-1. Passbands are set fiom 1.4 to 1.6 for the inertial band (in red) and 3.35 to 3.44 for (hemz band (in blue) Figure 5.7. Fiflh order elliptic band pass filtering of the magnitude of velocity squared from the time series fiom station ERA2. Passbands are set fiom 1.4 to 1.6 for the inenial band (in red) and 3.35 to 3.44 for thefmz band (in blue) Figure 6.1. The inner triangle of the principal domain Figure 6.2. Rotary bispectra and spectra of horizontal currents from station ERA2 for 292 days beginning July 26, Figure 6.3. Rotary bispectra and spectra of residual (tides removed) horizontal currents from station ERAî for 292 days beginning July Figure 6.4. Rotary spectra and bispectra of horizontal cunents from station CX-L over 286 days beginning October 1 1, Figure 6.5. Rotary bispectra and spectra of residual (tides rernoved) horizontal currents from station CX- 1 for 286 days beginning October 1 1, Figure 6.6. Rotary bicoherence-squared of horizontal currents from station ERA Figure 6.7. Rotary bicoherence-squared of residual (tides removed) horizontal currents fiom. station ERA List of Figures xüi Figure 6.8. Rotary bicoherence-squared of horizontal currents from station CX Figure 6.9. Rotary bicoherence-squared of residual (tides removed) currents from station CX- 1 Figure Rotary bicoherence-squared of horizontal currents from station ER42 over days Figure Rotary bicoherence-squared of horizontal currents from station ERAZ over days Figure 6.12 Rotary bicoherence-squared of horizontal currents from station ERA2 over days Figure 9.1 Rotary spectra from station CX- 1 showing diurnal, inertial and semidiumal peaks as well as notable peaks at jmz, M4, and 4-day period, but no peaks atjlki, or M Figure 9.2 Rotary spectra from station ERA2. ThejMz peak exist without any evidence of enhanced energy at M2-J Figure 9.3. Low fkquency multiple filter plots encompassing the fiequency ranges of/-ki and Mz-jfor stations CX- 1 and ERAZ. Colorbars indicate velocity in cm s Figure 9.4. Averaged rotary spectra of horizontal currents from 25 deployments at station QEP 1 with 1 O96 degrees of fieedom s howing smali peak near M2-J Figure 9.5. Rotary spectra of horizontal currents from station QEPl deployment 5 at 600 m depth; exhibithg prominent peak near M2$ Figure 9.6. Clockwise RMFT plots for horizontal currents from station QEPl Deployment 5, 600 m depth. Colorbars indicate rotary velocity in cm s-' List of Figures xiv Figure 9.7. Counterclockwise RMFT plots for station QEPl deployment 5, 600 m depth. Colorbars hdicate rotary velocity at cm s-' Acknowledgements 1 am extremely grateful for the opportunity afforded me by Dr. Richard Thomson to conduct oceanographic researc h. His invaluable scient i fic advice, boundless enthusiasm, and generous assistance have fueled me throughout the research project. 1 must also express appreciation for his good nature and his thouçhtful concern for my well king. 1 thank Dr. Susan Allen for easing my tnnsfer from music to science and the guidance, support, time and energy she generously provided me. 1 thank my committee rnembers Dr. Steve Pond, Dr. David Farmer and Dr. Brenda Burd for their guidance, support and critical evaluation of my work. My research experience has been greatly enhanced by my Russian cornrades Dr. Alexander Rabinovich and Dr. Evgueni Kulikov. Both gave fieeiy of their time and their inspiration was infectious. I must thank Dr. Alexander Rabinovich for allowing me access to his cornputer code for rotary spectra and harmonic tidal analysis and boih Dr. Rabinovich and Dr. Kulikov for introducing me to the multiple filter technique. 1 also thank Dr. Diane Masson for her generosity in time and ideas, her critical evaluation of the thesis, as well as her fiiendship and support. xvi To Dr. Mike Foreman, who generously provided me with tidal currents from the TOPEX/POSEIDON global inverse model, and to Dr. Howard Freeland, who allowed me acccss to his cunznt meter data, I express my shcere gratitude. For cornputer support, I am indebted to Kerry Kimersley, Edmand Fok, Joe Lhguanti and Koit Teng and for technical assistance in aspects of field oceanography, I thank George Chase, Tamas Juhisz and Reg Bigham. The work was funded through a Natural Sciences and Engineering Research Council Opentinç Grant to Richard Thomson. The Department of Fisheries and Oceans allowed me full access to their facilities at the Institute of Ocean Sciences, Sidney, BC. 1 am very grateful to both. On a more personal bel, 1 wish to commend my parents, Laszlo and Giselle, for the nuauring environment they were able to provide me and to thank hem for sacrifices they made to do so. The value of this to me, in addition to their unwavering love and suppon, cannot be oventated. 1 dedicaie this thesis to my nagymama, Gizella Torok, and to my godson and niece James and Sarah Fraser. A special thanks to rny sister Sue and brother-in-law John Fraser for moral suppon and love. My commdes in student toi1 also deserve recognition. 1 generously thank Ana Carrasco, Cnig McNeil, Kevin Paul and Marie-Claude Bourque for providing distractions, sharing scientific insight and eiendship. Finally, my deepest gratitude and love 1 give to Dana Sokalski, without whom this process would have been insurmouniable. Her steadfast and unconditional support, tolcrance and belief in my abilities was invaluable to my well king in stressful tirnes, 1 thûnk-you. Dedication xvii This thesis is dedicated to lhe rnemory Of my nagymama, Gizella Torok And to James Steven Fraser and Sarah Theresa Fraser Chapter 1 Introduction Oceanographic Context Deep ocean hing is crucial to the maintenance of the present dynamic nature of the world's oceans and hence plays an essential role in the oceans prodigious biological productivity. As pointed out by Munk and WurzscIz [1998], 'hithout deep mixing, the ocean would tum, within a few thousand years, into a stagnant pool of cold salty watrr . The energy needed to drive deep ocean mixing cornes from the winds and tides. A portion of this energy is convened directly into turbulent rnixing as it crosses the boundaries of the ocean. However, a significant fraction of this enrrgy proceeds into the ocean interior where it provides the energy for large-scale currents and the internal wavefield. Within the interior of the ocean, the physical mechanisms responsible for the conversion of the energy entering the ocean are not filly understood. It is thought that in the internal wavefield the energy cascades fiom large to small spatial scaies where it cm be utilized as the energy necessary for turbulent rnixing and eventually dissipated as heat in regions of high velocity shear. The mean vertical miuing rate necessary throughout the ocean interior for the maintenance of stratification is of order 1 O-' m2 s [Munk, 1966; Munk and Wunsch However, microstructure measurements give a pelagic rnixing rate (away from topography) of 10-~ m' s , which is confumed by dye release experiments [Ledwell et ai., This order of magnitude discrepancy underscores the need to understand the physical processes of energy conversion from the wind and tides to mixing in the deep ocean. The energy in the intemal wavefield is restricted to frequencies bctween the local Coriolis Frequency and the local Brunt-Vaisala frequency and is generally concentrated at the hrrtial and tidal Bequencies. The internal tidal energy is dominated by motions at the semidiunial M2 frequenc y. parallel. Diumal frequenc y internal waves are possible onl

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