On the Reflectance of Uniform Slopes for Normally Incident Interfacial Solitary Waves

1156 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 On the Reflectance of Uniform Slopes for Normally Incident Interfacial Solitary Waves DANIEL BOURGAULT Department of Physics and Physical Oceanography, Memorial University, St. John’s, Newfoundland, Canada DANIEL E. KELLEY Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada (Manuscript received 29 March 2005, in final form 2 September 2006) ABSTRACT The collision of interfacial solitary waves with sloping boundaries may provide an important energy source for mixing in coastal waters. Collision energetics have been studied in the laboratory for the idealized case of normal incidence upon uniform slopes. Before these results can be recast into an ocean parameterization, contradictory laboratory findings must be addressed, as must the possibility of a bias owing to laboratory sidewall effects. As a first step, the authors have revisited the laboratory results in the context of numerical simulations performed with a nonhydrostatic laterally averaged model. It is shown that the simulations and the laboratory measurements match closely, but only for simulations that incorporate sidewall friction. More laboratory measurements are called for, but in the meantime the numerical simulations done without sidewall friction suggest a tentative parameterization of the reflectance of interfacial solitary waves upon impact with uniform slopes. 1. Introduction experiments addressed the idealized case of normally incident ISWs on uniform shoaling slopes in an other- Diverse observational case studies suggest that the wise motionless fluid with two-layer stratification. The breaking of high-frequency interfacial solitary waves results of these experiments showed that the fraction of (ISWs) on sloping boundaries may be an important ISW energy that gets reflected back to the source after generator of vertical mixing in coastal waters (e.g., impinging the sloping boundary depends on the ratio of MacIntyre et al. 1999; Bourgault and Kelley 2003; Kly- the wavelength Lw and a length scale Ls characterizing mak and Moum 2003; Moum et al. 2003). Since mixing the sloping boundary (see Fig. 1 for a definition sketch). is important to many aspects of coastal ocean dynamics, Unfortunately, from the point of view of extrapola- these observations call for the development of a model tion to the ocean, the reflectance values differ signifi- capable of predicting ISW generation, propagation, and cantly between the experiments, as will be shown in dissipation in hydrographically complex oceans and section 3b. Helfrich (1992) reported reflectance values lakes. However, before such a generic model can be that were 0.1–0.4 lower than those of Michallet and established, a number of specific problems related to Ivey (1999) for similar impinging waves. Since the re- ISW dynamics must be solved. In particular for this flectance is bounded between 0 and 1, these differences paper, the reflectance of shoaling boundaries for nor- are large enough to merit further investigation. mally incident ISWs need to be better established. To shed light on this issue, we have attempted to The laboratory experiments of Helfrich (1992) and reinterpret the results of the laboratory experiment of Michallet and Ivey (1999) provide the best available Michallet and Ivey (1999). Our hypothesis is that side- insight into this problem of slope reflectance. These wall friction might have introduced a bias in the analysis of the measurements. Noting the lack of theories that describe the shoaling, breaking, and reflectance of Corresponding author address: Daniel Bourgault, Department of Physics and Physical Oceanography, Memorial University, St. large-amplitude ISWs on steep slopes (see, e.g., the re- John’s, NL A1B 3X7, Canada. cent review of Ostrovsky and Stepanyants 2005; Hel- E-mail: [email protected] frich and Melville 2006), we have addressed this prob- DOI: 10.1175/JPO3059.1 © 2007 American Meteorological Society Unauthenticated | Downloaded 09/25/21 03:25 PM UTC JPO3059 MAY 2007 BOURGAULT AND KELLEY 1157 where y is the across-tank spatial coordinate, B is the ␳ ␶ | tank width, 0 is a reference density, and s yϭB and ␶ | s yϭ0 are the stresses at the sidewalls. These stresses are parameterized following boundary layer theory (Kundu 1990, p. 312) as 1 1 ␶ | ϭ ϭϪ ␳ C |u|u and ␶ | ϭ ϭ ␳ C |u|u, s y B 2 0 D s y 0 2 0 D FIG. 1. Definition sketch showing an interfacial solitary wave of amplitude a and length L [defined by Eq. (6) in the text], propa- 0 w ͑2͒ gating in a two-layer fluid, with upper and lower undisturbed layer ϭ ϩ depths h1 and h2 (total depth H h1 h2), toward the sloping where u is the two-dimensional velocity field (i.e., along ϭ boundary of length Ls and slope s H/Ls. This definition for Ls tank and vertical) and CD is a drag coefficient whose is from Michallet and Ivey (1999), and Helfrich (1992) defined Ls from the base of the slope to the interface–bottom intersection. value was tuned in the simulations to test sensitivity to sidewall drag. Inserting Eq. (2) into Eq. (1) gives lem with a suite of fully nonlinear and nonhydrostatic |u|u F ϭϪC . ͑3͒ two-dimensional numerical simulations. In these simu- s D B lations the sidewall boundary condition is parameterized and is varied to mimic both laterally bounded labo- b. Simulations of laboratory experiments ratory domains and unbounded ocean domains. Simulations were performed with geometries and The model we used and the simulations we carried control parameters matching the experiments of out are described in the next section. After that, we Michallet and Ivey (1999). In these experiments mea- outline the results and discuss their interpretation, surements were made of the reflectance defined as highlighting the question of whether sidewall friction ϭ ր ͑ ͒ may have provided a bias in the laboratory results. We R ER E0, 4 conclude with recommendations for future laboratory where E is the total energy of the incoming ISW mea- studies, and with a tentative parameterization. 0 sured at the base of the slope and ER is the energy of the wave that is reflected back from the sloping bottom, 2. Methods also measured at the base of the slope. a. General Michallet and Ivey (1999) related the reflectance to the length ratio Lw /Ls (Fig. 1). More recently, Boeg- The analysis makes use of a numerical model that man et al. (2005) have argued that this parameter is not integrates the laterally averaged Boussinesq Navier– suitable for generalizing the laboratory results to field- Stokes equations. The model, described in detail by scale situations because it is independent of the bound- Bourgault and Kelley (2004), can simulate reasonably ary slope. These authors have proposed the use of the well the structure and velocity field of the first stages of Iribarren number, defined as a shoaling ISW when compared with the laboratory ␰ ϭ ր͑ ր ͒1ր2 ͑ ͒ observation of Michallet and Ivey (1999). As shown in s a0 Lw , 5 Bourgault and Kelley (2004), discrepancies between where s is the slope of the linear topography and a is model results and laboratory observations become ap- 0 the wave amplitude (see Fig. 1). We will adopt the parent during and after the transition to three-dimen- Boegman et al. (2005) recommendation and present sional turbulence caused by the wave breaking on the our results as a function of ␰ instead of L /L . slope. In the use of a two-dimensional model, we are w s Thirty-three numerical simulations were carried out thus working under the assumption that the reflection (Table 1). From each run the values of a , L ,E, and of normally incident waves does not depend on the 0 w 0 E were extracted, as explained in the following para- three-dimensional stages of a shoaling wave event. R graphs, and were compared with the laboratory mea- To address sidewall issues, we have incorporated a surements of Michallet and Ivey (1999). All simulations parameterization of sidewall friction by analogy to were carried out on a grid of horizontal and vertical common bottom-friction parameterizations used in spacing ⌬x ϭ 2.5 mm and ⌬z ϭ 1.25 mm, of maximum shallow water models. The laterally averaged drag depth H ϭ 15 cm and of constant width B ϭ 25 cm. The force per unit mass is represented as model is thus configured in a large-eddy mode, with a ␶ | Ϫ ␶ | ϭ s yϭB s yϭ0 ͑ ͒ grid fine enough to resolve the ISW breaking and some Fs ␳ , 1 0B of the collapse into smaller scales, but not the dissipa- Unauthenticated | Downloaded 09/25/21 03:25 PM UTC 1158 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 TABLE 1. Summary of numerical simulations. The densities of ␳ ␳ the top and bottom layers are 1 and 2 respectively. The experiment identifiers 1–16 match the identifiers used by Michallet and Ivey (1999, their Table 1), and the control parameters are set to match these laboratory experiments. The identifiers a and b refer to two different types of sidewall boundary conditions. In suite a, free-slip sidewalls are used; in suite b, no-slip sidewalls are used ϭ with CD 0.2 as inferred by calibrating the model to laboratory measurements (see section 3a). ␳ ␳ ϩ No. 1/ 2 h2/(h1 h2) Ls (cm) s 1 b 1.039 0.66 0 ϱ 2 a, b 1.040 0.82 0 ϱ 3 a, b 1.020 0.65 217 0.069 4 a, b 1.020 0.78 217 0.069 5 a, b 1.040 0.70 217 0.069 6 a, b 1.039 0.90 217 0.069 7 a, b 1.040 0.67 89 0.169 8 a, b 1.040 0.83 89 0.169 9 a, b 1.040 0.91 89 0.169 10 a, b 1.013 0.63 70 0.214 FIG.

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