@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Earth, Ocean and Atmospheric Sciences, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Mihaly, Steven Ferenc"@en ; dcterms:issued "2009-07-03T21:24:48Z"@en, "1999"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """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 semidiurnal and local inertial frequencies. The existence of this spectral peak is, in the absence of any known forcing at the sum frequency, sufficient to assume some form of nonlinear interaction between internal wave motions at the inertial and semidiurnal frequencies. 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 frequency (termed the 'ʃM2" frequency) 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 model for the nonlinear interaction. Although the validity of the weakness assumption used in deriving the coupling efficiency is in question, the physical mechanism of the coupling of two internal gravity waves via a resonant triad is well founded. The resonance condition prescribes geometric constraints on the propagation and wavelengths of a triad of interacting internal waves. The propagation and wavelengths of inertial and semidiurnal waves observed in the northeast Pacific are consistent with these constraints and the formation of resonant triads is likely. Analysis, based on the likelihood of the global formation of resonant triads between inertial and semidiurnal waves, indicates that higher latitudes are favored. An examination of two deep-ocean sites along Juan de Fuca Ridge shows that elevated energy is not usually coincident in the inertial, semidiurnal and ʃM2 frequency bands. When the energy in all three bands is coincident, motions in the ʃM2 band can be: 1) nonresonantly forced requiring the support of the inertial and semidiurnal motions or, 2) a result of the nonlinear transfer of energy from the inertial and semidiurnal motions. When the energy is not coincident, the energy in the ʃM2 frequency band can only result from 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 and ʃM2 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 for ʃMz waves while the CoAxial site is in the path of propagating ʃM2 wave energy from a remote source region."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/10099?expand=metadata"@en ; dcterms:extent "24627287 bytes"@en ; dc:format "application/pdf"@en ; skos:note "NONLINEAR INTERACTION OF INERTIAL AND SEMIDIURNAL CURRENTS IN THE NORTHEAST PACIFIC by STEVEN FERENC MIHALY B.Eng., The Technical University of Nova Scotia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Earth and Ocean Sciences We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1999 © Steven Ferenc Mihaly, 1999 UBC Special Collections - Thesis Authorisation Form http://www.library.ubc.ca/spcoll/thesauth.html In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the Un i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r sch o l a r l y purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada 1 of 1 5/25/99 10:47 A M 11 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 semidiurnal and local inertial frequencies. The existence of this spectral peak is, in the absence of any known forcing at the sum frequency, sufficient to assume some form of nonlinear interaction between internal wave motions at the inertial and semidiurnal frequencies. 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 frequency (termed the '7M2\" frequency) 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 model for the nonlinear interaction. Although the validity of the weakness assumption used in deriving the coupling efficiency is in question, the physical mechanism of the coupling of two internal gravity waves via a resonant triad is well founded. The resonance condition prescribes Abstract iii geometric constraints on the propagation and wavelengths of a triad of interacting internal waves. The propagation and wavelengths of inertial and semidiurnal waves observed in the northeast Pacific are consistent with these constraints and the formation of resonant triads is likely. Analysis, based on the likelihood of the global formation of resonant triads between inertial and semidiurnal waves, indicates that higher latitudes are favored. A n examination of two deep-ocean sites along Juan de Fuca Ridge shows that elevated energy is not usually coincident in the inertial, semidiurnal and JMi frequency bands. When the energy in all three bands is coincident, motions in the JM2 band can be: 1) nonresonantly forced requiring the support of the inertial and semidiurnal motions or, 2) a result of the nonlinear transfer of energy from the inertial and semidiurnal motions. When the energy is not coincident, the energy in the /IVI2 frequency band can only result from 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 and JM2 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 for JMz waves while the CoAxial site is in the path of propagating JM2 wave energy from a remote source region. Table of Contents Abstract ii List of Tables vii List of Figures ix Acknowledgements xv Dedication xvii CHAPTER 1. Introduction 1 1.1 Oceanographic Context 1 1.2 Structure of the Thesis .'...3 CHAPTER 2. Oceanographic Observations 5 2.1 Introduction 5 2.1.1 Measurements 6 2.1.2 Study Regions 7 2.1.2.1 Juan de Fuca Ridge 7 2.1.2.2 Quiet Eddy Moorings 11 2.1.2.3 Near-Coastal Region 13 2.2 Power Spectra 14 2.2.1 Motivation 14 2.2.2 Rotary Spectra 15 2.2.3 Regionally Averaged Rotary Spectra 18 2.2.3.1 Method : 18 2.2.3.2 Application 21 2.2.4 Pervasiveness of/M2 interactions 24 Table of Contents v 2.3 Summary and Conclusions 28 CHAPTER 3. Oceanic Nonlinear Interaction Mechanisms 29 3.1 Introduction 29 3.2 Background 30 3.3 Weak Resonant Interaction Mechanisms 32 3.3.1 Elastic Scattering 35 3.3.2 Induced Diffusion 36 3.3.3 Parametric Subharmonic Instability 36 3.3.4 Validity 37 3.4 Other Models 39 3.4.1 Eikonal Method 39 3.4.2 Direct Numerical Simulation 40 3.5 Resonant Interaction Theory 41 3.5.1 Discrete Interaction 41 3.5.2 Transport Theory 43 3.5.3 The Transport Equation 47 3.6 Summary and Conclusions 49 CHAPTER 4. Geometrical Considerations and Coupling Efficiency 50 4.1 Introduction 50 4.2 Geometric Considerations 51 4.2.1 Preliminary 51 4.2.2 Local Variability 59 4.2.3 Global Variability 62 4.3 Coupling Efficiency 70 4.4 Summary and Conclusions 78 CHAPTER 5. Spectral Analysis 80 5.1 Introduction 80 5.2 Observations 81 5.3 Harmonic Tidal Analysis 81 5.4 Rotary Spectra 83 5.4.1 Tidal Currents 84 5.4.2 Inertial Currents 88 5.4.3 JM2 and M 4 Spectral Peaks 91 5.5 Time Series Demodulation 92 5.5.1 Introduction 92 5.5.2 The Signal Model 93 5.5.3 Method 93 5.5.4 Multiple Filter Technique and Wavelet Analysis 96 5.5.5 Analysis of the Data 97 5.6 Discussion 109 CHAPTER 6. Bispectral Analysis 112 Table of Contents vi 6.1 Introduction 112 6.2 Higher Order Statistics 114 6.2.1 Moments and Cumulants 115 6.2.2 Cumulant Spectra 117 6.3 The Bispectrum 121 6.3.1 Estimation 123 6.3.1.1 The Model 123 6.3.1.2 Method of Estimation 124 6.4 Quadratic Phase Coupling 127 6.4.1 Biphase and Bispectrum Magnitude 128 6.4.2 Bicoherence 131 6.5 Statistical Issues 134 6.5.1 Bicoherence-squared 134 6.5.2 Biphase 135 6.5.3 Data Length 136 6.5.4 Stationarity and Trans ients 137 6.6 Analysis of Data 138 6.6.1 Goals 138 6.6.2 Plots of Bispectra Magnitude 139 6.6.3 Bicoherence-squared Plots 146 6.6.4 Biphase 155 6.7 Discussion 156 C H A P T E R 7. Conclusions 160 Nomenclature and Definitions 166 Bibliography 168 Appendix A : M2-/Interactions 177 Appendix B : Future Work 190 Vll List of Tables Table 2.1. Current meter deployments from Endeavour Ridge and CoAxial Ridge 11 Table 2.2. Current meter deployments from station QEP 13 Table 2.3. Current meter deployments from the Coastal Oceanic Dynamics Experiment 14 Table 2.4. Spectral density and rotary coefficient at tidal and inertial frequencies 22 Table 2.5. Variance and percentage of total variance at tidal and inertial frequencies 22 Table 4.1. Representative values from ERA2 and CX-1. Observed frequencies are the peak frequencies from rotary spectra with 50 degrees of freedom 52 Table 4.2. Propagating angle, (p , for inertial, semidiurnal and JMi frequency internal waves at three depths at stations CX-1 and ERA2 54 Table 5.1. Selected tidal constituents used for the harmonic analysis of tidal currents of the time series from stations ERA2 and CX-1 83 Table 5.2. Baroclinic tidal parameters from vertical mode decomposition for a 2100 m deep basin with a characteristic September stratification from a region near station ERA2. Rossby radius is calculated by df where c is phase speed of the mode and/ is the Coriolis parameter 86 Table 5.3. Tidal current velocities and rotary components for the four principal semidiurnal tidal constituents, derived using the global inverse tidal model, TPXO.2, oi Egbert et al. [1994] 99 List of Tables Table 6.1. Data length and resolution as a function of DFT length and sampling frequency 137 Table 6.2. Biphase of bispectral estimates nearest B(f,M2) of horizontal currents from ERA2 155 Table 9.1. Center frequencies of observed peaks and predicted values of inertial, tidal and derived bands 178 ix List of Figures Figure 2.1. Locations of current meter moorings used in this thesis 6 Figure 2.2. Current meter mooring stations at Endeavour Ridge. The red rectangular area represents the vent field. The map boundaries correspond to the rectangular region in Figure 2.1 9 Figure 2.3. Regionally averaged rotary spectra from three distinct oceanographic regions from the northeast Paci fic 20 Figure 2.4. Spectral density at the7M2 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 mooring region (QEP). Values are displayed as percentages 26 Figure 2.5. Spectral density of M 4 band normalized to the spectral density of the semidiurnal (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. 27 List of Figures x 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 35 Figure 4.1. The conic surfaces showing the possible propagation directions prescribed by two frequencies of wavevectors originating from the same point 53 Figure 4.2. Dispersion relations at station ERA2 for three values of Brunt-Vaisala frequency intersecting internal waves atf M 2 , and/M2 frequencies 55 Figure 4.3. The four limiting triads in phase space for sum interactions for the situation oo, 2000 m) oceanographic regions: a topographically rough oceanic ridge region (Endeavour Ridge and CX-1 of Figure 2.1), a topographically smooth abyssal plain region (QEP 1 of Figure 2.1) and a topographically smooth near-coast region (B03 and E05 of Figure 2.1). This chapter provides a description of the current meter measurement method, a description of the study regions, and explains the processing resulting in the fundamental observations of the thesis. Chapter 2. Oceanographic Observations 6 Figure 2 .1. Locations of current meter moorings used in this thesis 2.1.1 Measurements The measurement of horizontal currents was achieved with rotor and vane type current meter systems originally designed by Ivar Aanderaa [Aanderaa, 1964] and manufactured by Aanderaa Instruments, Bergen, Norway. For the deployments considered here, a typical mooring uses Aanderaa RCM series (RCM4, RCM5, and/or RCM7) current meters suspended beneath subsurface flotation. The mooring designs of the earlier deployments were based on general guidelines available in the oceanographic community for taught-line moorings [e.g. Berteaux, 1975]. Latter designs benefited additionally from model analyses using the computer program SSMOOR, [Berteaux, 1991]. Model analysis of a typical mooring design indicated a depression of only 90 cm for a steady 25 cm s\"1 flow. Root mean square flows for the study regions are only 4 cm s\"1 so that typical moorings would undergo small vertical excursions of Chapter 2. Oceanographic Observations 7 the order of 1-10 cm. The manufacturer specifies the accuracy of the current meters as ± 1 cm s\"1 or 2% of the actual speed, whichever is greater, with a threshold speed of 2 cm s\"1. Directions are specified accurate to ±5° for speed in the range of 5 to 100 cm s\"1. The deployment time of the current meters used here varies from approximately 3 months to a year. The sampling interval used for most of the longer deployments was 60 minutes and the sampling interval for the shorter deployments was 30 minutes. All the data have been processed keeping the original sampling interval. Since the speed is estimated by counting revolutions of the rotor, it is the average speed over the sample interval. The directions are obtained instantaneously at the end of each sample interval, at the same time as the \"counts\" are. This results in a time shift between speed and direction estimates equal to half the sampling interval and a direction estimate that has much less statistical certainty than the speed estimate. To reduce this error, consecutively recorded directions delineating a sample interval are averaged and paired with the estimate of speed over the sample interval. 2.1.2 Study Regions 2.1.2.1 Juan de Fuca Ridge The topographically rough oceanic ridge region is represented by a number of sites over Juan de Fuca Ridge. The ridge is a 525 km spreading center in the northeast Pacific spanning parallels from 44° through 48°. It is located about 300 km west of the coast of Washington Chapter 2. Oceanographic Observations 8 state and is oriented approximately 20° to the meridian. Considered a medium rate spreading center with a full spreading rate of 6 cm y\"1 [Riddihough, 1984], Juan de Fuca Ridge is an area of vigorous seismic activity and considerable bathymetric complexity. As part of the global Mid-Ocean Ridge system, a prominent feature of all the world's oceans with a total length in excess of 50 000 km, Juan de Fuca Ridge is not an isolated feature. The bulk of the current meter moorings in this region are located in waters over the second-most northern segment of the six segment Juan de Fuca Ridge system. This 90 km long segment, known as Endeavour Ridge, is shaped like an inverse \"S\" with a well-defined rift valley, striking at an azimuth of approximately 20° T, occupying the central region. A simplified view of the bathymetry has the rift valley floor at a depth of 2200 m with a width of 1 km and bounded on both sides by steep walls rising to rift crests at 2100 m depth. To the east of the ridge is Cascadia Basin with a depth of 2600 m directly adjacent to the ridge. Towards the west and northwest there are three seamount chains running perpendicular to the axial valley with the shallowest of the seamounts rising to within 1200 m of the ocean surface. Figure 2.2 shows the bathymetric complexity of the rift valley and indicates the placement of moorings at Endeavour Ridge. A total of 19 current meters have been deployed within the horizontal boundaries of the central rift valley (these deployments are termed \"on-axis\") over the period of ten years, 1985-1995. The on-axis deployments included one mooring which spanned the entire water column carrying six current meters. Additionally, there have been seven current meter deployments located outside of the bounds of the central rift valley (\"off-axis\") on the flanks of the ridge and adjacent plains. A further four deployments have been located directly over the rift crests Chapter 2. Oceanographic Observations 9 making a regional total of 30 deployments. Additional information on these moorings as well as other analyses of the current meter records can be found in Thomson et al. [1990], Allen and Thomson [1993], and Roth [1992]. 129°08'W 129°06'W 129°04'W Figure 2.2. Current meter mooring stations at Endeavour Ridge. The red rectangular area represents the vent Held. The map boundaries correspond to the rectangular region in Figure 2.1. Current meter data from the Juan de Fuca Ridge region are augmented by a single current meter (station CX-1 in Figure 2.1) located at a site on the 80 km CoAxial ridge segment (so named because of its proximity to Axial volcano). The area has the most prominent bathymetry of Juan de Fuca Ridge [Baker and Hammond, 1992], with Axial's central volcanic Chapter 2. Oceanographic Observations 1 0 dome having a relief of approximately 1400 m relative to the surrounding sea floor. The mooring site is located 56 km north of the volcano's caldera along the strike line (20°T) of an active rifting zone within a rift valley. The axial valley here is morphologically similar to the axial valley containing the on-axis moorings at Endeavour Ridge. It has rift crests shoaling to 2050 m, 200 m above the relief of the valley floor, but is substantially wider than Endeavour Ridge at 9 km. The single current meter that returned data from this mooring was at 2050 m depth. Table 2.1 lists the current meters used in this thesis from the Juan de Fuca Ridge region. These current meter records were selected from a larger data set of all the Juan de Fuca deployments in the IOS archive. The selection process was conducted by visually assessing the converted and de-spiked time series of speed and direction. Dropouts in the speed time series occur due to tape (for RCM 4/5 current meters) or digital storage (RCM 7 current meters) failure, mechanical failure from damage or biofouling, or to flow speeds below the threshold speed. Sporadic dropouts that were deemed to result from failure to reach threshold speed were replaced by the mean speed below the threshold level, 1 cm s\"1, to give a better representation of the low energy flow field. Records with significant dropouts due to instrument failure were discarded. The operation of the compass was investigated by histogram plots to insure that it did not \"stick\" and unnaturally favor directions and that it represented a physically feasible oceanic current field. As a result of the visual assessment, approximately a quarter of the data set were discarded. The final criterion for selection was the ability to produce high quality spectral plots and time series demodulation at inertial frequencies. To this end, only records longer than « 3 months were kept, with the exception of four current meters at station A2 (Table 2.1) which form part of a longer time series. Chapter 2. Oceanographic Observations 11 No Station Dep Latitude (•N) Longitude (•w) Depth (m) Start (d/m/y) End : (d/m/y) Records Sample Interval (min) Length (d) 1 A2 . 1 47 58.00 129 5.30 1506 27/09/85 14/05/86 11015 30 229 2 1 47 58.00 129 5.30 1706 27/09/85 12/06/86 12393 30 258 3 1 47 58.00 129 5.30 1707 27/09/85 12/06/86 12393 30 258 4 1 47 58.00 129 5.30 1956 27/09/85 12/06/86 12392 30 258 5 1 47 58.00 129 5.30 2056 27/09/85 05/06/86 12042 30 251 6 T1 1 48 11.00 129 15.00 1705 27/09/85 09/04/86 9279 30 193 7 T3 1 47 51.00 129 4.00 1513 27/09/85 08/05/86 10689 30 223 8 1 47 51.00 129 4.00 1971 27/09/85 27/04/86 10157 30 212 9 1 47 51.00 129 4.00 2070 27/09/85 14/06/86 12471 30 260 10 R1 1 47 58.20 129 6.20 1781 25/09/85 19/01/86 5555 30 116 11 R2 1 47 57.80 129 4.60 1699 25/09/85 12/06/86 4442 30 93 12 1 47 57.80 129 4.60 1955 25/09/85 12/06/86 12470 30 260 13 1 47 57.80 129 4.60 1956 25/09/85 12/06/86 12470 30 260 14 A3 1 47 56.40 129 6.20 1768 25/09/85 06/06/86 12509 30 261 15 1 47 56.40 129 6.20 2019 25/09/85 06/06/86 12509 30 261 16 1 47 56.40 129 6.20 2120 25/09/85 06/06/86 12509 30 261 17 1 47 56.40 129 6.20 2221 25/09/85 06/06/86 12509 30 261 18 A2 2 47 57.80 129 5.60 60 17/09/86 24/06/87 13459 30 280 19 2 47 57.80 129 5.60 486 17/09/86 15/06/87 13048 30 272 20 2 47 57.80 129 5.60 1686 17/09/86 14/06/87 13008 30 271 21 2 47 57.80 129 5.60 1936 17/09/86 10/06/87 12809 30 267 22 2 47 57.80 129 5.60 2036 17/09/86 16/06/87 13058 30 272 23 2 47 57.80 129 5.60 2136 17/09/86 16/06/87 13098 30 273 24 A2 3 47 57.10 129 6.30 1489 04/07/87 10/09/87 3247 30 68 25 3 47 57.10 129 6.30 1689 04/07/87 10/09/87 3247 30 68 26 3 47 57.10 129 6.30 1939 04/07/87 10/09/87 3247 30 68 27 3 47 57.10 129 6.30 2039 04/07/87 10/09/87 3247 30 68 28 CA 01 1 47 56.40 129 9.70 1854 21/07/94 14/05/95 7141 60 298 29 CB01 1 48 7.50 128 21.10 1786 16/07/94 30/01/95 4744 60 198 30 ERA2 1 47 57.00 129 5.70 1871 26/07/94 26/07/95 7025 60 292 31 CX-1 1 46 20.10 129 42.60 2050 11/10/93 24/07/94 6879 60 286 Table 2.1. Current meter deployments from Endeavour Ridge and CoAxial Ridge. 2.1.2.2 Quiet Eddy Moorings To examine the effects of topography on the current meter data collected at Juan de Fuca ridge, an offshore data set over smooth topography is required. Suitable offshore data from the IOS archive were limited to a single region located near line P at 49°33'N andl38°38'W, 720 Chapter 2. Oceanographic Observations 12 km west (285° T) of the Endeavour sites (QEP 1 of Figure 2.1). This mooring program was designed to measure kinetic energy in the eddy field of the northeast Pacific. The primary mooring carried current meters at depths chosen to give optimum separation of the barotropic and the first three baroclinic modes. These depths were: 200 m, 600 m, 1500 m, 2200 m, and 3000 m in 3890 m of water. The moorings were deployed consecutively seven times from May 30, 1989 to May 16, 1993. There were two variations of this scheme. First, in addition to the mooring at the primary site during the fourth deployment (August 29, 1990-May 2, 1991), three other moorings delineating an equilateral triangle were deployed. The array had an east-west oriented base of 89 km, with the primary site occupying the center of the triangle. Second, during the final year, May 20, 1993 to May 16, 1994, the current meters were deployed at: 2000 m, 2200 m, 2600 m, 2800 m, and 3000 m. Other information and analyses concerning these moorings can be found in Cummins and Freeland [1993], and Freeland [1993]. The current meters from these moorings are listed in Table 2.2 and were culled from a larger data set in the same manner as described for the current meters in Table 2.1 in the previous section. No Station Dep Latitude (•N) Longitude (•w) Depth (m) Start (d/m/y) End (d/m/y) Records Sample Interval (min) Length (d) 1 QEP1-A 1 49 32.73 138 37.32 200 30/05/89 09/09/89 6272 . 30 131 2 QEP1-A 1 49 32.73 138 37.32 600 30/05/89 09/09/89 6272 30 131 3 QEP1-A 1 49 32.73 138 37.32 1500 30/05/89 09/09/89 6272 30 131 4 QEP1-A 1 49 32.73 138 37.32 3000 30/05/89 09/09/89 6272 30 131 5 QEP1-B 2 49 32.90 138 38.00 200 10/10/89 14/05/90 5183 60 216 6 QEP1-B 2 49 32.90 138 38.00 600 10/10/89 14/05/90 5183 60 216 7 QEP1-B 2 49 32.90 138 38.00 1500 10/10/89 14/05/90 5183 60 216 8 QEP1-B 2 49 32.90 138 38.00 3000 10/10/89 14/05/90 5183 60 216 9 QEP1 4 49 33.04 138 40.13 200 29/08/90 02/05/91 5916 60 246 10 QEP1 4 49 33.04 138 40.13 600 29/08/90 02/05/91 5916 60 246 11 QEP1 4 49 33.04 138 40.13 1500 29/08/90 02/05/91 5916 60 246 12 QEP1 4 49 33.04 138 40.13 2200 29/08/90 02/05/91 5916 60 246 13 QEP1 4 49 33.04 138 40.13 3000 29/08/90 02/05/91 5916 60 246 14 QEP1 5 49 32.92 138 39.95 200 03/05/91 26/10/91 4235 60 176 15 QEP1 5 49 32.92 138 39.95 600 03/05/91 26/10/91 4235 60 176 Chapter 2. Oceanographic Observations 13 16 Q E P 1 5 49 32.92 138 39.95 2200 03/05/91 26/10/91 4235 60 176 17 Q E P 1 5 49 32.92 138 39.95 3000 03/05/91 26/10/91 4235 60 176 18 Q E P 1 6 49 39.99 138 35.23 600 28/10/91 13/09/92 7723 60 322 19 Q E P 1 6 49 39.99 138 35.23 1500 28/10/91 13/09/92 6240 60 260 20 Q E P 1 6 49 39.99 138 35.23 3000 28/10/91 13/09/92 7723 60 322 21 Q E P 1 7 49 33.19 138 38.21 200 15/09/92 19/05/93 5917 60 247 22 Q E P 1 7 49 33.19 138 38.21 600 15/09/92 19/05/93 5917 60 247 23 Q E P 1 7 49 33.19 138 38.21 1500 15/09/92 19/05/93 5917 60 247 24 Q E P 1 7 49 33.19 138 38.21 2200 15/09/92 19/05/93 5917 60 247 25 Q E P 1 7 49 33.19 138 38.21 3000 15/09/92 19/05/93 5918 60 247 26 Q E P 1 8 49 34.10 138 37.60 2000 20/05/93 16/05/94 8684 60 362 27 Q E P 1 8 49 34.10 138 37.60 2200 20/05/93 16/05/94 8684 60 362 28 Q E P 1 8 49 34.10 138 37.60 2600 20/05/93 16/05/94 8684 60 362 29 Q E P 1 8 49 34.10 138 37.60 2800 20/05/93 16/05/94 4278 60 178 Table 2.2. Current meter deployments from station QEP. 2.1.2.3 Near-Coastal Region A further two current meter sites are included to represent a near-coastal region. These moorings were deployed as part of the Coastal Oceanic Dynamics Experiment (CODE) [Thomson et al, 1985], during 1979-1981. The selected moorings were in deep (>2000 m) water seaward of the shelf break off Vancouver Island. Station B03, the most seaward current meter station on the B-line, is approximately 90 km offshore of Brooks Peninsula, and station E05, the outermost current meter station on the E-line, is 75 km offshore of Estevan Point (see Figure 2.1). A single deployment of the deepest current meters, at 1510 and 1025 m depth, respectively, were selected for analysis. Chapter 2. Oceanographic Observations 14 No. Station Dep Latitude CN) Longitude Cw) Depth (m) Start (d/m/y) End (d/m/y) Records Sample Interval (min) Length (d) 1 B03 1 49 38.60 128 47.10 1025 23/09/79 09/05/80 5482 60 228 2 E05 1 48 47.70 127 35.10 1510 16/09/79 18/01/80 5947 30 124 Table 2.3. Current meter deployments from the Coastal Oceanic Dynamics Experiment. 2.2 Power Spectra 2.2.1 Motivation In order to investigate nonlinearities in the data, the rotary power spectra of all current meter records were obtained. Although the power spectra do not directly indicate nonlinear interaction, statistically significant peaks in the frequency spectrum can be considered to 3 establish whether they conform to frequency triads of the form ^ soo(. = 0 , s - ± 1 (see section 3.3). If triads of this form do exist, and if there is no outside forcing at any one of the three frequency constituents, the triads existence provides evidence that the energy at the frequency which has no direct forcing is nonlinearly derived from the interaction of the other two frequencies. Chapter 2. Oceanographic Observations 15 2.2.2 Rotary Spectra The horizontal velocity components of mesoscale phenomena such as inertial oscillations, tides and eddies are often analyzed using rotary components. These motions are strongly rotational and are often better represented by rotary components than Cartesian (u,v) coordinates. The rotary spectrum is also invariant to coordinate rotation, facilitating the comparison of spectra of horizontal currents from different sites. The rotary approach is inspired by the optical principle of decomposing electromagnetic waves into two oppositely polarized components. For horizontal currents, the velocity vector of oscillating currents is represented by the summation of two oppositely rotating fixed magnitude vectors at each frequency band of the Fourier series of the original signal [Mooers, 1973; Gonella, 1972]. The decomposition is achieved by equating the equation of an ellipse to the complex value representation of the velocities at a specific frequency co , u(co)-r-iv(co) = w+(co)e1C0' +u (co)e (2.1) The Fourier coefficients a and b at frequency co have the form, w(co) = au(co)cosco£ + 6u(co)sincof v(co) = av (co) cos (at + bv (co) sin (at' (2.2) Solving for rotary components u+ and u , we get, Chapter 2. Oceanographic Observations 16 u u (2.3) The rotary spectra are defined as, S+(co) = E[(u+(co))\\u+((o))]/2 S\"(co) = £[(IT(G>))*(tr(©))]/ 2 ' (2.4) Here, E is the expectation operator and * signifies the complex conjugate. The sum of the clockwise and counterclockwise spectra over co , the total variance, is equal to the sum of the spectra derived from Cartesian coordinates (Sx,Sy). Once transformed into rotary components, statistical measures such as coherence and spectra are invariant to Cartesian coordinate rotation. Thus, effects, such as topographic steering or boundary reorientation of flow are suppressed and comparisons between widely separated sites can be made. Furthermore, much of the mesoscale flows have a predominant direction of rotation, facilitating the use of a single scalar to describe the two dimensional vector motions. In particular, flows at the inertial frequency in the Northern Hemisphere are strongly clockwise and can be well described by a single scalar, the clockwise rotary component. This also permits the use of the rotary characteristics of energy within the inertial frequency band from the rotary spectra to assess the quality of the data. Deviation from clockwise polarization of this energy (away from boundaries) is only likely to result from noisy or poor quality data, or instrument malfunction. Chapter 2. Oceanographic Observations 17 The partition of energy between counterclockwise (S*) and clockwise (S ) rotary spectra and hence the polarization of the currents are often described by the rotary coefficient, defined by Gonella [1972] as, S-(co)-r(co) r ( ( 0 ) =5-(co) + r (co)- ( 2- 5 ) For freely propagating internal waves, the polarization characteristics can be determined by linear theory. Following Gill [1982], the fraction of counterclockwise polarized energy to clockwise polarized is given by, S-^1J^I1. (2.6) 5-( 2000 m) at all sampled depths for both rough and smooth topography. A pair of deep near shore stations show that the spectral peaks exist in the deep near shore regime as well. Although the spatial sampling is somewhat limited by the availability of data, the spatial consistency of the spectral peaks at both the / M 2 and M 4 frequency within this data set implies that the nonlinear phenomena are a ubiquitous feature of the northeast Pacific. The existence of a significant energetic peak a precisely the sum of the inertial and M2 frequencies is, in the absence of any known forcing at that frequency, sufficient to assume some form of nonlinear interaction between these two oscillations. Likewise, the energetic peak at the M4 frequency is indicative of nonlinear self-interaction of the M 2 tide. Chapter 3 29 Oceanic Nonlinear Interaction Mechanisms 3.1 Introduction Well-established theory for finite amplitude, or strong interaction, among internal waves does not exist. However, significant progress has been made using weak interaction theory for the nonlinear interaction between internal waves. In this chapter, the mathematical models that are currently used to elucidate the physical processes underlying nonlinear interaction between oceanic waves will be introduced and examined. It is felt prudent to first conduct a complete review of the development of these theories in order to establish a basis for applicability. The most robust model for the particular nonlinearities is then chosen with the necessary caveats expressed for its shortcomings. The chapter begins with a quick overview of the development of thought on oceanic wave-wave interactions, proceeds to an introduction and discussion of Chapter 3. Oceanic Nonlinear Interaction Mechanisms 30 the prevalent models and mechanisms fo l l owed by a more detailed development o f the w e a k l y resonant wave-wave interaction theory. 3.2 Background The current body o f thought o n nonlinear interaction between water waves had its incept ion i n the 1950's . P r io r to that t ime, the study o f \"waves i n f lu ids\" was usual ly restricted to surface waves treated as inf in i tes imal waves for w h i c h the surface boundary cond i t ion c o u l d be l inear ized. Excep t for the examinat ion o f sol i tary waves i n sha l low water, nonlinear effects were general ly restricted to the re la t ively t r iv ia l dis tor t ion f rom a s inusoida l prof i le o f surface waves described b y Stokes i n the nineteenth century [Phillips, 1981]. The p r o b l e m o f energy transfer between spatial and tempora l scales for surface gravi ty waves i n deep water was first addressed b y Phillips [ I960] . H i s analysis proceeded b y assuming that the nonl inear coup l ing was sma l l and then expanding about the k n o w n linear solut ion us ing convent ional perturbation methods. The results demonstrated that the second-order perturbations gave rise to wave components w i t h wave number and frequency that were sum and differences o f the p r imary wave components. That is , two sinusoidal waves , exp{i (k ,x - c o / ) } and exp{i(k 2 x-co 20} w e a k l y forced terms o f the form exp{i[(k, ± k 2 ) x - ( c o , ± c o 2 ) f ] } . H o w e v e r , the phase ve loc i ty o f these secondary surface gravi ty waves never matched that o f a free inf in i tes imal wave o f the same wave number and hence the amplitudes stayed sma l l i n magnitude and were bounded i n t ime. W h e n the algebra was taken to the third-order perturbation, cases were produced where the phase ve loc i ty o f the Chapter 3. Oceanic Nonlinear Interaction Mechanisms 31 tertiary waves did match that of a free wave; that is, the waves satisfied the dispersion relation for surface gravity waves. Thus, when the following three conditions are satisfied: the tertiary wave grows linearly in time in a resonant manner and represents the existence of continuous energy transfer between discrete wave components. This nonlinear transfer mechanism is termed very weak resonant wave-wave interaction. While this indicated that there could be exchange of energy between wave scales, Phillips's approach of solving the equations of the time history (the growth) of the Fourier components proved too cumbersome for much further progress. However, two simple theoretical cases were solved following this method. Longuet-Higgins [1962] solved the discrete interaction case for which two of the primary wavenumbers were equal and Longuet-Higgins and Phillips [1962] solved the degenerate case where k, =k 3 , and k 2 =k 4 . The latter being the case where two wave trains run together and there is no transfer of energy, but rather, a simple modification of the phase velocities of each of the two participating primary waves. Benney [1962] regarded the three primary waves as slowly varying functions in time and could derive a complete set of interaction equations. This formulation was robust, even when the magnitude of the interaction product became comparable to that of the primary waves. Furthermore, the theory showed that if resonance was not attained, the cubic terms could still force an interaction product that remained an order of magnitude smaller. This theory was ± co, + co2 ± co3 = ±co, ±k, ± k 2 ± k 3 = ±k w,2 = g|k,.|, '4> (3.1) Chapter 3. Oceanic Nonlinear Interaction Mechanisms 32 modified by McGoldrick [1965] to include the interaction between gravity-capillary waves. The interaction here occurred at the quadratic expansion and had much simpler interaction equations with a much shorter interaction time scale. Ball [1964] successfully applied Benney's formulation to the quadratic interaction between two surface waves and one internal wave. The equations for the nonlinear transfer of energy in actual ocean conditions, that is, among many spectral components, were developed by Hasselmann [1962; 1966; 1967]. Hasselmann [1967] was able to consolidate all cases of nonlinear interaction under a general theory applicable to geophysical wave-wave interactions problems using the scattering formalism of quantum field theory. He accomplished this by bridging theory on interaction effects between secular random lattice vibrations in crystalline structures first proposed by Peierls [1929], with theory on nonlinear interactions of light waves and lattice vibrations proposed by Brillouin [1922] and Raman [1928] and, the theory of plasma wave interactions. With the limited computational ability of his time, he was able to show gradual transfer of energy from low wavenumber waves with high energy to high wavenumber waves with low energy. Empirical satisfaction came with the simultaneous publication of two papers, Longuet-Higgins and Smith [1966], and McGoldrick, Phillips, Huang, and Hodgson [1966], both showing the existence of resonant interactions and their conformation to theory. Subsequently interactions among internal waves were demonstrated experimentally by Martin, Simmons and Wunsch [1972]. Chapter 3. Oceanic Nonlinear Interaction Mechanisms 33 3.3 Weak Resonant Interaction Mechanisms Prior to 1972 the focus of work on nonlinear wave-wave interactions was in three areas: swell development and dissipation of surface waves in deep water, capillary waves, and the effect of internal (interfacial) gravity waves on surface gravity waves. The possibility of resonant nonlinear transfer between internal waves in a stratified fluid was shown to have a theoretical basis by Thorpe [1966], and by experiment by Davis and Acrivos [1967]. However, the major impetus to study interactions among internal waves came with the publication of the Garrett-Munk internal wave model of frequency and wavenumber spectra [Garrett and Munk, 1972]. The model is based on linear theory and observations, and, has since been upgraded to reflect newer data [Garrett and Munk, 1975; \\919; Cairns and Williams, 1976; Munk, 1981]. It is believed to accurately model the internal wave structure away from possible source regions such as topography, the upper thermocline and regions of strong mean flow. Its universality and specific shape invite explanation on dynamical bases such as the role played by nonlinear interactions. The dispersion relation for internal waves is such that perturbation of the nonlinear wave equation does admit solutions at the second-order and thus gives rise to triad interactions when the following conditions are met: co, ±co 2 = co 3 ' (3.2) k, ± k 2 =k 3 ' (3.3) Chapter 3. Oceanic Nonlinear Interaction Mechanisms 34 (3.4) co,. 2_f2m,2+N2(kt2+ll2) i = 1,2,3-Here, / is the local Coriolis frequency, N is the local Brunt-Vaisala frequency, and k(. = ki + /j + mk . Systematic calculation of energy transfer rates within the Garrett-Munk spectrum were conducted by McComas and Bretherton [1977] using numerical methods. It was found that much of the energy transfer could be understood in terms of three classes of limiting resonant triads. These three classes allowed for simplification of the governing equations (the transport equation), and a better physical understanding o f the mechanisms involved in the energy transfer. The three classes are termed: elastic scattering, induced diffusion, and parametric instability. A schematic representation in wavenumber space is given in Figure 3.1. Chapter 3. Oceanic Nonlinear Interaction Mechanisms 35 A m (a) E L A S T I C S C A T T E R I N G m (b) I N D U C E D D I F F U S I O N in (c) P A R A M E T R I C / S U B H A R M O N I C I I N S T A B I L I T Y F i g u r e 3.1. The three l i m i t i n g classes of resonant tr iads i n two dimensional wavenumber space. 3.3.1 Elastic Scattering Elas t i c scattering consists o f a ver t ica l ly propagat ing high-frequency wave ( shown i n F igure 3.1(a) as either k i or k?; higher frequency wavenumbers subtend smal ler angles w i t h the x-axis) backscattering f rom a low-frequency wave (k3). The backscattered wave has nearly the same frequency as the incident wave , the hor izontal wave numbers are s imi la r and the ver t ical wavenumbers are nearly opposite. The low-frequency component has nearly twice the ver t ical wavenumber and there is lit t le energy transfer to or from it. Ene rgy transfer occurs f rom the higher energy, high-frequency wave to the lower energy, high-frequency wave . The process effect ively damps out ver t ical asymmetry o f the energy flow i n the high-frequency por t ion o f the spectrum. Since the G M spectrum is ver t ica l ly symmetr ic , it is i n equ i l i b r i um w i t h respect to elastic scattering. In the ocean, the process is be l ieved to mode l the backscattering o f h igh-frequency waves f rom near-inertial waves [Miiller et al., 1986]. The s impl i f i ca t ion involves k h is the hor izonta l wavenumber a n d m is the vert ical wavenumber. Chapter 3. Oceanic Nonlinear Interaction Mechanisms 36 assuming that the wave act ion (the wave energy d i v i d e d by its intr insic frequency, E/o ) density is m u c h greater i n the l o w frequency wave (smal l a ), ^4(k 3 ) » A(k^) or A(k2), such that quadratic terms i n the govern ing transport equations i n v o l v i n g the two high-frequency waves can be ignored. Th i s s impl i f i ca t ion is not l i k e l y a realist ic assumption for the semidiurnal internal tide, since its intrinsic frequency is near that o f the inert ial frequency and it has, i n general, o n l y marg ina l ly less energy than inert ial osci l la t ions . 3.3.2 Induced Diffusion Induced dif fusion describes the interaction o f a high-frequency, h igh-wavenumber wave (ki or k2, i n F igure 3.1(b)) w i th a low-frequency low-wavenumber wave (kj), two waves strongly separated b y scale. The interaction produces a high-frequency, h igh-wavenumber wave (ki or k2) that is close i n frequency that o f the or ig ina l high-frequency wave . The high-frequency, h igh-wavenumber wave can be thought o f as propagat ing through a random low-frequency, large-scale wavef ie ld , where it experiences random perturbation o f its wavenumber . Since action, (E/o), is conserved a long a characteristic [Bretherton and Garrett, 1968], the perturbation induces di f fus ion o f wave act ion i n wavenumber space and results i n energy exchange w i t h the low-frequency, large-scale wavef ie ld . In the G M spectrum (and i n the real ocean, since the spectrum is based o n real ocean measurements) the spectrum o f ver t ical shear o f hor izonta l currents is the largest, thus induced dif fusion o f wave act ion is more dominant i n ver t ical wavenumber space. The s impl i f i ca t ion o f the governing equations results from assuming large spatial scale-separation between the interacting waves . F o r internal waves at Chapter 3. Oceanic Nonlinear Interaction Mechanisms 37 the loca l inert ial and semidiurna l frequency, it is not possible to close the tr iad equations o f frequency and wavenumber w h i l e satisfying the dispersion relat ion w i t h large spatial scale-separation. 3.3.3 Parametric Subharmonic Instability Parametric subharmonic instabi l i ty is the decay o f a large-scale higher-frequency wave (k3 i n F igure 3.1(c)) into two small-scale waves ( k i and k2) that have nearly opposite wavenumbers at h a l f the frequency o f the parent wave . The large-scale wave modulates the density field at twice the buoyancy frequency o f the small-scale waves analogous to a s imple pendu lum whose length is changed at twice its natural frequency. The instabi l i ty has been demonstrated in laboratory experiments b y McEwan and Robinson [1975], where it has been shown effective f rom large-scales through to fine microstructure scales. Since frequency resonance requires co, +co 2 =co 3 and sinceco,,co 2 > / the unstable wave must be at least If the mechan i sm is thought to p lay an important role i n transferring energy f rom large-scales i n the 2f to 4f frequency range to smal l scales i n the / to If range. Recent ly , Hirst [1996] has numer i ca l ly demonstrated that the subharmonic instabi l i ty o f the M 2 internal tide between the 28.8° parallels can be signif icant . 3.3.4 Validity A l t h o u g h almost a l l o f the current understanding o f nonl inear interactions among internal waves results f rom the appl ica t ion o f weak resonant theory, it has been c o n v i n c i n g l y argued Chapter 3. Oceanic Nonlinear Interaction Mechanisms 38 that oceanic internal waves are roughly t w o orders o f magnitude too energetic to be considered weak i n the context o f the theory [Holloway, 1980]. H o w e v e r , the development o f a strong interaction theory, such as direct interaction approximat ion , has been impeded b y the lack o f a sound theoretical basis and extreme computat ional d i f f icu l ty [Miiller et al. 1986]. Furthermore, strong interaction theory based on a turbulence mode l w o u l d lead on ly to d imens iona l sca l ing laws and ignore the observed w a v e l i k e behavior o f internal osc i l la t ions . Presently, both schemes are not i n a state to be di rect ly appl ied to the oceanic internal wavef ie ld . There, is also a question o f interaction t ime. It is general ly accepted that nonl inear interaction t ime scales must be larger or at least as large as the wave periods o f the interactants for the interaction to be considered weak [Holloway, 1982]. H o w e v e r , the re laxat ion t ime o f a band- l imi ted spectrum (a peak) superposed o n the G M spectrum indicates times that can be m u c h less than that o f the wave pe r iod over m u c h o f the spectrum [McComas, 1977, Holloway, 1982], thus seemingly contradict ing the weak interaction premise on w h i c h it was based. Miiller et al. [1986] c l a i m that the ratio o f interaction t ime to wave per iod is not necessari ly an adequate cri ter ion. T h e y c l a i m that the ratio is a measure o f the broadening o f the resonance surfaces, and, that i f the nonlinear interactions are \"coherent\" w i t h i n this broadened space, the va l id i t y o f weak resonant interaction is not v iola ted. That is, the interactions must be considered i n terms o f propagating dispersive wave packets and resonance occurs over a broadened v o l u m e o f phase space. W i t h respect to internal waves at inert ial and semidiurna l frequencies, i f a h i g h estimate o f wavenumber (100 m wavelength) is made, the interaction t ime w i t h i n a G M 7 6 spectrum [Cairns and Williams, 1976] is an order o f magnitude longer than that o f the wave periods. Thus , i n general , the interaction t ime cr i te r ion for the l o w -frequency, large-scale internal wavef i e ld is not v io la ted . Chapter 3. Oceanic Nonlinear Interaction Mechanisms 39 3.4 Other Models 3.4.1 Eikonal Method In react ion to questions o f the va l id i t y o f the weakness assumpt ion i n resonant wave-wave interactions, Henyey and Pomphrey [1983] and Henyey et al. [1986] use an e ikona l method that does not make a weakness assumption, but inherently emphasizes scale-separation. La rge scale separation confo rming to the resonance condi t ions put forth b y (3.2), (3.3), and (3.4) can occur i n the induced di f fus ion l imi t , where the two smal l waves must be separated i n frequency b y the frequency o f the large scale l o w frequency wave . H o w e v e r , the e ikona l technique is not based o n resonance and, as long as frequency condit ions imposed b y (3.2) are satisfied, can be appl ied to a l l scale-separated waves conforming to the dispers ion relat ion, (3.4). A l t h o u g h , it is not expected that inert ial and semidiurnal waves are strongly scale-separated, the actual scales are not k n o w n , and the l imits to w h i c h the scale-separation m a y be reduced w h i l e mainta ining the app l icab i l i ty o f the formulat ion are not clear. Hence , a b r i e f descr ip t ion o f the technique is inc luded here. E i k o n a l or ray tracing techniques have been often used to describe internal wave propagat ion i n low-frequency or steady geostrophic currents. Henyey and Pomphrey [1983] have extended the approach to assess the scale-separated induced di f fus ion l i m i t o f McComas and Bretherton [1977]. The technique consists o f mode l ing the small-scale por t ion as a superposit ion o f independent wave-packets, each centered o n a value o f k o c c u p y i n g a region near a point x . E a c h o f these packets move through the large-scale flow accord ing to the laws Chapter 3. Oceanic Nonlinear Interaction Mechanisms 40 o f part icle mechanics , thus their trajectories are g iven b y the solu t ion H a m i l t o n ' s ordinary differential equations o f mot ion . The H a m i l t o n i a n o f the set is g iven b y a d ispers ion relat ion H(k, x) = co = o~(k) + v 0 • k . The first term is the dispers ion relat ion i n absence o f large-scale f l ow , vo, and the second, a Dopp le r shift. Henyey and Pomphrey [1983] have shown that intr insic frequency term, a , can be de-coupled f rom the background buoyancy f ie ld and the interaction is entirely due to the Dopp le r shift term. The authors do not derive a transport equation, but rather per form numer ica l experiments w i t h 50 trajectories i n a background G M internal wavef i e ld to obtain estimates o f transport. The i r results for the induced dif fusion l imi t are i n variance w i t h those o f McComas and Bretherton [1977], w h o predict a net decrease i n frequency. The e ikona l method indicates a net increase i n frequency and hence an up-frequency f lux o f energy. Th i s result seems a bi t more realist ic and puts into doubt the va l id i ty o f w e a k l y resonant theory i n certain scale-separated high-frequency cases. 3.4.2 Direct Numerical Simulation The shortcomings o f the approaches that have been described w o u l d not occur w i th the direct numer ica l s imula t ion o f the hydrodynamic equations o f mot ion . Scale-separation and weakness o f interaction w o u l d not be l imitat ions. Furthermore, this approach w o u l d a l l o w interaction w i t h the \"vor t i ca l mode\" , the mode that carries potential vor t i c i ty at internal wave scales [e.g. Holloway, 1981, 1983; Miiller et al, 1986]. The vor t ica l mode has been heretofore ignored i n the Lagrang ian based studies r ev iewed i n this text since Lagrang ian methods assume on ly wave- l ike solutions. H o w e v e r , it is unclear h o w the inc lus ion o f potent ial vor t i c i ty w o u l d affect the observed fluctuations i n the internal wavef ie ld . Chapter 3. Oceanic Nonlinear Interaction Mechanisms 41 The major drawback, w h i c h incapacitates the pract ical appl ica t ion o f numer ica l s imula t ion , is the l imitat ions imposed b y computat ional ab i l i ty . A t present o n l y greatly s impl i f i ed , l o w resolut ion, or two-d imens iona l cases have been tractable. 3.5 Resonant Interaction Theory 3.5.1 Discrete Interaction A s was noted b y Phillips [1960], when equations o f the form (3.2) and (3.3) are satisfied and the dispers ion relat ion holds for the part icipat ing waves, the condit ions for resonance are met and there is systematic transfer o f energy among the waves. F o r the discrete interaction o f a single tr iad o f internal waves , there are many solutions to the resonance condi t ions. F o l l o w i n g the general approach o f Hasselmann [1966], the equations o f motions for a quadrat ical ly nonl inear sys tem may be cast in to the form, a 3 + ico 3 a 3 = - e ico 3 ^ r _ 3 1 2 a , a 2 . (3.5) k,+k2=k3 Here we have used the notation o f Mtiller et al. [1986], where a 3 = a 3 e \" \" ° 3 ' is the ampli tude o f the generated wave w i t h wavenumber k3, and frequency co 3 . T _ 3 1 2 is the coup l ing coefficient o f waves (k , ;co , ) and ( k 2 ; c o 2 ) to ( k 3 ; c o 3 ) and s is a smal l parameter characterizing the weakness o f the nonl inear interactions. The linear solut ion is obtained by setting e to zero (no Chapter 3. Oceanic Nonlinear Interaction Mechanisms 4 2 nonlinear interaction terms), a l l o w i n g a free l inear wave o f constant ampli tude to satisfy (3.5). A t higher orders, the nonlinear interaction terms act as forc ing to the zero order equation and (3.5) can be so lved by a wave w i t h ampli tude a,. = aft) that is a s l o w l y va ry ing complex funct ion o f t ime. Thus , dur ing resonance, the evolu t ion o f the amplitudes o f a discrete triad o f waves are governed by , d , = —£ ico , r *a 2 *a 3 , a 2 = - s ico 2 r*a,*a 3 , (3.6) d 3 = - e i c o 3 ra ,a 2 , where * signifies the complex conjugate and r = r_312. F o r (3.6) to be v a l i d , energy, E = a,a,.*, and momentum, P,. = (a,a,* / co,)k,. must be conserved. A p p l y i n g the condi t ions o f 3 3 resonance it can be s h o w n that ^ Ei and ^ P,. are indeed constant. ;=i 1=1 The equations govern ing the interaction o f waves are analogous to those o f interacting particles i n theoretical phys ics . In this analogy, the wave act ion, A - (a,a,.* /co,.), represents the number o f waves and l ike interacting particles is not necessari ly conserved (i.e. one particle interacts w i t h another single part icle to produce a third part icle) , but changes i n accordance to, S_AL = dAL = _d_A1 dt dt dt ' K ' } One uni t o f wave act ion o f wave 1 reacts w i t h one uni t o f wave ac t ion o f wave 2 to produce one unit o f wave act ion o f w a v e 3. Chapter 3. Oceanic Nonlinear Interaction Mechanisms 4 3 Var ia t ions o f (3.6) occur i n the literature and the coup l ing (or interaction coefficient) is described as an algebraic combina t ion o f contr ibut ing wavenumbers [Hasselmann, 1967; McComas and Bretherton, 1977; Bretherton, 1964]. E x p l i c i t expressions for the coup l ing coefficient for nonlinear transfer among three waves can be found i n Midler and Others, [1975]. T w o caveats apply to the discrete interaction equations, one, that the interaction must be sui tably weak; and two , the interaction must be str ict ly discrete. The more the former is v io la ted the less l i k e l y the interaction w i l l be discrete. Furthermore, these restrictions are l i k e l y to be v io la ted i n the real ocean, where, i n the case o f internal waves , w h i c h are dispersive and as such their dynamics preclude long interaction times among discrete wave packets, the net energy transfer is the result o f many resonant triad interactions. F o r the situation be ing examined i n this thesis, the frequency spectra can be v i e w e d as discrete l ine spectra at inert ial , semidiurnal and combina t ion frequencies superposed o n background spectra, w h i c h conforms to the G M spectrum. The wavenumber spectra are yet to be determined, and m a y not be determinable; however , understanding the effect o f var ia t ion o f wavenumber o n the eff ic iency o f energy transfer is sought. 3.5.2 Transport Theory In the previous section, the equations relat ing the g rowth o f the ampli tude o f one component based on the amplitudes o f the other components o f an interacting tr iad were presented. This describes the energy sharing among the three interactants, but does not establish the net amount o f energy transfer among many components . F o r the evaluat ion o f the energy transfer due to many nonlinear interactions among resonant triads through the Chapter 3. Qceanic Nonlinear Interaction Mechanisms 44 wavef ie ld , transport theory is used. T h e formulat ion is der ived b y c los ing the evolut ion equation for the energy density spectrum. A l t h o u g h there are more mathemat ical ly r igorous variat ions o f this der ivat ion by Hasselmann [1966; 1977], and Benney and Saffman [1966], the s impl i f i ed development o f Midler et al. [1986] is adhered to here i n order to c lar i fy the assumptions inherent i n the approximations. Substi tut ing a^Oe-\"0'' = where a,, is n o w a s l o w l y va ry ing function o f t, into (3.5) and rewr i t ing i n terms o f a 3 , produces, d 3 = - e ico 3 Jdk,dk 2ra,a 2e' A '8 (k , + k 2 - k 3 ) (3.8) where A represents co3 - co, - co2 and the con t inuum l imi t ^ -»J dk has been taken. k In order to evaluate (3.8) Muller et al. [1986] outline the f o l l o w i n g supposit ions. 1. T o integrate (3.8) assume that there is a sui tably s l o w t ime scale for the nonlinear interactions such that a , and a 2 can be considered constant over 0 < t < T. It is also necessary to a l l ow the existence o f near-resonant interactions to reveal the nature o f the transport equation as it nears resonance, as A —> 0 . (3.8) can then be integrated to g ive , At a 3 ( / ) - a 3 ( 0 ) = - e i c o 3|jk,cik 2 r a l ( 0 ) a 2 ( 0 ) e -|A( s i n -2 1 - 2 A / 2 5 ( k , + k 2 - k 3 ) (3.9) 2. T h e wavef ie ld is assumed to be Gauss ian , that is, a l l the mode amplitudes are statist ically independent over a t ime interval T, or that the correlat ion t ime o f the Chapter 3. Oceanic Nonlinear Interaction Mechanisms 45 wavef ie ld is short w i t h respect to T. The change i n energy can then be obtained by m u l t i p l y i n g each mode b y its complex conjugate and (3.9) becomes, . 2 sin — kW|)2 - ( | a 3 ( 0 ^ ) 2 ^(e(a)2\\dk,dk2\\T\\\\\\aX0)\\)\\\\a2(0f ^ T 6 ^ + k 2 ~ki) (3J0) and other s imi lar terms. The ( ) indicates ensemble averages. I f / is a l l o w e d to go to inf in i ty as resonance is neared, A —> 0 , and the replacement, ^ ^ f ( ^ —> no ( A ) is ( A / 2 ) ' exact. B u t accord ing to assumption 1 , / is l imi t ed to between 0 and T. 3. The replacement is o n l y l i k e l y to be accurate i f the f o l l o w i n g assumption is made. The var ia t ion o f |r|2^|a,(0)|^ (|a 2(0)|^ is smal l (the coup l ing coefficients and spectra are smooth) over T. The c losed evolu t ion equation for the energy densi ty spectrum (the transport equation) can then be wri t ten, 5(k,+k 2-k 3)., It is wor thwhi le to examine the basic statistical closure hypothesis used here, assumption 2. The assumption impl ies that, al though resonant interactions are cont inuously redis tr ibut ing energy in the wave spectrum, the essential independent statistical structure remains unchanged. Th i s does not readi ly seem self-consistent, however it has been p roven i n statistical mechanics Chapter 3. Oceanic Nonlinear Interaction Mechanisms 46 [Prigogine, 1962,], that for inf in i te ly weak nonlinear coup l ing among dispersive wave modes, the assumption is correct as long as the wavef ie ld is r igorous ly Gauss ian to beg in w i th . Hasselmann [1966] justifies this premise b y analogy to scattering i n part icle phys ics . H e reasons that when independent wave trains intersect and they satisfy resonance condi t ions , energy transfer occurs . D u r i n g this l imi ted pe r iod o f interaction, the coup l ing results in a weak statistical dependence. H o w e v e r , d i rect ly after the interaction the wave trains propagate into a new region, where they m a y interact w i t h resonant partners f rom a new wavef i e ld that is n o w assumed to be statistically independent. Th i s assumption can be made because a dispersive nonlinear wavef ie ld rap id ly attains r a n d o m phase (i.e. a Gauss ian state) [Prigogine, 1962]. Then , the net energy transferred is the result o f numerous interactions between stat ist ically independent components o f the wavef ie ld . The degree to w h i c h the oceanic internal wavef i e ld is Gauss ian has been investigated b y Briscoe [1977]. H i s conc lu s ion is \"most o f the t ime, but not a lways\" , and that there is l imi ted evidence that variance bursts or case o f strong energy input into the internal wavef ie ld lead to non-Gauss ian states. Cons ide r ing the energy spectra shown i n F igure 2.3, it is l i k e l y that the episodic inert ial energy as w e l l as the episodic internal tide lead to ocean states that are not r igorous ly Gauss ian . A l t h o u g h this violates the closure hypothesis, it is not k n o w n to what degree mino r variations from Gauss ian i ty w i l l affect the outcome o f energy transfer calculat ions us ing weak resonant theory. Chapter 3. Oceanic Nonlinear Interaction Mechanisms 47 3.5.3 The Transport Equation F o r oceanic internal waves the expl ic i t fo rm for the transport equation is , (3.12) It represents the energy transfer due to quadratic coup l ing through the evo lu t ion o f the wave act ion spectra, A. Th i s fo rm o f equation is often ca l led a B o l t z m a n n integral and was first der ived for interacting lattice vibrat ions b y Peierls [1929]. The transfer functions 1* and T~are often interpreted as posi t ive and negative c o l l i s i o n cross-sections and F e y n m a n n diagrams can be used to examine the interactions. H o w e v e r , there are some inconsistencies i n the interpretation o f these terms as c o l l i d i n g particles and/or anti-particles [McComas and Bretherton, 1977], and it may be more appropriate to consider them as s u m and difference interactions. F o r energy transfer among inert ial , semidiurna l and JM2, o n l y the s u m interactions need be considered. Further d i scuss ion us ing the particle phys ics analogy and F e y n m a n n diagrams may be found i n Hasselmann [1966, 1967]. S imp l i f i c a t i on and c lar i f ica t ion can result i n breaking up the transport equation into an energy balance equation o f the form, d_ dt 4 k ) = - 2 v , ( k ) 4 k ) + / ( k ) . (3.13) Chapter 3. Oceanic Nonlinear Interaction Mechanisms 48 This fo rm is k n o w n as the L a n g e v i n form where vp is defined as the L a n g e v i n rate. The rate that energy is scattered out o f wavenumber k is g iven by 2v p (k), wh i l e the i n c o m i n g rate o f energy due to wave-wave interactions is g iven by , ~^7~^ • V P an& are defined, Vp(k3) = | j Jdk I dk : T+b (k3 - k , -k2)5(co3 -co, - © 2 J ^ ( k , ) + A(k2)]-t IT'b (k3 - k , +k2)5(co3 -co, +co2)[^(k2)-^(k,)] (3.14) and /(k 3)=Jjdk,dk; r + 8 ( k 3 - k, - k 2 )5 (co3 - co, -co2) + 2T~b(k3 - k , + k 2 )5 (co3 -co, +co2) 4k,)4k2) (3.15) The L a n g e v i n rate can be der ived independent from the transport equation b y L a n g e v i n methods and does not require a r andom phase approx imat ion (i.e. an in i t i a l Gauss ian state)[Pomphrey, 1981]. T w o more quantities are defined as the Bo l t z rnann rate, 2v s (k) = ^ ^ ' ( k ) , (3.16) the rate o f change o f the spectrum, and the energy input to the wavenumber k, Chapter 3. Oceanic Nonlinear Interaction Mechanisms 49 / ( k ) = 2vF = 2vB + 2vp (3.17) 4 k ) 3.6 Summary and Conclusions A rev iew was conducted o f the development o f wave-wave interactions theories for oceanographic problems. A l t h o u g h there is no well-establ ished theory/model for finite ampli tude or strong interaction among internal waves , the bu lk o f the knowledge pertaining to the nonl inear interaction among internal waves has come from the appl ica t ion o f resonant weak wave-wave interaction theory. B o t h the geometr ical phys ics and eff ic iency o f resonance are w e l l grounded. Furthermore, the theory effectively explains m u c h o f the dynamics mainta in ing the universa l shape o f the Gar re t t -Munk spectrum and resonant interaction computations o f down-scale energy flux through the G M spectrum are compat ib le w i t h other estimates o f energy flux through the internal wave spectrum [Miiller et al., 1986]. H o w e v e r the range o f va l id i t y o f resonant interaction theory is not k n o w n and its appropriateness for the oceanic internal wavef i e ld is i n quest ion [Miiller et al., 1986; Holloway, 1980]. Chapter 4 50 Geometrical Considerations and Coupling Efficiency 4.1 Introduction In this chapter, I consider the geometr ical constraints imp l i ed b y triad resonance and establish a set o f rules for interacting waves. Th i s is f o l l owed b y an examinat ion o f the coup l ing eff ic iency w i t h i n these bounds. The general case is considered first, and then the specific case o f interacting inert ial , semidiurnal and JMi waves i n the northeast Pac i f i c . The g loba l var ia t ion is examined us ing a s imple mode l o f triad resonance i n w h i c h the inert ial wave frequency scales w i t h latitude. Chapter 4. Geometrical Considerations and Coupling Efficiency 51 4.2 Geometric Considerations 4.2.1 Preliminary The condi t ions for resonance, combined w i t h the dispers ion relat ion for internal gravi ty waves, prescribe a set o f geometric restraints w i t h w h i c h a nonl inear ly interacting triad o f waves must comply . The dispers ion relat ion for internal waves propagat ing in a cont inuously stratified med ium, (3.4), can be wri t ten as a function o f the angle, cp , the wavenumber vector (wavevector) makes w i t h the hor izonta l plane, as fo l lows , Here , / and /V are the loca l C o r i o l i s parameter and the B r u n t - V a i s a l a frequency, respectively. F r o m (4.1) it can be seen that, un l ike a surface gravi ty wave , the frequency o f an internal gravi ty wave , i n a cont inuous ly stratified m e d i u m , is independent o f the magnitude o f its wavevector . Thus , for a g iven stratification and latitude, the frequency is o n l y a function o f the angle the wavevector makes w i t h the gravitat ional vector. Solut ions for co are found f r o m / t o N, and us ing the geometr ical re lat ion, s i n 2 cp = 1 - c o s 2 cp , to solve e x p l i c i t l y for cp , g ives , co 2 = f2 s i n 2 cp + TV 2 co s 2 cp . (4.1) (4.2) Chapter 4. Geometrical Considerations and Coupling Efficiency 52 F o r each frequency, the fami ly o f possible solutions traces a ver t ica l ly oriented cone i n wavenumber space as i l lustrated i n F igure 4 .1 . U s i n g values for loca l C o r i o l i s , Brunt -Vaisa la ' and observed wave frequencies f rom station E R A 2 , over Endeavour R i d g e , the solut ion o f (4.2) for inertial , semidiurnal and /M2 frequency waves results i n the curves shown i n F igu re 4.2. Here , the B r u n t - V a i s a l a frequency (A7) is estimated from a representative B r u n t - V a i s a l a profi le constructed from a C T D cast i n the v i c i n i t y o f Endeavour R i d g e [Thomson et al, 1990]. The smoothed profi le has a m a x i m u m value o f N o f approximate ly 192 c p d at the top o f thermocl ine. B e l o w 400 m , the profi le varies nearly l inear ly w i t h depth, beg inn ing w i t h an N o f approximate ly 48 c p d and decreasing to an estimated 19.8 c p d at 1757m, the depth o f the current meter at station E R A 2 . The parameters used to derive the direct ion o f wavevectors for t h e / , M2, a n d7M2 frequencies at stations E R A 2 and C X - 1 for three states o f stratification are summar ized i n Tab le 4 .1 . Station L o c a l C o r i o l i s Observed Inertial Observed Observed 7M2 Frequency (/) Frequency Semid iurna l Frequency Frequency E R A 2 1.49cpd 1.54cpd 1.93cpd 3 .46cpd C X - 1 1.45cpd 1.50cpd 1.93cpd 3 .40cpd Table 4.1. Representative values from ERA2 and CX-1. Observed frequencies are the peak frequencies from rotary spectra with 50 degrees of freedom. Chapter 4. Geometrical Considerations and Coupling Efficiency 53 Figure 4.1. The conic surfaces showing the possible propagation directions prescribed by two frequencies of wavevectors originating from the same point. The wavevector di rect ion is a re la t ively strong function o f depth due to the var ia t ion o f /V w i t h depth (relative to a function o f latitude as a result o f the var ia t ion o f / ) . T h i s is i l lustrated Chapter 4. Geometrical Considerations and Coupling Efficiency 5 4 i n F igure 4.2 by the f ami ly o f curves JV=1.?.8 c p d (1757 m) , 7V=48.0 c p d (400m), and 7V=192 c p d (40 m). Th i s f ami ly o f curves is intersected by the o b s e r v e d / , M 2 , and / V I 2 frequencies at station E R A 2 . The curve /V=19.8 cpd , the dispersion relat ion at 1757 m , is surrounded by a dashed curve indica t ing a plus o r minus 10% varia t ion o f the B r u n t - V a i s a l a frequency. A l s o inc luded is the dispers ion curve wi thout rotation to illustrate the relative sensi t ivi ty o f the dispers ion relat ion to / and TV. The curve shows that rotation has about the same effect on propagat ion angle for the jM.2 frequency as a 10% var ia t ion o f N. H o w e v e r , at lower frequencies nearing the inert ial frequency,/, rotation plays a m u c h stronger role i n determining the propagation angle. The C T D cast used for these buoyancy frequency estimates was taken i n early September and reflects a re la t ively strongly stratified state. Ca lcu la ted values for wavevector angle, 9 , are summar ized i n Tab le 4.2 for stations E R A 2 and C X - 1 . Stat ion Dep th 9 / +/v cos (j) 1.5 0.5 X ^ X >v X ^ T § S. x \\ X I I = l ^ c p d X^ x X. s X . >v V \\<-A/=48 :pd : \\<^X^\\<)2_ co=3.46 cpd - : < X ' ' X -• • • •. x v V XJ x >JM , intere^ctibn>»tv6=81'i0 degrees ?V X. N X* X. X \\ . \\ N. X > X. v N. v. \\ x. x x s. . X co= 1.93 cpd-• ^ \\ >s-> M_, intersection at d)=86.3 degrees ^ x. X X ^*X. \" X X X ^X X. \\ \\ co=l .54 cpd - > / intersection at d>=88.9 degrees \\ X>-/=0, A^I9.8c l l I I x cpd 76 78 80 82 84 in degrees from the horizontal 86 88 90 Figure 4.2. Dispersion relations at station E R A 2 for three values of Brunt-Vaisala frequency intersecting internal waves ntf, M2, and/M2 frequencies. Since internal waves w i t h specific frequency generate ver t ica l ly oriented cones (Figure 4.1), a c losed resonant wavevector triad impl ies the intersection o f three cones; t w o cones or ig inat ing f rom the same point , and the third cutt ing both cones at the terminal points o f the wavevectors . The locus o f the intersection o f these three cones is a compl ica ted three d imens iona l curve. De te rmin ing the relationship between wavevectors o f a resonant triad m a y be s imp l i f i ed b y consider ing the extremes o f the possible combinat ions i n three-dimensions. In Chapter 4. Geometrical Considerations and Coupling Efficiency 56 Figure 4 .3 , three waves are depicted such that co, < co2 < co 3 . In this case, accord ing to the linear d ispers ion relat ion for internal waves , the angle 0 (the angle the wavevec tor makes w i t h the gravitat ional vector) satisfies the relat ion, 0, < 9 2 < 0 3 . Here , the wavevector w i th the highest frequency is set as the reference and is chosen to propagate upwards . D o w n w a r d propagat ion o f this highest frequency wavevec tor w o u l d result i n a mi r ro r image about the k-l (horizontal wavenumber plane) i n F igure 4.3. F r o m Figure 4.3 , a number o f important features can be ascertained for the interaction o f a triad o f waves w h i c h confo rm to co, < co2 < co 3 . • T h e two lower frequency wavevectors ( k i , k 2 ) o f a resonant tr iad must have ver t ical wavevectors o f opposite s ign. • There are two classes o f interactions that can then occur : 1) wavevector k i has the same ver t ical wavenumber s ign as wavevector k 3 , and 2) wavevector k 2 has the same ver t ical wavenumber s ign as wavevector k 3 . • W i t h i n each o f the two classes, there is a m i n i m u m (max imum) wavenumber ratio for 1^ ' I 1^2 I both - j— | and-j—r that occurs when the lowest frequency wavevector ( k i ) has the same F 3 I | k 3 | (opposite) hor izonta l wavevector di rect ion as the highest frequency wavevec tor ( k 3 ) . These extremes occur i n wavenumber ratio w h e n a l l three wavevectors and the gravitat ional vector are i n the same plane and correspond to the four triads, A , B , C , and D i n F igure 4.3. Chapter 4. Geometrical Considerations and Coupling Efficiency 57 • These extremes impose discrete w i n d o w s i n relative wavenumber magnitude w i t h i n w h i c h the closure o f resonant triads can occur . In order to solve for these discrete w i n d o w s o f relative wavenumber , four triads (triangles) are identif ied i n F igure 4.3. T h e y are denoted by A , B , C , and D at the vertex formed b y the intersection between the two lower frequency wavevectors k i and k 2 (or p r imary waves in the case o f s u m interactions) and termed the l i m i t i n g triads. K n o w i n g a l l the interior angles o f the triangles and us ing the L a w o f Sines, the relative magnitudes o f the three wavevectors par t ic ipat ing i n the tr iad may be established. In F igure 4-3, triangles A and B comprise the class i n w h i c h the lowest frequency wavevec tor ( k i ) has the same ver t ical wavenumber s ign (or ver t ical propagat ion direction) as the highest frequency wavevector ( k 3 ) . C and D correspond to the class where the midd le frequency wavevec tor ( k 2 ) has the same ver t ica l propagat ion di rec t ion as the high-frequency wavevec tor ( k j ) . Chapter 4. Geometrical Considerations and Coupling Efficiency Figure 4.3. The four l imi t ing triads i n phase space for sum interactions for the situation co, ••Ill l l l i l l l i l 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c lc (JM ) g g V 2 Group Velocity M 2 . S S ; / M 2 liHH i i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c g / C g ( /M 2 ) Figure 4.4. Range of wavelengths and group velocities of inertial and semidiurnal 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 between inertial and semidiurnal waves. Chapter 4. Geometrical Considerations and Coupling Efficiency 62 4.2.3 Global Variability In order to investigate the impact o f latitude o n the relative wavenumbers and group veloci t ies among interacting iner t ia l , semidiurnal and JM2 waves , a s imple m o d e l o f the var ia t ion o f wave frequencies w i t h latitude for the sum interaction, co, + co2 = co 3 , has been constructed. T h e frequency, co 2 , o f the semidiurnal wave is fixed for a l l latitudes at 1.93 cpd. The frequency o f the inert ial wave , co,, is determined by blue-shif t ing the loca l C o r i o l i s frequency by a factor o f 1.0333. The factor 1.0333 reflects the mean ratio o f observed inert ial wave frequencies and the C o r i o l i s frequency at stations E R A 2 and C X - 1 . The c r i t i ca l latitude for semidiurnal internal waves at the M2 frequency o f 1.93 c p d is 7 4 ° . A b o v e this latitude, internal waves at the M2 frequency become evanescent (i.e. cannot propagate as free waves) . M o r e important to consider here is the latitude at w h i c h the blue-shifted inert ial waves surpass the frequency o f 1.93 c p d and the inequali ty, co, = - i c o 2 r * a , * a 3> (4.8) r3a3 - - i c O j I T a ^ . dt These equations are essential ly (3.6) w i t h the smal l p a r a m e t e r s , absorbed b y the expl ic i t coup l ing coefficient , T. Chapter 4. Geometrical Considerations and Coupling Efficiency 73 Real T Triad A->B 300 0 30 60 90 120 150 180 Magnitude I~ Triad A-»B 270 0 30 60 90 120 150 180 Real T Triad C->D -300 0 30 60 90 120 150 180 horizontal angle 6H > between and 400 Imaginary Y Triad A-+B -400 320 0 30 60 90 120 150 180 Magnitude I\" Triad C—>D 400 0 30 60 90 120 150 180 Imaginary T Triad C-»D 0 30 60 90 120 150 180 horizontal angle 6H > between k^ and k^ Figure 4.8. The coupl ing coefficient, T , as a function o f horizontal wavevector angle between k i and kj as defined by Figure 4.3. The ver t ical propagation angles o f k, and k 3 are set by the frequencies o f / a n d / M 2 at station E R A 2 thus k, and k 3 are equivalent to k/ and at that station. T r i ads A and B correspond to/.ss./IVfc and triads D and C to M2 . s s . yM2. here T is spec i f ica l ly r_ 3 I 2 and includes the smal l parameter, e , shown i n (3.6). Th i s is the coup l ing coefficient used i n a l l subsequent plots . G r o w t h o f k3 requires that ti3 be real and posi t ive . H o w e v e r the values for a,, are complex and are set by the ambient wavef ie ld w h i c h is an u n k n o w n . Th i s makes it diff icul t or impossible to determine or predict the coup l ing eff ic iency due to ind iv idua l discrete interactions. Therefore, the calculat ions to fo l low on ly consider the coup l ing ef f ic iency as it relates to the spectral transfer function that is, the absolute Chapter 4. Geometrical Considerations and Coupling Efficiency 74 value o f the coup l ing coefficient , T. A s a result, the coup l ing ef f ic iency can be considered a funct ion o f stratification, C o r i o l i s frequency and hor izonta l angle. Furthermore, it is shown i n F igure 4.8 that Y is w e l l behaved between the l imits imposed b y the l i m i t i n g triads and the l i m i t i n g triads are at, or very close to, the m a x i m u m and m i n i m u m o f the range. Th i s smooth behavior between extremes is exhibi ted at a l l latitudes and permits the considerat ion o f o n l y the coup l ing coefficients o f the l i m i t i n g triads, A , B , C , and D as functions o f stratification and C o r i o l i s parameter, to characterize the coup l ing eff ic iency g loba l ly . Va lues for the four l i m i t i n g triads are plot ted i n F igure 4.9 and F igure 4.10 for four states o f stratification. T h e four states o f stratification are the three described earlier i n the chapter, A^=19.8 cpd , JV=48 c p d and JV=192 cpd , plus an estimate o f the lowest expected stratification for the region, iV=10 cpd . These figures indicate that the coup l ing ef f ic iency is inversely related to the B r u n t - V a i s a l a frequency, thus the thermocline has the lowest eff ic iency and the regions w i t h the weakest stratification, have the greatest eff iciency. T h e latitude dependency, also i l lustrated in F igure 4.9, and F igure 4.10, shows a tendency among a l l l i m i t i n g triads to have larger coup l ing coefficients at higher latitudes. The coup l ing coefficient, however , is a m u c h stronger function o f stratification, spanning an order o f magnitude for real ocean stratification. The transfer function g i v e n b y (4.4) characterizes the spectral energy transfer and scales w i t h the square o f the coup l ing coefficient indicat ing that regions w i t h weaker stratification are also regions w i t h enhanced energy transfer. The lat i tudinal dependency o f the transfer function, T, scales w i t h the square o f the coup l ing coefficient and the frequencies o f the interacting waves . Since two o f these frequencies increase w i t h latitude (one remains Chapter 4. Geometrical Considerations and Coupling Efficiency 75 constant), the tendency is to further enhance the energy transfer w i t h latitude. It is important to note, however , that the energy transfer is also d r iven by the relative energetics o f the interacting wavefie lds as indicated b y the overa l l spectral transport equation (3.12) and greater coup l ing eff ic iency does not necessari ly indicate that there is greater energy transfer. Chapter 4. Geometrical Considerations and Coupling Efficiency Figure 4.9. The magnitude of the coupl ing coefficient, Y, for the four l imi t ing triads as a function of latitude for /V=10 cpd and /V=19.8 cpd . T r i ads A and B correspond to the case, / S S . / M 2 , and triads C and D to the case, M2.ss.yM2. The red dotted l ine signifies the cut-off latitude for t r i ad interactions b e t w e e n / a n d M 2 . Chapter 4. Geometrical Considerations and Coupling Efficiency r Triad A, N=48 cpd T Triad A, N=192 cpd T (s/(m/m)) r (s/(m/m)) Figure 4.10. The magnitude of the coupling coefficient, Y, for the four limiting triads as a function of latitude for A=48 cpd and A=192 cpd. Triads A and B correspond to the case,/ss./M2, and triads C and D to the case, M2.ss.yM2. The red dotted line signifies the cut-off latitude for triad interactions between / and M 2 . Chapter 4. Geometrical Considerations and Coupling Efficiency 4.4 Summary and Conclusions 78 The geometr ical restraints o n triad interactions indicate a set o f rules for resonance for sum interactions o f inert ial and semidiurnal waves . The inertial and semidiurna l waves must be propagating i n opposite ver t ical directions to close a triad. T h i s restr ict ion sets up two cases, one, when the s u m 7M2 wave propagates i n the same d i rec t ion as the inert ial wave and, two , when it propagates in the same di rec t ion as the semidiurnal wave . Re la t ive wavelength (or wavenumber) bandwidths o f acceptance can be set up for each o f these cases and the knowledge o f one o f the wavelengths enables the determination o f the range for the other two components. T h i s is summar ized in F igure 4.4 for station C X - 1 , w h i c h m a y be considered representative o f the latitudes o f the data presented i n this thesis. The figure indicates that the resultant wavelength o f the /M2 wave is a lways greater than either that o f the p r i m a r y / and M2 waves, and that the ratio o f p r imary wavelengths must fal l into two bands to form triads. The bands are nar row and indicate that relative wavelengths o f the p r imary waves must be w i th in a factor o f two . F igure 4.4 shows that the group veloci t ies o f the resultant /M2 wave are a lways m u c h greater than that o f the p r imary waves (this is also true for the phase veloci t ies) . The structure o f resonant triads, that is , the relationships between the part icipat ing wavevectors i n magnitude and d i rec t ion are a function o f latitude and hence vary g loba l ly . A s imple mode l shows that the reg ion o f acceptable wavenumbers for resonant triads, F igure 4.6, is dist inct for the two cases, fssfM.2 and M2.ss.yM2. Th i s \" r eg ion\" for the /SS . /M2 case is o n overa l l , broader than the M2.SS./M2 case and increases w i t h latitude, whereas for the M2.SS./M2 case, the \" r eg ion\" has a sl ight tendency to decrease wi th latitude. It is further noted that for the M2.SS./M2 case, there is a great dispari ty between the wavelength o f the waves nearing the Chapter 4. Geometrical Considerations and Coupling Efficiency 79 equator such that the dispari ty m a y inhibi t the formation o f resonant triads. F o r the above reasons it is thought that the /SS. /M2 case is the dominant case for the transfer o f energy among inert ial , semidiurna l and JM2 waves Investigation o f the coup l ing eff ic iency at a single station reveals that there was little change w i t h var ia t ion o f hor izonta l angle for spectral energy transfer. I f a resonant tr iad is established, the hor izonta l propagat ion angles o f the par t ic ipat ing waves do not strongly influence the coup l ing eff iciency. Hence , the propagation directions o f inert ial waves created b y t rave l l ing w i n d fields and the semidiurnal internal wavef i e ld generated b y topographic effects w i l l not p lay an important role i n the eff ic iency o f energy transfer but rather i n the format ion o f resonant triads. Stratif ication as a function o f depth, o n the other hand, has a strong effect on coup l ing eff iciency, w i t h weaker stratification result ing i n greater coup l ing eff iciency. The other factor, the var ia t ion o f C o r i o l i s parameter w i t h latitude, has a weaker influence than stratification. In a l l cases, the c o u p l i n g coefficient is greater at h igher latitudes. O v e r a l l , the geometric structure o f resonant triads and the coup l ing eff ic iency both i m p l y that higher latitudes and weak stratification are favored for energy transfer among waves o f inert ial , semidiurna l and jM2 frequencies. Chapter 5 80 Spectral Analysis 5.1 Introduction In this chapter, current meter t ime series f rom t w o stations, E R A 2 and C X - 1 are examined i n detai l w i t h 2 n d order rotary spectral methods. H a r m o n i c analysis is first used to separate the barotropic and barocl in ic t ida l currents and then rotary spectral analysis is used to investigate the energetics i n the inert ial , t idal and interaction frequency bands. The \"interact ion band\" is the frequency band conta ining the /M2 and M 4 frequency peaks (3.20-4.20 cpd). Tempora l variations are estimated us ing a demodula t ion technique s imi la r to wavelet analysis for the rotary ve loc i ty components and by the direct filtering o f the t ime series o f hor izonta l currents. U s i n g the avai lable estimates o f wavenumbers for near-inertial waves , some o f the pr inc ip les developed i n Chapter 4 are appl ied. Chapter 5. Spectral Analysis 81 5.2 Observations The t ime series o f hor izonta l currents f rom stations C X - 1 and station E R A 2 have been chosen for detailed analysis because they exhibi t strong interaction peaks at the frequencies o f /M2 and M 4 . Station E R A 2 is located over the Endeavour segment o f Juan de F u c a R i d g e at 47°57 .0 ' N , 129°5.7 ' W at a depth o f 1757m, 350 m above the bot tom. Station C X - 1 , also over the Juan de F u c a R i d g e , is located on the C o - A x i a l segment at 4 6 ° 2 0 . 1 ' N , 1 2 9 ° 4 2 . 6 ' W at a depth o f 2050 m , 200 m above the bot tom. The time series were made up o f 60 m i n samples commenc ing J u l y 26, 1994 for station E R A 2 and October 11, 1993, for station C X - 1 . R e c o r d lengths were 7025 h and 6879 h , respectively. M o r e informat ion o n the study reg ion and the method o f measurement can be found i n Chapter 2. 5.3 Harmonic Tidal Analysis Since internal wave energy contributes o n l y to barocl in ic mot ions , the examinat ion o f internal waves is facil i tated b y the r emova l o f the variance o f hor izonta l currents due to barotropic motions. The as t ronomical ly forced barotropic t idal currents are determinist ic and have stationary per iodic i ty ; thus, they are predictable and can be effect ively r emoved from the current f ie ld. The internal t idal currents are also the result o f the as t ronomical forc ing but have other con t ro l l ing factors such as stratification and bot tom topography and they are presently not predictable. The drawback o f r emov ing the determinist ic barotropic tide is that often an indeterminate amount o f the internal t ida l currents are phase- locked to the barotropic t idal Chapter 5. Spectral Analysis 82 currents, and w i l l be removed as w e l l . Recent measurements i n the central northern Pac i f ic f rom an acoust ical tomographic array [Dushaw et al, 1995] indicate that a significant por t ion o f the dominant first mode semidiurnal internal tide is phase- locked to the ast ronomical barotropic forc ing. Th i s internal tide is thought to have traveled coherent ly over 2000 k m from the H a w a i i a n R i d g e . Nearer to the present study regions i n the northeast Pac i f i c , the analysis o f tomographic measurements b y Bracher and Flatte [1997] d i d not dis t inguish between barotropic and baroc l in ic modes. It d id reveal , however , that a first mode internal tide, t ravel l ing from the G u l f o f A l a s k a , cou ld account for the characteristic var ia t ion i n the acoustic data and w o u l d have a signif icant energy flux (800 W r a ' 1 ) associated w i t h it. F o r the data presented here, it is not k n o w n h o w m u c h o f the internal t idal current s ignal is phase- locked to barotropic internal t ide, and thus, the remova l o f the deterministic currents may result in an underest imation o f the internal semidiurnal t idal energy. The method o f t idal predic t ion used here is a least-squares fit o f the t idal harmonics . The frequencies o f the t ida l harmonic constituents are set by the as t ronomical forc ing. A subset o f these can then be chosen to predict the tide o n the bases o f their relative par t ic ipat ion i n the tide p roduc ing potential , OT, [see Gill, 1982] and the their reso lvabi l i ty w i t h respect to the length o f the t ime series. Tables o f the relative contr ibut ion o f t idal constituents to the equ i l i b r ium tide, O ^ / g , i n the form ampli tude factors can be found in Gill [1982] and, normal ized by the pr inc ipa l M2 t idal constituent, i n Emery and Thomson [1997]. The resolvabi l i ty o f the t idal constituents is based o n the R a y l e i g h cr i ter ion, w h i c h s i m p l y states that the m i n i m u m frequency difference between ne ighbor ing t idal constituents ( A / ) must not be less than the Chapter 5. Spectral Analysis 83 frequency that can be resolved b y the t ime series. The m a x i m u m resolut ion i n frequency o f a t ime series o f length L is ML, and the R a y l e i g h cr i ter ion can be wri t ten, AfL>\\. (5.1) The constituents used i n the harmonic analysis o f t ida l currents at the Endeavour ( E R A 2 ) and C o A x i a l ( C X - 1 ) sites are l is ted i n Table 5.1. T i d a l D a r w i n ' A m p l i t u d e R e c o r d Frequency P e r i o d Species Class i f ica t ion Fac tor L e n g t h (h) (cpd) (h) D i u r n a l Q , .0721 662 0.893 26.87 O i (Principal Lunar) .3769 328 0.929 25.82 P | (Principal Solar) .1755 4383 0.998 24.07 K | (Luni-solar) .5301 24 1.003 23.93 Semid iurna l N2 (Larger Lunar Elliptic) .1739 662 1.896 12.66 M2 (Principal Lunar) .9081 13 • 1.932 12.42 S2 (Principal Solar) .4225 355 2.000 12.00 K.2 (Luni-solar) .1150 4383 2.005 11.97 Table 5.1. Selected tidal constituents used for the harmonic analysis of tidal currents of the time series from stations ERA2 and CX-1. 5.4 Rotary Spectra Rotary spectra o f observed hor izonta l currents and residual hor izonta l currents f rom stations E R A 2 and C X - 1 are presented i n F igure 5.1. F O R T R A N code developed by A. B. ' Adapted from Gill, 1982 Chapter 5. Spectral Analysis 84 Rabinovich is used here for the harmonic analysis and the least-squares fit o f the t idal constituents. The t ime series formed by the least-squares fit is then removed f rom the or ig ina l t ime series to form the residual t ime series. De ta i l ed descriptions o f the least-square harmonic analysis method can be found i n Emery and Thomson [1997] or Foreman [1978]. The rotary spectra are estimated as described i n sect ion 2.2 at spectral densities ranging f rom 0.4 c p d to the N y q u i s t frequency o f 12 cpd . 5.4.1 Tidal Currents The currents are dominated b y diurnal and semidiurnal t ida l mot ions and inert ial osci l la t ions consistent w i t h the regional ly averaged spectra presented i n sect ion 2.2. The record lengths o f Z = 7 0 2 5 h and Z = 6 8 7 9 h f rom stations E R A 2 and C X - 1 , respectively, were suff icient ly l ong to separate a l l the major semidiurnal t ida l constituents l is ted i n Table 5.1. H o w e v e r , to estimate energy w i t h i n the spectral bands it is desirable to have a greater statistical certainty than can be obtained b y direct Four ie r transformation ( w h i c h has o n l y 2 degrees o f freedom per frequency estimate) o f the entire t ime series. U s i n g a Ka i s e r -Bes se l w i n d o w e d segment length o f 2048 h gives 10 degrees o f freedom per spectral estimate. W i t h this analysis, the variance w i t h i n the M2 frequency band exceeds, b y nearly a factor o f 4, the combined variance o f the N 2 frequency band and the S2 and K 2 frequency band (the S2 and K 2 t idal constituents are no longer separable because o f the shortened effective length o f the time series). Th is is substantially more than predicted by the ampli tude factors o f the semidiurnal 2 Adapted from Emery and Thomson, 1997 3 A. B. Rabinovich, P. P. Shirshov Institute of Oceanology, Moscow, Russia Chapter 5. Spectral Analysis 85 t idal constituents contr ibut ing to the equ i l i b r ium tide (Table 5.1) and makes the M2 t idal component c lear ly predominant . M o t i o n s at the M2 frequency at C X - 1 [Figure 5.1 (a)] are dominated by the counterc lockwise component w h i c h results i n a rotary coefficient o f r(co) = - 0 . 7 2 . The counterc lockwise component is also dominant i n the 4-day pe r iod spectral peak. Th i s is contrary to other deep Juan de F u c a R i d g e sites as described b y Cannon and Thomson [1996] and as shown i n F igure 2.3 i n this thesis, w h i c h indicate a predominant ly c l o c k w i s e rotary component for the 4-day osci l la t ions . Allen and Thomson [1993] provide a theory o f subinert ial osci l la t ions over s impl i f i ed ridge topography that demonstrates trapping and the preferential ampl i f i ca t ion o f the c l o c k w i s e rotary component . It is l i k e l y that the deviancy here o f the 4-day osci l la t ions and the semidiurnal currents at C o - A x i a l are a result o f the site's p r o x i m i t y to the massive A x i a l vo lcano , whose caldera, located 56 k m south, rises 1400 m o f f the sea floor. H o w e v e r , it is not k n o w n what actual phys i ca l mechanisms are p roduc ing these strong counterc lockwise rotary tendencies. T o characterize the spatial scale o f the barocl in ic tides i n the region, ver t ical mode decompos i t ion was conducted us ing stratification from nearby Cascad ia B a s i n . Results are summar ized i n Tab le 5.2. L i n e a r theory us ing an ocean depth representative o f the Endeavour R i d g e reg ion indicates that the barotropic tide w o u l d have a wavelength o f 9689 k m and a phase speed o f 144 ms\" 1 . Chapter 5. Spectral Analysis 86 M o d e N u m b e r H o r i z o n t a l Wave leng th (km) Phase Speed ( c m s\"1) Ros sby Rad ius (km) 1 137 307 18 2 84 190 11 3 54 122 7 4 41 92 5 5 34 77 4 6 28 62 3.7 7 24 54 3.2 Table 5.2. Baroclinic tidal parameters from vertical mode decomposition for a 2100 m deep basin with a characteristic September stratification from a region near station ERA2. Rossby radius is calculated by df, where c is phase speed of the mode and/ is the Coriolis parameter. A t station E R A 2 , the semidiurnal mot ions are near ly rect i l inear [Figure 5.1 (b)] w i t h a rotary coefficient o f r(co) = 0.15. F igure 5.1(c) and F igure 5.1(d) show the rotary spectra o f the residual series o f C X - 1 and E R A 2 , respectively, and indicate that little energy remains i n the d iurnal ( K i ) frequency band, thus ve r i fy ing the effectiveness o f the barotropic t idal r emova l . In the semidiurnal t ida l band, 17 .1% and 4 0 . 9 % o f the energy remain at C X - 1 and E R A 2 , respectively. Th i s remain ing M2 energy has a counterc lockwise to c l o c k w i s e energy ratio o f 0.11 and 0.03 for C X - 1 and E R A 2 , respectively, w h i c h comes close to the theoretical value, y « 0.02, predicted for a freely propagat ing semidiurnal internal wave at the latitudes o f the stations us ing, y = (co - f)21 (co + f)2 [Gill, 1982]. The observed energy ratio w i l l a lways be larger than the theoretical energy ratio i n the presence o f noise, w h i c h is equipart i t ioned w i t h an energy ratio o f y = 1. Chapter 5. Spectral Analysis 87 Figure 5.1. Ro ta ry spectra o f currents. B lue (red) curves are clockwise (counterclockwise) spectra w i th 50 degrees o f freedom, (a) 286-day record from C X - 1 ; (b) 292-day record from E R A 2 ; (c) and (d) as i n (a) and (b), respectively, for the residual time series. Chapter 5. Spectral Analysis 88 5.4.2 Inertial Currents A s required, the spectral densities i n the near-inertial frequency bands remain unaltered f o l l o w i n g the t idal r emova l . The inert ial bandwidth , as measured b y the r o l l - o f f o f the rotary coefficient f rom near-unity to 0.95, spanned 1.45 to 1.51 c p d at C X - 1 w i t h the m a x i m u m spectral densi ty at 1.50 cpd . A t E R A 2 , the near-inertial band had peak values i n the range 1.51 to 1.57 c p d w i t h the m a x i m u m at 1.52 cpd. F o r both C X - 1 and E R A 2 , the frequencies o f the spectral m a x i m a are greater than the loca l C o r i o l i s frequencies o f 1.45 and 1.49 cpd , respect ively. T h i s \"blue-shif t\" is c o m m o n to a l l near-inertial motions and has been reported to increase w i t h depth [c.f. Thomson et al, 1990; Thomson et al, 1998]. A n estimate o f the duration o f a s ignal , x , is the inverse o f the bandwidth o f the s ignal measured at the - 6 d B level [Emery and Thomson, 1997]. Th i s estimate gives x « 1 / (0.094cpd) = 10.6 days for C X - 1 and T « 1 / (0.107cpd) = 9.3 days, w h i c h is comparable to that found i n other reports for near-inert ial mot ions [e.g. Thomson etal, 1990; Thomson et al, 1998]. Estimates for near-inertial hor izonta l wavelengths have been reported for Queen Charlot te Sound , a semi-enclosed coastal area i n the northeast Pac i f i c , b y Thomson and Huggett [1981]. The i r measurements are centered o n 52° N w i t h a mean loca l C o r i o l i s frequency, s l igh t ly greater than that i n this study, o f 1.56 cpd . The i r wavelength estimates f rom two sites ranged f rom 85-95 k m and 300-700 k m i n the near-surface layer after the passage o f an eastward-m o v i n g cyc lone . DAsaro et al. [1995] report an asymmetr ica l t ime dependency o f the wavenumbers o f near-inert ial waves dur ing and after a \"s to rm\" event. T h e east-west and north-south hor izonta l wavenumbers , k and /, respectively, are estimated to be sma l l at the beg inn ing o f the storm event and correspond to a wavelength o f about 1700 k m . F o l l o w i n g the storm, the Chapter 5. Spectral Analysis 89 east-west wavenumber , /, increases w h i l e the north-south wavenumber , k, remains the same. Th i s is shown to be a result o f the p effect and can be expla ined as fo l lows . The solut ion to the l inear equations o f mot ion for inert ial currents when forc ing and diss ipat ion has been shut o f f is [e.g. Pond and Pickqrd, 1978], u + iv = Ue-'* (5.2) O n a be t a -p l ane , / varies as, f(y) = / 0 + Pv and substitution into (5.2) results i n a north-south wavenumber / = - fit. Th i s wavenumber increases negat ively w i t h t ime, resul t ing i n a shr ink ing hor izonta l wavelength [D'Asaro, 1989]. T h i s can be envis ioned as the t i l t ing o f the overa l l wavenumber vector w h i c h is s t r ic t ly ver t ical for pure inertial motions. Results from D'Asaro et al., [1995] show close agreement w i th this s imple l inear theory and indicate hor izonta l wavelengths o f 200 to 250 k m , 20 days after the storm's incept ion. The avai lable measurements o f near-inertial wavelengths i n the upper ocean are concentrated o n hor izonta l wavelengths and wavenumbers . Since near-inertial waves , b y def ini t ion, have a wavevector that is almost pure ly ver t i ca l ly oriented, hor izonta l wavevectors are very near to zero and thus hor izontal wavelengths are sensitive to minor variations i n the angle the wavevector subtends w i t h the gravitat ional vector. Measurements o f the ver t ical wavelength w o u l d not be subjected to this sensi t ivi ty, and w o u l d p rov ide a m u c h more reliable estimate o f wavenumber . In the measurements o f surface waves, for example, it is c lear ly best to measure the wavelength a long the d i rec t ion o f propagat ion not at smal l angles obl ique to the d i rec t ion o f the crests or troughs. A n y deviat ions in the plane wave form for surface waves w o u l d result i n large errors i n the wavelength estimate. Thus , it is the ver t ica l wavelength o f Chapter 5. Spectral Analysis 90 near-inertial waves that should be measured. H o w e v e r , measurements o f near-inertial ver t ical wavelengths at depth are few. Kunze and Sanford [1986] presented results f rom C a r y n Seamount ( 3 6 ° 4 0 ' N , 6 8 ° W ) , 2852 m depth, i n the Sargasso Sea, N o r t h At l an t i c . F o r waves apparently undergoing cr i t i ca l layer ampl i f ica t ion , they report ver t ical wavelengths i n the range 95 to 125 m and hor izonta l wavelengths o f 60 to 160 k m . Kunze and Sanford [1984], also report ver t ical wavelengths estimates o f 100 m and 500 m for two ind iv idua l near-inertial waves propagat ing i n the N o r t h Pac i f ic Subtropical Front (a permanent feature extending across the bas in at ~ 3 0 ° N ) . The measurements were taken over the depth range o f 100 to 800 m . A s was noted before, the increase i n frequency o f near-inertial waves w i t h depth has been w e l l documented. T h e increase i n mer id iona l wavenumber w i t h t ime due to the P effect w o u l d result i n an increase i n frequency o f the near-inertial wave as dictated b y the dispers ion relat ion as long as there were no change i n ver t ical wavenumber . The southward, -/ , propagation and hence, southward group ve loc i ty also contribute to the blue shift. Estimates o f ver t ical group ve loc i ty o f 25 m/day [Kundu, 1976] require about 80 days for the near-inertial energy to reach the depths o f the current meters at stations E R A 2 and C X - 1 . These estimates, however , are f rom the surface layer and vert ical group veloci t ies w i l l increase w i t h the blue shift and reduce the t ime required for the near-inertial waves to reach depths. A c c o r d i n g to linear theory, there is about a 27 % increase i n group ve loc i ty f rom the thermocline to the stratification at 1800 m . U s i n g p = 1.3 x l O ^ m f ' d a y \" 1 , an or ig inal size o f 1700 k m and a compromised 60 days to reach 1800 m , gives a hor izonta l wavelength o f about 80 k m , agreeing w e l l w i t h the measurements o f Kunze and Sanford [1986]. T h e ratio o f ver t ical to hor izonta l c o 2 - / 2 scale, y = T72 f> gives a ver t ical scale o f 30 m , w h i c h is somewhat lower than any the Chapter 5. Spectral Analysis 91 observations from the literature noted here. In order to be consistent w i t h as many observations as possible , a representative wavelength o f 100 m is chosen. T h e ratios o f inert ial to semidiurnal wavelengths that occur for resonance have been calculated for station C X - 1 in Chapter 4. F igure 4.4 indicates that the wavelength ratio range for X(f)/k(M2) is 0.62 to 0.80 for t h e / s s . / M 2 case and 1.49 to 1.91 for the M2.ss.JM2 case. V e r t i c a l mode decompos i t ion at the M2 frequency (Table 5.2) gives 137 k m and 84 k m for the hor izonta l wavelength o f the first two modes. U s i n g the X(f)/X(M2) ratios from Figure 4.4 gives, for the first mode, the hor izonta l ranges o f inert ial wavelengths as 85 to 110 k m for /SS.7M2 case and as 204 to 262 k m for the M2.SS./M2 case. T h e second mode yields the ranges 52 to 67 k m , for the f.ss.JM.2 case, and 125 to 160 k m for the M2.SS./M2 case. Since these predicted wavelengths are consistent w i t h avai lable measurements o f hor izonta l inert ial wavelength , the formation o f resonant triads is feasible w i t h ver t ica l mode decompos i t ion estimates o f M2 internal t idal wavelengths. 5.4.3/M 2 and M 4 Spectral Peaks O f part icular interest to this thesis are the significant spectral peaks at the exact s u m o f the inert ial and M2 frequencies ( t h e / M 2 frequency) and at twice the M2 frequency (the w e l l k n o w n M4 frequency, no rma l ly observed o n l y i n sha l low water). These peaks are 3 to 5 t imes more energetic than the background con t inuum whose rol l -off , 5 ~ co \" 2 , is i n general agreement w i t h that expected for the universa l internal wave spectrum away f rom source regions [Garrett and Munk, 1972]. The M 4 constituent has a nar row spectral peak w h i c h originates w i t h the narrow-Chapter 5. Spectral Analysis 92 band forcing o f the M 2 frequency. In contrast, the spectral peak o f the / M 2 frequency is re la t ively broad, reminiscent o f the broad inert ial frequency band. T h e percentage energy at the JMi frequency normal ized b y the total energies at the near-inertial and M 2 bands, RMi = SMj -(SfSM2) 2 , is 6 .2% at C X - 1 and 4 . 3 % at E R A 2 . The energy i n the M 4 band comprises 4 . 0 % and 2 . 3 % o f the total M 2 energy for C X - 1 and E R A 2 , respectively. These, compared to the Juan de F u c a region 's averages o f 3 .0% for the no rma l i zed JM2 frequency energy ratio and 1.9% for the normal ized M 4 energy ratio, indicate that the interactions at C X - 1 and E R A 2 are re la t ively strong. F o r compar ison , the energy ratios for a l l the stations are plot ted i n F igure 2.4 and F igure 2.5. 5.5 Time Series Demodulation 5.5.1 Introduction A mul t ip le filter technique is used to examine the nonstationary aspects o f the / , M 2 , jM.2, and M 4 osci l la t ions . The method was o r ig ina l ly developed b y Dziewonski et al. [1969] for the analyses o f transient seismic signals. The technique was designed to \"resolve complex transient signals composed o f several dominant periods that arrive at the recording station almost s imul taneously\" and has been effective i n the invest igat ion o f tsunami wave dispers ion [Gonzalez and Kulikov, 1993]. Chapter 5. Spectral Analysis 93 5.5.2 The Signal Model In order to proceed i n the development o f a rotary mul t ip le filter technique, it is helpful to understand the nature o f the s ignal investigated. Currents i n the ocean can be represented b y a s ignal mode l that is mixture o f a determinist ic part w i t h an addi t ive stochastic part. T h i s mode l can be wri t ten for a discrete s ignal as R s inusoids w i t h addit ive noise g(n), R V(n) = ^Arcos(wrn+^) + g(n), (5.3) where co r and § r are normal ized frequency and phase. 5.5.3 Method The method used here involves the separation o f the vector veloci t ies into c lockwise and counterc lockwise components based o n the pr inciples descr ibed i n section 2.2.2. Th i s is achieved i n practice for a discrete vector (complex-valued) series ( w ( n ) = u(n) + iv(n)) o f length L b y the appl ica t ion o f the D F T (Discrete Four ie r Transform). 1 L W(k) = -J^w(n)exp(-i2nnk/ L), 0^ 2 >X 3 ) = M4 (X 1 ^ 2 ^ 3 ) ~ W2 (T 1 (T 3 \" X 2 ) \" mi (X 2 ) W2 (X 3 (6-3) -m2(x3)m2(x2 - x 1 ) - / n l [ m 3 ( x 2 - x , , x 3 - T , ) + m3(x2,x}) + m 3 ( x 2 , x 4 ) + m 3 ( T 1 , T 2 ) ] + ( m 1 ) 2 [ m 2 ( x , ) + m 2 ( x 2 ) + m 2 ( T 3 ) + m 2 ( x 3 - x , ) + w 2 ( x 3 - x 2 ) + m 2 ( x 2 - x , ) ] - 6 ( / n 1 ) 4 . F r o m (6.3) it is apparent for a process o f zero mean, (rai=0), that the first three orders o f cumulants are s i m p l y equal to the moments. The zero mean fourth-order cumulant , however , becomes a function o f both second- and fourth-order moments. Higher-order cumulants w i t h zero t ime lags (i.e. x, = x 2 = x 3 = 0 ) also define other c o m m o n measures that describe the shape o f the probabi l i ty density dis t r ibut ion, for instance, the variance (the spread o f the probabi l i ty density dis t r ibut ion or pdf) is g i v e n by C2(0), skewness (asymmetry i n the probabi l i ty densi ty dis t r ibut ion or pdf) b y C3(0,0), and kurtosis (the sharpness o f peak o f the probab i l i ty densi ty dis t r ibut ion or pdf) b y C4(0,0,0). M u c h o f the research and applicat ions o f H O S to date have been i n engineering s ignal processing and use the fact that noise has a Gauss ian dis t r ibut ion and can be comple te ly characterized by its mean and variance. Therefore, the first- and second-order cumulants comple te ly characterize the noise and the noise s ignal w o u l d have ident ica l ly zero higher-order Chapter 6. Bispectral Analysis 111 cumulants . These higher-order cumulants str ict ly represent the non-Gauss ian por t ion o f the signal . Thus , i f the s ignal o f interest was non-Gauss ian , it cou ld be theoret ical ly reconstructed f rom the higher-order cumulants and separated f rom the noise. A l t h o u g h there are other interesting and important properties o f cumulants and moments result ing in useful applicat ions [see Mendel, 1991; Nikias and Petropulu, 1993], the properties described here are sufficient for the development o f a method o f measurement o f the nonlineari t ies i n wave-wave interactions that are central to this thesis. M o m e n t s and cumulants are t ime-domain measurements. T h e i r frequency-domain counterparts are n o w introduced. 6.2.2 Cumulant Spectra The frequency d o m a i n representation o f higher-order statistics are defined here i n terms o f discrete-time Four ie r transforms o f the cumulants. Cons ider a discrete process x(ri) w h i c h is real , s tr ict ly stationary, and has bounded (})-X*((£>i +co 2 +co3)], (6.10) where ^ ( c o ) = DFT(x(n)). U s i n g the fact that X ( - c o ) = X *(co) for a real-valued t ime series x(n), it can be seen i n a l l the above equations that the frequency indices sum to zero (i.e. J]co ( . = 0 ) . F o r the case o f the power spectrum, since the frequency indices are negatives o f each other, this property results i n the loss o f phase informat ion. The p o w e r spectrum is real-va lued and contains no informat ion about phase. The ca lcula t ion o f the bispectrum and tr ispectrum results, general ly, i n a complex value. The phase informat ion is retained between different frequency indices w h i c h sum to zero and is reflected i n the result ing phase o f the higher-order spectra. The bispectrum reveals phase informat ion between three different frequencies and the tr ispectrum reveals information between four frequencies. Th i s property can be extended to even higher orders, however pract ical constraints related to the calcula t ion Chapter 6. Bispectral Analysis 120 o f such higher orders as w e l l as the u t i l i ty o f such calculat ions has general ly l imi ted polyspectral calculat ions to order four. In oceanography, there have been several applicat ions o f the b ispec t rum and the tr ispectrum. The majori ty o f these applicat ions have been to surface waves , where the bispectrum has been calculated to investigate forced nonresonant interaction between three surface gravi ty waves [e.g. Hasselmann et al. 1963; Elgar and Guza 1985] and the trispectrum calculated to investigate resonant and nonresonant interaction between four waves [e.g. Elgar et al. 1995]. These applicat ions have been generally restricted to short waves. H o w e v e r , there has been at least one appl icat ion o f the b ispect rum to the analysis o f long waves i n the form o f surface tides. Marone and De Mesquita [1994] u t i l i zed bispectral analysis in an effort to differentiate between linear and nonlinear energy for a year- long real sea leve l record. In contrast, there has been m u c h less success w i t h the bispectral analysis o f internal waves. Neshyba and Sobey [1975] (hereinafter referred to as N S ) produced results f rom data w h i c h they interpreted as nonlinear interaction between ver t ica l ly separated internal wave traces. McComas and Briscoe [1985] (herein after referred to as M B ) revisi ted this paper and c o n v i n c i n g l y argued that the nonlinear interaction i n N S was a result o f their measurement methods and not actual interaction between the internal wave traces. In support o f this M B were able to construct a mode l that cou ld reproduce the major features o f the N S results. M o r e important ly, in re la t ion to the concerns o f this thesis, M B was able to show that the bispectra o f w e a k l y nonl inear resonant tr iad interactions i n a mode led Ga r r e t t -Munk ocean does not differ s ignif icant ly f rom zero. T h e y conclude that the l o w ut i l i ty is a result of, 1) a general insensi t iv i ty o f bicoherence to weak nonl inear i ty , 2) the p romiscu i ty o f the interactions, that is, the confusing effect o f many triads contr ibut ing to each component o f the internal wavef ie ld , Chapter 6. Bispectral Analysis 121 and 3) the lack o f strong interactions among the energy conta in ing components o f the internal wave field w i t h i n a Gar re t t -Munk ocean. B a r r i n g 1), w h i c h is a un iversa l statement, the deleterious effects o f both 2) and 3) are mit igated under the real ocean internal wave condit ions observed i n this thesis. The interactions here are restricted to discrete bands that l imi t the triads available to contribute to a nonlinear energy transfer and the evidence for interaction is contained in frequency bands w h i c h are among the highest energy conta ining bands in the power spectra o f the data analyzed. Thus , it is not unrealistic to expect better bispectral results than that were achieved i n M B w i t h the observed data o f this thesis. 6.3 The Bispectrum The bispectrum, 2?(co,,co 2), denoted by (6.9) is a two-d imens iona l c o m p l e x quantity, w i th magnitude, |£(<»,, co21, and biphase, Z2?(co,, c o 2 ) . It is doub ly per iod ic w i t h pe r iod 2 n and, due to a number o f symmetries that occur w h e n calcula t ing the triple correlat ion (the fourth-order cumulant sequence), is o n l y non-redundant i n a equilateral tr iangular region delineated by co2 > 0, co, > co 2 , co, + co2 < 7i . Th i s area is k n o w n as the inner triangle o f the p r inc ipa l domain and w i l l be the on ly reg ion o f concern w i t h regards to this thesis. F o r a discretely sampled s ignal , the p r inc ipa l doma in can be described i n terms o f sampl ing frequency, / „ as i l lustrated i n F igure 6.1. Chapter 6. Bispectral Analysis 122 Figure 6.1. The inner triangle of the p r inc ipa l domain In order to apply the bispectrum to investigate nonl inear properties o f oceanographic internal waves , a pract ical f ramework o f est imation needs to be developed. The development to this point has been i n terms o f a mathematical statistical v iewpoin t . T o take advantage o f the bispectrum it is necessary to relax some o f the mathemat ical ly r igorous assumptions that have been made. These are the same relaxations that are made i n convent ional second-order measures. The engineering concept o f a s ignal is adopted here i n favor o f the statistical \" random process\". That is , the current meter t ime series are mode led as stochastic signals w i t h strong Chapter 6. Bispectral Analysis 123 determinist ic components. Th i s mode l violates the assumption that the s ignal is pure ly random. I f the mode l includes determinist ic signals that are per iodic as w e l l (such as tides), the stationary assumption is also v io la ted (the mean is t ime va ry ing for a s inusoidal signal). Furthermore, the per iod ic i ty i n the s ignal impl ies that the cumulants associated w i t h it are not summable since a per iodic function has an infinite length. Thus , strict adherence to randomness, stationarity, and summable cumulants is not possible for the mode l proposed here. H o w e v e r , useful and usable measurements can s t i l l be made w i t h this m o d e l as l ong as the s ignal is cont inuous, differentiable and o f finite length. These restrictions, rough ly speaking, s i m p l y require the s ignal to be reasonably \" w e l l behaved\". T h e y are the same requirements as in the est imation o f second-order spectra. 6.3.1 Estimation A l t h o u g h the nature and properties o f bispectral estimates are m u c h different f rom those o f power spectral estimates, the approach is s i m p l y an extension o f the well-establ ished procedures used for power spectra. In this section, we introduce the s ignal mode l , and the method o f est imation o f rotary bispectra f rom a vector t ime series. 6.3.1.1 The Model The development o f H O S techniques in the literature has been based o n t w o models o f s ignal . A pure ly stochastic s ignal mode l and a mode l w h i c h is a mixture o f both deterministic and stochastic signals. A l t h o u g h both models can be appl ied to oceanographic measurements, Chapter 6. Bispectral Analysis 124 the latter is w e l l suited for the invest igat ion o f nonlinear interaction between discrete frequency bands and w i l l be the on ly mode l considered here. The mode l may be wri t ten for a discrete s ignal as R s inusoids w i t h addit ive noise g(n), R V(n) = X A cos(co,.« + ()),.) + g(n) ( 6 . 1 1 ) r=l where co r and § r are normal ized frequency and phase o f the rlh component . Th i s linear mode l can be designed to exhibi t the signatures o f nonl inear interaction. 6.3.1.2 Method of Estimation There are t w o methods that are currently i n use for b ispectrum est imation, direct and indirect [Nikias and Petropulu, 1993]. U t i l i z e d here is the direct method w h i c h is an extension o f the W e l c h per iodogram averaging technique for est imation o f power spectra [e.g. Press et al., 1992]. The first considerat ion (as is the case i n power spectra estimation) is the necessity for some form o f averaging to obtain a degree o f statistical certainty. T h i s is a result o f the addit ive noise i n the mode l and as w e l l as noise inherent i n the measuring process. I f a s ignal is pure ly determinist ic , the bispectra can be evaluated from a single rea l iza t ion [Nikias and Petropulu, 1993]. H o w e v e r , i n the no i sy case, the spurious effects o f noise can mask or distort the determinist ic component . These deleterious effects o f noise i n the s ignal can be reduced b y averaging. The f o l l o w i n g steps are undertaken to evaluate the rotary bispectra o f a two d imens iona l current t ime series. Chapter 6. Bispectral Analysis 125 • A c o m p l e x va lued series is constructed from hor izonta l ve loc i ty pairs u (east posi t ive, west negative) and v (north posi t ive , south negative), such that w(n)=u(n)+iv(n). • Th i s complex -va lued s ignal o f length N, (w(n), n = 0,---,N-\\) is d i v i d e d into K segments / = 1, 2,- ••, K o f length M. These segments m a y overlap and s t i l l main ta in statistical independence depending o n the nature o f the spectral w i n d o w used. • T h e complex-va lued mean, p ; , o f the ith segment is subtracted f rom each sample w i t h i n the segment. j M »'M?-rM ( 6 1 2 ) w, ' (n) = w , . ( « ) - p , ( n ) • Th i s zero-mean segment is then mul t ip l i ed by a suitable data w i n d o w sequence FdJji) to reduce the effects o f spectral leakage, w\"(n) = Fdw(n) x w / ( « ) • W i t h i n each segment, the Four ie r t ransform o f this w i n d o w e d complex -va lued t ime series w\"(ri)results i n a two-s ided spectra, o f w h i c h the posi t ive frequencies represent the counterc lockwise rotary spectra and the negative frequencies the c l o c k w i s e rotary spectra [Mooers, 1973]. The actual appl ica t ion o f the D F T creates a series o f spectral estimates o f w h i c h the second h a l f are the negative frequencies. Thus , dropping the p r ime notat ion for the mean removed , data w i n d o w e d s ignal , w(n), and app ly ing the discrete Four ie r transform ( D F T ) results in , Chapter 6. Bispectral Analysis 126 Wt(k)=-—]£w/(/i)exp(-i27i/i£/M)> 0,) + cos(co2n + 2) + 2 cos[(co, + co2 )n + (p, + (p2 ]. U s i n g t r igonometr ic identities, cos2 A +1 ; cosAcosB = —[cos(A + B) + cos(A-B)] 2 results i n , j 1 1 V (n) = ^ + 2 c °s(2co,« + 2(|),) + — cos(2co2/z + 2<|)2) + cos((co, + c o 2 ) « + (2, <|), + R2>xi>X2) = E[u(x, t)v(x + r,, / + x, )w(x + r 2 , / + x 2 ) ] , (6.22) where r and x are distance and time lag. It has been shown b y McComas [1978] that the bispectra due to nonlinear coup l ing us ing the usual assumptions o f weak resonant interaction is g iven by, |£(k,,k2,co,,co2| = co3 — = ©3|rf 5(k, +k 2 + k 3)0(0)3-co2 -co,) Ot ( 6 . 2 3 ) x [A(k2)A(k,)- ^(k 3M(k,) - A(k2)A(k3)], where, T, is a c o m p l e x interaction coefficient dependent o n wavenumber and, A = EI a , is the act ion density. Thus , it is possible g iven a mode l for the act ion density spectrum to predict the bispectra magnitude as a result o f weak resonant interaction and compare it to data. This has been done b y McComas and Briscoe, [1980] for a Gar re t t -Munk spectrum, they, however , considered the resul t ing bispectra too l o w to be statist ically significant. It has been shown that for theoretical quadratic phase c o u p l i n g the biphase is ident ical to zero. Since the real por t ion o f the b ispect rum has zero biphase, it has been used by some Chapter 6. Bispectral Analysis 131 researchers to indicate the degree o f coup l ing w i t h i n a s ignal [e.g. Elgar et al. 1995]. H o w e v e r , m u c h caut ion must be used as there is a somewhat tenuous relationship between the real b ispectrum and phase coup l ing . The real por t ion o f the bispect rum can be strongly and unpredictably modulated, i n the same w a y as is the bispectrum magnitude, b y characteristics o f the power spectrum. 6.4.2 Bicoherence In order to un l ink the bispectrum f rom unwanted dependence o n the magnitude o f the power spectrum, a normal iza t ion ca l led the bicoherence or, alternatively, bicoherence-squared has been suggested [e.g. Haubrich, 1965; Elgar and Sebert, 1989]. Bicoherence has been defined here as 6 2 ( co , , co 2 ) = |£[5(co , ,co 2 ) ] | 2 (6.24) E]\\W(ax)-W(a2f}E[S(ax+«>2)} This quanti ty can be estimated for the discrete case us ing, 1 K |2 K ZW) (6.25) Chapter 6. Bispectral Analysis 132 A s w i t h bispectra, c lockwise and counterc lockwise bicoherence-squared can be calculated by substituting the appropriate por t ion o f the D F T o f the t ime series w{n) as is described b y (6.13). Some o f the properties o f the bicoherence-squared are listed b e l o w w i t h d iscuss ion. • The bicoherence-squared is l imi ted to the range, 0 < b2 < 1. • The theoretical bicoherence-squared o f a Gauss ian random process is zero. It can be seen f rom (6.24) that the numerator o f the bicoherence-squared is the square o f the absolute value o f the bispectra, so that processes resul t ing i n zero bispectra also result i n zero bicoherence-squared. • The bispectra o f a s ignal is independent o f addi t ive Gauss ian noise, and this feature is one o f the cornerstones o f its u t i l i ty [e.g. Hinich, 1982]. H o w e v e r , for the bicoherence-squared, this is not general ly so, since the denominator is made up o f two second-order measures both o f w h i c h are sensitive to Gauss ian noise. Thus , as Gauss ian noise is added to the s ignal , the value o f the bicoherence-squared is biased towards zero. The bicoherence-squared is a normal ized measure o f the degree o f phase-coupl ing between three osci l la t ions at three frequencies related b y co, + co2 = co 3 . Cons ide r ing our s ignal mode l o f stochastic noise w i t h a strong deterministic component , three determinist ic components at frequencies co l ,co 2 ,co l +co 2 , w o u l d result i n a h i g h level o f bicoherence at co,,co2 without the presence o f any nonl inear i ty . H o w e v e r , i f w e mod i fy our m o d e l b y impos ing a l imi ta t ion that the determinist ic components can on ly be at the diurnal and semidiurnal frequency, elevated levels o f bicoherence-squared w o u l d be b y default indicat ive o f phase-coupl ing due to nonlineari ty . Therefore, i f there is s ignificant bicoherence-squared at the frequency indices / , Chapter 6. Bispectral Analysis 133 M2, and our assumption is that the motions at the /M2 frequency are not determinist ic, it indicates a nonl inear interaction between inert ial and semidiurnal (M2) motions. A final considerat ion is that the bicoherence-squared strongly relies on the number o f degrees freedom to obtain a stable and meaningful result. T o il lustrate, i f o n l y a single segment is used, that is K=\\, the value o f the bicoherence-squared, no matter what the nature o f the input s ignal is unity, That is, even a pure ly r andom Gauss ian s ignal w i l l result in a bicoherence o f uni ty, and signals w i t h litt le averaging w i l l be biased towards uni ty. The bicoherence-squared measure relies on the randomness i n phase and the stationarity between K segments. F o r an inf in i te ly l ong random Gauss ian process, each segment is equal ly weighted due to stationarity and has a random phase and therefore w i l l sum to zero. Devia t ions from zero bicoherence-squared result because o f a \"coherence\" i n t ime (or between segments) o f the phase relat ionship between osci l la t ions at co,,co2,cO| +co2 Therefore, even i f there is strong phase coup l ing between osci l la t ions, i f it is transient in nature, l o w estimates o f bicoherence-squared w i l l result. H o w e v e r , i f the length o f sampl ing for the bicoherence-squared estimate is shortened i n an attempt to \"capture\" the transient, the bias to un i ty must be taken into considerat ion. \\W(k) • W(lf\\\\V(k + if \\W(kf\\W(lf\\W(k + if (6.26) Chapter 6. Bispectral Analysis 134 6.5 Statistical Issues 6.5.1 Bicoherence-squared Since , for a finite length t ime series even a t ruly l inear random Gauss ian process w i l l produce nonzero levels o f bicoherence-squared, there needs to be a significant level o f bicoherence-squared defined w h i c h w i l l indicate devia t ion f rom Gauss iani ty . Haubrich [1965] has s h o w n that for a stationary Gauss ian process w i t h a large number o f degrees o f freedom the estimated bicoherence-squared, b , has an approximate chi-square (x ) d is t r ibut ion w i t h t w o degrees o f freedom (dof). Thus , s ignif icance levels o f bicoherence-squared w h i c h indicate non-Gauss iani ty w i t h a confidence coefficient a can be calculated f rom standard mathematical tables for percentage points o f chi-square dis t r ibut ion [e.g. Abramowitz and Stegun, 1970, pp. 984] . The chi-square dis t r ibut ion has also been shown to ho ld for non-zero values o f bicoherence-squared b y other researchers [e.g. Hinich, 1982, Elgar and Guza, 1988], and s h o w n to have bias and variance s imi la r to that o f ordinary coherence between two t ime series. One m a y use these results and a m a x i m u m l i k e l i h o o d scheme to achieve a better estimate o f the true value o f bicoherence-squared [Elgar and Sebert, 1989]. Al ternate ly , b m a y be measured in terms o f mult iples o f significant levels o f a n u l l hypothesis, Ho, where the hypothesis is that the s ignal is Gauss ian . L e t So be the significant level o f b1 under Ho, then the 1 0 0 ( l - a ) probabi l i ty , P, that b is less than So is g i v e n by , Chapter 6. Bispectral Analysis 135 P[i>2 < x2.a I dofj = 100(1 - a ) % (6.27) The final result here is in concurrence w i t h Haubrich's [1965] theoretical result. H o w e v e r , Haubrich has shown that this confidence level estimate is somewhat more pessimist ic than what is achieved v i a computer s imula t ion . F o r a confidence o f 9 5 % w i t h 21 to 42 dof, less than 6 /dof w h i c h is obtained for Sa f rom (6.27). 6.5.2 Biphase The statistics o f biphase are s imi la r to the statistics o f the phase o f the cross spectrum [Jenkins and Watts, 1968] and have been reported by Elgar and Sebert [1989]. The biphase has an approximate Gauss ian dis t r ibut ion, w h i c h is nonbiased w i t h a variance g i v e n by, I f za is the value o f a standardized two tailed Gauss ian (normal) dis t r ibut ion, the 100(1 - a ) percent confidence interval for zero biphase is g iven by Haubrich indicated that b2 is less than 4 /do f for a Gauss ian t ime series, w h i c h is marg ina l ly (6.28) ± z a / 2 V ° p ~ r ad ians , (6.29) Chapter 6. Bispectral Analysis 136 where za is obtained, k n o w i n g a -, f rom a table for areas under the standardized normal density funct ion [e.g. Bendat andPiersol, 1971, pp . 387] . 6.5.3 Data Length It has been general ly accepted that bispectral estimates have m u c h higher variance than power spectral estimates [e.g. Briscoe and McComas, 1980], and thus there is a need for greater degrees o f freedom for estimating the b ispec t rum than is needed to estimate the power spectrum to achieve s imi la r levels o f statistical stabili ty. F o r instance for h i g h l y nonlinear phenomena, such as w i n d tunnel turbulence, Helland et al. [1977] required 10 4 degrees o f freedom to achieve margina l statistical s ignif icance. Th i s requirement for greater degrees o f freedom for rel iable bispectral estimates translates into the need for greater data lengths as a result o f the uncertainty pr inc ip le and the necessity for adequate frequency resolut ion. Guide l ines for stationary stochastic signals have been presented by Hinich [1993] who suggests that reliable estimates can be achieved by requir ing that the number o f data segments, K, is as least as large as the length o f the record used i n the D F T (i.e. K> M). The data lengths necessary us ing this guidel ine are shown in Table 6.1 Chapter 6. Bispectral Analysis 137 D F T length, M S a m p l i n g Frequency Reso lu t ion Da ta L e n g t h 64 24cpd (hourly) 0 .375cpd 171d 48cpd (ha l f hourly) 0 .750cpd 85d 128 24cpd (hourly) 0 .187cpd 683d 48cpd (ha l f hourly) 0 .375cpd 341d 256 24cpd (hourly) 0 .094cpd 273 l d 48cpd (ha l f hourly) 0 .187cpd 1365d 512 24cpd (hourly) 0 .047cpd 10923d 48cpd (ha l f hourly) 0 .094cpd 546 l d 1024 24cpd (hourly) 0 .023cpd 4 3 6 9 l d 48cpd (ha l f hourly) 0 .047cpd 21845d Table 6.1. Data length and resolution as a function of D F T length and sampling frequency. The resolut ion is inversely propor t ional to D F T length, M, wh i l e the data length is proport ional to the square o f the D F T length (since for K = M, K x M - TV = M2). W i t h the current meter data used i n this thesis, the longer data lengths have hour ly sampl ing and have length in the range 250-300 days. F r o m Table 6.1 this indicates the need to use a D F T wi th a length o f 64, w h i c h results i n a frequency resolut ion o f 0.375 cpd . The peaks that are be ing analyzed are separated by 1.92 c p d ( M 2 ) - 1 . 5 5 c p d (/)=0.37 cpd , and w i t h a D F T length o f 64 values these frequency bands cannot be adequately resolved. Hence , a m i n i m u m D F T length o f 128 is used for bispectral estimates. 6.5.4 Stationarity and Transients The s ignal mode l under considerat ion here is that o f a determinist ic component w i th an addit ive stochastic component. The statistical considerations noted above are p r i m a r i l y necessary to insure the suppression o f the stochastic noise, that is, to insure that a random Chapter 6. Bispectral Analysis 138 element f rom noise w i t h no statistical s ignif icance cannot be responsible for a peak i n the estimated bispectra. The t idal components w i t h i n the s ignal are determinist ic but are not, i n a strict sense, stationary. H o w e v e r they va ry s l o w l y and smoothly , and the use o f second-order Four ie r analysis i n the study o f tides has been very successful so that it is not unreasonable to assume that third-order t idal analysis w i l l be successful as w e l l . The inert ial energy and the energies at the /M2 and M4 bands are intermittent and the diff icult ies associated wi th the analysis o f transient signals at the second-order w i l l be compounded at the third. The methods described above to reduce statistical uncertainty are not suited for the analysis o f transients. In fact, these statistical methods attempt to mitigate the effect o f transients o n the bispectra o f the s ignal . In order to study the bispectral characteristics o f transient signals, the statistical s tabil i ty o f greater degrees o f freedom must be g iven up to investigate shorter portions o f the s ignal . 6.6 Analysis of Data 6.6.1 Goals In this section the current meter t ime series f rom two stations exh ib i t ing strong peaks at the s u m o f the inertial (j) and semidiurnal (M2) frequencies are investigated i n terms o f their higher-order spectra. The goals are to establish whether either o f the t ime series have significant levels o f bicoherence-squared, whether they exhibi t indicators o f quadratic phase-coup l ing , and whether there are any higher-order statistical patterns w h i c h m a y be useful in understanding the processes at w o r k i n the ocean. Chapter 6. Bispectral Analysis 139 In order to achieve to achieve these goals, both the c l o c k w i s e and counterc lockwise rotary bispectra are calculated for the t ime series w i t h tides as w e l l as for the residual t ime series w i th tides removed . Firs t , the bispectra are presented to delineate, i n frequency space, areas where there is coincident elevated energy i n a triad o f frequency bands. Then , the bicoherence-squared and biphase are calculated to establish whether there is phase coup l ing and whether this coup l ing is consistent w i t h quadratic phase-coupl ing. 6.6.2 Plots of Bispectra Magnitude Figure 6.2 is a composi te plot o f the second-order power spectrum and the third-order bispectrum. B o t h counterc lockwise and c l o c k w i s e rotary components are displayed. The square contour plot o f rotary bispectral density is spli t a long a diagonal . T h e top/left triangular por t ion represents the counterc lockwise bispectra whereas the bottom/right triangular por t ion represents the c l o c k w i s e bispectra. The processing o f the hour ly sampled raw t ime series is conducted as described i n Chapters 2 and 5. A D F T length o f 128 data points is used to construct a l l the spectra represented. U s i n g a Ka i s e r -Bes se l w i n d o w , this results i n an estimated 218 degrees o f freedom for each spectral estimate. The frequencies o f the three major spectral peaks at K i , / , and M 2 , and as w e l l as the s u m peaks at JM2 and M 4 are h ighl ighted by red lines i n the power spectra and dashed red lines i n the bispectra. O n l y h a l f o f the total avai lable frequency band is shown because there are no significant bispectral features above 6 c p d and the bispectral and spectral informat ion can be represented i n a cohesive manner. The m a x i m u m bispectral density is i n the c l o c k w i s e triangle and is located at frequencies 1.50 and 1.88 cpd . The resolut ion o f the plots is g iven b y the sampl ing frequency Chapter 6. Bispectral Analysis 140 d iv ided by the length o f D F T used (fs/M) and is 0.1875 cpd . T h i s results i n quite a coarse gr id and the frequencies o f 1.500 c p d and 1.875 cpd correspond to the 8 t h and 10 t h discrete spectral estimate. H o w e v e r , the frequency values at the 8 t h and 1 0 t h spectral estimate are the closest frequencies to the actual values o f / and M 2 . B i spec t ra l estimates us ing the greater frequency resolut ion o f a 512 point D F T , Af = fj M =0.047 indicate that the bispectral m a x i m u m for this data set is at 1.55 and 1.92 cpd. Th i s higher resolut ion bispectral estimate is close to the power spectra estimate o f 1.52 cpd for the observed inertial peak and 1.93 c p d for the k n o w n frequency o f the M2 tide. The plots contained in F igure 6.3 have the same specifications as the plots in F igure 6.3. H o w e v e r , the eight largest d iurnal and semidiurnal constituents have been removed f rom the t ime series o f the latter us ing a least squares method (see Chapter 5). Spec i f i ca l ly , the harmonics used i n the least squares fit were, Q j , O i , P i , K i , N2, JVI2 S2, and K 2 . The bispectral m a x i m u m here has shifted to the discrete frequencies o f 1.50 and 1.50 cpd , reflecting the d imin i shed power at M2 frequency. Figure 6.2. Rota ry bispectra and spectra of horizontal currents f rom station E R A 2 for 292 days beginning J u l y 26,1994 Figure 6.3. Rota ry bispectra and spectra of residual (tides removed) horizontal currents f rom station E R A 2 for 292 days beginning J u l y 26 1994 Chapter 6. Bispectral Analysis 143 Presented i n F igure 6.4 and F igure 6.5 are the composi te spectral/bispectral plots for the station C X - 1 . T h e specifications are the same as for E R A 2 , except that a s l ight ly shorter times series length results i n fewer degrees o f freedom ( D O F = 214) . B o t h stations exhibi t s imi la r bispectra w i t h peaks occur r ing at B~(f,M2), B'(f,f), and, B'(M2,M2) [the c i rcumf lex over the frequency indices indicates that it is the nearest bispectral estimate to the denoted frequency], as w e l l as a peak at the lowest frequency and the inert ial frequency. T h e bispectra \" w i t h tides\" f rom C X - 1 (Figure 6.4) also has a notable peak at the diurnal frequency i n both the c l o c k w i s e and counterc lockwise bispectra. Th i s suggests that there is some fo rm o f mutual influence o n the strong semidiurnal barotropic t idal currents and the d iurnal t ida l currents i n the v i c i n i t y o f the station. A l t h o u g h it is the bispectral magnitude that is d i rect ly l i nked to nonlinear w e a k l y resonant theory b y 7.23, the bispectral magnitude alone c lear ly cannot be taken as an indicator o f nonlinear coup l ing . Its usefulness is i n indica t ing where there is s ignif icant coincident energy among three energy bands that warrant further investigation. The bicoherence-squared, sometimes ca l led the normal ized bispectrum is calculated next i n order to remove the power spectrum influence and to indicate the degree o f phase-coupl ing. Chapter 6. Bispectral Analysis 144 Figure 6.4. Rota ry spectra and bispectra of horizontal currents f rom station C X - 1 over 286 days beginning October 11,1993 Figure 6.5. Rota ry bispectra and spectra of residual (tides removed) horizontal currents f rom station C X - 1 for 286 days beginning October 11,1993 Chapter 6. Bispectral Analysis 146 6.6.3 Bicoherence-squared Plots Bicoherence-squared plots o f hor izonta l currents are presented for station E R A 2 for total currents i n F igure 6.6 and residual currents i n F igure 6.7. These same plots are repeated for station C X - 1 i n F igure 6.8 and i n F igure 6.9. The contour intervals are set as mul t ip les o f the 9 5 % signif icance level for a non-Gauss ian process w i t h the degrees o f freedom o f the bispectral estimate. Spectral w i n d o w , D F T length and degrees o f freedom are those used i n the bispectral calculat ions. Bicoherence-squared m a x i m a for station E R A 2 occur at frequency indices 1.50 and 2.06 c p d (i.e. at b2 (1.50,2.06)) for both the residual and total currents. Since the semidiurnal frequency straddles the 10 t h and 11 t h bispectral estimates (at frequencies 1.875 and 2.0625 cpd), the resolut ion is too coarse to define the semidiurnal frequency exact ly. H o w e v e r , higher resolut ion bispectra has indicated that the bispectral m a x i m a are at 1.55, 1.92 cpd , very close to the observed inert ial frequency and the M2 t idal frequency and thus I consider the value at 6 2(1.50,2.06) to be representative o f b2(f,M2). A c i rcumf lex over the bicoherence-squared and its frequency indices is used to denote that the value for bicoherence-squared is actual ly that o f the nearest bispectral estimates. F o r station C X - 1 there are no significant bicoherence-squared values at b (/,/), b ( M 2 , M 2 ) or b ( / , M 2 ) i n either the residual or total current plots . The residual p lo t shows o n l y isolated values w h i c h breach the So s ignif icance leve l , whereas for the total current p lo t there is a re la t ively strong peak at Z? 2(1.125cpd,1.125cpd). Chapter 6. Bispectral Analysis 147 Tempora l detail in the degree o f bicoherence-squared is p rov ided b y F igure 6.10, where o n l y the segments corresponding to days 136 to 178 are used i n the bicoherence estimate. D u r i n g this per iod , there is elevated energy i n both the inertial and JMi frequency bands. This pe r iod is compared to the first por t ion o f the record, days 1 to 135 (Figure 6.11), where there is energy i n the inertial band, w i t h little or no energy i n the 7M2 band and days 179-292 (Figure 6.12), where there is energy i n both inert ial and JM2 frequency bands. T h e versions o f these plots for the residual currents are not inc luded as they are qual i ta t ively s imi la r and show no new information. Chapter 6. Bispectral Analysis Figure 6.6. Rota ry bicoherence-squared o f horizontal currents f rom station E R A 2 . Chapter 6. Bispectral Analysis 149 Figure 6.7. Rota ry bicoherence-squared of residual (tides removed) horizontal currents f rom station E R A 2 . Chapter 6. Bispectral Analysis Figure 6.8. Rotary bicoherence-squared of horizontal currents from station CX-1. Chapter 6. Bispectral Analysis Figure 6.9. Rotary bicoherence-squared of residual (tides removed) currents from station CX-1 Chapter 6. Bispectral Analysis Figure 6.10. Rota ry bicoherence-squared of horizontal currents f rom station E R A 2 over days 136-178. Chapter 6. Bispectral Analysis Figure 6.11. Rota ry bicoherence-squared o f horizontal currents f rom station E R A 2 over days 135. Chapter 6. Bispectral Analysis Chapter 6. Bispectral Analysis 155 6.6.4 Biphase A A A Biphase is presented on ly for station E R A 2 at B(f, M 2 ) because it has the on ly t ime series where bicoherence-squared levels, above the inert ial frequency, warrant invest igat ion o f the biphase in order to establish whether there is evidence o f quadratic phase-coupl ing. Results are summar ized i n Tab le 6.3 Bispec t ra l Est imate W i t h Tides Res idua l W i t h Tides dur ing days 136-178 Res idua l dur ing days 136-178 Zf l (8 ,10) (degrees) -79.27 -61.52 -57.80 -34.67 Z 5 ( 8 , l 1) (degrees) -77.66 -67.34 -64.82 -42.27 Average angle (degrees) -78.46 -64.43 -61.31 -38.47 b2 0.2224 .1111 .5478 .3525 -I .0160 .0367 .0038 .0084 9 0 % confidence interval ± 11.92° ± 1 8 . 0 5 ° ±5 .81° ± 8.64° Table 6.3. Biphase of bispectral estimates nearest B(f,M2) of horizontal currents from E R A 2 The variance and confidence interval are calculated us ing (6.28) and (6.29). N o n e o f the estimates are w i t h i n the confidence interval and hence quadratic phase-coupl ing is not supported b y the data. Chapter 6. Bispectral Analysis 156 6.7 Discussion A l t h o u g h the magnitudes vary, there are s imilar i t ies between the spectral shapes and the bispectra topography for station E R A 2 and C X - 1 (Figure 6.2 through F igure 6.5). In contrast, the topography o f the bicoherence-squared between stations is marked ly different (Figure 6.6 through Figure 6.9). The bicoherence analysis f rom C X - 1 (Figure 6.8) suggests that phase-coup l ing occurs between the diurnal and semidiurna l components . There are peaks i n both the c l o c k w i s e and counterc lockwise components , but the counterc lockwise bicoherence-squared value is a factor o f 3 greater than the c l o c k w i s e value, and more than 6 times the 9 5 % significant level for Gauss ian noise. Th i s feature does not appear in the bicoherence-squared plot o f the residual series from C X - 1 (Figure 6.9) and this l i k e l y reflects the effectiveness o f the least squares method i n r emov ing the barotropic d iurna l t idal currents. Since both the diurnal and semidiurnal barotropic tide are as t ronomical ly forced and the semidiurna l frequency is roughly twice the diurnal frequency, a significant phase relat ionship is expected between them. T h i s is reflected i n the s ignif icant peaks at 6 2 ( K , * K , ) i n F igure 6.6 and F igure 6.8, w h i c h show the bicoherence-squared o f the t ida l ly inc lus ive currents f rom stations E R A 2 and C X - 1 , respect ively M o r e pertinent to this thesis is the lack o f significant bicoherence-squared at b (f,M2) and 6 2 ( M 2 , M 2 ) i n both the t ida l ly inc lus ive series (Figure 6.8) and the residual series (Figure 6.9) f rom station C X - 1 . Th i s observation is borne out at greater frequency resolut ion, as w e l l as when the t ime series is split to include o n l y regions o f enhanced energy i n the inert ial and JMz frequency bands, (days 30 to 110), and o n l y regions o f enhanced energy i n the M 2 and M 4 Chapter 6. Bispectral Analysis 157 frequency bands, (days 210 to 286, see F igure 5.4). These results indicate that there is no consistent phase c o u p l i n g between osci l la t ions at / , M2 and 7M2 frequency bands and no consistent phase coup l ing between the osci l la t ions at the M2 and M 4 frequencies. T h i s suggests that there is no nonl inear interaction between the observed wave trains at t h e / , IVk. / IVb, and M 4 frequencies observed at station C X - 1 . In marked contrast to station C X - 1 , both the bicoherence-squared plots , constructed us ing the t ime series \" w i t h tides\" and residual series, f rom station E R A 2 (Figure 6.6 and F igure 6.7) show the m a x i m u m bicoherence-squared to be i n the c l o c k w i s e inner triangle at 1.50 cpd , 2.06 c p d indicat ing phase-coupl ing between t h e / a n d M2 and JM2 frequency bands. The \"wi th tide\" analysis gives a bicoherence-squared o f 0.26, five times the significant level and twenty-f ive times the average over the c l o c k w i s e inner triangle. In the residual analysis , the bicoherence-squared drops to 0.13 at the same frequencies. A n explanat ion for this drop cou ld be that there is nonlinear interaction between the internal inertial wave and the barotropic tide. H o w e v e r , consider ing the great separation i n wavelengths between these waves, the mechanism o f resonant triads is not possible. It is more l i k e l y that us ing the least squares technique to remove the barotropic semidiurna l t idal currents also removes a por t ion o f the internal t ida l energy. Th i s w o u l d occur i f the internal tide were phase-coupled to the barotropic tide. A l t h o u g h there are no publ i shed observations o f bicoherence-squared between internal waves , compar i son o f the values here w i t h those obtained for forced secondary surface waves (e.g. Elgar et ai, 1995) indicate that this l eve l o f bicoherence-squared is very h igh , comparable o n l y to the highest levels observed dur ing storms w i t h shoal ing waves. Chapter 6. Bispectral Analysis 158 T o look at the temporal var ia t ion o f the bicoherence-squared at b (f,M2) f rom station E R A 2 , the rotary bicoherence squared is plot ted, w i th the consequential reduct ion o f statistical certainty, for the days 1-135 and the days 136-178. D a y s 136-178 correspond to the on ly coincidence i n t ime o f a spectral peak i n both t h e / a n d fM2 frequency bands f rom either station C X - 1 or E R A 2 . Th i s coinc idence i n t ime is i l lustrated best i n F igure 5.7. The c lockwise bicoherence squared plot over days 136-178 (Figure 6.10) shows a significant peak at b2(f,M2), c lear ly indica t ing phase-coupl ing between inert ial , semidiurna l and JM2 frequency bands. There is also a peak at b (fM2,jM2) suggesting nonl inear behavior o f the JM2 frequency wave . O n the contrary, the bicoherence-squared over days 1-135 (Figure 6.11) does not show an appreciable peak at b2(f,M2). O v e r the pe r iod from day 179 to the end o f the record, day 292, the rotary bicoherence-squared has a s ignif icant peak, al though not as narrow or strong as f rom days 136-178, at b2(f,M2), indica t ing nonl inear interaction between the inert ial and semidiurnal frequency bands. There is also significant bicoherence-squared act iv i ty at Zr ( M 4 , M 2 ) poss ib ly indica t ing the existence o f the M 6 t idal harmonic as a result o f the nonl inear interaction between the fundamental M2 and harmonic M4 t ida l currents. S ince there are no signif icant bicoherence-squared levels i n the internal wave f ie ld at C X - 1 , o n l y the biphase at E R A 2 is examined. B iphase is sought at the frequencies (f, M2) to indicate whether there is quadratic phase-coupl ing between the inert ial and semid iurna l frequency bands. The analysis shows that the biphase at E R A 2 does not fa l l w i t h i n a 9 0 % confidence level and that there is no evidence o f quadratic phase-coupl ing. It is possible that i n a no isy oceanic environment, the determination o f quadratic phase coup l ing v i a the biphase w i l l not be Chapter 6. Bispectral Analysis 159 possible. H o w e v e r , it should be noted that these are the first and o n l y measurements that show significant bicoherence-squared values between internal waves . The incidence o f quadratic phase-coupl ing m a y be uncovered i f examples o f persistent nonl inear interaction between internal waves is found. The results o f the bicoherence-squared analysis for station E R A 2 i m p l y that it is i n a region o f strong nonl inear act ivi ty , par t icular ly between inertial and semidiurna l waves and a l i k e l y generation region for internal waves o f JM2 frequency. Since the bicoherence-squared analysis for station C X - 1 shows no such phase coup l ing between inert ial and semidiurnal frequency bands, it is l i k e l y that the nonl inear i ty suggested by the rotary p o w e r spectral peaks at the 7M2 and M 4 from C X - 1 are remotely generated. Since internal waves are dispersive, remotely generated waves propagat ing into the v i c i n i t y o f C X - 1 at t h e / , M2 a n d y M 2 frequencies w o u l d not have been invo lved i n mutual interaction and, hence, exhibi t no phase-coupl ing. Chapter 7 160 Conclusions This thesis provides the first evidence for the nonl inear interaction between inert ial and semidiurnal currents. The observations o n w h i c h this evidence is based are from current meter moor ings i n water depths greater than 2 0 0 0 m i n the northeast Pac i f i c . S ince , I do not have access to observations f rom other oceanic regions, I cannot ascertain, direct ly, the g loba l extent o f the nonl inear interaction between inert ial and semidiurnal waves . H o w e v e r , there is no reason to assume that the nonl inear i ty is conf ined to the northeast Pac i f i c . It is l i k e l y that evidence for this nonlinear interaction w i l l be found in l ong time series o f currents f rom a l l mid-lat i tude regions o f the w o r l d ocean as long as there is sufficient internal wave energy at the inert ial and semidiurna l frequencies. In Chapter 2, the entire set o f observations was examined to determine the regional characteristics o f the nonlinear interaction. The observations are f rom three dist inct regions representing a topographica l ly smooth abyssal p l a in , a topographica l ly rough mid-ocean ridge and a near-coastal she l f region. The compar i son o f the averaged rotary spectra o f hor izonta l currents shows a remarkable s imi la r i ty among the three regions i n the t idal , inert ial , JM.2 and M 4 spectral bands. Spec i f ica l ly , the spectral estimate o f the 7M2 peaks were prac t ica l ly Chapter 7. Conclusions 161 identical for a l l three regions (1.66, 1.66, and 1.68 c m 2 s\"2 cpd\" 1 , respect ively) and compr i sed 3.0%, 3.6% and 2 .5%, o f the normal ized energy. Based o n the l imi ted data available to us, the energy at the JMi peak normal ized to the / and M2 peak energies d i d not vary strongly w i t h depth. Extens ive w o r k has been conducted on the nonlinear interaction o f internal gravi ty waves [e.g. Miiller et al, 1986]. N o n e o f this w o r k , however , has been di rect ly appl ied to oceanic wave measurements. In Chapter 3,1 have attempted to b r ing together the facets o f the available nonl inear interaction theory that I bel ieve m a y be appl icable to understanding the nonlinear interaction o f inert ial and semidiurnal internal gravi ty waves. The most prevalent theory is weak wave-wave resonant interaction. In a phys ica l sense, the interaction arises w h e n two internal waves cross paths and set up an interference pattern at the sum and/or difference frequencies o f the two waves. I f the wavenumber and frequency o f this interference pattern satisfies the dispers ion relat ion, the wave fo rm defined b y the interference pattern becomes a freely propagat ing wave and there is transfer o f energy to this new wave . U n d e r these condi t ions, the two waves and their interference pattern form a resonant tr iad. Th i s resonant triad concept is, I bel ieve, the most plausible mechan i sm for energy transfer among internal gravi ty waves . H o w e v e r , the diff icult ies arise i n quant i fying the energy transfer and obtaining the coup l ing coefficients among the members o f a resonant tr iad. The formulat ion, mathematical ly , involves the first order o f a perturbation expans ion made up o f the p r imary waves w h i c h forces the linear equation for the secondary wave . H o w e v e r , as pointed out b y Miiller et al [1986] there is no s imple expansion parameter. Since there are no measurements o f the energy transfer due to resonant triads, it is not possible to test the va l id i ty o f the der ived coup l ing coefficients. I do not derive coup l ing coefficients here, but use p rev ious ly publ i shed Chapter 7. Conclusions 162 derivations f rom Miiller and Olbers [1975] to examine the influence that the var ia t ion o f oceanic parameters, such as C o r i o l i s and B r u n t - V a i s a l a frequency, have o n the coup l ing eff ic iency. A c c e p t i n g the resonant triad as the mechanism o f energy transfer a l lows us to constrain some o f the characteristics o f interacting inert ial and semidiurnal waves. It is shown i n Chapter 4 that on ly p r imary waves propagat ing i n opposite ver t ical directions can form resonant triads. T h i s strong l imi ta t ion fits w e l l w i t h a mode l o f d o w n w a r d l y propagat ing w i n d -generated inertial energy and upward ly propagat ing barocl in ic t ida l energy generated by barotropic t ida l f l o w over topography. (In this study, the wave directions o f the inertial and semidiurnal waves are not k n o w n for certain, and the va l id i ty o f this result cou ld not be ascertained.) The JM2 wave can have a ver t ical propagat ion di rec t ion either coincident w i th the inert ial wave or the semidiurnal wave . I have termed these the f.ss.JM2 case (f waves have the same ver t ical propagat ion d i rec t ion as fM2 waves) and the M2.SS./M2 (M2 waves have the same vert ical propagat ion direct ion as 7M2 waves) . These two cases define regions o f acceptable wavenumbers for the format ion o f resonant triads. That is , for each case, there exists a range o f the relative wavelength o f the inert ial wave to the wavelength o f the semidiurna l wave such that, as long as the two waves intersect, they fo rm a resonant triad. The wavelength o f the resultantJM 2 wave depends on the hor izonta l angle between the inert ial and semidiurnal wave . X,(/) These ratios o f wavelength , ^ , are 0.62 to 0.80, for the/SS .7M2 case, and 1.49 to 1.91 for the M2.SS./M2 case at the latitudes o f the stations investigated i n this thesis. S ince the ocean can support many modes o f internal semidiurna l waves and there is a wide range o f wavenumbers reported for inert ial waves , it is very l i k e l y that the ratio between the wavelengths intersecting Chapter 7. Conclusions 163 inert ial and semidiurna l waves can fal l w i t h i n these two ranges o f ratios. W h e n this occurs there w i l l be transfer o f energy to the \" n e w \" wave at the s u m frequency. A l t h o u g h the propagat ion di rec t ion o f internal waves depend o n the stratification, the stratification affects a l l the wavevectors such that the two ranges o f ratios, or regions o f acceptable wavelengths, are not functions o f stratification. H o w e v e r , the frequency o f the inert ial wave is a function o f C o r i o l i s parameter the ranges o f acceptable wavelength do vary w i t h latitude. In general, the higher the latitude the greater l i k e l i h o o d o f the transfer o f energy. Th i s holds un t i l the latitude o f 68° where the frequency o f the near-inertial waves surpasses that o f the semidiurnal waves and propagating semidiurna l internal waves cease to exist. T o determine the coup l ing eff iciency, I examined the coup l ing coefficient used for spectral transfer o f energy f rom inert ial and semidiurnal waves to JM.2 waves. Th i s coefficient is not a strong function o f the hor izonta l angle made between the inert ial and semidiurnal waves at the latitude o f the observations i n this thesis. It is a somewhat stronger function o f this hor izonta l angle at l ower latitudes but I do not bel ieve that the hor izonta l angle is an important considerat ion i n the coup l ing eff iciency. The coup l ing eff ic iency is more s ignif icant ly l i nked to latitude, and at higher latitudes, i n general, the coup l ing coefficient is greater. H o w e v e r , the coup l ing coefficient, is a strong function o f stratification, w i t h the weakest stratification result ing i n c o u p l i n g coefficient an order o f magnitude greater than that i n the thermocl ine. A l t h o u g h observations o f the relative ampli tude o f the JM2 peak w i t h stratification are l imi ted , there is no indica t ion that there is any greater energy transfer at l ower stratification. Therefore, it is possible that there are other factors that are more important, or that the m o d e l o f coup l ing Chapter 7. Conclusions 164 is incorrect, or that the necessary observations are lack ing . It is uncertain from the observations in this thesis w h i c h is more l i ke ly . Second-order (power spectra) and third-order (bispectra) rotary spectral analysis are used to investigate the nonl inear i ty at two stations (see Chapter 5 and Chapter 6). P o w e r spectral analysis indicates that theJM.2 spectral peak is precise ly at the sum o f the observed peaks o f the inert ial and M2 (the dominant semidiurnal mot ion) spectral peaks. Th i s , i n the absence o f any other forc ing at that sum frequency, is sufficient to indicate that there is some fo rm o f nonlinear interaction between the inert ial and M2 waves. The range o f wavelengths to fo rm resonant triads is consistent w i t h estimates o f wavelengths o f inert ial and semidiurnal waves , indicat ing that there is a h igh l i k e l i h o o d that resonant triads are formed. T e m p o r a l var ia t ion o f rotary components from t w o stations shows that, i n general, enhanced inert ial energy, other than at seasonal t ime scales, is not coincident i n t ime w i t h enhanced energy i n t h e / M 2 frequency band. S i m i l a r l y , there is no coincidence between wave events in the M2 and JM2 frequency bands. W h e n there is co inc idence o f the three interactants, the interaction m a y be a forced nonresonant interaction or a resonant interaction. W h e n there is no coinc idence i n t ime w i t h elevated energy JM2 and energy i n the inert ial band, it signifies there was a nonl inear exchange o f energy to the JM2 frequency band. A l t h o u g h the necessary wavenumber observations are l ack ing to prove the resonant triad theory, it is p lausible that the energy exchange occurs as a result o f triad interactions. Rotary bispectral analysis can indicate the degree o f phase-coupl ing among wave components i n frequency bands related by co, + co2 = co 3 . Hence , it is idea l ly suited for ident i fying nonl inear interactions, since interacting waves w i l l be phase-coupled. The analysis Chapter 7. Conclusions 165 f rom t w o stations f rom Juan de F u c a R i d g e shows a remarkable d i ss imi la r i ty . A l t h o u g h the second-order rotary p o w e r spectra from both stations show wel l -def ined spectral peaks at the JMi frequency, there is o n l y s ignif icant bicoherence-squared (normal ized bispectra) at the frequency indices indicat ing phase coup l ing between waves at the / and M2 frequencies at one o f the stations. B o t h o f these stations are at s imi lar depths, near rough bot tom bathymetry. The phase-coupl ing at station E R A 2 impl ies that it is in a region where there is active coup l ing between inert ial and semidiurnal wave components and a l i k e l y source region for JM2 waves. The lack o f phase-coupl ing a m o n g / , M2 and /VI2 frequency bands evident at station CX-1 in the presence o f ample energy in these bands impl ies that the JM2 is not der ived f rom the / and M2 wave components present at CX-1 and has propagated into the v i c in i t y . That is , the /M2 waves were formed through wave interactions at some distal source and have propagated to the region o f the observations. 166 Nomenclature and Definitions 2?(co,, co 2 ) bispectrum b2 bicoherence-squared B(coi , co 2 ) bispectrum estimate at the nearest spectral estimate to co, and co 2 b (co, , co 2 ) bicoherence-squared estimate at the nearest spectral estimate to co, and co 2 cg group ve loc i ty c„ cumulant sequence o f order n Cn cumulant spectra o f order n E[ ] expectation operator / C o r i o l i s parameter JM2 sum o f inert ial and semidiurnal frequencies g(n) r andom noise K number o f data segments used to construct spectral or bispectral estimate k=(k,l,m) wavenumber vector mn moment sequence o f order n M„ moment spectra o f order n M4 twice the semidiurnal frequency M length o f data segment that is discrete Four ie r transformed n discrete t ime i n samples length o f t ime series, B r u n t - V a i s a l a frequency, order Nomenclature and Definitions 167 R(x) autocovariance function defined as R(z) = E[ {x(n) - p} {x(n + x ) - p ] r Lagrang ian distance lag r(a>) rotary coefficient 5 + ( c o ) , 5 \" ( c o ) counterc lockwise and c l o c k w i s e power spectral densi ty .ss. same s ign; same ver t ical wavenumber s ign u,v east-west and north-south ve loc i ty components u+,u counterc lockwise and c l o c k w i s e rotary ve loc i ty components x(n) discrete s ignal AT. frequency resolut ion CO radians per second Px autocorrelat ion function defined as R(x) / R(0) a 2 variance R(0) a standard devia t ion mean length o f data 168 Bibliography Aanderaa , I. 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Here , I br ief ly consider the difference interaction between the inert ial and semidiurnal currents at M2-/ Rotary mul t ip le filter plots o f hor izonta l currents have been presented in Chapter 5 for stations C X - 1 and E R A 2 for the frequencies / , M2, JMi and M 4 (Figure 5.2, F igure 5.3, F igure 5.4, and F igure 5.5). A t C X - 1 , the observed C o r i o l i s frequency f rom rotary spectral plots is 1.500 c p d w i t h a resolut ion o f 12 /256=047 cpd . The semidiurnal peak is observed at 1.922 c p d and a difference interaction w o u l d be expected at 0.422 cpd . A t this frequency range w i t h the resolut ion o f the spectra being 0.047 cpd , the difference frequency o f inert ial and diurnal motions should also be considered. The observed frequency peak for diurnal motions is at 0.984 cpd, therefore f-K\\ is 0.516 cpd . Table 9.1 presents the expected frequencies f o r / , K i , M2 as w e l l as the frequencies o f the observed peaks f rom the rotary spectra. Frequencies are calculated for JM2, M2-f and f-K\\ us ing the observed and the expected frequencies. Station Q E P 1, deployment 5, at 600 m depth, exhibits an apparent interaction peak at M2-/ and is compared to the averaged spectra from station Q E P 1 and the stations E R A 2 and C X - 1 . Appendix A. : M2-f Interactions 178 Station / M 2 Ki Mi M 2 - / / - K , 3 observed observed observed observed observed observed expected expected expected expected expected expected C X - 1 1.500 1.922 0.984 3.375 0.422 0.516 1.451 1.932 1.003 3.422 0.481 0.448 E R A 2 1.547 1.922 0.984 3.469 0.375 0.563 1.489 1.932 1.003 3.469 0.443 0.486 Q E P 1 1.547 1.922 0.984 3.516 0.375 0.563 D e p 5 1.526 1.932 1.003 3.469 0.406 0.523 600 m Q E P 1 1.547 1.922 0.984 3.469 0.375 0.563 A v e r a g e d 1.526 1.932 1.003 3.469 0.406 0.523 Table 9.1. Center frequencies of observed peaks and predicted values of inertial, tidal and derived bands F r o m the perusal o f F igure 9.1 and Figure 9.2, it is evident that the rotary spectra o f hor izonta l currents f rom stations C X - 1 and E R A 2 show no prominent peaks di rect ly at the M2-/ o r / - K | frequency. N o r is there any notable enhancement o f energy near the /+Ki frequency. B o t h stations are dominated b y inertial energy and b y the d iurna l and semidiurna l t idal currents, w i th s ignif icant peaks occur r ing at the JM2 and M4 frequency bands. The sub-diurnal region o f the spectra from C X I (Figure 9.1) has a dominant broad peak centered at about 0.23 c p d (the frequency o f the loca l m a x i m u m w h e n the spectrum is constructed us ing a 512 point F F T ) . Th i s broad band peak has been associated wi th w i n d - d r i v e n 4-day osc i l l a t ion trapped to Juan de F u c a R i d g e [Cannon and Thomson, 1996]. It is possible that this broad energy band w i l l mask any energy enhancement at the M2-/ frequency. In the sub-diurnal reg ion at E R A 2 4 Calculated from observed values of/and M 2 . 5 Calculated from observed values of/and K]. Appendix A. : M2-/Interactions 179 (Figure 9.2), there is a very broad elevated region encompass ing 0.2-0.5 c p d wi th no discernable discrete peaks. Results f rom analysis us ing a rotary mul t ip le filter technique ( R M F T ) for both o f these stations are shown i n F igure 9.3. F o r C X - 1 , a l l spectral peaks i n the c l o c k w i s e and counterc lockwise directions are located o f f the f-K\\ and M2-/ frequency bands. E n e r g y i n the inert ial arid /M2 bands (not shown in these plots) is concentrated i n the first 100 days. A t this station, it is evident that elevated energy levels at the JM2 frequency do not result i n elevated energy at the M2-/frequency. A t E R A 2 , some o f the spectral peaks c o u l d be construed to occur at the f r equenc ie s , / -K i and M2-/ but both the c l o c k w i s e and counterc lockwise rotary plots are s t i l l more consistent w i t h broadband noise. There is, however , the same seasonal e levat ion o f spectral energy beg inn ing around day 90 that is evident i n the inert ial and JM2 bands. It is concluded that at these stations, w h i c h both exhibi t strong f+M2 interactions, there are no accompanying M2-/interactions and no f-K\\ interactions. The reg ional ly averaged rotary spectra from station Q E P 1 (Figure 9.4) shows signs o f a smal l peak at the M2-/ frequency. Perusal o f the ind iv idua l spectra used to construct this average spectrum indicates that about h a l f the spectra have enhanced energy i n the 0.3-0.6 cpd range and h a l f show no sign o f enhancement. O f the spectra examined, Q E P 1 deployment 5 at 6 0 0 m (Figure 9.5) is chosen for further examinat ion because it is, re la t ively long (4235 samples), and exhibits a prominent peak at the M2-/ frequency. The data processed b y R M F T results i n the plots i n F igure 9.6 and Figure 9.7. A b o v e the inert ial frequency (the upper two plots i n both F igure 9.6 and Figure 9.7), the c l o c k w i s e energy dominates the counterc lockwise energy as expected. B e l o w the inertial frequency (the lower two plots in both F igure 9.6 and Appendix A. : M2-f Interactions 180 Figure 9 . 7 ) , the energy is more evenly distr ibuted between the counterc lockwise and c l o c k w i s e rotary components . The largest inert ial frequency peak begins at about day 150 i n the c l o c k w i s e rotary component ampli tude. It is f o l l owed by enhanced c l o c k w i s e rotary energy in the JM2 frequency band. H o w e v e r , there are no such c l o c k w i s e energy enhancements in the M 2 - / frequency band. The rotary components i n the M 2 - / frequency band tend to have c l o c k w i s e po lar iza t ion and do not appear to be associated w i t h any elevated energy i n any o f the other bands. B o t h the reg ional ly averaged and the single deployment rotary spectra from station Q E P 1 show two smaller peaks at about 5.06 and 5.44 cpd . These peaks correspond to the frequencies o f 2 / + M 2 and 2 M 2 + / ~ and suggest complex nonl inear behavior between the inert ial and semidiurnal currents inc lud ing higher harmonics . H o w e v e r , the statistical s ignif icance o f these peaks f rom non-averaged spectra f rom Q E P 1 is very l o w and no attempt to use demodulat ion has been made. Appendix A. : M 2 - / \" Interactions 181 10' Mooring CX-1 at 2050m, 1993 10 in | 10° E i _ -t—1 O CD a CO ^ o D_ 10\"' S(T) Total Spectrum S(+) Counterclockwise Spectrum S(-) Clockwise Spectrum 10\" 10u Frequency (cpd) DoF=50 Kaiser-Bessel Window N=512 Number of Records=6879 Time lnterval=60 min 101 Figure 9 . 1 . Rotary spectra from station C X - 1 showing diurnal, inertial and semidiurnal peaks as well as notable peaks at JM2, ML,, and 4-day period, but no peaks at f-Ku or M 2 - / Appendix A. : M2-f Interactions Mooring ERA2 at 1757m, 1994 10' 10' Q. O 10° E O cu Q. CO .« o i o n S(T) Total Spectrum S(+) Counterclockwise Spectrum S(-) Clockwise Spectrum 10' 10u Frequency(cpd) DoF=52 Kaiser-Bessel Window N=512 Number of Records=7025 Time lnterval=60 min 10' Figure 9.2. Rotary spectra from station ERA2. enhanced energy at M2-f. The JM2 peak exists without any evidence Appendix A. : M2-/Interactions Clockwise CX-1 Counterclockwise CX-1 Counterclockwise E R A 2 Days Figure 9.3. L o w frequency multiple filter plots encompassing the frequency ranges off-K\\ and M 2 - / f o r stations C X - 1 and E R A 2 . Colorbars indicate velocity i n c m s\"1. Appendix A. : M 2-/Interactions 184 Frequency (cpd) Figure 9.4. Averaged rotary spectra of horizontal currents from 25 deployments at station QEP1 with 1096 degrees of freedom showing small peak near M2-/ Also apparent are two higher frequency peaks at 2/+M2 and 2M2+/ Appendix A. : M2-/Interactions 10' Mooring QEP1-D5 at 600m, 1991 10' 110° E O CD Q . CO .„ 5 o PL 10 -2 S(T) Total Spectrum S(+) Counterclockwise Spectrum S(-) Clockwise Spectrum 10\" 10\" Frequency (cpd) DoF=30 Kaiser-Bessel Window N=512 Number of Records=4235 Time lnterval=60 min 10 Figure 9.5. Rotary spectra of horizontal currents f rom station QEP1 deployment 5 at 600 depth exhibi t ing prominent peak near M 2 - / . Appendix A. : M2-/Interactions 186 Clockwise Rotary Spectra QEP1-D5-600 Inertial and Semidiurnal Band Low Frequency Band Days from 3 May 1991 Figure 9.6. Clockwise R M F T plots for horizontal currents from station Q E P 1 Deployment 5, 600 m depth. Co lorbars indicate rotary velocity i n c m s\"1. Appendix A. : M2-/Interactions 187 Counterclockwise Rotary Spectra QEP1-D1-600m Inertial and Semidiurnal Band Low Frequency Band Days from 3 May 1991 Figure 9.7. Counterclockwise R M F T plots for station Q E P 1 deployment 5, 600 m depth. Colorbars indicate rotary velocity at c m s\"1. Appendix A. : M2-f Interactions 188 Appendix A. 1 Discussion F r o m the preceding Figures , there is no evidence f rom second-order spectral techniques that there is nonl inear energy transfer from super-inertial to subinertial osci l la t ions at Juan de F u c a r idge. H o w e v e r , at station Q E P 1 , there is elevated energy at the M2-ffrequency suggesting that there is nonl inear transfer f rom the semidiurnal t idal and inert ial osci l la t ions . Since there is no coincidence i n time between heightened inert ial energy and heightened energy at the M2-/ frequency discernable f rom the single t ime series f rom station Q E P 1 , it is not l i k e l y that the osc i l l a t ion is d i rect ly forced by the inert ial and semidiurna l wave motions, i m p l y i n g that it is a free wave . The counterc lockwise M2-/frequency peak at Q E P 1 has about 3.5 t imes the spectral densi ty o f the background l o w frequency cont inuum w i t h a bandwid th , AfBW « 0.14, measured at the h a l f p o w e r o f the peak. In compar ison , the JM2 peak from the same data has about 5 times the spectral density o f the background internal wave con t inuum w i t h a bandwidth , AfBI¥ « 0.18 cpd . These findings w o u l d seem to suggest that there is nonl inear transfer o f energy f rom super-inert ial to subinert ial osci l la t ions at some stations. H o w e v e r , there is no theory to describe such a transfer. Super-inert ial waves are considered waves o f the first class r e l y i n g p r i m a r i l y on gravi ty as a restoring force, whi le subinert ial waves are second class waves that re ly o n the C o r i o l i s force and the restoring torque associated w i t h the conservat ion o f potential vor t ic i ty for their existence [e.g. LeBlond and Mysak, 1978]. F o r this restoring force to be an effective wave mechanism, the waves must necessari ly be quite large. C o m m o n l y observed second class waves i n the ocean include planetary and topographic waves. Planetary waves have a m a x i m u m frequency at mid-lati tudes corresponding to periods o f about one year; they re ly on Appendix A. : M2-/Interactions 189 the beta effect (the variat ion o f the C o r i o l i s parameter w i t h latitude), w i t h length scales general ly i n excess o f 1000 k m . Eddies are sometimes descr ibed as nonl inear planetary waves, w i t h length scales substantially less than 1000 k m , the frequencies o f these motions, however , are a lways m u c h lower than that o f the observed M 2 - / osci l la t ions . Topographic waves can exist at a l l frequencies be low the loca l C o r i o l i s frequency but are not free waves and require the existence o f specific topography w h i c h i n turn establishes the frequency and wavelength o f the result ing waves. N o n e o f these wave types are viable candidates for the osci l la t ions at the M 2 - / frequency. Sub inert ial solutions to the l inearized equations o f mot ion i n a rotating f lu id do exist for the unstratified case [e.g. LeBlond and Mysak, 1978; Greenspan, 1968], however , except for their manifestation as T a y l o r co lumns , the author k n o w s o f no observations i n the ocean. 190 Appendix B: Future Work In order to test the va l id i ty o f tr iad resonance as the nonl inear interaction mechanism between inert ial and semidiurnal waves it is necessary to obtain rel iable estimates o f the wavenumbers o f the triad interactants. A s is shown i n this thesis, s ignif icant spectral peaks at the frequencies corresponding to / , M2 and JM2 are not necessari ly an indica t ion that the energies are phase coup led and that the JM2 spectral peak arises as a result o f the observed spectral energies at the / and M2 frequencies. H o w e v e r , I have identif ied a region over Endeavour R idge that exhibits strong phase-coupl ing among the / , M2 and fM.2 components. Th i s phase-coupl ing is indicat ive o f the nonl inear interaction among the frequency bands and i f the wavenumbers o f these interactants are measurable, it w o u l d greatly enhance our understanding o f the nonl inear transfer o f energy among internal waves . I f actual wavenumber est imation o f a l l three components is untenable, the wavenumbers or even just the wavenumber directions o f the two most energetic components (the inert ial and semidiurnal waves) , w o u l d be sufficient to test the va l id i t y o f the resonant triad mechan ism. Observations o f wavenumber cou ld be achieved by the m o o r i n g o f an u p w a r d facing acoustic D o p p l e r current prof i ler ( A D C P ) near the v i c i n i t y o f station E R A 2 ( 4 7 ° 5 7 . 0 ' N , 129° 5 . 7 ' W ) a long the Juan de F u c a R i d g e . A single A D C P at 75 k H z has a Appendix B. : Future Work 191 m i n i m u m rel iable range o f about 600 m , sufficient to resolve the ver t ica l wavelengths o f observed inert ial waves and waves w h i c h may be resonantly interacting w i t h it. The moor ing pe r iod w o u l d be over the winter months when there is sufficient energy i n the inert ial and 7 M 2 frequency bands. A n a l y s i s o f the measurements from the in i t ia l A D C P deployment can guide the second deployment o f a triangular array o f A D C P s . Th i s array w o u l d be designed to measure hor izonta l wavenumbers and propagat ion directions as w e l l as the extent o f the interaction vo lume. Further modif icat ions to the design o f the array us ing the second deployment cou ld be made so that the amplitudes o f the wavetrains before and after the interaction may be measured g i v i n g an estimate o f the coup l ing eff ic iency. "@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "1999-11"@en ; edm:isShownAt "10.14288/1.0099428"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Oceanography"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Nonlinear interaction of inertial and semidiurnal currents in the northeast pacific"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/10099"@en .