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The free oscillatory response of fjord-type multi-armed lakes Brenner, Samuel 2017

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The free oscillatory response offjord-type multi-armed lakesbySamuel BrennerB.A.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Samuel Brenner 2017AbstractThis study examines the structure and frequency of free seiche modes infjord-type multi-armed lakes in order to generalize features of the responseof those lakes. The effect of multiple arms on seiches within a lake is noteasy to predict. To do so, this study develops a simplified analytical model(SAM) based on idealized lake geometries. In addition, a characterization ofsurface (barotropic) modes is compared for two “Y-shaped” lakes: QuesnelLake in Canada, and Lake Como in Italy. Lake Como and Quesnel Lakeare studied through a combination of field observations and modelling, bothnumerically using a Finite Element Method (FEM) scheme and using SAM.SAM demonstrates that multi-armed lakes are subject to two classifica-tions of behaviour: a full-lake response, in which all arms are active; anda decoupled response, in which seiching is constrained to only two arms ofthe lake. A geometric parameter in each arm, which represents the traveltime of a progressive shallow-water wave in that arm, determines the rangeof behaviours expressed: each lake will either experience only a whole-lakeresponse or it will exhibit alternating whole-lake and decoupled modes.The behaviour predicted by SAM is consistent with modes observedand predicted in both Quesnel Lake and Lake Como. Modal periods areidentified from observed water level measurements using spectral analysis.FEM predicted periods agree with observations. SAM correctly reproducesthe periods of the lowest frequency modes in both lakes when a constant depthis used for each arm. Mode-shapes predicted by SAM qualitatively matchthose given by the FEM model. While all modes of Quesnel Lake are whole-lake modes, some of the modes in Lake Como exhibit a decoupled response.The results given here also support generalization of the fundamental modeas being inherently the same structure as Merian-type modes in simpleelongated lakes.While the study focusses on barotropic modes, SAM can be similarlyapplied to internal (baroclinic) modes, and so the general behaviours ob-served here are appropriate for describing both the barotropic and baroclinicresponses of multi-armed lakes.iiLay SummaryWind forcing along the surface of a lake pushes water to the downwind end.When the wind forcing is released, this water mass rocks back and forth in amotion called seiching. Accompanying this oscillation of the water surfaceare oscillations of the currents within the lake, which are important for theoverall transport of material. While these oscillations can be described verywell in simple lakes, when lakes have multiple arms we can no longer makegeneral statements about this seiching. This study uses a combination oftheoretical work and case studies to better describe seiching in multi-armedlakes. The results of the study identify that for certain geometries the seicheresponse will be constrained to only a subset of arms of the lake.iiiPrefaceThe work presented here represents original research carried out by theauthor under the supervision of B. Laval (University of British Columbia).The thesis is presented as two manuscripts (Chapters 2 and 3), with Chapter1 providing relevant background and motivation for both. Chapter 4 providesoverall conclusions and identifies opportunities for further work. A secondreader, G. Lawrence (University of British Columbia), provided valuablecomments on drafts of this work. The contributions of co-authors are asfollows:A version of Chapter 2 is being prepared for submission to a peer-reviewed journal as “Development of an analytical solution for seiche modesin fjord-type multi-armed lakes” by S. Brenner, and B. Laval. I was thelead investigator for this work, and completed all manuscript preparation. B.Laval is acting in a supervisory role and is providing critical feedback andongoing editing of the manuscript.Portions of Chapter 3 will also be submitted for publication; however,additional work will need to be completed before this is possible. As such,the final title and author list is still under discussion. In its current form, allanalysis and manuscript preparation in Chapter 3 was completed by myself.The field study of Quesnel Lake presented in Chapter 3 was developed anddeployed by B. Laval and S. Vagle (Institute of Ocean Sciences) as partof a wider study before the onset of this thesis work; however, I assistedwith retrieval of field data and performed all relevant analysis for this work.B. Laval is acting in a supervisory role in this endeavour and was involvedthroughout the project in concept formation and manuscript edits. Anadditional researcher, J. Shore (Royal Military College of Canada), performedmodelling work of Quesnel Lake. That work is not presented in the currentdocument, but comparisons have been made with the relevant observationsand modelling presented in Chapter 3 and so her work has helped to verifythe results here.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Review of relevant literature . . . . . . . . . . . . . . . . . . 21.2.1 Standing waves in basins of simple geometry . . . . . 21.2.2 Influence of complex geometry on seiching . . . . . . 91.2.3 Finite Element Method for solving the wave equationin an arbitrary two-dimensional domain . . . . . . . . 121.3 Objectives and organization . . . . . . . . . . . . . . . . . . 142 Development of an analytical solution for seiche modes infjord-type multi-armed lakes . . . . . . . . . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Model Development . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Governing equations . . . . . . . . . . . . . . . . . . . 162.2.2 Description of the multi-armed model . . . . . . . . . 182.2.3 Alternate method for calculating ωn . . . . . . . . . 232.2.4 Solutions for analytically defined bottom profiles . . . 242.2.5 Solutions for an arbitrary bottom profile . . . . . . . 252.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28vTable of Contents2.3.1 The use of Merian’s formula in multi-armed lakes . . 282.3.2 The impact of cross-sectional variation . . . . . . . . 292.3.3 Extensions of this model . . . . . . . . . . . . . . . . 322.3.4 Similarity to Neumann’s impedance method . . . . . 342.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Barotropic seiche modes in two fjord-type Y-shaped lakes 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 Site descriptions . . . . . . . . . . . . . . . . . . . . . 393.2.2 Field study . . . . . . . . . . . . . . . . . . . . . . . . 413.2.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.1 Lake Como . . . . . . . . . . . . . . . . . . . . . . . . 463.3.2 Quesnel Lake . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Accuracy of the simplified analytical model . . . . . . 553.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 General behaviours of multi-armed lakes . . . . . . . 563.4.2 The activation of higher modes . . . . . . . . . . . . . 593.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Summary and contributions . . . . . . . . . . . . . . . . . . 634.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.1 Additional study of Quesnel Lake and Lake Como . . 644.2.2 Forced response of multi-armed lakes . . . . . . . . . 664.2.3 The use of simplified analytical models in predictingseiche response . . . . . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70viList of TablesTable 2.1 Summary of modal frequency equations . . . . . . . . . 36Table 3.1 Details of triangular meshes used . . . . . . . . . . . . 45Table 3.2 Values of τi for each arm in Lake Como and Quesnel Lake 47Table 3.3 Modal periods of Lake Como . . . . . . . . . . . . . . . 48Table 3.4 Modal periods of Quesnel Lake . . . . . . . . . . . . . . 51Table 3.5 Comparisons of observed and predicted energy . . . . . 52viiList of FiguresFigure 1.1 Seiche schematic . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Schematic of a triangular FEM mesh . . . . . . . . . . 13Figure 2.1 Coordinate definitions for the simplified model . . . . 19Figure 2.2 “H”-shaped multi-armed geometry . . . . . . . . . . . 33Figure 3.1 Maps of Quesnel Lake and Lake Como . . . . . . . . . 40Figure 3.2 Lake Como mode-shapes . . . . . . . . . . . . . . . . . 49Figure 3.3 Spectral energy in Quesnel Lake . . . . . . . . . . . . 53Figure 3.4 Quesnel Lake mode-shapes . . . . . . . . . . . . . . . 54Figure 3.5 Schematic of a possible transient set-up of Quesnel Lake 60viiiAcknowledgementsI could not have completed this work without the help and support of myfriends and family. I am especially grateful to Angie, whose exceptionalpatience and understanding through the past two years has been instrumentalin my success.For helping with the study of Quesnel Lake, I’d like to thank Svein Vagle,Brody Granger, and Andrew Hamilton who spent a week with me on thelake performing fieldwork, and the researchers at the Quesnel River ResearchInstitute for hosting us. I’d also like to thank Jennifer Shore who providedfeedback and ideas about Quesnel Lake, who ran through many model runsbased on hunches I had, and who postponed her own publication submissionso that we could work together. Stephen De´ry provided atmospheric datawhen I was still hoping to link seasonal wind regimes to variations in QuesnelLake’s response, and his student, Hadleigh Thompson, is happily taking onthe task of trying to better describe the spatial wind patterns in the regionso that someone else might solve that problem.The inclusion of Lake Como in this thesis is due to the generosity ofGiulia Valerio and Marco Pilotti in sharing their data.I appreciate the discussions, the exchanges of ideas and problems, andthe camaraderie with the other students in both the Environmental FluidMechanics and the Physical Oceanography groups at UBC. In particular,David Hurley, Mark Sumka, Kelly Graves, and Brody Granger deserverecognition.I’ve benefited immensely from the support of professors and researchersin both Civil Engineering and Oceanography. In particular Greg Lawrence,Rich Pawlowicz, Ted Tedford, and Andrew Hamilton have helped me workthrough numerous questions. Of course, the professor to whom I owe the mostthanks is my supervisor, Bernard Laval, who read through many iterations ofthis work and provided a great deal of critical feedback and encouragement,and who supported me most by sharing my curiosity.This work has been partially supported by an NSERC CGS-M awardand Environment and Climate Change Canada’s Environmental DamagesFund.ixChapter 1Introduction1.1 MotivationBasin-scale standing wave oscillations in enclosed or semi-enclosed basins area ubiquitous feature in both natural and man-made water bodies. Standing-wave oscillations (or “seiches”) occur on both the water surface and theinternal temperature distribution, and both are accompanied by out-of-phaseoscillations of the horizontal velocity field. Many early studies in the fieldof physical limnology were attempts to better understand and explain thephenomenon of seiching. From the end of the 19th and beginning of the 20thcentury, a number of publications on the subjects of both surface and internalseiching (at that time called “temperature seiching”) in the Scottish Lochswere produced by the Scottish Lake Survey (e.g. Murray, 1888; Watson,1904; Wedderburn, 1907). These came at a time when oceanographers wereworking to better understand tidal waves in semi-enclosed seas (e.g. Defant,1918; Proudman, 1915). The connection between these problems allowedfor a great deal of understanding of these processes, and formed a basis forthe current theory of seiches. With modern computer technology and theadvancement of fully three-dimensional hydrodynamic models, researchersnow have the ability to accurately resolve many aspects of lake motion,including the seiche response. As a result, questions about seiches havelargely shifted from understanding the shapes and periods of these seiches tounderstanding their impacts on the processes such as mixing and transportof materials within lakes.Nonetheless, there are still aspects of seiching that are not fully under-stood. Despite being able to fully resolve wave modes and periods using two-and three-dimensional models, there is not always a satisfying explanationof why some features exist in the response of complex lakes. The earlystudies by the Scottish Lake Survey developed mathematical explanationsfor seiche motions in glacially-carved “fjord-type” lakes. These lakes aretypically steep-sided and deep, and, importantly, they are long and narrow.This simple geometry allows for the equations of motion to be applied alongthe longitudinal axis of the lake so transverse motions can be ignored. The11.2. Review of relevant literatureresulting reduction of dimension allows for the governing partial differen-tial equations (PDEs) to be converted to a more tractable set of ordinarydifferential equations (ODEs), which are simpler to both solve and explain.For basins whose surface geometry cannot be readily approximated as one-dimensional, the seiche response is necessarily predicted by a set of PDEsthat do not provide results that are intuitive, nor are they easy to generalize.The study of lakes that have features such as bays, sub-basins, or multiplearms are thus described in a case-by-case approach that doesn’t necessarilyprovide any predictive power or understanding of other lakes with similarfeatures.Here, a particular class of complex lakes is investigated: multi-armed fjord-type lakes. By considering fjord-type lakes specifically, this study will restrictthe range of possible motions to focus on the impact on the lake responseof multiple arms alone without having to consider transverse or rotationalmotions. In so doing, some generalized behaviours of this geometric featurecan be described. Before proceeding with the study, the following sectionprovides a review of literature describing the background and mathematicalformulation of the processes discussed through the remainder of the thesis.1.2 Review of relevant literature1.2.1 Standing waves in basins of simple geometrySeiching in lakes was perhaps first seriously described by Forel’s 19th centuryworks on Lake Geneva (Defant, 1960; Hutter et al., 2010; Wilson, 1972).However, this phenomenon was previously measured as far back as the 16thcentury (Wilson, 1972) and throughout both Europe and North Americafrom the 17th century onwards (Chrystal, 1905). Since these early accounts,seiching (particularly baroclinic seiching) has been considered to be one ofthe primary mechanisms for mass transport in lakes and has been the subjectof numerous studies. In addition to the many journal publications on thetopic, this physical process is also featured in sections of textbooks on thesubjects of both limnology (e.g. Wetzel, 1983) and oceanography (e.g Defant,1960; LeBlond and Mysak, 1978; Proudman, 1953), and in one case is thesubject of an entire textbook volume (Hutter et al., 2011). Given the historyand volume of work on this subject, it is infeasible to provide a full accountof all of these works. Presented in the sections below is a description of thephysical process and a summary of seminal works on both barotropic andbaroclinic seiching.Seiching refers to the standing-wave response of an enclosed or semi-21.2. Review of relevant literatureenclosed basin to some pressure imbalance. After the driving mechanismthat creates this imbalance is removed the lake attempts to return to someequilibrium position, but instead this position is overshot and a new pressureimbalance appears. This process repeats and the lake oscillates. In the case ofsurface (barotropic) seiching, this takes the form of an oscillation of the freesurface accompanied by an out-of-phase oscillation of the mean horizontalvelocity (Figure 1.1a). Most lakes exhibit seasonal thermal stratificationwith a sharp temperature gradient (the thermocline) separating the surface(epilimnion) and deep water (hypolimnion). This temperature interfaceresponds to a surface pressure imbalance by deflecting the opposite direction;when the pressure imbalance is allowed to relax, standing wave oscillationsof the interface occur. The out-of-phase two-layer circulation generated bythese internal (baroclinic) seiches creates a shear flow with mean epilimneticand mean hypolimnetic velocities having opposite sign (Figure 1.1b). Indescribing these processes, we first consider these oscillatory motions in lakesof “simple” geometry, which refer to lakes that are sufficiently long andnarrow such that transverse motions and effects from the Earth’s rotationare negligible compared to longitudinal motions, and these lakes are devoidof complicating factors such as bays, sub-basins, or multiple arms.u∗(a) Barotropic seichingu∗(b) Barolinic seichingFigure 1.1: Schematic of the evolution of (a) a surface (barotropic) seiche in a homoge-neous water body, and (b) an internal (baroclinic) seiche in two-layer system. After theforcing is removed, the surface/interface deflection and the water column velocity bothoscillate, but are out of phase both spatially and in time31.2. Review of relevant literatureBarotropic seichesWhile barotropic seiching is a response to some pressure imbalance or exci-tation, the driving force that creates the imbalance is not strictly defined.Defant (1960) lists a number of possible forcing mechanisms for seiches:1. The sudden return to its equilibrium of a surface previously disturbedby the passage of an atmospheric disturbance over a section of the lake2. The sudden oscillation back to a state of equilibrium of a watermasspreviously piled up by wind3. Sudden or rapid receding of an accumulation of water produced by anextremely rapid influx across a section of the lake (violent rainfalls).4. Shocks of rain drops falling on the water surface.5. Sudden changes in air pressure.6. Shock pressures of wind gusts on the lake surface.7. Subsiding of the electrical attraction on the surface by thunder clouds.These are roughly the same set of mechanisms previously identified byWedderburn (1922, conveying the findings of various works by Chrystal),though his list does not include mechanism (7) and it separates mechanism (6)into: “the effect of squalls”, “impact of wind gusts”, and periodic fluctuationsin the wind. Aside from it’s inclusion in the list, Defant (1960) provides noinformation for mechanism (7), and this does not appear in other prominentworks. In addition to the list provided by Defant (1960), other set-upmechanisms exist:8. Seismic activity (suggested by Chrystal (1905, 1908), and shown defini-tively in more recent works (e.g Barberopoulou, 2008; Pieters andLawrence, 2014))9. Landslides (e.g. Balmforth et al., 2009; Kulikov et al., 1996)10. Tidal forcing (especially in large lakes; e.g. Hamblin, 1974)Chrystal (1908) agrees with an earlier statement by Forel (as cited in Defant,1960) that changes in atmospheric pressure (mechanisms 1 and 5) are the mostfrequent causes of seiching, with the influence of wind (mechanisms 2 and 6)being less important. Wedderburn (1922) states that there is no evidenceof piling up of water at the downwind of the lake as would be required for(2), except for very shallow lakes. Despite this claim, modern theory oftensupports mechanism (2) as the cause of surface seiching. This shift in viewmay be due to the relationship of this mechanism to understanding of the41.2. Review of relevant literatureset-up of baroclinic seiching (Mortimer, 1952; Stevens and Imberger, 1996).The driving force determines which of the possible seiche modes are excited,but otherwise does not define the characteristic shapes or periods of thosemodes. The mode characteristics are a property of the lake geometry itself.Merian (1828, as cited in Chrystal, 1905) developed a formula for theperiods of longitudinal free oscillations in a rectangular basin of length Land constant depth H:Tn =2Ln√gH(1.1)where it is recognized that the term√gH in the denominator is the wavespeed of a shallow water wave. However, this formula was not connected toseiching until Forel (1876, as cited in Chrystal, 1905) suggested that it couldbe applied to predict the resonant periods of lakes. Merian’s formula is nowubiquitous in modern works.In the early part of the 20th century, many advances in the understandingof seiching were made by Chrystal working with the Scottish Lake Survey (e.gChrystal, 1905, 1908; Chrystal and Murray, 1907; Chrystal and Wedderburn,1905). In particular, Chrystal sought to describe the influence of cross-sectional variation on the sequence of modal periods and the location ofnodes. To this end, Chrystal (1905) presented an equation to describe thefree evolution of the depth-integrated horizontal velocity (q =∫ H0 udz) of alake in terms of variations in cross-sectional area, S(x), and width at thesurface, b(x):∂2Ξ∂t2= gσ(χ)∂2Ξ∂χ2,whereΞ = S(x)q(x, t), χ =∫ x0b(s)ds, and σ = S(x)b(x).He called the graph with χ on the x-axis and σ on the y-axis the “normalcurve” of a lake. In the case of constant width b(x) = b, and varying depthH(x), this reduces to the more familiar∂2q∂t2= gH(x)∂2q∂x2. (1.2)A corresponding equation can be developed for the deflection of the freesurface h(x, t):∂2h∂t2= g∂∂x[H(x)∂h∂x](1.3)51.2. Review of relevant literatureThese equations rely on the assumption that h H, and are thus “linearised”.For barotropic motions, this assumption is typically valid; however, it maynot be so for baroclinic motions.Chrystal (1905) then developed a series of mathematical functions whichhe named “Seiche-sine”, “Seiche-cosine”, and the “Lake Function” which solveEquation 1.2 for certain analytically defined bottom profiles. Halm (1905)shows that these functions are a part of a broader class of hypergeometricfunctions, and are not only of value in understanding lakes but can be usedto help solve other equations in the field of applied mathematics.Despite Chrystal’s success in solving Equation 1.2 for analytically definedprofiles, it was found that many lake geometries can not be easily defined bysuch profiles. As a result, researchers such as Proudman (1915) and Defant(1918, as cited in Defant, 1960; Mortimer, 1979) developed numerical methodsto solve Equations (1.2) and (1.3) for arbitrary profiles. Defant (1960)presents these together with the analytical method developed by Chrystal(1905) and a number of similar methods proposed by other researchers.Between these methods, reasonably accurate approximations for the periodsof free modes can be made for these basins.Equations (1.2) and (1.3) represent the free barotropic response of thelake with neither damping nor forcing considered. As mentioned at thebeginning of this section, there are many potential excitation mechanismsfor barotropic seiching. Some of these can be expressed as additional termsin Equation (1.2). A derivation of this equation from the basic equationsof motion and continuity gives rise to terms that represent two forcingmechanisms: wind shear on the lake surface, τs, and atmospheric pressuredeviations, pa (Wilson, 1972). If damping is assumed to act linearly withdamping coefficient K (e.g. Defant, 1960), then the governing equation forthe depth-integrated velocity becomes:∂2q∂t2+K∂q∂x− gH(x)∂2q∂x2= Fs(x, t) (1.4)where the forcing term,Fs(x, t) =1ρ(τs −H∂pa∂x).Equation (1.4) reduces to Equation (1.2) for K = Fs = 0. In the case of abasin of uniform depth, the free solutions (i.e. Fs = 0) to Equation (1.4) aregiven byq = Ae−Kt/2 sin(kx) sin(γt+ ),61.2. Review of relevant literaturewhere the resonant frequencies areγn = ωn√1− K2ωnand ωn = knc = npicL−1, n ∈ Z+, with the wave speed c = √gH. In theundamped case (K = 0), then γ = ω and the modal periods T = 2piγ−1are consistent with Merian’s formula. In real basins, K 6= 0, but dampingof barotropic seiching is typically small (Defant, 1960, Table 25) and themodal frequencies obtained by solving the free case (Equation 1.2) providereasonable estimates of the damped frequencies, i.e. γ ≈ ω.For a broader account of the subject of barotropic seiching, readers aredirected to the work of Wilson (1972) who provides a thorough summaryon the history of the subject as well as providing a great deal of detail onthe relevant mathematics (including the derivation of Equation 1.4 fromthe equations of motion, and a full treatment of the forcing term Fs foratmospheric pressure disturbances). While now more than a century old,Chrystal (1905) similarly provides a summary of early works on the subjectdating as far back as 1755, including a very thorough annotated bibliographycontaining 136 separate entries.Baroclinic seichesIn addition to studying oscillations of the free surface of lakes, the early studiesby the Scottish Lake Survey found evidence that the internal temperaturefield of Loch Ness also exhibits periodic motions (Murray, 1888). Watson(1904) recognized these oscillations as seiching of the interface between theepilimnion and hypolimnion and interpreted the forcing mechanism as aninternal reaction to a surface wind shear. Due to their added complexity,their increased likeliness to exhibit non-linear behaviour, and their arguablygreater importance compared to barotropic seiching, these baroclinic motionsdominate modern literature in the subject of wind-induced oscillatory motionin lakes. Nonetheless, the present study will largely focus on barotropicrather than baroclinic seiching. As such, this section will only present onemajor result: the decoupling of vertical modes.While a number of authors worked to build a mathematical frameworkfor baroclinic seiching (e.g. Aichi, 1918a,b; Priestley, 1909; Wedderburnand Williams, 1911), major advances were made by Longuet-Higgins (1952)and Heaps and Ramsbottom (1966). In particular, Longuet-Higgins (1952)developed linearised equations for the free oscillations of deflections of boththe water surface and the interface between density layers by starting from the71.2. Review of relevant literaturebasic equations of motion and continuity. His work showed that as a result ofthe small density differences between layers (∆ρρ−12 ∼ O(10−3)), barotropicand baroclinic motions can be effectively separated. The barotropic modesthen respond as the modes of the equivalent homogeneous body of water,whereas for a two-layered system the interface deflection of the baroclinicmodes (η(2)n ) is given byη(2)n = cos(knx)[An cos(ω(2)n t) +Bn sin(ω(2)n t)]where ω(2)n = knc(2) = npic(2)L−1, n ∈ Z, and the baroclinic wave speed c(2)isc(2) =√g′(H1H2H1 +H2). (1.5)The gravitational restoring force acts on density differences, so the wavespeed of internal modes depends on the “reduced gravity”, g′, which isexpressed in terms of the upper and lower layer densities ρ1,2:g′ = g(ρ2 − ρ1ρ2).Longuet-Higgin’s results produce a formula for modal periods analogous toEquation (1.1), and consistent with the one suggested by Watson (1904)much earlier.Compared with barotropic motions, the reduced gravity of baroclinicmodes results in these motions having amplitudes and periods that aretypically 2 to 3 orders of magnitude greater than barotropic modes. For asharp interface between layers, the two-layer circulation created by baroclinicmotions (Figure 1.1b) leads to the generation of considerable shear betweenlayers. This shear can act to dampen the baroclinic response (e.g. Horn et al.,2001; Imam, 2012; Imam et al., 2013b, 2017), or can lead to the growth offeatures such as Kelvin-Helmholtz instabilities or non-linear bores (Hornet al., 2001).Longuet-Higgins (1952) actually considered a three-layered medium andshowed a total of three “vertical” modes (one set of surface modes, andone set of modes for each of the two density interfaces). In general, thenumber of vertical modes present in a model will correspond to the numberof density layers considered, with a theoretical infinite number of modes fora continuously stratified water column. Provided the effects of the Earth’srotation are unimportant and ignoring damping, the interface deflection of81.2. Review of relevant literaturethe ith vertical mode (h(i)) is governed by∂2h(i)∂t2=[c(i)]2 ∂2h(i)∂x2where c(i) is the wave speed of the corresponding mode (e.g. c(2) given byEquation (1.5)), and in the form presented here c(i) is independent of x.Considering that the barotropic wave speed c(0) =√gH, this formula isconsistent with Equation (1.3).Imam (2012, Chapter 1) provides more detailed review of the baroclinicseiche process, and the variety of models (analytical and numerical) thathave been developed to consider both free and forced responses.Effects of the Earth’s rotationalFor basins that are large enough or of high enough latitude, the rotation ofthe Earth becomes important. This is reflected in the addition a Coriolisterm in the equations of motions. Csanady (1975) provides a detailed reviewon the modifications this term has on the seiche response of the lake. Theimportance of this term to the surface/internal seiche can be determined byconsidering the external/internal Rossby radius of deformation,R =cf, (1.6)where c is the wave speed of either barotropic or baroclinic waves, and f isthe Coriolis parameter, and comparing it to the transverse horizontal lengthscale of the lake. If R is much larger than the lateral horizontal scale of thebasin then rotational effects are unimportant, whereas if it is much smallerthen rotational effects dominate the response.In the classification of lakes that this study considers (fjord-type lakes),rotational effects are typically unimportant for barotropic seiching and whilethey may play a role in the baroclinic response, they rarely dominate thatresponse. With this in mind, no further review of this subject is presented.1.2.2 Influence of complex geometry on seichingThe simplicity of the geometry of the Scottish Lochs allowed for studiesthat provided a great deal of insight into the relevant mechanisms of bothbarotropic and baroclinic seiching. Of course, many basins don’t conformto the geometric constraints of narrow, elongated lakes devoid of islands,bays, sub-basins, or multiple arms. In most cases, studies of complex lakes91.2. Review of relevant literatureoccur on a case-by-case basis and so it is difficult to make generalizationsabout the impact of specific geometric features unless many studies havebeen completed. A body of literature exists regarding the seiche response oflakes with multiple basins that are separated by shallow sills or constrictions(e.g. Appt et al., 2004; Dorostkar and Boegman, 2013; Farmer, 1978; Imamet al., 2013a; Laval et al., 2008; Van Senden and Imboden, 1989), and sosome general concepts are understood in those geometries. Theory alsoexists for seiching in small bays and harbours of large lakes (e.g. Schwaband Rao, 1977; Wilson, 1972), but rarely are these considered in conjunctionwith the response of the whole lake (Kirillin et al., 2014, provides a notableexception). For other geometric features such as islands or multiple arms,far fewer studies exist.The lack of literature concerning seiching in multi-armed lakes maybe due in part to the added difficulty in providing a full explanation forthe spatial variability of the observed periods combined with the need toemploy some form of numerical solver to determine both periods and mode-shapes. Malinina and Solntseva (1972, as cited in Rudnev et al., 1995),among others, attempted to explain the barotropic periodicities observedin Lake Onega by applying Merian’s formula along some extent of the lake;however, as explained by Rudnev et al. (1995), this methodology incorrectlypredicts higher modes and fails to explain periodicities that seem to occuronly in specific arms. To fully describe this response, Rudnev employed anumerical method to solve the 2-dimensional depth-varying wave equations(see Section 1.2.3 below for more details). Numerical solvers are the normfor predicting seiche modes in multi-armed geometries. Similar techniquesto those employed by Rudnev were used to study barotropic seiching inLake Como by Buzzi et al. (1997), and later baroclinic seiching in the samelake by Guyennon et al. (2014). Carter and Lane (1996) applied a similarmethod in a few New Zealand Lakes, including multi-armed lake Te Anua.In a study of multi-armed Clear Lake, (Rueda and Schladow, 2002) presentthe relevant mathematics behind this method as applied on an unstructuredgrid, and in an online supplement provide their code for download; however,in studying internal wave dynamics, the same authors instead employed afully 3-dimensional hydrodynamic model (Rueda and Schladow, 2003). A3-dimensional hydrodynamic model was similarly employed in a study ofwind-forced baroclinic waves in two basins of Nechako Reservoir (Imam, 2012;Imam et al., 2017), though based on only the longest extent of the lake, asemi-analytical model also produced favourable results (Imam et al., 2013b).A two-dimensional, non-linear, forced model was applied to understand theseiche response to seismic activity in “Y”-shaped Lake Union (Barberopoulou,101.2. Review of relevant literature2008). Of the multi-armed lakes mentioned, only in Lake Como (Buzzi et al.,1997; Guyennon et al., 2014) and Nechako Reservoir (Imam, 2012; Imamet al., 2017) is characterization of mode periods and shapes for the fjord-like morphotype of interest given, and no relevant results were found withanalytical methods.Based on these few studies, little can be said about the general responseof multi-armed lakes. One interesting behaviour is evident in lakes withcomplex geometries when wave modes are developed in a 2-dimensionaldomain: many of these lakes have modes that are localized to individualbays or arms of a lake. In Flathead lake, where subsections of the lake aredistinct bays, a hydrodynamic model applied to the whole lake finds modesthat act as bay-modes with modal periods close to those predicted using anopen-mouthed application of Merian’s formula (Kirillin et al., 2014). In LakeOnega many higher modes act primarily within a single arm, with near-zerodeflections of the surface in the remainder of the lake (Rudnev et al., 1995);however, in this lake, the distinction between “arms” and “bays” is unclear,and these modes may be more in line with those predicted by (Kirillin et al.,2014). Lake Como is distinctly multi-armed, and both the barotropic andbaroclinic responses show an exaggerated form of this behaviour: the secondhorizontal mode has one arm that is essentially decoupled from the other two(Buzzi et al., 1997; Guyennon et al., 2014). However, multi-armed NatalkuzLake does not show a similar behaviour (Imam, 2012; Imam et al., 2017).Neumann (1944, as cited in Defant, 1960) developed a method for studyinginterconnected basins using the concept of impedance borrowed from thestudy of electrical circuits. In this method, the impedance of a system isdefined asZ =amplitude of pressurearea× amplitude of velocity ,and the modal frequencies are found by minimizing the impedance Z. Foran undamped system, this minimum is Z = 0. To account for the separatebasins, each basin is assigned an impedance factor, Zi based on its connectionsto other basins. For example, for a basin open at both ends,Z =iρcAtan(ωLc),and for a basin closed at one endZ =iρcAcot(ωLc).These impedances are added using rules similar to adding the resistors inelectrical circuits. For basins configured in series, S =∑Zi, for those111.2. Review of relevant literatureconfigured in parallel, P−1 =∑Z−1i . Then total resistance Z is the sum ofall basins connected in parallel or series. Both Wilson (1972) and Defant(1960) show examples of this method being applied to both fully and semi-enclosed basins with branched arms. This method provides perhaps the onlymethodology for predicting the seiche periods of a multi-armed lake withoutusing a numerical technique. However, despite the potential usefulness of thismethod, this author has been unable to find any example of it being appliedto a multi-armed lake. Furthermore, as will be discussed in Chapter 2, theimpedance method is unable to predict certain features of the response.1.2.3 Finite Element Method for solving the wave equationin an arbitrary two-dimensional domainGiven the prevalence of this technique and also due to it’s employment inChapter 3 of this thesis, I will briefly describe the development of a Galerkin-type Finite Element Method (FEM) numerical technique. Specifically, thissection will describe the use of this technique on an unstructured grid. Thiscan be seen as a summary of Rueda and Schladow (2002), though the use ofthis technique in describing seiche modes in lakes pre-dates that publication(e.g Carter and Lane, 1996; Hutter et al., 1982).In an arbitrary shaped 2-dimensional domain Ω with boundary ∂Ω,Equation (1.3) and the “no-flow” boundary condition become∇ · (H∇η) +(ω2g)η = 0,with (1.7a)∇η · nˆ = 0 on ∂Ω, (1.7b)where the operator ∇ is defined in along the two horizontal dimensions:∇ = (∂x, ∂y), and nˆ is the outward unit normal vector on ∂Ω.For simplicity, let Λ = ω2g−1. Then the weak formulation of Equa-tion (1.7) is attained by multiplying Equation (1.7a) by a test function ψ,integrating over Ω, expanding with Green’s first identity, and using Equa-tion (1.7b) to eliminate boundary terms. The result is−∫Ω(∇ψ) · (H∇η) dA+ Λ∫ΩψηdA = 0. (1.8)While Equation (1.8) is still exact, it is referred to as the weak form ofEquation (1.7) because it only satisfies the equation in an integral sense.While any solution of Equation (1.7) will satisfy Equation (1.8), the reverseis not true. To ensure that solutions to Equation (1.8) satisfy Equation (1.7),121.2. Review of relevant literaturewe need to satisfy Equation (1.8) with ψ(x, y) = δ(x, y) for every (x, y) ∈ Ω(which is an infinite number of test functions).The standard Galerkin procedure is to discretize Equation (1.8) onto thefinite dimensional subspace Sn of dimension N . In particular, we considera subspace Sn that can be described by an unstructured triangular mesh.Then each set of (x, y) coordinates maps to a single numbered node n(see Figure 1.2). In this discrete space, the continuous function η(x, y) isapproximated by η˜(xn, yn). If the set of basis functions {Ψj(x, y)}Nj=1 spanSn where N is the total number of nodes in Sn, then η˜ can be representedas a linear combination of those basis functions:η˜(xj , yj) =N∑jη˜jΨj .123456789101112131415161718192021222324252627Figure 1.2: Schematic of the type of unstructured triangular mesh used in the FEManalysis. Each set of (x, y) coordinates at each node is defined by a single number n.By choosing appropriate functions in the discrete subspace for the testfunctions ψ, the the solution to Equation (1.7) on the continuous domain Ωis approximated on a discrete domain by the matrix problemKη˜ = ΛMwhere the matrices K and M, called the “stiffness” and “mass” matricesrespectively. Taking the basis functions for the test functions ψ = Ψi, thesematrices are given byKij =∫ΩH(∇Ψi)(∇Ψj)dA,Mij =∫ΩΨiΨjdA,131.3. Objectives and organizationand are dimension N ×N . In the simplest case, the basis functions are the“pyramids”:Ψk(xl, yl) ={1 if k = l0 if k 6= l k, l = 1, . . . , N.Then the matrices K and M can be constructed using simple geometricarguments (e.g. Alberty et al., 1999).The eigenvalues Λ of the original problem are then the eigenvalues ofthe matrix M−1K, and the modeshapes are given by the correspondingeigenvectors. In general, both M and K will be sparse, and a number oftechniques exist for approximating the eigenvalues of M−1K, such as theLanczos procedure (Schwab, 1980). Some modern computer programs, suchas Matlab have inbuilt solvers for this (in Matlab, the eigenvalues of thismatrix can by found by using the eigs function).1.3 Objectives and organizationThis study examines the impact of surface geometry of a lake on the expectedmodes of the free oscillatory response. In particular, focus is placed onfjord-type multi-armed lakes. Fjord-type lakes are typically elongate andnarrow, so considering this classification of lakes allows for the study toaddress the importance of multiple arms without having to address additionalcomplexities associated with rotational effects or interactions with transversemodes. Rather than present a single case, the study attempts to make generalstatements about the impact of this geometric feature.The remainder of this document is organized as follows:Chapter 2 takes a theoretical approach to develop a simplified analyticalmodel that can predict mode shapes and periods for standing waves inmulti-armed lakes with an arbitrary number of arms. The implication ofmodel parameters on the solution space is discussed.Through a combination of observations and modelling, Chapter 3 de-scribes the barotropic seiche response of two multi-armed lakes of similargeometry. Both Quesnel Lake (located in British Columbia, Canada) andLake Como (located in the Lombardy region of Italy) are “Y”-shaped fjord-type lakes. By comparing the responses of these two lakes, further insightis gained into the impact of this geometric feature. The analytical modeldiscussed in Chapter 2 is considered in the context of these lakes.14Chapter 2Development of an analyticalsolution for seiche modes infjord-type multi-armed lakes2.1 IntroductionDespite the broad analytical work in the subjects of both barotropic andbaroclinic seiching done by oceanographers and limnologists studying simplelakes (e.g. Chrystal, 1905; Defant, 1960; Longuet-Higgins, 1952; Wedderburnand Williams, 1911), the application of their techniques to more complexgeometries has required the use of numerical techniques (e.g. Rudnev et al.,1995; Schwab and Rao, 1977). While these numerical solvers are valuabletools and may be simple to use (Rueda and Schladow, 2002), they must still beapplied on a case-by-case basis. Thus, in order to understand the behaviourinherent in a particular geometric feature, a wide enough body of literatureneeds to be established to make inferences from those results. Because manytypes of geometric complexity are fundamentally two-dimensional features(e.g. constrictions, bays, or multiple arms), it is difficult to approach theseproblems using an analytical framework.This chapter focuses on the development of a simplified analytical modelfor predicting the free mode periods and mode shapes for fjord-type multi-armed lakes. The model will consider only the defining geometric consid-eration of these lakes: their multi-armed nature. The oscillatory responseof real lakes will be modified by many factors; by removing all of thesecomplexities except the fundamental geometric feature, understanding canbe gained about which behaviours can be attributed the interaction of thearms of the lake.152.2. Model Development2.2 Model Development2.2.1 Governing equationsWe consider the linearised equations of motion and continuity for a hy-drostatic, homogeneous fluid, ignoring the Coriolis terms, frictional effects,and external forcing. Further, motion is assumed to primarily along thelongitudinal axis of the lake, so transverse motions are neglected. Then theequations of motions are:Momentum:∂q∂t+ gH∂h∂x= 0 (2.1a)Continuity:∂h∂t+∂q∂x= 0 (2.1b)where H(x) is the still water depth, h(x, t) is the surface deflection (thetotal depth is given by H + h), and q(x, t) is the depth-integrated horizontalvelocity (q =∫ H0 udz). To arrive at these we have assumed a constantwidth b(x) = b and that deflections are small compared to total depthh(x, t) H(x). Moving forward, it will be useful to non-dimensionalize xand H(x) as follows:x∗ =xLH∗ =H(x)H0In general we will consider these equations within the domain x ∈ [0, L], sox∗ ∈ [0, 1]. H0 is some characteristic depth so H∗ is O(1).Making these substitutions, we combine Equations (2.1a) and (2.1b) andassume an oscillatory response q(x, t) = φ(x)eiωt, and h(x, t) = η(x)eiωt.This results in two ordinary differential equations for the mode shapes:ddx∗(H∗dηdx∗)+ λ2η = 0, (2.2a)H∗d2φdx∗2+ λ2φ = 0 (2.2b)where the non-dimensional eigenvalue λ is given by λ = ωL(gH0)−1/2. Itwill be convenient to define a parameter, called “wave travel time”, asτ = L(gH0)−1/2, so λ = ωτ .From Equations (2.1a) and (2.1b) we have that the mode shapes η and φare related byφ = i(√gH0λ)H∗dηdx∗, and η = i(1λ√gH0)dφdx∗(2.3)162.2. Model Developmentwhere the inclusion of the imaginary number i =√−1 simply indicates thath and q are 90o out of phase in time.Sturm-Liouville Boundary Value ProblemsBefore considering the solutions, it will be useful to place Equations (2.2a)and (2.2b) in the context of a broader mathematical theory which allow us tomention some known properties of these equations. Through the early 19thcentury, a number of advances were made in understanding a classificationof boundary value problems that often arise in when considering separablePDEs. These problems are now called “Sturm-Liouville” (SL) problems(Boyce and DiPrima, 1986) and have the formddx[p(x)dydx]+ [λw(x)− q(x)]y = 0 (2.4)over the finite interval [a,b] together with the boundary conditionsα1y(a) + α2dydx(a) =0 (2.5a)β1y(b) + β2dydx(b) =0 (2.5b)for some prescribed constants α1, α2, β1, β2. The theory also requires thatp(x), w(x) > 0 on (a, b), p−1, q, w ∈ L1loc([a, b])In these equations, non-null solutions are only allowed for certain discretevalues of the parameter λ. These eigenvalues, λn, are found by solvingΠ(λ) = 0Where Π(λ) is the equation obtained by substituting the general solutionto Equations (2.4) and (2.5a) into Equation (2.5b). If Y (x, λ) solves Equa-tions (2.4) and (2.5a), thenΠ(λ) = −(β2β1)(1Y (b, λ)dYdx(b, λ))The central theorem of Sturm’s first memoir (Everitt, 2005; Hinton, 2005)gives thatddλ(pydydx)(x, λ) < 0 (2.6)172.2. Model Developmentwhere y is a solution to Equations (2.4), (2.5a) and (2.5b).And sosign(dΠdλ)= −sign(β2β1).Following from this, the theorem predicts the following two properties of SLequations that we will make use of.1. Eigenvalues are real and form an infinite sequence of increasing magni-tude:λ1 < λ2 < . . . < λn < . . .→ +∞2. Eigenvalues are simple. That is, for each eigenvalue λn, there is a single,unique (up to a constant), eigenfunction yn that satisfies Equation (2.4).These functions can be written asyn(x) = Any(x, λn)for some mode-dependant constant An.A great number of other properties of these equations are also known (fora list of theorems see Hinton, 2005), but these will not be invoked in thepresent study.2.2.2 Description of the multi-armed modelWhile there are a number of different geometries that can fall under thedescription of “multi-armed lakes”, we consider the case of N arms radiatingoutward like spokes from one central confluence point. To solve for thewave response, a coordinate system is set up in each arm such that thelocal x-axis, x∗i , coincides with that arm’s primary longitudinal direction.Some of the algebra will be simplified by using a coordinate system withpositive x∗i moving inwards towards the confluence (see Figure 2.1). Thus, thetwo-dimensional domain is converted to a set of N coupled one-dimensionaldomains, and Equations (2.2a) and (2.2b) are used to predict the longitudinalvariation in ηi and φi along each arm.Equations (2.2a,b) are applied in each arm with boundary conditionsapplied at the endpoints x∗i = 0, 1 as follows:BC1: φi(0) = 0 (i.e. no flow condition),BC2: ηi(1) = ξi, andBC3: φi(1) = σi,182.2. Model Development000x∗1x∗ 2x ∗3Figure 2.1: Schematic representation and coordinate definitions of the multi-armedsystem being considered in the present study for the case N = 3.where ξi and σi are prescribed constants (however, the values of theseconstants will be found through the coupling procedure discussed belowrather than strictly prescribed). Making use of Equation (2.3) allows thecombination of BC2 and BC3 into a boundary condition in SL form:ξiφi(1) +(σiλ√gH0,i)dφidx∗i(1) = 0. (2.7)A similar result can be shown for ηi. Here, the eigenparameter λ exists in theboundary conditions, along with σi and ξi which will be shown to both bemode-dependant values. While this is not necessarily consistent with classicSL problems, Schneider (1974) has shown that the relevant SL propertiesstill hold.The set of N ODEs is coupled at the junction by imposing conditions onξi and σi. Strictly, these conditions act as a set of boundary conditions onEquations (2.1a) and (2.1b), but we will refer to them separately as “couplingconditions”. They are given as follows:CC1: Continuity of surface height: ξ1 = ξ2 = . . . = ξNCC1: Conservation of mass:∑Ni=1 biσi = 0for bi the width of the ith arm at the junction. For simplicity, for theremainder of this document we will assume b1 = b2 = . . . = bN , and sodrop the bi from this conditionCC1: Preservation of these properties through time: ω1,n = ω2,n = . . . = ωN,n.Within each arm, the solutions to Equations (2.2a,b) depend on the formof H∗. Nonetheless, the conditions necessary to couple the equations can be192.2. Model Developmentapplied for the general form and thus the derived solutions and behaviourbecome largely independent of the actual form of H∗.We take advantage of the SL property (2) that eigenfunctions are simpleand write the eigenfunctions for each mode, n, and arm, i, asηi,n = Ai,nη(x∗i , λi,n)φi,n = αi,nφ(x∗i , λi,n)where the functions η(x∗i , λi,n) and φ(x∗i , λi,n) are known, and determinedby satisfying Equations (2.2a) and (2.2b) together with BC1, and Ai,n andαi,n are some mode dependant constants for each arm that are yet to bedetermined. These constants will be linearly related through Equation (2.3),so αi,n = CiAi,n, where Ci are also known. In general Ci depend on the formof H∗, but for all the situations considered in the present study, Ci =√gH0,i.To apply BC2, we set the value of ηi,n at the confluence to be equal tothe defined constant ξi, so ξi = Ai,nη(1, λn). To proceed, it is necessary toconsider two cases for ξi: either ξi 6= 0 or ξi = 0. As we will show immediately,the former condition allows for the mode shapes to be determined with ease,but creates a more complicated expression to determine the frequencies. Inthe latter case, the expression for ωn is straightforward, but more effort mustbe applied to arrive at the mode shapes.Case 1: ξi 6= 0For the case where ξi 6= 0, BC2 can be rearranged to give Ai,n in terms ofξi. Because Ai,n is a mode dependant constant based on initial conditions,ξi is also mode dependant: ξi = ξi,n. Furthermore, the application of CC1indicates that ξi,n is independent of which arm it is being applied to: ξi,n = ξn.Thus, we obtain the following expressions for η and φ:ηi,n(x∗i ) = ξnη(x∗i , λi,n)η(1, λi,n),φi,n(x∗i ) = ξnCiφ(x∗i , λi,n)η(1, λi,n).Then BC3 gives:σi = ξnCiφ(1, λi,n)η(1, λi,n). (2.9)The values λi,n can be determined by solving this equation if ξn and σi areprescribed. However, these values are not prescribed but instead given by202.2. Model Developmentimposing the “coupling conditions”. Substituting Equation (2.9) in CC2,and factoring out ξn yieldsf(ωn) =N∑i=1Ciφ(1, ωnτi)η(1, ωnτi)= 0. (2.10)If τ1 6= τ2 6= . . . 6= τN (denoted the TNE case), the modal frequenciesωn are found by finding values of ω that satisfy f(ω) = 0. However, ifτi = constant = τ (denoted the TE case), Equation (2.10) simplifies tof(ωn) = φ(1, ωnτ) = 0If zφn are the roots of φ(1, z), then the modal frequencies ωn are given byωn = zφnτ−1.Case 2: ξi = 0In the second case that ξi = ξn = 0, BC2 demands that Ai,nη(1, λi,n) = 0, soto avoid trivial solutions, we require that η(1, λi,n) = 0. The results presentedfor ξi 6= 0 would thus imply division by zero and are invalid. Instead, if zηnare the roots of η(1, z), then ωn are given simply by ωn = zηnτ−1i . However,if τi conform to the TNE case (τ1 6= τ2 6= . . . 6= τN ), then this will producedifferent frequencies for each arm. In order to satisfy Equations (2.1a,b) forall time, ωn must be equal for all arms (CC3). This indicates that ξn cannotequal zero except in the TE scenario that τi = constant.Then BC3 gives σi,n = αi,nφ(1, zηn). Recognizing that φ(1, zηn) are equalacross all arms and can be factored out, CC2 is reduces toN∑i=1αi,n = 0.There are an infinite number of values of αi,n that satisfy this condition,but if we separate αi,n into a mode-dependent component, ζn, and an arm-dependent component, γi, so that αi,n = ζnγi, then we can construct γi sothat it forms a basis. For γi given byγi =1 if i = j and i 6= k−1 if i = k0 otherwisej = 1, . . . , (N − 1)k = (j + 1), . . . , N212.2. Model Developmentthen all αi,n can be expressed as a linear combination of ζnγi, and CC2 issatisfied. Then equations ηi,n and φi,n are given byηi,n(x∗i ) =ζnγiCiη(x∗i , zηn)φ(1, zηn)(2.11a)φi,n(x∗i ) = ζnγiφ(x∗i , zηn)φ(1, zηn)(2.11b)Physically, the values of γi indicate that in these modes all the flowexiting one arm will enter exactly one different arm (for each mode, exactly 2arms will be active all the others will be quiescent). Here, each eigenvalue isrepeated with multiplicity given by the number of possible combinations ofγi, which is given by the binomial coefficient NC2 (i.e. for a three armed lake,eigenvalues will have multiplicity 3C2 = 3; for a four armed lake, eigenvalueswill have multiplicity 4C2 = 6; etc.). This presents difficulties in determiningthe values ζn; in order to solve for these values using the initial conditions,the values of γi will need to be specified. Without this specified, there maybe no unique solution for ζn. Indeed, this set of basis functions are notuniquely defined and other bases can also produce viable results; however,if another set of basis functions was specified then a linear combination ofthose functions would yield eigenfunctions that match the current choice andso this decoupled response is not an artefact of the choices made here.If Equation (2.10) is naively used when the wave travel times for eacharm are equal, then the zηn wave modes and their repetitions will be missed.Thus, having τi = constant leads to two separate equations that define ωn:ωn =zηnτ(2.12a)ωm =zφmτ. (2.12b)Equation (2.12a) corresponds to a node existing at the confluence (and hasthe multiplicity discussed above), and Equation (2.12b) corresponds to aanti-node existing at the confluence (and has no multiplicity). As a resultof the first SL property, the function η(1, λ) is oscillatory in λ and willhave an infinite number of roots. Sturm’s oscillation theorem then givesthat the roots of dη(1,λ)dλ will occur between subsequent roots of η (Boyceand DiPrima, 1986). By using Equation (2.3) it can be shown that for thegeometries considered in Section 2.2.4, φ(1, λ) ∝ dη(1,λ)dλ , so the roots zφm willoccur between subsequent values of zηn (e.g. zη1 < zφ1 < zη2 ), and zηn 6= zφm forall m,n.222.2. Model DevelopmentFollowing the methods described in this section, it can be shown that themixed case (denoted TM) where there are a total of M arms with the samevalue of τi (i.e. τi = τi+1 = . . . = τM ), but M is fewer than the total numberof arms (1 < M < N), then there will be a repetition of the eigenvaluesassociated with only those arms with equal values of τi, and so eigenvaluesthat correspond to ωn = zηnτ−1M will have multiplicity of MC2, and all othereigenvalues will have multiplicity 1. To calculate the full set frequenciesωn of the TM case, it is necessary to take both the frequencies predictedby Equation (2.10), and an additional set of frequencies predicted usingEquation (2.12a) with τ = τM . As before, the set of frequencies calculatedwith Equation (2.12a) will be those that exhibit decoupled behaviour.2.2.3 Alternate method for calculating ωnIn the case that τi = constant, the modal frequencies ωn are determinedsimply by ω = {zηnτ−1, zφnτ−1}, n ∈ Z+. When τi 6= constant, there is noclosed form solution for ωn but the values zηn still have a bearing on thesolution. When ωnτi = zηn, then the function η(1, λi,n) = 0. Because thisappears in the denominator of Equation (2.10), any time ωnτi → zηn± thenf(ω)→ ±∞. By using Equation (2.6), it can be shown thatddωf(ω) > 0.So between each set of asymptotes, there will be a single root of f(ω).Consider a sequence Ω constructed asΩ = sort(zηnτi−1 : i = 1, . . . N, n ∈ Z+)where the operator sort arranges elements monotonically from smallest tolargest, and the sequence retains repeated values. Then the values of Ωcorrespond to subsequent ω such that f(ω) diverges (i.e. Ω are the orderedsequence of asymptotes of f(ω)). Then the values of ωn are constrained bythe values of Ω:Ωn ≤ ωn ≤ Ωn+1.In fact, this behaviour is retained in the case that τi = constant and in theTM case.Because f(ω) diverges rapidly away from ω = ωn, these modal frequenciescan be approximated byωn ≈ Ωn + Ωn+12. (2.13)232.2. Model DevelopmentThe advantage to this methodology is that if τi = τj (repetition of one ormore values of τ), then for some n, we will have Ωn = Ωn+1 = zηnτ−1(i,j), sothe correct value of ωn will be calculated. In addition, this method will yieldthe correct multiplicity of eigenvalues.As will be discussed further in Sections 2.3.1 and 2.3.2, by consideringthis pattern, additional insight can be gained into response of the lake.2.2.4 Solutions for analytically defined bottom profilesThe solution to Equations (2.2a) and (2.2b) depend on the form of H∗.For certain analytically defined bottom depth profiles these equations havefunctional solutions. The model developed here will be applied for the twosimplest geometries of depth variation: a constant bottom profile (H(x) =H0), and a linearly varying profile (H(x) = (H0/L)x). Solutions can similarlybe developed for a limited number of more complicated functions H∗ (e.g.Chrystal (1905) considers a parabolic bottom profile1: H(x) = H0[1 −(2xL−1)2]).The general solutions to Equations (2.2a) and (2.2b) for a constantbottom are given simply by trignometric functions:η(x∗) = A cos(λx∗) +B sin(λx∗)φ(x∗) = α sin(λx∗) + β cos(λx∗).For linear bottom variation of the form H(x) = (H0/L)x, a variable substi-tution χ = 2λ√x∗ will convert Equation (2.2a) to Bessel’s equation, so thefunctions η and φ are given in terms of Bessel functions:η(x∗) = AJ0(2λ√x∗)+BY0(2λ√x∗)φ(x∗) = α√x∗J1(2λ√x∗)+ β√x∗Y1(2λ√x∗).In order to satisfy BC1 we require B = β = 0 for both cases of depth1Chrystal (1905) solved Equation (2.2b) using a standard power series expansion of theform φ(χ) = a0 + a1x + a2x2 + . . ., and found two linearly independent series solutionsfor φ which he named “Seiche-cosine” and “Seiche-sine”. In this case, a transformation ofEquation (2.2a) yields Legendre’s equation, so a closed form solution exists in terms ofLegendre Polynomials which are closely related to the functions Chrystal developed.242.2. Model Developmentvariation. Then Equation (2.10) becomesConstant Bottom:N∑i=1√gH0,i tan(ωnτi) = 0, (2.14)Linear Bottom:N∑i=1√gH0,iJ1(2ωnτi)J0(2ωnτi)= 0, (2.15)2.2.5 Solutions for an arbitrary bottom profileA general solution for the SL equation does not exist (Hinton, 2005). To solveEquations (2.2a) and (2.2b) for an arbitrary function H∗ it is necessary touse some approximate approach. The most accurate approximate approach isthe use of a numerical scheme which will produce solutions of arbitrarily highprecision provided a small enough step size, but such an approach defeatsthe spirit of the present study and so it will not be employed here. Othermethods of approximation include breaking the bathymetry into smallersections which can each be fitted with an polynomial curve (Chrystal, 1905;Chrystal and Wedderburn, 1905), or the use of an asymptotic approach.The WKB method is one such asymptotic approach appropriate forequations of this form. This approach gives approximate solutions to theequationd2ydx2+ λ2f(x)y = 0accurate for large eigenvalues, as λ → ∞+ (Bender and Orszag, 1999).While the approach is perhaps more popular in fields such as optics, ithas nonetheless been applied to understanding the nature of water waves(LeBlond and Mysak, 1978), and in a theoretical framework has even beenapplied to the problem of seiches (Ortiz et al., 2013). The method isoccasionally referred to as the Liouville-Green (LG) method for work thatthose mathematicians did in the 19th century; in fact, the work of Green(1837) in developing ideas associated with this approach involved wavepropagation in an elongated channel with varying depth and width.The WKB method assumes a solution of the form y(x) ∼ eλS(x) and,following the standard perturbation approach, S(x) is expanded in an infiniteseries around a small parameter (chosen to be λ−1):S(x) =∞∑n=01λnSn(x).252.2. Model DevelopmentThis expansion is substituted back into the original equation and the S0(x)and S1(x) terms are retained and computed. Then, if f(x) > 0 for all x theresult isy(x) ∼ α[f(x)]1/4sin(λ∫ x√f(s)ds+ θ)for arbitrary amplitude α and phase θ. This formula can be applied toEquation (2.2b) directly to find the asymptotic behaviour of φ. A leadingorder solution for η can be found by either applying Equation (2.3) to theresult, or using a variable substitution η = Y (x∗)(H∗)−1/2 in Equation (2.2a)and thus converting it to a form that the WKB approximation can beapplied. Both methods produce the same result, but first it is necessary toneglect some additional higher-order terms. We obtain the following WKBapproximations for φ and ηφn(x∗) ∼ αn(H∗)1/4 sin [λnΨ(x∗) + θn] (2.16a)ηn(x∗) ∼ An(H∗)1/4cos [λnΨ(x∗) + θn] (2.16b)where Ψ(x∗) =∫ x∗0 [H∗(s)]−1/2ds, and αn = −An√gH0.To apply the multi-armed model, Equations (2.16a) and (2.16b) aresubstituted into Equation (2.10) to giveN∑i=1√gHi(1) tan(ωnτi∫ 10ds√H∗i (s)+ θn)= 0 (2.17)where the value of θn is found by applying BC1 Equation (2.16a). IfH∗(0) 6= 0, then θn = 0; however, if H∗(0) = 0 then x∗ = 0 is a singularpoint and additional work is necessary to determine θn and ensure consistency.Endpoint singularitiesFor H∗(0) = 0, the point x∗ = 0 is a singular point. If this singularity isintegrable, then a modified turning point analysis can be used for φ (seeHinch, 1991, p. 133-134). An “inner solution”, φI, is developed near thesingular point, while an “outer solution”, φII (given by Equation (2.16a)),exists through the rest of the domain. Then these solutions are matched bysettinglimx∗→∞φI(x∗) = limx∗→0φII(x∗).262.2. Model DevelopmentTo find the inner solution, H∗(x∗) is expanded in a Taylor series aboutx∗ = 0. The first non-zero termH∗(x∗) ∼ x∗[dH∗dx∗]x∗=0as x∗ → 0 (2.18)is substituted into Equation (2.2b), and φI is the solution to the resultingODE. Here, this gives φI(x∗) in terms of the zero-order Bessel function, J0(z).The behaviour limx∗→∞φI(x∗) is given by the ‘large argument’ expansion ofJ0(z).The x∗ → 0 behaviour of φII(x∗) is given by substituting Equation (2.18)into Equation (2.16a) and calculating the corresponding function Ψ. Matchingthese solutions results in θn = −pi/4 for all n. Because Equation (2.16b)can be derived from Equation (2.16a), this value of θn can be used inboth equations, and so the multi-armed model can be used with the WKBapproximation accounting for endpoint singularities by using Equation (2.17)with θn = −pi/4. However, with this approximation, η(x∗) is still singular atx∗ = 0. While this doesn’t necessarily invalidate the use of Equation (2.17),it would be desirable to have η bounded as x∗ → 0. To account for thisbehaviour it is necessary to construct a higher order approximation for η(x∗).If η = Y (x∗)(H∗)−1/2, then Equation (2.2a) becomesd2Ydx∗2+(λ2H∗+14H∗2)Y = 0.In Equation (2.16b), the final term Y (4H∗2)−1 was neglected as λ→∞+,but if H∗(0) = 0, then the two coefficients of Y can be of comparable ordernear x∗ = 0 and so that term should not be neglected. Retaining this termgivesηII ∼ An(λ2H∗ + 14)1/4 cos(∫ x∗0√λ2H∗(s)+14H∗(s)2ds+ ψn)(2.19)in the outer region. To determine ψn, a similar matching procedure as abovecan be employed. However, the integral inside of the cosine in Equation (2.19)will not converge if H∗ ∝ x∗ as x∗ → 0, so the expansion Equation (2.18)cannot be used. Instead, assumeH∗(x∗) ∼ H∗1/2√x∗as x∗ → 0where H∗1/2 is some constant. Making this substitution in Equation (2.2a)allows for solutions in terms of Bessel functions of order 1/3. Then matchinggives ψn = −5pi/12 as λ→∞. However, for small λ (which is the scenarioof interest) the inner and outer solutions do not match.272.3. Discussion2.3 Discussion2.3.1 The use of Merian’s formula in multi-armed lakesIt is the intuition of some researchers that a first approximation of modalperiods in a multi-armed lake can be made by applying Merian’s formulaalong the longest longitudinal extent of the lake and ignoring other arms (e.g.Caloi and Spadea, 1958; Imam et al., 2013b; Laval et al., 2008; Malinina andSolntseva, 1972). While Rudnev et al. (1995) suggests that this approachmay be inaccurate for higher modes, the present model indicates that insome cases this formula may approximate the correct results for the firstmode. Given the TE conditions on τ , the first mode will exhibit a decoupledresponse in which only two of the arms of the lake are active in the response;in this case it is clear that Merian’s formula applied along those active armsshould produce modal periods that agree with this simplified model.Surprisingly, a case can also be made for the “first-guess” accuracy ofMerian’s formula in the TNE or TM cases. To explain the success of thisapproach, consider the behaviour described in Section 2.2.3: frequenciesωn will be located between successive asymptotes of Equation (2.10) whenconsidered over a range of ω. For example, for the flat bottomed case,asymptotes will be located at all (2n − 1)pi(2τi)−1, n ∈ Z+, so the thelocation of the first mode will be constrained topi2τ1< ω1 <pi2τ2(2.20)where τ1,2 are the two largest values of τi. Consider the case H0,1 = H0,2 =H0, then τ1 > τ2 implies L1 > L2. The frequency predicted by Merian’sformula applied along total length L = L1 +L2 (the longest extent of the lake)is ω = pi√gH0(L1 + L2)−1. Substituting these values into Equation (2.20)producespi√gH02L1<pi√gH0L1 + L2<pi√gH02L2,or equivalently2L1 > L1 + L2 > 2L2.We see that for L1 > L2 this is consistent, and so the frequency predictedby Merian’s formula provides a good approximation of the fundamentalfrequency predicted given by Equation (2.14).282.3. Discussion2.3.2 The impact of cross-sectional variationDepth variationThe classic approach to estimating modal periods using Merian’s formulasuggests that modal periods, Tn, follow a harmonic progression given byTn = T1(1,12,13,14, . . .)=T1n.Chrystal (1905) showed that in simple lakes, these ratios of modal periodsdo not follow this harmonic sequence when depth variation is accounted for.However, his work did not explicitly discuss the impact of depth variation onthe fundamental mode. By considering an approach similar to that employedin Section 2.3.1, it is possible to investigate the impact of depth variation inmulti-armed lakes.As described in Sections 2.2.3 and 2.3.1 The frequency of the fundamentalmode is constrained between the first two asymptotes: Ω1 ≤ ω1 ≤ Ω2. Bycomparing the constant bottom (subscript C) model and the linearly varying(subscript `) models, it is evident that the locations of these asymptotes(Ω1,2 = zη1τ−11,2 where τ1,2 are the two largest values of τi) are relativelyinsensitive to changes in depth variation. IfHi =H`Lixthen(H0,i)` = H`(H0,i)C =1Li∫ L0H`Lixdx =12H`and(τi)` =Li√gH`(τi)C =Li√g(12H`) = √2(τi)`As in Section 2.3.1, for the constant bottom (zη1)C = pi/2; for the linearly292.3. Discussionvarying bottom (zη1 )` is the first zero of J0(2z), so (zη1 )` ≈ 1.202. So(Ω1,2)` ≈ 1.202(τ1,2)`(Ω1,2)C =pi2√2(τ1,2)`≈ 1.111(τ1,2)`.Thus, (Ω1,2)C ≈ (Ω1,2)`. Because the first modal frequency ω1 will be con-strained by these asymptotes, we see that the prediction of the fundamentalmode is not overly sensitive to depth variation.In contrast, as predicted in simple lakes by Chrystal (1905), higher modesare expected to be much more sensitive to depth variation. The harmonicsequence of modal periods predicted by Merian’s formula is a result of theeven spacing between successive zeros zφm when a constant bottom bathymetryis used. For lakes of varying depth, these zeros will not be uniformly spacedand so the periods of higher modes deviate from the harmonic sequence. Inmulti-armed lakes modal periods are predicted based on an interaction ofthe zeros zηm and so the impacts of depth variation will compound.The spatial structure of the mode-shapes of higher modes will also beimpacted by depth variation. Specifically, for forms of H∗ that shoal towardsthe ends of the lake (H∗(0)→ 0 and dH∗dx∗ > 0), such as the linearly-varyingdepth case, the amplitude of the mode-shape function η decreases withincreasing distance from the origin. This is visible in the analytically definedsolution to the linearly varying depth case (η(x) ∝ J0(λ√x∗)), but also inthe case of arbitrary depth variation Equation (2.16b) (η(x) ∝ [H∗]−1/4).As a result, the highest predicted peak in η occurs at x∗i = 0 (i.e. at theends of the arms), and as the mode number increases more of the domainis governed by lower amplitude waves. This will lead to a behaviour wheredeflections are increasingly localized to only the very tips of the domain.The WKB approximation that was put forward as a way of estimatingthe response in lakes of arbitrary depth variation allows for some furtherinsight into the impact of depth variation on the response of the lake. Themodel presented here indicates that relative values of the parameter τi =Li(gH0,i)−1/2, called the “wave travel time” predicts which of the the twoclasses of behaviour will be exhibited: either a whole-lake response or adecoupled response. When considering an arbitrary bottom profile alongeach arm, the WKB approximation (Equation (2.17)) suggests that thecondition τi = constant is replaced by∫ Li0ds√gHi(s)= constant. (2.21)302.3. DiscussionIn fact, this change in condition confirms the intuition that the time of travelof a progressive wave in a given arm has an influence on whether or not adecoupled response should be expected. In a channel of varying depth, thebarotropic wave speed is c(x) =√gH(x). If a wave in such a channel travelsa distance dx with a speed c(x), then the time of travel is dt = dx[c(x)]−1,and the total travel time over the domain x is t =∫dt =∫dx[c(x)]−1,which is the quantity described by Equation (2.21). This indicates thatEquation (2.21) generalizes the condition τi = constant to an arbitrary depthprofile.Unfortunately, the WKB method faces limitations in it’s ability to cor-rectly predict the modal periods of multi-armed lakes. In simple systemsthe WKB approximation can provide very good estimates of the eigenvaluesof an ODE, particularly for higher modes (Bender and Orszag, 1999). Formulti-armed lakes, both η and φ need to be estimated. While Equation (2.2b)is in a form compatible with the WKB approximation, in order to approx-imate the solution to Equation (2.2a) additional steps must be taken. ηhas to be determined either from φ through Equation (2.3), or through avariable substitution to convert Equation (2.2a) into a WKB form. In bothof those cases, the λ→∞+ condition is used to neglect higher-order termsin order to produce a tractable result. In the variable substitution method,additional terms can be retained to preserve accuracy (Equation (2.19)),but if H∗(0) = 0 (i.e. if there are endpoint singularities), then an additionalapproximation must still be made in the η equation in order to match theinner and outer solutions near the singular point. Because of the additionalapproximations necessary for the η function, the level of accuracy of theequations for φ and η will not match. Modal periods are predicted based ona ratio of these functions (Equation (2.10)) so this accuracy mismatch indi-cates that the condition CC2 is unable to guarantee total mass conservationacross the lake; this effect will be most pronounced for lower modes. Thefunctions φ and η may provide independently good approximations for themode-shapes if the correct frequencies ωn are supplied, but it is unlikely thatthe WKB method will accurately predict those frequencies for the lowestmodes.Width variationAs developed, the model assumes a constant width along arms of the lakeb(x) = b, so cross-sectional variability occurs solely due to variations inlake depth. In many fjord-type lakes, this is a reasonable approximation forbarotropic modes as arm width may vary much more gradually than total312.3. Discussiondepth. However, it is trivial to re-write Equation (2.2a) (and similarly Equa-tion (2.2b)) in terms of the total cross-sectional variation S(x) (Proudman,1953):ddx(S(x)dηdx)+ λ2b(x)η = 0. (2.22)Accounting for total cross-sectional variation in the model developed inthe present study would likely increase accuracy of the results, and could bedone with minimal adaptation. However, it is expected that the behaviourand important parameters predicted by such a modified model would notchange from the present results. And while adapting the mathematicallydescription may be relatively simple, defining physical parameters such ascross-sectional area in the vicinity of the confluence may be difficult in actuallake systems.2.3.3 Extensions of this modelMore complex geometriesOf interest is the application of the techniques described here to more complexgeometries. For example, more complex branching multi-armed lakes canalso be thought of as a series of one-dimensional reaches that are coupled byusing boundary conditions at a variety of connection points. Unfortunately,the solutions for these more systems can quickly become cumbersome. For an“H”-shaped geometry (Figure 2.2), the equation analogous to Equation (2.14)is1√gH0,11 tan(ωnτ11) +√gH0,12 tan(ωnτ12)+1√gH0,21 tan(ωnτ21) +√gH0,22 tan(ωnτ22)+√gH0,M tan(2ωnτM ) = 0 (2.23)Predictably, in this case it is not straightforward to understand possibledecoupled responses (which might occur if there are nodal lines at either ofthe two confluences). So while these ideas may have merit, it would likely beeasier to predict the response of branched multi-armed basins using numericalmethods.In complex geometries other than multi-armed lakes, some analogousprocedure as discussed here may be appropriate. If Equations (2.2a,b) (ortheir two-dimensional equivalents) can be simplified by considering only the322.3. Discussion0000x ∗11x∗ 12x∗M x∗ 21x ∗22x∗MFigure 2.2: Schematic representation of an “H”-shaped multi-armed geometryoverriding geometric feature then insight may be gained into the impact ofthat feature on the response. In the present study, no other geometries areconsidered.Baroclinic ModesLonguet-Higgins (1952) (among others) have shown that for a perfectlytwo-layer system, the barotropic and baroclinic modes can be effectivelyseparated. The barotropic modes will respond as per the modes of anequivalent homogeneous body of water. The baroclinic modes will be governedby a set of equations of the same form as the barotropic modes, but with amodified wave speed given by c(2) = (g′Heff)1/2. Here, the “reduced” gravity,g′, is dependent on the relative densities of the upper and lower layers, ρ1,2respectively:g′ = g(ρ2 − ρ1ρ2).The “effective depth”, Heff, is given by the harmonic mean of the upper andlower layer thicknesses H1,2 (which add to the total depth H1 +H2 = H(x)):Heff(x) =H1H2H1 +H2= H1[1− H1H(x)].For baroclinic seiching, if the non-dimensional depth H∗ and eigenvaluesλn are defined as H∗ = Heff(x)(H0)−1 and λn = ωnL(g′Heff)−1/2, thenEquations (2.2a) and (2.2b) can be used to describe the mode shapes ofinterface deflection η = η(2) and bottom-layer flow φ = φ(2), and so themodel described in this study can be applied to predict the frequencies andmode-shapes of baroclinic modes.Fjord-type lakes are typically very deep with steep side-walls, so thetotal depth H(x) H1 through most of the lake, and therefore Heff ∼ H1.332.3. DiscussionThis is important because, while barotropic seiching is sensitive to thedepth variation H(x), the effective depth Heff used in the formulation of thebaroclinic wave equations is essentially constant except near the shore. Achoice for the characteristic depth H0 to be equal to the upper layer thickness,H0 = H1, results in H∗ ∼ 1. This suggests that using a constant bottomdepth variation in the simplified analytical model should be appropriatefor most cases of baroclinic seiching, regardless of the actual form of H(x).Adapting Equation (2.14) to the parameters that describe baroclinic modesgivesN∑i=1tan(ωnLi√g′Heff)= 0 (2.24)The lack of variation in Heff has an additional effect on the applicationof this model. Because H0 = Heff ∼ H1, and H1 is typically spatiallyconstant, H0,i = constant. So the TE condition of τi = constant simplifiesto Li = constant for the decoupling of baroclinic waves.In contrast to barotropic modes, whose higher horizontal modes aredifficult to predict due to sensitivity to depth variation, it is expected thatEquation (2.24) will accurately represent higher baroclinic modes. Of course,those modes are still modified by variation of the cross-sectional area alongthe thalweg (Mortimer, 1979), so a truly accurate representation of thosehigher modes would have to modify the model to consider width variation.2.3.4 Similarity to Neumann’s impedance methodAs discussed in Section 1.2.2, the impedance theory developed by Neumann(1944, as cited in Defant, 1960) provides an analytical methodology forcalculating modal frequencies of connected basins in lakes and bays. Defant(1960) includes an example of this method using a three-armed geometryanalogous to that investigated here (see Figure 76 of his text). For theclosed system, this results in the following equation for predicting the modalfrequencies:b1c1 tan(ωL1c1)+ b2c2 tan(ωL2c2)+ b3c3 tan(ωL3c3)= 0,where ci is the wave speed in arm i, ci = (gHi)1/2, and bi is the width ofarm i. If b1 = b2 = b3, this is identical to Equation (2.14). This techniquesimilarly reproduces Equation (2.23) for the geometry described by Figure 2.2.However, the impedance method does not properly account for the possibilityof nodal lines positioned at the location of the confluence. Because of this342.4. Conclusionslimitation, the impedance method is unable to reproduce the decoupledmodes predicted by the present model, which is one of the central results.Due to the simplicity of the impedance method, for multi-armed orinterconnected basins of any geometry in which a decoupled response is notsuspected to occur this method will likely yield a reasonable approximationof the longest mode periods without the need of a numerical approach. Thesensitivity of higher modes to depth variation (which is not included in thismethod) would still preclude accurate predictions of those modes. However,through this model, both depth and width variation can be accounted for bysplitting a single basin into a number of sub-basins each with it’s own depthand width (i.e. assuming a “stepped” bottom bathymetry); such a proceduredoes increase the complexity of the resulting equation for determining ωn.2.4 ConclusionsThis chapter develops a simplified analytical model to identify the keygeometric parameters responsible for the sieche response of fjord-type multi-armed lakes. In simple elongated lakes, the period of the fundamentalmode is given by Merian’s formula, in which the length of the basin andthe shallow-water wave speed are both key parameters. As expected, theseparameters are similarly important in multi-armed lakes; however, they aretaken individually for each arm of the lake. The model suggests two differentclasses of behaviour based on the relative values of a parameter τ , definedas τi = Li(gH0,i)−1/2, which represents the time of travel of a progressiveshallow-water wave moving longitudinally in the ith arm of the lake. Thesebehaviours differ based on the case of τi = constant (denoted the TE case)or τi 6= constant (denoted the TNE case).The TNE may be expected to be more common in real water bodies.In this scenario, all of the arms of the lake are active in every mode; thebehaviour observed by Buzzi et al. (1997) and Guyennon et al. (2014) inLake Como in which the eastern arm does not act for some modes, is notpresent for the TNE case. Due to this structure, this case is referred to asa “whole-lake mode”. In this case, the modal frequencies ωn are given bythe solution to a transcendental formula, Equation (2.10), whose specificform is based on the form of depth variation of the arms. In contrast, twoequations exist for the modal frequencies for the TE case, both of whichare simple and analytic provided that the zeros zη,φn are known. The set ofmodal frequencies given by these two formula correspond to either a nodeor an anti-node appearing at the confluence point. The modes that have352.4. ConclusionsTable 2.1: Summary of modal frequency equations for the TE and TNE cases, as developedin Section 2.2.2. In the TE model, the ξn = 0 case corresponds to nodes located at theconfluence point, while ξn = 0 corresponds to anti-nodes being located at the confluencepoint. In the TNE model, ξn must be non-zero.TE TNEξn = 0 ωn =zηnτ-ξn 6= 0 ωn = zφnτN∑i=1Ciφ(1, ωnτi)η(1, ωnτi)= 0a node occurring at the confluence exhibit a decoupled behaviour in whichonly 2 of the arms of the lake are active, while the remainder of the lake isquiescent. These “decoupled modes” have an eigenvalue multiplicity givenby the binomial combination NC2, where N is the total number of arms ofthe lake. If a subset of the arms have equal wave travel times, a “mixed-case”(TM) can occur, which is an amalgamation of the two results. The whole-lakemodes will be calculated as normal using Equation (2.10) while additionaldecoupled modes are calculated using Equation (2.12a) and exist for only thearms with repeated τi. The equations for determining the modal frequenciesfor the TE and TNE cases are summarized in Table 2.1.The whole-lake versus decoupled behaviour is developed for the generalform of the underlying equations, and so is independent of the actual formof the depth variation of the lake. While general solutions do not existfor an arbitrary bottom profile, the WKB asymptotic method is used toapproximate solutions in this case. While limitations of the WKB methodmay prevent accurate prediction of lower modes, this method reiterates thatthe ratios of the wave travel time along the longitudinal extents of each ofthe arms is the parameter of interest in predicting a decoupled or whole-lakeresponse.Interestingly, despite the complexities associated with having multiplearms, this model suggests that the first fundamental period can still beestimated with reasonable accuracy by assuming a constant depth in eacharm and applying Merian’s formula along the two arms that create thelongest extent. More strictly, the formula should be applied across the armsthat have the two highest values of τi. This behaviour is retained for boththe TE and TNE cases of τi. Contrary to the fundamental mode, whichappears to be relatively independent of depth variation, the periods of highermodes are very sensitive to the form of H(x).362.4. ConclusionsWhile developed for barotropic motions, the separation of vertical modessuggests that this model is equally valid for baroclinic seiching. In fact, in anidealized two-layered system, the limited spatial variability in Heff suggeststhat it suffices to take only the constant-bottomed variation of this model.Then the changes in wave travel time between arms of the lake is determinedby the length of arm, so the condition that determines whether a decoupledresponse is expected is simply if Li = constant. While additional complexitymay exist for baroclinic modes (such as the influence of the Coriolis force,or the increased likelihood of non-linear effects), the present model shouldstill improve on the current ability to make quick, rough estimates of modalperiods and shapesThe present model predicts behaviours only for a specific type of surfacegeometry. Nonetheless, it exemplifies the level of additional insight that canbe gained through the application of a simplified approach. Before now, therewas no consensus on whether a decoupled response was a general feature ofmulti-armed geometries. Now, not only is this decoupling explained, but acriteria exists to predict if it may happen in a given lake. By reducing thetwo-dimensional domain of the lake into a set of coupled one-dimensionaldomains, the model considers only the defining geometric feature of multi-armed lakes. A similar approach may be taken to consider the influence ofother geometric features in different lake classifications.37Chapter 3Barotropic seiche modes intwo fjord-type Y-shapedlakes3.1 IntroductionIn lakes with low hydraulic throughflow, water quality is modulated byphysical processes that induce currents and mixing in the lake. Given theimportance of seiching in driving currents that lead to the transport of massand materials through the lake, these events are of particular importance inlake systems. For complex or multi-armed lakes, prediction of seiche mode-shapes and periods typically relies on case-by-case application of numericalmodels or detailed field studies, and so generalizations are difficult to make.The simplified analytical model developed in Chapter 2 provides a frameworkfor understanding the role of geometry in fjord-type multi-armed lakes whichcan be used to provide context for observed or numerically modelled results.One behaviour Chapter 2 predicts is that the first mode of a multi-armedlake should conform to simple linear oscillation of the longest extent of thelake which could be roughly estimated by applying Merian’s formula alongthat extent. This justifies the assumptions made by Malinina and Solntseva(1972, as cited in Rudnev et al., 1995) in Lake Onega, Laval et al. (2008)in Quesnel Lake, Caloi and Spadea (1958) in Lake Como, and Imam et al.(2013b) in Nechako Reservoir. However, the model indicates that additionalarms and complexities cannot be neglected for higher modes.Perhaps the most interesting result in Chapter 2 is that it is possiblefor multi-armed lakes to exhibit a response in which one or more arms aredecoupled from the main body of the lake and do not oscillate. A combinationof field and numerical study of internal (baroclinic) seiching within LakeComo found such a result, where for some modes the eastern arm of thelake is inactive in the response (Guyennon et al., 2014). A numerical schemeused by Buzzi et al. (1997, hereafter BGS) showed that the same holds true383.2. Methodsfor surface (barotropic) seiche modes in Lake Como. Rudnev et al. (1995)finds that barotropic oscillations in some arms or bays of Lake Onega act inisolation, with near-zero deflection in the remainder of the lake; however, itis difficult to commensurate the geometry of Lake Onega with the systemstudied in Chapter 2. In contrast, Imam (2012) and Imam et al. (2017) findno such decoupling in multi-armed Nechako Reservoir, nor do Carter andLane (1996) find any decoupling in multi-armed Lake Te Anau. For radialmulti-armed lakes (where all of the arms meet at a single confluence point)such as “Y”-shaped lakes, the model in Chapter 2 provides a geometriccriteria to predict whether a decoupled behaviour is possible; however, themodel is developed only for an idealized case. It will be instructive to considerthe application of the simplified model to real lake geometries.This chapter investigates two “Y”-shaped lakes: Lake Como, located inNorthern Italy; and Quesnel Lake, located in Western Canada. While thelakes are different sizes, they have a number of similarities. Both lakes arefjord-type lakes with a three-armed geometry. As is typical of fjord-typelakes, the arms of both of these lakes are narrow and elongated with steepside walls. In the context of the model developed in Chapter 2, these twolakes exemplify two different classes of behaviour. As identified by BGS andGuyennon et al. (2014), Lake Como appears to exhibit decoupling of theeastern arm. Similar studies have not been conducted in Quesnel Lake andso it is not yet known if a decoupled behaviour exists; however, based on thesignificant differences in length and depth of the arms, the simplified modelpredicts only whole-lake modes. The two classifications allow for verificationof the parameters and behaviours predicted in Chapter 2. The results willbe validated using a combination of field observations, and a Finite ElementMethod (FEM) numerical scheme (see Section 1.2.3). A comparison of thepredicted response of both lakes will provide further ability to make generalstatements about standing wave modes in multi-armed lakes.3.2 Methods3.2.1 Site descriptionsQuesnel LakeQuesnel Lake (Figure 3.1a) is a fjord-type lake, and consistent with thatmorphotype, it is narrow-armed and deep, with a mean depth of 157 m anda maximum depth of 511 m (Laval et al., 2008). It has an east-west span of81 km and a north-south span of 36 km. The lake is roughly “Y”-shaped,393.2. MethodsM1M3M5M10NNorth ArmEast ArmWest ArmWestBasinCariboo Island(a) Quesnel LakeLeccoComo0 5 10 kmNNorth ArmComo Branch(West Arm)Lecco Branch(East Arm)(b) Lake ComoFigure 3.1: (a) Map of Quesnel Lake showing the locations of the moorings ( ); (b) Mapof Lake Como showing the locations of the communities of Lecco (where the limnographdiscussed in Section 3.2.2 was deployed) and Como. In both maps, contour lines are shownevery 100 m, and red dashed lines indicate the local thalweg for each arm used in themodel discussed in Section 3.3.3. The distance bar in (b) applies to both figures, whichare presented to the same scale.403.2. Methodswith three arms (the North, East, and West Arms) extending outward froma central junction. At the end of the West Arm, Cariboo Island creates aconstriction and shallow sill that separates a distinct sub-basin, the WestBasin. This sill is an important feature for understanding baroclinic seiching,and the exchange of deep water between the basins Laval et al. (2008).Quesnel Lake follows a seasonal stratification pattern consistent withdimictic lakes, modified slightly by pressure effects on the temperature ofmaximum density due to the great depth of the lake Laval et al. (2012).During summer, the lake is strongly stratified with a warm eplimnion overly-ing a cold (∼ 4◦C) hypolimnion; in the winter it experiences weak reversestratification with cold (< 4◦C) water overlying warmer (∼ 4◦C) water. De-spite some subtle details caused by the pressure effects, the lake experiencesregular seasonal turnovers of the entire water column occurring roughly inDecember (“fall” turnover), and May (spring turnover) (Laval et al., 2012).Lake ComoLake Como (Figure 3.1b) is located in the Lombary Region of Northern Italy,and extends into the southern edge of the Alps. It is smaller than QuesnelLake, with an east-west span of 26 km and a north-south span of 40 km.Like Quesnel Lake, Lake Como is also “Y”-shaped and fjord-like. One ofits three arms extends north from the confluence point, while the other twobifurcate into south-east and south-west pointing arms. The city of Lecco islocated near the tip of the western of these two arms while the city of Comois located near the tip of the eastern arm, so these are occasionally referredto as the “Lecco Branch” and the “Como Branch”. Despite its smaller sizecompared to Quesnel Lake, Lake Como is also very deep with a maximumdepth of 425 m and a mean depth of 151 m (Guyennon et al., 2014).In contrast to Quesnel Lake, Lake Como does not experience full seasonalturnover; instead it is considered oligiomictic and fully turns over onlyoccasionally (Guyennon et al., 2014; Morillo et al., 2009; Salmaso and Mosello,2010). Typical winter turnover only extends to ∼150 m depth (Morillo et al.,2009).3.2.2 Field studySurface seiching is characterized by consistent oscillations of the water surface.The natural periods of the oscillatory modes in a lake can be determinedthrough spectral analysis of these water level signals. Peaks in spectralenergy will correspond to modal periods.413.2. MethodsIn evaluating and characterizing the barotropic response of Lake Como,this study builds on the work of BGS, and will draw on the results presentedby those authors; no additional field study has been performed. BGS collectedwater level readings in Lake Como using a limnograph mounted near thecommunity of Lecca (on the south-east arm; see Figure 3.1b). The limnographrecorded water levels on three separate dates: Feb. 17, Feb. 18, and Mar. 1,1996. The recording intervals were ∼9.5 hours, ∼7.5 hours, and ∼11.7 hours,respectively, and and the limnograph measured water elevation at a rateof 1 sample per minute. With these data BGS were able to identify modalperiods using spectral analysis.To identify surface seiche modes in Quesnel Lake, data are presented fromfour moorings that were installed in the lake in Nov. 2014 (see Figure 3.1a).Each of the moorings was equipped with an RBR duo T.D. in the upperwater column. These instruments recorded total pressure at a samplingperiod of 4 seconds then these data were averaged over 1-minute. In order toperform frequency analysis, the longest continuous pressure record betweenmooring servicing was used, which extends from Oct. 2, 2015 to Sept. 15,2016.The pressure sensors installed at the moorings measured total pressure;in order to relate total pressure to water level through hydrostatic pressure,the barometric pressure signal must first be subtracted. However, thespectral analysis considers the periodicity of data fluctuations, and so actualwater level depth is unimportant. Provided that the barometric pressurefluctuations do not have an oscillatory signature that can be confused withthe barotropic response, the identification of seiche modes from total pressurewill be relatively insensitive to the barometric pressure. A weather stationinstalled at the Quesnel River Research Centre (QRRC), approximately1.4 km downstream from the outflow of Quesnel Lake, has intermittentmeasurements of barometic pressure throughout the period of record of themoorings. These data are available at a 15-minute sampling period so theycan be used to rule out the effects of barometric pressure on any oscillatorymotion with a period greater than 30 minute (corresponding to the Nyquistfrequency).3.2.3 ModellingFollowing convention, we investigate the free-response by considering thehomogenous linear shallow-water (LWS) equations. Based on mean depth,the external Rossby radius at Quesnel Lake (latitude 52.5◦ N) is ∼340 km.The Rossby radius at Lake Como (latitude 46.2◦ N) is ∼370 km. These423.2. Methodsvalues are both much larger than the scale of each of the lakes, so Corioliseffects are neglected. In the absence of Coriolis forces, the LSW equationsare given in a two-dimensional domain as (e.g. Rudnev et al., 1995):Momentum:∂q∂t+ gH∇h = 0 (3.1a)Continuity:∂h∂t+∇q = 0 (3.1b)where H(x, y) is the still water depth, h(x, y, t) is the surface deflection (thetotal depth is given by H + h), q(x, y, t) is the depth-integrated horizontalvelocity (q(x, y, t) =∫ H0 udz), and the operator ∇ is defined in along thetwo horizontal dimensions: ∇ = (∂x, ∂y).Assuming an oscillatory response, q(x, y, t) = φ(x, y)eiωt and h(x, y, t) =η(x, y)eiωt, Equations (3.1a) and (3.1b) are combined to give the Helmholtz-like equations:∇ · (H∇η) +(ω2g)η = 0, (3.2a)∇2φ+(ω2gH)φ = 0. (3.2b)Equations (3.2a) and (3.2b) are eigenvalue problems and as such havean infinite number of eigenfunction solutions. These solutions representthe undamped free response modes of the lake surface deflection (η) andhorizonal current (φ). The true response of a lake to some external forcingwill be composed of these free response modes and, because damping ofbarotropic seiching is small (Defant, 1960), the observed oscillatory periodsshould match the predicted periods to a high degree of accuracy.Two methods are used to determine the solutions to these equations: anumerical solver, and the simplified analytical model developed in Chapter 2.For clarity, this study will present the mode-shapes ηn obtained by solvingEquation (3.2a) using these models. While the φn will not be shown, theseare obtained from ηn through Equation (3.1a); that is φn = −iωngH∇ηnwhere the imaginary number i =√−1 simply indicates that h and q occur90◦ out of phase in time. Thus the mode-shape deflections have correspondinghorizontal velocities.Numerical SolverIn a two-dimensional domain, Equation (3.2a) can be solved numericallyusing the Finite Element Method (FEM; see Section 1.2.3). In FEM, the433.2. Methodsmodal frequencies and shapes are given by solving for the eigenvalues andeigenvectors of the discretized problem:Kη˜ = ΛM.The matrices K and M that result from from discretization are large andsparse, so the matrix M−1K is large and dense. While the exact eigenvaluesand eigenvectors can be computed for such matrices, the size of these matricesmay lead to computer memory constraints that limit the ability to solvefor them exactly. In this case, approximation methods such as the Lanczosmethod are employed to estimate eigenvalues (Schwab, 1980). The accuracyof this approximation is determined by the number of iterations (and thereforealso has a computational cost).The numerical scheme employed by BGS used a regular rectangulargrid, and considered grid spacings of both 250 m and 500 m. The domainwas defined by 2399 active elements in the case of the 250 m grid spacing,and 596 active elements in the case of 500 m grid spacing. The study alsoconsidered different values for the “truncation number”, NF, used in theLanczos procedure. The results show that the modal periods are sensitive toboth grid size and NF (see Table 10 in their paper), with the fundamentalmode seeing a 2.8 minute change from the lowest predicted result (38.9 min)to the highest (41.7 min). Given the increase in computational power nowavailable since BGS studied barotropic seiching on Lake Como, we repeatthe numerical experiments in an attempt to increase the accuracy of theresults and reach numerical stability.As described in Section 1.2.3, the method employed in the present studyuses an unstructured grid made up of triangular elements. These elementsrange in size, with small side-lengths and close spacing where the surfacegeometry changes more rapidly, and larger side-lengths farther from theboundaries. The FEM method is used for predicting the barotropic periodsof both Lake Como and Quesnel Lake. In each lake, the FEM was run ontwo different mesh densities (“low-density” denoted LDM, and “high-density”denoted HDM) to demonstrate stability of the results. Details of thesemeshes can be found in Table 3.1. Eigenvalues of the resulting mass andstiffness matrices are determined using the inbuilt Matlab function eigs,which approximates eigenvalues using Arnoldi iteration (a generalization ofthe Lanczos method).443.2. MethodsTable 3.1: Details of the two different mesh geometries (LDM and HDM) used in theFEM analysis for Lake Como and Quesnel Lake.Lake Como Quesnel LakeLow-Density Mesh (LDM)nodes 16051 26143triangular elements 28580 45744min. side-length 5.26 m 1.0 mmax. side-length 451.3 m 665.1 mmedian side-length 76.4 m 75.6 mHigh-Density Mesh (HDM)nodes 31796 43205triangular elements 56655 78150min. side-length 3.61 m 0.64 mmax. side-length 394.2 m 530.4 mmedian side-length 50.3 m 58.9 mSimplified Analytical ModelIn addition to the numerical model, we employ the simplified numericalmodel (SAM) developed in Chapter 2. The basis of this model is that eachof the arms of the lakes is considered to be a one-dimensional domain, withseiche motions occurring only along the longitudinal extent of that arm.This conversion transforms the the two-dimensonal vector φ(x, y) into aone-dimensional scalar quantity φi(xi) in each arm, with direction givenby the geometry of the arm; similarly η(x, y) → ηi(xi) (i = 1, 2, 3). Thus,the PDEs in Equations (3.2a) and (3.2b) are converted into a set of ODEscoupled at the junction of the three arms. This coupling is achieved byimposing the following boundary conditions:1. continuity of surface height: η1(L1) = η2(L2) = η3(L3); and2. conservation of mass:3∑i=1φi(Li) = 0.SAM’s ability to accurately predict the solutions to Equations (3.2a)and (3.2b) is limited by the simplification of the two-dimensional domainas coupled one-dimensional domains, and by the methods by which depthvariation is accounted for (discussed further in Chapter 2 and below). This453.3. Resultsmodel is not expected to predict the solutions with the same level of accuracyas the numerical solver (Section 1.2.3), but will instead provide additionalcontext for those results.The model predicts two different classes of behaviour dependent on therelative values of a parameter, τi, within each arm of the lake. This parameteris defined as τi = Li(gH0,i)−1/2, where Li is the along-thalweg length ofeach arm i and H0,i is some characteristic depth for that arm; physically,τi represents the time of travel of a progressive shallow-water wave in theith arm of a lake. The model predicts two different set of formulae fordetermining both the modal frequencies (ωn) and their spatial structure(ηi, φi) for the two conditions: τ1 = τ2 = . . . = τN (denoted TE) andτ1 6= τ2 6= . . . 6= τN (denoted TNE). In the TE case alternate modes havea multiplicity of eigenvalues and a response in which some arms can beeffectively decoupled from the remainder of the lake. In the TNE case allarms are active in each response. A mixed case (denoted TM) is also possiblewhen τi is equal only across some of the arms. In this case a subset ofthe modes can exhibit arm decoupling and need to be predicted using anadditional formula.In the present study we will consider two approximations of depth vari-ation within each arm of each of the lakes: a constant depth in each arm(SAM-CB), and a linearly varying depth in each arm (SAM-LB). In SAM-CB,the characteristic depth for each arm is given by the mean along-thalwegdepth of that arm:H0,i =1Li∫ Li0Hi(xi)dxiIn SAM-LB, the characteristic depth for each arm is given by a least-squaresfit of the equation HLB = (H0,iL−1i )xi to the depth variation Hi(xi). Basedon the values of H0,i determined in each of these methods, the values of τi aregiven in Table 3.2. As shown in the table, for Quesnel Lake, the behaviour isgiven by the TNE case and so no decoupled response is expected. However,in Lake Como the North Arm and the Lecco Branch (i = 2, 3) share thesame value of τi for both the SAM-CB and SAM-LB models, so we need toconsider the TM response.3.3 Results3.3.1 Lake ComoThe limnographs employed by BGS show oscillations with amplitudes of∼ 1−2 cm (see their Figure 5). We report the results of only the limnograph463.3. ResultsTable 3.2: Values of τi for each i arm for the SAM-CB and SAM-LB models for LakeComo and Quesnel Lake.Lake Como Quesnel Lakei Arm τi Arm τiSAM-CB 1 Como Branch 560 s West Arm 1203 s2 Lecco Branch 505 s North Arm 965 s3 North Arm 505 s East Arm 900 sSAM-LB 1 Como Branch 474 s West Arm 847 s2 Lecco Branch 373 s North Arm 688 s3 North Arm 373 s East Arm 716 sthat measured surface displacement on Feb. 17, 1996, which had both themost energetic and also the most distinct spectral peaks. The periodsmeasured on the other dates are tabulated in their work and show a highlevel of agreement.The FEM analysis was applied to Lake Como using both the LDM andHDM meshes (see Table 3.1). The modal periods predicted by the twomeshes differed by ≤ 0.1 minutes so these results are views as numericallystable,and we will report only the results of the FEM applied with the HDM.Table 3.3 compares the modal periods predicted using the FEM and SAM tothose observed and predicted by BGS.As seen in Table 3.3, the numerical predictions given by BGS are greaterthan the modal periods predicted by the FEM analysis performed here, withthe most pronounced effect in the fundamental mode. Compared to theobserved periods for modes greater than two, the present FEM analysisperforms better than BGS’s results. It is worth noting that despite theapparent improvement in results given by the present model, the analysisof BGS is still a reasonable measure of the observed periods. The first andsecond modes can not be spectrally distinguished in the observational datadue to the width of the spectral peak centred at ∼35 minutes. BGS predicta mode-2 period of 35.3 minutes so they conclude that this peak simplyobfuscates the fundamental mode. Considering the results of the presentFEM, we believe that the peak cannot be attributed to either one of thefirst two modes more strongly than the other and represents a combinationof the energy of both modes. Re-analysis of the water level time seriesusing different spectral techniques (such as different choices in windowingor pre-whitening) may help improve the separation between these signals,473.3. Resultsbut given the similarity between the two periods (36.8 min and 32.1 min), itis unlikely that it would be possible to resolve them completely without amuch longer time series or higher frequency recording.Table 3.3: Modal periods, Tn, in minutes for Lake Como. The periods observed byBGS correspond to the measurement period on Feb. 17 and the period modelled by BGScorrespond to the 250m grid spacing with NF=2000. SAM-CB and SAM-LB periodsmarked with an asterisks (n = 2, 5) correspond to periods estimated using the TM responseso these modes have some decoupled behaviour. The wide spectral peak in observed datamakes it impossible to separate the modal periods of the first and second mode; the n = 2value of 35.3 min was attributed by BGS to the second mode, but this may be inaccurate.Buzzi et al. (1997) FEM SAMn Observed Modelled HDM SAM-CB SAM-LB1 - 40.8 36.8 35.6 37.92 35.3 35.7 32.1 33.7∗ 32.5∗3 20.9 22.4 20.3 17.7 22.34 15.8 17.0 15.5 11.9 16.45 12.8 13.7 12.3 11.2∗ 14.2∗6 10.2 10.8 9.9 8.8 12.3Because the limnograph measured water elevation at only one locationon the lake, it is not possible to verify the mode-shapes predicted by themodels. Instead, the consistency between observed and FEM predictedperiods is taken as an indicator of the success of the model. This criterionis typically sufficient for studies of this nature (e.g. Carter and Lane, 1996;Hutter et al., 1982) and so the mode-shapes predicted by the FEM are takenas correct. Figure 3.2 shows the normalized surface deflections in Lake Comocorresponding to the first four modes as predicted by both the FEM andSAM (-CB and -LB) analyses.Notwithstanding the slight difference in predicted modal periods betweenthe present FEM model and the numerical model employed by BGS, there ishigh fidelity between the mode-shapes of the first two modes predicted witheach of these methods (compare Figure 3.2a, n = 1, 2 with Figures 3 and 4from BGS). However, whereas it is clear from the present FEM results thatthe second mode, T2, has two nodal lines in its response, BGS only labelsone of these as a nodal line in their results. The second nodal line is shownbut misrepresented as a surface displacement contour. Interestingly, there isalso good agreement between the present mode-shapes and the V1 baroclinicmodes presented by Guyennon et al. (2014).483.3.Resultsn=1n=2n=3n=4(a) FEMxxxxxxxxxxxxxxxx xxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxx xx xx xx xx xx xxxxxxxxxxxxxxxx(c) SAM-LB-1-0.500.51Normalized AmplitudeFigure 3.2: Mode-shapes in Lake Como predicted by (a) The FEM model, (b) SAM-CB, and (c) SAM-LB corresponding to thelabelled mode numbers, n. Deflections are normalized between -1 (blue) and 1 (red), and nodes are indicated by black contours or dots.The plain black lines in the western arm in (b) and (c) for n = 2 represents to the decoupling of that arm for the second mode.493.3. ResultsAs seen in Table 3.3, in Lake Como both SAM-CB and SAM-LB producereasonable values of the modal periods when compared to the FEM orobserved results. For SAM-CB, these results become increasingly inaccuratefor higher modes, whereas SAM-LB is able to retain some level of accuracy inthat range. Because higher modes are more sensitive to depth variation, it isnot expected that SAM-CB would be accurate for those modes. Despite theerror in predicted periods, both SAM-CB and SAM-LB produce qualitativelyreasonable mode-shapes (see Figure 3.2). For modes 1, 3, and 4, the correctnumber of nodal lines, and their approximate locations are well represented.The mode-2 response in Lake Como corresponds to the decoupled modepredicted by the TM response of SAM. SAM predicts that this mode is aco-oscillation of the North Arm with the Lecco Branch, with a single nodeat the confluence point and zero deflection in the Como Branch. As seenin the FEM results, this prediction is a reasonable representation of theresponse. The FEM predicted mode-shape has two nodal lines, both verynear each other and in the vicinity of the confluence point. The maximumdeflection (normalized to range from -1 to 1) between these nodes is ∼+0.05before entering the eastern arm. The deflection at the tip of the Como Armis ∼-0.06. Due to the very small deflections in this arm, Guyennon et al.(2014) and BGS both describe the second horizontal mode as exhibiting adecoupled response of the Como Branch. The mode-shapes predicted byGuyennon et al. (2014) show that the fifth mode also shows decoupling ofthe Como Branch. In SAM, an additional decoupled response also occurs forthe fifth mode (see Table 3.3).Despite the success of SAM in predicting the decoupled response in theComo Arm, the actual choice of the location of the confluence point (and thusthe values of τi) is subjective. An alternate choice of confluence point in LakeComo results in a complete TNE case, where the values of τi in each armare similar but unequal. This indicates that the criterion that determineswhether or not decoupling is expected to occur is sensitive to choices made bythe individual researchers. Fortunately, the results predicted by SAM werenot overly sensitive to this choice. When this analysis was repeated witha different confluence point, the predicted modal periods did not shift byany greater than 1 minute and the mode-shapes were not strongly affected.While the Como Branch no longer experienced a zero-deflection, it did havevery small values for deflection (similar to the FEM model). This indicatesa robustness of SAM to such subjective choices.503.3. Results3.3.2 Quesnel LakePressure signals from all stations across Quesnel Lake show consistent high-frequency, low amplitude (less than approximately 0.05 dbar) oscillations forthe entire period of record. Larger, transient surface displacements (pressurevariations up to approximately 0.15 dbar that last from 12-24 hours) occurredoccasionally throughout the record. Both types of oscillatory signals areattributed to surface seiching; however, the transient oscillations are likelytriggered by wind storms and are more consistent with the typical schematicrepresentation of seiching. Spectral analysis of these pressure and water leveldata (Figure 3.3) reveal a number of distinct peaks, which are tabulated inTable 3.4.Spectral analysis was also performed on the barometric pressure signal(not shown). This analysis showed peaks at the diurnal and semi-diurnalfrequencies with a steady decay towards the Nyquist frequency. A smallpeak at the third harmonic of the diurnal signal (T = 8 hr) appears but isnot statistically significant. No energetic peaks exist within the frequencyrange corresponding to barotropic motions in the lake.As with Lake Como, the difference in the periods predicted by the FEManalysis using either the HDM or LDM grids were ≤ 0.1 minutes and so onlythe HDM results are presented. The FEM predicted periods correspond tothe peaks in spectral response; however, in all cases the solver overestimatesthe periods compared to the observations (see Table 3.4). This may be due,in part, to a more limited spatial coverage of bathymetric data available forQuesnel Lake compared to Lake Como, particularly in the deeper sections ofthe lake (such as the East Arm). Despite these differences, the FEM appearssuccessful in its ability to identify the observed periods.Table 3.4: Modal periods, Tn, in minutes for Quesnel Lake as measured from observationaldata, and model results.Observed FEM SAMn HDM SAM-CB SAM-LB1 75.3 79.0 75.1 70.92 61.1 62.4 63.2 60.63 46.0 47.8 33.3 40.64 33.5 35.4 24.8 30.8There are a number of significant peaks in Figure 3.3 at all mooringstations at higher frequencies (30-minute periods and below). The periods of513.3. Resultsthese modes are less than the 30-minute period corresponding to the Nyquistfrequency of the barometric pressure signal. While there is no reason to expectthat the atmospheric pressure would have any oscillatory structure withfrequencies in this range, it is not possible to separate hydrostatic pressurefrom atmospheric pressure for these frequencies. Furthermore, higher modesare much more sensitive to bathymetic variation and so prediction of thesemodes can be less accurate; note that for the modes presented in Table 3.4 theFEM model has errors of ∼ 2− 4 min, and the difference between periods ofthese higher modes can be < 1 min. For these reasons, the present discussionwill focus only the lowest frequency modes.The η-eigenmodes predicted by the FEM are shown in Figure 3.4, wherethey are compared to those predicted using both SAM-CB and SAM-LB.The spatial coverage of mooring data in Quesnel Lake is insufficient todeduce the spatial structure of the mode-shapes from observational dataalone. However, the number of moorings and their distribution does permitsome validation of the predicted mode-shapes. Table 3.5 compares the energyassociated with the spectral peaks at the different station locations andthe associated energy at the same locations from the numerical predictions.These results show a good level of agreement between observed and modelledmode-shapes.Table 3.5: The energy associated with each n mode, relative to the energy measured atstation M1. The observed values are taken from the spectral peaks shown in Figure 3.3.The predicted values are taken from the displacements given by the FEM analysis shownin Figure 3.4; energy is proportional to displacement squared. No observed value is givenfor M5 for n = 2 because there is no evident spectral peak to measureObserved FEMn M1 M3 M5 M10 M1 M3 M5 M101 1.00 0.73 0.09 0.02 1.00 0.71 0.06 0.012 1.00 0.63 - 3.36 1.00 0.56 0.01 3.073 1.00 0.37 0.31 0.01 1.00 0.32 0.36 0.004 1.00 0.11 0.86 0.57 1.00 0.02 0.86 0.60In Quesnel Lake, SAM-CB provides a good estimate of the first twoperiods of oscillation, but is unable to accurately predict the periods ofhigher modes (Table 3.4). While SAM-LB does improve on the accuracy ofthe results for modes 3 and 4, the increase is only marginal and is offset bya decrease in accuracy in modes 1 and 2. Along-thalweg depth variation inQuesnel Lake is poorly described by a linear model, particularly in the East523.3. Results100101102Period[min]10010210410610810101012Spectral Energy [dbar2]M1M3M5M10Figure 3.3: Spectral energy computed for pressure recorded at the moorings. For clarity,the lowest signal (M1) is shown with true values, while subsequent signals are each shiftedvertically by a factor of 103. Significant energy is also contained in periods longer than 500minutes but no peaks of significance or interest are contained in that range; the choiceof axis limits is made to highlight the spectral peaks associated with barotropic modes.Vertical grey bands highlight the peaks tabulated in Table 3.4 and discussed through thetext. 95% confidence bounds are shown by dashed lines along the bottom of the panel.533.3.Resultsn=1n=2n=3n=4(a) FEMxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx(b) SAM-CBxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx(c) SAM-LB-1-0.500.51Normalized AmplitudeFigure 3.4: Mode-shapes in Quesnel Lake predicted by (a) The FEM model, (b) SAM-CB, and (c) SAM-LB corresponding to thelabelled mode numbers, n. Deflections are normalized between -1 (blue) and 1 (red), and nodes are indicated by black contours or dots.543.3. ResultsArm.As with Lake Como, SAM does produce qualitatively accurate mode-shapes with both SAM-CB and SAM-LB depth variations in Quesnel Lake(Figure 3.4), though these are less accurate within Quesnel Lake than theywere in Lake Como. Whereas in Lake Como, there was very little differencebetween the positions of the nodes between the FEM, SAM-CB, and SAM-LB models, in Quesnel Lake the node locations do experience some travelbetween models, particularly in the second mode. The FEM model predictsthat the second mode has one node that is located on the sill near CaribooIsland, and a second node near the entrance of the East Arm. In bothSAM-CB and SAM-LB, the Cariboo Island node has moved eastward and islocated midway along the West Arm; in SAM-CB the movement of the nodenear the entrance to the East Arm has caused it to shift around the cornerinto the entrance of the North Arm, thus incorrectly predicting the phaserelationship between the terminus of the North and West Arms.The differences between observed and predicted modal periods and modeshapes suggest that the modelled response of Quesnel Lake is more sensitive todepth variation than Lake Como. One possible explanation for this increasedsensitivity is the influence of the sill and constriction at Cariboo Island.Partial wave reflections at this location may act to modify the standing wavemodes of the lake. The present study does not have a mechanism to directlyinvestigate the impact of a sill on the barotropic response, but it is notedthat neither of the instances of SAM (-CB or -LB) include the influenceof the sill, whereas the FEM does. Additionally, the second mode, whereSAM and FEM mode-shape results have the greatest difference, the nodalline predicted by the FEM is directly over the sill; it is unclear if this is acoincidence or not.3.3.3 Accuracy of the simplified analytical modelDespite the fact that predictions are less accurate in Quesnel Lake than theyare in Lake Como, overall the mode-shapes predicted by SAM showed goodagreement with the FEM model in both lakes. In most cases, SAM accuratelypredicted the phase relation between arms, and the number and approximatelocation of all nodal lines. The primary difference between mode-shapespredicted by SAM-CB versus SAM-LB is the amplitude of the deflectionsaway from the terminus of the arms. In both lakes, SAM-LB predictedlower deflections away from the ends of the arms which more accuratelycaptured the variation in deflection amplitude exhibited by the FEM predictedmode-shapes when compared to SAM-CB model. This behaviour can be553.4. Discussionattributed to the spatially decreasing amplitude of oscillations in the functionsη when the lake depth H(x) is increasing away from the ends of the arms(see Section 2.3.2 in Chapter 2 for more details). Note in particular thedifferences between the results of SAM-CB and SAM-LB in the deflection atthe anti-nodes in the third and fourth modes in both lakes. The fact thatSAM reproduces this mode localization is taken as another indicator of itssuccess.As expected, SAM was also able to predict the periods of the lowestmodes, which are relatively insensitive to depth. The inability of SAM-CBto predict the periods of higher modes is a reflection of the simplificationsmade to account for depth variation. It is expected that a model that is ableto include depth variation would more accurately predict mode-shapes andperiods of higher modes; however, energy is typically contained in the lowestmodes and so accurate representation of higher modes may not be a majorconcern in most lakes.3.4 Discussion3.4.1 General behaviours of multi-armed lakesThere are a number of strong similarities between the barotropic mode-shapespredicted for both Lake Como and Quesnel Lake. From theses similarities,some general statements are made about the response of fjord-type multi-armed lakes.In both lakes, the first mode is governed by high deflection and a singlenode in the arm with the largest value of τi (the i = 1 arm; coincidentally, thewestern arm of each lake). Out-of-phase deflections are distributed betweenthe remaining two arms of the lake. In Quesnel Lake, the value of τ1 in theWest Arm is greatest because that arm is relatively shallow compared to theother two; the much longer extents of the East and North Arms (i = 2, 3)results in low deflections in those arms in order to conserve mass. In LakeComo, where all of the arms have similar lengths, the negative deflectionin each of the North and East Arms (i = 2, 3) accounts for roughly half ofthe positive deflection in the Como Branch (i = 1). The horizontal anglesformed by the arms as they extend from central junction slightly obfuscatesthe fact that in both of these lakes the fundamental behaviour is roughly aneast-west rocking of the surface.Section 2.3.1 suggests that the period of the fundamental mode can beestimated with Merian’s formula applied along the longest extent of the lake.This implies that the fundamental mode should be largely described as a563.4. Discussionsimple oscillation of the two arms that form that extent, though other armswill still be included in the response. The nodal line for the first mode isexpected to occur in the arms with the highest value of τi which is consistentwith the results in both Lake Como and Quesnel Lake. This mode is can bethought of as a linear co-oscillation of that arm out-of-phase with the rest ofthe lake.A similar response is seen in the first horizontal mode for other multi-armed and complex lakes. In a three-layer model, both the V1H1 andV2H1 baroclinic responses of Nechako Reservoir are characterized by linearoscillations in which Natalkuz Lake is out-of-phase from Knewstubb Lake(an overall east-west oscillation). In Lake Onega the first mode is marked bya high deflection in Povenetskiy Bay with a nodal line in that arm. All otherarms and bays of the lake oscillate together with the main body of the lake(Rudnev et al., 1995). The oscillation appears to be a simple north-southrocking of the entire lake. Lake Te Anau (Carter and Lane, 1996) and ClearLake (Rueda and Schladow, 2002) have first modes that are also describedas simple linear oscillations despite the complex shape of those lakes.In Lake Como, the second mode corresponds to the mixed TM typeresponse predicted by SAM, in which only the North Arm and the LeccoBranch are active. Because Quesnel Lake is not expected to have a decoupledresponse, we don’t necessarily expect strong agreement between the twolakes for this mode. Indeed, in Quesnel Lake the second mode has non-zerodeflections across all arms of the lake; however, the overall response is stilllargely characterized as an oscillation of only two arms. For both LakeComo and Quesnel Lake, the two arms that characterize the second modeare i = 2, 3; that is, the two arms most active in the mode-2 response arethe ones that were not dominant in the mode-1 response. It is further notedthat in both lakes, the two nodal lines appearing in the second mode do notoccur within the same arm.The third mode in both lakes is a radial mode with an anti-node nearthe geometric centre of the lake. For a lake with a geometry that conformsto the TE case, SAM predicts that the first whole-lake mode will have ananti-node at the confluence point. Neither of these lakes is described by theTE case, and the anti-node is not directly at the confluence point. Giventhat within each of the two lakes, all of the arm lengths are of the same orderof magnitude, it is unsurprising that a response with a central anti-nodemay appear as the third mode. Nechako Reservoir, on the other hand, doesnot seem to have any radial mode of the same structure (Imam, 2012; Imamet al., 2017). In Nechako Reservoir, sidearms are relatively short comparedto the total east-west extent of the lake so the mode-2 radial-like response is573.4. Discussionsimply analogous to the mode-2 response in simple elongated lakes.In the fourth mode (and higher modes not shown), deflections becomemore complex and harder to describe. In both Lake Como and Quesnel Lake,the fourth mode has four nodal lines, and in both cases the distribution ofthese nodes is consistent: two nodes exist in the i = 1 arm (with the highestvalue of τi), and one node exists in each of the other two arms. An additionalfeature of these higher modes is that deflections are increasingly localizedto the tips of the arms. This localization is actually expected, and can beexplained in terms of the analytical model (see Section 2.3.2 in Chapter 2).In simple elongated lakes, there is an expected one-to-one correspondencebetween mode number and the number of nodal lines. For the lakes describedabove, this is true for the first mode which contains a single nodal line in eachcase. For some complex geometries, this one-to-one correspondence is nottrue for higher modes. For example, Rudnev et al. (1995) labels two nodallines in both the second and third modes of Lake Onega, and only threenodal lines exist in the fourth mode. However, the geometry of Lake Onegashould not be classified strictly as a multi-armed lake due the the ambiguitybetween “arms” versus “bays”, so the modes predicted may represent theinfluence other geometric features. In Nechako Reservoir, which conforms tothe classification of fjord-type multi-armed geometry, the number of nodallines observed in each baroclinic mode matches the horizontal mode numberfor the majority of modes shown (the only exception being the upper interfacein the V1H5 response) (Imam, 2012; Imam et al., 2017) . In the presentstudy, the one-to-one correspondence also holds.The results suggest predictability of the locations of these nodes in lakesof this geometric class. As described, we observe that the first mode is ageneral Merian-type oscillation with a single node located in the the armwith the highest value of τi. The Nth-mode response of an N -armed lakewill be a radial mode with a single modal line in every arm. The results herefurther suggest that the modes numbered 1 ≤ n ≤ N seem to have 0 − 1nodes per arm, and modes numbered N ≤ n ≤ 2N will have 1− 2 nodes perarm, etc. These patterns likely arise because the values of τi are unequal butof the same order of magnitude for all arms. It is likely that in a case whereone arm is disproportionately long or small, these descriptions would notbe valid; in those cases, the longest extent of the lake may behave similarlyto a simple elongated lake with limited modification by the smaller arm(s).Furthermore, in a fully decoupled case (the TE case), the placement andnumber of nodes will not follow this pattern; however, in that case it will bereasonably easy to predict where these nodes will occur. For high enoughmode numbers it is expected that transverse modes and harbour-constrained583.4. Discussionmodes may erode these patterns just as they would in simple geometries.3.4.2 The activation of higher modesIn both the FEM and SAM results, the distribution of barotropic seicheamplitude within the lake has considerable spatial variation. While thehighest deflections are always present at one extremity of the lake for eachof the modes, the distribution of out-of-phase deflections between arms ofa lake may result in relatively low deflections in certain arms, even if adecoupled response is not present. For example, in the first mode of LakeComo the FEM predicts a maximal normalized deflection of +1 at the tip ofthe Como Branch, but the deflections at the tips of the Lecco Branch and theNorth Arm are only -0.39 and -0.52, respectively. Similar results are shownin Quesnel Lake. This energy distribution creates interesting questions interms of which of the modes will be active in the lake response. In simplelakes, the dominant response is typically consistent with the fundamentalmode for both barotropic and baroclinic modes (Mortimer, 1952). Whilethis result has been attributed to resonance between temporal fluctuationsof the wind forcing and the V1H1 period (e.g. Hutter et al., 1983) or theminimal damping of the V1H1 mode (e.g. Mortimer, 1952), Imam et al.(2017) suggests that instead, it is the spatial uniformity of the wind fieldthat results in that mode being activate. In general, it is expected that theenergy imparted to a given mode is related to the similarity of that modeto the spatial structure of the forcing mechanism (Guyennon et al., 2014;Shimizu et al., 2007). As a result, for multi-armed lakes in which the localtopography may result in non-uniform wind fields, higher modes may becomethe dominant response.An extension of this result is that wind storms localized to a single arm ofa multi-arm lake have the potential to generate currents and displacementsin other arms. The linearized problem is an initial-condition eigenvalueproblem, so any initial wind set-up, F0(x, y), will be constructed by a linearcombination of eigenmodes:F0(x, y) =∞∑n=1Anηn(x, y).When forcing subsides and the surface is allowed to relax, these modes willoscillate with their distinct frequencies ωn. In the TNE case, all modesηn(x, y) have deflections across the entire domain; at t = 0 these deflectionsmay add to zero in some sub-domain, but because the modes all oscillate593.5. Conclusionswith different frequencies, at some time t > 0 the modes will separate and soh(x, y, t) 6= 0 everywhere.While not a general feature of multi-armed lakes, we speculate that theconstriction near Cariboo Island may also contribute to the activation ofhigher modes in Quesnel Lake. In a study of Lake Winnipeg, Einarsson andLowe (1968) discuss the impact of constrictions in that lake on barotropicset-up. Those authors suggest that constrictions in Lake Winnipeg induce atime-lag on the steady-state set up of the lake. The transient set-up wouldinclude a state in which each individual basin of the lake undergoes itsown set up (see Figure 3.5). Imam et al. (2017) notes similar local tilts ofthe metalimnion of separated basins in Nechako Reservoir. The processesdescribed by both Einarsson and Lowe (1968) and Imam et al. (2017) havethe potential to lead to an initial set-up condition in Quesnel Lake thatmatches Figure 3.5. Relaxation of a set-up that consists of basin-specific tiltsof the water surface is likely to impart energy to higher modes in the samemanner that those modes are activated when their mode-shapes reflect thespatial variation of forcing. This may account for the high energy is modeswith periods ≤ 30 minutes observed in Figure 3.3.West BasinMain lake bodySillFigure 3.5: Schematic representation of a possible transient set-up of Quesnel Lake. Thesolid black line represents the undisturbed water surface, while the dashed line shows theinitial stage of set-up.3.5 ConclusionsBy comparing the spatial structure of the mode-shapes of both QuesnelLake and Lake Como, we’ve been able to infer some general patterns in thestructure of the modes in Y-shaped lakes, or multi-armed lakes with anynumber of arms. The results justify the use of a simplified analytical model(SAM). As seen here, the first fundamental mode in these lakes does notdiffer substantially from simple elongated lakes and acts as a simple linearoscillation between the arm with the highest value of τi and the remainderof the lake. However, due to the varied directions at which arms radiate603.5. Conclusionsfrom the confluence point, it may be difficult to recognize this mode as asimple co-oscillation without considering the results of SAM. The locationsof nodes and corresponding spatial distribution in surface deflection in highermodes may not be easy to intuit, but the SAM provides a relatively easymethod by which to predict these mode-shapes. The results show that theone-to-one correspondence between mode number and the number of nodallines observed in elongated lakes appears to also be a feature of multi-armedlakes of the TNE and TM classes, though the inclusion of harbour-typemodes and transverse modes will degrade that pattern at high enough modenumbers. Numerical modelling shows the localization of deflections to theterminus of arms at higher mode numbers for both lakes. This behaviour isexplained analytically by SAM, and is expected to be a general feature ofboth simple and multi-armed lakes.One important result of SAM is the possibility of a decoupled response inone or more arms of the lake. Specifically, the relative values of a parameterτi = Li(gH0,i)−1/2, which is the travel time of a shallow water wave in theith arm, determines if such a behaviour will occur. If τ1 6= τ2 6= τ3 (calledthe TNE case), then no decoupling will occur, and all modes will act aswhole-lake modes. Alternatively, if τi = τj for a given subset of arms of thelake (called the TM case) then there will be some modes that act only acrossthe arms i and j with a node at the confluence point, and all other armswill be decoupled from the response. When all arms have equal values ofτi (τ1 = τ2 = τ3, called the TE case), multiple sets of decoupled modes willoccur. In Lake Como, the wave-travel-times in the North Arm and LeccoBranch are equal, and the mode-2 response is an oscillation of only thesetwo arms with the Como Branch absent from the response. No decouplingis observed in Quesnel Lake which corresponds to a TNE case. BecauseSAM provides a geometric criterion to predict the possibility of decoupledarms, such predictions are possible for a given lake before a more detailednumerical analysis is completed.Even in lakes where decoupling or arms is not expected to occur, thisstudy shows that seiche modes in multi-armed lakes do exhibit considerablespatial variation in modal amplitudes (and corresponding velocities). Thisspatial heterogeneity of mode-shapes of both barotropic and baroclinic modeshas implications for water quality within multi-armed lakes due to the roleof horizontal velocity in both resuspension and transport. Due to their shortperiods, horizontal velocities generated by the barotropic response can becomparable in magnitude to those caused by baroclinic seiches (Lemmin andMortimer, 1986), and both have the capacity to resuspend bottom sedimentsin lakes (Bloesch, 1995; Chung et al., 2009; Gloor et al., 1994). These bottom613.5. Conclusionssediments are able to act as stores of nutrients or of harmful toxins (Chunget al., 2009), so resuspension is an important for both water quality andecology. The spatial variation of seiche-induced horizontal velocities in multi-armed lakes (with decoupling of arms being an extreme example of spatialvariation) can then lead to localized areas of resuspension. Baroclinic seichingis responsible for the transport of material through a lake (Hodges et al.,2000; Mortimer, 1952), so if there is a localized nutrient or pollution loadinginto a lake (i.e. from a river inflow or due to localized areas of resuspension),the spatial variation in internal currents and the decoupling of arms willdetermine whether that loading has the ability to impact the entire lake oronly some subsection. In an extreme case, high loadings into an arm that isdecoupled from the rest of the lake in the seiche response can limit mixingor flushing in that arm, and create localized pollution or eutrophicationconcerns.62Chapter 4Conclusions4.1 Summary and contributionsThis study examined the free oscillatory response of lakes with multiplenarrow, elongated arms. This was achieved through the use of an analyticaldescription in an idealized case (Chapter 2), and a comparison of case studiesconducted to two Y-shaped lakes (Chapter 3). The results of the study allowfor some general statements to be made about this response.Chapter 2 presents the development a simplified analytical model (SAM)for idealized, multi-armed lakes. This model considered bathymetric variationby describing the results in the cases of a constant bottom and a linearlyvarying bottom, and presenting the results of an asymptotic approximationfor an arbitrary bottom. It was found that regardless of the specific form ofdepth variation being considered, the relative values of a single parameter τi,which represents the travel time of a progressive shallow-water wave alongthe ith arm of the lake, can be used to classify the lake into two behaviouralregimes: lakes with decoupled modes, and lakes with only whole-lake modes.While these idealized results were valuable, it was also necessary to exam-ine realistic bathymetries. Chapter 3 compared both field results, numericallymodelled results, and SAM results of two Y-shaped lakes: Quesnel Lake andLake Como. The results provide justification of the ability of the SAM toboth predict and explain the seiche modes of these lakes.Of particular interest in this study is the explanation in Chapter 2 of thepotential decoupled response in multi-armed lakes. While this behaviour waspreviously observed in Lake Como (Buzzi et al., 1997; Guyennon et al., 2014),it was absent from Knewstubb and Natalkuz Lakes (Imam, 2012), leadingto questions about when such a behaviour might be expected to occur ingeneral, or whether Lake Como presented an a-typical case. Not only doesthe SAM provide an explanation for why this decoupling happens, but alsogives a criteria for predicting whether it will occur in a given lake. If thewave travel time, τi, is equal in two arms of a multi-armed lake, the resultingstanding wave mode produced by those two arms will have a node at theconfluence point of all of the arms. With zero-deflection at the junction,634.2. Future workall other arms will be absent (decoupled) in the response for that mode.When applied to Lake Como in Chapter 3, this prediction is consistent withobserved behaviour; conversely, consistent with the SAM predictions, QuesnelLake is not expected to exhbit any decoupled behaviour.The results of this study also provide some basis for predicting the mode-shape of the fundamental mode. As demonstrated by Malinina and Solntseva(1972), Laval et al. (2008) and Imam et al. (2013b), the intuition of someresearchers in trying to predict the fundamental period of a multi-armedlake is to apply Merian’s formula along the longest continuous extent of thelake. The results of the SAM suggest that even for lakes in which none ofthe arms are decoupled, applying Merian’s formula as suggested may providea reasonable “first guess” of the fundamental seiche period. This first modewill then conform to a general “linear” oscillation in which the arm with thehighest value of τi exhibits motion out-of-phase from the remaining armsof the lake in a back-and-forth rocking analogous to simple elongated lakes.The studies of Lakes Como and Quesnel in Chapter 3 confirm that the firstmode is well described by this general linear oscillation, with a single nodalline in the arm that has the highest value of τi.An additional interesting result of this study is the observations inChapter 3 of the localization of deflections for higher modes. In this geometrywhere motions are primarily longitudinal, this localization resulted in highdeflections near the terminus of different arms, with a relatively inert lakebody. The SAM provides an analytical explanation for this in terms ofthe structure of mode-shapes attributed to shoalling bathymetries versusflat-bottomed. An extension of the SAM to include width variation (seeSection 2.3.2) would cause an exaggeration of these results. In more complextwo-dimensional domains, this mode localization may be related to the bayand harbour modes observed in locations such as Lake Onega (Rudnev et al.,1995), or Flathead Lake (Kirillin et al., 2014).4.2 Future work4.2.1 Additional study of Quesnel Lake and Lake ComoBarotropic modesBuzzi et al. (1997) provides a detailed study of barotropic modes withinLake Como. The modelling work performed by those authors is repeatedin Chapter 3 with a higher mesh density, and more context is provided toexplain the structure of the predicted modes. While this study appears644.2. Future workconsistent with the modes observed by Buzzi et al. (1997) during a fieldstudy, the width of the spectral band centred near 35-minutes in the observeddata obstructs the ability to differentiate between the first and second modes.Furthermore, because of the the lack of spatial coverage of field data, it is notpossible to verify the predicted mode-shapes. In order to fully validate themodel output, it would be valuable to perform an additional field study ofLake Como. Such a study would entail a spatial array of water level sensors(at minimum, one at the terminus of each arm); these sensors would need torecord at a high enough frequency and for a long enough interval to allowfor the separation of these modes in the spectral response.The present work provides a characterization of the barotropic modes inQuesnel Lake that is consistent with both the observed oscillatory periodsand the spatial distribution of spectral energy signals. Nonetheless, thereare still unanswered questions regarding the barotropic response of the lake.In particular, it would be interesting to better understand the role of theconstriction at Cariboo Island on seiche modes. Chapter 3 provides somecircumstantial evidence that the form of the free modes is modified by theinclusion of the constriction, possibly due to partial reflections. A more robustanalysis of this effect can be undertaken using the finite element method(FEM) numerical model used in Chapter 3. An artificial lake bathymetry canbe constructed in which Cariboo Island is removed and the sill is smoothedout, effectively removing the constriction; a comparison of the appliction ofthe FEM model to this constructed bathymetry with the results presentedin Chapter 3 would help explain the impact of the sill on the mode-shapesand periods of the free modes.As described in Chapter 3, the sill may also have an impact on theactivation of higher barotropic modes. Given that the observed spectralresponse of Quesnel Lake agrees with the periods predicted by the FEM model,this time lag may be negligible compared to the period of the barotropicmodes. Even if the time lag does not impact the periods of the barotropicmodes, the transient form of the set-up depicted in Figure 3.5 may occur inreaction to short-duration wind gusts. Some of the energetic peaks at higherfrequencies in the spectral response of Quesnel Lake (Figure 3.3) may beattributed to a reaction to such a partially set-up surface.Baroclinic modes in Quesnel LakeChapter 3 provides a characterization of barotropic modes in Quesnel Lake.While these are interesting in explaining the behaviour of multi-armed lakes,barotropic motions may not contribute to mass transport to the same degree654.2. Future workas baroclinic motions. Laval et al. (2008) found that baroclinic motions canhave a profound impact on the exchange of deep water between basins ofthe lake, but as of yet no characterization has been made of these internalmodes. While it would be of interest to apply the FEM and SAM modelsdiscussed in the present study to the baroclinic response of Quesnel Lakedirectly, it may first be necessary to further understand the role of the sill atCariboo Island in modifying baroclinic modes.The transient surface set-up in lakes that contain constrictions describedby Einarsson and Lowe (1968), and depicted schematically in Figure 3.5 maysimilarly apply to the baroclinic set-up. The so-called “split layer” observedin Lake Constance by Appt et al. (2004) is likely an extreme example ofsuch a set-up, in which non-linear effects cause the interface to surface in thevicinity of a sill. Imam (2012, Chapter 6) further discussed the possibility ofa split layer in Knewstubb and Natalkuz lakes, and argues that the conditionto describe it’s occurrence should be Wd < 1 < Wu, where Wu,d are theWedderburn numbers in the upwind (u) and downwind basins (d). In LakeConstance, the subsistence of the wind forcing during the split layer event wasassociated with the generation of an internal surge. It would be instructiveto consider that result in the context of the degeneration regimes describedby Horn et al. (2001); it is possible that if the upwelling was less drastic, theset-up would have relaxed instead into a decoupled set of standing wavesconstrained to each of the two basins of the lake. If the existence of aconstriction can lead to the decoupling of baroclinic seiche modes betweenbasins, it would have a strong impact on the choices of geometry used inpredicting the free response modes.4.2.2 Forced response of multi-armed lakesIn basins of simple geometry, energy is primarily contained in the fundamentalhorizontal mode for both barotropic and baroclinic seiching (Mortimer, 1952).In more complex lakes, this may not be the case. It is believed that forlakes of complex geometry, the energy imparted to a particular barotropicmode will be based on the similarity of the corresponding mode-shape to thespatial distribution of the forcing pattern (Guyennon et al., 2014). Imam(2012) suggests that the activation of baroclinic modes will be related toresonance conditions between the wind timeseries and the free modes of thelake. His study, however, did not account for the full spatial variation of thewind field.Whether through a hydrodynamic model such as ELCOM (Hodges et al.,2000) for FVCOM (Chen et al., 2003) or a semi-numeric model such as the664.2. Future workimpulsively forced TVC developed by Imam (2012); Imam et al. (2013b), itwould be interesting to evaluate the the forced response of a multi-armed lakewith the specific goal of understanding the role of the different arms on theresponse and activation of different modes. Such a study would likely rely ona more complete picture of the spatial variability of the wind forcing. Lavalet al. (2008) suggests that the mountainous terrain surrounding QuesnelLake acts to channel the wind along the local thalweg of each arm. Aspart of the present study, this author did attempt to analyse data collectedat lakeshore meteorological stations at Quesnel Lake in an effort to betterdescribe the spatial and temporal variation of the wind forcing, and linkthose results to observed temporal variation in the spectral energy of differentbarotropic modes within the lake. While those results provided evidencethat agreed with the assertions of Laval et al. (2008), the lack of wind datathat was coincident with recordings of water level in the lake prevented arobust analysis. No definitive conclusions were made.Given that fjord-type lakes typically sit in deep valleys carved by thesame glaciers that formed the lakes themselves, the impact of topographyon local wind patterns is likely to be an underlying question in all force-response models of these lakes. Using an array of meteorological stations,Ludwig et al. (2004) provides one framework for evaluating these datausing Empirical Orthogonal Functions (EOFs), which may be of value indescribing topographic channelling. A more complete study of topographicwind channelling over a lake and the resulting effects on baroclinic circulationhas recently been completed on Lake Iseo (Valerio et al., 2017). Thoseauthors simulated the wind field using the Weather Research and Forecasting(WRF) atmospheric model, and their results indicate spatial variation of thewind field had an impact on the energy imparted to baroclinic modes.In multi-armed lakes, a more complete understanding of the spatialvariation of the wind (through modelling or field study) may reveal windforcing that is locally constrained or may be channelled in different directionsalong different arms of the lake. Such patterns could result in some modeother than the fundamental mode being the primary energetic mode of thelake.674.2. Future work4.2.3 The use of simplified analytical models in predictingseiche responseAppliction of the SAM to other lakesIt would be very interesting to apply the SAM to other multi-armed lakes.Two potential contenders for which the SAM may provide valuable insightare Shuswap Lake in British Columbia, and Keuaka Lake in New York.Shuswap Lake has four arms extending from a central junction. Aconstriction at the junction separates the lake into two connected longitudinalextents. Assuming that depth doesn’t vary drastically through the lake, thelengths of the arms can be used to infer whether τi would be expected to beequal (the TE case) or unequal (the TNE case), or a mixture of the two (theTM case). In fact, a cursory investigation suggests a TM case may occur.Given the constriction, and the potential for some arms decoupled from theresponse, it would be interesting to see if all of the possible free modes arerealized in the lake.Like Lake Como and Quesnel Lake, Keuka Lake is Y-shaped. The lakehas two shorter arms that extend continuously in a roughly north-southdirection, and a longer arm that branches out almost perpendicular to theother two before turning in a north-east direction. While the SAM does notconsider the relative angles of the arms as they extend from the junction, itis reasonable to expect that response of the lake may favour the activationof a mode along a continuous extent, compared to one that needs to turna corner. Based on the results presented in Chapter 3, one might expectthat the fundamental mode in Kueka Lake would be a co-oscillation of thenorth-east arm with the main body of the lake in an overall east-west rockingmotion. However, a field study may find that the second mode, which islikely a north-south oscillation of the two shorter arms with relatively lowdeflection in the north-east arm (possibly even a a fully decoupled north-eastarm), may be the dominant mode within the lake.Development of models for other geometric featuresThe present SAM considers the effects of multiple arms on the seiche responseof a lake. While this is of interest, there are other geometric features thatmay be of similar interest to other researchers. By considering the processused to develop the model described in Chapter 2, it may be possible toconstruct SAM’s for other geometries as well.Ultimately, the development of the SAM for multi-armed lakes is limitedin its application by the number of multi-armed lakes of interest or importance.684.2. 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