Experimental investigation of nonlinear internal waves in deep water with miscible fluids
Roberto Camassa, Matthew W. Hurley, Richard M. McLaughlin, Pierre-Yves Passaggia, Colin F. C. Thomson
EExperimental investigation of nonlinear internal waves in deep waterwith miscible fluids
Roberto Camassa ∗ Matthew W. Hurley † Richard M. McLaughlin ‡ Pierre-Yves Passaggia § Colin F. C. Thomson ¶ May 31, 2018
Abstract
Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW)propagation in ‘ deep water ’ configuration with miscible fluids. The results are validated againstdirect numerical simulations and traveling wave exact solutions where the effect of the diffusedinterface is taken into account. The waves are generated by means of a dam break and theirevolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV).In particular, data collected in a frame moving with the waves are presented here for the first time.Our results are representative of geophysical applications in the deep ocean where weakly nonlineartheories fail to capture the characteristics of large amplitude ISWs from field observations.
Diurnal and semi-diurnal ocean tides generate large amplitude ISWs, as observed for instance on theHawaiian ridge (Rudnick et al. , 2003), near the Luzon strait (Duda et al. , 2004; Alford et al. , 2015),in the gulf of Alaska (Churnside & Ostrovsky, 2005), in the Sulu sea (Apel et al. , 1985), off the coastof California (Pinkel, 1979), and more recently in the Tasman sea (Johnston et al. , 2015). These largeamplitude ISWs propagate for very long distances from their sources (Hosegood & van Haren, 2006;Kunze et al. , 2012; Simmons & Alford, 2012) across the deep ocean (see Helfrich & Melville (2006)for a review on ISWs), have been observed to reach amplitudes as large as 300m (Rudnick et al. ,2003) with nonlinearity α = η max /h ≈
5, where h is an appropriate reference depth scale (Stanton &Ostrovsky, 1998). Such large amplitude internal solitary waves induce localized regions of strong shearflows that contribute to a large amount of mixing near the bottom (Kunze et al. , 2012; van Haren,2013) and across the thermocline (Liu et al. , 1985; Brandt et al. , 2002). Internal solitary waves arealso likely to be amplified by interaction with mesoscale eddies (Xie et al. , 2015). ISWs play a centralrole in ocean circulation as they provide a means for mixing and dissipation in the pycnocline of both ∗ Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Car-olina, Chapel Hill, NC 27599, USA. Email: [email protected] † Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Car-olina, Chapel Hill, NC 27599, USA. Email: [email protected] . ‡ Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Car-olina, Chapel Hill, NC 27599, USA. Email: [email protected] . § Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA. Email: [email protected] . ¶ Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Car-olina, Chapel Hill, NC 27599, USA. Email: [email protected] . a r X i v : . [ phy s i c s . f l u - dyn ] M a y he deep ocean (Grimshaw et al. , 2003; Grimshaw & Helfrich, 2012) and the continental shelf (Lamb,2014; Alford et al. , 2015; Bourgault et al. , 2016; Passaggia et al. , 2018).While the dynamics of ISWs in shallow configurations has arguably received most of the attentionin the literature (see, e.g., Helfrich & Melville (2006) for a review, and Zhao et al. (2016) very recentdevelopments) its deep water counterpart has been extensively studied mainly from a mathematicalperspective, and this mostly in the weakly nonlinear and two-layer regimes modelled by, respectvely,the Korteweg-de-Vries (KdV) (Grimshaw, 1981), the Intermediate Long Wave (ILW) (Davis & Acrivos,1967; Kubota et al. , 1978) and the Benjamin-Ono (B-O) equations (Benjamin, 1986; Kalisch & Bona,2000). More recently, Choi & Camassa (1999) derived a fully-nonlinear model for internal waves intwo-layer deep water configuration that makes no assumptions on the amplitude of the wave. Thesystem of equations in the depth and the depth-averaged fluid velocity of the thin layer reduces theILW and BO models in the appropriate weakly nonlinear, unidirectional limit, but can accommodatewaves in the nonlinear regime as well. Mathematically, the intermediate long wave equation andits infinite-depth limit, the Benjamin-Ono equation, are completely integrable models for internalwaves in deep water, and in principle allow for exact solutions to be computed. While these arecertainly a desirable property of a physical model, the weak nonlinearity assumption excludes internalwave phenomena commonly seen in nature. In particular, wave amplitudes are observed to be ofmagnitudes several times the thickness of the upper layer of water, which falls well outside the weaknonlinearity assumption. The next step towards an accurate comparison can consist of modeling ISWsby solving the Dubreil-Jacotin-Long (DJL) equation (Dubreil-Jacotin, 1934; Long, 1953), which givessteady-state solutions to the Euler equations in the moving frame of the wave. While this approachdoes not have the appeal of analytical-type methods previously cited, it is not restricted to simplestratifications and provides an intermediate theoretical tool between direct, time- dependent numericalsimulations and integrable or analytically tractable models.Experimental and field observations are routinely compared with numerical solutions to supportresults extracted from field experiments and extrapolate integral quantities such as wavelength andwave speed (Vlasenko et al. , 2000; Preusse et al. , 2012 b , a ; Lien et al. , 2014). Recently, Passaggia et al. (2018) used a DJL solution to compute waves corresponding to the field measurements of Moum et al. (2003) and provided an accurate representation of both waves and instability dynamics. Amongthe open problems in ISW modelling, the role played in the dynamics by turbulent wakes not fullyseparated by the laminar wave motion is yet to be studied in detail. In what follows, we indirectlyaddress this issue by a systematic comparison of experimental data with analysis of the DJL equationsand direct numerical simulations, with solutions and observations of waves both close and far fromtheir generation site.Experiments in the miscible and shallow regime have been extensively carried out (see, e.g., Grue et al. (1997, 1999); Carr et al. (2008); Fructus et al. (2009); Carr et al. (2011, 2017)). However, forthe deep water regime, results of interfacial waves from experiments mostly used immiscible fluids(e.g. using water/petrol or water/silicon oil) see, e.g., Michallet & Barth´elemy (1998); Kodaira et al. (2016). To the best of our knowledge, the results presented herein are the first to be performedwith miscible fluids on much larger scales than other experiments, both for the horizontal and/or thevertical length scales. The use of salt as a stratifying agent, plus the size of the domain, positivelyindicates that the results are representative across several length scales and indicate scalability to theextreme case of oceanic ISWs. This assertion is confirmed in the conclusions of the present study,where our experimental, numerical and theoretical results are compared with observations from fieldmeasurements taken near the Luzon strait for large amplitude internal solitary waves (Ramp et al. ,2004; Huang et al. , 2016).In order to compare wave properties, we collect local profiles of fluid velocities and density fromParticle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF); these tools allow computa-2a) W = 0 . m Long section top layer (red)bottom layer (blue)laser mounted on cartgatedam Deep section separating gate L = 27 m H = m H = 1 . m (b) L = 9 m laser mounted on cartdamgate H = m gate closedseparatingtop layer (red) ρ t = 0 .
998 g/ccbottom layer (blue) ρ b = 1 .
020 g/cc
Figure 1: Schematic showing the two section of the of the modular wave tank for the long setup (a)and the deep setup (b).tion of amplitude, speed, and wavelengths from a dataset. These properties are then compared withDJL solutions and direct numerical simulations of the Euler equations. We show that experimentsand simulations do capture the dynamics of internal solitary waves, and, in particular, we show thatexperiments closely reproduce the DJL predictions, thus also assessing the effect of a diffused interface.Direct numerical simulations of two-dimensional time evolution governed by the Euler equations arein good agreement with wave quantities. In contrast with DJL solutions, the time dependent waveprofiles clearly highlight a lack of fore-aft symmetry, which could be caused by dynamical interactionof the emerging (main) solitary wave with the wake from the generation site for traveling distancescommensurate to those of the experiments.Specifically, the paper is organized as follows: Section § § § § Experiments were conducted at the Joint Applied Mathematics and Marine Science Laboratory inChapel Hill, North Carolina. Our approach is to generate internal solitary waves under stable con-tinuous stratification in the modular wave tank of the Joint Fluids lab. The dimensions of our tank,27m long, up to 3m high and 0 .
75m wide (see figures 1(a,b)) allow us to reach larger scales than3reviously achieved in laboratory experiments, such as those reported in Koop & Butler (1981); Grue et al. (1999); Michallet & Barth´elemy (1998); Kodaira et al. (2016); Carr et al. (2017).The modular tank was setup in two different ways: a long and shallow experiment where the wavewas propagated down the full length of 27m, and a deep but shorter experiment where the wave prop-agated over 9m in the 2 .
25m deep section. In the deep setup, the base layer was 2 .
15m deep whereasthe top layer was 10cm. In the case of long and shallow runs, the adjustable gate between the long andthe deep sections remained fully open, for a total fluid depth of 77 cm consisting of a top and bottomlayer of 15 cm and 62 cm respectively (see figure 1a). This particular choice was made to match the ex-periments of Grue et al. (1999) in order to proved a term of comparison for our experimental data withthe shallow configuration. The laser and the imaging system, consisting of a double set of a ParticleImage Velocimetry (PIV) camera and Laser Induced Fluorescence (LIF) cameras, all equipped withnarrow band-pass optical filters – respectively at 532nm for the PIV and 583nm for the LIF – weremounted on a computer-controlled motorized cart. The full setup was either kept in a fixed positionor towed along the tank to track the spatio-temporal evolution of the waves as they traveled down thelong section (see figure 1b). In the case of the deep, short runs, the adjustable gate was closed to createa water tight seal between the deep and the long section. It will be shown later that these conditionsare sufficient to generate nonlinear internal waves in the deep regime in a salt-stratified environment.The laser was mounted under the tank, and a single PIV/LIF setup was mounted on a fixed platformat the level of the top layer. In both cases, an 80cm tall gate formed an “internal” dam that was eitherpulled up during the long runs or down during the deep runs in order to release the trapped fresh water.Stratifications were generated by combining salt and fresh water to a desired salinity, and thenpumping it into the flume from the adjacent storage and processing unit. This allowed for profileswithin the tank ranging from a nearly two-layer structure to continuous stratifications of general form.All tank sides, including the bottom, consist of tempered glass panels, which allow for visualizationfrom all directions. The flow was illuminated using a Litron ® Nd:Yag diode pumped double cavitylaser with 100mJ per pulse at 532nm wavelength used for both PIV and LIF (cf. figure 1(a,b)).PIV combined with LIF data were collected with a custom built camera setup using up to four GigEBobcat B3320 Imperx ® ® DG535 pulse generator and acquired simultaneously on a computer using three Ethernetcards. The analysis of the experimental data was performed using the Matlab ® -based open-sourcePIV software DPIVSoft (Meunier & Leweke, 2003; Passaggia et al. , 2012) to process the PIV and theLIF images. In addition to the stationary camera set-up, a moving camera mount followed, and movedslightly faster than, the wave in order to capture their evolutions over a longer distance.Initial density measurements were acquired using an Orion TetraCon ®
325 conductivity probeconnected to a WTW Cond ® ® Paar DM35 densitometer whosecalibration was verified to the fourth digit. Initial density profiles were measured before each ex-periment using the Orion TetraCon ®
325 conductivity probe mounted on a Velmex ® linear stage.Measurements of conductivity and temperature were collected every centimeter and interpolated tocalculate the corresponding density profile. These initial vertical profiles of density are shown in figure2(a-c) for both the deep and the long section. In particular, with ρ ( z ) the density at the verticallocation z , and ∆ ρ the total density variation from bottom to top (with reference density ρ being theminimum density), we denote hereafter the renormalized density profile by B ( z ), where B ( z ) = ρ ( z ) − ρ ∆ ρ ; (1)this is shown in figure 2(a) in the case of the deep configuration and 2(b) for the long/shallow config-uration. 4a) − − . − − .
50 0 0 . . . . ( H − z ) / h B ( z ) (b) − − . − − .
50 0 0 . . . . ( H − z ) / h B ( z ) (c) Density (g/cm ) D e p t h ( c m ) Figure 2: Background density profiles extracted from the experiments and used in both numericalsimulations and DJL calculations. (a) Initial profiles normalized with respect to the reference profile B ( z ) = (tanh(4( x +1)) − / µ L per liters of water using BrightDyes ® Rhodamine WT.Lavision ® nearly neutrally buoyant polydisperse polyethylene PIV particles with diameters in therange [10:100] µ m were mixed in a separate tank, sprayed over the free surface and left to slowlysettle across the layers to seed the experiment. Note that the settling speed was several orders ofmagnitude smaller than the speed of the waves. The laser was double-pulsed at 3Hz while image pairswere recorded by mean of double exposure with a 20ms difference between two successive frames.The camera set-up was centered at 2 .
46m from the end of the tank, or 15m from the end of the longexperiments. The purpose of this location was to capture a well-developed (i.e. sufficiently separatedfrom the wake) solitary wave while minimizing end effects.In our experimental facility, the salt water is recycled after each experiment. It is sent to a reverseosmosis unit which separates the salt component, and is subsequently filtered using a carbon and UVfilters. The recycled water out of the reverse osmosis unit is then sent to 15 storage tanks containingup to 23m of brine (six tanks) and fresh water (nine tanks). We consider the dynamics of an inviscid incompressible fluid governed by the Euler equations, restrictedto two-dimensions which are written u t + uu x + wu z = − p x ρ , (2a) w t + uw x + ww z = − p z ρ − g, (2b) ρ t + uρ x + wρ z = 0 , (2c) u x + w z = 0 . (2d)In the notation above, ( u, w ) are the horizontal and vertical fluid velocities, respectively, ρ is the(variable) fluid density, and p is the pressure. Throughout the present manuscript, subscripts indicatepartial derivatives. These equations are discretized and solved using the VarDen algorithm (Almgren et al. , 1998) with initial conditions chosen to match those in the experiments. The numerical strategyof the VarDen code uses a second-order accurate projection method and a second-order predictor-corrector scheme for time integration. Both time-step and spatial mesh are adaptive with a base meshdiscretization of ∆ x ≈ . x ≈ . ρ ( x, z ) = ρ (cid:20) z − H gate (cid:0) W gate − x ) (cid:1)(cid:21) , (3)where ρ ( z ) is the background density, W gate the width of the gate (fixed at 56 cm in both experimentsand simulations) and H gate is the depth of the fluid behind the gate beyond the rest state. Thebackground density ρ is assumed to be a hyperbolic tangent function and it is best-fit to match thedensity profile measured before each experiment. Boundary conditions are taken to be free-slip, no-fluxwalls including the top of the domain; this rigid lid assumption could be viewed as the most significantdeparture of the simulations from the experiments. However, this is expected to cause negligible effectsgiven the bounds on density differences, although our results do show a legacy of weak free-surfaceeffects (cf. § ∇ η + N ( z − η ) c η = 0 , (4)6n which η is the vertical displacement of a given isopycnal (constant ρ ) line, N ( z − η ) = − gρ (cid:48) ( z − η ) /ρ ( z − η ) is the Brunt-V¨ais¨al¨a frequency, and c the speed of the traveling wave. The DJL equation isa reduction of the Euler equations to a steady waveframe x → x − ct , and here we used the Boussinesqapproximation version, whereby density variations are neglected for the inertial forces. The DJLequation has a natural variational formulation, so by using an iterative method that minimizes theenergy traveling wave solutions can be computed efficiently (Stastna & Lamb, 2002). The presentresults are evaluated using the multi-grid strategy of Dunphy et al. (2011), with the DJL solutionsobtained by continuation with increasing energy using fixed point iterations. The domain sizes for theDJL solver were [ L, H ] = [9 , . L, H ] = [9 , . n x , n z ] =[256 , n x , n z ] = [2048 , x → ± L/ ρ ( z ) = ρ + ∆ ρ (cid:18) (cid:18) z − z δ (cid:19)(cid:19) , (5)where ρ = 0 . T = 23 o C, ∆ ρ is the maximum density difference measured between the top and the bottom of thetank, δ is the parameter that defines the thickness of the pycnocline, and z is the height of theinflection point of the density profile. Direct numerical simulations of the initial value problem for the dam-break-type initial solution arecompared with experimental results and traveling-wave solutions computed from the DJL equations.In what follows, we first report a comparison for wave properties between the three methods, andshow that onset of nonlinear internal solitary wave propagation is observed in deep water from fixedlocation measurements. Next, horizontal wave profiles from fixed measurements are analyzed for bothdeep and long configurations. Finally, in § The results were extracted from the LIF and PIV data as illustrated in figure 3(a-c) for the deepsetup and figure 3(d-f) for the long setup. The spatio-temporal evolution of the waves in both thedeep and long experiments are shown by means of density identified from the LIF, used as proxyand velocity computed using PIV. Figure 3(d-f) shows the largest amplitude wave generated in thelong tank, and replicates the waves measured by Grue et al. (1999). The particular wave shownin figure 3(a) is compared with a much larger wave obtained in the deep tank in figure 3(c). Theexperiments in the long configuration were repeated multiple times to check for the consistency ofthe generation mechanism, using similar initial density profiles as shown in figure 2, and the resultsare displayed for both experimental data, DJL predictions and DNS simulations in figures 5(a-d).Instantaneous profiles of horizontal velocity and density (through the proxy by LIF) are shown infigure 4(a) at x/h = 70 from the dam gate for the deep case and figure 4(b) at x/h = 80 for the longcase. These measurements are compared with the companion DNS of the initial value problem andthe DJL results. The error measured between the simulation and the corresponding experiments at7a) ( H − z ) / h (b) ( H − z ) / h (c) ( H − z ) / h (d) ( H − z ) / h (e) ( H − z ) / h (f) ( H − z ) / h Figure 3: Example of internal wave data in experiments for the long/shallow setup: (a-c) horizontalvelocity field u (( x − ct ) /h vs. ( H − z ) /h ), vertical velocity w (( x − ct ) /h vs. ( H − z ) /h ) and densityfield ρ (( x − ct ) /h vs. ( H − z ) /h ), respectively, for a dam height d/h = 6 ± δ/h ≈ .
55, corresponding to c/ √ g (cid:48) h ≈ . a/h ≈ .
718 and λ/h ≈ .
04. (d-f) Similar to (a-c)but for a dam height d/h ≈ ± δ/h ≈ .
4, corresponding to a wave withcharacteristics c/ √ g (cid:48) h ≈ . a/h ≈ .
544 and λ/h ≈ . η max is less than 10% between the simulation and the DJLprofile and 4% between the experiment and the direct numerical simulation. Both the amplitude andthe velocity are closely predicted for the deep and the long experiments. Here, the DJL solutions werecomputed by matching the amplitude with the experiment. The overall agreement is satisfactory, andthe results, including measurements from other waves, show a slight improvement of the comparisonwith increasing amplitude. Note that near the surface the density profile inferred from LIF datadeparts from the simulation, due to either light absorption by the dye (cf. figure 4(a)) or uneven8a) − − − − − − . − . . . . . . ( H − z ) / h ( U ( z ) /c , B ( z )) | η max U ( z ) /c exp. B ( z ) exp.DJL U ( z ) /c sim. B ( z ) sim. (b) − . − − . − − . − − . − . − . . . . . . ( H − z ) / h ( U ( z ) /c , B ( z )) | η max U ( z ) /c exp. B ( z ) exp.DJL U ( z ) /c sim. B ( z ) sim. (c) − . − − . − − . − − . − − . − . . ( H − z ) / h ( U ( z ) /c , B ( z ) | η max DJL U ( z ) /c sim. B ( z ) sim. Figure 4: Example of velocity profiles u ( z ) and density profiles b ( z ) extracted from the PIV andLIF data at the center (maximum pycnocline displacement) of a wave, collected along the wavetankat different distances x from the generation gate; (b) long/shallow configuration, x/h = 85 ; (c)long/shallow configuration, x/h = 65.concentration due to fading at this particular location in the tank (cf. figure 4(b)). The size of thefacility, and the length of time for the set up of each experiment, made the dye intensity difficult tocontrol.It is also interesting to notice that the vertical wave profile for the larger amplitude waves exhibitsvertical oscillations, shown in figure 3(b), which appear to be synchronized with the oscillations of thefree surface (not shown here). The wavelength of this oscillation is much shorter than the wavelengthof the internal solitary wave, and seem to be dominant near the the front of the wave, a feature thatagrees with the oned recently reported by Kodaira et al. (2016) for large amplitude interfacial ISWsin immiscible fluids. Wave characteristics defined by amplitude a , wave speed c and effective wavelength λ/h are shown infigure 2(a-d), where the full symbols represent the results extracted from the two-dimensional Eulersimulations, and the hollow symbols with error bars depict the experimental results. All results arenon-dimensionalized by the characteristic depth h given by the mean-density isoline in the rest state,and the characteristic velocity c = g ( ρ max /ρ min − h . The wave speed c was measured using twotemporal mean-density displacement profiles located at fixed locations in the tank and separated by adistance ∆ x ≈ τ between the9wo temporal profiles, with the wave velocity calculated using c ≈ ∆ x/τ . Once determined, thespeed of the wave was used to transform the temporal scale t to the spatial scale X through thetransformation X = x − ct . The effective wavelength λ/h is defined as λ = 1 a (cid:90) ∞ ζ ( X ) dX (6)with ζ ( X ) = ζ ( x − ct ) the displacement of the mean-density isoline, but computed in practice byintegrating in time along the transformed spatial axis X from the beginning of the experiment untilthe crest of the wave is observed in both the experiments and the numerical simulations.The amplitude, both experimentally and numerically, of each ISW as a function of the dam heightis shown in figure 5(a), where the response of the ISW amplitude a/h to increasing dam height d/h remains essentially linear. This relationship indicates the consistency of the generation mechanismand the clear relation between dam height and resulting solitary wave amplitude. This confirms thatthe present experiments essentially lie within the deep regime, because no saturation of amplitudewith dam height was observed, as it would be expected for the shallow regime when approaching aconjugate state. The normalized effective wavelength λ/h as a function of the wave speed is shown infigure 5(b), which shows that our measurements are well within the transition from linear to nonlinearinternal solitary waves. The DJL solutions, depicted by means of solid and dashed black lines for thethicker and thinner pycnoclines in figures 5(b-d), both describe a minimum effective wavelength λ/h as a function of the wave speed c/ √ g (cid:48) h . The critical point found in the vicinity [ λ/h, c/ √ g (cid:48) h ] ≈ [4 , (cid:78) )). This critical point is also captured by the experiments and associatedDNS, although the thicker pycnoclines suggest that this minimum shifts to slightly lower values ofboth λ/h and c/ √ g (cid:48) h .The normalized speed c/ √ g (cid:48) h is displayed as a function of the amplitude a/h in figure 5(b),where the DJL prediction is found to closely predict the speed simulated by the nearly two-layerstratification shown by the red triangles. The ISWs’ speed are well within the bounds predicted bythe DJL calculations except for the three larger amplitude experimental waves, which appeared to befaster than both the DJL prediction and the respective numerical simulations. However a closer lookat the PIV and LIF data shows that these waves were strongly asymmetric, with what appeared tobe a recirculating core, from which large amplitude billows were observed developing downstream.Figure 5(c) depicts the effective wavelength λ/h as a function of the ISWs’ amplitude a/h , where agood collapse between theory, numerical simulations and experiments is further confirmed. The DJLresults show the effect of pycnocline thickness, parametrized in nondimensional form by δ/h , on thewave properties. As δ/h increases, the critical amplitude a/h at which the minimum of the effectivewavelength λ ( δ/h ) occurs also increases, while λ/h appears to remain essentially independent on thethickness in the range of pycnoclines measured in our experiments, from δ/h = 0 . δ/h = 1 . λ/h is reported asa function of the non-dimensional wave speed c/c . The collapse between simulations and experimentsis well within the DJL predictions where the thickness of the pycnocline does not seem to play animportant role. The errors in the collapse between the DJL wave properties, the experiments and the numericalsimulations suggest that the wave profiles provide more insights on wave details such as symmetry andthe effects of wakes. The DJL solutions of the Euler equations are symmetric, traveling, steady states10a) .
11 1 10 a / h d/hH = 0 . H = 2 . H = 0 . H = 2 . (b) . . . . . . . . a / h c/c (c) . . . . λ / h a/hH = 2 . δ/h = 0 .
1, 2.15mDJL, δ/h = 1 .
45, 2.15mDJL, δ/h = 1 .
45, 6mDJL, δ/h = 0 .
5, 0.77mDJL, δ/h = 0 .
5, 0.77m (d) . . . λ / h c/c Figure 5: ISW characteristics, summary of all simulations and experiments: (a) ISW’s amplitude a/h as a function of the dam height d/h , (b) amplitude a/h as function of wave speed c/ √ g (cid:48) h , (c) reducedwavelength λ/h as function of wave amplitude a/h , and (d) reduced wavelength λ/h as function ofwave speed c/ √ g (cid:48) h . The full symbols show the numerical simulations, the hollow symbols with errorbars refer to the experiments, and the solid and dashed curves are the theoretical predictions from theDJL equation at various locations and diffused interface thicknesses, as labeled in the legends.waves in the moving frame of reference X → x − ct . In contrast, we are interested in comparing waveprofiles for both experiments and numerical simulations, which are developing solitary waves that havenot yet reached an asymptotic steady state (which of course could only be achieved adiabatically withrespect to the slow decay due to the presence of viscosity in real experiments). Profiles of solitarywaves are shown in figure 6(a) for the long configuration and figure 6(b) for the deep experimentstracking isopycnals from the experimental data, by matching the dye intensity with the correspondingvalue of the density. Values of the thirtieth and seventieth percentile of the density are reported infigure 6(a,b), where the amplitude of the DJL solution was matched with the experimental results.The collapse between experiments and the DJL profile is particularly striking for the deep case. While11 a) − . − − . − . − . − . − − . − . − − ( H − z ( B i , t )) / h ( x − ct ) /h expDJLsim (b) − . − − . − − . − − . − − − − ( H − z ( B i , t )) / h ( x − ct ) /h Figure 6: Height of constant density z ( B i , t ), where B i is a given value with i = [1 , x − ct ) /h . The dashed linesare iso-density of the thirtieth (blue) and seventieth (red) percentile of the density, corresponding to B = 0 . B = 0 . (a) . . . . . . . − − δ ( x − c t ) / h ( x − ct ) /h expDJLsim (b) . . . . . − − − δ ( x − c t ) / h ( x − ct ) /h Figure 7: Thickness δ ( x − ct ) of the diffused interface, measured by the difference between the thirtiethand seventieth percentile of the density reported in figures 6(a,b) for: (a) the long configuration, and(b) the deep case in the time-space coordinate.the numerical simulation are in good agreement with DJL profiles, apart from a slight asymmetrywith respect to the trough of the wave in figure 6(b), the experimental and numerical results seem12 a) − . − . − . − . − . − − m i n [ ( ∂ u ) / ( ∂ z ) ] ( x − ct ) /h expDJLsim (b) − . − . − . − . − . − . − . − . . − − m i n [ ( ∂ u ) / ( ∂ z ) ] ( x − ct ) /h Figure 8: Minimum value of the shear ∂u ( x − ct, z ) /∂z computed along the waves reported in figures6(a,b) for the long (a) and deep (b) cases in the time-space coordinate.to degrade with respect to the DJL solution for the smaller amplitude wave shown in figure 6(a).Despite the small mismatch in amplitude, the asymmetry between the leading and the trailing edgeof the wave seems to be more important than for its large amplitude counterpart. The most likelyscenario for this discrepancy seems to be related to the lack of separation from the wave’s wake,wherein large amplitude, two-dimensional structures are seen to persist in the numerical simulations.This asymmetry is a consequence of the generation mechanism, which we show in the next section tobe less apparent as the ISW travels farther along the wave tank. Of course, to increase the fidelityof the simulations with respect to experiments the former would have to be implemented in fullythree-dimensional settings, substantially increasing the computation cost.The difference between the two isopycnals difference ∆ δ/h above is shown in figure 7(a,b) for boththe deep and the long case, respectively, where we aim at confirming that the expansion and com-pression of the isopycnals induced by the wave can be observed in the experiment, and is captured bythe DJL and direct numerical simulations. In the long/shallow case, figure 7(a) shows this compari-son between the DJL solution, the simulation and the experiment. At the leading side of the wave,the isopycnals undergo an expansion, followed by a compression at the trough and a re-expansionalong the lee side. The comparison between the DJL waves and the simulations is in good qualitativeagreement for both amplitude and locations of compression and expansion. The comparison with theexperimental data is less precise, and it is barely captured for the experiments in the long configura-tion (see figure 7a) where the variations are only of the order of ≈ .
5% of h . The comparison andagreement in the deep case and a larger amplitude wave is more striking (see figure 7b). Variationsin the pycnocline thickness are found to be of the order of ≈
5% of h for the numerical simulation aswell as for the DJL wave and the experiment.Finally, we report the minimum value of the shear computed from the DJL solution, the experimentand the numerical simulation in figure 8(a,b) for the short/deep and the long/shallow cases. Thecollapse between experiments, DJL solutions and numerical simulations is well predicted for bothcases. This plot tests the quality of our experimental and numerical approaches. In the next section13e extend the present analysis beyond the scales previously reported by Grue et al. (1999) for thelong/shallow regimes, and investigate the evolution of these ISWs by tracking their spatio-temporalevolution. To the best of our knowledge, this is the first time that such measurements with datacollected in the co-moving frame of the wave has been attempted. This helps determine the trendof our waves in eventually reaching a traveling (adiabatic) steady state form farther down the longsection. The discrepancy for the measurements performed at 12m from the gate in the long section led us totrack several ISWs by following them with the moving motorized cart shown on top of the long sectionin figure 1. Once the ISWs were seen to pass the first location at x = 12m, the cart was started with avelocity 10% greater than the traveling wave speed for a KdV solitary wave (Choi & Camassa, 1999) c = c (cid:18) − a ρ ( H − h ) − ρ h ρ h ( H − h ) + ρ h ( H − h ) (cid:19) . Tracking ISWs remained however a difficult task due to the variability of wave speed immediatelyafter the generation. We report only a limited number of measurements in figures 9(a-d) for 3 ISWswith initial dam heights d = [15 , , x/h . The amplitude of these waves was found to decay slowly as they travelled along the 27 m of the long section of our modular tank; this decay became less pronounced as the waves separatedfrom their wakes. It is interesting to notice that both numerical simulations and experiments showa decay of the wave amplitude as the ISWs travel farther down the long section (Grimshaw et al. ,2003). We remark that the simulation solves the Euler equation and does not include viscosity, theonly source of dissipation being induced by the numerical scheme itself. Also, we remark that thelarger amplitude ISWs measured in the experiment seem to decay faster than the amount predictedby the simulation. This could be attributed to the vibrations induced by the cart carrying the laser,cameras, and power supplies, which can inject noise in the fluid flow, as discussed below.Measurements of wave profiles inferred from the LIF are shown in figure 9(b) for two representativeamplitudes and compared with DJL solutions for the lee-side of the waves, where an overshoot wasoriginally observed at x/h = 80. The collapse between experiments and theory appears to improvesignificantly, except for the near-wake region. Here transient growth of finite amplitude noise, inducedby the vibrations of the cart, may be responsible for the transition to Kelvin-Helmholtz-type instabil-ities which are seen as oscillations at both the trough, the lee and more strikingly in the wake of thewave (Camassa & Viotti, 2012; Passaggia et al. , 2018). Note that stationary measurements, which didnot induce such vibrations, displayed no noticeable growing transient inside the wave region.The good agreement in figure 9(b) suggests that the ISWs are becoming close to a traveling wave-type solution. To support this conclusion, profiles of density measured at the trough of the ISWs,inferred from the LIF for the density and horizontal velocity measured from PIV, are compared infigure 9(c,d) for two representative waves. Both profiles appear to be in very good agreements betweenDJL and the simulations. Furthermore the overshoot observed at x/h ≈
80 has now disappeared, andwhile horizontal velocity near the bottom show a slight mismatch of the order of 8% – probably dueto finite tank effects – the overall agreement is excellent. This suggests that our ISWs have to travela distance of x/h ≈
120 from their generation site in order to more closely approach a travelingwave-type state.The possible application of our results in the context of open ocean measurements and theoreticalmodels is further discussed next, together with our conclusions on the present study.14a) . . . . . a / h x/h d = 15cm d = 20cm d = 30cm d = 15cm d = 20cm d = 30cm (b) − − . − . − . − . − − . − − − − − − − η ( x − c t ) / h ( x − ct ) /h exp, d = 30 cmDJL, d = 30 cmsim d = 30 cmexp, d = 20 cmDJL, d = 20 cmsim d = 20 cm (c) − − − − − − . − . . . . . . z / h ( U ( z ) /c , B ( z )) | η max U ( z ) exp B ( z ) exp B ( z ) simu U ( z ) simuDJL (d) − − − − − − . − . . . . . . z / h ( U ( z ) /c , B ( z )) | η max U ( z ) exp B ( z ) exp B ( z ) simu U ( z ) simuDJL Figure 9: Wave properties from moving frame: (a) Normalized ISW amplitude a/h as a function ofthe non-dimensional cart position x/h tracked in the long section for three representative dam height.(b) Profiles of the fiftieth percentile of the density inferred from the LIF measured at x/h ≈ d − h ) /h ≈ d − h ) /h ≈ B = ( ρ ( z ) − ρ min ) / ( ρ max − ρ min ) (blue-green) inferred from the LIF at the position of maximum displacement η max compared with the DJL solution (lines) computed for the same amplitudes at ( d − h ) /h ≈ .
33 (c)and ( d − h ) /h ≈ The present study reports a clear experimental evidence of nonlinear ISWs in the deep regime andmiscible fluids. Results are systematically compared with DJL solutions, direct numerical simulationsof the dam-break initial value problem, and the associated experiments. Measurements are reportedfor the fully nonlinear regime in a deep configuration, past the critical point, which is confirmed by theDJL theory where the nondimensional effective wavelength increases with respect to both the wave15a) λ a/h DJL, δ/h = 0 . δ/h = 1 . δ/h = 1 .
45, 6m deepStanton & Ostrovsky (1998)Liu et al. (2004)Ramp et al. (2004)Huang et al. (2016)1101000 .
01 0 . (b) . . . . a / h c/c Figure 10: Effective wave length λ/h as a function of wave amplitude a/h for the present results insalt=stratified water ompared with previous experimental results from Koop & Butler (1981) (blacksymbols) for the case of immiscible fluids (water and freon), the KdV prediction (thin dashed line (- --)), two-layer model Grue et al. (1997) (thin dashed-dotted line − · − · − ), fully nonlinear model ( —– )Choi & Camassa (1999, 1996) in the long/shallow case, the field data of Liu et al. (2004) (large hollowtriangles), Ramp et al. (2004) (large hollow circles) and Huang et al. (2016) (large hollow pentagon).Thick lines are the DJL predictions shown in figure 5(c). (b) same as figure 5(c) but compared withfield data from the above references.amplitude and wave speed. These results are supported with local measurements of LIF and PIV,which are further compared with the DJL waves and the direct numerical simulations with an overallsatisfactory agreement. In addition, we explore the effect of the interface thickness on the dynamics,and show that thick pycnoclines modify both wave amplitude and wave speed. Larger thicknessesincrease the wave amplitude while decreasing the speed and the wavelength, an effect that does not16eem to be experimentally studied in the same depth of other properties of internal waves.Vertical and horizontal wave profiles are also compared between numerical simulations, experimentsand DJL solutions. For these properties, such velocity and density measurements, the agreementis excellent. Horizontal profiles of isopycnal displacement demonstrate the overall good agreementbetween the DJL solutions, numerical simulations and experiments. We show in the long/shallowconfiguration that waves measured too close to their region of generation tend to be asymmetric,as they have not yet fully separated from their wakes. This effect is however not observed fartherdown the long section and shows that relatively large experimental facilities are necessary in order toobserve the approach to true traveling waves in the mathematical sense of models such as KdV, DJL,and Camassa-Choi (Korteweg & De Vries, 1895; Dubreil-Jacotin, 1933; Choi & Camassa, 1999). Suchasymmetries have also been reported in oceanic observations and are known to play an important rolein the redistribution of nutrients and chlorophyll, which are essential drivers of primary production inthe euphotic zone (Dong et al. , 2015).The present results are gathered in figure 10(a) together with the results from Koop & Butler(1981) where interfacial ISWs were generated in immiscible fluids (water and liquid freon) and thelinear/nonlinear theories for the deep and shallow cases. In addition, we report large amplitude ISWswave characteristics from field measurements in the deep regime, in the case of a train of internalsolitary waves propagating westward off the Oregon Shelf with very thin stratification (Stanton &Ostrovsky, 1998) ( (cid:5) ). We also compare with ISWs propagating westward from Luzon straight inthe South China Sea from Liu et al. (2004) ( ◦ ) and Ramp et al. (2004) ( (cid:52) ), and an extreme wavedeveloping eastward in the Northern China Sea (Huang et al. , 2016) ( (cid:68) ). Effective wavelengths λ/h for both DJL, experimental, and field measurements are reported in figure 10(a) as a function ofwave amplitude a/h . As displayed by this figure, our measurements fill the gap between the two-layerexperiments of Koop & Butler (1981) and Michallet & Barth´elemy (1998) and field measurementsof large amplitude ISWs in deep waters. At this point, a remark is in order: while experimentsproducing nonlinear ISWs could be performed in tabletop set-ups, our experiments bridge the gapbetween tabletop experiments (subject to small-scale effects such as viscosity, surface tension, andMarangoni convection at the surface (Luzzatto-Fegiz & Helfrich, 2014) and field observations. Instead,we observed large-scale effects such as transient growth and free-surface motions. The consistency ofresults across length-scales suggests that laboratory ISW characteristics faithfully represent ISWsmeasured from field observations. Wave speeds from field measurements are also compared with theDJL solutions, experimental data, and numerical results in figure 10(b), which shows an overall goodagreement. These data support the idea that large amplitude ISWs in the nonlinear regime from fieldobservations in the ocean (Stanton & Ostrovsky, 1998; Huang et al. , 2016) or in lakes (Preusse et al. ,2012 a , b ) are only accurately modelled by DJL type solutions (Dunphy et al. , 2011) and nonlineartheories such as the Choi-Camassa two-layer model (Choi & Camassa, 1999).PIV results and images from experiments indicate the presence of small surface waves travelingwith the ISW for the larger amplitude ISWs. The resonance between capillary and gravity waves hasbeen recently explored experimentally and theoretically for interfacial ISWs in immiscible fluids byKodaira et al. (2016). Our numerical code is based on a rigid lid approximation, and thus cannotaccommodate free surface motions. The present experimental results could however provide an inter-esting starting point to study surface-internal wave interactions in the case of miscible fluids, whichdeserves more investigation and represents an important application for the dynamics and predictionof large amplitude internal solitary waves (Alford et al. , 2015).17 cknowledgements RC, RMM, and CT acknowledge the support by the National Science Foundation under grants RTGDMS-0943851, CMG ARC-1025523, DMS-1009750, DMS-1517879, and DURIP N00014-12-1-0749.PYP acknowledges the support by the National Science Foundation under grant NSF OCE-1155558and NSF OCE-1736989. RC, RMM, and CT thank David Adalsteinsson for helpful comments on thepost-processing of the numerical results using DataTank.
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