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UBC Theses and Dissertations

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UBC Theses and Dissertations

Physical modelling of tidal resonance in a submarine canyon Le Souëf, Kate Elizabeth 2013

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PHYSICAL MODELLING OF TIDAL RESONANCE IN A SUBMARINECANYONbyKate Elizabeth Le Soue?fB.E., The University of Western Australia, 2006B.Sc., The University of Western Australia, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Oceanography)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2013c? Kate Elizabeth Le Soue?f, 2013AbstractThe Gully, Nova Scotia (44?N) is unique amongst studied submarine canyonspoleward of 30? due to the dominance of the diurnal (K1) tidal frequency, whichis subinertial at these latitudes. Length scales suggest the diurnal frequency maybe resonant in the Gully. A physical model of the Gully was constructed in atank and tidal currents were observed using a rotating table. Resonance curveswere fit to measurements in the laboratory canyon for a range of stratifica-tions, background rotation rates and forcing amplitudes. Resonant frequencyincreased with increasing stratification and was not affected by changing back-ground rotation rates, as expected. Dense water was observed upwelling ontothe continental shelf on either side of the laboratory canyon and travelled at leastone canyon width along the shelf. Most of this upwelled water was pulled backinto the canyon on the second half of the tidal cycle. Friction values measured inthe laboratory were much higher than expected, possibly due to upwelled watersurging onto the shelf on each tidal cycle, similar to a tidal bore. By scaling ob-servations from the laboratory to the ocean and assuming friction in the ocean isalso affected by water travelling onto the shelf, a resonance curve for the Gullywas created. Resonance curves explain why the diurnal frequency dominatesover the semi-diurnal (M2) frequency throughout the year at the Gully, even ifstratification at the shelf break varies.iiPrefaceThis thesis contains details of experiments and analysis designed and undertakenprimarily by the author, Kate Le Soue?f. Susan Allen was the supervisor on thisproject and was involved in concept formation, interpretation of results andmanuscript edits. This work is previously unpublished, although a manuscriptbased on Chapter 2 will be submitted for publication in the future.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Submarine canyons . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Internal waves in submarine canyons . . . . . . . . . . . . . . . . 21.3 Studying submarine canyons . . . . . . . . . . . . . . . . . . . . . 41.4 Oscillating flow in submarine canyons . . . . . . . . . . . . . . . 61.5 The Gully, Nova Scotia . . . . . . . . . . . . . . . . . . . . . . . 71.6 Tidal resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Analytical models of the Gully . . . . . . . . . . . . . . . . . . . 111.8 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Physical modelling of resonance . . . . . . . . . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iv2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Scaling analysis . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Rotating table . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Canyon insert . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Stratifying the tank . . . . . . . . . . . . . . . . . . . . . 182.2.5 Light sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.6 Running an experiment . . . . . . . . . . . . . . . . . . . 212.2.7 Flow visualisation . . . . . . . . . . . . . . . . . . . . . . 212.2.8 Image processing . . . . . . . . . . . . . . . . . . . . . . . 222.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 General circulation over tidal cycle . . . . . . . . . . . . . 272.3.3 Resonance curve fitting . . . . . . . . . . . . . . . . . . . 312.3.4 Amplification and phase lag . . . . . . . . . . . . . . . . . 342.3.5 Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Experimental considerations . . . . . . . . . . . . . . . . . 442.4.2 Factors affecting resonant frequency . . . . . . . . . . . . 452.4.3 Structure of resonance . . . . . . . . . . . . . . . . . . . . 472.4.4 Friction in the laboratory . . . . . . . . . . . . . . . . . . 492.4.5 Resonance in the Gully . . . . . . . . . . . . . . . . . . . 502.4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Implications and future research . . . . . . . . . . . . . . . . . . 613.3 Broader context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62vReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A Non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . 67B Additional laboratory methods . . . . . . . . . . . . . . . . . . . 68B.1 Canyon insert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68B.2 Neutrally buoyant particles . . . . . . . . . . . . . . . . . . . . . 69C Standard error calculation . . . . . . . . . . . . . . . . . . . . . . 70D Separate resonance curves . . . . . . . . . . . . . . . . . . . . . . . 71viList of Tables2.1 Parameters and non-dimensional parameters . . . . . . . . . . . . 172.2 List of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Fit parameters for each stratification (N) . . . . . . . . . . . . . 362.4 Fit parameters for each background rotation (f) . . . . . . . . . 362.5 Fit parameters for each forcing amplitude (?f) . . . . . . . . . 362.6 Fit parameters for three locations along canyon . . . . . . . . . 432.7 Fit parameters for repeated experiments . . . . . . . . . . . . . 432.8 Parameters for resonance curves using representative stratifica-tions from three different depths in the Gully . . . . . . . . . . . 542.9 Parameters for resonance curves at different stratifications in theGully . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56viiList of Figures1.1 Location of the Gully, Nova Scotia . . . . . . . . . . . . . . . . . 82.1 Isobaths in the Gully and the laboratory canyon . . . . . . . . . 192.2 Density profile and buoyancy frequency in the tank . . . . . . . . 202.3 Velocity field for Experiment 3 (? = 0.359s?1) . . . . . . . . . . 252.4 Velocity field for Experiment 3 (? = 0.8s?1) . . . . . . . . . . . . 262.5 General circulation in the laboratory model, with tidal flow op-posite to rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 General circulation in the laboratory model, with tidal flow inthe same direction as rotation . . . . . . . . . . . . . . . . . . . . 292.7 Photographs of dye transported onto shelf . . . . . . . . . . . . . 302.8 Amplification and phase difference for each stratification . . . . . 362.9 Amplification and phase difference for each stratification against?/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.10 Amplification and phase difference for each background rotation 382.11 Amplification and phase difference for each forcing amplitude . . 392.12 Amplification and phase difference for three locations along canyon 412.13 Amplification and phase difference for repeated experiment . . . 432.14 Amplification and phase difference in the Gully calculated usingstratification chosen at three depths . . . . . . . . . . . . . . . . 54viii2.15 Amplification in the Gully for a range of stratifications averagedin the canyon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.1 Non-dimensional parameters for each experiment . . . . . . . . . 67B.1 Laboratory tank . . . . . . . . . . . . . . . . . . . . . . . . . . . 68D.1 Amplification and phase difference for first stratification . . . . . 71D.2 Amplification and phase difference for second stratification . . . 72D.3 Amplification and phase difference for third stratification . . . . 72D.4 Amplification and phase difference for first stratification against?/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.5 Amplification and phase difference for second stratification against?/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.6 Amplification and phase difference for third stratification against?/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74D.7 Amplification and phase difference for first background rotation . 74D.8 Amplification and phase difference for second background rotation 75D.9 Amplification and phase difference for first forcing amplitude . . 75D.10 Amplification and phase difference for second forcing amplitude . 76D.11 Amplification and phase difference for canyon head . . . . . . . . 76D.12 Amplification and phase difference for mid canyon . . . . . . . . 77D.13 Amplification and phase difference for canyon mouth . . . . . . . 77ixAcknowledgementsI have been extremely lucky to work in Susan Allen?s lab. Susan is a patient,enthusiastic and understanding supervisor who always makes time for her grad-uate students. Susan, thank you for all the advice, support and assistance inthe lab, during our meetings and every time I have come by your office. Inparticular, thank you for your unshakeable belief that we would make the tablespin again!I would like to thank my supervisory committee members, Douw Steynand Greg Lawrence, for their time, suggestions and enthusiasm throughout myproject. And thank you to Douw for helping me formulate my research plan inEOSC571 and for the chats about skiing.Thank you to David Jones for constant, patient assistance with the elec-trical wiring and programming of the table, and also for preventing me fromelectrocuting myself, and to Doug Latornell for working evenings and weekendsto repair the table. Thanks to David Jessop for being a great lab buddy in thedark second basement.I consider myself very fortunate to have shared the Waterhole with so manyexcellent companions, incredible MATLAB advisors and wonderful friends.Thank you to my wonderful parents and siblings for their ongoing support,to Az for convincing me to return to university and to Vince for constant en-couragement and love.xChapter 1Introduction1.1 Submarine canyonsSubmarine canyons are topographic coastal features that cut into the continentalshelf. Roughly 20% of the western North American shelf edge between Alaskaand the Equator is interrupted by canyons [Hickey , 1995]. Submarine canyonsare common in many regions of the world. They can vary greatly in size buthave typical cross canyon length scales of approximately 10 kilometres, withtopographic slopes along the longitudinal axis of up to 45 ? [Mirshak and Allen,2005].In general, exchange between the deep ocean and the continental shelf is lim-ited. Homogeneous, geostrophic flow cannot change its depth and is restrictedto follow isobaths along the continental shelf, such that deep ocean exchangeoccurs only when ageostrophic dynamics occur [Allen and Durrieu de Madron,2009]. Interruptions in the continental shelf, such as submarine canyons, disruptalong-shelf flow and can cause mixing, internal waves and upwelling [Hickey ,1995]. The presence of a canyon transports deep water further onto the shelf1than would occur for a straight continental shelf and provides enhanced mixing,particularly due to tides [Allen and Durrieu de Madron, 2009].Submarine canyons play an important role in regional ecosystems and canbe locations of enhanced species diversity and biological productivity [Hickey ,1995]. In many canyons, the accepted reason for this diversity is upwelling,which provides a nutrient source that increases phytoplankton and zooplanktondensity [Hickey , 1995]. For example, canyon upwelling at Barkley Canyon,British Columbia, can be observed close to the surface at 10 metres depth andzooplankton species are passively advected by the currents in and around thecanyon [Allen et al., 2001]. Submarine canyons can therefore affect coastalbiological processes.Mixing in submarine canyons can be intense. For example, the energy ofinternal wave fields in canyons is elevated compared to the open ocean [Kunzeet al., 2002] leading to enhanced breaking and kinetic energy dissipation, whichcan be up to 102 times higher than in the open ocean [Gregg et al., 2005].Canyons are regions of enhanced turbulence, upwelling and internal wave gen-eration, as well as increased transport of material across shelf/slope [Boyer et al.,2000]. The physical processes in submarine canyons are therefore an importantpart of the coastal zone, particularly when considering sinks and sources ofenergy.1.2 Internal waves in submarine canyonsSubmarine canyons can generate internal gravity waves with tidal frequencies,known as internal tides, by the scattering of barotropic tides over sloping topog-raphy [Baines, 1983]. In stratified shelf systems, internal tides are generated bytidal flow over abrupt shelf break topography [Wunsch, 1975]. If the tidal forc-ing frequency (?) is superinertial (? > f where f is the Coriolis frequency) and2below the buoyancy frequency, N , the oscillations may propagate away from thegeneration site as plane internal waves [Petruncio et al., 1998]. Forced motions,such as tides with ? < f , can produce subinertial internal waves, but thesewaves will be bottom trapped, decay exponentially away from the seafloor andwill only propagate on topography [Swart et al., 2011]. Internal wave generationis strongest where the bottom slope is critical for a given frequency [Petruncioet al., 1998].Freely propagating internal waves travelling outside the canyon can also be-come trapped in a canyon. Submarine canyons usually become narrower as theybecome shallower, which causes geometric funneling and linear amplification ofenergy towards the head [Swart et al., 2011]. Internal wave energy incidentfrom the ocean can be further focused by successive reflections off supercriticalcanyon walls, creating an ?internal wave trap? in the canyon [Gordon and Mar-shall , 1976]. Observations from submarine canyons have shown an increase inenergy toward the bottom and head of canyons, due to geometric focusing andfunneling [Swart et al., 2011].There are three reflection conditions possible for internal waves normallyincident on a sloping seabed, such as the bottom of a submarine canyon. Theangle of the incoming internal waves to the horizontal (?) is determined by theinternal wave frequency, density profile and latitude [Cacchione et al., 2002]. Ifthe bottom slope (?) is less than the angle of the incoming internal waves, theninternal waves can be transmitted [Cacchione et al., 2002]. If the bottom slopeis greater than the angle of the incoming internal waves, waves are reflected[Cacchione et al., 2002]. For a critical bottom slope and a critical frequency,the incident ray will be reflected parallel to the sloping bottom, and will beunable to carry energy away [Eriksen, 1982]. This critical condition (?/? = 1)leads to maximum bottom velocities and shear stresses [Cacchione et al., 2002].3Many submarine canyons that have been studied (e.g. Monterey Canyonat 37 ?North) are at latitudes where the frequency of the semi diurnal M2 tide(?M2) is super inertial (?M2 > f) but the frequency of the diurnal K1 tide(?K1) is subinertial (?K1 < f). The M2 tide at these latitudes can travelas a freely propagating internal wave and experiences geometric focusing andfunneling in a submarine canyon. The subinertial K1 tide at these latitudesmay generate bottom trapped internal waves in a canyon, which can also befurther amplified by geometric focusing and funneling. Available studies fromsubmarine canyons with latitudes poleward of 30 ?, such as Monterey Canyon[Petruncio et al., 1998], have found that the M2 tide tends to be amplified insubmarine canyons due to geometric focusing of the freely propagating internalwaves from outside the canyon. Harmonic analysis of measured data at GaopingCanyon, Taiwan (23?N), also shows that the superinertial semi diurnal (M2)tidal constituent is dominant, although the generation mechanism at Gaopinghas not been conclusively determined and both the M2 and K1 frequencies aresuperinertial [Wang et al., 2008].1.3 Studying submarine canyonsSubmarine canyons have been effectively studied using observations, numeri-cal models, analytical models and physical scale models. Field observations insubmarine canyons are among the most difficult in the ocean to make [Hickey ,1995]. Successfully mooring instruments on steep canyon walls can be challeng-ing. Field measurements are expensive to collect and are necessarily limitedin space and time. The spatial scales of processes in canyons are also typi-cally small, requiring many instruments to delineate spatially coherent signals[Hickey , 1995]. Field observations are further limited by the number of physicalprocesses acting simultaneously, which makes it difficult to link a specific cause4to an observed effect [Boyer et al., 2000].The steep and abrupt topography of submarine canyons also presents diffi-culties for numerical models. Early numerical models could not deal with thecomplexities of canyons, such as the presence of an open boundary, steep slopesand abrupt changes in isobar orientation [Hickey , 1995]. A major advance wasmade when numerical models were developed that included realistically steepand abrupt topography (e.g. Allen [1996]). More recent numerical models ofcanyons are still limited by computational capacity to resolve boundary layers,advection and larger flow [Allen and Durrieu de Madron, 2009]. Models suchas MITgcm can be used to study canyons, but require sufficient horizontal res-olution to achieve a smooth representation of the steep canyon side slopes andsufficient vertical resolution to resolve the wind driven Ekman layer at the sur-face [Dawe and Allen, 2010]. Truncation errors in numerical models can lead toerrors in vertical advection of approximately 70% [Allen et al., 2003]. Numeri-cal models are further limited by lack of sufficient measured data available formodel development and comparison.Analytical models can be developed to explain simplified canyon systems.For example, Swart et al. [2011] used two simplified 1D models to study crosscanyon and along canyon flow in the Gully, Nova Scotia. These models providedmeaningful information about dynamic processes in the canyon. However, thesituations that can be studied using an analytical model are limited and theassumptions that are required (e.g. 1.5 layer ocean where top layer is stagnant)are often overly restrictive.The last method for studying submarine canyons is to utilize a physical scalemodel. Physical models have been used to study many physical processes insubmarine canyons. Laboratory models can isolate individual physical processesand can be a cost-effective way of guiding the development of numerical models5[Boyer et al., 2004]. To use a physical model, non-dimensional parameters arecalculated from field observations to represent dominant forces affecting flow.The non-dimensional parameters are then matched to the laboratory model toensure dynamic similarity between field and laboratory.Laboratory models are cheaper and faster than field observations but have anumber of limitations. Length scales are limited for practical reasons such thatlarge Reynolds number flows (turbulent flows) cannot be modelled. Frictionmay also dominate laboratory flows. The bathymetry and shape of submarinecanyons in the laboratory are also simplified for practical reasons and to limitfriction.1.4 Oscillating flow in submarine canyonsSeveral studies have modelled flow in laboratory canyons. Boyer et al. [2000]performed a series of experiments modelling an oscillating flow along a shelfbreak near a single isolated canyon in a laboratory tank. The backgroundoscillating flow was driven by modulating the rotation rate of the rotating table.Boyer et al. [2000] found that a mean downstream flow was generated along theshelf, as well as a mean flow from deep ocean up the canyon. The mean flow wassignificantly stronger for subinertial oscillations [Boyer et al., 2000]. A numericalmodel by Kampf [2009] suggests that purely oscillatory flow does not generateupwelling, but that upwelling is instead caused by the steady component of theflow.Comparisons between the laboratory model of Boyer et al. [2000] and anumerical model showed good agreement in the structure and magnitude ofcurrents at the shelf break, although there was less agreement between labora-tory and numerical models higher and lower than the shelf break, where residualcurrents were weaker [Perenne et al., 2001]. A scaling argument was applied6to these data by Boyer et al. [2004] to characterise the strength of the horizon-tal component of the time mean velocity in the laboratory. This relationshipwas derived for laminar flow in a small tank with a 1 metre diameter. Subse-quent experiments in the 13 metre diameter Grenoble facility showed that thisrelationship did not apply to large Reynolds number flows for transitional andturbulent flow, but that the behaviour of the large and small laboratory modelswas qualitatively similar [Boyer et al., 2006].Oscillatory flow in canyons has been observed to form pools of dense water onthe shelf on either side of the canyon (Haidvogel [2005], Kampf [2009]). Thesepools can be several times the area and volume of the canyon itself and theonshelf transport of dense water increases with increasing forcing period andstrength [Haidvogel , 2005]. Kampf [2009] found that these pools are caused bythe steady component of the oscillating flow, rather than by purely oscillatingflow.Due to rotation of the earth, flow within a submarine canyon is asymmetric.When the oscillating flow acts in the same direction as rotation, flow divergesfrom the upstream wall of the canyon and exits from the canyon mouth nearthe downstream wall [Boyer et al., 2006]. When the oscillating flow acts againstthe direction of rotation, flow follows the upstream canyon wall into the canyonand upwells along the downstream wall [Boyer et al., 2006].1.5 The Gully, Nova ScotiaThe Gully is a broad, deep submarine canyon near Sable Island, Nova Scotia(Figure 1.1). The Gully is the largest submarine canyon on the eastern NorthAmerican coast and is approximately 35 kilometres in length and 15 kilometreswide at the mouth. The maximum depth of the Gully is around 2200 metres.The canyon cuts into the continental shelf, which is between 200 and 500 metres7  80oW   72oW   64oW   56oW   48oW   24oN   30oN   36oN   42oN   48oN   54oN InsetCanadaUSA20020020020020020020020020050050050010001000100020002000200030003000 3000  66oW   64oW   62oW   60oW   58oW   56oW   43oN   44oN   45oN   46oN   47oN SableIslandPEINovaScotiaThe GullyFigure 1.1: Location of the Gully, Nova Scotiain depth in this region. The top of the canyon is approximately defined bythe 500 metre contour. The steep walls of the Gully are reflective for tidalfrequencies and the floor may be transmissive, critical or reflective, dependingon the stratification [Swart et al., 2011].The Gully is an important habitat for deep sea corals and Northern bottlenosed whales and has been a Marine Protected Area since 2004. The Gully is themost important habitat for cetaceans on the Scotian Shelf and is used by up to13 different cetacean species [Mann, 2002]. Elevated concentrations of nutrientshave been observed at the canyon head [Strain and Yeats, 2005], and the Gully isknown as a rich feeding ground for groundfish and pelagic fish [Mann, 2002]. Anunderstanding of the physical oceanography of the region has been recognizedas providing the ?framework for addressing various interdisciplinary and appliedissues in the region? [Han et al., 2002]. Until recently, the process responsible for8the observed vertical nutrient transport was not clear as large scale circulationalong the coastline makes the Gully a downwelling canyon [Swart et al., 2011].Sandstrom and Elliott [2002] found that semi diurnal (M2) tides were amplifiedthreefold in the canyon relative to the adjacent shelf.To further investigate which processes may be contributing to the verticalnutrient transport, Greenan et al. [2013] collected one year of current meter ob-servations at four moorings in the Gully. One mooring was placed 5 kilometresfrom the canyon head and the other three moorings were 20 kilometres fromthe canyon head across the width of the canyon. The original moorings weredesigned to withstand currents of approximately 20-30 cm s?1; however, actualcurrents in the canyon significantly exceeded these speeds, resulting in excessivemovement of the moorings. Maximum current velocities along the longitudi-nal axis of the canyon were approximately 80 cm s?1. Some of the mooringsexperienced pressure variations of approximately 500 db from knock-down bythe fast moving currents. Greenan et al. [2013] analysed these observations tocalculate a total transport of 35,500 m3 s?1 towards the canyon head from adepth of 200 metres down to the seabed and estimated mixing in the Gully tobe approximately 20 times greater than on the adjacent shelf.Currents in the Gully were significantly higher than expected. Swart et al.[2011] performed Fourier analysis on current data from the Gully and waterlevel data from Halifax, which is approximately 200 kilometres south east ofthe Gully. This analysis revealed that the dominant frequency in the Gully wasthe diurnal (K1) frequency, whereas at Halifax the dominant frequency was thesemi-diurnal (M2) frequency. Although the M2 constituent dominates tides inthis region, some process is occurring in the Gully to amplify the K1 frequency.Shan et al. [2013] used a 3-dimensional ocean circulation numerical modelto study the circulation of the Gully, and showed that wind significantly influ-9enced circulation above the canyon rim, while deeper in the canyon, shelf scaleflow and tides were more important. On an annual mean timescale at depthsbelow 200 metres, wind and other processes cause off-shelf transport, but over-all transport is approximately 0.003Sv on-shelf due to the tidal residual. Thismodel was unable to produce the bottom intensified tidal ellipses measured bySwart et al. [2011], due to insufficient spatial resolution. This study emphasizesthe importance of the tide-topography interaction in the canyon.The Gully is the first canyon where the diurnal K1 tidal dominance has beenidentified, despite the high latitude (44 ? North) where K1 tidal frequencies aresubinertial (?K1 < f). Other studies of submarine canyons at similar latitudes(e.g. Monterey Canyon) show the dominance of the superinertial M2 tidal fre-quency. Although both K1 and M2 are superinertial at Gaoping Canyon, the M2frequency still dominates observations [Wang et al., 2008]. Swart et al. [2011]suggested that tidal resonance of the K1 frequency in the Gully could be causingthe K1 frequency to dominate tidal currents.1.6 Tidal resonanceResonance occurs when a forcing excites a natural mode of a system. In theocean, tides can produce tidal resonance when the frequency of the tide matchesthe natural frequency of a semi-enclosed body of water. At resonance, the mag-nitude of a system?s response is amplified relative to the forcing magnitude.Amplification refers to the ratio of the magnitude of the response to the magni-tude of the forcing. The phase lag is the difference in phase between the forcingand the response. The quality factor (Q) is used to represent the width of theresonance curve. A large Q factor indicates a large response at a narrow bandof frequencies, while a smaller Q indicates a smaller magnitude response at abroad range of frequencies.10The Bay of Fundy between New Brunswick and Nova Scotia is one of themost famous examples of tidal resonance in the world. The period of the semi-diurnal M2 tide matches the period of the free mode of oscillation in the Bay,creating large tides with a tidal range of over 15 metres [Garrett , 1972]. TheQ factor for the Bay of Fundy is estimated to be approximately 5.25 and phaselag was observed to increase with the forcing frequency [Garrett , 1972].Sutherland et al. [2005] studied tidal resonance in the Juan de Fuca Strait andthe Strait of Georgia to find the resonant period and quality factor. The tidalelevation gain from the Pacific Ocean to the Strait of Georgia was calculatedfor observed data. This system was found to have a relatively low Q value ofapproximately 2, indicating a highly dissipative system and highlighting theneed for numerical models to include large bottom friction coefficients in thisregion [Sutherland et al., 2005].Internal tidal resonance has been observed in many areas of the ocean.For example, Dushaw and Worcester [1998] observed a resonant internal wavetrapped between the shelf north of Puerto Rico and the turning latitude for theK1 and O1 frequencies. The turning latitude is the latitude at which each tidalfrequency is equal to the inertial frequency. These resonant internal waves cov-ered a distance of approximately 1100 kilometres. Internal tidal resonance wasalso observed by de Young and Pond [1987] in Indian Arm, British Young and Pond [1987] found that the resonance period of the fjord in-creased over winter, until an internal resonant response was observed at the K1frequency in late winter.1.7 Analytical models of the GullyAs described above, Swart et al. [2011] developed two one-dimensional analyticalmodels of the Gully to describe cross canyon and along canyon flow. Although11these models are simplifications of the real system, important physical dynamicswere represented and the possibility of tidal resonance of the K1 frequency wasinvestigated.The first model was a cross shelf model of an infinite trench to study the lowerlayer baroclinic response to barotropic forcing along the shelf. For simplicity,this model neglected topographic variation in the longitudinal direction (y).Swart et al. [2011] showed that along canyon velocity in a triangular trenchdepended on three parameters: (1) ratio of canyon width to Rossby radius(W/R), (2) trench depth relative to lower layer depth on the shelf (?H/H0)and (3) the temporal Rossby number (?/f). Along-shelf barotropic forcing atnear inertial frequencies could initiate a baroclinic response flow of the samemagnitude in the canyon. As barotropic flow crossed from the shallow shelf intothe trench, water depth increased, stretching occurred and cyclonic vorticity wasgenerated. As flow exited the trench on the other side, water depth decreased,squashing occurred and anti-cyclonic vorticity was generated. When barotropicforcing was positive, this process resulted in a positive baroclinic v velocityinside the trench and when barotropic forcing was negative, a negative baroclinicv velocity inside the trench.The second analytical model by Swart et al. [2011] considered motion forcedin the along-canyon direction. For this case, a 1.5 layer channel was used, witha closed boundary at the head of the channel. This model showed that thevelocity profile of a resonant wave in the canyon could be significantly changedby stratification, forcing frequency and basic geometry changes in the along-canyon direction.The length scales in the Gully are consistent with tidal resonance of theK1 tidal component and the diurnal frequency was very near to resonant forthe stratification observed by Swart et al. [2011]. The role of the resonance12in the Gully is therefore to amplify the effects of the geometric funneling onthe along-canyon baroclinic flow [Swart et al., 2011]. The analytical modeland observations both showed the same pattern of increase in velocity towardthe head of the canyon, providing a possible mixing mechanism to explain theelevated nutrient and organic carbon levels observed in the Gully [Swart et al.,2011].1.8 Research questionsThe Gully presents an interesting challenge in the study of physical oceano-graphic processes in submarine canyons. Swart et al. [2011] made many ad-vances in the study of the physical oceanographic processes in the Gully. Cur-rent measurements were collected, the K1 tidal dominance was observed and ananalytical model was developed that captured many of the important dynamicalprocesses in the canyon.The next step in the study of the Gully is to further analyze the tidal reso-nance in the canyon, by determining the factors that influence resonance and thestructure of the resonance. The rotating table at UBC provides the opportunityto study the possibility of tidal resonance in the Gully.This study attempts to answer the following research questions:1. Can tidal resonance observed in the Gully be reproduced in a physicalmodel?2. Which conditions will create resonance (e.g. stratification, forcing fre-quency)?3. What is the three dimensional structure of resonant tides in the Gully?4. What is the spatial extent of the influence of the canyon beyond the Gully?135. How do measurements from the physical model compare to measurementscollected from the Gully by Swart et al. [2011]?6. What is the Q factor of the Gully?14Chapter 2Physical modelling ofresonance2.1 IntroductionThe objective of the current study is to reproduce tidal resonance observed inthe Gully in a laboratory model to determine which factors affect resonanceand to characterise the structure of the resonance in the Gully. This sectiondescribes how the physical model was used to study tidal resonance.Firstly, laboratory methods are described, including descriptions of scal-ing, equipment and procedures for running an experiment. Results from thelaboratory are presented next and analysed with a theoretical description ofresonance. Factors affecting tidal resonance are discussed and laboratory re-sults are compared to observations from the Gully. Through scaling argumentsand assumptions about friction, results from the laboratory are applied to theocean to create a resonance curve for the Gully. In the third chapter, resultsare interpreted with reference to the original research questions.152.2 Methods2.2.1 Scaling analysisA scaling analysis was undertaken to ensure dynamic similarity between theocean and the physical model. Four non-dimensional parameters were chosento encompass the dynamic processes that are of fundamental importance tothis system, including the temporal Rossby number (Rot = ?/f , where ? isthe frequency of oscillation and f is the Coriolis frequency (f = 2? where? is the rotation rate of the table)), Rossby number (Ro = U/(fRc), whereU is the along shelf velocity and Rc is the radius of curvature of the canyon,which is measured along the shelf break isobath on the flank of the canyon),Burger number (Bu = (NHs)/(fLc), where N is the buoyancy frequency (N2 =?(g/?0)(??/?z)), g is gravity, ?0 is the reference density, ? is density, z is thevertical co-ordinate, Hs is the depth of the shelf break and Lc is the length ofthe canyon) and Froude number (Fr = U/(NHs)).Non-dimensional parameters were calculated from field measurements at theGully reported in Swart et al. [2011] (Table 2.1). Note that canyon width is mea-sured at the shelf break. The frequency of oscillation (?), Coriolis frequency(f), amplitude of forcing (?f) and buoyancy frequency (N) were varied inthe laboratory to match the field non-dimensional parameters as closely as waspractical. Experiments were performed to cover a range of non-dimensional pa-rameters around the observed values at the Gully (Table 2.2, shown graphicallyin Appendix A).2.2.2 Rotating tableA physical model of a submarine canyon was used for this experiment. Thisexperimental set-up has been used previously to study submarine canyons (e.g.16Table 2.1: Parameters and non-dimensional parametersVariable Symbol The Gully ModelShelf break depth Hs 500 m 2.1 cmShelf length Ls 70 km 22.5 cmCanyon length Lc 35 km 14 cmCanyon width (mouth) W 15 km 10.1 cmRadius of curvature Rc 16 km 6 cmCoriolis frequency f 1.01? 10?4 s?1 0.5? 0.75 s?1K1 frequency ?K1 7.27? 10?5 s?1 0.36 s?1 (f = 0.5 s?1)0.54 s?1 (f = 0.75 s?1)M2 frequency ?M2 1.41? 10?4 s?1 0.70 s?1 (f = 0.5 s?1)1.04 s?1 (f = 0.75 s?1)Buoyancy frequency N 3.16? 10?3 s?1 0.86? 1.55 s?1(at shelfbreak)Incident velocity U 0.1 m s?1 0.05? 0.15 cm s?1Rossby number Ro 0.06 0.02? 0.05Temporal Rossby number Rot 0.72 0.4? 1.6Burger number Bu 0.45 0.17? 0.46Froude number Fr 0.06 0.03? 0.09Table 2.2: List of experimentsExperiment f ?f N ? (s?1)(s?1) (s?1) (s?1)1 0.5 0.04 0.83 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.82 0.5 0.04 1.11 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.83 0.5 0.04 1.53 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.84 0.75 0.04 0.83 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.25 0.75 0.04 1.11 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.26 0.75 0.04 1.53 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.27 0.5 0.08 0.83 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.88 0.5 0.08 1.11 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.89 0.5 0.08 1.53 0.2/0.3/0.359/0.4/0.498/0.6/0.714/0.810 0.75 0.08 0.82 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.211 0.75 0.08 1.11 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.212 0.75 0.08 1.53 0.3/0.45/0.539/0.6/0.747/0.9/1.071/1.217Mirshak and Allen [2005], Waterhouse et al. [2009]). General tank topography isshown in Dawe and Allen [2010]. A cylindrical Plexiglass tank with a diameterof 1 m is co-axially placed upon a rotating table. The tank has a deep abyssalplain in the centre and a shallower continental shelf around the outside. Canyonshapes can be inserted into a 22? removable section of the continental shelf (seephoto in Appendix B). The table can be programmed to rotate at constant andvarying speeds.2.2.3 Canyon insertA canyon insert for the cylindrical tank was built in fiberglass based on multi-beam bathymetry of the Gully (see Appendix B). The Gully is a complexshape, with a curved centreline and numerous ridges that would affect refrac-tion of short waves in the canyon. However, this study is primarily concernedwith longer wavelength waves that are not easily modified by the presence ofcurves and ridges. A simplified bathymetry was therefore created by smoothingthe original bathymetry until it was symmetric (Figure 2.1).Smoothing the canyon also minimised friction of water passing over thecanyon in the tank. Friction in the canyon insert was further minimized byexaggerating the canyon width by a factor of 1.5. Finally, the canyon wasstraightened so the longitudinal axis of the canyon would be aligned with aradial line from the centre of the tank to the outside of the tank.2.2.4 Stratifying the tankThe two bucket method [Oster , 1965] was used to create salt stratification inthe tank. Fluid slip rings in the centre of the rotating table allow the tank to befilled whilst rotating. The rotation rate of the table was constant (f = 0.5 s?1or f = 0.75 s?1) during filling. Filling the tank to a depth of 9.1 cm took18 12?   6?   59oW  54?  48?  42?  42?  48?  54?   44oN   6?   0510 km(a) The Gully [m]?2500?2000?1500?1000?5000x [cm]y [cm](b) Laboratory canyon [cm]  ?5 0 5?5051015?9?8?7?6?5?4?3?2?10Left ? ? RightFigure 2.1: (a) Isobaths in the Gully in m. Contours are 240 m apart, (b)Isobaths in the laboratory canyon in cm. Contours are 1 cm apart. ?Left? isdefined as x < 0, ?Right? is defined as x > 0 and y = 0 represents the shelfbreak.approximately 90 minutes. Once the tank was full, the table was covered andleft to rotate for a further 3 hours to ensure solid body rotation was achieved.Density measurements were taken before and after an experiment at a rangeof depths to determine the actual stratification created using this filling method.Density measurements were made with an Anton Paar DSA5000. The measuredand expected density and buoyancy frequency profile are shown in Figure 2.2before and after an experiment. The expected density was calculated from thedensity measured in the buckets prior to the experiment (Bucket 1 is the bucketwith more salt in the method described by [Oster , 1965]) and took into accountdiffusion of salt over 4.5 hours. The amount of mixing that occurred over thecourse of an experiment is within the error of the density measurements.The buoyancy frequency at the shelf break was taken as the average betweenthe ?before experiment? and ?after experiment? fitted lines at 2.1 cm depth. Table2.2 shows the N value used for each experiment.191.04 1.045 1.05 1.0550123456789density [kg/L]Depth [cm]  0 0.5 1 1.5 2 2.50123456789N2 [s?2]Depth [cm]CalculatedBucket 1Bucket 2Before exp.Before exp. fitAfter exp.After exp. fitDensity and buoyancy frequency (N2) with depth for 0.44kg of saltFigure 2.2: Density profile (left) and buoyancy frequency (right) in the tank.Asterisks indicate density samples taken from buckets at beginning of experi-ment. Solid lines indicate calculated density and buoyancy based on densitiesmeasured in buckets. Triangles and circles indicate measured density samplestaken before and after an experiment respectively. Dashed lines are fitted tothe measured data.2.2.5 Light sheetThe table has a superstructure for attaching a slide projector and a video camera1.8 m above the tank. Power was supplied to the projector and camera viaelectrical slip rings. The projector provided intense, focussed light that wasreflected by a mirror onto the canyon to illuminate a horizontal sheet of lightinto the canyon. For each experiment, a horizontal sheet of light approximately2 cm thick was illuminated by the projector, with the top of the light sheet at20the shelf break.2.2.6 Running an experimentOnce solid body rotation was achieved for a given N and f , oscillation of thetable was initiated on the control computer and the video camera began to film.The rotation rate of the table (ft) varied such that ft = f + ?fsin(?t), wheret is time. Each frequency of oscillation (?) was filmed for a period of 50 tidalcycles, similar to methods described in Boyer et al. [2000]. After 50 tidal cycles,the video was switched off, a new profile with a different ? was selected on thecomputer and filming began again.The overall stratification of the tank after each experiment was assumed to beclose to the stratification at the beginning of the experiments. This assumptionwas supported by the small difference in measured density before and after anexperiment (Figure 2.2). This allowed multiple oscillating frequencies to be runconsecutively for each combination of N and f .2.2.7 Flow visualisationEach experiment was filmed on a video camera (Canon HDR-CX190), attachedto the superstructure above the table and focussed on the canyon region. Thecamera recorded at a resolution of 1920 ? 1080 pixels, at a rate of 30 framesper second. Every 1 second of video was converted into 10 images (i.e. 0.1 sbetween images).Flow was observed in the tank by particle tracking. Particles were composedof varying amounts of wax and titanium dioxide to create a range of particledensities which were neutrally buoyant at different depths in the water column[Reuten, 2006]. Particles ranged in size from 0.5 mm to 1.5 mm. To reduce sur-face tension and prevent particles clumping together, the particles were mixed21with water and a surfactant before being added to the tank (see Appendix B).Particles were added to the tank immediately prior to beginning an experiment.Dye was used in addition to particles in some experiments to visualise threedimensional flow.2.2.8 Image processingParticle Image Velocimetry (PIV) was used to analyse images and estimatevelocity fields. The tool ?MatPIV? developed by J. Kristian Sveen and availableonline was applied to the jpeg images [Sveen, 2004]. PIV tools compare animage of particles at one time point to a subsequent image to determine howparticles have moved between shots and thus to calculate particle speeds. A twodimensional vector field can then be created to represent velocities for every pairof images.A number of MatPIV filters were chosen to remove erroneous results causedby low image quality or a lack of particles. A single pass window of 64 pixelswas chosen, with an overlap of 50% between windows. Using this window size,velocities could be calculated over the area of the canyon with a resolution ofapproximately 1 cm by 1 cm, which was large enough to avoid errors from lackof particles while still providing reasonable spatial resolution. Recommendedvalues in MatPIV were used for the threshold for signal to noise filtering (1.3),global filtering (4) and local median filtering (2.5), and outliers were interpolatedusing linear interpolation.Velocity results were averaged at 8 points of the tidal cycle over the 50 tidalcycles. The amplitude of the forcing and response was approximated by fittingsinusoids to velocity data averaged across a number of grid points outside andinside the canyon, respectively. Amplification inside the canyon was calculatedas the amplitude of sine wave fitted to the response divided by the amplitude22of the sine wave fitted to the forcing outside the canyon. Phase difference wasalso calculated between the sine wave fitted to the response and the sine wavefitted to the forcing.232.3 Results2.3.1 Velocity fieldsTwo representative velocity fields calculated using PIV are shown in Figure 2.3(? = 0.359 s?1) and Figure 2.4 (? = 0.8 s?1) for Experiment 3. These Figuresshow averaged velocities in the canyon at 8 points in the tidal cycle at the depthof the horizontal light sheet (2.1 cm-3.6 cm depth). The shelf break is at 2.1 cmdepth. Mass is not conserved in each picture because each velocity field onlyshows velocities at one depth (i.e. velocities are not depth integrated). Canyoncontours are shown underneath for reference. The ?left? flank is defined as theflank on the left when looking from deep ocean up the canyon towards the shelf(x < 0) and the ?right? flank is the flank on the right (x > 0) (Figure 2.1).Comparison of along-canyon velocity in the two velocity fields shows theeffect of varying ?. Figure 2.3 represents a near-resonant condition. When? = 0.359 s?1, velocities within the canyon are amplified relative to velocitiesoutside the canyon. Flow into and out of the canyon is stronger on the leftflank of the canyon (x < 0) than the right flank of the canyon (x > 0) and theresponse within the canyon varies with distance from the canyon mouth. Forexample, halfway through the tidal cycle (t = T/2, where t is time and T is tidalperiod), velocity at the canyon head is out of the canyon, velocity midway alongthe canyon is very low and velocity at the canyon mouth is into the canyon.This pattern is reversed at the end of the tidal cycle (t = T ).In contrast, Figure 2.4 represents a non-resonant condition. When ? =0.8 s?1, velocities inside the canyon are much less than velocities outside thecanyon. There is still variation in direction and velocity with distance from thecanyon mouth, but velocities are generally low within the canyon.24?505?10?5051015x [cm]y [cm]t=T/80.5cm/st=T/4t=3T/8t=T/2t=5T/8t=3T/4t=7T/8t=TTidally averaged velocity (mean subtracted), ? = 0.359 s?1 , 02?19?2013Figure 2.3: Averaged velocity field around shelf break, for Experiment 3 (? = 0.359 s?1) at 8 points throughout the tidal cycle(period T). Canyon contours (1 cm apart) are shown in grey. A reference vector of 0.5 cm s?1 is shown on the first panel.25?505?10?5051015x [cm]y [cm]t=T/80.5cm/st=T/4t=3T/8t=T/2t=5T/8t=3T/4t=7T/8t=TTidally averaged velocity (mean subtracted), ? = 0.8 s?1 , 02?19?2013Figure 2.4: Averaged velocity field around shelf break, for Experiment 3 (? = 0.8 s?1) at 8 points throughout the tidal cycle(period T). Canyon contours (1 cm apart) are shown in grey. A reference vector of 0.5 cm s?1 is shown on the first panel.262.3.2 General circulation over tidal cycleGeneral circulation in and around the canyon was sketched from observationsof particle and dye movement during one experiment (Experiment 11, ? =0.359 s?1). This frequency was chosen as the system was close to resonance,with amplified velocities inside the canyon compared to outside the canyon.The circulation sketched during this experiment was similar to observationsmade of general circulation at resonant frequencies in other experiments andcan therefore be taken to represent a generic resonant condition.Circulation in the canyon when tidal flow was in the opposite direction asbackground rotation (phase 1) is shown in Figure 2.5 and when tidal flow wasin the same direction as background rotation (phase 2) in Figure 2.6. Arrowlength represents approximate relative velocity. Photographs showing the extentof dye travelling onto the shelf at three points in the tidal cycle are also shownin Figure 2.7.In both phases, velocities in the along canyon direction were amplified, withrelatively large velocities along the longitudinal axis of the canyon when flowentered and exited the canyon. Upwelling was observed on the flanks of thecanyon during phase 1 and downwelling was observed during phase 2. Dyeexperiments showed that water within the canyon could be transported ontothe shelf and that shelf water could be pulled down into the canyon.27  Background rotationTidal flowFigure 2.5: General circulation in the laboratory model, with tidal flow oppositeto rotation. Orange depths indicate the continental shelf, blue depths indicatedepths below shelf break.28  Background rotationTidal flowFigure 2.6: General circulation in the laboratory model, with tidal flow in thesame direction as rotation. Orange depths indicate the continental shelf, bluedepths indicate depths below shelf break.29t = T/8 t = T/4 t = 3T/810 cmFigure 2.7: Photographs of dye transported onto shelf at three points in the tidal cycle (period T). General flow in eachphotograph corresponds to velocities shown at the same time points in Figure 2.3. Dye was injected outside canyon atapproximately 4 cm depth a few tidal cycles earlier. White line indicates approximate location of shelf break. Note that thewhite circle on the right side of the canyon mouth is a surface bubble.30Upwelled water was observed to extend approximately 10-15 cm (1 to 1.5times the canyon width) horizontally along the shelf on either side of the canyonwalls. Much of this water was downwelled back into the canyon during thesecond half of the tidal cycle. A small proportion of water from within thecanyon remained on the shelf between tidal cycles, but there did not appear tobe a constant flow or jet of water onto the shelf. Water upwelled from withinthe canyon onto the shelf was not observed to reach the surface. The shallowestupwelled water was estimated from observations to have reached a depth ofapproximately 1 cm (half the shelf break depth), however this height was notdirectly measured.As with the velocity fields created with PIV, the general circulation diagramsshow asymmetrical patterns in velocities on the canyon flanks close to the canyonmouth due to the effect of rotation. Stronger velocities were observed on theleft side of the canyon as water entered and exited the canyon.2.3.3 Resonance curve fittingFollowing the resonance fitting approach of Sutherland et al. [2005], curves werefit to measured amplification and phase lag data to determine the resonantfrequency (?0) and linear friction coefficient (?) of the laboratory canyon foreach experiment, or set of experiments.Firstly, the canyon is assumed to be a rectangular box of length L. Thebox is assumed to be a 1.5 layer system, with an infinitely deep upper layerand a lower layer of constant thickness h, so reduced gravity (g?) was used(where g? = g??/?0, where g is gravity, ?? is the difference in density betweenthe upper layer and the lower layer and ?0 is reference density). The internalRossby radius of deformation in the laboratory is approximately 22 cm, whichis larger than the width of the canyon (10.1 cm) so velocities in the x direction31are assumed to be negligible within the canyon. The basic equations for thesystem then become:?i?v + g????y+ ?v = 0 (2.1)?i?? + h?v?y= 0 (2.2)where ? is the frequency of the oscillation, ? is the elevation of the surfacebetween the two layers and v is velocity in the along canyon direction. Frictionin the laboratory is non-turbulent and a coefficient for linear friction in thelaboratory canyon could be approximated by:? =??h(2.3)where ? is viscosity of water and ? is depth of the Ekman layer.Boundary conditions ?(0) = a at canyon mouth and v(L) = 0 near thecanyon head are used to solve (2.1) and (2.2) for ? and v:?(y) =a cos(k(y ? L))cos(kL)(2.4)v(y) =i?a sin(k(y ? L))hk cos(kL)(2.5)wherek2 =?2g?h(1 +i??) (2.6)Expanding (2.6) and assuming for now that ? > ?, so that smaller termscan be neglected, gives:kL =?L?g?h(1 +i?2?) (2.7)32Note that even if ? ? ?, this approximation results in an error of approxi-mately 12.5% in the real term and 6.25% in the imaginary term. This is assumedto be acceptable for our purposes, because the approximation allows for a con-venient simplification of the relationship.The frequency of quarter wavelength resonance in a rectangular bay is givenby:?0 =pi2?g?hL(2.8)such that:kL =??0pi2+pi4??0i= ? + i(2.9)where? =??0pi2;  =pi4??0(2.10)The theoretical amplification in the canyon is approximated byAei? =?(L)?(0)(2.11)where A is amplification and ? is phase lag.Combined with (2.9) and using expansions for cosh and sinh, this simplifiestoAei? ?1cos ? ? i sin ?(2.12)Using approximations, the behaviour of this function as the frequency ap-proaches the resonant frequency (? ? ?0) is given byAei? =1(1? ??0 ?12 i??0)(2.13)33If the quality factor is defined as Q = ?0/?, then (2.13) becomesAei? =1(1? ??0 ?12 i1Q )(2.14)2.3.4 Amplification and phase lagEquation (2.14) was split into real and imaginary components, which were thensolved simultaneously for ?0 and ? by using a non-linear least squares fit tomeasured amplification and phase lag results. Errors associated with each fit-ted component were used to compare various experiments, stratifications, back-ground rotations and forcing amplitudes. Standard error is reported besidefitted ?0 and ? values in the following sections, as well as the root mean squareerror (RMSE) for each fit. Appendix C contains details of standard error cal-culations. Phase difference between forcing and response was required to be0 degrees at the resonant frequency. Appendix D shows each fitted curve andmeasured data separately.Effect of stratificationFirstly, Equation (2.14) was fit to measured amplification and phase differencedata for each stratification (N). Figure 2.8 shows fitted curves for each of thethree stratifications. Amplification was greatest for the largest stratification(A = 2.97 for N = 1.53 s?1).Table 2.3 shows the resonant frequency and friction coefficient calculatedfor each stratification. Resonant frequency and amplification within the canyonboth increased with increasing stratification. Fitted resonant frequencies weresignificantly different between each of the stratifications (P < 0.01). Frictioncoefficients were not significantly different between each stratification.For all three stratifications, the ratio of resonant frequency to stratification34(?0/N) was between 0.28 and 0.30, demonstrating there is a relationship be-tween stratification and resonant frequency. Amplification and phase differencedata were therefore fit against this new governing parameter ?/N instead ofsimply ? for all remaining analyses to remove the effect of stratification. Thefitted curves for each stratification plotted against ?/N are shown in Figure 2.9.Effect of background rotationEquation (2.14) was then applied to measured amplification and phase differencedata for each background rotation (f). Figure 2.10 shows the fitted curvesto each background rotation rate, plotted against ?/N , and Table 2.4 showsparameters calculated for each of the fitted curves. There was no significantdifference between the resonant frequencies and friction coefficients fitted to thetwo background rotation rates.Effect of forcing amplitudeNext, Equation (2.14) was fit to measured amplification and phase differencedata for each forcing amplification (?f). Figure 2.11 shows the fitted curvesto each forcing amplitude, plotted against ?/N , and Table 2.5 shows param-eters calculated for each of the fitted curves. The resonant frequency fittedto the lower forcing amplitude (?f = 0.04 s?1) was significantly higher thanthe resonant frequency fitted to the higher forcing amplitude (?f = 0.08 s?1)(P < 0.01). The friction coefficient for the higher forcing amplitude was alsosignificantly higher than for the lower forcing amplitude (P < 0.01).Variation with distance from canyon mouthTwo dimensional velocity fields showed that flow response inside the canyonvaried with distance from the canyon mouth. Equation (2.14) was therefore fit toamplification and phase difference data at three locations in the canyon to show350 0.5 1 1.501234567omega0 = 0.4579s?1, lamda = 0.3081s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)N=0.83s?1 fitN=0.83s?1 dataN=1.11s?1 fitN=1.11s?1 dataN=1.53s?1 fitN=1.53s?1 dataFigure 2.8: Amplification and phase difference for each stratification (N) for Ex-periments 1-12. Curves were fit to measured amplification and phase differencedata simultaneously for each stratification.Table 2.3: Fit parameters for each stratification (N)N (s?1) ?0 (s?1) ? (s?1) Q (-) A (-) RMSE RMSEat ?0 for A (-) for ? (?)0.83 0.23? 0.013 0.22? 0.034 1.05 2.1 0.82 32.61.11 0.33? 0.013 0.28? 0.032 1.18 2.4 0.78 37.71.53 0.46? 0.011 0.31? 0.023 1.49 3.0 0.81 24.0Table 2.4: Fit parameters for each background rotation (f)f (s?1) ?0/N (-) ?/N (-) Q (-) A (-) RMSE RMSEat ?0 for A (-) for ? (?)0.5 0.30? 0.0082 0.22? 0.018 1.36 2.7 1.02 27.10.75 0.28? 0.0072 0.24? 0.018 1.17 2.4 0.49 36.4Table 2.5: Fit parameters for each forcing amplitude (?f)?f (s?1) ?0/N (-) ?/N (-) Q (-) A (-) RMSE RMSEat ?0 for A (-) for ? (?)0.04 0.31? 0.0052 0.18? 0.011 1.72 3.6 0.80 36.20.08 0.25? 0.0084 0.32? 0.023 0.78 1.6 0.34 25.0360 0.5 1 1.501234567omega0 = 0.30003s?1, lamda = 0.20185s?1?/N (?)Amplification (?)  0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)N=0.83s?1 fitN=0.83s?1 dataN=1.11s?1 fitN=1.11s?1 dataN=1.53s?1 fitN=1.53s?1 dataFigure 2.9: Amplification and phase difference for each stratification (N) against?/N for Experiments 1-12. Curves were fit to measured amplification and phasedifference data simultaneously for each stratification.370 0.5 1 1.501234567omega0 = 0.2843s?1, lamda = 0.23966s?1?/N (?)Amplification (?)  0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)f=0.5s?1 fitf=0.5s?1 dataf=0.75s?1 fitf=0.75s?1 dataFigure 2.10: Amplification and phase difference for each background rotation(f) for Experiments 1-12. Curves were fit to measured amplification and phasedifference data simultaneously for each background rotation value.380 0.5 1 1.501234567omega0 = 0.25309s?1, lamda = 0.31924s?1?/N (?)Amplification (?)  0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)? f=0.04s?1 fit? f=0.04s?1 data? f=0.08s?1 fit? f=0.08s?1 dataFigure 2.11: Amplification and phase difference for each forcing amplitude (?f)for Experiments 1-12. Curves were fit to measured amplification and phasedifference data simultaneously for each forcing amplitude.39how the response varied. Only one stratification was considered (N = 1.11 s?1).The canyon head was defined as 7cm < y < 10cm, mid canyon was defined as4cm < y < 7cm and mouth was defined as 1cm < y < 4cm. The canyonhead region deliberately did not extend to the end of the canyon (y = 14 cm)because this shallower region was not well illuminated in every experiment.Velocities at all three locations in the canyon were predominantly rectilinear, soalong-canyon velocities were used instead of speed. Figure 2.12 shows the fittedcurves to measured data, with fit parameters shown in Table 2.6.No significant difference was found between resonant frequencies at eachlocation along the canyon, however amplification increased towards the canyonhead. The friction coefficient (?) calculated at the mouth of the canyon wassignificantly greater than the coefficient calculated at the canyon head (P <0.01).2.3.5 RepeatabilityExperiment 3 was replicated to determine the repeatability of the results. Res-onance curves were fit to measured amplification and phase difference data forthe original experiment (3a) and the repeat experiment (3b). Resonance curvesare shown in Figure 2.13 and fit parameters in Table 2.7. There was no sig-nificant difference between the resonant frequencies for Experiments 3a and 3b(P > 0.01). There was, however, a significant difference between the two frictionfactors.Figure 2.13 clearly shows a large difference in amplification at resonant fre-quency between Experiments 3a and 3b. Differences between curves fit to Ex-periments 3a and 3b may have been due to a number of factors. For fittedcurves shown in the preceding sections, resonance curves were fit to a numberof different experiments, whereas the curves for these repeats were only fit to400 0.5 1 1.501234567omega0 = 0.31215s?1, lamda = 0.44701s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Canyon head fitCanyon head dataMid canyon fitMid canyon dataCanyon mouth fitCanyon mouth dataFigure 2.12: Amplification and phase difference for three locations along canyon,while N = 1.11 s?1 for Experiments 2, 5, 8 and 11. Curves were fit to measuredamplification and phase difference data simultaneously for each experiment. If curves are fit to only one experiment, we expect the shapeand peak of the resonance curve to be strongly influenced by each data point,which may not be an accurate representation of the response in the canyon atthat frequency. For example, the amplification measured at ? = 0.50 s?1 forExperiment 3b was the highest amplification measured in all the experiments,however, other than at resonance, measured amplification data for Experiments3a and 3b look similar. A possible explanation for the difference between thetwo experiments at resonance could be fewer particles in the canyon at the res-onant frequency for Experiment 3a. This will be discussed in more detail inthe next section. Stratification may also have differed slightly between the twoexperiments.Table 2.7 also shows parameters for fitting one curve to both experiments41(?3a+3b?). Standard errors for ?0 and ? on this curve are smaller than errorsfor 3a, but larger than errors for 3b. This implies that the source of variation isbetween experiments, rather than within experiments and further confirms theneed to fit curves to more than one experiment at a time.Nevertheless, for the purposes of comparing resonant frequencies, the re-peated experiments gave a similar resonant frequency. The repeated experi-ments showed that the actual magnitude of amplification may differ from ex-periment to experiment. This error can be reduced by fitting resonance curvesto a number of experiments. This approach was taken for the resonance fits forstratification, forcing frequency, background rotation and location along canyon.42Table 2.6: Fit parameters for three locations along canyonLocation ?0 (s?1) ? (s?1) Q (-) A (-) RMSE RMSEat ?0 for A (-) for ? (?)Head 0.33? 0.013 0.28? 0.032 1.18 2.4 0.78 37.7Mid 0.34? 0.013 0.30? 0.032 1.12 2.2 0.76 27.2Mouth 0.31? 0.018 0.45? 0.052 0.70 1.4 0.54 25.50 0.5 1 1.501234567omega0 = 0.49683s?1, lamda = 0.16153s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Exp 3a fitExp 3a dataExp 3b fitExp 3b dataFigure 2.13: Amplification and phase difference for two repeats of Experiment3. Curves were fit to measured amplification and phase difference data simul-taneously for each experiment.Table 2.7: Fit parameters for repeated experimentsExperiment ?0 (s?1) ? (s?1) Q (-) A RMSE RMSEat ?0 for A (-) for ? (?)3a 0.45? 0.027 0.36? 0.059 1.25 2.4 0.55 28.73b 0.50? 0.010 0.16? 0.013 3.12 6.2 0.49 12.33a + 3b 0.48? 0.014 0.25? 0.029 1.92 3.8 0.90 23.3432.4 Discussion2.4.1 Experimental considerationsIn this study, neutrally buoyant particles were used to observe velocities. Parti-cle density was difficult to control, as experiments were up to 100 minutes long,during which time particles could sink or move out of the region of interest.Also, during experiments when velocities in the canyon were very low, particlescould not be advected into the canyon. Adding more particles directly into thecanyon during an experiment was avoided because if particles sink, they create awhite patch on the video and particle movement above these sunk particles canno longer be observed. Particle density was therefore not constant, particularlyat lower velocities at lower stratifications, which may have increased differencesbetween resonant and non-resonant conditions.However, even when particle density and velocities were low, the frequencyof particle oscillations still matched the frequency of oscillation of the table.Also, amplification used in this study is a relative ratio of observed velocitiesinside and outside the canyon, rather than an absolute measure of velocities. So,as shown in the Results section, the magnitude of the amplification may varybetween repeated experiments but the resonant frequency will likely remainsimilar. Wherever possible, resonance curves were therefore fit to results ofmultiple experiments.An important finding of Swart et al. [2011] was that observed currents inthe Gully were bottom intensified. However, measurements of currents at thebottom of the laboratory canyon were not made in this study. When filmingparticles from above the canyon and using a white horizontal light sheet, it isnot possible to distinguish between particles at different depths. Coloured hori-zontal light sheets can be used to light particles of different depths with differentcolours, however the colour of underwater particles is difficult to resolve with44a standard video camera in low light conditions. Manual examination of videofootage shows that particle colour can be misidentified by automated processing[Bowie, 2006]. Filming only those particles travelling along the sloping bottomis therefore not possible with the current setup. For this reason, dye was usedto qualitatively sketch the three dimensional flow to provide an indication ofapproximate relative velocities in three dimensions.Swart et al. [2011] found that canyon shape had a significant effect on alongcanyon and cross canyon velocity. This study has only considered one canyonshape because constructing canyons for the laboratory tank is time consum-ing and challenging. The along canyon depth profile of the laboratory canyonmatched the Gully profile and the laboratory cross canyon profiles were similarto the Gully, but the canyon was straightened. We assume that this did notaffect resonance in the canyon as testing further canyon shapes was beyond thescope of this study.2.4.2 Factors affecting resonant frequencyResonant frequency in the laboratory canyon increased with increasing strat-ification. Swart et al. [2011] identified the resonant frequencies in the Gullyas:?0Lcc= (n+12)pi for n = 0, 1, 2, .... (2.15)where c is wave speed and Lc is canyon length. Since c = Nh (h is the depth ofthe lower layer) and Lc/h is constant, the ratio ?0/N is expected to be constantfor resonance. An increase in stratification increases wave speed, which decreasesthe time for a wave to travel into the canyon and out again, which increasesthe frequency at which resonance occurs. In the laboratory canyon, h = 9.1 cmand Lc = 4 ? (14 cm) for resonance of the quarter wavelength, so the expected45ratio is ?0/N = 0.26, which is close to our measured ratio of 0.28 to 0.3. Ouraverage measured ratio of ?0/N corresponds to a quarter wavelength resonancein a canyon of approximate length 12.3 cm. This could suggest fluid is notresonating right up to the canyon head, due to the canyon shallowing towardsthe head.Background rotation did not affect resonant frequency in the laboratory, sothe temporal Rossby number (?/f) does not determine the frequency of reso-nance. This was as expected because the internal Rossby radius of deformationis greater than the width of the canyon and f is therefore not present in theequation derived by Swart et al. [2011] to describe the resonant frequencies.An inverse relationship was found between forcing amplitude and resonantfrequency. When forcing amplitude increased in the laboratory, the forcing ve-locity outside the canyon increased. Velocity inside the canyon was thereforeexpected to increase. However, speed inside the canyon did not appear to in-crease when forcing amplitude increased, so amplification values were smallerwhen forcing amplitude increased. The higher forcing amplitude had a signif-icantly higher friction factor, implying that friction in the canyon increases asforcing velocity increases. Returning to equation (2.6) and taking g?h and k asconstants, gives the following:? = ??c2k2 ??24?i?2(2.16)This relationship shows that as friction (?) increases, frequency (?) de-creases. Therefore, as forcing amplitude increases, friction factor also increases,which causes the resonant frequency to decrease as observed in the laboratoryresults.462.4.3 Structure of resonanceThe results presented show that a barotropic forcing was generating a baroclinicresponse in the canyon, because current direction varied with depth and overallresponse at resonance varied with stratification. Oscillating flow in the labora-tory canyon created asymmetrical flow patterns, with stronger currents on theleft side of the canyon. This was observed by Boyer et al. [2006], who describehow strong anticyclonic vorticity is generated when tidal flow is in the same di-rection as the mean flow. Vorticity generated when tidal flow is in the oppositedirection to the mean flow is weaker, which leads to an asymmetrical flow field,with flow following the shelf break. Stronger inflow and outflow are thereforeexpected on the western flank of the Gully. Swart et al. [2011] observed moreenergetic tidal currents on the eastern side of the canyon compared to the west-ern side. However, these observations were measured close to or above the shelfbreak (316 m on the eastern flank and 550 m on the western flank), whereaslaboratory observations were from deeper within the laboratory canyon.As described above, direct measurements of bottom velocities were not madein the laboratory canyon so bottom intensification observed in the Gully was notdirectly observed in the laboratory. However, the three dimensional sketches offlow show that, at least qualitatively, velocities increased towards the bottomof the canyon.Tidal pulses of dense, nutrient-rich water onto the shelf could have biologicalimplications for organisms living above and around the Gully. Swart et al.[2011] suggested that some fluid resonating within the Gully was spilling ontothe continental shelf possibly at the head of the canyon and our results supportthis. Water that was upwelled onto the flanks of the canyon resembles the densepools described by Haidvogel [2005]. The modified Rossby radius (R?) describedby Allen and Thomson [1993] (R? =?g?h/(f2 ? ?2)) suggests that the diurnal47tide may have a spatial influence of between 6 km and 25 km beyond the canyon,depending on the choice of g?.At resonance, upwelled water moved approximately 10-15 cm either side ofthe laboratory canyon horizontally, which represents a distance of approximately17-25 km in the ocean (note an exaggeration of 1.5 in the x direction, so 1 cmin the laboratory represents 1.7 km in the ocean). This estimate is within therange suggested by the modified Rossby radius. Much of the upwelled water isthen pulled back down into the canyon over one tidal cycle, which is 24 hoursin the Gully. Upwelled water was estimated to reach a shallowest depth of 1 cmfrom the surface, which represents a depth of approximately 240 m in the ocean.This conclusion is supported by Greenan et al. [2013], who found that the upper200 m of water above the Gully is not affected by the presence of the canyon.The effect of K1 resonance in the canyon may therefore be experienced up to25 km either side of the Gully and up to a shallowest depth of 240 m at thecanyon mouth.Current velocities measured by Swart et al. [2011] increased along the canyonand doubled from the canyon mouth to the canyon head. Our results show thatamplification increased from the canyon mouth to mid canyon. Figure 2.12shows that velocities at the head were approximately 1.7 times velocities atthe mouth at resonance, with an amplification of 1.4 at the canyon mouth and2.4 at the canyon head. Unlike the observations, amplification was similar atmid canyon and the canyon head. This may be because the laboratory canyonis experiencing more friction than the Gully, which may be slowing velocitiestowards the canyon head in the laboratory.Swart et al. [2011] also measured Greenwich phase along the canyon, which isthe phase of the local response compared to the phase of the forcing at a referencelongitude 0?, so that a larger angle indicates wave crests are arriving later48[Pawlowicz , 2002]. Swart et al. [2011] suggested the lack of a phase differencebetween moorings along the canyon indicated a long horizontal wavelength inthe canyon. The lower panel of Figure 2.12 show that the three locations inthe laboratory canyon have approximately the same phase lag from the forcing,consistent with a standing wave.2.4.4 Friction in the laboratoryWe suggested that friction in the laboratory (?L) could be estimated by equation(2.3), based on Ekman pumping. Taking h = 3 cm as the depth of moving fluid,? = 10?6 m2 s?1 and ? = 2 mm as the Ekman depth (? =?(2?/f) withf = 0.5 s?1) gives ?L ? 2 ? 10?2 s?1. This estimated friction factor is anorder of magnitude less than friction factors found by fitting resonance curvesto measured data, suggesting this theory is underestimating frictional processesthat are occurring in the laboratory.In this calculation, the depth (h) was assumed to be the depth of movingfluid in the canyon. However, water surging onto the shelf over the flanks ofthe canyon will create a very shallow layer of dense water on the shelf andthis layer could be experiencing a different amount of friction than water inthe canyon. The dense water on the shelf may in fact be experiencing a muchlarger amount of friction than water in the canyon, which would account for theunderestimation described above, and we now make an attempt to quantify thefriction due to this dense water.Water surging out of the canyon onto the shelf will be denser than thesurrounding fluid on the shelf. The movement of the fluid can therefore beconsidered as a gravity wave where Fr = 1, or Nd = u where N is buoyancyfrequency, d is the depth of the dense fluid on the shelf and u is the velocity ofthe dense fluid as it moves onto the shelf. Taking N ? 1 s?1 and u ? 0.1 cm s?149gives d = 1 mm.This depth estimate can be tested by considering the area that the denselayer will occupy on the shelf. First, the volume of fluid travelling into thecanyon from the open ocean (V ) is estimated by taking the approximate velocityin the canyon at maximum flood tide (u = 0.5 cm s?1), which enters the canyonthrough an area of approximately 5 cm wide and 3 cm deep at the canyon mouth,to give an approximate volume of V = 30 cm3 over a quarter of a tidal cycle(T/4). Assuming half of this volume travels onto the shelf and that this poolhas a depth of 1 mm, the fluid will occupy an area of approximately 150 cm2.If the fluid surges onto the shelf along the whole length of the canyon (14 cm),the area of the fluid will be approximately 11 cm wide. This suggests the fluidtravels approximately 11 cm either side of the canyon on each tidal cycle, whichis similar to the distance observed in dye experiments. Therefore, a depth of1 mm for the dense fluid on the shelf seems reasonable.We then assume that the Ekman depth (?) is the same as the fluid depth(i.e. ? = d) and recalculate (2.3) to give ?shelf = 1 s?1. Assuming that half thewater that travels into the canyon on a tidal cycle travels onto the shelf gives?L = ?shelf/2 = 0.5 s?1, which is the same order of magnitude as the frictionfactors found by fitting resonance curves to measured laboratory data.2.4.5 Resonance in the GullyTo extend the findings from the laboratory to the ocean and hence generateresonance curves for the Gully, a resonant frequency (?G) and a friction factor(?G) are required.Firstly, to estimate resonant frequency in the Gully, (2.17) and (2.18) wereused to scale between the laboratory and the Gully, where resonant frequencyin the laboratory (?L) is a function of buoyancy frequency (NL), depth (hL)50and canyon length (LL), and ? is a constant. Similarly in the ocean, resonantfrequency in the Gully (?G) is a function of buoyancy frequency (NG), depth(hG) and canyon length (LG). For both the laboratory and the Gully, canyondepth was taken as the depth from shelf break to canyon bottom, as this wasthe depth estimated to be affected by resonance.?L = ?NLhLLL(2.17)?G = ?NGhGLG(2.18)Measured data from the laboratory were used to estimate ?. Taking ?L/NL =0.29, hL = 7 cm and LL = 14 cm gives ? = 0.58. Then taking NG =3.16? 10?3 s?1, hG = 1, 600 m and LG = 35, 000 m gives ?G = 8.4? 10?5 s?1.Next, the friction factor in the Gully (?G) is estimated by the approximationused by Sutherland et al. [2005]:?G =CD|u|h(2.19)where CD is the dimensionless friction coefficient, u is velocity and h is depth ofthe lower layer. Assuming that this relationship applies to the water resonatingin the Gully, we can take u ? 0.1 m s?1 based on typical values from Swart et al.[2011] and h ? 1, 600 m, which is the depth of water estimated to be moving inthe Gully. Typical values of the coefficient CD range from 0.001?0.01. However,this results in ?G ? 10?7 s?1, which gives a large Q value of approximately 102,which seems unrealistic since it suggests that at resonance, velocities inside thecanyon are at least 2? 102 times the forcing velocity.Therefore, the actual friction factor is expected to be much larger than thisestimate. Above, we formulated friction in the laboratory by assuming that51friction was generated from water surging onto the shelf and similar processesare expected to occur in the ocean. Legg and Klymak [2008] found tidal boresgenerated at tall, steep ridges could travel upslope as internal hydraulic jumpswith Fr = 1. Additionally, Cossu and Wells [2013] found that large internalseiches in Lake Simcoe, Ontario, caused a bore to propagate up the slopinglakebed with Fr ? 1. These bores are assumed to similar to the bores thatcould be travelling up the flanks of Gully as water resonates in the canyon. Soif Fr = 1, then N = u/h, and ?G = CDN .The friction coefficient is chosen to be large (CD = 0.01) compared to manyoceanic applications to give reasonable amplification values. Other authors haveused this value to represent tidal flow in channels [Mullarney et al., 2008] andstraits [Foreman et al., 1995] where friction is high. So taking N = 3.16 ?10?3 s?1 at the shelf break and CD = 0.01, gives ?G = 3.2?10?5 s?1. Note thatthis method of calculating friction gives a value that is two orders of magnitudelarger than the first estimate of friction above.Above, the friction factor and resonant frequency were calculated using strat-ification at the shelf break. However, since the shape of the stratification curvediffers between the laboratory and the Gully, stratification at another depth maybe more appropriate and this will change the friction factor and resonant fre-quency. For this reason, stratification is chosen from three depths (shelf break,mid canyon and halfway between shelf break and mid canyon) and friction factorand resonant frequency are calculated at each depth to illustrate our uncertaintyabout our estimates. For the Gully, average stratification values at each depthwere taken from Swart et al. [2011] and for the laboratory, stratification valuesat different depths were taken from Figure 2.2. Resonance curves for each ofthe three depths are shown in Figure 2.14, with parameters shown in Table 2.8.From Figure 2.14, we can estimate that the resonant frequency for the Gully52is between 6.6 ? 10?5 s?1 and 8.4 ? 10?5 s?1, with a Q factor between 2.7and 2.9. For comparison, the Bay of Fundy has a high Q factor of approxi-mately 5 [Garrett , 1972] and the Salish Sea has a low Q factor of approximately2 [Sutherland et al., 2005]. The Q value for the Gully is therefore relativelylow, indicating a dissipative system with broad resonance. Velocities inside thecanyon are 5 to 6 times faster than velocities outside the canyon at resonance.The ratio of amplification at K1 to amplification at M2 varies from approxi-mately 3 to 6, depending on the depth where the representative stratificationis chosen. Regardless of choice of depth for representative stratification, theK1 frequency is still expected to dominate the current record for the averagestratification values.Figure 2.14 shows that the depth where stratification is chosen is importantfor these resonance curves. We now choose the representative stratification asthe observed stratification averaged over the top half of the canyon because weassume this is the most reasonable parameter to represent resonating fluid inthe canyon. Resonance curves can then be created to show how resonance inthe Gully would change if stratification inside the canyon were to vary. Stratifi-cation in the Gully varies seasonally and is influenced by a number of factors onthe Scotian Shelf, including outflow from the St. Lawrence River, the equator-ward Labrador Current and the poleward Slope Water Jet [Shan et al., 2013].Observations show that averaged buoyancy frequency (N) averaged from depthsof approximately 200 m to 1250 m inside the canyon typically varies from ap-proximately 2.1? 10?3 s?1 to 2.5? 10?3 s?1 over the year [Swart et al., 2011].Resonance curves for this range of stratifications in the canyon are shown inFigure 2.15. Parameters for each curve are shown in Table 2.9. Resonant fre-quency and friction factor increase with increasing stratification but the Q factorremains constant as stratification increases (Q = 2.9).530 0.5 1 1.5 2 2.5 3x 10?40123456Frequency [s?1]Amplification [?]  0 0.5 1 1.5 2 2.5 3x 10?4?100?50050100Frequency [s?1]Phase lag [?]mid canyonbetween mid canyon and shelf breakshelf breakK1 frequencyM2 frequencyFigure 2.14: Amplification and phase difference in the Gully, calculated usingrepresentative stratifications chosen at three different depths inside the canyon,with CD = 0.01. K1 and M2 frequencies are shown.Table 2.8: Parameters for resonance curves using representative stratificationsfrom three different depths in the GullyDepth of representative N ?0 (s?1) ? (s?1) Q (-) A at ? = ?0Mid canyon 6.6? 10?5 2.2? 10?5 2.9 5.8Halfway between mid 7.2? 10?5 2.6? 10?5 2.8 5.6canyon and shelf breakShelf break 8.4? 10?5 3.2? 10?5 2.7 5.454Figure 2.15 shows the location of the K1 and M2 tidal frequencies for eachstratification. Amplification at K1 is 5 to 6 times larger than amplificationat M2 for all stratifications. The curve with N = 2.3 ? 10?3 s?1 representsaverage stratification observed in the canyon and could therefore be consideredthe average resonance condition. The K1 frequency is close to resonance in theGully for the average stratification. The observed ratio of K1 velocity to M2velocity in the Gully was approximately 3 [Swart et al., 2011], which is slightlylower than the ratio for the average stratification curve. This difference may bebecause instruments were not at the location of maximum velocity, or perhapsbecause we are still slightly underestimating friction in the ocean.Nevertheless, if stratification in the canyon were to increase above average,resonant frequency would increase and amplification at K1 would increase, pro-viding the stratification did not increase above N = 2.5? 10?3 s?1 (2 standarddeviations above average). If stratification were to decrease below average, am-plification at K1 would decrease. However, for all stratification values observedin the Gully over the year, amplification at K1 is expected to be greater thanat M2, so K1 is expected to dominate over M2.Our conclusions are supported by observations from the Gully, where the K1frequency is observed to dominate year round [Swart et al., 2011]. Observationsof power calculated from current velocities and power calculated from sea levelvariations at Halifax show that the magnitude of amplification at the Gullyvaries throughout the year, with slightly higher amplification from Septemberto January and slightly lower amplification from February to April [Swart et al.,2011]. These variations in observed amplification could reflect slight shifts inthe location of the resonant frequency for the Gully, resulting from varyingstratification.550 0.5 1 1.5 2 2.5x 10?40123456Frequency [s?1]Amplification [?]  0 0.5 1 1.5 2 2.5x 10?4?100?50050100Frequency [s?1]Phase lag [?]N=2.1?10?3s?1N=2.3?10?3s?1N=2.5?10?3s?1K1 frequencyM2 frequencyFigure 2.15: Amplification in the Gully for a range of observed stratificationsaveraged in the canyon (CD = 0.01). K1 and M2 frequencies are shown. N =2.3? 10?3 s?1 is the average stratification.Table 2.9: Parameters for resonance curves at different stratifications in theGullyN (s?1) ?0 (s?1) ? (s?1) A at K1:A at M22.1? 10?3 6.1? 10?5 2.1? 10?5 5.12.3? 10?3 6.6? 10?5 2.3? 10?5 5.82.5? 10?3 7.2? 10?5 2.5? 10?5 5.6562.4.6 ConclusionA physical model was used to investigate tidal resonance in a submarine canyon.The principal findings of this study are the identification of factors that affecttidal resonance in the laboratory canyon, the characterization of the generalstructure of tidal resonance in the canyon and the production of resonancecurves for the Gully. Similar to Swart et al. [2011], a baroclinic response tobarotropic forcing was observed in the canyon. Dense water resonating withinthe laboratory canyon was upwelled onto the shelf on both sides of the canyonand much of this water returned to the canyon on the second half of the tidalcycle. The friction factor was characterised according to water surging overonto the flanks of the canyon. This friction factor proved to be very importantfor understanding the dynamics in both the laboratory and the ocean, andparticularly important for estimating the Q value in the Gully. We concludethat the K1 frequency is close to resonance in the Gully. Seasonal changes instratification at the shelf break could change the magnitude of the amplificationexperienced at K1 frequency, but the K1 frequency would still be expected todominate over the M2 frequency throughout the year.57Chapter 3Conclusion3.1 Research questionsThe results of this study can be summarised by returning to each of the ResearchQuestions posed in Chapter 1 and addressing how each question was answered.1. Can tidal resonance observed in the Gully be reproduced in a physicalmodel?Resonance curves fitted to measured data from the laboratory show thattidal resonance was likely reproduced in the laboratory canyon. There wasan amplified response inside the canyon compared to outside the canyon,and this response varied with frequency in a similar way to resonancecurves in other systems. The scaling from the ocean to the laboratorytherefore seems to have conserved the most important aspects of the res-onance system.2. Which conditions will create resonance (e.g. stratification, forcing fre-quency)?58Resonance is caused by a combination of stratification, forcing frequency,forcing velocity and canyon geometry. Canyon geometry was not var-ied in this study. As expected, forcing frequency had a profound effecton whether or not resonance occurred. This was illustrated by the peaksobserved in the resonance curves fitted to measured data. Increasing strat-ification was shown to increase the resonant frequency and a lower forcingamplitude had a higher resonant frequency than a higher forcing ampli-tude. Background rotation did not affect the resonant frequency.In the Gully, our scaling suggests the conditions are close to resonance forthe K1 frequency. Increasing or decreasing stratification could move thesystem towards or away from resonance at the K1 frequency, dependingon the magnitude of the stratification change.3. What is the three dimensional structure of resonant tides in the Gully?Sketches were made to illustrate the three dimensional structure of reso-nant tides in the laboratory canyon. PIV was also used to observe flowin two dimensions around the shelf break. From these observations, theresonant tide in the laboratory canyon was characterised by strong along-canyon velocities on the incoming and outgoing tide, with stronger flowon the left flank of the canyon than the right flank. The response withinthe canyon varied with distance from the canyon mouth, with strongeramplification towards the middle and head of the canyon.Surging bores of dense water were also observed travelling onto the shelfon the incoming tide and back into the canyon on the outgoing tide. Lab-oratory observations suggest that the high friction factor may be due todense bores travelling up and down the flanks of the laboratory canyonat resonant frequencies. A large friction factor was also calculated for theGully, indicating that the same process may be occurring at resonance59in the ocean. This explanation allowed for the generation of resonancecurves for the Gully.4. What is the spatial extent of the influence of the canyon beyond the Gully?Dye experiments were used to investigate the influence of the canyon be-yond the Gully. The most important finding from these experiments wasthat water was upwelled from within the canyon approximately 10-15 cmalong the shelf either side of the canyon. This was estimated to representa distance of approximately 17-25 km in the ocean. Much of this waterwas downwelled back into the laboratory canyon on the second half of thetidal cycle. Dense water is estimated to reach a shallowest depth of 240 mat the canyon mouth.The influence of the resonance in the deeper part of the laboratory tankwas not investigated, since the shelf area was assumed to be of moreinterest for biological processes.5. How do measurements from the physical model compare to measurementscollected from the Gully by Swart et al. [2011]?Qualitative comparisons between measurements in the ocean and labo-ratory were made. A near resonant response was observed around theexpected scaled frequency, suggesting the scaling analysis was sound. Ve-locities in the laboratory canyon did increase from the canyon mouth tomid canyon, as observed in the ocean, but were not observed to increasemuch between the mid canyon and the canyon head. Two dimensionalplots of velocities at resonance in the laboratory agree with patterns ob-served at moorings in the Gully. The ratio of K1 to M2 frequencies pre-dicted from laboratory results was similar to the ratio observed in theGully.606. What is the Q factor of the Gully?Estimates of resonant frequency and friction factor in the Gully indicatethat the Q factor is likely to be between 2.6 and 2.9, which represents arelatively broad resonance. As stratification varies over the year, resonantfrequency and amplification at K1 and M2 are also expected to vary. How-ever, for the range of stratifications observed over the year at the Gully,the K1 frequency is expected to dominate over the M2 frequency.3.2 Implications and future researchA number of research questions remain for further analysis. High friction factorssuggest that the pool of dense water transported onto the continental shelf maybe relatively large in area and could play an important role in the dynamics ofthe resonance. Further investigation of this dense pool could include charac-terising dimensions and velocities within the pool, as well as the time the poolspends on the shelf and the long term evolution of the pool after many tidal cy-cles. This may have implications for the management of the Gully because theMarine Protected Area around the canyon currently extends approximately 5kilometres either side of the canyon, but as shown, water may be travelling muchfurther than this along the continental shelf. Assuming that biological activityon the shelf is affected by the presence of these bores, the biological impact ofresonance in the Gully may extend beyond the Marine Protected Area.Observing flow in the lower part of the canyon presented challenges, as de-scribed above. Progress could be made by refining video processing methods toconsistently distinguish between colours so that coloured light sheets could beused to observe the bottom layer.Numerical models are powerful tools for studying ocean systems. Shan et al.[2013] constructed the first numerical 3-dimensional model of the Gully, how-61ever the model could not resolve the bottom intensified currents observed inthe Gully. Increasing spatial resolution of this model may allow internal waveprocesses and tidal resonance to be resolved in the model. The performance ofthe laboratory model could then be compared to the numerical model.Tidal resonance is clearly not the only process acting in the Gully. For exam-ple, wind and shelf scale circulation influence circulation above the canyon rim[Shan et al., 2013]. Considering all processes together is necessary to understandthe nature of circulation within the Gully.3.3 Broader contextThis project is part of the ongoing effort to understand the dynamics of the Gullysystem. Understanding the physical oceanography of the canyon has been recog-nised as providing the underlying framework for addressing multi-disciplinaryissues in the region [Han et al., 2002], such as fisheries and cetacean manage-ment and climate change impacts. This study complements the recent work ofShan et al. [2013] and Greenan et al. [2013] and provides information about tidalresonance, which may contribute to mixing in the Gully and therefore impactecosystems in the region.In the broader sense, this work also contributes to the growing body of lit-erature concerning submarine canyons and internal tides. Submarine canyonsare increasingly recognised as important in the energy budget of the world?soceans due to the increased internal wave energy dissipation within canyons.The propagation of internal wave bores, which may be occurring on the shelvesof the Gully, may contribute significantly to energy dissipation along the conti-nental shelf. The dissipation of energy from tides within the global ocean hasimportant implications for ocean circulation and climate change.62ReferencesAllen, S. (1996), Topographically generated, subinertial flows within a finitelength canyon, Journal of Physical Oceanography, 26, 1608?1632.Allen, S., and X. Durrieu de Madron (2009), A review of the role of submarinecanyons in deep-ocean exchange with the shelf, Ocean Science, 5, 607?620.Allen, S., and R. Thomson (1993), Bottom-trapped subinertial motions over mi-docean ridges in a stratified rotating fluid, Journal of Physical Oceanography,23, 566?581.Allen, S., C. Vindeirinho, R. Thomson, M. Foreman, and D. Mackas (2001),Physical and biological processes over a submarine canyon during an upwellingevent, Canadian Journal of Fisheries and Aquatic Sciences, 58, 671?684.Allen, S., M. Dinniman, J. Klinck, D. Gorby, A. Hewett, and B. Hickey (2003),On vertical truncation errors in terrain following numerical models: Compar-ison to a laboratory model for upwelling over submarine canyons, Journal ofGeophysical Research, 108, C03003, doi:10.1029/2001JC000978.Baines, P. G. (1983), Tidal motion in submarine canyons - a laboratory experi-ment, Journal of Physical Oceanography, 13, 310?328.Bowie, A. (2006), Upwelling in short submarine canyons, MSc thesis, Universityof British Columbia.Boyer, D. L., X. Zhang, and N. Perenne (2000), Laboratory observations ofrotating, stratified flow in the vicinity of a submarine canyon, Dynamics ofAtmospheres and Oceans, 31, 47?72.Boyer, D. L., D. B. Haidvogel, and N. Perenne (2004), Laboratory-numericalmodel comparisons of canyon flows: a parameter study, Journal of PhysicalOceanography, 34, 1588?1609.Boyer, D. L., J. Sommeria, A. S. Mitrovic, V. C. Pakala, S. A. Smirnov, andD. Etling (2006), The effects of boundary turbulence on canyon flows forcedby periodic along-shelf currents, Journal of Physical Oceanography, 36, 813?826.63Cacchione, D. A., L. F. Pratson, and A. S. Ogston (2002), The shaping ofcontinental slopes by internal tides, Science, 296, 724?727.Cossu, R., and M. Wells (2013), The interaction of large amplitude in-ternal seiches with a shallow sloping lakebed: Observations of ben-thic turbulence in Lake Simcoe, Ontario, Canada, PLoS ONE, 8, doi:10.1371/journal.pone.0057444.Dawe, J., and S. Allen (2010), Solution convergence of flow over steep topog-raphy in a numerical model of canyon upwelling, Journal of Geophysical Re-search, 115, C05008, doi:10.1029/ Young, B., and S. Pond (1987), The internal tide and resonance in IndianArm, British Columbia, Journal of Geophysical Research, 92, 5191?5207.Dushaw, B., and P. Worcester (1998), Resonant diurnal internal tides in thenorth atlantic, Geophysical Research Letters, 25, 2189?2192.Eriksen, C. (1982), Observations of internal wave reflection off sloping bottoms,Journal of Geophysical Research: Oceans, 87, 525?538.Foreman, M., R. Walters, R. Henry, C. Keller, and A. Dolling (1995), A tidalmodel for eastern Juan de Fuca Strait and the southern Strait of Georgia,Journal of Geophysical Research, 100, 721?740.Garrett, C. (1972), Tidal resonance in the Bay of Fundy and Gulf of Maine,Nature, 238, 441?443.Gordon, R. L., and N. F. Marshall (1976), Submarine canyons: internal wavetraps?, Geophysical Research Letters, 3, 622?624.Greenan, B., B. Petrie, and D. Cardoso (2013), Mean circulation and high-frequency flow amplification in the Sable Gully, Deep Sea Research Part II:Topical Studies in Oceanography, doi:10.1016/j.dsr2.2013.07.011.Gregg, M. C., G. S. Carter, and E. Kunze (2005), Corrigendum, Journal ofPhysical Oceanography, 35, 1712?1715.Haidvogel, D. (2005), Cross-shelf exchange driven by oscillatory barotropic cur-rents at an idealized coastal canyon, Journal of Physical Oceanography, 35,1054?1067.Han, G., P. Roussel, and J. Loder (2002), Seasonal-mean circulation and tidalcurrents in the Gully, in Advances in understanding the Gully ecosystem: Asummary of research projects conducted at the Bedford Institute of Oceanog-raphy (1999-2001), edited by D. Gordon and D. Fenton, p. 39, Fisheries andOceans Canada, Ottawa.64Hickey, B. M. (1995), Coastal submarine canyons, in Proceedings of the Uni-versity of Hawaii ?Aha Huliko?a Workshop on Flow Topography Interactions,edited by P. Muller and D. Henderson, p. 95, SOEST Special Publication,Honolulu.Kampf, J. (2009), On the interaction of time-variable flows with a shelfbreakcanyon, Journal of Physical Oceanography, 39, 248?260.Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg (2002), Internalwaves in Monterey submarine canyon, Journal of Physical Oceanography, 32,18901913.Legg, S., and J. Klymak (2008), Internal hydraulic jumps and overturning gen-erated by tidal flow over a tall steep ridge, Journal of Physical Oceanography,38, 1949?1964, doi:10.1175/2008JPO3777.1.Mann, K. (2002), Overview of results, in Advances in understanding the Gullyecosystem: A summary of research projects conducted at the Bedford Instituteof Oceanography (1999-2001), edited by D. Gordon and D. Fenton, p. 72,Fisheries and Oceans Canada, Ottawa.Mirshak, R., and S. Allen (2005), Spin-up and the effects of a submarine canyon:applications to upwelling in Astoria Canyon, Journal of Geophysical Research,110, C02013, doi:10.1029/2004JC002578.Mullarney, J., A. Hay, and A. Bowen (2008), Resonant modulation ofthe flow in a tidal channel, Journal of Geophysical Research, 113, doi:10.1029/2007JC004522.Oster, G. (1965), Density gradients, Scientific American, 213, 70?76.Pawlowicz, R. (2002), Observations and linear analysis of sill-generated internaltides and estuarine flow in Haro Strait, Journal of Geophysical Research, 107,doi:10.1029/2000JC000504.Perenne, N., D. B. Haidvogel, and D. L. Boyer (2001), Laboratory-numericalmodel comparisons of flow over a coastal canyon, Journal of Atmospheric andOceanic Technology, 18, 235?255.Petruncio, E. T., L. K. Rosenfeld, and J. D. Paduan (1998), Observations ofthe internal tide in Monterey Canyon, Journal of Physical Oceanography, 28,18731903.Reuten, C. (2006), Scaling and kinematics of daytime slope flow systems, un-published PhD thesis, University of British Columbia, Vancouver, 284p.Sandstrom, H., and J. Elliott (2002), Tidal mixing and the Gully ecosystem,in Advances in understanding the Gully ecosystem: A summary of researchprojects conducted at the Bedford Institute of Oceanography (1999-2001),edited by D. Gordon and D. Fenton, p. 48, Fisheries and Oceans Canada,Ottawa.65Shan, S., J. Shenga, and B. Greenan (2013), Physical processes affecting circu-lation and hydrography in the Sable Gully of Nova Scotia, Deep Sea ResearchPart II: Topical Studies in Oceanography, doi:10.1016/j.dsr2.2013.06.019.Strain, P. M., and P. A. Yeats (2005), Nutrients in the Gully, Scotian shelf,Canada, Atmosphere-Ocean, 43, 145?161, doi:10.3137/ao.430203.Sutherland, G., C. Garrett, and M. Foreman (2005), Tidal resonance in Juande Fuca Strait and the Strait of Georgia, Journal of Physical Oceanography,35, 1279, doi:10.1175/JPO2738.1.Sveen, J. K. (2004), An introduction to MatPIV v.1.6.1, eprint series ISSN0809-4403, Department of Mathematics, University of Oslo.Swart, N., S. Allen, and B. Greenan (2011), Resonant amplification of subin-ertial tides in a submarine canyon, Journal of Geophysical Research, 116,C09001, doi:10.1029/2008JC004956.Wang, Y., I. Lee, and J. Liu (2008), Observation of internal tidal currents inthe Kaoping Canyon off southwestern Taiwan, Estuarine, Coastal and ShelfScience, 80, 153?160.Waterhouse, A., S. Allen, and A. Bowie (2009), Upwelling flow dynamics inlong canyons at low Rossby number, Journal of Geophysical Research, 114,C05004, doi:10.1029/2008JC004956.Wunsch, C. (1975), Internal tides in the ocean, Reviews of Geophysics, 13, 167?182.66Appendix ANon-dimensionalparametersFigure A.1 shows non-dimensional parameters (Burger, Rossby, Froude and tem-poral Rossby numbers) for each laboratory experiment and non-dimensionalparameters calculated from measured data in the ocean (see Table 2.2).Sheet2Page 1B a c k g r o u n Bd BB Badd Bd ad cd kd gtiTlfTw?i??fT?? ? ? ? f????fT? ? ? ?w?i ?fTtiTlfTw?i??fTB a c k g r o u n Bd BB Badd dad dkd drd dud Bd Bad Bk?T?i?fw?i??fT?? ? ? ? f????fT? ? ? ?w?i ?fT?T?i?fw?i??fTB a c k g r o u n Bd BB Badd dad dkd drd du??? ? ??w?i??fT?? ? ? ? f????fT? ? ? ?w?i ?fT??????w?i??fT?a ?c ?k ?g ?r ?o ?u ?ndd gBB ga?f???T??w??? ? ??w?i??fT?? ? ? ? f???? ?? ?f?f???T??w??????w?i??fTFigure A.1: Non-dimensional parameters for each laboratory experiment. Pro-files v2-v9 represent varying ? values used in each experiment.67Appendix BAdditional laboratorymethods  Backkgroourgnd tdindTdigroklTrf  tdindTdigrokrtuT gdctdw??rrnd?oi?aT? ? ?Figure B.1: Laboratory tank, canyon painted white for a dye experiment.B.1 Canyon insertThe canyon insert was constructed from fiberglass using a rubber mould. Firstly,canyon contours created in MATLAB as pdfs were cut from 1/8 inch mahogany68plywood using a laser cutter. The plywood contours were glued together tocreate a canyon replica. Plaster of Paris was smoothed over the canyon replicato smooth any edges and the plaster was sealed with a coat of spray paint toprevent the rubber from absorbing into the plywood.Brushable rubber was painted over the canyon replica to create a preciserubber mould of the canyon. The rubber mould was placed in a plastic tuband supported on all sides by Plaster of Paris. The rubber mould was thenfilled with fiberglass resin to create a fiberglass canyon, because fiberglass canwithstand salt water conditions. Once set, the fiberglass canyon was removedfrom the mould and inserted into the continental shelf of the tank (see FigureB.1). Edges were sealed with a watertight caulk and the whole canyon area waspainted with matte black paint to reduce reflection during filming.B.2 Neutrally buoyant particlesTo reduce surface tension and prevent particles clumping, particles were mixedwith Kodak Professional Photo-Flo Solution to create a slurry (3:1 ratio of waterto Photo-Flo).69Appendix CStandard error calculationStandard errors reported in Tables 2.3 to 2.7 were calculated in the followingway. Firstly, the MATLAB function lsqcurvefit was used to fit resonance curvesto sets of measured data. Using the Jacobian from lsqcurvefit, the 95% confi-dence interval (95CI) was found with nlparci for each fitted resonant frequency(?0) and friction coefficient (?). Standard errors (SE) for each ?0 and ? werethen calculated from the confidence intervals (e.g. SE = (95CI??0)/1.96).70Appendix DSeparate resonance curves0 0.5 1 1.501234567 omega0 = 0.2273s?1, lamda = 0.21684s?1Frequency (s?1)Amplification (?)  N=0.83s?1 fitN=0.83s?1 data0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Figure D.1: Amplification and phase difference for first stratification (N) forExperiments 1-12710 0.5 1 1.501234567 omega0 = 0.33241s?1, lamda = 0.28263s?1Frequency (s?1)Amplification (?)  N=1.11s?1 fitN=1.11s?1 data0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Figure D.2: Amplification and phase difference for second stratification (N) forExperiments 1-120 0.5 1 1.501234567 omega0 = 0.4579s?1, lamda = 0.3081s?1Frequency (s?1)Amplification (?)  N=1.53s?1 fitN=1.53s?1 data0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Figure D.3: Amplification and phase difference for third stratification (N) forExperiments 1-12.720 0.5 1 1.501234567 omega0 = 0.27485s?1, lamda = 0.26219s?1?/N (?)Amplification (?)  N=0.83s?1 fitN=0.83s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.4: Amplification and phase difference for first stratification (N) against?/N for Experiments 1-12.0 0.5 1 1.501234567 omega0 = 0.30006s?1, lamda = 0.2552s?1?/N (?)Amplification (?)  N=1.11s?1 fitN=1.11s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.5: Amplification and phase difference for second stratification (N)against ?/N for Experiments 1-12.730 0.5 1 1.501234567 omega0 = 0.30003s?1, lamda = 0.20185s?1?/N (?)Amplification (?)  N=1.53s?1 fitN=1.53s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.6: Amplification and phase difference for third stratification (N)against ?/N for Experiments 1-12.0 0.5 1 1.501234567 omega0 = 0.30152s?1, lamda = 0.22218s?1?/N (?)Amplification (?)  f=0.5s?1 fitf=0.5s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.7: Amplification and phase difference for first background rotation (f)against ?/N for Experiments 1-12.740 0.5 1 1.501234567 omega0 = 0.2843s?1, lamda = 0.23966s?1?/N (?)Amplification (?)  f=0.75s?1 fitf=0.75s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.8: Amplification and phase difference for second background rotation(f) against ?/N for Experiments 1-12.0 0.5 1 1.501234567 omega0 = 0.31385s?1, lamda = 0.17648s?1?/N (?)Amplification (?)  ? f=0.04s?1 fit? f=0.04s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.9: Amplification and phase difference for first forcing amplitude (?f)against ?/N for Experiments 1-12.750 0.5 1 1.501234567 omega0 = 0.25309s?1, lamda = 0.31924s?1?/N (?)Amplification (?)  ? f=0.08s?1 fit? f=0.08s?1 data0 0.5 1 1.5?100?50050100150?/N (?)Phase difference (?)Figure D.10: Amplification and phase difference for second forcing amplitude(?f) against ?/N for Experiments 1-12.0 0.5 1 1.501234567 omega0 = 0.33241s?1, lamda = 0.28263s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Canyon head fitCanyon head dataFigure D.11: Amplification and phase difference for canyon head, while N =1.11 s?1 for Experiments 2, 5, 8 and 11.760 0.5 1 1.501234567 omega0 = 0.34254s?1, lamda = 0.30481s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Mid canyon fitMid canyon dataFigure D.12: Amplification and phase difference for mid canyon, while N =1.11 s?1 for Experiments 2, 5, 8 and 11.0 0.5 1 1.501234567 omega0 = 0.31215s?1, lamda = 0.44701s?1Frequency (s?1)Amplification (?)  0 0.5 1 1.5?100?50050100150Frequency (s?1)Phase difference (?)Canyon mouth fitCanyon mouth dataFigure D.13: Amplification and phase difference for canyon mouth, while N =1.11 s?1 for Experiments 2, 5, 8 and 11.77


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