EXPERIMENTS WITH WITHDRAWAL-INDUCED RESONANT INTERNAL WAVESbyGARY ALLANB Sc, Simon Fraser University, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the required standard.THE UNIVERSITY OF BRITISH COLUMBIAOctober, 1993© GARY ALLAN, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^(c.;r-ef)The University of British ColumbiaVancouver, CanadaDate^-ct DE-6 (2/88)AbstractControl of natural and artificial reservoirs by means of outlet structures (specifically,dams) has been considered by Imberger (1980). Such control has been proposed onKootenay Lake in eastern British Columbia, as a means to modify the ecology of the lake.This study considers the feasibility of pumped control of a reservoir and the practicalproblems associated with application on the lake.The generation of internal waves in a rectangular tank by a pump with sinusoidallymodulated flow has been investigated via laboratory experiments. Theory appropriate tointernal waves in two layer systems has been presented with an extension to deal with theeffect of viscosity. The study shows that modulated pumping is a viable method ofgenerating internal waves experimentally and may have applications in controlledreservoirs on which outflow could be modulated.The period of resonant waves increases with interfacial thickness as does waveamplitude: observed periods ranged from 1.01 to 1.05 times greater than theoreticallydetermined periods for two-layer systems. Theoretical determination of the period isbased on consideration of the density structure and boundary conditions of the systemunder study; one would expect real conditions to produce some variation in the observedperiod versus the theoretical period. The result above is borne out by observation of theamplitude response curves generated by the data and by calculations of the beatfrequencies associated with the forcing and generated waves and agrees approximatelywith the data of Hyden (1974) and Thorpe (1968). There is some indication that as theinterface widens the amplitude response curves sharpen, implying that if one hoped togenerate a resonant wave in a system with a thick interface one would have to be closer tothe resonant period than in a system with a thinner interface to achieve a similar response.Calculation indicates that, for the peak response, the wave with maximum amplitude, 1.7to 4 % of the energy input per cycle is present in the wave. Energy input per cycle isdetermined by integrating instantaneous kinetic energy of the water that is pumped over acycle. Fourier transforms of the data show response, in some cases, in the 6th harmonicmodes of the systems, with a decline in energy from the 1st to 2nd mode of 20 to 25 %,typically. Estimates are presented for the damping coefficients for the systemsinvestigated: the average value is 0.0020 s-1.A brief review of problems associated with a practical application of resonant internalwaves forced by flow modulation is presented. Several technical problems exist whichneed further consideration before the application is undertaken, on Kootenay Lake ineastern British Columbia. It may be practicable to try generating internal waves on asmaller lake or reservoir controlled by a dam as a means of testing the feasibility.111Table of ContentsAbstract^ iiTable of Contents^ ivList of Tables vList of Figures viSymbols^ viiAcknowledgments^ ix1. Introduction 12. Literature Review^ 32.1^Internal Resonant Waves: Theory and Experiment^ 32.2^Seiches in Lakes 72.3^Kootenay Lake 83. Theory of Resonant Internal Waves^ 113.1^Problem Definition 113.2^Basic Solutions^ 123.3^Relaxation of Restrictions 144. Experimental Description, Materials , Methods, and Error Analysis^ 214.1^Materials^ 224.2^Methods 254.2.1^Preparation^ 254.2.2^Testing 264.3^Sources of Error 274.3.1^Calculation of the Resonant Periods ^ 274.3.2^Timing^ 284.3.3^Errors in the Voltage to Density Conversion^ 284.3.4^Error Estimates from the Data 295.^Results 385.1^Observations on the Raw Data^ 385.2^Density Variations^ 425.3^Resonance and Interfacial Thickness 435.4^Amplitude Response 465.5^Energy^ 475.6^Beating 485.7^Fourier Transforms^ 495.8^Free Oscillations 516^The Practicality of a Forced Internal Seiche on Kootenay Lake^ 547^Discussion and Conclusions^ 58References^ 62Figures 65ivList of Tables1 KC1 Standards: conductivities @ 25 C^ 242.1 Experimental Parameters and Results Experiment 1^ 312.2 Experimental Parameters and Results Experiment 2, day 1^ 322.3 Experimental Parameters and Results Experiment 2, day 2 332.4 Experimental Parameters and Results Experiment 2, day 5 342.5 Experimental Parameters and Results Experiment 3, day 1^ 352.6 Experimental Parameters and Results Experiment 3, day 3 362.7 Experimental Parameters and Results Experiment 3, day 5 373 Ratio of Half Width to Height of Resonance Curves of figure 21 ^ 464 Energy of the Peak Observed Waves^ 485 Amplitude Decay for Experiments 2 and 3 52VList of Figuresla Physical arrangement and parameters for two layer systems^ 65lb Physical arrangement and parameters for systems with constantBrunt-Vaisala frequency^ 65lc Physical arrangement and parameters for systems with constantdensity gradient between two homogeneous layers^ 652a From Thorpe (1968), experimental apparatus 662b From Thorpe (1968), experimental results^ 663 From Hyden (1974), experimental apparatus 674a From Ting (1992), experimental apparatus 684b From Ting (1992), experimental results 685 Experimental Apparatus^ 696 Sensitivity of the Theoretical Period^ 707 Interface Oscillation Data ... Experiment 1, runs 1-5^ 718 Interface Oscillation Data ... Experiment 1, runs 6-10 729 Interface Oscillation Data ... Experiment 1, runs 11-15 7310 Interface Oscillation Data ... Experiment 2, day 1, runs 1-7 ^ 7411 Interface Oscillation Data ... Experiment 2, day 1, runs 7-14 7512 Interface Oscillation Data ... Experiment 2, day 2, runs 1-10 ^ 7613 Interface Oscillation Data ... Experiment 2, day 2, runs 11-17 7714 Interface Oscillation Data ... Experiment 2, day 3, runs 1-10 ^ 7815 Interface Oscillation Data ... Experiment 2, day 3, runs 11-17 7916 Interface Oscillation Data ... Experiment 3, day 1, runs 1-7 8017 Interface Oscillation Data ... Experiment 3, day 2, runs 1-7 ^ 8118 Interface Oscillation Data ... Experiment 3, day 3, runs 1-7 8219 Density Profiles for All Experiments^ 8320 Variation of Resonant Period with Interfacial Thickness: Data^ 8421 Amplitude Response for All Experiments 8522 Spectra: Experiment 1^ 8623 Spectra: Experiment 2, day 1, runs 1-4^ 8724 Spectra: Experiment 2, day 2, runs 1-5 8725 Spectra: Experiment 2, day 5, runs 1-5 8826 Spectra: Experiment 2, day 1, runs 5-7 8827 Spectra: Experiment 2, day 2, runs 6-10^ 8928 Spectra: Experiment 2, day 5, runs 6-10 8929 Spectra: Experiment 2, day 1, runs 7-14 9030 Spectra: Experiment 2, day 2, runs 11-17 9031 Spectra: Experiment 2, day 5, runs 11-17^ 9132 Spectra: Experiment 3, day 1^ 9133 Spectra: Experiment 3, day 3 9234 Spectra: Experiment 3, day 5 9235 Free Oscillation Data: Experiment 2 9336 Free Oscillation Data: Experiment 3^ 9337 Spectra of Free Oscillations: Experiment 2 9438 Spectra of Free Oscillations: Experiment 3 9439 Kootenay Lake and South-Eastern British Columbia^ 95viSymbolsa, wave amplitudec,, internal wave velocity, = g hun(x,t), function describing internal wave shapeE =^Po Pof„, wave frequencyf c, the Coriolis frequencyFR' resonant frequency (table 2)g, acceleration due to gravity, reduced gravity, = egH, lake or tank depth, = h + hihu,k, upper and lower layer thicknesses27rk, the wave number = —AL, lake or tank lengthA, wave length of interfacial wavesn, wave mode numberN, Brunt — Vaisala frequency,^iflapp dzv, coefficient of viscositypu,pi upper and lower layer densitiespo, average densityAp, density difference, = p1 — pu01, lower layer velocity potential functioncpu, upper layer velocity potential functionT n , nth mode period = —2LncviiSymbols (cont.)T0' observed period (table 2)TR' resonant period (table 2)TT' theoretical period (table 2)con, angular frequency = 271f nviiiAcknowledgmentsThis work was funded by a grant from BC Hydro. The support of Dr. Greg Lawrenceand the technical staff in the environmental and fluid mechanics laboratories in CivilEngineering at UBC is much appreciated. I wish to thank family and friends for theirmoral support and patience, especially my mother and Annette Rohr, Zora Knezevic andTim Ma. Finally, I wish to thank Dr. Michael Isaacson and Dr. Peter Ward for kindlyreading and commenting on my thesis.ixli^IntroductionThis laboratory project was intended to investigate the physics of internal wavesincluding the effect of a finite thickness interface and viscous damping, and todemonstrate the efficacy of generating a seiche, a resonant internal wave, in a stratifiedtank by pumping. This research is motivated in part by the possibility of controllinginternal lake and reservoir dynamics by modulated withdrawals. Imberger (1980)suggested a 'pump-back scheme' to modify the thermal regime within a reservoir bycontrolled withdrawal and inflow, perhaps using a storage system. He suggested that thebasin modes could thus be made to resonate. It is the possibility of internal resonancedriven by pumping, as suggested by Imberger, that is explored in this report. Thepossibility of controlling the internal dynamics of a lake or reservoir is especiallyapplicable in British Columbia, with its heavy reliance on hydro-electric powergeneration, and numerous sites on which withdrawal might be controlled and the internalfluid dynamics effected.The original motivating factor for this project was the great decline in the Kokaneefishery on Kootenay Lake in eastern British Columbia; a brief literature review ofKootenay Lake limnology is presented in chapter 2. Also motivating this project hasbeen the desire to demonstrate the usefulness of pumping to generate resonant internalwaves, the interest in exploring the effect on the resonant period of interfacial layers offinite thickness, and the desire to calibrate and test an existing computer model of internalwave behaviour.Chapter 2 consists of a literature review on experiment and theory of internal waves,internal waves in lakes, and the limnology of Kootenay Lake. Resonant internal waves1are described mathematically in chapter 3. The experimental design, materials, methodsand error analysis are described in chapter 4. The results are presented and analyzed inchapter 5. Chapter 6 considers problems one might expect to encounter in an applicationof forced resonance on Kootenay Lake. Chapter 7 summarizes the work and presentsrecommendations for further study.2Literature Review2.1 Internal Resonant Waves: Theory and ExperimentStanding internal waves, also known as internal seiches, are commonly found in lakesand have been studied by a number of people, notably Wedderburn (1911), and Mortimer(1952). The theoretical study of standing waves dates back to a paper by Stokes in 1847in which first order linear solutions for both progressive surface and interfacial waveswere found. Lamb, in several editions, the latest in 1944, presented the theory of internalresonant waves on two layer systems; this development served as the model fortheoretical development in chapter 3. Lamb considers the effect of viscosity, v, onresonant frequency and decay of amplitude. Harrison (1908) extended Lamb's analysisof the effects of viscosity for systems with one and two infinite layers and shows that1internal waves decay approximately as —,---V vThorpe (1968) has made extensive studies, theoretical and experimental, of standinginternal waves at the interface of two fluids and in a continuously stratified fluid. Thorpe(1968) presented finite amplitude solutions as power series expansions with respect to thewave slope as an expansion parameter. The ratio of the coefficient of the second-orderterm to that of the first order term represented a measure of non-linear effects in theinternal waves. Thorpe (1968) showed that the presence of the upper fluid reduced theamplitude of the higher harmonics in the wave profile. Voisin (1991) considered thegeneration of Boussinesq and non-Boussinesq internal waves on stably stratified fluids interms of Greens functions; in Boussinesq approximations, variations in density areignored where they affect inertia, but retained in buoyancy terms, in the Navier-Stokesequation. Voisin defines stably stratified systems to be those of constant Brunt-Vaisala311frequency N = –1–Ed , where g, p, z are the acceleration due to gravity, density andp dzvertical dimension, respectively. Figures la, lb and lc, define the physical situations fortwo layer systems, for systems with constant Brunt-Vaisala frequency, and for systemswith constant density gradient between two homogeneous layers, respectively.Krauss (1966), Phillips (1966), Thorpe (1968) and Keller and Munk (1970) allconsidered the problem of a finite thickness interfacial layer, that is a gradual densitytransition. Krauss (p 34-36) looked at the hyperbolic tangent density profile(p = po exp --e-A tanh (2 d-1(–z – h1)) , where p, A), Ap, d, h1 are the density, average2p,density, density difference, interface thickness and depth of the lower layer, respectively.Phillips (p 211-214) considers a constant density gradient between two homogenousregions, figure lc. Thorpe did not specify the density regime he considered, but it mightbe assumed to be like the one examined by Phillips. Keller and Munk derived thedispersion relation for the exponential density profile: p = pi, ex{p .1L-2 (Zi - z) in theginterface where pie N, z1, and z are the density above the interface, the Brunt-Vaisalafrequency, the depth to the top of the interface and the depth respectively (where theupper surface of the system is taken as the zero of depth). Below the interfacial layer, of{thickness d, the density is p = p„ exp —N2 d . The dispersion relations obtained for thegvarious distributions are presented in chapter 3 and their predictions for wave periods arepresented in section 5.3.4Wedderburn (1911) reported observations of internal waves with periods approximately5% above the values predicted by theory. Wedderburn used a rocking table to inducewave motion.In 1968 Thorpe investigated the resonant response of two-layer systems. Hisexperimental apparatus consisted of a tank (30" x 14" x 8") fitted with a plunger at eachend. The plungers were operated in simple harmonic motion 1800 out of phase. Due todifficulties sealing the plungers, Thorpe chose to encase the fluid in polyethene bags,fitted to the tank (see figure 2a).As to experimental parameters, Thorpe included density differences in his results but didnot record the depths of the layers in his work. He stated only that the undisturbedinterface was usually arranged to lie level with the bottom of the plungers which were 8"from the bottom of the tank. Some of Thorpe's results are included as figure 2b. Thorpeworked with density differences of 4.7, 10.0, 15.0 and 22.8 kg/m3 in the second resonantmode. His data indicates that the peak amplitude of oscillation due to forcing increasedas the density difference increased. Also, the ratio of the half widths of the responsecurves, in figure lb, to their heights is constant with no tendency to vary with density, at0.64 ± 0.04 s-1. Assuming hi = 8" and k = 22" these results indicate that for secondmode, which is what Thorpe studied, the observed periods of peak response were1.13 ± 0.05 times greater than the theoretical periods. For a two layer system the exacttheoretical equation for the period isyv2^T 2L = 2L^(pi –P.)g nc n (k(pl coth(khd+pu coth(kh.)))where n is the mode number, in this case, 2. However, in this case, the approximation= 2L = 2L (g' —huh/1/2(2)^nc n^H(1)5was used rather than the more accurate equation 1 as Thorpe did not report the actualdensities of the layers, and the layer depths are only approximate. Theoreticaldetermination of the period is based on consideration of the density structure andboundary conditions of the system under study; these equations are derived, in generalform, in chapter 3. Thorpe's results will be compared to the results for this project insection 5.4.Thorpe (1968) also calculated an estimate for the rate of viscous damping q usingobservations of the time for internal waves to decay, assuming an exponential decayenvelope e-q`. For his system, he estimated q to be 0.01 s-1•Hyden (1974) conducted extensive experiments with the forced resonance of two layersystems. His apparatus, see figure 3, consisted of a 18.22 m tank, 1.2 m high and 0.6 mwide. The tank was divided into a 1.5 m pumping zone and 16.72 m flume part. In thepumping zone a vertically reciprocating sealed piston operating at 0.2 to 2 radians perminute with a displacement of 0.29 m3. supplied the energy for the internal waves drovethe internal waves.Hyden's results, from his table 1, are graphed along with comparable data from thisproject in figure 20 and discussed in section 3.3. Based on the parameters Hydenreported, his calculated theoretical periods are slightly low. Equation 1 which is exact fora system of two layer density structure, produces almost identical results to the above,values were obtained approximately 1% higher than those Hyden reports. Usingequation 2, which is often used to estimate the internal wave period on large scalesystems, similar values were calculated, about 1% above Hyden's numbers. Apparently6Hyden used his data in a slightly different manner compared to that used in this report, orhe made a systematic error.Ting (1992) investigated forced internal waves in a rectangular trench, induced bynormally incident surface waves. Figure 4 shows Ting's experimental design, taken fromhis figure 3. Ting's experiments were conducted in a wave tank 36.6 m long, by 0.61 mdeep by 0.394 m wide with glass walls. A horizontal false bottom of plywood was placedin the tank to create a rectangular trench 0.6 m wide and 0.152 m deep. The upstreamend of the trench was located 19.15 m from a bulkhead wave generator. The portion ofthe tank downstream of the trench was not used in the study and was sealed off from thetrench by a vertical wall.Ting developed a linear model to describe small-amplitude, simple harmonic wavemotions, assuming a linear density variation between two homogeneous, finite layers. Hefound the natural frequency of the system decreased as the density interface thicknessincreased and as the depth of the lower fluid decreased. His three-layer viscous theorymatches his experimental results well. Figure 3b, Ting's figure 11, shows part of Ting'sHresults demonstrating the match of theory to experiment. The ratio^is the internalH3wave amplification factor (the meanings of H3 and H4 individually are undefined in thepaper), while kl is the wave number of the surface wave times the channel width.Z,2^Internal Resonant Waves: Seiches in LakesStanding internal waves on lakes (seiches) have long been observed. Watson (1904)observing 30 m fluctuations in Loch Ness, gave the correct interpretation of them asbeing standing waves on the interface between two fluids of different densities. He found7reasonable agreement between observation and prediction (using equation 13, chapter 3).Wedderburn (1911) confirmed this interpretation and advanced the theory, withobservations of Loch Ness and the Madusee, in Germany.Mortimer (1952, 1953, 1961, 1974, 1975) summarized the theory of internal resonantwaves in terms of Kelvin and Poincare waves, the Coriolis effect and non-linear effects.He also summarized the observations on several lakes and applied the basic theories ofinternal seiches to them. Csanady (1972, 1982) deals theoretically with seiches on verylarge lakes, the Great Lakes. Csanady (1973, 1975) deals with the topic of wind inducedbarotropic motions on large lakes: Poincare waves on the thermocline, coastal jets andKelvin waves, topographic gyres and the effects of Coriolis and friction. Thorpe (1971,1974) developed theory to fit data for the internal surge and seiche on Loch Ness and acriterion for the applicability of linear theory to internal seiche activities.Stratified systems and lakes possess a range of internal oscillatory modes and may bemodeled as viscously damped mechanical oscillators (Lamb, 1944; Wilson, 1972).Typically, large lakes are not strongly damped (Mortimer, 1974). Hutchinson (1974)reports observations of 150 cycles in the decay of a seiche on Loch Earn. The modesdepend on basin morphometry and density stratification. The modes are characterized bywavelength: first mode implies a wavelength of 2L where L is the basin length and, in. 2Lgeneral, for the nth mode, the wavelength is —. On small to medium sized lakes onlythe first mode is usually excited, or observed, (Mortimer, 1952, 1953). Csanady (1975)reports 7th mode waves on Lakes Ontario and Michigan. In general, higher modeseiching is more important on large lakes where the forcing and damping due to windmay vary from site to site over the surface of the lake and the local density structure andmorphometry may vary as well. In lakes of regular morphometry, the observed period8usually matches the predicted values from equation 15, chapter 3, quite well(Wedderburn, 1911; Wetzel, 1975).13 Kootenay LakeConsiderable work has been done Kootenay Lake with respect to biology, chemistry andphysics. Zyblut (1969) and Cloern (1976) wrote about changes in the limnology ofKootenay Lake over the previous 10-20 years. They presented data concerningmacrozooplankton masses, lake chemistry and nutrients and flow. Northcote (1972a,1973) presented concise historical details of human impact on the lake over the last 80-100 years and related the morphometric changes in the main fish species. Daley et al(1981) presented a very detailed review of Kootenay Lake limnology with emphasis onthe effects of damming the inlet and outlet rivers.Chapter 6 considers the applicability of forcing resonant seiches on Kootenay Lake viathe dams on the outlet the lake. Forced resonant seiches were originally proposed as ameans of mitigating an ecological problem in the lake, the decline of the Kokanee fisherydue to decreased productivity and competition by the lake shrimp, Mysis relicta. Sparrow(1964) detailed the introduction of Mysis relicta, a fresh-water shrimp, into KootenayLake in 1949 and 1950, and its observation in 1961 and 1962. Mysis relicta is a seriouscompetitor of the Kokanee species which is in decline; it has been proposed that atransverse internal seiche on Kootenay Lake, driven by downstream dams, might increasethe export of Mysis relicta and help the Kokanee. At a time when the Kokaneepopulation was burgeoning and Kootenay Lake was suffering eutrophication, Northcote(1972) considered the effects of the introduction of Mysis relicta to Kootenay Lakefinding the introduction generally favourable. Lasenby, Northcote, and Fiirst (1986)9considered the theory, practice and effect of Mysis relicta introduction in North Americaand Scandinavia at a time when the Kokanee in Kootenay Lake were disappearing, the1970s. Lake productivity and nutrification was considerably decreased through thatperiod due to the inlet dams and improved practices at a large upstream fertilizer plant(Daley et al, 1981). They urged no further introductions without considerable caution.Martin and Northcote (1991) considered Kootenay Lake with regard to Mysis relictaintroduction, concluding that the introduction on Kootenay Lake was a poor model forintroductions elsewhere, due to the effects of widely varying productivity. Northcote(1991) considered the history of Mysis relicta introductions, problems with same, andpotential methods to control the shrimp.The circulation and internal dynamics of Kootenay lake are detailed by Carmack andGray (1982). The work includes the effects of riverine inputs, vertical mixing andseasonal effects. Dissolved matter (salts) distributions are treated in Hamblin andCarmack (1980) while looking at flow patterns; a model is developed to explain thedistribution of dissolved salts traced by conductivity profiles. Wiegand and Carmack(1981) describe a wintertime temperature inversion deep (-60-80 m) within KootenayLake and look at the effect of convective mixing and riverine influences on its stabilityand eventual decay. Carmack et al (1986) deal with various mechanisms of water massdistribution on Kootenay Lake: riverine circulation, mixed-layer dynamics, and internalwaves. They report 15 m amplitudes for internal seiches on Kootenay Lake. Crozier andDuncan (1984) present a mass of data, chemical, physical, and biological, summarizingwhat was known and observed about Kootenay Lake to that time. Walters et al (1991)present an adaptive model of the ecology of Kootenay Lake accompanied by a computerprogram of the same. The program was used to predict the response of the Kokanee tochanges in other major elements of the ecology of the lake.103.^Theory of Resonant Internal Waves: 11 Problem DefinitionFor simplicity, consider a rectangular basin of length L, width W , and depth H inwhich a layer of light fluid of density p ^thickness k rests upon a layer of fluid ofdensity p, > A, and thickness k, such that k + hi = H. Refer to figure la.The goal is to develop expressions for the period and shape of the wave, on the interfacebetween the layers, traveling in the x direction, along the length of the tank, as in figurela. This section and section 3.2 are derived mainly from Lamb (1944). Initially,consider the fluids immiscible and inviscid, and the walls, bottom and top of the tank asbeing perfectly rigid and that any waves which develop are infinitesimal. These criteriawill be relaxed somewhat after initial development of the theory.The geometry may be considered 2-dimensional, as the waves are confined to the xzplane, along the length of the tank, as in figure la. Now define O. and Oi as the velocitypotentials in the upper and lower layers, respectively; these must satisfy Laplace'sequation: V2Ø = 0. Also, define n(x,t) as the elevation of the interface from itsequilibrium level, where we will define z = 0.The boundary conditions are, for vertical velocities:dOat z = -4 , --I-W=0 , andazaat z = h., --Ø14 (t) = 0dz(3)11and for horizontal velocities:at x = 0, dA(t) = 0 , Plu- (t) . 0 , and^(4)ax^axat x = L , –±Pia (t)= 0, --1)E(t)= 0ax^ax^3.2^Basic SolutionsFunctions Ou and Oi satisfying the criteria of equations 3 and 4 are:Ou = Du cosh[k(z – hu)lexp i(kx + cot)^(5)01 = D, cosh[k(z + hi)] exp i(kx + ax) (6)where Di and Du are constants, k is the wave number, —271. , where A is the wavelength,Aand co is the angular frequency. Definingn(x,t) = a exp i(kx – wt)^ (7)where a is the amplitude, then the vertical velocity of the interface isan = –icoaexpi(kx– cot)^ (8)atAt the interface, the expressions for the vertical velocitiesak (0,0= Ad (0,0= ±id (0, t)dz^dz^dt^(9)^Therefore,^–kAi sinh(khi) = kA, sinh(khu) = –icoa^(10)Solving equation 10, for the constants A, and A,„ produces–icoa^, andAu = k sinh(khu)icoa Ai = ksinh(khu)Pressure at the boundary is given in the upper and lower layers, respectively, by12a0i,=^+ gz), and—191(4 +gz)L dt(12)where p, p1, and pis are the pressure, and the lower and upper layer densities,respectively. These forms of Bernoulli's equation for the pressure are derived in Phillips(1966, p 19) from the momentum equation in terms of total pressure for homogeneousfluids in inertial frames of reference:dU — -+coxu+v —P +1u2 + gz = vV274^(13)dt^po 2where co, and -LC are the angular velocity and velocity vectors, respectively and the othersymbols have already been defined. Assuming irrotational flow, and ignoring viscosity,we have —co = V x u =0 and then i4 may be represented as the gradient of a scalarfunction, 14 = VØ ; essentially, this assumption was made in assuming the existence of thefunctions Ou and 01 earlier. Assuming an incompressible fluid, V •Tt = 0 and againV20 =0 Then, the momentum equation becomesV —+—+—u +gz} = 0dt po 2^(14)do p 1 21 2^iIgnoring —2u , as it s negligible in most situations, it is possible to writep dø—pc+ —at+ gz =0which can be rearranged with a suitable change of subscripts to give the equations abovefor pressure.Pressure continuity at the interface requires pu = p, which leads to(—icoA, cosh(kh,) — ga) = p.(—icoAu cosh(khu) — ga)^(16)Substituting A, and Au from equation 11 into equation 16 gives the angular frequency:13(15)coth(kk) + coth(kk)^(17)— p)^)gk^1/2co=( u The wave velocity, c, is then)112c co ^(P1 P.)g = =k (k(pi coth(khd + p coth(kik)))and the period of oscillation, T, for a tank of length L, is)-1/2(Pi 13,4)g T = —2L =2Lc^(k(picoth(khi) + pi, coth(khu)))for the mode n = 1. Other modes beyond the fundamental, are possible, such that2 L= —T , where the nth mode has a wavelength of —, which leads to equation 1,presented in chapter 2.At lake scale, the length, L, is typically much greater than the depth, H = h + h1, and the(P —P.)^relative density difference between the layers e = i ^«1 , on the order of 0.001.Since, for very small values, coth x —1 , when considering internal waves on lakes,equations 18 and 19 may be approximated asc (g' huk )1/2L H )T 2L = 211g, huh/ y1/2andL H )where g' = eg, the reduced acceleration of gravity; from which we get equation 2, shownin chapter 2, for the general case of the n th mode.3.3^Relaxation of RestrictionsIn experimental work, and on lakes, the upper surface is free. A free upper surface leadsto wave like solutions similar to those derived for the interface in which:(18)(19)(20)(21)14cs = (—g tanhk(k +10)1/2(22)but leaves the interfacial solution unchanged (Lamb, 1944, p371).(Lamb (1944, p372) shows the ratio of the amplitudes of the two waves is — eL —1 forPuh1, h « A , which, in typical lakes and reservoirs, will be on the order of 0.001 to 0.002;the negative sign in the ratio implies a 180° phase shift between the surface and internalstanding waves.The preceding analysis assumed that the wave amplitude, a, was much smaller than thewavelength A.. A more accurate expression from Thorpe (1968) takes account of finitewave amplitudes and shows that the wave amplitude effects the wave frequencyco =^ 1+ ^+9^9^18^6^6 +^(23)2 gk(pi _^/^a2k2r3c,43AC/ +p„C^32^C42 C'12while the wave speed isco = ^g(p, - pu) (1+ a2k2C13C 9^9^18^6^6^8k^k(piCi+ p.C.)^32^C12 c/ cu C13C. C;;C? jjj(24)to third order which reduce expressions derived earlier as a goes to zero, where CI andCu are coth(khi) and coth(kh), respectively. This provides a means of estimating theeffect of observed waves amplitudes on the theoretical periods.In the two layer model presented the density changes discontinuously at the interface. Inreality the density profile continuously varies, and the interfacial layer is of finitethickness. Diffusion processes control the growth and structure of the interfacial layer.gc,, c,34ci ci3c„ cu2c121/215An estimate of the effect of a finite thickness interface on the first mode frequency isgiven in Phillips (1966); it is valid for relatively sharp interfaces, that is systems in whichthe interface is a relatively small portion of the total depth of fluid. Assume the interfaceextends from z1 down to z2, a distance d, and refer to figure lc. The vertical modes ofthe internal seiche obey the following differential equation (Phillips, 1966):d2 w ± { N2 (z) _ (02 }k2W = 0^ (25)dz2^co 2^(where N(z) = --g —ap 112 is the Brunt-Vaisala , or stability, frequency and W is ap azsolution to the differential equation, the internal wave function, and k is the horizontalwave number. For the lowest internal mode, assuming N to be negligible outside theinterfacial layer, then W(z) is almost constant across the thermocline and integration ofequation 25 over the interface thickness d gives the change in W (z)z,^N2f W" dz E k2W(k)f {1— --27}dz^ (26)d^a)22leading toW(hi+) – w (hi) a k2W(h1){d – co2 po^ (27)where Ap is the density difference from the lower to the upper layer and p, is the densityat the centre of the interface, the average density.Functions satisfying the boundary conditions of zero vertical velocity at the bottom of thetank and the water surface are:W(z) = sinh(k(z – ha)), 0 < z < h.sinh(kh") sinh(k(z + k)), o>z >—k;sinh(khi)Substituting 22 into 21 gives a relation for the lowest mode frequency:)112co=(Apgk 1 p, coth(khi) + coth(kh.) + kd(28)(29)16where po is the density at the centre of the density profile. This reduces to equation 17 asthe interface sharpens, i.e. as kd —> 0.As noted in chapter 2, other prescriptions exist to determine the effect of a finiteinterfacial layer on wave dynamics. These include numerical determinations directlyfrom equation 25. Also, from Krauss (1966, p 34-36), the following dispersion relationis derived for the hyperbolic tangent density profile presented in chapter 2:^24-' (2n + 1)(&2—^co^cl-2[(2n + 1)2 —1}(co2 —k2 +^ k + = 0 ,^(30)co2^Ap (02 gd_i Ap^P Powhere f is the Coriolis frequency, and n, the mode number, is defined by Krauss tobegin at 0 for the first mode. Keller and Munk (1970) and Krauss (1966) derived thedispersion relation for an exponential density profile:"N2 — w 2 In [tanh(hi^tanh(k(Z2 1-1))]tan[kd( N2 — ct)2 ) 1/2 1^w2 = 0(02(1+ 2( N —2 w2) tanh(kzi)tanh(k(z2 +1-1)))^(31)cowhere the upper surface is taken as z = 0. Finally, Thorpe (1968) proposes thisdispersion relation:2co2 = co2[1 — — kdk2 tanh( kill —°^3for an upper layer of infinite depth. All the proposed dispersion relations will beexamined in light of the data obtained in this project, and with respect to the data ofHyden (1974), in section 5.3.The effect of viscosity on wave dynamics, particularly with respect to the period orfrequency, needs to be considered. As the viscosity of water is very small, viscous(32)17dissipation is significant only in the boundary layers adjacent to the sidewalls and thebottom of the tank and at the interface. The effect of viscous dissipation on propagatingwaves is to slowly decrease the amplitude with distance traveled by the wave, and todecrease the phase speed and the wavelength. Ting (1992) shows that the attenuation dueto viscosity and the wave number are related by k =^+ (1+ Ok* where ks is theAattenuation rate, which he derives in terms of the angular frequency, the viscosity and thewave number.The effect is considered in two parts by Thorpe (1968) and the result is simple. At theboundaries, the change in Co due to viscosity isAct)^coo vr 2( L+147)2)^LW )where coo is determined from equation 17. At the interface, the change in co due toviscosity isA^—k (coov)112LAco= —2 2Thus, from Thorpe (1968) the total viscous effect on the angular frequency isAco^)112(—k +2(^2L+W))^2^LWWaves in a system will interact with each other and with the natural modes of oscillationof the system. For example, a wave within a bounded system, such as a lake orexperimental tank, will reflect from the boundaries. The wave and its reflection willinterfere, constructively or destructively, depending on how close the wave is to aresonant mode of the system. Only waves at or near a resonant mode frequency co arealways in phase with their reflections and interfere constructively with themselves.Therefore, only waves at or near resonant mode frequencies will survive for long withinthe system.(33)(34)(35)18Turner (1973), and Davis and Acrivos (1967) deal in detail with the interference ofinterfacial waves. This discussion is primarily adapted from Turner (1973, p 39-44).Consider any system which can be described on linearized theory as the sum ofundamped waves sin(IcX — Cot) and for which the dispersion relation f(k, co) = 0 isknown, where 17 and X are the wave number and position vectors, respectively. Thedefinition of these vectors represents an extension to three dimensions; up to this point,the theoretical development has been in two dimensions, but I have chosen to beconsistent with the source from which the argument was obtained (Turner, 1973). Now,given two waves, (Icl, col) and (172,(02), of small-amplitude such that they may bedescribed by linear theory, there will be a cross-term of the formsin(ri — (olt)sin(k2X — (020. This may be considered a forcing disturbance on thelinearized system: the interaction consists of the response to this weak forcing.The above product term may be expressed as the sum and difference waves with wavenumbers and frequencies k3 =171± k2, and (03 = (01 ± (02. Each combination with one ofthe wave numbers k3 will tend to generate secondary waves at frequencies co, such thatf(173, w) 0. Only if (03 = (0,, however, will the waves remain in phase and the energytransfer become significant; this is not the general case. Energy transfer is limited byhow closely the frequencies match, the amount of energy available in the primary waveand viscosity. Three waves related by there wave numbers as above and separatelysatisfying the dispersion relation are known as a resonant triad. It is known that energytransfer from primary to secondary modes occurs over much longer periods than those ofthe waves involved and is inversely proportional to the amplitude of the interactingwaves (Phillips, 1966).19Thus, when a system, with resonant mode frequencies con , is forced at frequency a,there will be a transfer of energy from the forcing wave to the resonant modes. Due todestructive interference only waves at or near the resonant mode of the system are likelyto exist with significant amplitude for any length of time. However, energy from a forcedwave at a non-resonant frequency will be transferred to resonant modes.Thus, even when stimulating at or very near a resonant mode of the system, we shouldexpect energy to show up, at lower levels, in all resonant modes, especially if we waitlong enough. In particular, energy should be expected in the difference frequency of theprimary and the first resonant mode: this the "beat" frequency and effectively representsthe lowest available mode for energy on the system which will satisfy the dispersionrelation.204.^Experimental Description, Materials , Methods, and Error AnalysisThe work in this project consisted of experiments on two-layer systems. The purposewas to demonstrate the efficacy of pumping as a means forcing internal waves and toinvestigate the resonant response of two layer systems as to the effect of an interface offinite thickness on the amplitude and period of motion.The experiments took place in tank 2.40 m long by 0.60 m high by 0.60 m wide, seefigure 5. The tank was divided into 1.235 m test section and a reservoir occupying thebalance of the tank. At one end of the tank pumping occurred through a slit 2 mm wideacross the width of the tank at a height of 0.30 m. The water removed was pumped to thereservoir and allowed to flow back to the test section over a wall 0.525 m high. Thus, thetest section was 1.235 m x 0.525 m x 0.60 m. This constant, slow refill of the test sectionmaintained constant layer depths throughout each experiment and allowed longexperiments to be undertaken. The rate at which the water returned to the test sectionmay be calculated based on the flow through the pump (6 to 27 11min) and the area abovethe wall (0.075 m x 0.60 m = 0.0.045 m2) and is approximately 2 to 10 mm/s. In theexperiments conducted, with a typical lower layer depth of 160 mm this return flow hadno visible turbulent effect on the interface. The interface was maintained well below thepumping slit to preclude pumping dense lower layer fluid out and having it mix with theupper layer fluid on return over the wall.Withdrawal through a line sink from a two layer system, with the sink above the interfacewill cause the interface to be drawn up. In withdrawal theory, it is established that onethe^(r,2 )1/3can determine e up draw using An 1.5 11— -,^(Turner, 1973) where An is theg21elevation change, and q is the flow per metre in —m3is . Based on the pumping rate of 6-m27 1/min for the pump used (see section 4.1) and up draw of 0.009 to 0.024 m wasexpected.In the experiments the parameters were approximately:- 160 nunk 365 mmpi -1000 –1001 kg/ m3.--,1004 –1008 kg/ m3Exact values of all parameters used, and error estimates, are to be found in table 2,inserted at the end of this chapter.4i MaterialsThe equipment, as in figure 5, consisted of a tank (2.40 m x 0.60 m x 0.60 m), two 450litre reservoirs, a pump under computer control, a conductivity sensor mounted on atraverse such that it may be raised and lowered to desired depths within the tank. Thetank is subdivided into experimental and reservoir subsections by an aluminum wall0.525 m high, 1.235 m from the end wall.The pump is a Masterflex 7549-50 with two Masterflex 7529 pump heads. The hoseused was Masterflex #82 with a maximum flow of 13.5 litres per pump head per minute.The pump proved to be reliable and accurate over the range of 20 to 100% of maximumflow, based on observation of pumping volumes into a tanked over timed tests.22The conductivity sensor is a "Microscale Conductivity Instrument" from PrecisionMeasurement Engineering, mounted on a traverse using a Proscale Gauge, from AccurateTechnologies, to determine depth. The sensor is rated accurate to ±1% over the range 5mS/cm to 800 mS/cm. The gauge is rated accurate to ±0.01 mm, but observation throughuse indicates that the reliable error is about ±0.2 mm.The computer used is an IBM 386 clone with math co-processor. Two hardware boardsinterfaced between the computer and other equipment. An A/D (DAS8) card, fromStrawberry Tree Inc, was used to convert analog voltages received from the conductivitysensor to machine usable digital data. An analog output board, from Cole-Parmer Inc.,controlled by software written in 'C' and IBM Assembly language, supplied 4-20 mA tomodulate the pumping rate.Other equipment used from time to time included volumetric flasks, accurate to 1 ppt, aSartorius analytic balance, accurate to ±0.0001 gms, and a conductivity meter.The software performed numerous functions. Before actual testing the software was usedto record conductivity profile information on the systems. During testing, it controlledthe pump signaling 1000 times per cycle to modulate the flow in a sinusoidal manner.The software monitored the conductivity probe recording information every half second.From the voltages recorded, conductivities were determined by linear interpolation usingtable 1 for KC1 standards for calibration.23Table 1^KCI Standards: conductivities @ 25 CConcentrations^Conductivities @ 25 C^0.05 M 6668 mS/cm0.10^129000.20 248200.30^35889At temperatures other than 25° C, the formulaK(7) = (1+ 0. 0191(T(° C) — 25))K(25° C)^(36)(Standard Methods, 1989) was used for calculation of conductivities. Conversion ofconductivity data to salinity and then to density involves other more complex but well-known formulae, not reproduced here. These may be found in Standard Methods (1989),pages 2-62, and 2-64. The centre of the density profile was determined by interpolation,and the value was used to locate the probe and in calculating resonant periods.As well, the software followed the free oscillations of the interface using a 'window' ofconductivity observations 50 ms apart. This allowed the software to distinguish theoccurrence of a peak reading by comparing first, middle and last elements of a 50element array of observations. This permitted a reliable way of estimating the true freeperiod of the tank with each test.24I/ Methods 4.2.1 PreparationFirst, the reservoirs were filled 48-72 hours prior to filling the experimental tank, to allowthe water to reach room temperature. Second, the water in the reservoirs was salted toroughly 2.5 to 3 ppt, with 3.5 to 4 grams NaC1 (research grade material) per litre, toensure that the conductivity of the water is within the linear region of the conductivityprobe (5 mS/cm to 800 mS/cm). Third, the experimental tank was filled to a depth of 50-51 cm from the reservoirs, the centre wall was installed, sealed with foam between it andthe walls and bottom of the tank, and braced with an aluminum plate to prevent anyflexing and ensure a square fit. Fourth, the remaining 175 to 200 litres of water in thereservoirs were salted a further 3 to 7 ppt by adding 4 to 10 grams NaC1 per litre anddyed pink with Rhodamine WT dye. Fifth, the salted, dyed dense water was introducedinto the test side of the tank, over a 4 to 6 hour period, to form a two layer stratifiedsystem with a relatively sharp interface.4.2.2 DALh_igFirst, during the filling process, samples of the upper and lower layer waters were takenfor density and conductivity measurements. Second, the conductivity probe wascalibrated using KC1 standards and table 1. The error in the concentrations of thestandards is estimated as 0.1%. Third, the density profile of the waters within theexperimental tank was determined, moving the probe very slowly through the depth tominimize disturbance. The reading at a given depth appeared to depend on the directionof approach of the probe implying that the probe may drag fluid with it or in some other25way influence the interface. The probe was kept on throughout the experiments asturning it off contributes to measurement drift (personal communication, Michael Head,Precision Measurement Engineering). Fourth, the probe was positioned at the centre ofthe density profile as reported by the computer, based on the density profile informationgathered in the third step. Fifth, tests were run generating waves on the interface.The pump rate Q , in us, was varied sinusoidallyQ=Asin(2%)+U,^ (37)1 i^1where A , T, and^ iare the amplitude n —, period (s) and mean pumping rate n —,respectively. As noted on page, the pump proved most reliable over the range from 20 to100 % of its maximum rate. Therefore Q was set to 0.6 of the maximum and A to 0.4of maximum producing a range in Q from 0.2 to 1.0. The period was stepped througha series of values varied above and below the theoretical values for one or more resonantmodes. To allow for the lengthening of the period due to various effects (discussed inchapter 1, section 2), the calculated resonant period was multiplied by 1.04. Pumpingoccurred across a line sink 300 mm above the bottom of the tank. The water pumped outwas returned to the reservoir half of the tank and flowed over the centre wall to maintainthe overall depth in the tank. The depth of water above the centre wall was typically 65to 75 mm; this depth was observed to vary from day 1 to day 5, presumably due toevaporation. When necessary, small volumes of water were added to the reservoir side ofthe tank to replace that which had evaporated. At an average flow rate of about 16 l/min(60% of the 27 l/min maximum rate), the flow velocity over the wall was about 6 mm/s.The sixth step was to determine the natural period from the free, unforced oscillations ofthe system. The natural period was monitored by a hand-held stop-watch and also by thecomputer which followed a window of 50 samples in 2.5 s as noted earlier.264.3^Sources of Error4.3.1 Calculation of the Resonant PeriodsCalculation of resonant periods accurately is important in this thesis. This calculation issensitive to variation or error in the variables it depends on. Figure 6 shows the variationof theoretical period T with k, & h, , L, and Ap.The depths and the positioning of the probe are controlled to 0.01 mm, and the length isknown to within ±2 mm. The profile may be out by up to 2 mm, based on myobservation that readings vary depending on the approach direction. Therefore, theaccumulated error in T due to depth and length errors may be 0.1 to 0.2 s.More serious is the problem of density difference. Even accuracy of 1 part in 10,000means an error of 0.2 kg/m3 in the density difference or 0.8 s in T. To minimize thissource of error, multiple density measurements were performed on each sample, toproduce a reliable technique and reduce the error by averaging.A 0.1°C variation in water temperature or a similar error in measurement of temperatureresults in a 0.01 kg/m3 error in the density bases on data from the CRC Handbook ofChemistry and Physics, 66th Edition. To minimize error here, all temperatures were readto 0.05°C; water temperature was assumed to being invariant over testing periods. Aswell, the KC1 standards were floated in the tank to equilibrate temperatures, and the waterin the reservoirs equilibrated with room temperature for two days before starting anexperiment.274.3.2 TimingThere were two minor problems with the timing within the computer program. The firstis that the timing control functions available were adequate to only 0.05 s and the secondis that the instructions the computer performs take time. Neither alone would havemattered, but together they deserved consideration. The instructions performed by thecomputer to sample the conductivity probe between signals to the pump amounted toapproximately 0.5% of the pumping period on average; combined with the coarse timingaccuracy noted above, they could not be compensated for. However, by recording theactual pumping periods, averages with reliable error values were obtained and arereported in chapter 5.4.3.3 Errors in the Voltage to Density ConversionThe development of density structure information depends on the probe, A/D card andthe formulae used.The probe is calibrated at 5 points, using KC1 standards. The probe tends to drift;Michael Head, of Precision Measurement Engineering, indicated that the best procedureis to keep the probe powered up throughout an experiment, which was done. The actualerror in measurement is estimated as 1%, based on the specifications.28The DAS8 is a 12 bit card, and has therefore 4096 divisions for the 10V range used. Thisis 0.00244 V/division, for a probable error of ±0.00122V in a reading. Typically, thereadings covered a range of voltage of about 0.8 to 1.1 V, so the error is roughly 0.1%.The actual equations used come from Standard Methods (1986) and from the CRCHandbook (1989). These were calibrated against conductivity/density data in the CRCHandbook for NaCl.4.3.4 Error Estimates from the DataError estimates are reported in tables 2.1 to 2.7, which include relevant data regardingparameters of the experiments and some results, and their determination is outlined here.The parameters and data are discussed fully in chapter 5 on results. The error estimatesare only shown in the first table of the set (2.1) and may be assumed to be the same for allsubsequent figures in the set except where variations are noted in the tables. For thedepths of the layers the error estimate was based the cumulative effect of measurementerrors (±0.2 mm) and an estimate of the inaccuracy in locating the centre of the interface(1-2 mm). For the layer densities, the error estimate was based on the standard deviationof 32 measurements of masses for upper and lower layer liquids. The error in thetheoretical period was based on the estimated errors in layer thicknesses and densities.The error estimate in interfacial width was based on a visual estimate from figure 19. Forthe forcing period the error allowance was based on the standard deviations over theforcing periods for the experimental day concerned; in experiment 2, where 3 distinctgroups of forced period exist for each test day, 3 estimates are made for each day. Theerror estimate for the forcing frequency was based on the error estimate for forcingperiod. The error allowance for the normalized period was based on the error estimates29for the forcing and theoretical periods. The estimate of error in the amplitude was basedfirst on the criterion for starting a new test that any interfacial motion should be less than5 mm (15 minutes of idle time between tests being sufficient for this) and second anallowance for local turbulence and instrument drift and anomalous measurements, a fewof which are observable in the data, having magnitudes of roughly 0.5 - 1 mm (seefigures 10, 12, and 14). Error in estimated beat periods was based on visual estimatesfrom figures 7 to 18. The error in the beat frequency was based on the estimate for errorin beat period. The error in the resonant frequency was estimated based on the estimatederrors in beat frequency and forcing frequency. The estimated error in the observedresonant period was based on the averaged standard deviations of 10-30 measurementsfor those tests which produced high amplitude free oscillations (that is, those tests inwhich the forced oscillations were also large, at or near the resonant period).30Table 2.1^ Experimental Parameters and ResultsUpper layer (mm)Run#Forcingperiod,TF (s)Experiment 1Forcing freq (Hz) Non-dim.^Amplitude, Beatperiod,^(pi)) (inun)^Period (s)TF/TTBeat freq (Hz) Resonant freq.(Hz)363 ± 2 1 27.7 ± 0.2 0.0361 ± 0.0003 0.89 ± 0.02 7.7 ± 1.0 190 ± 20 0.0053 ± 0.0005 0.0308 ± 0.0008Lower layer (mm) 2 28.2 0.0355 0.90 7.6 220 0.0045 0.0310162 ± 2 3 28.7 0.0348 0.92 7.1 260 0.0038 0.03104 29.2 0.0342 0.94 7.7 290 0.0034 0.0308Densities (kg/m3) 5 29.8 0.0336 0.96 8.7 325 0.0031 0.0305upper: 1000.32 ± 0.05 6 30.3 0.0330 0.97 9.3 430 ± 40 0.0023 ± 0.0002 0.0307 ± 0.0005lower: 1006.94 7 30.7 0.0326 0.98 10.1 530 0.0019 0.0307difference: 6.88 8 31.3 0.0319 1.00 13.8 710 0.0014 0.03059 31.8 0.0314 1.02 20.5 NETheor. period TT (s) 10 32.2 0.0311 1.03 23.6 NEcba•-- 30.2 ± 0.3s 111232.833.20.03050.03011.051.0623.628.5NENEInterfacial^width (mm) 13 33.8 0.0296 1.08 15.7 NE72 ± 3 14 34.3 0.0292 1.10 9.2 NE15 34.8 0.0287 1.12 7.1 625 0.0016 0.0303Average Resonant frequency FR 0.0307 ± 0.0006 HzResonant period TR (=1/FR) : 32.6 ± 0.6 sObserved resonant period To: 32.8 ± 0.4sUpper layer (mm)368 1 8.70 ± 0.2 0.115 ± 0.003 0.29 1.3Lower layer (nun) 2 9.70 0.103 0.32 0.9157 3 10.6 0.094 0.35 1.04 11.6 0.086 0.38 1.3Densities (kg/m3) 5 14.0 ± 0.3 0.0714 ± 0.0015 0.46 5.3upper: 1001.44 6 15.0 0.0666 0.49 1.2lower: 1008.25 7 16.0 0.0625 0.52 0.6difference: 6.81 8 29.6 ± 0.4 0.0338 ± 0.0005 0.97 11.09 30.8 0.0325 1.01 18.1Theor period TT (s) 10 31.7 0.0315 1.04 28.830.4 11 32.2 0.0311 1.06 10.412 32.5 0.0308 1.07 7.1Interfacial width (mm) 13 33.6 0.0298 1.10 1.664 14 34.5 0.0290 1.12 0.70.0024 0.03140.0020 0.03180.0023 0.0313Table 2.2^ Experimental Parameters and ResultsExperiment 2Day # 1^Run^Forcing^Forcing freq^Non-dim. Amplitude, Beat^Beat freq^Resonant#^period, TF (Hz)^period,^(p-p) (mm) period (s)^(Hz)^freq (Hz)(s) TF/TTAverage Resonant frequency FR 0.0315 ± 0.0006 HzResonant period TR (=l/FR): 31.7 ± 0.6 sObserved resonant period To: 31.9 ± 0.9sUpper layer (mm)369 1 9.2 ± 0.2 0.109 0.30 0.6Lower layer (mm) 2 10.2 0.0980 0.33 0.4156 3 10.8 0.0926 0.35 0.54 11.3 0.0885 0.37 0.5Densities (kg/m3) 5 12.3 0.0813 0.40 0.6upper: 1001.51 6 14.7 ± 0.3 0.0680 0.48 1.5lower: 1008.14 7 15.6 0.0641 0.51 0.6difference: 6.63 8 16.1 0.0621 0.53 0.69 16.7 0.0599 0.55 0.8Theor period TT (s) 10 17.6 0.0568 0.58 0.4t.,..)(4.) 30.7 111229.7 ± 0.431.10.03360.03220.971.0210.313.5Interfacial width (mm) 13 31.7 0.0315 1.03 19.584 14 32.3 0.0310 1.05 33.515 32.5 0.0308 1.06 25.016 33.6 0.0298 1.10 12.317 34.8 0.0287 1.14 2.4^0.0028^0.03080.0021^0.0308Table 2.3^ Experimental Parameters and ResultsExperiment 2Day # 2^Run Forcing^Forcing^Non-dim. Amplitude, Beat^Beat freq^Resonantperiod, TF freq (Hz)^period,^(p-p) (nun) Period (s)^(Hz)^freq (Hz)(s) TF/TTAverage Resonant frequency FR 0.0308 ± 0.0006 HzResonant period TR (=111'R) : 32.5 ± 0.6 sObserved resonant period To: 32.3 ± 0.2sUpper layer (mm)368 1 9.4 ± 0.2 0.106 0.30 3.0Lower layer (mm) 2 10.4 0.0962 0.34 1.1157 3 10.9 0.0917 0.35 1.14 11.4 0.0877 0.37 3.2Densities (kg/m3) 5 12.4 0.0806 0.40 2.5upper: 1001.62 6 14.8 ± 0.3 0.0676 0.48 4.9lower: 1008.12 7 15.9 0.0629 0.51 5.6difference: 6.50 8 16.4 0.0610 0.53 2.29 16.8 0.0595 0.54 2.0Theor period TT (s) 10 17.8 0.0562 0.58 2.1t.")41,30.9 111230.8 ± 0.431.80.03250.03141.001.0310.019.0Interfacial width (mm) 13 32.5 0.0308 1.05 20.7132 14 33.1 0.0302 1.07 24.015 33.6 0.0298 1.09 32.516 34.3 0.0292 1.11 34.117 35.6 0.0281 1.15 15.20.0029 0.03060.0018 0.02960.0018 0.0299Table 2.4^ Experimental Parameters and ResultsExperiment 2Day # 5^Run^Forcing^Forcing^Non-dim. Amplitude, Beat^Beat freq^Resonantperiod, TF freq (Hz)^period,^(p-p) (mm) Period (s)^(Hz)^freq (Hz)(s) TF/TTAverage Resonant frequency FR 0.0300 ± 0.0006 HzResonant period TR (=I/FR) : 33.3 ± 0.6 sObserved resonant period To: 33.5 ± 0.4sTable 2.5 Experimental Parameters and ResultsExperiment 3Day # 1 Run Forcing Forcing freq Non-dim. Amplitude, Beat Beat freq Resonant# period, TF(s)(Hz) period,TF/TT(p-p) (mm) Period (s) (Hz) freq (Hz)Upper layer (nun)365 1 35.3 ± 0.2 0.0283 ± 0.0003 0.91 ± 0.02 7.1 NELower layer (mm) 2 37.3 0.0268 0.96 16.3 450 0.0022 0.0246160 3 38.8 0.0257 1.00 28.7 NE4 39.7 0.0252 1.03 37.3 NEDensities (kg/m3) 5 40.3 0.0248 1.04 28.3 NEupper: 1000.12 6 42.2 0.0237 1.09 17.5 NElower: 1004.32difference: 4.207 44.2 0.0226 1.14 8.3 400 0.0025 0.0251Theor period TT (s)c.4^38.7UiInterfacial width (mm)60Average Resonant frequency FR 0.0248 ± 0.0006 HzResonant period TR (=1/FR) : 40.3 ± 0.9 sObserved resonant period To: 39.6 ± 0.4sTable 2.6^ Experimental Parameters and ResultsExperiment 3Day # 3^Run Forcing^Forcing^Non-dim. Amplitude, Beat^Beat freq^Resonant#^period, TF freq (Hz)^period,^(p-p) (mm) Period (s)^(Hz)^freq (Hz)(s) TF/TTUpper layer (nun)369 1 37.1 0.0270 0.95 10.3 300 0.0033 0.0237Lower layer (mm) 2 39.0 0.0256 1.00 12.4 600 0.0016 0.0240156 3 40.8 0.0245 1.05 44.1 NE4 41.0 0.0244 1.05 45.7 NEDensities (kg/m3) 5 41.2 0.0243 1.06 48.7 NEupper: 1000.15 6 43.2 0.0231 1.11 26.0 900 0.0011 0.0242lower: 1004.25difference: 4.107 45.1 0.0222 1.16 13.1 NETheor period TT (s)(44^39.2Interfacial width (mm)74Average Resonant frequency FR 0.0240 ± 0.0006 HzResonant period TR (=1/FR) : 41.7 ± 0.9 sObserved resonant period To: 40.9 ± 0.4sTable 2.7 Experimental Parameters and ResultsExperiment 3Day # 5 Run Forcing Forcing Non-dim. Amplitude, Beat Beat freq Resonantperiod, TF(s)freq (Hz) period,TF/TT(p-p) (mm)^Period (s) (Hz) freq (Hz)Upper layer (mm)369 1 37.1 0.0270 0.95 8.1 300 0.0033 0.0237Lower layer (mm) 2 40.1 0.0249 1.03 9.2 750 0.0013 0.0236156 3 41.7 0.0240 1.07 31.6 NE4 42.2 0.0237 1.08 68.1 NEDensities (kg/m3) 5 43.1 0.0232 1.11 49.2 1700 0.0006 0.0238upper: 1000.20 6 45.2 0.0221 1.16 9.0 NElower: 1004.20difference: 4.007 47.2 0.0212 1.21 8.1 NETheor period TT (s)39.6Interfacial width (mm)98Average Resonant frequency FR 0.0237 ± 0.0006 HzResonant period TR (=111'R) : 42.2 ± 0.9 sObserved resonant period To: 41.7 ± 0.4sResultsThe results of three experiments, comprising 84 tests, are reported. Areas of interestconsidered include the effect of interfacial thickness on resonant period and damping,estimates of damping coefficients and viscosity, the partitioning of energy amongst theavailable resonant modes, and the interference of the forcing and resonant waves.The first experiment consisted of one day only of testing, while the other 2 experimentsconsisted of 3 days of testing over 5 day periods. The test parameters, and results, aresummarized in figures 7 to 38, in appendix 1, and in tables A2.1 to A2.7, in appendix 2.Table A2.1 concerns experiment 1, while tables A2.2 to A2.7 concern experiment 2, days1, 2, and 5, and experiment 3, days 1, 3, and 5, respectively.Tank temperatures averaged 18.8 ± 0.5 °C for the three experiments.The corrections for non-linearity are small; for example, for k = 360 mm, h1 = 160 mm,L = 1.235 m, and a = 35 nun, the ratio of wave speeds calculated using equations 24 and18 is 0.985. For a = 15 mm, the ratio is 0.997. Thus, the corrections range from 3 to 15parts in one thousand, the period increasing with amplitude.5.1^Observations on the Raw DataFigures 7 - 18 depict the raw data collected for all experiments. Experiments 2 and 3sampled at the rate of 2 Hz; for experiment 1 the rate was 4 Hz. The variation ofinterface elevations which are reported were inferred from the density profiles obtained,38as in figure 19, by linear interpolation. Linear interpolation is justified as the densitystructure is highly stable during the testing; this is borne out by the profiles reported infigure 19 taken before and after testing. Further comment on the interface stability ismade in section 5.2.Regarding experiment 1 (see figures 7 to 9), the growth in amplitude as the forcing periodapproaches the resonant frequency is clear (traces 10 and 11 have the maximumamplitude). As well, the beat envelope extends as the resonant frequency is approached.Runs 11 and 12 show marked amplitude variation and appear much less smooth than theother runs. This may be due to pre-existing motion on the interface. That is, if theoscillation of run 10 had not sufficiently decayed before run 11 began, then the data ofrun 11 might reflect this. And similarly, run 12 might show the effect of run 11. Fifteenminutes were allowed between runs, which translates into approximately 27 oscillationperiods. Subsequently, estimates will be provided indicating decay coefficients of about-0.002 s4: in 15 minutes this would result in a decay to about 16% of the originalamplitude or about 5 mm if the original amplitude was 30 mm. By the time the first 400seconds of the next run had occurred, approximately 13 periods and half the length of therun, the amplitude of the previous run would be down to about 7% of the original or 2-3mm for an original amplitude of 30 mm. Thus, while a carryover of amplitude from onetest to the next existed, in general the effect should have been small. In retrospect, alonger delay between runs may have been appropriate, however the preliminary testingthat was done indicated 15 minutes would be adequate.It appears that the probe was located a few millimetres above the centre of the densitydistribution in experiment 1. The centres of all the traces are offset from the probelocation by 2-3 mm. This offset may also be the cause of the visible asymmetry of the39traces which are much sharper on the upside of an oscillation and rounder on the lowside.In experiment 1, consider the third wave groups in runs 1, 2, and 3. These groups are setsof 6 or 7 oscillations whose amplitudes are modulated by a wave of longer period,approximately 250 s. The oscillations in these groups have a common feature: "knees".That is, from the peak of an oscillation the wave height appears to decline slowly forsome time. Then the rate of decline sharply increases, after approximately half theoscillation. The point at which the rate sharply varies is the "knee". A similar "knee"appears in the traces as they move from the troughs back the peaks. For the oscillationsconsidered here, this results in a profile like broad triangular mountains separated by "U"shaped valleys. This knee may be due to some error in the density profile obtained forthe experiment, or may be a real feature of the density profile, or may in fact represent theway in which the interface moves in response to the pumping or some combination ofthese effects. As similar knees are apparent in experiments 2 and 3, it is less likely it isan error in the profiles. A knee is observable in all the runs, for at least some oscillations,although it becomes less noticeable for runs with large amplitudes. The persistence of theknee, if it is due to some feature of the density structure, indicates the stability of thedensity structure despite pumping.For experiment 2 the data is grouped by days. Figures 10 and 11 are for day 1, figures 12and 13 for day 2, and 14 and 15 for day 5. In each pair of figures, the first shows data forforcing frequencies at or near the second or third harmonics, while the second shows datafor forcing frequencies at or near the first harmonic mode. Run lengths varied in thisexperiment and in experiment 3 depending on how close the forcing frequency was to thecalculated natural frequency. This choice was made to minimize time spent gathering40data at frequencies distant from the resonant frequency and to minimize the disturbanceon the interface.Figures 10, 12, and 14, showing amplitude response for high forcing frequencies, eachshow occasional "spikes" in the traces of 1-2 mm amplitude, against typical signalamplitudes of 0-4 mm. These spikes are evident in only one of the larger amplitudetraces (trace 13, figure 11) and their origins are unknown. Perhaps they are due to someoccasional anomalous behaviour of the probe, or to "noise". Perhaps their rarity in thelarger amplitude traces is due or to the 10-fold difference in scale used for those tracesand the small amplitude traces. These three figures (10, 12 and 14) representing responseon different days to similar forcing frequencies, are very different in appearance from oneanother, but the traces for each day do resemble one another. That is, in figure 10 for day1, the traces all show obvious long and short period components, except 5 which appearsto be the closest to a resonant frequency. In figure 12, for day 2, the traces all appear tobe primarily noise superimposed on long period waves. The traces of figure 14 are inbetween the other two in appearance, and have long and short wave components withsuperimposed noise. This difference in appearance and the lower than expectedamplitudes on day 2 are unexplained.Figures 11, 13, and 15 represent the results of forcing at or near the first mode. They arequite similar in appearance. Apparently the probe was offset a similar amount for eachday's runs as the asymmetry of the traces is consistent from day to day. Also there is aknee, like that mentioned for experiment 1, observable in some oscillations of some runseach day.41Experiment 3 is represented by figures 16 to 18. Again, the probe offset and knee areobservable. Also, a beat envelope is observable in some cases (see day 2, runs 1 and 2,and day 3, runs 1 and 2). Where an envelope is not observed, this can be attributed inpart to not collecting data for a sufficient time to observe the envelope. Possibly anundecayed wave or waves due to earlier runs, which was alluded to earlier, also interferesto hide the beat envelope.5.2^Density VariationsFigure 19 shows the density profiles determined during the 3 experiments. Samplingoccurred at 1-2 mm intervals over the ranges in which the densities changed. Forexperiments 2 and 3, note that the profiles tend to intersect at a single point: forexperiment 2, this point is at an elevation of 150 mm, while for experiment 3 theelevation is approximately 148 min. If this intersection point is taken as the natural levelof the interface then the probe was typically set 8-10 mm high. As well, the calculatedperiods for the systems, based on the elevation at which the probe was set would beincorrect. The sensitivity analysis of chapter 4.3 indicates that the effect of this error, if itindeed is an error, would be less than 0.25 s in the calculated value of the theoreticalperiod.The profiles taken immediately before and after testing, for experiments 2 and 3, indicatethat testing had no major effect on the profile thickness and thus little effect on the naturalperiod of the systems over the course of testing. The stability of the system in the face offorced oscillations is borne out by the value of the local Richardson number. The localRichardson number J is the square of the ratio of the Brunt-Vaisala frequency to the rate42(of shear: this becomes J = (—N – —a))ka)-2 (Phillips, 1966, page 214) at the point wherew Nshear is maximal, and takes values in the range 7-30, for experiments 2 and 3. This iswell above the generally accepted stability cutoff value of .14- (Phillips, 1966, page 214).5.3^Resonance and Interfacial ThicknessThe non-dimensional, normalized periods of the runs with peak response for each test dayof each experiment are shown in figure 20a against the normalized interfacial thickness.Normalized period is defined as the forcing period TF divided by the theoretical periodTT (see table 2, appendix 2). These ratios are shown in the fourth column of table 2. Thenormalized interfacial thickness is defined as the interfacial thickness divided by the totaldepth. The error bars plotted for experiment 3, day 1 are the same for all other points andindicate that the data could support an intercept variation of ±0.02 and a slope variation of±0.2.The interfacial thickness was estimated graphically from figure 19. For each densityprofile in figure 19, the vertical portions of the graph were extended and a line was drawntangent to the profile at its centre. The points where the line and the verticals met wereassumed to be the extent of the interface.For comparison, data from Hyden (1974) is included on figure 20a. Hyden used anAp equivalent method to estimate interfacial thickness d: d^ap) . There is a largedz )maxapparent difference between the results of this project and of Hyden. This difference isaccentuated in appearance by the scale of the graph. It is apparent that Hyden's43experiments were in a lower range of normalized interfacial thickness than those of thisproject. One might expect the two sets of results to be complementary, but Hyden'ssuggest a much stronger increase in period with interface. Possibly, the much longerwavelengths of Hyden's internal waves (33.44 m) compared to the those in this project(2.470 m) has had an effect on the results. Another potential explanation lies in thedifferent boundary conditions of the experiments. Thorpe's system had essentially zerohorizontal velocity in the lower layer while the upper layer was oscillated back and forthby the pistons (figure la). Hyden's system had zero horizontal velocity in each layer atone end and non-zero, opposing flows in the layers at the other end. The system used inthis project featured zero horizontal velocity in the lower layer, zero velocity in the upperlayer except through a narrow slit (2 mm thick) at one end and the return flow from thereservoir. This overflow would result in a non-zero surface velocity, which was notmeasured but was earlier estimated as 2 to 10 mm/s, in chapter 4.Methods to predict the behaviour of period with interface exist and were presented insection 3.2. These are, for reference: first, the direct numerical solutions of thedifferential equationd2W + {N2(z)– c°21k2w =0 (equation 25),2de^cosecond, Phillips' (1966) dispersion relation for a linear density distribution between twohomogeneous layersco=(Apgk^1^1/2po coth(kizi)+ coth(kk) + kd(equation 29),third, Krauss' (1966) solution to the hyperbolic tangent density profile2d-I(2n +1)(co2 – f2)1I2 co^d-2[(2n +1)2 –1}(co2 – f2)+^ kh+^– gd-1 —Ap – gd-^1 ^= 0 (equation 30),co2 APco2P. P.44fourth, Keller and Munk's (1970) solution to the exponential density profile:1N2 — cv2 )112 ritanh(kz,) — tanh(k(z2 + LIMtan[kd(N2 —2 (02 )112 ] ^CO2CO (1+ (N2 — 6)2) tanh(kz, )tanh(k(z2 + H))) = 0(equation 31),(02and fifth, Thorpe's (1968) formulaco2 = co2[1— —2 kci(2 tanh(khi) —1)] (equation 32).^°^3The numerical predictions and the predictions of the formula for the exponentialdistribution are shown on figure 20, with the data. All results are presented plottedagainst normalized interface thickness as defined above based on the total system depthused in this project: 0.525 m. The only dispersion relation which takes into account thetotal depth of the system is that for the exponential density profile. The numericalsolutions of equation 25, one for each test day in this project are courtesy Mr. Gu Li.The various predictions of the other formulae seriously conflict with the those presentedand with the data and are not shown. The most reasonable results are given by theformula for the exponential profile, which lie between Hyden's results and the results forthis project and have been shown as well on figure 20. The numerical calculationslikewise lie above the experimental results they are based on in all but one case and seemto confirm the results for the exponential profile.455,4 Amplitude ResponseFigure 21 shows the amplitude response for the 3 experiments. In all 3 experiments, thepeak response is at a period (the resonant period) greater than the theoretical period, asdetermined by equation 19. The resonant period increases in real terms and relative to thetheoretical period as the interface widens. This effect is consistent over experiments 2and 3. It is not observable in experiment 1, as experiment 1 consisted of 1 day of testingonly.The amplitude of the peak response increases with decreasing density difference. Thepeak amplitude is greatest for experiment 3, with a density difference of 4.2 kg/m3 andleast for experiments 1 and 2 which had similar density differences of approximately6.8 kg/m3. The ratio of half widths of the response curves in figure 21 to their heights isshown in table 3 below. It is difficult to see a conclusive trend in this data, however theresults of experiment 3 suggest that as the interface widens the resonance peak sharpens,that is the response falls off much more rapidly as the forcing frequency diverges fromthe resonant frequency.Table 3^Ratios of the Half Widths to the Heights ofthe Resonance Curves in figure 21 (mm - 1)Experiment 1^ 0.27Experiment 2, day 1^0.22Experiment 2, day 2 0.21Experiment 2, day 5^0.39Experiment 3, day 1 0.31Experiment 3, day 3^0.20Experiment 3, day 5 0.0946Resonance peaks are observable in Thorpe's (1968), figure 2b, similar to the resultsreported here. As noted in chapter 2, Thorpe's results are 1.13 ± 0.05 times greater thanthe theoretical period. Hyden's results (figure 20) range from 1.06 to 1.12 times greaterthan the theoretical periods. The results reported here are at values from 1.01 to 1.05times greater than the theoretical periods. A potential explanation of the discrepancies inthe experimental results of Thorpe, Hyden and this project lie in the differentexperimental scales used, although the discrepancies do not increase with increasingwavelength or depth. Hyden's system had the longest wavelengths and greatest depths,and Thorpe's the shortest wavelengths and least depths, so one might expect the results ofthis project to be intermediate between those of the other two but from the resultsreported in this paragraph, this is not the case.5,5 EnergyThe potential and kinetic energy stored in an interfacial wave for the system used in thisproject is1^2,rEs = TrgApLWa2 f cos2 Od0 7.27Apa2,0based on the dimensions of the tank given earlier in chapter 3. The instantaneous kineticenergy input isEK(t)=-i1 m v2^ (39)where m is the mass of water moved and v is the velocity. The total kinetic energy inputin a single cycle isEK = EK(t)dt^ (40)(38)47The mass moved per unit time is pQ(t) where Q(t) is the flow rate(7-25 11min, or0.117(10-3) - 0.417(10-3) m3/s, in all the experiments). The water velocity v(t),assuming uniform velocity across the slit of area As , 0.0012 m2, is^or 0.0975 toA,0.348 m/s. ThenEK =^Q3 (t)dt 0.010T F^ (41)The total energy pumped into the system in each cycle was approximately 0.31 to 0.33 Jin experiments 1 and 3, and approximately 0.40 to 0.43 J in experiment 3. Thus, thestored energy represents 1.7 to 4.1 % of the energy added each cycle.The energy in the peak waves observed for each day of testing is given in table 4. Theresults indicate that as the interface widens, more energy is channeled into wave motionon the interface. This is obviously of relevance in reservoirs or lakes in which theinterface may be significant fraction of the total lake depth. In Kootenay Lake, forexample, the interface may be 30 m out of a total depth of 120 m or 25 %.Table 4^Energy of the Waves of MaximumAmplitude (J)Experiment 1^0.0051 ± 0.0002Experiment 2, day 1^0.0051Experiment 2, day 2 0.0068Experiment 2, day 5^0.0069Experiment 3, day 1 0.0053Experiment 3, day 3^0.0091Experiment 3, day 5 0.0177485.6 BeatingEstimates of the beat periods and frequencies for each test in which a reasonable estimatecould be made visually are shown in table 2. By combining the forcing frequencies andthe beat frequencies, as sums or differences as appropriate, estimates of the resonantfrequencies are obtained in the final column of table 2. The notation 'NE' indicates NoEstimate. The average values of the resonant frequencies are converted to resonantperiods and compared with directly observed resonant periods. Direct observation wasmade, during free oscillation, with a stop-watch for experiment 1 and, for experiments 2and 3, by computer (programmed to follow the variation in interface elevation at intervalsof 0.05s and report when the peak), and with a stop-watch, by hand.The resonant periods reported directly and by inference are consistent with one another,and always within error limits. As well, the resonant periods always increase withincreased interface thickness (table 2, and observable in figure 21), which conforms to theresults noted in section 5.4 and with the theoretical predictions.5.7 Fourier TransformsFigures 22 to 34 show the Fast Fourier Transforms (FFTs) of the data collected. Forclarity, 3 or 4 representative transforms are shown for each day's testing. To calibrate thetransforms, the arbitrary total amplitude of the signal reported by each FFT, was relatedto the energy of the wave based on the observed amplitude, as shown in table 4.49By the sampling theorem, if the sampling period for the data is ts all frequencies atf^ = Luse will show up in an FFT (Lathi, 1983). The sampling period, ts, actuallytsused in the experiments 2 and 3 was 7 s, so f„ff = 2 hz ; for experiment 1 thesampling period was Y4 s, and the cutoff frequency was 4 hz. The forcing and resonantfrequencies are about 74 hz each, while the beat frequency is considerably less than that.The higher resonant modes most likely to be observed, for n= 2, 3, are at frequencies ofabout X0 , and 3 hz. Thus the sampling frequencies were sufficiently high.The FFT is a discrete transform, thus the energy of a range of frequencies will be mappedto a single frequency. The range is given by the sampling frequency divided by thenumber of samples used in the transform. For experiments 2 and 3, the samplingfrequency was 2 hz, while for experiment 1 it was 4 hz. However, in experiments 2 and 3512 data points were used in the transform while in experiment 1, 1024 were used. Thus,the range in all cases was 72 5 6 = 0.0039 s-1.All the transforms of runs pumped at or near the first harmonic of the system (otherwisereferred to as the resonant mode), figure 22, and figures 29 to 34, show most of theenergy in the first harmonic: for experiments 1 and 2 this is at a frequency of 0.031 Hz (aperiod of 32s), while for experiment 3 the energy is at 0.025 Hz (40s). These results areconsistent with the observed resonant frequencies noted in table 2, considering that thefrequency resolution of the transforms is 0.0039 Hz. These figures, 22, and 29 to 34, alsoshow significant, though much less, energy in the higher harmonics, at 2, 3 and 4 timesthe frequency of the first harmonic; the second harmonic contains approximately X to 7of the energy in the fundamental mode. Energy can be observed into the 6th harmonic infigure 33, experiment 3, day 1. Very small energy peaks are observed at very low50frequencies in the FFTs, that is in the beat modes. The transforms for experiment 3,figures 32 to 34 seem to indicate greater amounts of energy in the higher resonant modeswhen the interface is thinnest (for figure 32). The data for experiment 2, for pumpingnear the first harmonic frequency, do not indicate any trend. The transforms of the freeoscillation data for experiment 3, in figure 38, to be discussed further in section 5.8,contradict the indications of figures 32 to 34: they indicate the energy in higher resonantmodes was a greater proportion of the total energy later in the experiment when theinterface was thickest.The transforms of runs pumped at higher frequencies (figures 23 to 28) showconsiderable energy, relative to the total energy in the interfacial waves, in the beat mode.In figures 23 to 25, as above, the first harmonic of the system receives a considerableproportion of the energy and higher harmonics relatively less. Of note, in figures 23 to25 are peaks at the pumping frequencies themselves. Thus, we can infer that the forcedwaves do exist in the system, though they are rapidly attenuated. In figures 26 to 28 theenergy peaks in the second harmonic, presumably because the pumping frequenciesclosely approximate that of the second harmonic.5.8^Free OscillationsFigures 35 and 36 depicts free oscillation data collected during the course of experiments2 and 3. Below, in table 5, are results extracted from the data concerning the rate atwhich signals on the interface decay. From this data the apparent viscosity has beenestimated using equation 35; the values obtained correspond well with the accepted valueof molecular kinematic viscosity, approximately 10-6 m2/s. Estimates of decay weremade from each oscillation and averaged; this averages out the observable variability in51decay from period to period. The estimate of error in the decay rate and in the viscosityare based on the statistics for the sample; 15 to 33 % in the decay rate, 40 to 70 % in theviscosity. Possibly, the rate of decay increases as the interface widens, (however themeasurement error is large and no conclusion may be drawn) and the rate of decay seemsgenerally lower at the lower density difference of experiment 3 versus experiment 2. Anincreased rate of amplitude decay might be expected with increased interfacial width asthere are more available modes of oscillation. With more available modes for energytransfer the overall rate of energy transfer may be much increased without any increase inthe rate of energy transfer to any particular mode.Table 5^Amplitude Decay for Experiments 2 and 3Amplitude^Elapsed^Exponential^Apparentrange^Time^Decay Factor^Kinematic(max;min (mm)) (cycles; s) (s - 1)^Viscosity(10 m2/s)Exp 2, day 1 29±1 - 6±1 23; 733±5 -0.0021±0.0005 1.1 ±0.6Exp 2, day 2 33-7 23; 744 -0.0021±0.0003 1.1 ±0.4Exp 2, day 5 42 - 8 22; 737 -0.0023 ±0.0008 1.3 ±1.0Exp 3, day 1 35 - 12 18; 705 -0.0015 ±0.0004 0.7 ±0.4Exp 3, day 3 40 - 11 17;695 -0.0019±0.0004 1.1 ±0.4Exp 3, day 5 50 - 11 17;710 -0.0021±0.0004 1.4 ±0.6Comparing the results in table 4 to information on oscillation persistence reported byHutchinson (1975) for Loch Earn, over 150 oscillations, if we take a value of 0.002 s-1,the time for a wave on the tank to decay to 1 % of its initial amplitude would be about2300 s, or 60 to 70 periods. Apparently, on lakes internal waves are more persistent than52in the tank. The more rapid attenuation of waves in the tank is due to side, bottom andend wall friction being of proportionately much greater importance than in a lake.Figures 37 and 38 show the transforms of the free oscillation data of figures 35 and 36respectively, with energy normalized to 1. The energy is normalized as the actual energylevels of the free oscillations depend on the amplitude of the forced waves whichimmediately preceded them, and so are not comparable. The figures show thedistribution of energy in the fundamental and higher modes. Energy levels in thefundamental mode are approximately 3-5 times the levels in the second harmonic. Thereis no consistent trend of relative energy levels from one test day to next over the twoexperiments: that is, the energy in the fundamental mode, from greatest to least, runssecond, first, third in experiment 2 in experiment 3, and first, third, and second day.53fi^The Practicality of a Forced Internal Seiche on Kootenay LakeTo assess the practicality of controlling the internal dynamics of a lake or reservoir usingthe method of this project, a brief critical look at the situation which originally inspiredthis project is presented. The decline of the Kokanee fishery on Kootenay Lake ineastern British Columbia (see map, figure 39) provides one possible application forcontrolled internal lake dynamics. The decline is dealt with in detail in Walters et al(1991), Northcote (1973a&b, 1974, 1991), and Martin and Northcote (1991). The fresh-water shrimp Mysis relicta is thought to be a major competitor of the Kokanee, andincreased export of the shrimp from the lake might help the Kokanee. Using dams,located west of Nelson, on the outlet river of Kootenay Lake and by matching theperiodicity of the dam operation to the periodicity of resonant internal waves across thelake, large scale internal waves across the lake might be generated. Timing themaximum outflow from the lake due to the internal wave to coincide with the diurnalvertical migration of Mysis relicta through the water column (Northcote, 1991), onemight induce greater export of the shrimp from the lake.The proposal is to force an internal seiche by increasing and decreasing withdrawal ratesabout the normal mean flow which is about 1000 m3/s. By synchronizing the variationof the rate of withdrawal with the natural period of the transverse seiche across KootenayLake a large motion might be induced on the thermocline. The natural period of theinternal transverse seiche at the point known as Queen's Bay, the location of the riveroutlet of Kootenay Lake, is roughly 6 hours based on the cross-channel width of 4 kmand epilimnetic and hypolimnetic depths of 30 and 120 m, respectively. Kootenay Lake'sepilimnion, at 14-18°C, and the hypolimnion, at 4-6°C, are separated typically by ametalimnion of 20-25m thickness.54Mysis relicta rises 120 m from the muddy bottom each night to feed under cover ofdarkness from midnight to 4 AM between 10 and 30 m from the surface (Northcote,1991). Their migratory behaviour effectively keeps them safe from significant predationby Kokanee or the Gerrard trout in the main lake, as the fish hunt by sight. By timingthe enhanced internal seiche so that the thermocline is rising near the outlet betweenmidnight and 4 AM, when Mysis relicta is near the surface, large numbers of the animalsmight be drawn out of the lake.There are several potential problems with in-field application. First, the outlet ofKootenay Lake is a 47 km long series of wide and often deep basins connected by narrowriver sections. Many points are very shallow and the flow might become super-critical atthose points precluding the possibility of generating a wave at the dam-site and expectingit to propagate upstream to the lake.Second, considerable refraction of the wave might be expected. The waves would enterthe lake with wavelengths much greater than the width of the outlet river at KootenayLake, approximately 200 m. Thus, a large portion of the wave energy may deflect intothe north or south arms of the lake. Also, the geometry of the lake is such that a linedrawn from the outlet Queen's Bay strikes the eastern boundary of Kootenay Lake at anoblique angle rather than at 90 ° , thus the major portion of the wave energy might not bereflected back.Third, Mysis relicta is a reasonable swimmer (5-10 cm per second) (Northcote, 1991)and may be able to counter the increased current near the outlet by diving. Mysis relicta55might also adapt to the altered current by simply allowing the current to carry it up tofeeding depths.Fourth, Kootenay Lake is 107 km long, and the outlet is 200 m wide, a ratio of over500:1. Assuming an even distribution of Mysis relicta through the lake, which may beincorrect, we might expect Mysis relicta to move into Queen's Bay from the north andsouth to replace those exported. However the shrimp are likely to move horizontally atthe drift velocity of the lake and the flushing time for the lake is 2 years. The fact is thatit took many years, 15, before they were actually observed after their introduction(Sparrow, 1964; Northcote, 1972a)). Based on that time period for colonization anddistribution, Mysis relicta is unlikely to move rapidly into Queen's Bay. Thus, anincreased export of mysids might be observed representing the local population ofQueen's Bay, but distant areas of the lake might be unaffected for extended periods oftime. This period might, as well, exceed the replication period for Mysis relicta. It isdistinctly possible that export at Queen's Bay would be enough to benefit the Kokaneeand that export there would induce rapid export throughout the lake and Walter's et al,(1991) predicts that such increased export would be valuable to the Kokanee, but thereremains some uncertainty.Finally, the generation of forced periodic seiches across Kootenay Lake, with the periodicincrease and decrease in outflow, would require environmental review with respect topotential impacts on the Kootenay River system. As well, as the Kootenay River system,and the Columbia River system into which the Kootenay system drains, fall under thepurview of the International Joint Commission, the United States would have to beconsulted as to their possible objections; the actual seiches would only effect theKootenay Lake and its West Arm but release changes might have downstream effects.56Thus, there is considerable work to be done before proceeding onto Kootenay Lake withthe idea of enhancing the Kokanee environment. None of the concerns mentioneddisqualifies the concept of generating internal waves within Kootenay Lake using thedownstream dams or the potential value this might have for the Kokanee. A goodstrategy might be to locate a small reservoir controlled by a dam and test the strategy. Byinstalling chains of thermistors at various locations within the reservoir and operating theoutlet cyclically around the usual average flow, over two or three seasons a body of datamight develop to indicate the internal dynamics of the reservoir and the response toforcing.57it^Conclusions and RecommendationsSinusoidally modulated pumping has been demonstrated as a viable means of generatinginternal waves. Waves have been generated under a variety of conditions and usingforcing frequencies at or about the first, second and third harmonic frequencies.Increased interfacial thickness has been seen to result in increased resonant period(figures 20 and 21). Comparison of the data obtained with that of Hyden (1974) showsgeneral agreement in trend (an increase of resonant period with interfacial thickness) butno agreement for the rate of increase. Further, various theoretical methods of calculatingthe variation of period with interfacial width, have been found to diverge in their results.Equation 29, from Phillips (1966) seems to best match the data obtained here but notHyden's data. The formulae of Krauss (1966) and Thorpe (1968) seem to be unrelated toPhillips predictions, or the numerical predictions, or the data. The most reasonablepredictions seem to be provided by the dispersion relation for the exponential profile:these lie between Hyden's results and the this project's results. Therefore equation 31, thedispersion relation for the exponential density profile, is a reasonable choice forestimating the effect of interface thickness in designing further experiments.It might be helpful to devise experiments that consider a wider range of normalizedinterfacial thickness, perhaps from 0 to 0.5. Such experiments could be repeated forvarious density differences, perhaps 5, 10 and 20 kg/m3. Using the same nominal layerthicknesses, such that the interface could grow both up and down without interference ofthe surface or bottom, data could be obtained that might shed more light on the behaviourof period and amplitude with interfacial thickness and density difference. Neither this58experiment nor Hyden's covered a wide range of normalized interfacial thickness. Itwould be beneficial if a wide range of normalized interface thickness were covered bythe same experiment to discount any effect variations in setup and procedure might haveon the results.Increased interface thickness appears to increase the decay coefficient of the system.That is, waves are less persistent as the interface widens, possibly due to a greater rangeof available modes for energy to be partitioned into, see table 4. The results allowedestimation of the viscosity which are consistent with known values of water's viscosity,m21.1(10-6) - at 20° C.The data are insufficient to make many conclusions regarding the relation of responsecurve width and interface thickness, but it seems apparent that to get significant response,the forcing frequency should be within 2-5 % of the resonant frequency (refer to figure21 and table 3). Calculation of the resonant frequency is uncertain, but it is reasonable tobase the calculation on density profiles. One can proceed in either of two ways:1) Assume the centre of the profile indicates the location of the interface forthe purpose of equation 19, to determine the period for a two layer system. Thenone would determine the effect of the finite interface by estimating it from thedensity profile, normalize it (by dividing by the total depth) and determine the ratioof the expected period to the one calculated by equation 19 from the graph of thedispersion relation for the exponential density profile.2) Estimate the extent of the interfacial region and the upper and lowerboundaries of it, and use this directly in the equation for the dispersion relation forthe exponential density profile.59A beat envelope is observed in many of the raw data traces and energy is observed in atvery low frequencies in all the FFTs, frequencies associated with the envelopes. Thebeating corresponds exactly to expectations, within the limits of error. The beating maybe used in other experiments as a means of detecting the actual resonant period of thesystem by adding or subtracting the beat frequency to the forcing frequency as has beendone in table 2 for the data of this project.Considerable energy goes into higher harmonics of the system (approximately 20-25%),even when the systems are pumped very close to the resonant period. This significant"loss" from the first mode might be important in an on-site application as first modebehaviour is likely to be that which is sought, and the energy leakage may make thedifference between success and failure. There is no consistent trend of increased ordecreased energy in higher modes as the interface thickens, as noted in section 5.7. Thisis an area of interest that might be explored more closely in subsequent work. Oneimprovement easily attainable is increased resolution in the FFTs: by increasing thenumber of data points taken by increasing the sampling frequency and duration ofsampling and using more points in the transforms better resolution may be obtained.This might help determine the ratios of the energy in various modes to the total withprecision.Further work in this area could include a closer look at the relation of interfacial thicknesson amplitude response, resonant period, and energy dissipation as these are all critical inany application at Kootenay Lake or elsewhere. A range of density differences might beexplored. Different geometries of the experimental system might be explored, with thewithdrawal closer to the top of the tank and with the light upper layer being less thick60than the dense lower layer, somewhat closer to the situation in a typical deep lake orreservoir.Finally, it appears reasonable to consider investigations on small reservoirs controlled bydams to further test the effect of internal wave generation by forced resonance, based onthe positive results obtained during this project in the generation of internal waves bypumping.61ReferencesCarmack, EC and CBJ Gray 1982, Patterns of circulation and nutrient supply in amedium residence-time reservoir, Kootenay Lake, BC, Can. Water Res. J., 7:1.Carmack, EC, RC Wiegand, RJ Daley, CBJ Gray, S Jasper, and CH Pharo 1986,Mechanisms influencing the circulation and distribution of water mass in amedium residence-time lake, Limnol. Oceanogr., 31(2):249.Cloern, JE 1976, Recent limnological changes in southern Kootenay Lake, BritishColumbia, Can. J. Zool., 54:1571.CRC Handbook of Chemistry and Physics, 66th Edition 1986, (pub: CRC Press, BocaRaton, Fla).Crozier, RJ, and WFA Duncan 1984, Kootenay Lake 1984: compilation and synopsisof physical, chemical and biological data from 1968 to 1984, Province of BC,Min. of Environment.Csanady, GT 1972, Response of large stratified lakes to wind, J. Physical Oceanogr., 2:3Csanady, GT 1973, Wind induced barotropic motions in long lakes, J. PhysicalOceanogr., 3:439.Csanady, GT 1975, Hydrodynamics of large lakes, Ann. Rev. Fluid Mechanics, 7:357.Csanady, GT 1982, Circulation in the coastal ocean, (pub: D Reidel, Boston).Daley, RJ, EC Carmack, CBJ Gray, CH Pharo, S Jasper, and RC Weigand 1981,The effects of upstream impoundments on the limnology of Kootenay Lake, BC,Scientific Series #117, Nat'l Water Res. Inst., Inland Waters Directorate.Davis, RE, and A Acrivos 1967, The stability of oscillatory internal waves, J. FluidMechanics, 30:723.Hamblin, PF, and EC Carmack 1980, A model of the mean field distribution of adissolved substance within the riverine layer of a fjord lake, Water Res. Res.,16(6):1094.Harrison, WJ 1908, The influence of viscosity on the oscillations of superposed fluids,Proc. Lond. Math. Soc. (2), 6:396.Hutchinson, GE 1974, A treatise on limnology, (pub: John Wiley and Sons Inc, NY).Hyden, H 1974, Water exchange in two layer stratified waters, ASCE, Hydraulics J Div,100(HY3):345.Imberger, J 1980, Selective Withdrawal: a Review, Second International Symposium onStratified Flows, 1:381.62Keller, JB and WH Munk 1970, Internal wave wakes of a body moving in a stratifiedfluid, Physics of Fluids, 13(6):1425.Krauss, W 1966, Methoden und Ergebnisse der Theoretischen Ozeanographies II InterneWellen, (pub: Gebriider Borntraeger, Berlin-Nikolassee).Lamb, H, 6th Edition, 1944, Hydrodynamics, (pub: Dover Pub, New York)Lasenby, DC, TG Northcote and M Hirst 1986, Theory, practice and effects of Mysisrelicta introductions to North American and Scandinavian lakes, Can. J. Fish.Aquat. Sci., 43:1277.Lathi, BP 1983, Modern Digital and Analog Communications, (pub: Holt, Tinehart andWinston, New York).Martin, AD, and TG Northcote 1991, Kootenay Lake: and inappropriate model forMysis relicta introduction in north temperate lakes, American Fisheries SocietySymposium, 9:23.Mortimer, CH 1952, Water movements in lakes during summer stratification; evidencefrom the distribution of temperature in Windermere, Phil. Trans. Roy. Soc. B,236:355.Mortimer, CH 1953, The resonant response of stratified lakes to wind, Schwiez. Zeit.Hydrol., 15:94.Mortimer, CH 1961, Motion in therrnoclines, Verh. Internat. Verein. Limnol., 14:79.Mortimer, CH 1974, Lake hydrodynamics, Mitt. Internat. Verein. Limnol., 20:124.Mortimer, CH 1975, Substantive corrections to SIL communications (IVLMitteilungen) Numbers 6 And 20, Verh. Internat. Verein. Limnol., 19:60.Northcote, TG 1972a, Kootenay Lake: man's effects on the salmonid community, J.Fish. Res. Bd. Can., 29(6):861.North cote, TG 1972b, Some effects of mysid introduction and nutrient enrichment on alarge oligotrophic lake and its salmonids, Verh. Internat. Verein. Limnol.,18:1096.Northcote, TG 1973, Some impacts of man on Kootenay Lake and its salmonids, TechReport 25, Great Lakes Fishery Commission.Northcote, TG 1991, Success, problems, and control of introduced Mysid populations inlakes and reservoirs, American Fisheries Society Symposium, 9:5.Phillips, OM 1966, Dynamics of the upper ocean, (pub: Cambridge University Press,London).63Sparrow, RAH, PA Larkin and RA Rutherglen 1964, Successful introduction ofmysis relicta into Kootenay Lake, British Columbia, J. Fish. Res. Bd. Can.,21:1325.Standard Methods, 17th Edition, 1989, (pub: APHA, AWWA, WPCF, Washington,DC).Thorpe, SA 1968, On standing internal gravity waves of finite amplitude, J. FluidMechanics, 32(3):489.Thorpe, SA 1971, Asymmetry of the internal seiche in Loch Ness, Nature, London,231:306.Thorpe, SA 1974, Near-resonant forcing in a shallow two-layer fluid: a model for theinternal surge in Loch Ness?, J. Fluid Mechanics, 63(3):509.Ting, FCK 1992, On forced internal waves in a rectangular trench, J. Fluid Mechanics,235:255.Turner, JS 1973, Buoyancy effects in fluids, (pub: Cambridge University Press,Cambridge).Voisin, B 1991, Internal wave generation in uniformly stratified fluids. Part 1. Green'sfunction and point sources, J. Fluid Mechanics, 231:439.Walters, C, J Digisi, J Post, J Sawada 1991, Kootenay Lake Fertilization ResponseModel, Dept. Resource Ecology (UBC).Watson, ER 1904, Moments of the waters of Loch Ness as indicated by temperatureobservations, Geogr. J., 24:430.Wedderburn EM 1911, The temperature seiche, Trans. Roy. Soc. Edin., XLVII part IV(#22):93.Wetzel, RG 1975, Limnology, (pub:WB Saunders Co:Toronto).Wiegand, RC and EC Carmack 1981, A wintertime temperature inversion in KootenayLake, British Columbia, J. Geophys. Res., 86(C3):2024.Wilson, BW 1972, Seiches, Advan. Hydrosci., 8:1.Zyblut, ER 1970, Long term changes in the limnology and macrozooplankton of a largeBritish Columbia lake, J. Fish. Res. Bd. Can., 27(7):1239.64den41ss^IFrofi 1ge4^pe^ee eu-I+N consfmnfc)^co els4- ck.nf "-*^= /00 k Z0.0/CD^ et a Coryli a '1^is ireta ca-7—cc/iv^9-7 Mal reTcrit4 A rCsirin4 1,1 1.'5^i ce-7rFigure 1 - physical arrangement and parameters for(a) two layer systems/c, = " ex+ Sikr^e Z = 0(b) systems with constant Brunt-Vdisdla frequency(c) systems of constant density gradient between two homogeneous layers•IMI•65To cluster0140.12010000=-<G060040120-6^0.7 0.9^14^PI^11^1-3^14^P5^14a.Figure 2 - From Thorpe (1968)(b)Pim= 10. The apparatus; (a) side view. (b) detail in plan of the plunger drive.(a) Experimental apparatusnor= 12. Response curves for different density differences and constant total plungeramplitude 0.3$ cm. Total wave amplitude, Coy, divided by wavelength, A, is plotted againstplunger frequency a measured in radians per second. The curves from left to right are fordensity differences of 4.7, 10-0, 15.0 and 22•8, 10-3g/c.c. respectively.(b) Experimental results66for internal wavesgrid for velocitymeasurementconductivity meters1(udjustable level)I^wave gage for internal waveswave gage0.25-0.5m0.5mIi•freshwatersaltwater maw; 36m16.72m1 5mFigure 3 - From Hyden (1974), experimental apparatusFIG. 4.--Laboratory Set-Up (1 ft = 0.305 m)67^L.. 19.15 mT = Locations of wave measurementsFIGURE 3. Experimental arrangement and locations of wave measurements.Figure 4 - From Ting (1992)(a) Experimental apparatus25IL% 20H,1510_0.1^0.3^0.4^0.5^0.6^0.7klFtcutut 5. Response curves for internal waves for a sharp density interface. The theoretical resultsare computed using 111 = 22.15 cm. At = 8.95 cm, J - 1.3 cm, and 1,2/p1. 1.05. a- 0.0035<- fis/h <0.01; x. 0.012< Halt <0.016; LI, 0.019< HdA <0.025: —. three-layer viscous theory:---, three-layer inviscid theory.(b) Experimental results0PumpPCConductivity Probe^ 1.235m^CentreWallFigure 5 - Experimental ArrangementI = inflow to tanks_O.= outflow from tankR = return to tank2.4MPLAN" 10.525Mob, RMEND MN. GM.EquilibriumInterface0.300.160ELEVATION69Sensitivity of the Theoretical PeriodCurves^Legend(a) shows variation of theoretical period for 0.1 kg/m3variations in the density difference when the upperdensity is 1000 kg/m3and the lower density rangesfrom 1003.9 kg/m3 to 1006.9 kg/m3(other parameters: hu=365mm, h1=160mm and L=1235mm)(b) shows variation of theoretical period for 1mm variationsin the difference of the upper and lower layer depths,hu-hl, such that150<hl<165 and hu+h1=525mm(also, L=1235mm)(c) shows variation of theoretical period for 1mm variationsin length L such that 1225<L<1245 (also, hu=360mm, h1=160mm)Note: for (b) and (c),(1) density difference = 4.0 kg/m3(2) density difference = 6.8 kg/m370Interface Oscillation Data ... Experiment 1, runs 1-5see key, table 2.1EEco=cc>oiu/l•^= probe locationWV\iy.^.I^.vYTIr1rrrvr"vYTT rrr7771^Trrrrvv\r-vvvV\.^A^A^. I^•^..Mr y T Tr TVIIPWV 7 y V.- - '- y In r^'irWW 1111 T yA.^•^A^• •lOMM'V 7 T y y^Ty r-V.WWT rrnrerl. y y rvirwv0^II^11111111^11110^0^0^0^00 0 0 0 04-^C■I^V)^NI'^InTime (s)II0^0^00 0 0CO^n^coFigure 771vvVi^IIIirgy*T^v."1-fyy!! TT.Alit^iAl.vyy .i1/1/1ATTYYMYYVvvv1 1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^110mm0^0^0^0^0^0^0^0^0^00 0 0 0 0 0 0 0 0COTime (s) Figure 8Interface Oscillation Data ... Experiment 1, runs 6-10(see key, table 2.1) 72Interface Oscillation Data ... Experiment 1, runs 11-15(see key, table 2.1) 73o^o^oo o oin co N.o^00^0^00^0 0 0Time (s) Figure 101 m mI I^, . ..A A I•^11^I^:n.vrv-v-vyvvvol'9i-T,I.1 I^ 5f\j4V\AN'V./1‘NI\AVvvv\i v-\AA,,1 1^1^1^1^1^1^1^1^1^1^I IInterface Oscillation Data ... Experiment 2, day 1, runs 1-7(see key, table 2.2)74Interface Oscillation Data ... Experiment 2, day 1, runs 7-14see key, table 2.2------1 4----.-"-- ---1 31 2VE 1 iEcofa>ww(\(\r\i\\11:0\r\ "1\9 \, 1 ommII^I^I^I^I^I^I^i il^III0 00 0 0 0^0^00^0^0^0^0 0 0.,— cy v Tr LO (0 N.Time (s) Figure 1175Interface Oscillation Data ... Experiment 2, day 2, runs 1-10see key, table 2.3I.Ecor.g3.oLaIS'''.4.1w4%wriek^ 1 0E-N.,r \ re\ ,i. rportilrhgn"r"."44".^741...,,.n. au.^...44rTnTymeliiinerrrt-rn-vrlurr1-ni'YN-..49rt.nr."--.4""u‘'f"'#'1""8 wrn"teiraw*Frr4-J,"1"mw.4141—+".1".""""47 ffl4.5""rALPVM,Prk,-mdriwn‘..(V\AX-xi \IL\ti\r"A'1\1\\PV\Gi■ 1 mm._.....eitnelinsLcAppsicom,,ct±,••01111111111111100^0^0^0^0^00^0 0 0 0 0 0— cv co ‘t ,c, (0 N.Time (s) Figure 1276Akiliiiiii1 1^I^I^IP2Aiv ^1 7 v^1 0 m m1 6vTI\^\'\ \\^ v10^0^0^0^0^0^0^00 0 0 0 0 0 0CV CDTime (s) Figure 13Interface Oscillation Data ... Experiment 2, day 2, runs 11-17(see key, table 2.3)77AlA10^aIfiVilllffillAPTATIL ALAk.11811111Mil1111,11111111MIRM ‘01111rv, ^'111Mil Ty Fr ii^r^R Aall1210111111111111111FOINM6110101l!'t 1111 11 mmI 11IIIA a kL^L, LI+ it^• I^," TT lr^r, ir I" TT rTITTY91I"P^a. i^L^.. A ALA& ANA&^I.&-up-^•ATIATZTA„ATIPEL.1^rillINI1911111111I I^I^I 1^1^1^1^1^1^I!0°Time (s)^Figure 14Interface Oscillation Data ... Experiment 2, day 3, runs 1-10(see key, table 2.4)78Interface Oscillation Data ... Experiment 2, day 3, runs 11-17(see key, table 2.4)79Interface Oscillation Data ... Experiment 3, day 1(see key, table 2.5)80Interface Oscillation Data ... Experiment 3, day 2, runs 1-7(see key, table 2.6)A VAWJAVV vvvalFAVAVATIPOSPOPOS lOmm1 ^,^/Wirt y iiMAI1111111/^vvrifiritilififiyArittavas A A,0^0^0^0^0^0^0^0^00 0 0 0 0 0 0 0inTime (s)^Figure 1781Interface Oscillation Data ... Experiment 3, day 3, runs 1-7(see key, table 2.7)82Density Profiles for All Experiments83Variation of Resonant Period with interfacial Thickness: Data84Amplitude Response for All ExperimentsECDTS=0.E<ALCO0o.ioAleeua.60-40-20-____—_Experiment 3Day1rid0 ,,.,,'.,;,.EDay 2,3Day 5^ _--20-_—e0Experiment 2'.-r ?c, „..^,.,'20-Experiment 1I^I^I^1^I0.210.310.4^0.5^0.6^0.7^0.8^0.9^1Non-dimensional Period (Tf/Tr)1^11.1^1.2Figure 211.385Spectre: Experiment 10.00140.001272"^0.00113oii 0.0008O....>, 0.0006 -cntiCuj 0.0004 -0.0002 ----20.031Hz (T=32.0s)o-'N 0i^•c,,^coOild I-8,13^12416.111■..._N^r... i^•1- 00 ,-10.1i_o --_000.02^0.04^0.06^0.08Frequency0.1^0.12^0.14^0.16^0.18^0.2^(Hz) Figure 2286Spectra: Experiment 2, day 1, runs 1-40.0000350.00003 -1-70.000025'007.w^0.00002 -12.....>, 0.000015 -E°4,)C^0.00001 -LLI0.00000500-1CZN 0)X^•c7) Z.;0^II6 F- NXco......^COCA 0N CV cix^•0 '4"F._ 0.^ii \o /- \Ilico•.-7"1.7.\0.08FrequencyN1Tr8ci17')r..„•07/-NXcoo•o0.111Cin-0)I--0.12(Hz)n, .--X 00coII7 1-0^.....0.14^0.16^0.18^0.2Figure 230.02^0.04^0.06Spectra: Experiment 2, day 2, runs 1-50.00006150.00005"Ili- 1,3,4.13^0.000040 N S..= X 7m 0 7•■ 0.00003>.N T,-^ dx 0,1;i'N011I-""eZNF.0 0.00002cv^z .7) c7)^X^•o■II^I-^cm CVCO 7c.)X •CO 0CO 7C O 0^i^i Oiliii 0b I0.000014t,^AVIMAIL,411"-- -•-,--.4*-----^......1-..,4..4,..-A^....^a......t. -.-----.... -_-- --....* _0 0.02^0.04^0.06^0.08^0.1^0.12 0.14 0.16^0.18 0.2Frequency (Hz) Figure 2487Spectra: Experiment 2, day 5, runs 1-50.000080.00007-40.00006 -5 ""cr)N.0 1 2^•o1:2 0.000054)0)^..-N 17;2 cb^ 0 , II^7ri^. 1- N cvCL0.00004-CI^• 'CZ^° ''''c%) (7)^- , -N "I^•CO °••••.. 0 j! Z..,i c■i '-Chco^II>1 5 d :7,^8 7 61-E) 0.00003\6 I-N.Cla 0.00002 -32 2 co^•o coil:::"; 1 _0.00001 4‘Oakail*,44_4114 it01-140.-r,A,_,,,___,---.•-k 1A--4-- -0 0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18 0.2Frequency (Hz) Figure 25Spectra: Experiment 2, day 1, runs 5-100.000140.070Hz (T=14.2s)0.00012 -';-‘0.0001 -^0.031Hz (T=31.9s)"ao'170 0.00008 -^1a-.. 5)., 0.00006 -4) 0.144Hz (T=6.9s)c 0.00004 -Ili0.105Hz (T=9.5s)0.00002 -41 70 0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18^0.2Frequency (Hz)^Figure 2688Spectra: Experiment 2, day 2, runs 6-100.000070.00006".70.00005o7.0 0.00004O.....>, 0.00003 -I?•= 0.00002 -ILI0.00001 -0---0.0313Hz (Tr=31.9s)0.0703Hz7\8 .(Tr=14.2s)0.101Hz (Tr=9.9s),0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18^0.2Frequency (Hz) Figure 27Spectra: Experiment 2, day 5, runs 6-100.000120.0703Hz (T=14.2s)0.0001 --)0.0313Hz (T=31 .9s)-13^0.000080 -T.)0....^0.00006 ->.e0 0.00004 -6^61C 0.133Hz (T=7.5s)Ill0.094Hz (T=10.7s)0.00002 -76^I60,,,garSi h.^-..-__^_.,- ill■-■^.... ....0^0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18^0.2Frequency (Hz) Figure 2889Spectra: Experiment 2, day1, runs 7-140.00140.0313Hz (Tr=31.9s)0.0012 --, -^ill1:1oi=n)a.0.0008 - 17')cv\^ d■ 1 0 I>,F2)0.0006 -^7 ^I-N a,i^• u, N•CLU0.0004 —CDT0^II 2i— coo ...-^choö0.0002 - Q• I 1.,;44 NI_^10_^1.......,,,,..._00^0.02^0.04^0.06^0.08^0.1^0.12 0.14 0.16^0.18 0.2Frequency (Hz) Figure 29Spectra: Experiment 2, day 2, runs 11-170.00251 40.031 Hz (T=31 .9s)-J0.00211-^1513013lia0.0015 --0-.... N02^•gCV CO0.001 - 0 •-■:' 0^IIC N^,....i^•Ill nr 00.0005 -cn '-0^II1 4 6 I-I 530 I i0^0.02^0.04^0.06 0.08^0.1^0.12 0.14 0.16^0.18 0.2Frequency (Hz) Figure 3090Spectra: Experiment 2, day 5, runs 11-170.00140.0012-)0.001oi 0.00080....).. 0.0006EnWIll0.0002 -o0-1::-C 0.0004--^154#1-0.031Hzi 14,^16xcsLod15,Oil(T=31.9s)To--is-..- NI-^xoa)0.16^oa')7,-st7-A0.02^0.04^0.06 0.08Frequency^-0.1^0.12^0.14^0.16^0.18^0.2(Hz) Figure 31Spectra: Experiment 3, day 10.00070.0006-,0.0005or_. 0.0004O....\>, 0.0003 -CA4-)1110.0001 -0-*a--N_____220.020.025HzNIir-)d0.04c 0.0002-Oil)■...._kedAL`(T=40.0s)-core,6-IZ.^,4.i0.06Nx 7,a) u,ts0 6°7,^COoiN. 70. 1.-o ..--0-1incoClibNxcvo6_0.12(Hz)_IL-c-i'cobNxco,--0^in0.14...._70).'1bNXó0.08^0.1Frequency0.16^0.18^0.2Figure 3291Spectra: Experiment 3, day 30.00250.023Hz (T=42.6s)0.002 —1713o•eua.0.0015 —...>.2)0 001 —' 0.047 - 0.051Hz (T=19.7-21.3s)CDC1110.0005 — 0.078Hz (T=12.8s)R .....0 0.02^0.04^0.06^0.08^0.1^0.12 0.14 0.16^0.18 0.2Frequency (Hz) Figure 33Spectra: Experiment 3, day 50.0080.007.7_,,J 0.0061:3°-^0.005C0O.....^0.004 —>+IT 0.003—OCLti^0.002 —0.001 —0———,- -0.023Hz0.047Hz_A,4,5(T=42.6s)_ __,.._(T=21.3s)0.070Hz (T=14.2s)40^0.02^0.04^0.06^0.08^0.1^0.12Frequency (Hz)0.14 0.16^0.18Figure 340.292Free Oscillation Data: Experiment 2-gOilithiTivAiriripiwpar-,_,A,A,A,k,4■,•.,■,..,..,-,-co4;tLLIlillitgraginalinlikUlnaUMW I T 1 " " " " " • • •iiiiiii^■^a^...11111111MMEIVIIIIIIMEMEIMON111p1KIMAISWIIIIINISIP w 1Day; 1 1•7A.1•'k'■'A-4Ili^1^1^1^1^I^1 I0^0^0 0 00 0^0^0,-- CV CI •ctTime (s)00101^I^1^10 00^0CO 1••••Figure 35Free Oscillation Data: Experiment 3mrfirwmmonnuirmAyipqrAvivAwkwirm-I irwv.T.DayiI. Iiiii,,,,I INJIMILININIAMIWAIRVIATAVAVAVACIWAVAVS111 liTYVVVV^vDayv3 1>0u,lommIiA^1^I^I^I^1^1^1^A^ A^A^•^•^•^•^•IIMIIIIIIIIIIMEMBRIVIIIIIMM •wwwv/If,^",50^0^0^0^0^.^.^.. . 0 . . . .,— cv c.3 .4' 1.0 CD 1,-Time (s) Figure 3693Spectra of free oscillations: experiment 2110a...)..12)CDCLii1:7a)NToE0z0.250.2-0.150.1 -I-0.0313Hz:^(T=31.9s)- - -1,--- 0.0293Hz (T=34.1s)Day 1^Day 2^Day 5iN^Nii^=intil^71)^N0.^0NX^=o .2 ci^co N^cr, ^inA^co (3)^c>.^0,.,..^ A0^0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18Frequency (Hz) Figure 370.2Spectra of free oscillations: experiment 30.250I^0.2O......>aE) O. 1 50CII• o. 1N•E...^0.05 -0Z---■•......,1.J•0.0254Hz(T=39.8s)0.0234HzNNIIa) coco 03Tr -*0006 dA- - Day 1^Day 3^Day 5(T=42.7s)^-NIco0u-)^N^N = =d^co cyC■I^CO1.--^).... 06 ciI^Itt0^0.02 0.04^0.06^0.08^0.1^0.12^0.14^0.16^0.18Frequency (Hz) Figure 380.294o ,0^tO SO^. "PULOmE I RESDUNCAN k^CO‘')DAM ■4AR GEIS T•DUNCANLAKEpAkc.4.11,,CRKIMBERLEY•"44..CORRAUNN DAMNELSONS cw7.5.1ED STATESFERN'SFigure 39 - Kootenay Lake and South-Eastern British Columbia95
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Experiments with withdrawal-induced resonant internal waves Allan, Gary 1993
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Title | Experiments with withdrawal-induced resonant internal waves |
Creator |
Allan, Gary |
Date Issued | 1993 |
Description | Control of natural and artificial reservoirs by means of outlet structures (specifically, dams) has been considered by Imberger (1980). Such control has been proposed on Kootenay Lake in eastern British Columbia, as a means to modify the ecology of the lake. This study considers the feasibility of pumped control of a reservoir and the practical problems associated with application on the lake. The generation of internal waves in a rectangular tank by a pump with sinusoidally modulated flow has been investigated via laboratory experiments. Theory appropriate to internal waves in two layer systems has been presented with an extension to deal with the effect of viscosity. The study shows that modulated pumping is a viable method of generating internal waves experimentally and may have applications in controlled reservoirs on which outflow could be modulated. The period of resonant waves increases with interfacial thickness as does wave amplitude: observed periods ranged from 1.01 to 1.05 times greater than theoretically determined periods for two-layer systems. Theoretical determination of the period is based on consideration of the density structure and boundary conditions of the system under study; one would expect real conditions to produce some variation in the observed period versus the theoretical period. The result above is borne out by observation of the amplitude response curves generated by the data and by calculations of the beat frequencies associated with the forcing and generated waves and agrees approximately with the data of Hyden (1974) and Thorpe (1968). There is some indication that as the interface widens the amplitude response curves sharpen, implying that if one hoped to generate a resonant wave in a system with a thick interface one would have to be closer to the resonant period than in a system with a thinner interface to achieve a similar response. Calculation indicates that, for the peak response, the wave with maximum amplitude, 1.7 to 4 % of the energy input per cycle is present in the wave. Energy input per cycle is determined by integrating instantaneous kinetic energy of the water that is pumped over a cycle. Fourier transforms of the data show response, in some cases, in the 6th harmonic modes of the systems, with a decline in energy from the 1st to 2nd mode of 20 to 25 %, typically. Estimates are presented for the damping coefficients for the systems investigated: the average value is 0.0020 s-1. A brief review of problems associated with a practical application of resonant internal waves forced by flow modulation is presented. Several technical problems exist which need further consideration before the application is undertaken, on Kootenay Lake in eastern British Columbia. It may be practicable to try generating internal waves on a smaller lake or reservoir controlled by a dam as a means of testing the feasibility. |
Extent | 5311133 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-10-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050486 |
URI | http://hdl.handle.net/2429/2592 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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