Limnol. Oceanogr., 50(5), 2005, 1620±1637 ᭧ 2005, by the American Society of Limnology and Oceanography, Inc.

The degeneration of internal waves in lakes with sloping topography

L. Boegman,1 G. N. Ivey, and J. Imberger Centre for Water Research, The University of Western Australia, Crawley, Western Australia 6009, Australia

Abstract In a laboratory study, we quanti®ed the temporal energy ¯ux associated with the degeneration of basin-scale internal waves in closed basins. The system is two-layer strati®ed and subjected to a single forcing event creating available potential energy at time zero. A downscale energy transfer was observed from the wind-forced basin-scale motions to the turbulent motions, where energy was lost due to high-frequency internal wave breaking along sloping topography. Under moderate forcing conditions, steepening of nonlinear basin-scale wave components was found to produce a high-frequency solitary wave packet that contained as much as 20% of the available potential energy introduced by the initial condition. The characteristic lengthscale of a particular solitary wave was less than the characteristic slope length, leading to wave breaking along the sloping boundary. The ratio of the steepening timescale required for the evolution of the solitary waves to the travel time until the waves shoaled controlled their development and degeneration within the domain. The energy loss along the slope, the mixing ef®ciency, and the breaker type were modeled using appropriate forms of an internal Iribarren number, de®ned as the ratio of the boundary slope to the wave slope (amplitude/wavelength). This parameter allows generalization to the oceanographic context. Analysis of ®eld data shows the portion of the internal wave spectrum for lakes, between motions at the basin and buoyancy scales, to be composed of progressive waves: both weakly nonlinear waves (sinusoidal pro®le with frequencies near 10Ϫ4 Hz) and strongly nonlinear waves (hyperbolic±secant-squared pro®le with frequencies near 10Ϫ3 Hz). The results suggest that a periodically forced system may sustain a quasi-steady ¯ux of 20% of the potential energy introduced by the surface wind stress to the benthic boundary layer at the depth of the pycnocline.

The turbulent benthic boundary layers located at the pe- by the relative magnitudes of the period Ti of the horizontal rimeter of strati®ed lakes and oceans serve as vital pathways mode-one internal seiche and the characteristic time scale of for diapycnal mixing and transport (Ledwell and Hickey the surface wind stress, Tw (see Stevens and Imberger 1996, Ͼ 1995; Goudsmit et al. 1997; WuÈest et al. 2000). Within the their table 1 for typical values). When Tw Ti /4, an ap- littoral zone, this mixing drives enhanced local nutrient ¯ux- proach to quasi-equilibrium is achieved, with a general ther- es and bioproductivity (e.g., Sandstrom and Elliott 1984; Os- mocline tilt occurring over the length of the basin (Spigel trovsky et al. 1996; MacIntyre et al. 1999). Indirect obser- and Imberger 1980; Stevens and Imberger 1996, and Fig. vations suggest that turbulent benthic boundary layers are 1b, c). From this initial condition, the resulting internal wave energized by internal wave activity. In lakes, the turbulence ®eld may be decomposed into a standing seiche, a progres- occurs as a result of (1) bed shear induced by basin-scale sive nonlinear surge, and a dispersive high-frequency inter- currents (Fischer et al. 1979; Fricker and Nepf 2000; Gloor nal solitary wave (ISW) packet (Boegman et al. 2005). For K et al. 2000) and (2) breaking of high-frequency internal some lakes, Tw Ti, and the progressive surge is believed waves upon sloping topography at the depth of the metal- to originate as a wind-induced wave of depression at the lee imnion (Thorpe et al. 1972; MacIntyre et al. 1999; Michallet shore (see Farmer 1978, and Fig. 1d). This wave of depres- and Ivey 1999). While the former process is relatively well sion will progress in the windward direction, steepening as understood, the latter remains comparatively unexplored. it travels, and eventually forming a high-frequency ISW In lakes where the effects of the Earth's rotation can be wave packet. In the absence of sloping topography, as much neglected, the internal wave weather may be characterized as 20% of the available potential energy (PEo)Ðresulting from a wind forced tilt of the isopycnals at the basin scaleÐ may be found in the ISW ®eld (Boegman et al. 2005). This 1 To whom correspondence should be addressed. Present address: downscale energy ¯ux has signi®cant implications for hy- Coastal Engineering Laboratory, Department of Civil Engineering, drodynamic modeling. The high-frequency waves typically Queen's University, Kingston, Ontario K7L 3N6, Canada. have wavelengths of order 100 m (Boegman et al. 2003), Acknowledgments much smaller than the feasible grid spacing of ®eld-scale The authors thank K. G. Lamb, B. R. Sutherland, E. J. Hop®nger, hydrodynamic models (Hodges et al. 2000). If the high-fre- J. J. Riley, G. W. Wake, and two anonymous reviewers for com- quency waves break upon sloping topography, between 5% ments and/or discussions on this work. We also thank Herve Mich- and 25% (Helfrich 1992; Michallet and Ivey 1999) of the allet for supplying the laboratory data shown in Fig. 10. The Lake incident solitary wave energy (1% to 5% of the PEo) may Pusiano data were collected by the Field Operations Group at CWR be converted by diapycnal mixing to an irreversible increase and the Italian Water Research InstituteÐin particular we thank in the potential energy of the water column. Diego Copetti. The wavelet software was provided by C. Torrence and G. Compo. The generation and propagation of the progressive surge L.B. was supported by an International Postgraduate Research and ISW packet in natural systems with variable wind stress Scholarship and a University Postgraduate Award. This research and complex topography are not clearly understood. Many was funded by the Australian Research Council and forms Centre long, narrow lakes and reservoirs are characterized by ver- for Water Research reference ED 1665-LB. tical and sloping topography at opposite ends (e.g., at dam 1620 Internal waves in closed basins 1621

talimnion, and hypolimnion. If the vertical density gradient is abrupt through the metalimnion, the lake may be approxi-

mated as a simple two-layer system of depth h1 and density ␳ ␳ ϭ ϩ 1 over depth h2 and density 2, where H h1 h2 is the total depth and L denotes the basin length (e.g., Heaps and Ramsbottom 1966; Thorpe 1971; Farmer 1978). Internal waves may be initiated within a strati®ed lake by an external disturbance, such as a surface wind stress, ␶ (e.g., Fischer et al. 1979, p. 161). This stress advects surface water toward the lee shore, thus displacing the internal layer interface through ␩ a maximum excursion o as measured at the ends of the basin. The excursion is dependent on the strength and duration of the wind event. A steady-state tilt of the interface is achieved Ͼ ␩ for Tw Ti/4, allowing o to be expressed in terms of the ϭ ͙␶ ␳ shear velocity u* / o as Lu*2 ␩ ഠ (1) o Ј g h1

␳ Јϭ ␳ Ϫ ␳ ␳ where o is a reference density and g g( 2 1)/ 2 is the reduced gravity at the interface (see Spigel and Imberger 1980; Monismith 1987). The internal response of the waterbody to a forcing event can be gauged by the ratio of the wind-disturbance force to Fig. 1. (a) Schematic diagram of the experimental facility (not the baroclinic restoring force (Spigel and Imberger 1980). to scale). WG denotes ultrasonic wave gauge. See Experimental Thompson and Imberger (1980) quanti®ed this force with methods section for a detailed description. (b) Initial condition at t ϭ 0 with upwelling at the slope. (c) Initial condition at t ϭ 0 with the Wedderburn number, which is given for our two-layer downwelling at the slope. (d) Initial condition for the experiments system in terms of Eq. (1) as Ͼ by Michallet and Ivey (1999). (b±c, and d) Representative of Tw ␩ Ͻ Ϫ1 o Ti/4 and Tw Ti/4, respectively. W ϭ (2) h1 walls and river mouths, respectively). The sloping bathym- Weak initial disturbances (WϪ1 Ͻ 0.3) will excite a stand- etry is believed to favor breaking of high-frequency waves ing seiche that is well described by the linear wave equation (Thorpe et al. 1972; Hunkins and Fliegel 1973) as well as (Mortimer 1974; Fischer et al. 1979; Boegman et al. 2005). increasing the production of bed shear (Mortimer and Horn The periods of this seiche for a two-layer system are 1982), thus introducing signi®cance to both the direction and 2L magnitude of wind-stress, which sets up the initial thermo- (n) ϭ T i (3) cline displacement. For a complete understanding of the sys- nco tem, this directionality must be coupled with the T , T , and w i where n ϭ 1,2,3, etc., denotes the horizontal mode (herein the steepening timescale, Ts (de®ned following). In this study, laboratory experiments and ®eld observa- the fundamental timescale; Ti, without superscript, is used to ϭ ϭ tions are used to examine the dynamics of the large-scale represent the gravest mode, where n 1) and co ͙ Ј ϩ wave-degeneration process in a rectangular, strati®ed basin (g h1h2)/(h1 h2) is the linear long-wave speed. Ϫ with sloping topography. Our objectives are to quantify the Moderate forcing (0.3 Ͻ W 1 Ͻ 1.0) results in the devel- energy loss from the breaking of the ISW packets as they opment of a nonlinear surge and dispersive solitary wave shoal and to cast the results in terms of parameters that are packet (Thorpe 1971; Horn et al. 2001). The temporal de- external to the evolving sub±basin-scale ¯ow (e.g., wind velopment of the surge may be quanti®ed by the nonlinearity speed and direction, boundary slope, quiescent strati®cation, parameter de®ned by Boegman et al. (2005) as etc.). This will facilitate engineering application and param- a␣ 3␩ ͦh Ϫ h ͦ eterization into ®eld-scale hydrodynamic models. We ®rst ϳ o1 2 (4) present the relevant theoretical background. The laboratory co122 hh experiments are then described, followed by a presentation ϳ ␩ where we have used the wave amplitude scaling a o (e.g., of results. Finally, the results are summarized and placed ␣ ϭ within the context of what is presently known about the en- Horn et al. 1999) and the nonlinear coef®cient (3/2)co(h1 Ϫ ergetics of strati®ed lakes. h2)/(h1h2) (Djordjevic and Redekopp 1978; Kakutani and Yamasaki 1978). If the interface is at middepth, ␣ vanishes, Theoretical background steepening cannot occur, and there is no production of sol- itary waves. As the progressive nonlinear surge steepens, its During the summer months, a strati®ed lake will typically length scale decreases until nonhydrostatic effects become possess a layered structure consisting of an epilimnion, me- signi®cant and the wave is subject to dispersion (see Ham- 1622 Boegman et al.

→ ϭ mack and Segur 1978). This occurs as t Ts, the steepening where Ls denotes the slope length and x 0 at the toe of timescale (Horn et al. 2001) the slope. The location of the turning point on the slope is ϭ given in nondimensional form by letting h2(x) h1 L T ϭ (5) Ϫ s ␣␩ xh21h o ϭ (10) LHs Steepening is eventually balanced by dispersion and the As a wave travels along the slope, it steepens, a increases, surge degenerates into a high-frequency ISW packet. Internal and the streamlines approach vertical. During steepening, the solitary waves may be modeled to ®rst order by the weakly maximum horizontal ¯uid velocity in the direction of wave nonlinear Korteweg±de Vries (KdV) equation propagation umax will increase more rapidly than c and a limiting amplitude may be achieved where the velocities in 3␩ץ ␩ץ ␩ץ ␩ץ ϭ ϩ ␣␩ ϩ ␤ ϭ the wave crest become equal to the phase velocity. Here, this co 0 (6) x3 limit is de®ned as the breaking limit and the location on theץ xץ xץ tץ slope where this limit is observed is the breaking point. ␩ where (x,t) (positive upward) is the interfacial displacement The type of internal wave breaking that occurs at the ␤ ϭ and the dispersive coef®cient (1/6)coh1h2. A particular breaking point may be inferred by analogy to surface break- solution to Eq. (6) is the solitary wave equation (Benney ers. Of the continuum of surface breaker types that exists, 1966), four common classi®cations are used (see Komar 1976). Spilling breakers occur on mildly sloping beaches where x Ϫ ct steep waves gradually peak and cascade down as white wa- ␩ Ϫ ϭ 2 (x ct) a sech΂΃␭ (7) ter. Plunging breakers are associated with steeper beaches and waves of intermediate steepness, the shoreward face of where the phase velocity, c, and horizontal length scale are the wave becomes vertical, curling over and plunging for- given by ward as an intact mass of water. Surging breakers occur on steep beaches where mildly sloping waves peak up as if to 1 plunge, but the base of the wave surges up the beach face. c ϭ c ϩ ␣a (8a) o 3 Collapsing breakers, which mark the transition from plung- ing to surging, begin to curl over and then collapse upon 12␤ themselves with some of the water mass surging forward up ␭2 ϭ (8b) a␣ the slope. For surface waves, Galvin (1968) and Battjes (1974) Note the functional dependence between ␭ and a for the found that the ratio of the beach slope S to the wave slope nonlinear waves. (a/␭) was suitable for classifying the breaker type. This ratio Breaking of high-frequency ISWs may occur if the waves is expressed as either off-shore or near-shore forms of the propagate along the density interface into coastal regions Iribarren number ␰, with sloping topography. Off-shore, the interface is typically S positioned such that h Ͻ h and ␣ Ͻ 0, resulting in ISWs ␰ ϭ (11a) 1 2 b ␭ 1/2 of depression. As a bounded slope is approached, the shoal- (ab/ ϱ) ϭ ing waves will encounter a turning point, where h2 h1 and S ␣ ϭ Ͼ ␣ Ͼ ␰ϱ ϭ (11b) 0. Beyond the turning point, h1 h2 and 0, causing 1/2 (aϱϱ/␭ ) the ISW to change polarity and become waves of elevation. Analytical KdV theories (see Miles 1981 for a review) have where the subscripts ϱ and b refer to off-shore and near- been extended to model the evolution and propagation of shore wave properties, respectively (Fig. 2a,b). The breaker

solitary waves when the topography and background ¯ow height ab is measured when the wave face ®rst becomes ver- are slowly varying (e.g., Lee and Beardsley 1974; Djordjevic tical in the surf zone. and Redekopp 1978; Zhou and Grimshaw 1989). Horn et al. The dynamics of internal wave breaking on sloping to- (2000) further extended these theories to a strong space±time pography have been classi®ed according to the ratio of ␭ϱ

varying background, but propagation through the turning to the slope length, Ls (e.g., Michallet and Ivey 1999; Bour- point and wave breaking were necessarily avoided. First- gault and Kelley 2003). This parameter retains no knowledge order analytical models do not allow transmission of the of the actual boundary slope and consequently may not be ISWs through the singular point at ␣ ϭ 0. used to generalize results. This is shown in Fig. 2c,d, where, ␭ ± The location of the turning point may be easily obtained in both diagrams, ϱ/Ls is equivalent, yet H1/Ls H2/Ls and for a quiescent two-layer ¯ow with sloping topography in different breaking dynamics are expected as the internal the lower layer. De®ne the thickness of the lower layer along waves shoal. The utility of the ratio of the beach slope to the slope in x as the internal wave slope has been suggested (e.g., Legg and Adcroft 2003), but a formal classi®cation based on internal wave data has yet to be performed. Perhaps this stems from H ϭ Ϫ Ϫ the dif®culty in measuring the parameters a , ␭ϱ, and aϱ for h21(x) H h ΂΃x (9) b Ls internal waves. It is preferable to recast ␰ in terms of readily Internal waves in closed basins 1623

Experimental methods

The experiments were conducted in a sealed rectangular acrylic tank (600 cm long, 29 cm deep, and 30 cm wide) into which a uniform slope of either 1/10 or 3/20 was in- serted, extending the entire height of the tank and positioned at one end (Fig. 1). The tank was ®lled with a two-layer strati®cation from reservoirs of fresh and saline ®ltered wa- ter (0.45-␮m ceramic ®lter). For visualization purposes, the lower layer was seeded with dye (Aeroplane Blue Colour 133, 123). Prior to commencing an experiment, the tank was rotated to the required interfacial displacement angle. From this condition, the setup and subsequent relaxation from a wind-stress event was simulated through a rapid rotation of the tank to the horizontal position, leaving the interface in- clined at the original angle of tilt of the tank. Depending on Fig. 2. Schematic diagram of wave and slope properties: Off- the initial direction of rotation prior to commencing an ex- shore wave amplitude, aϱ, off-shore wavelength, ␭ϱ, near-shore periment, the resulting inclined interface at t ϭ 0 could be breaker height, ab, boundary slope, S, slope-length, Ls, and slope-

height, H1 and H2. characterized as either upwelling on the slope (Fig. 1b) or downwelling on the slope (Fig. 1c). The ensuing vertical displacements of the density interface ␩(x,t) were measured measured variables (e.g., wind speed, quiescent ¯uid prop- using three ultrasonic wave gauges (Michallet and BartheÂ- 2 erties). This is accomplished for ISWs with a sech pro®le lemy 1997) distributed longitudinally along the tank at lo- ␭ ϳ ␣␩ by noting that the wave slope aϱ/ ϱ o/co, resulting in cations A, B, and C (Fig. 1). The wave gauges logged data S to a personal computer at 10 Hz via a 16-bit analog-to-digital ␰ ϭ (12) sech (␣␩ /c )1/2 converter (National Instruments PCI-MIO-16XE-50). The oo experimental variables considered in this study, together ␩ where o is easily estimated using (1). Similarly, for sinu- with the resolution with which they were determined, are ␭ ϳ ␩ soidal waves aϱ/ ϱ f o/co giving given in Table 1. S To visualize the interaction of the internal wave ®eld with ␰ ϭ (13) the sloping topography, digital images with a resolution of sin ( f␩ /c )1/2 oo 1 pixel per mm were acquired at a rate of 5 Hz from a In the following sections, progressive sinusoidal waves are progressive scan CCD camera (PULNiX TM-1040) addressed and the wave frequency, f, is discussed. equipped with a manual zoom lens (Navitar ST16160). In-

Table 1. Summary of experimental runs. The experimental variables together with the resolution with which they were determined: the Ϯ ␩ Ϯ slope, S, the interface depth, h1 ( 0.2 cm), the maximum interface displacement, 0 ( 0.2 cm), the horizontal mode one basin-scale internal ⌬␳ ഠ Ϫ3 Ϯ wave period (Ti), the steepening time scale (Ts), and the density difference between the upper and lower layers, 20 kg m ( 2kg Ϫ3 ␩ ϩ Ϫ m ). The 0 values were measured along the vertical end wall, where the and symbols denote an initial displacement as measured above and below the quiescent interface depth, respectively (i.e., ϩ as in Fig. 1c and Ϫ as in Fig. 1b).

Ϫ1 Ϫ1 Run Sh1/HWTi (s) Ts (s) Run Sh1/HW Ti (s) Ts (s) 1 3/20 0.19 Ϫ0.39 130 146 19 1/10 0.19 Ϫ0.26 130 218 2 3/20 0.18 Ϫ0.86 133 66 20 1/10 0.20 Ϫ0.81 128 70 3 3/20 0.21 Ϫ0.93 126 61 21 1/10 0.19 Ϫ0.85 130 67 4 3/20 0.29 Ϫ0.38 113 167 22 1/10 0.30 Ϫ0.23 112 283 5 3/20 0.29 Ϫ0.58 113 109 23 1/10 0.30 Ϫ0.50 112 130 6 3/20 0.29 Ϫ0.90 113 71 24 1/10 0.27 Ϫ0.92 115 66 7 3/20 0.47 Ϫ0.35 102 862 25 1/10 0.51 Ϫ0.29 102 2,880 8 3/20 0.47 Ϫ0.73 102 413 26 1/10 0.51 Ϫ0.59 102 1,416 9 3/20 0.48 Ϫ0.90 102 493 Ð Ð Ð Ð Ð Ð 10 3/20 0.20 ϩ0.43 128 132 27 1/10 0.20 ϩ0.15 128 379 11 3/20 0.19 ϩ0.62 130 92 28 1/10 0.20 ϩ0.85 128 67 12 3/20 0.20 ϩ0.86 128 66 29 1/10 0.20 ϩ1.0 128 57 13 3/20 0.30 ϩ0.34 112 191 30 1/10 0.27 ϩ0.25 115 244 14 3/20 0.30 ϩ0.60 112 108 31 1/10 0.30 ϩ0.50 112 130 15 3/20 0.30 ϩ0.86 112 76 32 1/10 0.30 ϩ0.91 112 72 16 3/20 0.49 ϩ0.29 102 2,997 33 1/10 0.51 ϩ0.29 102 2,880 17 3/20 0.50 ϩ0.41 102 Ð 34 1/10 0.51 ϩ0.66 102 1,265 18 3/20 0.50 ϩ0.82 102 Ð Ð Ð Ð Ð Ð Ð 1624 Boegman et al.

Fig. 3. Video frames showing the wave ®eld evolving from the initial condition in (a). The surge and ISW packet are propagating to the right in (b) and to the left in (c), (d), (f), and (g). Wave breaking is shown to occur upon the slope. For this experiment, h1/ ϭ ␩ ϭ H 0.29 and o/h1 0.90. The apparent dye-free layer near the tank bottom is a spurious artifact of light re¯ection. dividual frames were captured in a LabVIEW environment from a digital frame grabber (National Instruments PCI- 1422) and written in real-time to disk. The tank was illu- minated with backlight from twenty-eight 12-V, 50-W hal- ogen lamps (General Electric Precise MR16; Fig. 1). To remove ¯icker associated with the AC cycle, a constant DC supply was maintained to the lamps via four 12-V car bat- Fig. 4. (a) Time series and (b) continuous wavelet transforms teries. The images were corrected for vignetting, variations showing the temporal evolution of the internal surge and solitary in illumination intensity, dust, and aberrations within the op- wave packet for the case of no slope (case 1). (c,d) Same as (a) and (b) except for the initial condition of upwelling on the 3/20 slope tical system and pixel gain and offset following the methods (case 2). (e,f) Same as (a) and (b) except for the initial condition of Ferrier et al. (1993). This procedure required ¯at-®eld ␩ of downwelling on the 3/20 slope (case 3). For all experiments, o/ composite images of (1) the room darkened and the lens cap ഠ ഠ ഠ h1 0.7, h1/H 0.2, and Ts 70 s. The arrows in (c) and (e) on the camera (dark image), as well as (2) the tank ®lled denote the direction of wave propagation relative to the tank sche- with dyed saline water from the reservoir (blue image), and matics shown in Fig. 1c,d, respectively. (3) ®ltered tap water (white image). From the three ¯at-®eld image composites, a linear regression was performed on the gray scale response at each pixel location versus the mean density interface. Characteristic of a standing horizontal pixel response across the CCD array to yield slope and in- mode-one (H1) seiche, the lower layer moved toward the tercept arrays. The experimental images were subsequently downwelled end of the tank with a corresponding return ¯ow corrected at each pixel by subtracting the dark image, sub- in the upper layer. Progressing initially from left to right, an tracting the intercept array, and dividing by the slope array internal surge and ISW packet were also evident (Fig. 3b). (e.g., Cowen et al. 2001). Note that a correction for light These re¯ected from the vertical end wall and progressed attenuation was not required as a result of the backlight ge- left toward the slope (Fig. 3c). The ISWs subsequently ometry. The gray scale pixel response within each image shoaled upon the topographic slope and wave breaking was was then calibrated to the ¯uid density by associating mea- observed (Fig. 3d,e). Not all wave energy was dissipated at sured densities in the ¯uid layers with the mean corrected the slope, as a longwave was re¯ected (Fig. 3e, right side of pixel response of the blue and white images and mapping tank). This was re¯ected once again from the vertical wall the intermediate values according to the log-linear dye ver- and a secondary breaking event was observed (Fig. 3g). Af- sus concentration relationship, as determined from an incre- ter the secondary breaking event, the internal wave ®eld was mental calibration procedure in a test cell. relatively quiescent (Fig. 3h). The frequency content of the internal surge and ISW pack- Results et, as well as the effects of the directionality associated with the initial forcing condition (i.e., upwelling or downwelling Flow ®eldÐTo illustrate the qualitative features of the on the slope), were examined through a time±frequency ¯ow, let us concentrate on the behavior of a particular ex- analysis carried out using continuous wavelet transforms periment (run 6, Table 1). From the initial condition, as (Torrence and Compo 1998). Results from wavegauge B, a shown in Fig. 3a, the ¯ow was accelerated from rest by the nodal location for the H1 seiche, are shown for three rep- baroclinic pressure gradient generated by the initial tilted resentative experiments (Fig. 4), where the forcing ampli- Internal waves in closed basins 1625

␩ ഠ ഠ tude, o/h1 0.7, and layer thickness, h1/H 0.2, were similar. Case 1 is a reference case with no slope (Fig. 4a,b). The majority of the energy was initially found in the pro- Ն gressive surge with a period of Ti/2. For t/Ts 1, the energy appeared to be transferred directly from the surge to solitary waves with a period near Ti /16; there was no evidence of energy cascading through intermediate scales (cf. Horn et al. 2001; Boegman et al. 2005). The ISW packet persisted until Ͼ t/Ts 6, the wave period gradually increasing from Ti/16 to Ti/8 as the rank-ordered waves in the packet separate ac- cording to Eq. (8a). The waves were eventually dissipated by viscosity. In case 2, the initial condition of upwelling on the slope (Fig. 4c,d) is as shown in Fig. 1b. The ISW packet initially evolved in the same manner as case 1, for a small Ͻ Ͻ time. However, over 2 t/Ts 3, the fully developed ISW packet interacted with the sloping topography and rapidly lost its high-frequency energy to wave breaking (see Fig. 3). The re¯ected long wave, shown progressing to the right at ഠ t/Ts 3, traveled the tank length and continued steepening. A second ISW packet was observed to develop and was Ͻ Ͻ observed to break over 4 t/Ts 5. Case 3 is the initial condition of downwelling on the slope (Fig. 4e,f), as shown in Fig. 1c. In this experiment, the ISW packet initially in- ഠ teracted with the slope at t/Ts 1. At this time, the wave packet was in the early stages of formation and only a weak breaking event, with little apparent energy loss from the wave, was observed. The re¯ected wave packet then contin- ued to steepen as it propagated to the right and re¯ected off the end wall. Again, a second breaking event was observed Ͻ Ͻ over 3 t/Ts 4, with more signi®cant energy loss from the wave. From these results in closed systems, it is clear that the nature of the surge and ISW packet life history is dependent not only on the bathymetric slope but also on the direction of the forcing relative to the position of the topo- Fig. 5. Schematic illustration showing the evolution of the spa- graphic feature. tial wave pro®le as observed in the laboratory. (a±f) Evolution from This concept is illustrated in Fig. 5, where visualization the initial condition in Fig. 1c, g±l. Evolution from the initial con- of the laboratory experiments (e.g., Fig. 3) has allowed the dition in Fig. 1b. Illustrations are representative of Ti/2 time inter- wave degeneration process, from the initial conditions vals. Arrows denote wave-propagation direction. shown in Fig. 1b,c, to be conceptually depicted. A longwave initially propagates from the upwelled ¯uid volume (Fig. 5b and h). This wave steepens as it travels, thus producing high- frequency ISWs. The wave groups re¯ect from the vertical signal over the wave/packet period. Note that net downscale wall with minimal loss and break as they shoal along the energy ¯ux through wave breaking at the boundary may not bathymetric slope. The breaking process is expected to be be quanti®ed by this method because we cannot account for governed by ␰. the observed upscale energy ¯ux to re¯ected longwaves. Re- sults are presented in Fig. 6, showing contours of ENS /PEo Internal solitary wave energeticsÐIn the preceding sec- (Fig. 6a) and EISW/PEo (Fig. 6b±f). In each panel, the vertical ␣␩ tion, a qualitative description of the wave±slope dynamics axis ( o/co) indicates the relative magnitudes of the linear in a closed basin was presented. To quantify the temporal and nonlinear components of the internal wave ®eld (from evolution of the ISW energy and the energy loss from these Eq. (4)), while the horizontal axis (Ti) reveals how the sys- waves as they interact with the topographic boundary slope, tem evolves in time. The times at which the solitary wave we applied the signal processing methods of Boegman et al. packet shoals upon the sloping beach and Ts are also indi- (2005). This technique requires the energy in the various cated (see caption). internal modes to be discrete in frequency space (as dem- For case 1, where there is no slope, the surge energy (ENS / onstrated in Fig. 4), thus allowing the individual signals to PEo) is shown to increase during an initial nonlinear steep- Ͻ Յ be isolated through selective ®ltering of the interface dis- ening phase (0 t Ts) and retain up to 20% of the PEo ഠ Ͼ placement time series. By assuming an equipartition between at t Ts (Fig. 6a). When t Ts, the ENS/PEo decreased from kinetic and potential forms of wave energy, the energy in this maximum, with a corresponding increase in ISW energy, the surge (ENS) and ISW wave packets (EISW ) was deter- until eventually all of the surge energy was transferred to mined from the ®ltered time series as the integral of the the solitary waves with little loss (Fig. 6b). In the absence 1626 Boegman et al.

Fig. 6. General evolution of internal wave energy normalized by the PEo introduced at the initial condition: (a) Surge energy (i.e.,

ENS/PEo) and (b) ISW energy (i.e., EISW/PEo) for the case of no slope. The contours in (a) and (b) are least squares ®t to the data (see Boegman et al. 2004, 2005). ISW energy for the case of (c) upwelling along the 3/20 slope and (d) upwelling along the 1/10 slope. ISW energy for the initial condition of (e) downwelling along the 3/20 and (f) downwelling along the 1/10 slope. Contours are ϭ presented as a percentage of the PEo introduced at t 0 and are compiled using the data in Table 1 and Boegman et al. (2005). The Fig. 7. Images of mixing as the layer interface advances along contour interval is 5%. The ratio Ts/Ti is indicated (solid line) for the slope for t Ͼ 0. Data from experimental run 6. The horizontal ␣␩ a particular o/co. The times at which the solitary wave packet window length is 1 m and the aspect ratio is 1:1. The upper layer ␳ ഠ Ϫ3 ␳ ഠ shoals upon the sloping beach are also shown (dashed line). density 1 1,000 kg m , and the lower layer density 2 1,020 kg mϪ3. of sloping topography, the energy in the ISW packet was ultimately lost to viscosity on time scales of order 3Ti to 5Ti. Figure 6c,d shows results for case 2, upwelling on bound- EISW/PEo energy levels were about 5±10%, and this energy ary slopes of 3/20 and 1/10, respectively. In both panels, the was lost from the ISW ®eld at the boundary. As with the ␣␩ ഠ evolution of EISW/PEo was initially analogous to case 1 with case of upwelling at the slope, for o/co 0.5, a secondary no slope; however, now the waves shoaled after traveltimes breaking event may occur at 2Ti. of 0.5Ti, 1.5Ti, and 2.5Ti. The ®rst wave/slope interaction It is clear from Fig. 6 that the forcing direction (upwelling occurred at 0.5Ti, prior to the development of any high-fre- or downwelling) determines the travel time of the ISW pack- Ͻ quency ISWs (i.e., t Ts). At this time, there was no dis- et until it shoals, as a multiple of Ti/2. Similarly, the growth cernible breaking of the basin-scale surge (with a mild wave of the ISW packet is governed by Ts. The key nondimen- slope) on the relatively steep boundary slope. The character sional parameter, relative to the traveltime, is Ts/Ti (Fig. 6). of the second wave-slope interaction was dependent on the From Eqs. (3) and (5), this parameter is ␣␩ ␣␩ Ͼ Ͻ magnitude of o/co. For o/co 0.5, we ®nd Ts 1.5Ti, T 1 hh which results in nearly all ISW energy (as much as 25% of s12ϭ Ϫ ␩ (14) PEo) being lost near the boundary or re¯ected during the Ti12o3(h h ) 1.5T shoaling event. For some experiments where ␣␩ /c ഠ i o o ഠ 0.5, secondary breaking events were observed at 2.5T . Ex- We note that in Fig. 6c,d, the maximum EISW/PEo 20% i ␣␩ ഠ perimental visualization revealed that these events transpired occurred at o/co 0.5; whereas in Fig. 6b, the maximum ഠ ␣␩ ഠ when the ISW packet had not fully developed before the EISW/PEo 20% occurred at o/co 1. Experimental vi- primary breaking event (i.e., T ഠ 1.5T ); hence, subsequent sualization suggests that the slight reduction in wave-®eld s i ␣␩ → packet development occurred prior to 2.5Ti. energy as o/co 1 for the experiments with upwelling In Fig. 6e,f, results are shown for case 3, the initial con- along the topographic slope may have resulted from en- dition of downwelling at boundary slopes of 3/20 and 1/10, hanced energy transfer to mixing and dissipation as the den- respectively. Again, the evolution of EISW/PEo is initially sity interface accelerated along the slope from the initial rest similar to the case with no slope but the waves then shoaled condition (Fig. 7). This mixing did not occur during the ex- ഠ around Ti and 2Ti.Byt Ti, wave steepening had begun, periments with vertical end walls. Internal waves in closed basins 1627

Breaker observationsÐFollowing the success of classi- tion of the breaker type in terms of ␰ and the measured ϭ fying surface breakers according to the Iribarren number, we re¯ection coef®cient, R Er/Ei, where Er and Ei are the found it instructive to test these classi®cations for internal energy in the re¯ected and incident wave packets, respec- wave breaking. Measurements of aϱ and ␭ϱ were obtained tively, calculated from the wave-gauge B signal. The signal from the wave-gauge data at wave-gauge B. Images from is not ®ltered in order to allow for the observed upscale

the CCD camera were used to estimate ab and classify the energy transfer from the breaking high-frequency wave breaker type. In Figs. 8 and 9, internal waves are shown to packet to a re¯ected longwave, which scales as the packet spill, plunge, and collapse, suggesting that a form of the length (re¯ected long-wave energy would be removed by the Iribarren classi®cation may be suitable for internal wave ®lter employed in Fig. 6). Our experimental methodology breaking. did not permit calculation of the mixing ef®ciency ⌫ for a Spilling breakers were observed on the milder slope (S ϭ particular wave breaking event. However, ⌫ may be obtained ␣␩ Ͼ 1/10), when the wave steepness is high ( o/co 0.8). Two from published observations of breaking lone solitary waves examples of spilling breakers are presented in Fig. 8a±f and (Michallet and Ivey 1999, their ®g. 10), when expressed as g±l. In both examples, the rear face of each ISW in the a function of ␰. ␰ ␰ ␰ packet breaks as a wave of elevation in a spilling manner. In Fig. 10a±c, we plot R versus b, ϱ, and sech, respec- Mixing and dissipation appear to be very localized and on tively. For small ␰, the wave slope is steep relative to S and occasion (e.g., Fig. 8c), baroclinic motions shear off the crest the wave propagation over the slope approaches that of a of the wave. These observations are similar to those by Hel- wave in a ¯uid of constant depth, spilling breakers are ob- frich (1992, his ®gs. 3±5). served, R → 0, and viscosity dominates, with minimal wave Plunging breakers were observed on both slopes for mod- energy converted to an increase in the potential energy of Ͻ ␣␩ Ͻ erately steep waves (0.4 o/co 0.8). Plunging occurred the system through diapycnal mixing (Fig. 10d,e). Converse- ␰ primarily during the breaking events at 1.5Ti in Fig. 6c,d. ly, for waves with very large , the time scale of wave±slope The initial condition of upwelling on the slope allowed a interaction is small, R → 1, and the collapsing breakers again

long travel time (1.5Ti) and a fully developed wave packet. induce minimal mixing. The inertia of these waves appears Recall that, in these breaking events, up to 20% of the PEo unable to overcome the stabilizing effect due to buoyancy. is lost from the high-frequency ISW ®eld. Two examples of Intermediate to these extremes, plunging breakers develop spilling breakers (Fig. 8m±r and s±x) suggest these events with gravitational instabilities that drive mixing ef®ciencies, are more energetic than the spilling class. In each example, peaking near 25%. As the incident waves contain as much

an intact mass of ¯uid from the rear face of each solitary as 25% of the PEo, it implies that 6% of the PEo may be wave of depression was observed to plunge foreward (e.g., converted by diapycnal mixing to an irreversible increase in Fig. 8o and u), becoming gravitationally unstable and form- potential energy. These results are consistent with those for ing a core of mixed ¯uid, which propagates upslope. These lone solitary waves by Michallet and Ivey (1999), which observations are similar to numerical results by Vlasenko have been recast in terms of ␰ and are also presented in Fig. and Hutter (2002, their ®g. 6). 10b,c. The present results, however, have distinctly lower R Collapsing breakers were also observed to occur on both values. This may be attributed to the observation that de- boundary slopes, when the incident wave slope was milder caying turbulence and wave re¯ection associated with the than that required for plunging. Two examples of collapsing leading wave generally interferes with subsequent waves in breakers (Fig. 9a±f and g±l) show that the breaking events the packet, increasing ¯uid straining and therefore dissipa- were less energetic than plunging and were characterized by tion within the breaker. For this same reason, the breaker a mass of ¯uid beginning to plunge forward (e.g., Fig. 9d type classi®cations strictly apply to only the leading wave and i), and then collapsing down upon itself (e.g., Fig. 9e, in each packet. Some differences, between these results and and j±k). Compared with plunging breakers, little mixing those of Michallet and Ivey (1999), might also arise from occurred. These observations are similar to those by Mich- the presence of a no-slip boundary condition at the top sur- allet and Ivey (1999, their ®g. 3). face (Redekopp pers. comm.). ϭ ␣ ϭ For experiments where h1 h2, 0, and ®rst-order Interpretation of our results, in terms of the mixing model KdV type solitary waves are prohibited from forming. Suf- by Ivey and Imberger (1991), suggests that, in the spilling ␩ Ͼ ®ciently strong forcing ( o/h1 0.4) may, however, generate and collapsing regions, mixing is suppressed by viscosity supercritical ¯ow conditions and an internal undular bore and buoyancy, respectively; consequently ⌫Ͻ15%. In the (see Horn et al. 2001). The large baroclinic shear associated plunging region, inertia dominates, suggesting that the scale with this forcing was capable of generating Kelvin±Helm- of the most energetic overturns approaches the Ozmidov holtz instability, both in the tank interior (Horn et al. 2001, scale. As a result, the potential energy available for mixing their ®g. 8) and at the boundary (Fig. 9m±r and s±x). Break- is maximized and ⌫Ͼ15%. Such large-scale overturning is ing of the undular waves through shear instability was ob- also observed for Kelvin±Helmholtz billowing (Imberger served to result in strong mixing of the water column (Fig. and Ivey 1991) and critical internal wave breaking (Ivey and 9r,x). These observations are qualitatively similar to oceanic Nokes 1989). ®eld observations by Moum et al. (2003, their ®g. 14) and Orr and Mignerey (2003, their ®g. 9). The breaking pointÐNow that breaker classi®cation guidelines have been established and the energy loss at the Breaker classi®cation and re¯ection coef®cientÐThe slope has been quanti®ed, it is useful to estimate the position breaker visualizations, as presented above, allow classi®ca- along the slope where breaking occurs (breaking point). 1628 Boegman et al.

Fig. 8. False color images of spilling breakers: (a±f) run 28, (g±l) run 32. False color images of plunging breakers: (m±r) run 6, (s±x) ␳ ഠ Ϫ3 run 12. The horizontal window length is 1 m and the aspect ratio is 1:1. The upper layer density 1 1,000 kg m , and the lower layer ␳ ഠ Ϫ3 density 2 1,020 kg m . Internal waves in closed basins 1629

Fig. 9. False color images of collapsing breakers: (a±f) run 14, (g±l) run 11. False color images of Kelvin±Helmholtz breakers: (m±r) ␳ ഠ Ϫ3 run 17, (s±x) run 18. The horizontal window length is 1 m and the aspect ratio is 1:1. The upper layer density 1 1,000 kg m , and ␳ ഠ Ϫ3 the lower layer density 2 1,020 kg m . 1630 Boegman et al.

Fig. 10. Re¯ection coef®cient (R), mixing ef®ciency (⌫), and breaker type classi®ed according ␰ to the various forms of the Iribarren number: (a) Breaker type and R versus b, (b) breaker type ␰ ␰ ⌫ ␰ and R versus ϱ, (c) breaker type and R versus sech, (d) breaker type and versus ϱ, and (e) breaker ⌫ ␰ type and versus sech. The dashed lines demarcate the breaker classi®cations and are inferred in (d) and (e) from (b) and (c), respectively. The ⌫ observations in (d) and (e) are from Michallet and Ivey (1999).

Most studies to date have been concerned with the mechan- Armi 1999; Sveen et al. 2002). Relevant to the present study, ics (e.g., Kao et al. 1985; Lamb 2002; Legg and Adcroft shoaling ISWs on closed slopes have been examined in the 2003) and location (e.g., Helfrich and Melville 1986; Vla- laboratory by Wallace and Wilkinson (1988) and Helfrich senko and Hutter 2002) of wave breaking upon the slope± (1992). However, in the former study, a turning point was Ͻ shelf topography characteristic to the coastal ocean. Com- not encountered as h2 h1. Helfrich (1992) determined a paratively fewer studies have considered the propagation and breaking criterion in terms of the undisturbed lower-layer

breaking of ISWs over a topographic ridge (e.g., Farmer and depth at the breaking point hb and the distance from the Internal waves in closed basins 1631

Fig. 11. The breaking point (criterion) as a function of the un- disturbed lower layer depth (hb). Li is the distance from the begin- ning of the slope to the undisturbed interface±slope intersection. The experiments with slope, S ϭ 0.034, 0.050, and 0.067 are from Helfrich (1992, his ®g. 6).

Fig. 12. Theoretical turning point versus measured breaking point. Various breaker types are shown: spilling, collapsing, and beginning of the slope to the undisturbed interface±slope in- plunging. Error bars denote uncertainty in determining the breaking tersection, Li. In Fig. 11, the breaking points for the present point due to parallax (shows maximum and minimum position). experiments (as observed from the CCD camera) are shown to complement these results. The tendency for aϱ/hb to in- ␭ Field observations crease as ϱ/Li decreases is found to also be valid for steeper slopes, supporting the observation that larger waves tend to propagate into slightly shallower water (relative to their ini- Lake PusianoÐTo assess the applicability of our results, tial amplitude) before breaking (Helfrich 1992). An increase we applied our analysis to some recent observations from a ␭ lake with suitable ®eld data and topography (Lake Pusiano, in bed-slope with ϱ/Li is also evident. The excursion of the interface along the slope and enhanced interfacial shear as- Italy; Fig. 13). This small, seasonally strati®ed lake, with sociated with the basin-scale seiche are likely responsible near vertical and gradually sloping bathymetry on the north for the observed scatter in the breaking points of the present and south shores, respectively, is subject to pulses of high study, relative to (Helfrich 1992). The line of best-®t to the wind stress from the northerly direction (Fig. 14a). The ep- isodic wind events, on days 203 and 205, introduce energy data Ͼ to the lake at the basin scale. For these events, Tw Ti/4

aϱ 0.14 and a general thermocline tilt is expected. Using the ob- ϭϪ0.03 (15) Ϫ1 ϭ ␩ 0.52 served wind velocity, W o/h1 was calculated (Table 2) h (␭ϱ/L ) b i from Eqs. (1) and (2). Following Stevens and Imberger allows generalization of the results. (1996), the wind stress during each wind event was averaged An analytical breaking criterion was derived from Eq. (6), and integrated over Ti /4. The response of the internal wave using the variable coef®cients ␣ and ␤ that are dependent ®eld to these wind events, as observed at station T, is con- on the local ¯uid depth (e.g., Djordjevic and Redekopp 1978). This ®rst-order weakly nonlinear form of KdV theory does not permit wave breaking (Whitham 1974) and is also limited to relatively steep slopes, where breaking occurs rap- idly; thus, higher order nonlinear effects predominate over a relatively short transition zone. For details of the analysis, the reader is referred to Boegman (2004). The observations presented thus far show the internal sol- itary waves of depression to ultimately break as waves of elevation. Before breaking, these waves must change polar- ity by passing through the turning point (beyond which ␣ Ͼ 0). This is shown to occur in Fig. 12, where the observed location of the breaking point is consistently after the the- oretical prediction for the location of the turning point. Note Fig. 13. Map showing the position of Lake Pusiano in Italy (48.80ЊN, 9.27ЊE) and the lake bathymetry. A thermistor chain (in- that Eq. (10) is strictly appropriate for an undisturbed two- dividual sensors every 0.75 m between depths of 0.3 and 24.3 m) layer strati®cation and does not account for the excursion of and weather station (2.4 m above the surface) were deployed at the interface along the slope and enhanced interfacial shear station T during July 2003. Data were sampled at 10-s intervals. associated with the basin-scale seiche. Isobaths are given in meters. 1632 Boegman et al.

Fig. 14. Observations of wind speed and direction and isotherm displacement from Lake Pus- iano at T. (a) Mean hourly wind direction (dots), and 10-min average wind speed (solid line), corrected to 10 m above the surface; (b) isotherms at 2ЊC intervals calculated through linear inter- polation of thermistor data at T (10-min average); (c) detail of shaded region in (b) (1-min average). Note change of time base. The bottom isotherm in (b) and (c) are 8ЊC and 10ЊC, respectively. (d) ␳ Ϫ ␳ ␩2 Њ Normalized instantaneous energy calculated as cog( 2 1) 20 from the displacement of the 20 C ␩ isotherm ( 20) in (c) and the integral of this quantity shown as a fraction of the PEo over Ti/2 ␩ intervals as shown (see Table 2). PEo calculated assuming a linear 20 tilt of maximum excursion given by Eq. (1). (e) Same as (d) except for the layer interface from experimental run 31 rather ␩ than 20. Letters b through f denote corresponding panels in Fig. 5.

␩ sistent with other published ®eld observations (e.g., Hunkins frequency, by low-pass ®ltering and detrending 20. The total and Fliegel 1973; Farmer 1978; Boegman et al. 2003). Spe- energy was then integrated over the period of a progressive

ci®cally, the generation and subsequent rapid dissipation of wave packet Ti /2 high-frequency internal waves was observed (Fig. 14b). To toiϩT /2 directly compare these observations to the laboratory results, ϭ ␳ Ϫ ␳ ͵ ␩2 Etotalcg o( 2 1) 20(t) dt (16) an approximation to a two-layer strati®cation was made by t separating the epilimnion from the hypolimnion at the 20ЊC o ␩ isotherm ( 20). The nonlinear waves were isolated from con- (see Boegman et al. 2005). The observed temporal evolution ␩2 tamination, due to motions at the basin scale and buoyancy of 20 and Etotal (Fig. 14c,d) are similar to laboratory data Internal waves in closed basins 1633 are from 0

␩ vations of were estimated from Fig. 10b±e. Holloway (1987) Holloway (1987) Ziegenbein (1969) Boegman et al. (2003) Stanton and Ostrovsky (1998) Bourgault and Kelley (2003) Boegman et al. (2003) Boegman et al. (2003) Hunkins and Fliegel (1973) Halpern (1971) Halpern (1971) Sandstrom and Elliott (1984) Apel et al. (1985) Thorpe et al. (1972) Farmer (1978) Farmer (1978) Unpubl. data (2001) Unpubl. data (2001) Wiegand and Carmack (1986) Fig. 14 Fig. 14 Wiegand and Carmack (1986) ⌫ and ⌫ Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð R 5±10 5±10 5±10 5±25 5±15 5±15 (%) Source 15±20 20±25 15±20 R Ð Ð Ð Ð Ð Ð Ð Ð Ð 20 50 28³ 26³ 40 40 20 Ð Ð Ð 6±8 (%) 10±20 10±20 ϱ

␰ Ð Ð 0.01 0.01 0.3 0.2 0.03 0.03 0.01 0.1 N/A N/A 0.9 0.4 0.02 0.02 0.04 0.3 1 0.6 0.4 0.8 * ² sin

␰ sech в в в в в в в N/A N/A в в ␰ Ð* 0.05² 0.09² 0.03² 0.08² 0.004² 0.004² 0.6* 0.5* 0.4* 0.5* or 0

␩ N/A N/A N/A Ð N/A N/A N/A N/A N/A N/A Ð 3.4³ 3.0³ N/A Ð (m) 12 10 15 15 12 10 10 30 30 ϱ 910 830 Ð 460 Ð 210 422 317 124 ␭ 880 460 460 250 (m) 1,100 6,500 2,520 2,130 1,150 1,260 6,300 5 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 f ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ (Hz) 1.4 1.5 2.7 2.8 4.2 4.2 4.2 5.8 8.3 1.1 1.5 1.6 2.1 2.8 5.3 6.3 6.3 2.3 1.7 6.5 8.3 3.7 ϱ 5 2 2 5 2 3 a 10 20 10 10 30 16 30 35 40 12 21 65 12 10 15 20 (m) 2 75 65 17 17 65 68 75 75 61 17 20 80 75 38 75 h 100 960 140 115 147 147 (m) 3,300 1 7 7 7 7 9 7 h 30 30 30 20 50 15 12 43 48 40 12 35 20 20 25 (m) 170 S N/A N/A N/A Ð 1/100 1/20 1/50 1/50 1/20 1/16 1/200 1/200 1/200 1/25 3/2,000 3/2,000 1/20 1/11 1/20 1/300 1/300 1/100 Location Table 2. Tabulated list of ®eld observations together with the calculated parameters used in this study. With the exception of Lake Pusiano, the obser Bodensee Kootenay Lake Lake Pusiano (day 203) Lake Pusiano (day 205) Kootenay Lake N/A denotes a parameter that is not applicable. The Ð symbol denotes that data, while applicable, were not available. (Horn et al. 2001, their table 1). Parameters calculated directly from observations in Lake Pusiano (Fig. 14) are denoted by ³. Seneca Lake Massachusetts Bay Massachusetts Bay Scoatian Shelf St. Lawrence Lake Kinneret Lake Kinneret North West Shelf North West Shelf Strait of Gibraltar Lake Biwa Oregon Shelf Sulu Sea Loch Ness Babine Lake Babine Lake Bodensee 1634 Boegman et al. from an experiment (run 31) with comparable initial and stabilities, generated by the surface wind forcing and the boundary conditions as shown in Fig. 14e (downwelling on baroclinic shear of the basin-scale motions in the lake inte- ഠ Ͻ ␩ Ͻ Ͻ Ͻ ϳ Ϫ2 slope, h1/H 0.3 and 0.4 o/h1 0.5). During 0 t rior, with frequencies 10 Hz. The third feature is the ഠ 0.5Ti, the ®eld and laboratory data show Etotal /PEo 20% intermediate portion of the spectrum, which is shown to be ഠ and Etotal/PEo 40%, respectively. The lower energy levels dominated by freely propagating nonlinear wave groups ca- in the ®eld appear to be attributed to the absence of a pro- pable of breaking at the lake perimeter (depending on ␰). gressive nonlinear surge, which, in the laboratory, carried Moderate forcing (0.3 Ͻ WϪ1 Ͻ 1), and perhaps topography Ͻ Ͻ the high-frequency waves. During 0.5Ti t 1.5Ti, the in rotational systems (Saggio and Imberger 1998; Wake et high-frequency oscillations in both systems contain approx- al. 2004), is required for excitation of these waves, which ϳ Ϫ4 imately 15% of Etotal/PEo, and this energy rapidly decreases are observed to have sinusoidal pro®les at 10 Hz (Fig. Ͼ 2 ϳ Ϫ3 to approximately 5% as t 1.5Ti. 14c) and sech pro®les at 10 Hz (Saggio and Imberger Figure 5 allows the wave degeneration process in Lake 1998; Boegman et al. 2003). It is interesting to note that Pusiano to be conceptually depicted. The initial condition, a Lake Kinneret, while strongly forced, does not appear to northerly wind and sloping southern shore, was as shown in generate a strong nonlinear internal wave response; yet the Fig. 5a. Upon termination of the wind stress, the system spectral peak resulting from shear instability near ϳ10Ϫ2 Hz evolves schematically as shown in Fig. 5b±f. A long wave is enlarged relative to Biwa and Pusiano (Fig. 15a). Drawing initially propagates from the upwelled ¯uid volume toward on examples found in the literature, the behavior of the spec- the sloping topography (Fig. 5b). This wave re¯ects from trum in the nonlinear 10Ϫ4 to 10Ϫ3 Hz bandwidth may be the slope and travels back toward the vertical wall (Fig. 5c), generalized according to the ratio of the wave height and the ␩ ϭ Ϫ1 re¯ecting once more and steepening as it travels. Upon re- depth of the surface layer a/h1 (or equivalently o/h1 W ϳ ␩ turning toward the slope, a high-frequency wave packet has because a o). This ratio is commonly used to gauge the ഠ formed with Etotal/PEo 15% (Fig. 5d). These waves break nonlinearity of progressive internal waves (e.g., Stanton and ϳ upon the slope losing 10% of the PEo to local turbulent Ostrovsky 1998). A consistent trend is found throughout ob- dissipation and mixing (Fig. 5e). servations from a variety of natural systems with differing The observed energy loss at the slope in Pusiano may be scales. Figure 15b shows sinusoidal waves near 10Ϫ4 Hz and 2 Ϫ3 Ͻ Ͼ compared with the predicted energy loss using the internal sech waves near 10 Hz when a/h1 0.4 and a/h1 0.4, Iribarren model put forth in Fig. 10. The nondimensional respectively. The waves associated with shear instability in ␰ ␰ Lake Kinneret are mechanistically and observationally in- parameters ϱ, sech, and Er/Ei were calculated from the ®eld ഠ ϳ Ϫ2 data on days 203 and 205. In Fig. 10b, the parameter ␰ϱ is consistent with this model (a/h1 0.2 and 10 Hz). The revised spectral model for strati®ed lakes is summarized in shown to be an accurate measure of Er/Ei in both the ®eld and in the laboratory. Spilling and plunging breakers are Table 3. predicted, with the ®eld measurements being grouped among Combination of the observations in Fig. 15b and the ␰ the laboratory data collected in this study and that from model presented in Fig. 10 suggest that, in lakes with mod- ␰ erate to steep boundary slopes, 10% Յ R Յ 50% and 5% Michallet and Ivey (1999). Conversely, the parameter sech is shown to overestimate the energy loss upon the slope in Յ ⌫ Յ 25%, in the Sulu Sea, R ϳ 40% and 15% Յ ⌫ Յ ϳ Pusiano relative to the laboratory model in Fig. 10c. Inspec- 20% and, in the St. Lawrence estuary, R 6% to 8% (Table tion of Fig. 14c and Table 2 reveals that the wind forcing is 2). These calculations may differ by as much as 50% from moderate (i.e., WϪ1 ഠ 0.4); consequently, the high-frequency those calculated using the ratio of the wave length to the waves in Pusiano are sinusoidal in pro®le and thus will pos- slope length (e.g., Michallet and Ivey 1999; Bourgault and sess a greater wave length than that given for a sech2 pro®le Kelley 2003) and are only applicable to waves shoaling into by Eq. (8b). In Fig. 10c, underestimation of the actual wave a closed slope at the depth of the pycnocline. length (and hence overestimation of the wave slope) will In strati®ed waterbodies, internal waves provide the cru- ␰ cial energy transfer between the large-scale motions forced result in a spurious decrease in sech and corresponding over- estimation of E /E . For these waves, ␰ is more suitable. by winds and tides and the small-scale turbulent dissipation r i sin and mixing along sloping boundaries. This energy ¯ux oc- curs through downscale spectral transfer and shoaling of An interpretation of the wave spectrum high-frequency internal waves along topography suited for Ͻ ␩ wave breaking. In lakes, moderate wind forcing (0.3 o/ Ͻ A general model for the internal wave spectrum in lakes h1 1) excites sub±basin-scale nonlinear wave groups that was proposed by Imberger (1998). This model is analogous propagate throughout the basin. These wave groups are also to the Garrett±Munk spectrum found in the oceanographic tidally generated in coastal regions. In both systems, when Ͻ ␩ ϳ Ͻ literature. We use the results presented herein and the recent 0.3 o/h1 a/h1 0.4, the waves have frequencies near Ϫ4 Ͻ ␩ work by Antenucci and Imberger (2001) and Boegman et al. 10 Hz and a sinusoidal pro®le whereas, when 0.4 o/h1 ϳ Ͻ Ϫ3 (2003) to update this model for large lakes with a seasonal a/h1 1, the waves have frequencies near 10 Hz and thermocline and a discernible frequency bandwidth between a sech2 pro®le. the motions at the basin and buoyancy scales (bandwidth Ͼ Using laboratory experiments, we have quanti®ed the 102ϳ103 Hz). The three main features of the internal wave down-scale spectral energy ¯ux for closed basins with a two- spectrum are shown in Fig. 15a. First is the presence of layer strati®cation. High-frequency internal solitary waves Ն discrete natural and forced basin-scale seiches at frequencies evolve from the basin-scale motions as t Ts and contain between zero and ϳ10Ϫ4 Hz. Second is the presence of in- up to 20% of the available potential energy introduced by Internal waves in closed basins 1635

Fig. 15. (a) Spectra of the vertically integrated potential energy signal (see Antenucci et al. 2000) from Lake Pusiano (Fig. 14b) and lakes Kinneret and Biwa (see Boegman et al. 2003). The data were sampled at 10-s intervals and the spectra have been smoothed in the frequency domain

to improve con®dence at the 95% level, as shown by the dotted lines. Nmax denotes the maximum buoyancy frequency. (b) Observations of wave nonlinearity, pro®le, and frequency from the liter- ature. Sources are as given in Table 2. For buoyancy period calculations, we have assumed N ϳ ϳ Ϫ2 Nmax 10 Hz.

the wind. As Ti/2 is the characteristic wave travel time over slope to the boundary slope. This de®nition is easily gen- L, the ratio Ti /Ts describes the wave evolution and represents eralized to waves shoaling into a closed slope at the depth a balance between high-frequency wave growth through of the pycnocline in lakes, oceans, and estuaries. For closed nonlinear steepening and high-frequency wave degeneration basins, knowledge of the wave pro®le allowed ␰ to be recast through shoaling at the boundary. using properties of the quiescent ¯uid and the forcing dy- Wave breaking was observed at the boundary in the form namics. of spilling, plunging, collapsing, and Kelvin±Helmholtz Further study is required, as the laboratory facility did not breakers. A single wave breaking event resulted in 10% to permit examination of extremely mild slopes, such as those 75% of the incident wave energy being lost to dissipation commonly found in many lakes and coastal oceans. These and mixing, the remaining energy being found in a re¯ected results will facilitate the parameterization of nonhydrostatic long wave. The energy loss was dependent on the breaker and sub±grid-scale wave processes into ®eld-scale hydro- type. Comparison with the results by Michallet and Ivey dynamic models (e.g., Boegman et al. 2004). (1999) suggests that the mixing ef®ciency was also depen- dent on the breaker type and ranged between 5% and 25%. References These boundary processes were modeled in terms of an in- ternal Iribarren number ␰, de®ned as the ratio of the wave ANTENUCCI,J.P.,AND J. IMBERGER. 2001. On internal waves near the high frequency limit in an enclosed basin. J. Geophys. Res. 106: 22465±22474. Table 3. A revised spectral model for strati®ed lakes. Summa- ,J.IMBERGER, AND A. SAGGIO. 2000. Seasonal evolution of rized from the data presented in Fig. 15. the basin-scale internal wave ®eld in a large strati®ed lake. Limnol. Oceanogr. 45: 1621±1638. Frequency Buoyancy APEL, J. R., J. R. HOLBROOK,A.K.LIU, AND J. J. TSAI. 1985. The Wave (Hz) periods Sulu Sea internal soliton experiment. J. Phys. Oceanogr. 15: 1625±1650. Linear basin scale Ͻ10Ϫ4 Ͼ100 BATTIES, J. A. 1974. Surf similarity. Proc. Coastal Engineering Con- Nonlinear sinusoidal pro®le ϳ10Ϫ4 ϳ100 ference, ASCE, 14: 466±479. Nonlinear sech2 pro®le ϳ10Ϫ3 ϳ10 BENNEY, D. J. 1966. Long nonlinear waves in ¯uid ¯ows. J. Math. Shear instability ϳ10Ϫ2 ϳ1 Phys. 45: 52±63. Maximum buoyancy frequency ϳ10Ϫ2 ϳ1 BOEGMAN, L. 2004. The degeneration of internal waves in lakes 1636 Boegman et al.

with sloping topography. Ph.D. thesis, Centre for Water Re- Internal wave evolution in a space-time varying ®eld. J. Fluid search, The University of Western Australia. Mech. 424: 279±301. ,J.IMBERGER,G.N.IVEY, AND J. P. ANTENUCCI. 2003. HUNKINS, K., AND M. FLIEGEL. 1973. Internal undular surges in High-frequency internal waves in large strati®ed lakes. Limnol. Seneca Lake: A natural occurrence of solitons. J. Geophys. Oceanogr. 48: 895±919. Res. 78: 539±548. ,G.N.IVEY, AND J. IMBERGER. 2004. An internal solitary IMBERGER, J. 1998. Flux paths in a strati®ed lake: a review, p. 1± wave parameterization for hydrodynamic lake models. Proc. 18. In J. Imberger [ed.], Physical processes in lakes and oceans. 15th Australasian Fluid Mechanics Conference. Coastal and Estaurine Studies. V. 54. , , AND . 2005. The energetics of large-scale , AND G. N. IVEY. 1991. On the nature of turbulence in a internal wave degeneration in lakes. J. Fluid Mech. 531: 159± strati®ed ¯uid. Part 2: Application to lakes. J. Phys. Oceanogr. 180. 21: 659±680. BOURGAULT, D., AND D. E. KELLEY. 2003. Wave-induced mixing IVEY,G.N.,AND J. IMBERGER. 1991. On the nature of turbulence in a partially mixed estuary. J. Marine Res. 61: 553±576. in a strati®ed ¯uid. Part 1: The energetics of mixing. J. Phys. COWEN, E. A., K.-A. CHANG, AND Q. LIAO. 2001. A single-camera Oceanogr. 21: 650±658. coupled PTV-LIF technique. Exp. Fluids 31: 63±73. , AND R. I. NOKES. 1989. Vertical mixing due to the breaking DJORDJEVIC,V.D.,AND L. G. REDEKOPPP. 1978. The ®ssion and of critical internal waves on sloping boundaries. J. Fluid Mech. disintegration of internal solitary waves moving over two-di- 204: 479±500. mensional topography. J. Phys. Oceanogr. 8: 1016±1024. KAKUTANI,T.,AND N. YAMASAKI. 1978. Solitary waves on a two- FARMER, D. M. 1978. Observations of long nonlinear internal waves layer ¯uid. J. Phys. Soc. Japan 45: 674±679. in a lake. J. Phys. Oceanogr. 8: 63±73. KAO, T. W., F. S. PAN, AND D. RENOUARD. 1985. Internal solitons , AND L. ARMI. 1999. Strati®ed ¯ow over topography: The on the pycnocline generation, propagation, shoaling and break- role of small-scale entrainment and mixing in ¯ow establish- ing over a slope. J. Fluid Mech. 159: 19±53. ment. Proc. R. Soc. Lond. A 457: 3221±3258. KOMAR, P. D. 1976. Beach Processes and Sedimentation. Prentice- FERRIER, A. J., D. R. FUNK, AND P. J. W. ROBERTS. 1993. Appli- Hall. cation of optical techniques to the study of plumes in strati®ed LAMB, K. G. 2002. A numerical investigation of solitary internal ¯uids. Dyn. Atmos. Oceans 20: 155±183. waves with trapped cores formed via shoaling. J. Fluid Mech. FISCHER, H. B., E. J. LIST,R.C.Y.KOH,J.IMBERGER, AND N. H. 451: 109±144. BROOKS. 1979. Mixing in inland and coastal waters. Academic. LEDWELL,J.R.,AND B. M. HICKEY. 1995. Evidence for enhanced FRICKER,P.D.,AND H. M. NEPF. 2000. Buoyancy effects and in- boundary mixing in the Santa Monica basin. J. Geophys. Res. ternal seiche decay. Proc. International Symposium on Strati- 100: 20665±20679. ®ed Flow 5: 313±318. LEE,C.Y.,AND R. C. BEARDSLEY. 1974. The generation of long GALVIN, C. J. 1968. Breaker type classi®cation on three laboratory nonlinear internal waves in a weakly strati®ed shear ¯ow. J. beaches. J. Geophys. Res. 73: 3651±3659. Geophys. Res. 79: 453±462. GLOOR, M., A. WUÈ EST, AND D. M. IMBODEN. 2000. Dynamics of LEGG, S., AND A. ADCROFT. 2003. Internal wave breaking at con- mixed bottom boundary layers and its implications for diapyc- cave and convex continental slopes. J. Phys. Oceanogr. 33: nal transport in a strati®ed, natural water basin. J. Geophys. 2224±2246. Res. 105: 8629±8646. MACINTYRE, S., K. M. FLYNN,R.JELLISON, AND J. R. ROMERO. GOUDSMIT, G.-H., F. PETERS,M.GLOOR, AND A. WUÈ EST. 1997. 1999. Boundary mixing and nutrient ¯uxes in Mono Lake, Cal- Boundary versus internal diapycnal mixing in strati®ed natural ifornia. Limnol. Oceanogr. 4: 512±529. waters. J. Geophys. Res. 102: 27903±27914. MICHALLET, H., AND E. BARTHEÂ LEMY. 1997. Ultrasonic probes and HALPERN, D. 1971. Observations of short-period internal waves in data processing to study interfacial solitary waves. Exp. Fluids Massachusetts Bay. J. Marine Res. 29: 116±132. 22: 380±386. HAMMACK,J.L.,AND H. SEGUR. 1978. Modelling criteria for long , AND G. N. IVEY. 1999. Experiments on mixing due to in- water waves. J. Fluid Mech. 84: 359±373. ternal solitary waves breaking on uniform slopes. J. Geophys. HEAPS,N.S.,AND A. E. RAMSBOTTOM. 1966. Wind effects on the Res. 104: 13467±13477. water in a narrow two-layered lake. Phil. Trans. R. Soc. Lond. MILES, J. W. 1981. The Korteweg±de Vries equation: A historical A 259: 391±430. essay. J. Fluid Mech. 106: 131±147. HELFRICH, K. R. 1992. Internal solitary wave breaking and run-up MONISMITH, S. G. 1987. Modal response of reservoirs to wind on a uniform slope. J. Fluid Mech. 243: 133±154. stress. ASCE J. Hydraul. Eng. 113: 1290±1306. , AND W. K. MELVILLE. 1986. On nonlinear internal waves MORTIMER, C. H. 1974. Lake hydrodynamics. Mitt. Internat. Verein. over slope-shelf topography. J. Fluid Mech. 167: 285±308. Limnol. 20: 124±197. HODGES, B. R., J. IMBERGER,B.LAVAL, AND J. APPT. 2000. Mod- , AND W. H ORN. 1982. Internal wave dynamics and their eling the hydrodynamics of strati®ed lakes. Proc. Hydroinfor- implications for plankton biology in the Lake of Zurich. Mitt. matics Conference 4: 23±27. Internat. Verein. Limnol. 127: 299±318. HOLLOWAY, P. E. 1987. Internal hydarulic jumps and solitons at a MOUM, J. N., D. M. FARMER,W.D.SMYTH,L.ARMI, AND S. VA- shealf break region on the Austrailan North West Shelf. J. Geo- GLE. 2003. Structure and generation of turbulence at interfaces phys. Res. 92: 5405±5416. strained by internal solitary waves propagating shoreward over HORN, D. A., J. IMBERGER, AND G. N. IVEY. 1999. Internal solitary the continental shelf. J. Phys. Oceanogr. 33: 2093±2112. waves in lakesÐa closure problem for hydrostatic models. ORR,M.H.,AND P. C. MIGNEREY. 2003. Nonlinear internal waves Proc. Aha Huliko Hawaiian Winter WorkshopÐInternal Grav- in the South China Sea: Observation of the conversion of de- ity Waves II 11: 95±100. pression internal waves to elevation internal waves. J. Geo- , , AND . 2001. The degeneration of large- phys. Res. 108: 3064. Doi:10.1029/2001JC001163. scale interfacial gravity waves in lakes. J. Fluid Mech. 434: OSTROVSKY, I., Y. Z. YACOBI,P.WALLINE, AND I. KALIKHMAN. 181±207. 1996. Seiche-induced mixing: Its impact on lake productivity. ,L.G.REDEKOPP,J.IMBERGER, AND G. N. IVEY. 2000. Limnol. Oceanogr. 4: 323±332. Internal waves in closed basins 1637

SAGGIO, A., AND J. IMBERGER. 1998. Internal wave weather in strat- VLASENKO,V.,AND K. HUTTER. 2002. Numerical experiments on i®ed lakes. Limnol. Oceanogr. 43: 1780±1795. the breaking of solitary internal waves over a slope-shelf to- SANDSTROM, H., AND J. A. ELLIOTT. 1984. Internal tide and solitons pography. J. Phys. Oceanogr. 32: 1779±1793. on the Scotian Shelf: A nutrient pump at work. J. Geophys. WAKE, G. W., G. N. IVEY, AND J. IMBERGER. 2004. The temporal Res. 89: 6415±6426. evolution of baroclinic basin-scale waves in a rotating circular SPIGEL,R.H.,AND J. IMBERGER. 1980. The classi®cation of mixed- basin. J. Fluid Mech. 515: 63±86. layer dynamics in lakes of small to medium size. J. Phys. WALLACE,B.C.,AND D. L. WILKINSON. 1988. Run-up of internal Oceanogr. 10: 1104±1121. waves on a gentle slope in a two-layered system. J. Fluid STANTON,T.P.,AND L. A. OSTROVSKY. 1998. Observations of high- Mech. 191: 419±442. ly nonlinear internal solitons over the Continental Shelf. Geo- WHITHAM, G. B. 1974. Linear and nonlinear waves. Wiley. phys. Res. Lett. 25: 2695±2698. WIEGAND,R.C.,AND E. C. CARMACK. 1986. The climatology of STEVENS, C., AND J. IMBERGER. 1996. The initial response of a internal waves in a deep temperate lake. J. Geophys. Res. 91: strati®ed lake to a surface shear stress. J. Fluid Mech. 312: 39± 3951±3958. 66. WUÈ EST, A., G. PIEPKE, AND D. C. SENDEN. 2000. Turbulent kinetic SVEEN, J. K., Y. GUO,P.A.DAVIES, AND J. GRUE. 2002. On the energy balance as a tool for estimating vertical diffusivity in breaking of internal solitary waves at a ridge. J. Fluid Mech. wind-forced strati®ed waters. Limnol. Oceanogr. 45: 1388± 469: 161±188. 1400. THOMPSON, R.O.R.Y., AND J. IMBERGER. 1980. Response of a nu- HOU, X., AND R. GRIMSHAW. 1989. The effect of variable currents merical model of a strati®ed lake to wind stress. Proc. Intl. Z Symp. Strati®ed Flows 2: 562±570. on internal solitary waves. Dyn. Atmos. Oceans 14: 17±39. IEGENBEIN THORPE, S. A. 1971. Asymmetry of the internal seiche in Loch Z , J. 1969. Short period internal waves in the Strait of Ness. Nature 231: 306±308. Gibralter. Deep-Sea Res. 16: 479±487. ,A.HALL, AND I. CROFTS. 1972. The internal surge in Loch Ness. Nature 237: 96±98. Received: 3 August 2004 TORRENCE, C., AND G. P. COMPO. 1998. A practical guide to wavelet Accepted: 18 April 2005 analysis. Bull. Amer. Meteor. Soc. 79: 61±78. Amended: 30 May 2005