Shoaling Internal Solitary Waves B

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Shoaling Internal Solitary Waves B. R. Sutherland1, K. J. Barrett2 and G. N. Ivey3

Abstract. The evolution and breaking of internal solitary waves in a shallow upper layer as they approach a constant bottom slope is examined through laboratory experiments. The waves are launched in a two layer fluid through the standard lock-release method. In most experiments, the wave amplitude is significantly larger than the depth of the shallow upper layer so that they are not well described by Korteweg-de Vries theory. The dynamics of the shoaling waves are characterized by the Iribarren number, Ir, which measures the ratio of the topographic slope to the square root of the characteristic wave slope. This is used to classify breaking regimes as collapsing, plunging, surging and non-breaking for increasing values of Ir. For breaking waves, the maximum interface descent is predicted to depend upon the topographic slope, s, and the incident wave’s amplitude and width, Asw and Lsw, respectively. In agreement with experiments we find the maximum ⋆ descent is Hi √4sAswLsw. Likewise, we apply simple heuristics to estimate the speed of interface descent,≃ and we characterize the speed and range of the consequent upslope flow of the lower layer after breaking has occurred.

1. Introduction to the shallow-layer depth (Benney [1966]; Whitham [1974]; Sutherland [2010]). The experiments of Grue et al. [1999] Large amplitude surface waves in relatively shallow water indicate that KdV is useful for amplitudes up to 40% the are generally classified as solitary waves, in homage to the depth of the shallow layer. However, oceanic internal soli- first reported single hump-shaped surface wave generated by tary waves are frequently observed to have vertical displace- a towed barge on the River Severn (Russell [1844]). Within ments from tens to hundreds of meters with amplitudes estuaries and the coastal oceans, internal solitary waves are many times that of the shallow-layer depth (Helfrich and manifest as large amplitude sinusoidal undulations of the Melville [2006]). Under these circumstances, the wave struc- pycnocline, which separates a surface layer from relatively ture deviates significantly from that predicted by KdV the- more dense underlying fluid. They can be generated by river ory and the dependence of speed and width upon amplitude plumes (Nash and Moum [2005]) and by tidal flow over steep is qualitatively different. For example, in laboratory experi- topographic features such as underwater sills and the con- ments Stamp and Jacka [1995] and Grue et al. [1999] found tinental shelf (Osborne and Burch [1980]; Apel et al. [1985]; that the crest of large amplitude internal solitary waves New and Pingree [1992]; Farmer and Armi [1999]; Pinkel tended to flatten and their lateral extent increased with am- [2000]; New and DaSilva [2002]; Zhao and Alford [2006]; plitude. These dynamics are well-captured by the Dubreil- Klymak et al. [2006]; Li and Farmer [2011]; Alford et al. Jacotin-Long (DJL) equation (Dubreil-Jacotin [1937]; Long [2011]). Though studied in part for their (admittedly rela- [1953, 1956]; Lamb [2002]; White and Helfrich [2008]) al- tively small) contribution to ocean mixing on a global scale, though, except in special circumstances, this equation does internal solitary waves have a substantial impact upon ma- not have analytic solutions that explicitly reveal the rela- rine ecosystems where they shoal (Xu et al. [2012]), and their tionship between the speed and width of solitary waves upon associated stresses can degrade the support of pipelines that the amplitude. link offshore oil drilling sites to refineries at the coast. KdV theory, its weakly nonlinear relatives and DJL the- The prototypical solitary wave results from a balance be- ory assume a rectilinear geometry in which the depth of the tween weak nonlinearity, which tends to steepen it, and dis- ambient is constant. Although the KdV equation can be persion, which tends to flatten it. The combination of these modified to include slow variations of the background prop- effects is captured by the Korteweg-de Vries (KdV) equation erties (Helfrich et al. [1984]; Helfrich and Melville [1986]; (Korteweg and de Vries [1895]) which, assuming quiescent Grimshaw et al. [1998]; El et al. [2007, 2012]), an under- upstream and downstream conditions, predicts that the in- standing of shoaling, overturning and breaking internal soli- terfacial displacement due to the waves takes the form of a tary waves lies beyond the capacity of analytic theory. In- squared hyperbolic secant. stead, insight into these dynamics have been gained through Whether for a two-layer fluid or for Boussinesq waves in a combination of laboratory experiments and fully nonlinear continuously stratified fluid, KdV theory requires the am- numerical simulations. plitude to be large, but not so large that it is comparable Early laboratory studies into internal solitary wave shoaling examined solitary waves in an approximately two-layer fluid approaching a shelf, passing over a slope in the transi- 1 tion from deep to shallow ambient fluid. In the experimental Departments of Physics and of Earth & Atmospheric set-up of Kao et al. [1985] the upper layer was always more Sciences, University of Alberta, Edmonton, Alberta, Canada. 2 shallow than the lower layer whether in the deep ambient Department of Physics, University of Alberta, or over the shelf, so that a solitary wave of depression could Edmonton, Alberta, Canada. 3 exist in both the deep and shallow ambient. They observed School of Environmental Systems Engineering (SESE) and UWA Oceans Institute, University of Western Australia, wave steepening in the lee of the approaching wave and con- Perth, Australia. sequent partial reflection and transmission of the wave from the shelf, with some irreversible energy loss due to mixing. The experiments of Helfrich and Melville [1986] likewise ex- Copyright 2013 by the American Geophysical Union. amined solitary waves shoaling onto a shelf, but their set-up 0148-0227/13/$9.00 was designed to examine the transition of the wave from one 1 X - 2 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES of depression in the deep ambient (in which the upper layer et al. [2005a] classified the character of internal solitary wave fluid was relatively shallow) to one of elevation over the shelf breaking in terms of the (internal) Iribarren number, Ir: (in which the upper layer fluid was relatively deep). Beyond s the prediction of KdV theory adapted for slowly varying to- Ir . (1) pography (Helfrich et al. [1984]), their experiments revealed ≡ Asw/Lsw instabilities and mixing when the waves transformed from p depressed to elevated form. Their study assumed a KdV structure for the incident waves Experiments examining the approach of a solitary wave of −1/2 to define the wave extent Lsw = Lsw,KdV Asw . With depression toward a uniform slope (Helfrich [1992]) showed this definition, they found that solitary waves∝ formed plung- that the lower layer ambient was drawn downslope while the ing breakers if 0.45 . Ir . 0.75, and collapsing breakers if lee of the wave steepened. Eventually this evolved to create & “boluses” of dense fluid that propagated upslope in simi- Ir 0.75. lar manner to the production of boluses by solitary waves Consistent with Michallet and Ivey [1999], they found of elevation incident upon a slope (Wallace and Wilkinson that the mixing efficiency was greatest for plunging break- [1988]). However, the downslope flow induced by the leading ers. They also formulated an empirical prediction for the in- edge of the wave that opposed the wave’s advance permit- terface descent, Hi, at the maximum breaking depth. This ted multiple boluses to form from just a single incident wave. was given in terms of Asw, Lsw,KdV and the horizontal ex- The study by Helfrich [1992] focused upon the number and tent of the slope in the lower layer, L1 (L1 = H1/s, in which size of boluses formed by the shoaling solitary wave. Rel- H1 is the depth of the lower layer). In particular, for rela- evant to the experiments presented here, he also compared tively shallow slopes they found (adapted from eq. (15) of the depth of the location of maximum breaking to the wave’s Boegman et al. [2005a]) width, finding that the wave amplitude relative to the breaking depth was approximately 0.4 0.1, exhibiting a small 0.52 ± Lsw,KdV decreasing trend as the width of the incident wave increased Hi 7.1 Asw . (2) relative to the horizontal extent of the slope in the lower ≃ L1 layer. Because the solitary waves were generated by a flap mechanism the incident wave amplitudes were not so large Numerical simulations restricted to two dimensions fur- compared to the shallow layer depth, and so resembled KdV- ther explored the dynamics of internal solitary waves shoal- like waves. The range of slopes examined was also limited ing on constant slopes between s = 0.01 and 0.3 (Aghsaee to s =0.034, 0.050 and 0.067. et al. [2010]). The initial conditions were set by solutions Larger amplitude internal solitary waves in two-layer fluid of the DJL equation for a solitary wave over a flat bot- were created in laboratory experiments using a lock-release tom having amplitudes from a fifth to double the (shallow) mechanism (Michallet and Barthélemy [1998]; Grue et al. upper-layer depth. Although the simulations could not cap- [2000]). This method was used to examine the energetics of ture the effects of mixing in fully three dimensional flows shoaling waves on a constant slope by comparing the ener- (Fringer and Street [2003]; Venayagamoorthy and Fringer gies of the incident and reflected waves and by measuring [2007]), they could be used to classify the breaking regimes the change in the potential energy of the stratified ambi- and to gain insight into the instability mechanisms during ent (Michallet and Ivey [1999]). The experiments examined breaking. shoaling on three slopes: s = 0.069, 0.169 and 0.214. They Aghsaee et al. found that up to 25% of the energy was lost due to mixing, [2010] delineated the regimes for surging, with the greatest loss occurring when the width of the wave collapsing and plunging breakers by comparing the time was approximately half the total length of the slope. scales for the downslope flow, separation-bubble formation Using a tilted tank to generate a solitary wave train from and leeward interface steepening. Comparing these time- an internal seiche (Boegman et al. [2005b]), the shoaling of scales helped to explain why a particular breaking mech- the wavetrain on a constant slope with s =0.097 and 0.145 anism occurred. But they did not predict how an incident was examined (Boegman et al. [2005a]). They noted that wave would break knowing its structure and the structure of the mechanism for wave breaking was different depending the topography it approaches. Analysis of their simulations upon the relative magnitudes of the topographic slope and revealed more complicated relationships of breaking regimes the incident wave slope. upon topographic and wave slopes. Generally they observed Non-breaking and different breaking regimes are illus- incident solitary waves plunged, collapsed or surged upon a trated in Figure 1. At very steep topographic slopes waves slope as Ir increased. However, the range separating the reflect with no evidence of overturning at the slope or aloft regimes depended upon the magnitude of the topographic (Figure 1a). For moderately more shallow slopes, overturning can occur at the slope itself forming a “surging breaker” slope. (Figure 1b). Peculiar to internal waves, the downslope flow They also determined an empirical relationship for the beneath the leading flank of the approaching wave can be- breaking depth. Defining the wave extent, Lsw,A, to be the area divided by amplitude and assuming Lsw,A L1 their come so strong relative to the lower layer flow beneath the ≪ trailing flank of the wave that the flow separates from the expression reduced to (adapted from eq. (5.1) of Aghsaee boundary and reconnects lower down. This forms an across- et al. [2010]) slope vortex with counterclockwise vorticity for a rightward- 0.28 propagating incident wave. If the interface in the lee of the Lsw,A wave is not vertical when this “separation bubble” occurs, it Hi 7.1Asw (3) ≃ L1 is classified as a collapsing breaker (Figure 1c). If the slope is very shallow, the leeward interface can overturn within the The discrepancy in the exponent between (3) and (2) may body of the fluid even as the bottom fluid continues to flow be attributed to the explicit measurement of L in the downslope toward the incident wave. This is referred to as sw,A a plunging breaker (Figure 1e). As our experiments show, a numerical simulations and also because the simulations were separation bubble at the boundary can develop at the same designed so that only a single wave of depression impacted time that the trailing flank of the wave overturns. This we upon the slope. refer to here as a collapsing-plunging breaker (Figure 1d). The breaking depth predictions (2) and (3) both depend Defining s to be the topographic slope and Lsw and upon the length of the slope, L1, a quantity that is diffi- Asw to be the characteristic width and vertical displace- cult to estimate in realistic oceanographic circumstances for ment amplitude, respectively, of the incident wave, Boegman which the bottom topography varies in slope from the abyss SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 3 to coast. One aim of this study is to formulate a predic- allow salt water to pass easily underneath it. Fresh water tion for breaking depth that depends upon more accessible was then added to the surface water between the gate and parameters as might be measured, for example, in field sit- endwall so increasing the extent of the surface fresh water to uations. a depth Hℓ. When done, the total depth of the ambient was In this work we perform laboratory experiments to re- measured to be H. In these experiments, H ranged between examine the problem of internal solitary wave shoaling on 0.43 and 0.46m with the upper-layer ambient ranging from a constant slope with the intent to broaden the parameter H0 =0.05 to 0.15m. range explored earlier in both experiments and simulations. In the long-tank experiments the lower layer was dyed before clear fresh water was layered on top. In these ex- We generate waves with amplitudes up to 3.5 times the shal- 3 low layer depth, well beyond the KdV regime. So that the periments, poppy seeds (of mean density 1065g/m and di- results can be readily applied to interpret ocean observa- ameter 0.001m) where sprinkled on the slope in order to tions, we classify the wave breaking regimes in terms of in- visualize boundary layer separation. After filling, a false cident solitary wave characteristics that can be measured by endwall was inserted to full depth mid-way along the tank, surface observations and vertical time series constructed for a distance L from the slope-end of the tank. The distance from the false endwall to the start of the slope was fixed to example from traversing CTD probes or moored thermistor be L Ls =2.60m and the length of the lock was set to be chains (e.g. van Haren and Gostiaux [2012]). Our analy- − either Lℓ =0.40 or 0.80m. The total ambient depth ranged ses focus upon predicting the maximum breaking depth and between H = 0.16 and 0.18m with the upper layer depth the consequent upslope propagation of the lower layer as a ranging between H0 =0.01 to 0.04m. bolus. Before the start of an experiment, the ambient density Section 2 describes the experimental set-up, in which soli- profile was measured using a conductivity probe (MSCTI, tary waves are generated by lock-release in an approximately Precision Measurement Engineering), which was calibrated two-layer fluid. The qualitative results and analysis meth- by measuring the conductivity of prepared salt-water solu- ods used to examine the properties of the incident solitary tions whose densities were measured using an Anton Paar wave and its shoaling characteristics are given in 3. In 4 we DMA5000 densitometer. These measurements were used to examine how Asw and Lsw depend upon the characteristics provide an accurate measurement of the fresh water layer of the lock and we show how the structure of the gener- depth, H0, and the thickness of the interface between salt ated solitary wave differs significantly from that predicted and fresh water. Specifically, H0 was determined to be the by KdV theory. Whereas Michallet and Ivey [1999] focused distance from the surface to the depth in the fluid where upon measuring the energetics of the incident and reflected the density was the average of the fresh and salt-water: solitary wave from a fixed slope and Helfrich [1992] focused (ρ0 + ρ1)/2. The thickness of the interface 2Ht was mea- upon the number boluses that resulted from breaking, here sured as the distance over which the density changed from we focus upon the deepening of the shoaling solitary wave (3ρ0 + ρ1)/4 to (ρ0 +3ρ1)/4. In the experiments reported and consequent slope run-up as it depends upon the incident upon here, Ht < 0.004m. In the classification of wave break- wave properties and the magnitude of the slope. Based upon ing, Ht was as large as 0.02m. observations of the breaking structure, we formulate a pre- At the start of an experiment, the gate was rapidly ex- diction for the breaking depth, which compares well with extracted vertically. The fresh water layer in the lock then periments. We also examine the time-scale for breaking and slumped upward into the shallower fresh water ambient thus we measure the upslope advance of the leading bolus. The launching a hump-shaped internal solitary wave of depres- implications for interpreting ocean observations and predict- sion which propagated rightward toward the slope. ing wave-breaking locations are discussed in 5. Up to four runs were performed in the long-tank experiments before the two-layer ambient was recreated as described above. After two runs the fresh water layer had 2. Experiment Setup deepened and the interface had thickened as a result of mixing by the shoaling solitary waves. Between the second and Internal solitary waves were generated in an approxi- third run the false endwall was temporarily extracted so that mately two-layer salt-stratified fluid through the standard the relatively deep fresh water layer in the test section flowed lock-release method (Michallet and Ivey [1999]; Grue et al. rearward as a bore into the previously sealed-off section of [2000]). Experiments were performed in short and long the tank. Before the bore returned into the test section, the tanks. One was L = 1.97m long, 0.49m high and 0.18m false endwall was re-inserted. In this way the surface layer wide. The other was the same tank used by Michallet and shallowed by as much as half and the interface between fresh Ivey [1999] in their study of internal solitary wave gener- and salt water was thinner. ation and dissipation. Its interior was 6.90m long, 0.25m In the short-tank experiments a single camera (Canon high and 0.25m wide. In both tanks a constant slope of EOS Rebel T3i) was situated 3m in front of the tank with length L and height H was established at one end of the the lens (Canon EFS 18-35mm) situated at midpoint of the s s tank. This recorded movies of the evolution of the genera- tank, as illustrated in Figure 2. The slopes examined ranged tion, propagation and shoaling waves. between s Hs/Ls =0.143 to 0.417. ≡ In the long-tank experiments two digital cameras sepa- The tanks were first filled with salt water of density ρ1. rately recorded the steadily propagating the incident soli- Fresh water of density ρ0 was then poured slowly into a tary wave and the shoaling wave on the slope. The incident sponge-float creating a layer of depth H0 on top of the salt- wave was recorded by a camera (Nikon D90) situated 2.5m water. In all experiments the density difference between the from the side of the tank with the lens (Nikon DX SWM salt and fresh water ranged from ∆ρ ρ1 ρ0 = 10 to 3 ≡ − ED Aspherical) at the same height as the interface. In most 20 kg/m . experiments the midpoint of the field of view was 1m to the In the short-tank experiments the interface was marked right of the gate. The second camera (Canon EOS Rebel by dye as the fresh water was first poured into the sponge T3i) was situated 2.5m from the side of the tank in front float on top of the salt-water layer. In some of these exper- of the midpoint of the slope with the lens (Canon EFS 18- iments, fine kaolin clay was deposited on the slope to help 35mm) situated at the height of the slope at its midpoint. visualize boundary layer separation. After filling, a gate was The movies were processed and analyzed using Matlab. inserted a fixed distance from the endwall opposite the slope The methodology used to measure the properties of the forming a lock of width Lℓ = 0.286m. The bottom of the propagating and shoaling solitary waves are described in the gate was situated a few centimeters above the bottom to next section. X - 4 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

3. Qualitative Results counterclockwise. This gives rise to substantial resuspension of clay particles on the slope. Here we present snapshots and time series images ex- In an experiment with the same slope but with steeper in- tracted from typical experiments. These were chosen to il- cident wave (Figure 4c), the separation of the flow from the lustrate how quantitative measurements were taken and to slope, characteristic of collapsing breakers, is more clearly show how the breaking mechanism of the shoaling wave de- evident. In this case the rear flank of the wave simultane- pended upon the magnitude of the slope and incident wave ously overturns within the body of the fluid, characteristic amplitude and extent. of a plunging breaker. Such a collapsing-plunging breaker 3.1. Solitary Wave Properties acts to mix the upper and lower layer fluid substantially between the slope and the depth of the undisturbed interface. Figure 3a shows a snapshot of a rightward-propagating At later times this mixed patch acts to thicken the interface solitary wave of depression. The field-of-view ranges over between the surface and deep ambient. the ambient fluid depth and in the horizontal over a distance If the slope is shallower still (Figure 4d), the rear flank of between 0.70 and 0.30m to the left from the base of the the wave overturns even as the lower-layer fluid continues to − − slope (where the slope reaches the bottom of the tank). In be drawn downslope beneath the leading flank of the inci- this experiment the significant deepening of the fresh water dent wave. It resembles plunging surface waves in that the layer from H0 =0.016m to H0 + Asw =0.063m is evidence lee of the incident wave “pours” over the interface to make of a large amplitude wave whose description lies beyond the a tube-like structure. For this plunging breaker, mixing oc- range of validity of KdV theory (Stamp and Jacka [1995]; curs initially within the body of the fluid. However, at later Grue et al. [1997, 1999]). Indeed, the snapshot reveals a flattened trough whose structure differs qualitatively from times a separation bubble can still form on the slope and the squared hyperbolic secant structure of KdV waves. In the resulting turbulent upslope flow can result in substan- the quantitative analysis that follows we also show that the tial mixing and sediment resuspension. (e.g see Figure 8 of dependence of width upon amplitude significantly deviates Boegman and Ivey [2009]). from KdV behaviour. These classifications of the breaking regimes may be de- The corresponding vertical time series in Figure 3b is lineated in terms of the Iribarren number (1), as described constructed by stacking from left to right vertical slices in 4.2. at x = 0.70m taken through a succession of snapshots. To examine the deepening of the interface and consequent Here, t =− 0 corresponds to the time when the gate is ex- run-up along the slope of the lower layer fluid after maxi- tracted. As is the case in most experiments with lock-length mum deepening occurs, we constructed vertical and along- Lℓ =0.40m, the leading solitary wave had a smaller trailing slope time series. wave. We found that the number of trailing waves increases The development and consequent evolution of a with lock length, consistent with Maxworthy [1980]. In ex- collapsing-plunging breaker in a long-tank experiment is periments with a lock length of Lℓ = 0.80m, up to three shown through a sequence of snapshots in Figure 5. As the trailing waves were observed. wave front approaches the slope, the interface between fresh The solitary wave amplitude A , was determined to be sw and salt water forms a sloping front parallel to the slope. the difference between the maximum interface depth and the This front descends as fresh water piles into the right corner depth, H0, of the interface before the experiment began, as indicated in Figure 3a. of the tank over the slope and salt water flushes along-slope As well as vertical time series, we constructed horizontal downward and to the left (Figure 5b). The confluence of the times series (not shown) by stacking from bottom to top downslope bottom flow with the incoming flow aloft leads to successive horizontal slices at a depth midway between the turbulent separation of the flow where the interface touches initial interface depth and the maximum interface depth (i.e. the bottom slope at the location of its steepest descent (Fig- a distance H0 +Asw/2 below the interface). By tracking the ure 5c). Thereafter the interface rises back up the slope with rightward advance of the interface in these time series, we a front resembling that of a gravity current (indicated by an computed the speed, Csw, of the solitary wave. arrow in Figure 5d). The half-width, Lsw of the solitary wave was measured Particles seeded on the slope are observed to be resus- from the vertical time series by determining the time for the pended into the fluid column near the separation point both interface to deepen from H0 + Asw/2 to H0 + Asw and mul- at the position of maximum interface descent and during the tiplying this result by Csw. This is illustrated in Figure 3b. consequent upslope flow. Sediment transport and resuspen- For the solitary wave shown in Figure 3, we found Asw = sion is the subject of a separate research. Here our interest 0.047 m, L =0.150m and C =0.070m/s. sw sw is with the occurrence of boundary layer separation and the 3.2. Shoaling Solitary Waves characteristics of the interface descent and the upslope flow that follows. Figure 4 shows snapshots from four experiments in which Along-slope and vertical time series were constructed incident solitary waves were incident upon relatively steep from movies of the flow, as shown in Figure 6. The along- and shallow slopes compared with the incident wave slope. slope time series was constructed by taking diagonal slices These illustrate four distinct behaviours. If the slope is very through images a short distance (typically 0.005 to 0.010m) steep relative to the slope of the incident wave reflects with above the slope and stacking these in time. These revealed little to no overturning of the interface. In Figure 4a, the the rapid downslope advance of the interface while the wave wave develops into a surging breaker in which the interface is approached the slope. Figure 6a shows that it moved downs- vertical at the slope itself at the maximum breaking depth. lope by approximately 0.20m to d = 0.13m relative to In this case a small amount of mixing occurs at the slope mn itself as the interface begins to propagate back upslope. the base of the slope at d = 0. This time series also showed If the topographic slope is moderately smaller (Fig- the relatively slow upslope advance of the interface after ure 4b), the lower layer separates from the boundary as it the point of maximum descent was reached. The interface flushes downslope under the leading flank of the incident moved upslope at constant speed for some distance before wave. This collapsing breaker forms a separation bubble decelerating and ultimately halting. In some experiments, below the point where the interface intersects the boundary. multiple boluses of lower-layer fluid were observed to ad- As is clearer in movies of the experiment (provided as sup- vance up the slope. In these cases, we focus our analysis plementary material), the flow in the separation bubble is upon the leading front alone. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 5

Fitting a line to the location of the interface in the along- that pass through the points in Figure 8a,b reveal that slope time series as it initially rises upslope gives a measure of the upslope interface position versus time, as shown in Asw 0.45( 0.02) (Hℓ H0) and Lsw 1.68( 0.06) (Hℓ H0). Figure 7. The best-fit line to the points along the front at ≃ ± − ≃ ± (5)− early times is given by d =0.0266t 0.0626. The points show A and L are both approximately proportional to − sw sw that the current rises at constant speed, Cu 0.0266 m/s, H H0, and so are approximately proportional to each ≃ ℓ until time t 13s after which it decelerates at an ap- other:− L 3.7A . In part for this reason, we conclude ≃ sw sw proximately constant rate as given by the best-fit quadratic that the generated≃ waves were generally of such large ampli- d = 0.00248t2 +0.0911t 0.483. − − tude that they are not well described by KdV theory. This From the along-slope distance of maximum interface de- predicts that the width, Lsw,KdV (defined as area over am- scent, dmn, we determined the location at which to construct plitude), depends upon its amplitude, A, according to (e.g. the vertical time series. For example, the time series in Fig- see eq. (4.85) and accompanying text in Sutherland [2010]) ure 6b was constructed by stacking vertical slices taken at 2 the horizontal position xmn dmn/√1 + s , in which s is ≡ H¯ 3 the slope. The time series shows that the interface descends Lsw,KdV =4 . (6) at an almost constant speed until reaching the slope at its r 3A point of maximum descent, a distance 0.09m below the orig- That is, the width of a KdV solitary wave decreases as the inal interface height. Fitting a line to the location of the inverse square root of the amplitude. A departure of KdV interface in the vertical time series as it descends gives a measure of the vertical descent speed, W . By understand- theory is expected because in most of our experiments the i solitary wave amplitude is comparable to or larger than the ing what parameters determine the value of Wi, we are able ¯ to predict the time scale for breaking. characteristic depth H. As shown in Figure 8c, the speed of very large amplitude solitary waves is typically larger than the shallow water lin- 4. Quantitative Results ear wave speed given by

We performed over 60 experiments examining shoaling ′ C0 = g H,¯ (7) solitary waves on a slope. We begin by examining the prop- q erties of the incident solitary wave measured 0.5-1.0m to ′ in which g g(ρ1 ρ0)/ρ1 is the reduced gravity. Quali- the left of the base of the slope. This distance was suffi- ≡ − ciently close to the slope itself that dissipation of the prop- tatively, this is consistent with KdV theory, which predicts agating wave due to viscosity was considered negligible be- the speed, Csw,KdV , exceeds C0 by an amount that increases tween the measurement and breaking locations (Michallet with the wave amplitude: and Ivey [1999]). The amplitude, Asw, half-width at half- maximum, Lsw, and speed, Csw, are measured as they de- 1 A Csw,KdV = C0 1 + . (8) pend upon the lock parameters. 2 H¯ Thereafter we present the analyses of solitary wave shoaling. So that the results may be applied to ocean observa- However, a best-fit line through our data gives tions, the breaking regimes, maximum interface descent and speed and the consequent upslope flow are characterized in Hℓ H0 Asw Csw C0 1+0.052( 0.004) − C0 1+0.12 , terms of the characteristics of the incident solitary waves ≃ ± H¯ ≃ H¯ and not the lock parameters. (9) in which the last expression makes use of the first expres- 4.1. Solitary Wave Properties sion in (5). Thus the solitary wave speed is relatively slower than the corresponding KdV wave speed. This results from The structure of the solitary wave was set by the depth, the non-negligible flow in the lower layer that opposes the H , and horizontal extent, L , of the upper layer lock-fluid ℓ ℓ advance of the solitary wave. as well as by the characteristic ambient depth, as measured The properties of the solitary waves in our experiments by the harmonic mean of the upper and lower layer depths Stamp and Jacka Grue et al. of a two-layer fluid: are consistent with [1995] and [1999]. They serve as a reminder that it is erroneous to H0(H H0) infer the wavepacket width and speed from its amplitude H¯ − . (4) using KdV theory if the wave amplitude is comparable to ≡ H or larger than the shallow-layer depth. For oceanic solitary In practice we found that the amplitude, width and speed waves observed by means of a thermistor chain, the ampli- of the wave were set by the difference of the lock-depth and tude can be measured directly. The width is best estimated the depth of the upper layer fluid: Hℓ H0. The lock- as we have done in these experiments by measuring the time length determined the number of generated− waves. For ex- for the interface to descend from half to full maximum and periments with Lℓ =0.28 and 0.40m, a single hump-shaped multiplying this by the wave speed. In the ocean the speed disturbance was generated with a small trailing wave. With itself can be determined by the advance of smooth and rough Lℓ =0.80m, three to four waves were generated. surface signatures induced by the waves, which can be ob- Figure 8 shows how the amplitude, width and speed of served onboard ship, by satellites or by pairs of moorings the solitary wave depended upon the depth of the upper situated a known distance apart. layer lock-fluid. The errors in the measurements result from In the analysis that follows, the measured solitary wave uncertainty in locating the middle of the interface both in properties are used to characterize the dynamics of shoaling the lock fluid and in the ambient as the wave passed the on a slope. test section. To reduce these errors, here we plot the results for experiments in which the initial interface half-thickness 4.2. Breaking Regimes was less than 0.004m. Despite some scatter in the data, we found that the amplitude and width of the solitary wave vary The qualitative dynamics of the shoaling wave depend approximately linearly with the depth of the upper layer upon the bottom slope relative to the slope of the incom- lock-fluid. Explicitly, the best-fit lines through the origin ing solitary wave front. This is shown in Figure 9. For X - 6 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES experiments with prescribed topographic slope, s, and mea- [2010]. In both, the shoaling depth was assumed to depend sured incident wave slope Asw/Lsw, Figure 9a plots differ- upon the horizontal distance, L1 = H1/s, from the base of ent symbols depending upon whether the waves reflect with the slope to the point where the undisturbed interface inter- no significant breaking or whether they break as surging, sected the slope. But one should not expect the dynamics plunging or collapsing waves. Superimposed on this plot are of shoaling to depend upon the length of the slope in the lines of constant (internal) Iribarren number (Boegman et al. bottom layer: in the limit of an infinitely long slope, an in- [2005a]), defined by (1). We found that waves reflected with cident solitary wave would still act to deepen the interface little to no overturning if Ir> 1.5. Collapsing and plung- to a finite extent relative to its incident amplitude. ing breakers formed for lower Ir. Predominantly collapsing As has been done by Helfrich [1992], Boegman et al. breakers occurred if Ir> 0.7 and plunging breakers occurred [2005a] and Aghsaee et al. [2010], Figure 10b plots the ra- predominantly if Ir< 0.3. tio of incident wave amplitude to maximum interface dis- For comparison with the numerical simulations of Agh- placement depth against the ratio of the wave extent to the extent of the slope in the lower layer. For best compari- saee et al. [2010], the breaking regime data are replotted in son with Aghsaee et al. [2010], we define the extent of the Figure 9b. Qualitatively we find the same trend in which wave to be L = 2L . Consistent with earlier studies, we waves change from collapsing to plunging to surging as Ir w sw find that Asw/Hi holds values in a range about 0.6 0.1 for increases. However, the critical Iribarren numbers separat- ± small Lw/L1. However, we do not find a decreasing trend ing the breaking regimes are different in our experiments. for larger Lw/L1. In part, this is because our study focused In part, this is because the wave extent, Lsw,A, defined in upon larger amplitude incident waves and in many of our the simulations was based upon the area of the wave divided experiments the slope was steeper than s =0.2. by the amplitude. This value is larger than the half-width at half maximum, Lsw, defined in our experiments. Par- 4.4. Vertical Descent Speed tially to correct for this, the Iribarren number, Ir2, plotted along the abscissae in Figure 9b is defined in terms of Vertical time series taken at a location along the slope Lw 2Lsw Lsw,A. Consistent with Aghsaee et al. [2010], where the interface descends deepest showed that the ini- ≡ ≃ we find collapsing and plunging breakers for Ir2 in a range tial descent speed was constant almost until the interface between 0.4 and 1.2 for slopes below s =0.30. Because our descended to the slope itself. We predict this speed by fol- classification of collapsing and plunging breakers was based lowing the reasoning used above to estimate the maximum upon observations of the interface shape at the point of max- interface descent. We estimate the characteristic time for descent based upon the time for the incoming solitary wave imum breaking, we were unable to delineate the different to traverse its whole extent (approximately 2αL , with α regimes as rigorously as could be done from the examina- sw being an order unity constant) while moving at speed Csw. tion of time scales in the simulations. Hence, using (10), an estimate of the descent speed is 4.3. Breaking Depth ⋆ Hi 1 Asw Wi = = Csw s . (11) Particularly in experiments with plunging-collapsing 2αLsw/Csw α r Lsw breakers, we found that the shoaling solitary wave evolved so that the interface was nearly vertical where it intersected Figure 11 compares the measured descent speed, Wi, ⋆ the slope at its point of maximum descent (e.g. see Fig- against Wi for a range of experiments restricted to those on ures 4c and 5c). We use this observation to estimate the slopes shallower than s =0.42 and with initial ambient inter- maximum vertical displacement, Hi, of the interface. As- face half-thickness less than 0.004m. Although there is some ⋆ suming the fresh water that pools in the rightmost corner scatter, we find Wi 1.01( 0.05)Wi if we set α =1.6. ≃ ± of the tank has a right-angled triangular shape, the volume per unit width of this fluid below the initial depth of the 4.5. Upslope Bolus Propagation interface is HiLi/2, in which Li Hi/s is the horizontal ≡ Finally, we measured the along-slope evolution of the distance from the point of maximum descent to where the leading bolus that propagates upslope after the interface de- undisturbed interface intersects the slope. This we equate to scends to its maximum depth. In the experiments of Helfrich the approximate area (volume per across-slope width) of the [1992], the leading bolus was found to propagate initially at leading incident solitary wave, AswLw 2AswLsw. Thus we a speed, C 0, proportional to the shallow water speed com- ≃ b predict the maximum interface descent to be puted for the undisturbed ambient at the depth of maximum interface descent, C0,brk. Explicitly, the shallow water speed ⋆ Hi = √4sAswLsw. (10) is given by (7) in which the characteristic depth at the breaking point is H¯ = H0Hi/(H0 + Hi). Whereas Helfrich [1992] Figure 10a compares this prediction with the measured found that Cb0 0.6 C0,brk, this relation does not hold in ≃ maximum depth of descent of the interface, Hi. Points are our experiments with larger amplitude waves and steeper plotted only for those experiments with initial interface half- slopes. This is illustrated in Figure 12a, which shows Cb0 thickness smaller than 0.004m. Considering the crude ap- holds a range of values between 0.02 and 0.03m/s, with no proximation that the incident solitary wave area is given by obvious dependence upon C0,brk. 2AswLsw, (10) well predicts the breaking depth for wave When it reaches its maximum depth, the interface asso- shoaling on both moderate and shallow slopes. The predic- ciated with plunging and surging breakers is approximately tion succeeds less well for steep slopes (s > 0.4) because the vertical. Ignoring the ambient motion, we consider the upper-layer fresh water does not pool in the corner with a extent to which the bolus may be treated as an upslope- right-angled triangular shape in advance of surging breakers propagating gravity current. We imagine the bolus origi- and non-breaking reflected waves. For shallow slopes, which nates from a partial-depth lock-release at the point of maximum breaking in which the lock-depth is H and the total are more representative of the ocean, the prediction works i depth is H + H0. The initial speed of a gravity current of quite well because in most experiments incident waves shoal i depth h in a relatively deep ambient is g′h/2. If we as- as plunging or collapsing breakers. sume the current depth is half the depthp of dense fluid in The prediction (10) differs qualitatively from the empiri- the imagined lock, then cal prediction (2) determined by the laboratory experiments of Boegman et al. [2005a] and the empirical prediction (3) 1 C = g′H . (12) determined by the numerical simulations of Aghsaee et al. gc 2 i p SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 7

This is the speed of a current moving along a horizontal not invoke friction, Richardson numbers or shape factors, as bottom. Crudely, we correct for the fact that the current in Wallace and Wilkinson [1988], examines the dependence ′ is moving upslope by computing the component of speed in of Cb upon the incident bolus speed Cb0 and the along- the upslope direction. Thus we propose that the gravity cur- slope distance between where deceleration begins and the rent speed up a slope s which originates from a partial-depth surface. Figure 13b shows that these quantities are related lock-release of depth Hi is approximately by

′ ′ 2 1 g Hi C = . (13) Cb 1.2( 0.1) Cb0 /db. (15) gc,s 2 r 1 + s2 ≃ ± There is significant scatter in the data, which is to be ex- Figure 12b plots the bolus speed against Cgc,s, with both pected because detailed physical processes have been explic- quantities being given relative to the shallow water speed itly neglected. Implicit in (15) are the influences of buoyancy C0 given by (7), in which H¯ is given by (4). Plotting the results in this way does reveal a tentative relationship between (determined through Cb0), and the upper layer return flow the observed bolus speed and the predicted gravity current and gravity (determined through db and not Hb). Arguably, speed, although the bolus speed is found to be about half friction does not play a significant role at least during the the gravity current speed. early stages of deceleration. Particularly for collapsing and plunging breakers, the dynamics of the bolus are significantly different from those of a gravity current. In particular, during the generation of the 5. Discussion and Conclusions bolus, significant across-slope vorticity is clearly associated with it, whereas a gravity current head is assumed to be In light of recent ocean observations revealing internal irrotational. solitary waves of great amplitude (many times the depth of Nonetheless, comparing the results of Figures 12a and b, the surface layer) at the continental shelf, we have performed we conclude that the upper-layer depth (and hence the shal- laboratory experiments examining shoaling large amplitude low water speed) plays a negligible role in determining the solitary waves on slopes from shallow to steep. The study initial speed of the bore. This makes intuitive sense par- and analyses have been performed in a way to assist in ticularly for shoaling large amplitude waves for which the the interpretation of ocean observations, particularly ver- deepening of the shallow layer is so large at the breaking tical time series constructed from moored thermistor chains point that all memory of the original shallow layer depth is or CTD profiles (Klymak and Moum [2003]; Bourgault et al. erased. [2007]; Scotti et al. [2008]; Nash et al. [2012]). In their analysis of boluses shoaling upslope onto a shelf, Being generated by lock-release in an approximately two- Wallace and Wilkinson [1988] proposed that the speed of layer fluid, solitary waves of depression were created with the bolus should depend upon upslope distance as a power amplitudes up to 3.5 times the shallow layer depth. Con- law with exponent φ. Their expression was rewritten by Stamp and Jacka Grue et al. Helfrich [1992] in the form sistent with [1995]; [1999], we confirmed that KdV theory should not be used to infer the φ width of the waves from their amplitude, if Asw is compa- Cb = Cb0(1 d/db) , (14) − rable to or larger than the characteristic depth H¯ . Instead, the width is best determined as the half-width at half maxi- in which db is the along-slope distance from the surface to the position where the bolus begins to decelerate, with de- mum, Lsw, from measurements of the horizontal wave speed celeration beginning at d = 0. In their experiments, Wallace (observed at the surface or by comparing a wave passing and Wilkinson [1988] found φ =0.60 0.05 for slopes with through two closely spaced mooring sites) and the time for s =0.030. For boluses on a constant slope± with a relatively the interface to descend from half to maximum vertical dis- more shallow upper layer, Helfrich [1992] found better agree- placement. This measure of wave extent is more practical ment between theory and experimental measurements using than its definition in terms of area over amplitude, which φ 0.25. That said, the theoretical prediction underesti- requires longer time series and which assumes the wave is ≃ mated the observed speeds soon after bolus formation and indeed solitary with no breaking in the lee, as has been ob- it overpredicted the speeds as the bolus came to halt. served in the lab for very large amplitude waves (Carr et al. In our analysis of along-slope time series, we accurately [2008]). and continuously measured the along-slope bolus position Our analysis of breaking regimes classified in terms of the versus time, as illustrated in Figures 6a and 7. The results consistently showed that the bolus did not immediately be- Iribarren number was consistent with earlier experiments gin to decelerate upon its formation. Instead it propagated (Boegman et al. [2005a]) and numerical simulations (Agh- along-slope at constant speed until it had risen from a depth saee et al. [2010]), although the critical Iribarren numbers Hi to a depth Hb. Only then did deceleration commence. distinguishing between collapsing, plunging and surging dif- The constant-speed regime preceding deceleration is also ev- fered because we defined Ir in terms of the half-width Lsw. ident in Figure 13 of Helfrich [1992]. We developed a simple theory predicting the maximum From the along-slope position at which the bolus first be- interface descent at breaking. Although qualitatively differ- gan to decelerate, we determined its corresponding depth ent in form than the empirical equations of Boegman et al. Hb. For example, in Figure 7, the along-slope distance for [2005a] and Aghsaee et al. [2010], it’s predictions were con- transition, db, occurs where the linear and quadratic curves sistent with experiments. are tangent. Knowing db, Hb is easily determined by geome- Significantly, whereas the empirical predictions rely on a try. Figure 13a plots the rise height Hb Hi versus the max- − constant slope from base to interface, the prediction (10) imum interface depth, Hi. The only experiments reported requires only a comparison between the area of the solitary upon here had an interface half-thickness less than 0.004m and the incident wave slope was sufficiently large to form a wave to the area of the lower layer fluid between where the plunging breaker. The plot shows that the bolus propagates interface initially meets the slope and where it ultimately upward more than half the depth of the maximum interface descends most. For the former we approximated the area descent before it begins to decelerate. by the product of amplitude and twice the half-width. For The along-slope time series shows that when the bolus the latter, because our lower boundary had constant slope, does begin to slow down it does so with constant deceler- we took the area to be that of a right-angled triangle. This ′ ation, Cb . A simple analysis of the deceleration that does formula may readily be applied to observations of solitary X - 8 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES waves shoaling on the continental shelf. For example, Ta- El, G. A., R. H. J. Grimshaw, and W. K. Tiong, Transforma- ble 1 computes the maximum descent and run-out of the tion of a shoaling undular bore, J. Fluid Mech., 709, 371–395, thermocline for shoaling solitary waves in the Sulu Sea, the doi:10.1017/jfm.2012.338, 2012. Australian North West Shelf and the Scotian Shelf. The re- Farmer, D. M., and L. Armi, The generation and trapping of sults show the interface can descend from tens to hundreds solitary waves over topography, Science, 283 (5399), 188–190, of meters with run-out lengths on the order of kilometers. 1999. These calculations assumed constant topographic slope. Fringer, O. B., and R. L. Street, The dynamics of breaking pro- More generally, if the topographic height above the abyss is gressive interfacial waves, J. Fluid Mech., 494, 319–353, 2003. given by η(x), then one can estimate the horizontal position, Grimshaw, R. H. J., E. Pelinovsky, and T. Talipova, Solitary wave transformation due to a change in polarity, Stud. App. Math., xb, where breaking occurs through the implicit relation 101 (4), 357–388, 1998. x0 Grue, J., H. A. Friis, E. Palm, and P. O. Ruso as, A method for 2AswLsw = H1 η dx. (16) computing unsteady fully nonlinear interfacial waves, J. Fluid Zxb − Mech., 351, 223–252, 1997. Grue, J., A. Jensen, P.-O. Ruso as, and J. K. Sveen, Properties of Here H1 is the depth below the thermocline in the open large-amplitude internal waves, J. Fluid Mech., 380, 257–278, ocean and x0 is the position where the undisturbed ther- 1999. mocline intersects the topography (i.e. where η(x0) = H1). Grue, J., A. Jensen, P.-O. Ruso as, and J. K. Sveen, Breaking Using (16) to find xb, the maximum interface descent is sim- and broadening of internal solitary waves, J. Fluid Mech., 413, ply H1 η(xb). 181–217, 2000. The− stress and turbulence associated with solitary wave Helfrich, K. R., Internal solitary wave breaking and run-up on a breaking can resuspend sediments and nutrients into the uniform slope, J. Fluid Mech., 243, 133–154, 1992. fluid column. As such, to manage marine ecosystems and Helfrich, K. R., and W. K. Melville, On long nonlinear internal to assess potential hazards for the offshore-oil industry, it waves over slope-shelf topography, J. Fluid Mech., 167, 285– is important to predict the location of breaking sites and 308, 1986. the along-slope range and stresses associated with boluses Helfrich, K. R., and W. K. Melville, Long nonlinear internal that run upslope following a breaking event. The diagnos- waves, Annu. Rev. Fluid Mech., 38, 395–425, 2006. tics presented here give crude order of magnitude estimates Helfrich, K. R., W. K. Melville, and J. W. Miles, On interfacial of breaking locations and the evolution of the leading bo- solitary waves over slowly varying topography, J. Fluid Mech., lus. Examining the impact of these processes upon bedload 149, 305–317, 1984. transport and sediment resuspension is the subject of work Holloway, P. E., Internal hydraulic jumps and solitons at a shelf in preparation. break region on the Australian North West Shelf, J. Geophys. Res., 92, 5405–5416, 1987. Acknowledgments. The authors are deeply grateful to Kao, T. W., F.-S. Pan, and D. Renouard, Internal solitons on the Marco Ghisabalberti and Nicole Jones for their assistance in pro- pycnocline: Generation, propagation, and shoaling and break- viding equipment for the experiments in the School of Environ- ing over a slope, J. Fluid Mech., 159, 19–53, 1985. mental Systems Engineering at the University of Western Aus- Klymak, J. M., and J. N. Moum, Internal solitary waves of el- tralia. Part of this research was conducted by Sutherland as a evation advancing on a shoaling shelf, Geophys. Res. Lett., Gledden Senior Visiting Fellow at the University of Western Aus- 30 (20), 2045, doi:10.1029/2003GL017706, 2003. tralia. Klymak, J. M., R. Pinkel, C. T. Liu, A. K. Liu, and L. David, Prototypical solitons in the South China Sea, Geophys. Res. Lett., 33 (11), L11,607, 2006. Korteweg, D. J., and G. de Vries, On the change of form of long References waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag., 39, 422–443, 1895. Aghsaee, P., L. 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Osborne, A. R., and T. L. Burch, Internal solitons in the An- White, B. L., and K. R. Helfrich, Gravity currents and internal daman Sea, Science, 208, 451–460, 1980. waves in a stratified fluid, J. Fluid Mech., 616, 327–356, 2008. Pinkel, R., Internal solitary waves in the warm pool of the western Whitham, G. B., Linear and Nonlinear Waves, 636 pp., John equatorial Pacific, J. Phys. Oceanogr., 30, 2906–2926, 2000. Wiley and Sons, Inc., New York, USA, 1974. Russell, J. S., Report on waves, Brit. Assoc. Rep., 311, 1844. Sandstrom, H., and J. A. Elliott, Internal tide and solitons on the Xu, J., J. Xie, Z. Chen, S. Cai, and X. Long, Enhanced mixing Scotian Shelf: A nutrient pump at work, J. Geophys. Res., 89, induced by internal solitary waves in the South China Sea, 6415–6426, 1984. Cont. Shelf Res., 49, 34–43, 2012. Scotti, A., R. C. Beardsley, B. Butman, and J. Pineda, Shoaling Zhao, Z., and M. H. Alford, Source and propagation of internal of nonlinear internal waves in Massachusetts Bay, J. Geophys. solitary waves in the northeastern South China Sea, J. Geo- Res., 113, C08,031, doi:10.1029/2008JC004726, 2008. phys. Res., 111, C11,012, 2006. Stamp, A. P., and M. Jacka, Deep-water internal solitary waves, J. Fluid Mech., 305, 347–371, 1995. Sutherland, B. R., Internal Gravity Waves, 378 pp., Cambridge B. R. Sutherland, Departments of Physics and of Earth & At- University Press, Cambridge, UK, 2010. mospheric Sciences, University of Alberta, CCIS 4-183, Edmon- van Haren, H., and L. Gostiaux, Energy release through internal ton, AB, Canada, T6G 2E1. ([email protected]) wave breaking, Oceanography, 25 (2), 124–131, 2012. K. J. Barrett, Department of Physics, University of Al- Venayagamoorthy, S. K., and O. B. Fringer, On the formation berta, CCIS 4-183, Edmonton, AB, Canada, T6G 2E1. (kjbar- and propagation of nonlinear internal boluses across a shelf [email protected]) break, J. Fluid Mech., 577, 137–159, 2007. Wallace, B. C., and D. L. Wilkinson, Run-up of internal waves G. N. Ivey, School of Environmental Systems Engineering on a gentle slope in a two-layered system, J. Fluid Mech., 191, (SESE) and UWA Oceans Institute, University of Western Aus- 419–442, 1988. tralia, Perth, Australia, 69007 ([email protected]) X - 10 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

Table 1. Extract of data from Table 2 of Boegman et al. [2005a] giving observed ambient conditions including the upper layer depth (H0), the lower layer depth (H1) and the topographic slope (s), and observed amplitude (Asw) and total horizontal extent (λ∞ ≃ 4Lsw) of an approaching solitary wave. From these parameters, we determine the Irib- arren number (1), eq. (10) is used to predict the maximum ⋆ interface descent (Hi ), and we ﬁnd the horizontal distance ⋆ ⋆ (Li = Hi /s) that the interface travels downslope to the maximum descent position. All length values are given in meters. ⋆ ⋆ Location s H0 H1 Asw λ∞ Ir Hi Li Source NorthWestShelf 3/2000 43 80 30 910 0.0041 6.4 4,300 Holloway [1987] NorthWestShelf 3/2000 48 75 35 830 0.0037 6.6 4,400 Holloway [1987] ScotianShelf 1/200 50 100 30 1,100 0.015 13 2,600 Sandstrom and Elliott [1984] SuluSea 1/11 170 3,300 65 6,300 0.45 193 2,100 Apel et al. [1985] SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 11

a) No Breaking b) Surging Breaker

c) Collapsing Breaker

d) Collapsing-Plunging Breaker

e) Plunging Breaker

Figure 1. Schematics illustrating diﬀerent breaking mechanisms for solitary waves of depression approaching a constant slope. Sketches are representative of the ﬂow around the time of maximum breaking depth. X - 12 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

L false gate endwall

ρ0 H0

Hℓ ρ0

Hs H ρ1

Lℓ Ls Figure 2. Experiment set-up showing the deﬁnition of length scales and and densities used to characterize the control variables of the experiment. Note that the length of the tank is much longer than the height, as indicated by the zig-zag at the tank bottom at the mid-section of the schematic. The slope is indicated in black, salt water in dark gray and fresh water in light gray.

a) Snapshot b) Vertical Time Series 0.15 Lsw/Csw Asw 0.1 [m] z 0.05

0 0.7 0.6 0.5 0.4 0.3 0 5 10 15 20 − − − − − x [m] t [s]

Figure 3. a) Snapshot and b) vertical time series of an internal solitary wave taken from an experiment in the 3 long tank with ρ1 = 1012.1 kg/m , H = 0.164 m, H0 = 0.016 m, Hℓ = 0.10m and Lℓ = 0.40m. The horizontal position in a) is shown with respect to the location of the base of the slope oﬀ-image to the right. Time in b) is given with respect to the time at which the gate is extracted. The vertical axes in a) and b) are the same. Note, the vertical black section between x = 0.45 and 0.41m in a) is the result of a supporting column− outside −the tank blocking the back-lighting. The horizontal white dashed line in a) indicates the depth of the undisturbed interface and the amplitude of the wave is indicated by the arrow. The white arrow in b) indicates the time for the interface to deepen from Asw/2 to Asw. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 13

a) Ir=1.42, t=17s 0.4 0.3

[m] 0.2 z 0.1 0 b) Ir=1.13, t=19s 0.4 0.3 [m] 0.2 z 0.1 0 c) Ir=0.95, t=16s 0.4 0.3

[m] 0.2 z 0.1 0 d) Ir=0.64, t=15s 0.4 0.3

[m] 0.2 z 0.1 0 0 0.5 1 1.5 x [m] Figure 4. Snapshots from four experiments in the short tank distinguishing diﬀerent breaking mechanisms according to the indicated Iribarren number: a) surging breaker (vertical interface at boundary), b) collapsing breaker (boundary layer separation), c) collapsing- plunging breaker (boundary layer separation and overturning aloft) and d) plunging breaker (overturning aloft). Note, movies of each of these experiments can be viewed as supplemental material to this paper. X - 14 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

0.15 a) t=0s

0.1 [m] z 0.05

0.15 b) t=4 s

0.1 [m] z 0.05

0.15 c) t=8 s

0.1 [m] z 0.05

0.15 d) t=12s

0.1 [m] z 0.05

0 0 0.1 0.2 0.3 x [m] Figure 5. Successive snapshots from the experiment with parameters given in Figure 3. The slope is s =0.417 and the incident wave has amplitude and width Asw = 0.047m and Lsw = 0.150m, respectively. So Ir 0.75, corresponding to a plunging breaker. The vertical≃ white arrow in d) indicates the front position of the upslope- propagating lower layer. Note, a movie of this experiment can be viewed as supplemental material. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 15

a) Diagonal time series 0.01 m above slope

0.3

0.2 [m] d 0.1

b) Vertical time series above maximum descent 0.15

0.1 [m] z 0.05

0 0 10 20 t [s]

Figure 6. a) Vertical time series taken above the position of maximum interface deepening and b) diagonal time series taken along a slice parallel to the slope 0.010m above it. Both are shown for the experiment shown in Figure 5 with parameters given in Figure 3 and with slope s = 0.417. Time t = 0 corresponds to the moment the interface maximum displacement passes over the base of the slope and d measures the distance upslope from its base. X - 16 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

Position of upslope interface advance 0.4

0.3 [m] d 0.2

0.1 5 10 15 20 t [s]

Figure 7. Position of upslope-propagating bolus following the maximum interface descent determined from a diagonal time series taken 0.5cm above the slope for the experiment shown in Figure 5. Circles indicate front positions located at different times. The long-dashed line is the best-fit line to the earliest 10 points. The short-dashed line is the best-fit quadratic to the last 5 points. The horizontal solid line indicates the along-slope distance from the bottom of the slope to the position where the underdisturbed interface intersects the slope (at d =0.385m). The surface is at d =0.426 m. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 17

a) Amplitude 4

3 ¯ H / 2 sw A

slope= 0.47 0.02 0 ± b) Half-width 20 ¯ H / 10 sw L

slope= 1.71 0.08 0 ± c) Speed 2 0 slope= 0.052 0.004 /C 1 ± sw C

0 0 5 10 (H H )/H¯ ℓ − 0 Figure 8. Relative a) amplitude, b) width and c) speed of the incident solitary wave plotted against the depth of upper-layer lock ﬂuid relative to the characteristic depth of the ambient, H¯ . Symbols denote initial lock conditions: Hℓ < 2H0 (triangles), 2H1 Hℓ 4H1 (squares) ≤ ≤ and Hℓ 4H1 (circles); Lℓ 0.40m (open symbols), ≥ ≤ Lℓ 0.80m (closed symbols). The typical error bars are shown≥ in the top-left of each plot. X - 18 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

a) Topographic slope vs Wave Slope b) Topographic Slope vs Iribarren Number 0.8

0.6

Ir=1.5 s 0.4

Ir=1.0 Plunging 0.2 Collapsing Ir=0.7 Surging No Breaking Ir=0.3 0 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 2.5 3 Asw/Lsw Ir s/ A /2L 2 ≡ q sw sw

Figure 9. a) Breaking regimes showing breaking characteristics observed in several experiments as they depend upon the topographic slope s and the characteristic incident wave slope Asw/Lsw. The dashed lines show values of constant Iribarren number, as indicated. b) The breaking regimes are plotted on a regime diagram with topographic slope and Iribarren number (after Figure 17 of Aghsaee et al. [2010]). The symbols indicating the breaking regimes for both plots are indicated in the legend in b). Collapsing-plunging breakers have a circle and cross superimposed. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 19

a) Maximum Interface Descent 8 s< 0.2 0.2 s< 0.4 ≤ 6 s 0.4 ≥ ¯ H /

i 4 H

0 0 2 4 6 8

√4sAswLsw /H¯

b) Relative Amplitude versus Wave Extent 1 i

/H 0.5 sw A

0 0 0.5 1 (2Lsw)/L1 Figure 10. a) Observed maximum interface displace- ⋆ ment, Hi, versus prediction, Hi √4sLswHsw, based upon observed incident solitary wave≡ properties and plotted with diﬀerent symbols depending upon the slope as ⋆ indicated. Both Hi and Hi are plotted relative to the initial characteristic ambient depth H¯ . The dashed line ⋆ indicates where Hi = Hi . b) Incident wave amplitude relative to maximum interface displacement plotted versus the wave extent relative to horizontal length of the slope in the lower layer. (After Figure 6 of Helfrich [1992], Figure 11 of Boegman et al. [2005a]. and Figure 18 of Aghsaee et al. [2010]). Diﬀerent symbols are plotted as indicated in a). X - 20 SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES

Interface Vertical Descent Speed 0.02

[m/s] 0.01 i W

slope= 1.01 0.05 ± 0 0 0.01 0.02 W ⋆ 1 C sA /L [m/s] i ≡ α sw q sw sw Figure 11. Speed of vertical descent of the interface toward its maximum displacement depth. Symbols are the same as those in Figure 8. The dashed line is the best-ﬁt line through the points that passes through the origin.

a) Upslope Bolus Speed 0.05

0.04

0.03 [m] 0 b

C 0.02

0.01

0 0 0.02 0.04 0.06 0.08 C0,brk [m] b) Bolus vs Upslope Gravity Current Speed 0.8

0.6 0 /C

0 0.4 b C

0.2 slope= 0.48 0.02 ± 0 0 0.5 1 1.5 Cgc,s/C0 Figure 12. a) Speed of the upslope-propagating leading bolus plotted against the shallow water speed evaluated at the location of maximum interface descent. b) Speed of the upslope bolus plotted against the predicted speed of an energy conserving gravity current released from a lock of partial depth Hi. Both values are given relative to C0, the shallow water speed of the ambient upstream of the slope. In both plots the symbols indicate the magnitude of slope as in Figure 10. SUTHERLAND, BARRETT AND IVEY: SHOALING INTERNAL SOLITARY WAVES X - 21

a) Depth Traversed by Bolus at Constant Speed

0.06 [m]

i 0.04 H − b

H 0.02 slope= 0.59 0.04 ± 0 0 0.05 0.1 Hi [m] b) Bolus vs Upslope Gravity Current Speed 0.006

] .

2 0 004 slope= 1.2 0.1

[m/s ± ′ b

C 0.002

0 0 0.002 0.004 0.006 2 2 Cb /db [m/s ] Figure 13. a) Height of rise upslope of the bolus while it moves at constant speed compared with the maximum interface descent. b) Magnitude of along-slope deceleration of the bolus after its constant-speed phase compared with the characteristic deceleration based upon its upslope speed and along-slope distance to the surface. In both plots the symbols indicate the magnitude of slope as in Figure 10.