Bolus formation from fission of nonlinear internal waves over a mild slope

Amin Ghassemi; Saeid Zahedi; Leon Boegman

doi:10.1017/jfm.2021.1033

Bolus formation from fission of nonlinear internal waves over a mild slope

Published online by Cambridge University Press: 15 December 2021

Amin Ghassemi

Saeid Zahedi and

Leon Boegman

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Amin Ghassemi*: Affiliation:
Environmental Fluid Dynamics Laboratory, Department of Civil Engineering, Queen's University, 58 University Avenue, Kingston, Ontario K7L 3N6, Canada
Saeid Zahedi: Affiliation:
Environmental Fluid Dynamics Laboratory, Department of Civil Engineering, Queen's University, 58 University Avenue, Kingston, Ontario K7L 3N6, Canada
Leon Boegman: Affiliation:
Environmental Fluid Dynamics Laboratory, Department of Civil Engineering, Queen's University, 58 University Avenue, Kingston, Ontario K7L 3N6, Canada
*: † Email address for correspondence: amin.ghassemi@queensu.ca

Article contents

Abstract
Introduction
Materials and methods
Results
Discussion
Conclusions
References

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Abstract

Breaking nonlinear internal waves (NLIWs) of depression on boundary slopes drives mixing in the coastal ocean. Of the different breaker types, fission is most commonly observed on mild slopes of continental margins. However, fission on mild slopes has rarely been investigated in the laboratory owing to limitations on flume length. In the present work, a train of NLIWs of depression is generated in an 18.2 m wave flume and shoaled upon a mild uniform slope. During fission, each NLIW of depression scatters into one or two NLIWs of elevation, which transforms into a bolus at the bolus birth point, where shear instability occurs through the pycnocline. The bolus propagates upslope, decreasing in size until it degenerates by shear and lobe-cleft instability, while losing volume to a return flow along the bed. The location of the bolus birth point, bolus propagation length scale, initial size and the number of boluses from each incident wave are parameterized from the wave half-width and the wave Froude number associated with the incident NLIW. These are compared with the characteristics of boluses generated by other breaking mechanisms on steeper slopes. Some bolus characteristics (height to length ratio, change in size and velocity field) are similar for boluses generated by fission, collapsing sinusoidal waves and internal solitary waves of elevation; however, the number of boluses, their birth point and initial height differ. The boluses formed by fission have more initial energy and no reflection. Further research is required to better quantify bolus-driven mixing on continental margins.

JFM classification

Geophysical and Geological Flows: Coastal engineering Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows

Information

Type: JFM Papers
Information: Journal of Fluid Mechanics , Volume 932 , 10 February 2022, A50
DOI: https://doi.org/10.1017/jfm.2021.1033 [Opens in a new window]
Copyright: © The Author(s), 2021. Published by Cambridge University Press

1. Introduction

Nonlinear internal waves (NLIWs) are ubiquitous phenomena of stratified lakes and coastal oceans (Jackson Reference Jackson2007, their figure 1; Osborne & Burch Reference Osborne and Burch1980; Boegman & Ivey Reference Boegman and Ivey2009; Boegman & Stastna Reference Boegman and Stastna2019) where they can freely travel for hundreds of kilometres before losing their energy through turbulent mixing and viscous dissipation during shoaling and breaking (Apel Reference Apel2002; Boegman, Ivey & Imberger Reference Boegman, Ivey and Imberger2005; Shroyer, Moum & Nash Reference Shroyer, Moum and Nash2009; Barad & Fringer Reference Barad and Fringer2010). The formation of NLIWs in oceans, through tide–topography interactions (Apel et al. Reference Apel, Holbrook, Liu and Tsai1985; Lamb Reference Lamb1994), is different from that in lakes, where they form by nonlinear steepening of basin-scale internal seiches (Horn, Imberger & Ivey Reference Horn, Imberger and Ivey2001; Boegman & Ivey Reference Boegman and Ivey2009). Shoaling and breaking of NLIWs on sloping boundaries lead to localized mixing of the background density stratification (Wunsch Reference Wunsch1971; Wüest, Piepke & Van Senden Reference Wüest, Piepke and Van Senden2000; Allshouse & Swinney Reference Allshouse and Swinney2020), drive vertical biogeochemical fluxes (Haury, Briscoe & Orr Reference Haury, Briscoe and Orr1979; Sandstrom & Elliott Reference Sandstrom and Elliott1984; Scotti & Pineda Reference Scotti and Pineda2007) and energize sediment resuspension and transport along continental margins (Southard & Cacchione Reference Southard and Cacchione1972; Pomar et al. Reference Pomar, Morsilli, Hallock and Bádenas2012; Ma et al. Reference Ma, Yan, Hou, Lin and Zheng2016).

Experimental and numerical studies on NLIW shoaling and breaking on slopes can be categorized into three groups: (1) breaking over idealized continental slope-shelf topography (Helfrich & Melville Reference Helfrich and Melville1986; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Lim, Ivey & Jones Reference Lim, Ivey and Jones2010; Lamb & Xiao Reference Lamb and Xiao2014); (2) breaking over a uniform slope (Wallace & Wilkinson Reference Wallace and Wilkinson1988; Boegman & Ivey Reference Boegman and Ivey2009; Aghsaee, Boegman & Lamb Reference Aghsaee, Boegman and Lamb2010; Nakayama et al. Reference Nakayama, Sato, Shimizu and Boegman2019; Xu & Stastna Reference Xu and Stastna2020); and (3) breaking over realistic topography (Vlasenko & Stashchuk Reference Vlasenko and Stashchuk2007; Lamb & Warn-Varnas Reference Lamb and Warn-Varnas2015; Masunaga et al. Reference Masunaga, Arthur, Fringer and Yamazaki2017; Rivera-Rosario et al. Reference Rivera-Rosario, Diamessis, Lien, Lamb and Thomsen2020; Guo, Zhan & Hoteit Reference Guo, Zhan and Hoteit2021). Uniform slope topography (figure 1) is common to the margins of lakes, fjords and some coastal regions, e.g. Mono Lake (MacIntyre et al. Reference Macintyre, Flynn, Jellison and Romero1999), Lake Biwa (Boegman et al. Reference Boegman, Imberger, Ivey and Antenucci2003), Lake Michigan (Hawley Reference Hawley2004), the Oregon continental shelf (Klymak & Moum Reference Klymak and Moum2003), the Alboran slope (Puig et al. Reference Puig, Palanques, Guillén and El Khatab2004), the Australian North West Shelf (Holloway, Chatwin & Craig Reference Holloway, Chatwin and Craig2001; Jones et al. Reference Jones, Ivey, Rayson and Kelly2020), the Faeroe–Shetland Channel (Hosegood, Bonnin & Van Haren Reference Hosegood, Bonnin and Van Haren2004), the St Lawrence River estuary (Bourgault et al. Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014), Monterey Bay, CA (Walter et al. Reference Walter, Squibb, Woodson, Koseff and Monismith2014) and the continental slope of the northern South China Sea (Reeder, Ma & Yang Reference Reeder, Ma and Yang2011; Ma et al. Reference Ma, Yan, Hou, Lin and Zheng2016).

Figure 1.

Schematic showing the propagation of an NLIW of depression in a two-layer stratification system toward a uniform boundary slope. Here the total depth (H) consists of a thicker (h ₂) and commonly denser (ρ ₂) lower layer which is overlaid by a thinner (h ₁) and lower denser (ρ ₁) layer. An NLIW of depression with amplitude a and wavelength $\lambda = 2{L_w}$ propagates towards a planer slope with steepness S. Here, L_i is the pycnocline length above the slope.

Laboratory and numerical studies, to examine NLIW breaking on uniform slopes (Boegman et al. Reference Boegman, Ivey and Imberger2005), identified three different breaking mechanisms (spilling, plunging and collapsing), based on an internal form of the Iribarren number:

(1.1)

\begin{equation}Ir = \frac{S}{{\sqrt {{S_{wb}}} }},\end{equation}

where S is the boundary slope, ${S_{wb}} = {a / \lambda }$ is the wave slope, a is the wave amplitude and $\lambda $ is the wavelength (figure 1).

Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010; their figure 6) further classified breaking by performing high-resolution fully nonlinear two-dimensional (2-D) direct numerical simulations (DNSs) over a wider range of bottom slopes. They categorized breaking of internal solitary waves (ISWs) into four distinct processes (collapsing, plunging, surging and fission) as a function of S and ${S_w} = {a / {{L_w}}}$, where ${L_w} = 0.5\lambda $ is the wave half-width. An ISW is an idealized ocean wave, defined as an isolated nonlinear dispersive wave that propagates without changing shape (Lamb Reference Lamb2014); whereas NLIW is a general term that includes ISWs. Sutherland, Barrett & Ivey (Reference Sutherland, Barrett and Ivey2013) combined the results of Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) and Boegman et al. (Reference Boegman, Ivey and Imberger2005) with their own experimental data to generally classify breaking in terms of the Iribarren number. Nakayama et al. (Reference Nakayama, Sato, Shimizu and Boegman2019) performed further three-dimensional (3-D) simulations to show that $S$ and ${S_w}$ can separate fission and surging breakers, but not collapsing and plunging breakers (figure 2). They introduced a new criterion to classify these two specific breaker types (see their (12)).

Figure 2.

Different breaking mechanism regimes for incident NLIWs, classified according to wave slope $({S_w})$ and bottom slope $(S)$ based on the results by Nakayama et al. (Reference Nakayama, Sato, Shimizu and Boegman2019).

Most wave-breaking events (e.g. Oregon shelf, New Jersey shelf, South China Sea) occur on mild slopes $(0.001 < S < 0.01)$, where fission is the predominant breakdown process as incident waves of depression change polarity to waves of elevation during shoaling (Shroyer et al. Reference Shroyer, Moum and Nash2009; Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Lamb & Xiao Reference Lamb and Xiao2014). Fission requires a gradual slope in a large domain, and so has only been computationally modelled over slopes where $S \le 0.05$ (e.g. Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Arthur & Fringer Reference Arthur and Fringer2014, Reference Arthur and Fringer2016; Lamb & Warn-Varnas Reference Lamb and Warn-Varnas2015; Nakayama et al. Reference Nakayama, Sato, Shimizu and Boegman2019; Guo et al. Reference Guo, Zhan and Hoteit2021); typical experimental boundary slopes (e.g. Boegman et al. Reference Boegman, Ivey and Imberger2005; Sutherland et al. Reference Sutherland, Barrett and Ivey2013) being too short (~2–6 m) for fission to be observed (table 1; figure 2). Given that Helfrich (Reference Helfrich1992) had a long flume with a gentle slope ($S = 0.03$; table 1), it is likely that he observed fission, as suggested by Michallet & Ivey (Reference Michallet and Ivey1999).

Table 1.

Comparison between boundary slopes used in experimental studies of shoaling NLIWs and observed boundary slopes in the coastal ocean. Note that the experimental slopes are typically orders of magnitude greater than those observed in the ocean, which leads to differing breaking mechanisms in the laboratory compared with the ocean.

Fission is the transformation of an ISW into a wave train as a result of a change in the environmental conditions (Boegman & Stastna Reference Boegman and Stastna2019). In our experiments, this occurs owing to a change in total depth, which causes an incident NLIW of depression to pass through a turning point (figure 1), where the lower fluid layer becomes thinner than the upper fluid layer, and leads to fission of the incident wave into a rank-ordered packet of NLIWs of elevation (Grimshaw, Pelinovsky & Talipova Reference Grimshaw, Pelinovsky and Talipova1998; Nakayama et al. Reference Nakayama, Sato, Shimizu and Boegman2019; Xu & Stastna Reference Xu and Stastna2020).

Considering the common gradual slopes in lakes (~0.01) and coastal oceans (~0.001; Cacchione, Pratson & Ogston Reference Cacchione, Pratson and Ogston2002 and table 1), it can be deduced that fission would be the most common breaker type, as shown by field observations (Orr & Mignerey Reference Orr and Mignerey2003; Shroyer et al. Reference Shroyer, Moum and Nash2009; Ma et al. Reference Ma, Yan, Hou, Lin and Zheng2016). Despite the common occurrence of fission, the limitations of small domains in both experimental and numerical set-ups have resulted in a research focus on steeper slopes and the associated breaking mechanisms (e.g. plunging and collapsing) that are less common in the field (table 1; also see Boegman et al. (Reference Boegman, Ivey and Imberger2005), their table 2). The numerical studies that have been performed on fission have been analytical (Grimshaw et al. Reference Grimshaw, Pelinovsky and Talipova1998), 2-D (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Xu & Stastna Reference Xu and Stastna2020), a 3-D simulation that did not examine a wide parameter space (Arthur & Fringer Reference Arthur and Fringer2014, Reference Arthur and Fringer2016) and one that did not provide detailed information about the final stages of shoaling (Nakayama et al. Reference Nakayama, Sato, Shimizu and Boegman2019).

Shoaling and breaking of internal waves can result in the formation of a bolus (a turbulent vortex with a circulation region within its interior) that progresses along the bed and may become detached from the pycnocline. Boluses may transport mass and bed material upslope in a (quasi-)trapped core (Helfrich & Melville Reference Helfrich and Melville1986; Helfrich Reference Helfrich1992; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Vieira & Allshouse Reference Vieira and Allshouse2020). Observations of boluses in the ocean have been reported in field studies (e.g. Klymak & Moum Reference Klymak and Moum2003; Carter, Gregg & Lien Reference Carter, Gregg and Lien2005; Moum et al. Reference Moum, Klymak, Nash, Perlin and Smyth2007; Walter et al. Reference Walter, Woodson, Arthur, Fringer and Monismith2012; Richards et al. Reference Richards, Bourgault, Galbraith, Hay and Kelley2013; Jones et al. Reference Jones, Ivey, Rayson and Kelly2020). Bolus-induced transport influences coastal ecosystems, where dense, cold and nutrient-rich water is transported upslope. This can be critical for the maintenance of coral reefs and kelp forests (Leichter et al. Reference Leichter, Wing, Miller and Denny1996), where the cool ambient water reduces coral bleaching (Leichter et al. Reference Leichter, Wing, Miller and Denny1996; Goreau et al. Reference Goreau, Mcclanahan, Hayes and Strong2000; West & Salm Reference West and Salm2003). Breaking of NLIWs and the resultant formation of boluses can act as a nutrient pump, which supplies the required nutrients to the euphotic zone on the shelf (Sandstrom & Elliott Reference Sandstrom and Elliott1984; Bogucki, Dickey & Redekopp Reference Bogucki, Dickey and Redekopp1997; Scotti &Pineda Reference Scotti and Pineda2004). Boluses may also resuspend and transport sediments upslope (Southard & Cacchione Reference Southard and Cacchione1972; Bourgault, Kelley & Galbraith Reference Bourgault, Kelley and Galbraith2005; Richards et al. Reference Richards, Bourgault, Galbraith, Hay and Kelley2013; Vieira & Allshouse Reference Vieira and Allshouse2020) with the finer grains transported offshore by the return flow to form an intermediate nepheloid layer (Ivey &Nokes Reference Ivey and Nokes1989; McPhee-Shaw Reference Mcphee-Shaw2006; Nakayama et al. Reference Nakayama, Shintani, Kokubo, Kakinuma, Maruya, Komai and Okada2012; Pomar et al. Reference Pomar, Morsilli, Hallock and Bádenas2012) and the coarser sediments settling to make bedforms (Karl, Cacchione & Carlson Reference Karl, Cacchione and Carlson1986; Reeder et al. Reference Reeder, Ma and Yang2011; Boegman & Stastna Reference Boegman and Stastna2019). Resuspension and transport can act on settled larvae and cysts, injecting them into the water column and transporting them upslope inside trapped cores (Scotti & Pineda Reference Scotti and Pineda2004).

The formation of bolus-like flow features has been studied experimentally (Wallace & Wilkinson Reference Wallace and Wilkinson1988; Helfrich Reference Helfrich1992; Michallet & Ivey Reference Michallet and Ivey1999; Moore, Koseff & Hult Reference Moore, Koseff and Hult2016; Allshouse & Swinney Reference Allshouse and Swinney2020) and numerically (Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Arthur & Fringer Reference Arthur and Fringer2014, Reference Arthur and Fringer2016; Bourgault et al. Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014; Vieira & Allshouse Reference Vieira and Allshouse2020). Wallace & Wilkinson (Reference Wallace and Wilkinson1988) observed the formation of one bolus for each incident wave, as the final stage of shoaling of ISWs of elevation over gentle slopes (S = 0.03, 0.054; table 1). In comparison, Helfrich (Reference Helfrich1992) observed the formation of several boluses (3–10; their figure 9) during ISW shoaling on uniform slopes (table 1). Michallet & Ivey (Reference Michallet and Ivey1999) also studied breaking of NLIWs of depression on steep slopes and observed a single bolus after breaking (table 1). The most comprehensive laboratory study (Moore et al. Reference Moore, Koseff and Hult2016) observed bolus formation as the final stage of periodic internal wave train shoaling in a quasi-two-layer system on a uniform slope (S = 0.2; table 1). They categorized boluses into four types, based on the breaking mechanism (collapsing, plunging and two types of boluses for surging). They did not observe fission, as their slope was too steep.

Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) performed 2-D DNS of bolus generation from collapsing, plunging and surging breakers. They proposed that ISWs of elevation, with trapped cores, are the final stage of shoaling during fission. Bourgault et al. (Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014) and Masunaga et al. (Reference Masunaga, Arthur, Fringer and Yamazaki2017) simulated the 2-D field-scale sediment movement and resuspension by boluses from fission and surging breaking of internal waves, respectively. These were in good visual agreement with field observations. Venayagamoorthy & Fringer (Reference Venayagamoorthy and Fringer2007) simulated propagation of boluses over slope-shelf topography and concluded that 2-D simulations could accurately reproduce bolus dynamics. Three-dimensional boluses were simulated to occur from fission by Arthur & Fringer (Reference Arthur and Fringer2014) and were strikingly similar in structure to the observations of Helfrich (Reference Helfrich1992) for the same wave amplitude and bathymetric slope. More recently, Vieira & Allshouse (Reference Vieira and Allshouse2020) performed numerical simulations that found the thickness of the pycnocline influenced the bolus characteristics resulting from surging of internal waves over steep slopes $(S = 0.105- 0.231)$. This was validated experimentally by Allshouse & Swinney (Reference Allshouse and Swinney2020) who observed three ball, hook and sliver boluses to evolve from surging breakers.

Given the lack of experimental data on shoaling of boluses generated by fission of NLIWs, there is a need for further laboratory-based research on this topic. The objectives of the present study are to characterize experimentally the final stage of breaking by fission, to parameterize bolus dynamics (number of boluses, bolus size, propagation distance) in terms of readily measurable incident wave parameters and to compare boluses generated from fission to those generated by other mechanisms. This knowledge will help oceanographers better understand NLIW shoaling and breaking, better choose mooring sites, better interpret the observed data, and will help validate and improve numerical simulations at both the laboratory and field scales.

2. Materials and methods

2.1. Theoretical background

To parameterize our results and compare our data with existing parameterizations, we present the relevant theoretical background. In stratified lakes and oceans, when the pycnocline is much thinner than the total depth H = h ₁ + h ₂, a density stratified fluid can be approximated as a quasi-two-layer system with a lower layer depth h ₂ and density ρ ₂ which is overlaid by an upper layer with depth h ₁ and density ρ ₁.

Although many of their waves were strongly nonlinear (e.g. $a > {h_1}$ Stanton & Ostrovsky (Reference Stanton and Ostrovsky1998); $a/{h_1} > 0.22$, Cui, Dong & Wang Reference Cui, Dong and Wang2021), Helfrich (Reference Helfrich1992) and Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) parameterized their results in terms of the weakly nonlinear Korteweg–de Vries (KdV) equation for an inviscid fluid under the Boussinesq approximation (Osborne & Burch Reference Osborne and Burch1980; Apel Reference Apel2002; Ostrovsky & Stepanyants Reference Ostrovsky and Stepanyants2005):

(2.1)

\begin{gather}{\eta _t} + {c_0}{\eta _t} + \alpha \eta {\eta _x} + \beta {\eta _{xxx}} = 0,\end{gather}

(2.2a,b)

\begin{gather}\alpha = \frac{{3{c_0}}}{2}\frac{{{h_1} - {h_2}}}{{{h_1}{h_2}}},\quad \beta = \frac{{{c_0}}}{6}{h_1}{h_2},\end{gather}

(2.3)

\begin{gather}{c_0} = \sqrt {\frac{{g^{\prime}{h_1}{h_2}}}{H}} ,\end{gather}

(2.4)

\begin{gather}c = {c_0} + \left( {\frac{{\alpha a}}{3}} \right), \end{gather}

(2.5)

\begin{gather}{\lambda _{KdV}} = \sqrt {\frac{{12\beta }}{{\alpha a}}} ,\end{gather}

(2.6)

\begin{gather}\gamma = \frac{{|\alpha |a}}{{{c_0}}} = \frac{3}{2}a\frac{{|{h_1} - {h_2}|}}{{{h_1}{h_2}}}.\end{gather}

Here $\eta$ is the vertical displacement of the pycnocline, α is the nonlinearity coefficient, β is the dispersion coefficient, c ₀ is the linear long-wave speed, c is the nonlinear wave phase speed, a is the wave amplitude (negative for a wave of depression), $g^{\prime} = g({\rho _2} - {\rho _1})/{\rho _2}$ is the reduced gravity owing to stratification, ${\lambda _{KdV}}$ is the characteristic length scale of a KdV-type solitary wave and $\gamma $ is the non-dimensional nonlinearity parameter (Boegman et al. Reference Boegman, Ivey and Imberger2005).

The applicability of weakly nonlinear KdV theory to characterize our observations was assessed by an $a/{h_1}$ criterion applicable for density stratifications $0.11 < {h_1}/{h_2} < 0.43$ (Cui et al. Reference Cui, Dong and Wang2021; their table 5). In KdV theory, waves of depression exist only when h ₁ < h ₂, causing fission to occur as the wave passes through the turning point where the nonlinearity coefficient vanishes $(\alpha \to 0)$. Equations (2.2) and (2.5) show a and $\alpha $ to have the same sign and, consequently, the change to a wave of elevation as $\alpha > 0$ whence h ₁ > h ₂ after the turning point (Grimshaw et al. Reference Grimshaw, Pelinovsky and Talipova1998). We differentiate our lab-observed waves as narrow or broad-crested according to $0.1 < \gamma < 3$, or equivalently $a < ({h_2} - {h_1})/2$, from (2.6). Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) applied this criterion to show that the majority of ISWs observed in the field were narrow crested.

To calculate the observed NLIW half-width, we followed Michallet & Ivey (Reference Michallet and Ivey1999):

(2.7)

\begin{equation}{L_w} = \frac{c}{a}\int_{{t_0}}^{{t_1}} {\eta (t)\,\textrm{d}t} ,\end{equation}

where the corresponding wave period is

(2.8)

\begin{equation}T = \frac{{2{L_w}}}{c}.\end{equation}

We compared our results with the parameterizations introduced by Hult, Troy & Koseff (Reference Hult, Troy and Koseff2010) and Moore et al. (Reference Moore, Koseff and Hult2016). These embody both the incident wave and ambient fluid properties, which are readily measurable in the field. They include wave steepness (ka), wave Reynolds number (Re) and wave Froude number (Fr):

(2.9)

\begin{gather}ka = \frac{{2{\rm \pi} a}}{\lambda },\end{gather}

(2.10)

\begin{gather}Re = \frac{{\omega {a^2}}}{\upsilon },\end{gather}

(2.11)

\begin{gather}Fr = \frac{{\omega a}}{{\sqrt {{{g^{\prime}}_c}H^{\prime}} }},\end{gather}

where $k = 2{\rm \pi}/\lambda $ is the wavenumber, $\omega = 2{\rm \pi}/T$ is the wave frequency, ${g^{\prime}_c} = 2g({\rho _2} - {\rho _1})/({\rho _2} + {\rho _1})$ is the reduced gravity for this set of equations, $H^{\prime} = {h_1}{h_2}/H$ in an equivalent depth and υ is the kinematic viscosity.

2.2. Experimental set-up and measurement techniques

The experiments were conducted in the Queen's University Internal Wave Flume, a glass-walled rectangular wave flume (figure 3, 18.2 m long, 0.75 m wide and 0.58 m deep). A planer Perspex slope (S = 0.04) extended upward from the bed to a height of 40 cm at the end wall. This was the mildest practical slope that could be installed in the flume and expected to generate fission (figure 2). This slope was close to the oceanic case (S = 0.035; table 1), where Richards et al. (Reference Richards, Bourgault, Galbraith, Hay and Kelley2013) observed boluses. At the other end of the flume, an aluminium flapping airfoil-type horizontal wave paddle (Thorpe Reference Thorpe1978; Nakayama & Imberger Reference Nakayama and Imberger2010) was installed to generate internal waves with different amplitudes and periods. The wave paddle was as wide as the flume and consisted of a 33 cm long airfoil connected with a piano hinge to a 93 cm long flapping plate (figure 3). The plate was driven by an electric motor connected with an eccentric drive shaft.

Figure 3.

Schematic of experimental set-up. Periodic NLIWs of depression are generated within the two-layer fluid by flapping the wave paddle with an electric motor connected through an eccentric drive shaft. The upper fluid layer is fresh water and the lower fluid layer is saline. See table 2 for a list of the experimental parameters.

The flume was filled to a total depth of H = 40 cm using two different layer densities (ρ ₂ = 1020 or 1040 kg m⁻³) and heights (h ₂ = 25 or 30 cm) chosen based on the limitations of the wave paddle (size and practical movement) and also the wave gauge (operational range in saline water). The density stratification was prepared by adding fresh water to the flume to the desired depth (h ₂), which was measured with transparent ruler tape (±2 mm), and then adding salt (Cargill Calcium Magnesium Free) to reach the desired density (ρ ₂), which was measured with a hydrometer (Fisherbrand™ 11555G; ±2 kg m⁻³). The lower layer was seeded with food dye to enable flow visualization. Fresh water (ρ ₁ = 998 kg m⁻³) was slowly added over the lower layer, using a floating sponge to minimize mixing between the two layers, until the appropriate upper layer thickness (h ₁) was achieved. This method created a density stratification with a thicker lower saline layer underlying a thinner fresh layer separated by a thin pycnocline $\delta \approx 2\;\textrm{cm} \pm \textrm{2}\;\textrm{mm}$ (Troy & Koseff Reference Troy and Koseff2005), commonly known as a ‘quasi two-layer’ stratification. The stratification profile can be theoretically reproduced as a hyperbolic tangent function and supports NLIWs of depression, which is typical oceanic stratification (Helfrich & Melville Reference Helfrich and Melville2006; Jackson, Da Silva & Jeans Reference Jackson, Da Silva and Jeans2012; Allshouse &Swinney Reference Allshouse and Swinney2020).

The wave paddle was positioned at the interface between the two fluid layers. To generate NLIWs of depression, the plate was flapped downward at the end wall to a desired depth (approximate wave amplitude) and then driven upward to the horizontal (i.e. the location of the quiescent density interface) using the electric motor. Measured wave amplitudes were typically 70 % of the flapping plate displacement along the end wall. Periodic movement of the paddle forced both the upper and lower layers generating a continuous train of NLIWs of depression. The wave amplitude was increased, in successive experiments, until the paddle movement created excessive localized mixing. In coastal regions, observations of ~3–11 NLIW packets (Jackson Reference Jackson2007), with ~10 rank-ordered NLIWs in each packet (e.g. Stanton & Ostrovsky Reference Stanton and Ostrovsky1998; Shroyer et al. Reference Shroyer, Moum and Nash2009; Zulberti, Jones & Ivey Reference Zulberti, Jones and Ivey2020), are more common than lone solitary waves. Therefore, we followed the approach of Wallace & Wilkinson (Reference Wallace and Wilkinson1988), Helfrich (Reference Helfrich1992) and Moore et al. (Reference Moore, Koseff and Hult2016) and generated periodic trains of NLIWs. This also facilitated piece-wise measurement of the velocity profile over successive wave shoaling events.

The applicability of KdV theory to describe our experimentally generated waves was not known a priori. The NLIW half-width was expected to scale with the length of the flapping plate and, for KdV waves, would be coupled to the wave amplitude according to (2.5). Therefore, the wave amplitude was first set by the vertical excursion of the flapping plate and then (2.5) was applied to compute the wavelength, which was taken as $\lambda \approx 9{\lambda _{KdV}}$ following the observations by Aghsaee & Boegman (Reference Aghsaee and Boegman2015). Using this modified wavelength, the wave period (2.8) was then determined and set as the period of rotation of the motor. Regardless of whether or not the NLIWs are accurately described by KdV theory, similar to lone waves formed by lock release, the resultant periodic NLIWs will evolve to their natural form once generated by an interfacial disturbance.

For this study, eight experiments with different wave amplitudes and density stratifications were performed (table 2). To measure a and T, a capacitance-type surface wave probe was customized to measure internal waves, using foam insulation to prevent spurious oscillations of the free surface from contaminating the internal wave signal (Wallingford HRIA-1018). The wave probe was installed 3 m downstream from the wave paddle (figure 3). The flow was visualized with backlight from fluorescent tubes illuminating translucent Perspex diffusing sheets, and recorded with three high-definition video cameras, two Canon 650D recorded the upslope wave propagation from the side, and a Canon VIXIA HV30 recorded the overhead plan view. The physical characteristics of the observed phenomena were measured with transparent ruler tape affixed to the flume and by digitizing the videos in Grapher software. The bolus propagation distances were recorded manually during each experiment.

Table 2.

Experimental parameters, where a is the wave amplitude (negative means wave of depression), ρ ₂ is the lower layer fluid density, h ₂ is the lower layer thickness, γ is non-dimensional nonlinearity parameter (2.6), T is the wave period (2.8), c is the nonlinear wave phase speed (2.4), L_w is the wave half-width (2.7), S_w is the wave slope, ka is the wave steepness (2.9), Re is the wave Reynolds number (2.10) and Fr is the wave Froude number (2.11) (the boundary slope S = 0.04 and the total height H = 0.4 cm are constant for all the tests).

Time series of velocity profiles within the boluses were recorded with three acoustic Doppler velocity profilers (ADVPs; Nortek Vectrino II; 1 mm s⁻¹ precision and ±5 % measurement error). One ADVP was positioned where the pycnocline intersects the slope (interaction point; figure 1) and the other two ADVPs were positioned both 90 cm upstream and downstream of the interaction point (figure 3). All three ADVPs were set to measure 3-D velocity profiles at 50 Hz over a 3 cm vertical profile (4 cm to 7 cm from the ADVP probe) with 1 mm resolution. At the beginning of each test, the ADVPs were 6 cm above the bottom, which caused 1 cm of the profile to be lost beneath the bed, so as to locate the region of maximum signal-to-noise ratio (SNR) or ‘sweet spot’ directly at the bed and minimize acoustic reflection. The SNR was also increased by local seeding with talcum powder (Ahmari Reference Ahmari2013). During each experiment, the velocity profile was measured by iteratively moving the ADVP vertically through the water column on pointer gauges (±0.1 mm). At each depth, five consecutive NLIWs of depression were measured (from five rotations of the motor) with the middle wave being chosen to be representative of an NLIW in a shoaling packet. The first and last waves had different characteristics than those in the middle, consistent with the observations by Wallace & Wilkinson (Reference Wallace and Wilkinson1988), Helfrich (Reference Helfrich1992) and Moore et al. (Reference Moore, Koseff and Hult2016).

A sensitivity analysis showed that the measurements were invariant for packets consisting of ten consecutive waves. After each cycle of five waves, the ADVPs were moved 2 cm upward and the procedure was repeated to capture new velocity profiles, with 1 cm overlap, until the ADVP reached the surface. A maximum of seven runs was required to capture the entire velocity profile (for h ₂ = 0.25 m; table 2). Diapycnal mixing from shoaling boluses was observed to be minimal over the very long pycnocline (16.25 m to 18.5 m; figure 1), consistent with numerical simulations by Arthur & Fringer (Reference Arthur and Fringer2014), (2016). Experiments were terminated before appreciable variation in the bolus characteristics or thickening of the pycnocline was observed (e.g. Ivey & Nokes Reference Ivey and Nokes1989; Michallet & Ivey Reference Michallet and Ivey1999; Boegman & Ivey Reference Boegman and Ivey2012).

Only velocity data with average correlation > 60 % and SNR > 15 dB were retained for analysis (McLelland & Nicholas Reference Mclelland and Nicholas2000; Nikora & Goring Reference Nikora and Goring2000; Chanson Reference Chanson2008). The data were despiked using a phase-space thresholding technique (WinADV 2.031) and the final profiles were stitched together in Matlab. As the flow was unsteady, mean velocity profiles $(\bar{U},\bar{V},\bar{W})$ were obtained by applying a low-pass filter to the instantaneous velocity data $(u,v,w)$ with a spectral cutoff between peaks associated with the NLIWs and small-scale turbulence (Aghsaee & Boegman Reference Aghsaee and Boegman2015). The velocity signals were then decomposed into mean and turbulent components using a Reynolds decomposition (e.g. $u = \bar{U} + u^{\prime}$). We were unable to measure the velocity profile within 4 cm of the free surface and 0.5 cm of the bed, owing to the probe configuration and SNR limits close to the solid boundary, respectively (Chanson Reference Chanson2008; Aghsaee & Boegman Reference Aghsaee and Boegman2015).

3. Results

3.1. Flow field

Periodic movement of the paddle generated a train of NLIWs of depression that travelled toward the slope passing through the wave probe (figure 4). The wavelength (table 2), measured by the wave probe (figure 4 and (2.7)) was larger than the characteristic KdV wavelength from (2.5); $\lambda = 2{L_w} \approx 7{\lambda _{KdV}}$. It is well known that for large amplitude NLIWs, the wavelength from (2.5) is too narrow (Wallace & Wilkinson Reference Wallace and Wilkinson1988; Boegman & Stastna Reference Boegman and Stastna2019). For example, field observations show $\lambda \approx 3.6{\lambda _{KdV}}$ (Holloway Reference Holloway1987; Boegman et al. Reference Boegman, Imberger, Ivey and Antenucci2003) and lab experiments show $\lambda \approx 9{\lambda _{KdV}}$ (Aghsaee & Boegman Reference Aghsaee and Boegman2015). Therefore, our NLIWs were not well described by KdV theory (e.g. when $a > {h_1}$, Stanton & Ostrovsky Reference Stanton and Ostrovsky1998). We found $a/{h_1} > 0.22$ (Cui et al. Reference Cui, Dong and Wang2021), which confirmed that weakly nonlinear KdV theory was not applicable to our observations and extended KdV would be more approptiate.

Figure 4.

Time series of the pycnocline displacement ($\eta$) for the run 6 recorded by the wave probe. The resultant wave amplitude (a) and wave period (T) are shown.

The non-dimensional nonlinearity parameter (2.6) confirms that all the observed waves were narrow crested as $0.1 < \gamma < 3$ (table 2). Moreover, in all the experiments, ${L_w} < 0.35{L_i}$, where ${L_i}$ is the pycnocline length above the slope (figure 1). Therefore, a positive tail was expected to form behind the rear face of the wave of depression over the slope (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010). Shear-induced breaking was not observed prior to shoaling, as confirmed by $a < 2.24\sqrt {{h_1}\delta } (1 + \delta /{h_1})$ (Fructus et al. Reference Fructus, Carr, Grue, Jensen and Davies2009).

3.2. Wave shoaling and bolus formation

When the train of NLIWs of depression shoaled over the mild slope, each incident NLIW of depression (figure 5b; blue arrow) degenerated through fission into one or two NLIWs of elevation as it passed through the turning point (figure 5; green line). In this process, the leading face of the NLIW of depression became elongated, parallel to the slope and the rear face steepened. A positive tail (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010) formed behind the rear face of the wave, which eventually evolved into one or more NLIWs of elevation (figure 5d; blue arrow), as in the observations by Orr & Mignerey (Reference Orr and Mignerey2003) and Ma et al. (Reference Ma, Yan, Hou, Lin and Zheng2016).

Figure 5.

Snapshots in time for run 2 (each row represents a time step), which show fission for a typical NLIWs of depression over a mild slope and bolus formation when there is one bolus for each incident NLIW of depression. The blue arrows in b, d, g and k show the incident NLIW of depression, the evolving NLIW of elevation after the turning point, the bolus and the point where the bolus has completely dissipated, respectively. Time period between snapshots is Δt = 6 s and the locations of the ADVPs are shown with red dots. All three image panels are acquired simultaneously with the same camera.

In figure 5 (run 2, table 2), for each NLIW of depression, there was only one NLIW of elevation after the turning point. Each NLIW of elevation then transformed into a bolus at the ‘birth point’ (figure 5g; blue arrow), which was between the turning point and the interaction point (figure 1; see also §§ 4.1 and 4.2). The boluses propagated upslope, decreasing in size until they degenerated (figure 5k; blue arrow). Each bolus preceded the next bolus formed from the subsequent NLIW of depression in the train (bolus downslope of the blue arrow in figure 5k). There was no evidence of reflection, as in the observations by Haury et al. (Reference Haury, Briscoe and Orr1979).

The transformation from an NLIW of elevation into a bolus is difficult to observe with discrete oceanographic moorings (Jones et al. Reference Jones, Ivey, Rayson and Kelly2020); this transformation is shown in figure 6 (run 6; table 2). The NLIW of elevation formed after the turning point, which is outside the field of view (figure 6b), and progressed upslope as the rear face steepened (figure 6c). At the bolus birth point (figure 6d; blue arrow), the rear face steepness was maximum and shear instabilities, in the form of Kelvin–Helmholtz billows, were observed to develop through the density interface, particularly along the rear face. The bolus formed as a turbulent vortex, with a thin trailing return flow of lower layer fluid attached to the pycnocline (§ 4.1). The bolus moved upslope while gradually deceasing in size until it degenerated (run 2, figure 5). The boluses were all similar in shape (e.g. Wallace & Wilkinson Reference Wallace and Wilkinson1988; their figure 7); hook- and sliver-type boluses simulated on thicker pycnoclines and steeper slopes were not observed (Vieira & Allshouse Reference Vieira and Allshouse2020).

Figure 6.

Snapshots in time for the run 6 which show typical transformation of an NLIW of elevation to a bolus when there is one bolus for each incident NLIW of depression. The blue arrow shows the location where the rear face of the NLIW of elevation reaches maximum steepness, and shear instability in the form of Kelvin–Helmholtz billows forms through the rear face. We consider this to be the bolus birth point. Thereafter, the bolus progresses upslope, decreasing in size until it degenerates. The time interval between snapshots is Δt = 5 s and the locations of the ADVPs are shown with red dots. Each image panel is acquired with a different camera.

For some experiments, two boluses formed from each incident NLIW of depression (§ 4.3.4); thin return flow in the lower layer, both in front and behind, connecting the boluses (§ 4.3.3). As an example of this, figure 7 shows snapshots for run 1 (table 2), where an NLIW of depression has passed the turning point. The first NLIW of elevation formed behind the rear face of the incident wave of depression, which had steepened (figure 7b). The first NLIW of elevation evolved into a bolus (as for the lone bolus in figure 6d) but remained attached to the pycnocline with a tongue of lower layer fluid (figure 7d; blue arrow). As the first bolus propagated away, a second NLIW of elevation formed where the rear face continued to shoal (figure 7e; orange arrow). A second bolus was similarly generated after forming another positive tail (figure 7f; green arrow). The creation of more boluses was prevented by elongation of the leading face of the next NLIW of depression in the wave train (figure 7h).

Figure 7.

Snapshots in time for run 1 which show typical bolus formation mechanism when there are two boluses for each incident NLIW of depression. The blue arrow shows the tongue of the lower layer fluid attaching the first bolus to the pycnocline. The orange arrow shows the second NLIW of elevation and the green arrow shows the second bolus (see supplementary movie S1). The time interval between snapshots is Δt = 4 s and the locations of the ADVs are shown with red dots. The third ADVP is outside the field of view.

The waves of elevation, generated by fission, were rank-ordered and there was no evidence of interaction between the first wave of elevation (or bolus) generated by a trailing wave of depression and the last wave of elevation (or bolus) generated by the preceding wave of depression. Each bolus completely degenerated before the arrival of the trailing bolus (figure 7 and supplementary movie S1 available at https://doi.org/10.1017/jfm.2021.1033). These observations (figure 7) are consistent with the simulations by Arthur & Fringer (Reference Arthur and Fringer2014) (their figure 18).

3.3. Bolus instability

Shear instability was observed through the bolus pycnocline and lobe-cleft instability at the leading edge (Simpson Reference Simpson1972; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Arthur & Fringer Reference Arthur and Fringer2014). The instabilities through the pycnocline, particularly near the crest and through the rear face, formed from strong shear in the parallel flow (§ 4.1; movie S3 in the supplementary material). At times the instabilities were in the form of Kelvin–Helmholtz billows, for example at the front of the bolus (just above the nose) in figure 8(a). Similarly, Jones et al. (Reference Jones, Ivey, Rayson and Kelly2020) also observed shear instability through the rear face of near-bed NLIWs of elevation in the ocean.

Figure 8.

Images showing run-up of a bolus after its formation (run 4). The bolus is travelling from left to right in the upslope direction. (a) Side view (see supplementary movie S2), (b) top view of two boluses as indicated by arrows: the green arrow shows the top view of the bolus in panel (a); the yellow arrow shows the preceding bolus during its final stage of shoaling as it degenerates (see supplementary movie S3). In this run, there is one bolus for each NLIW of depression. The bolus in panel (a) has a height of ~4 cm and the smaller-scale shear instabilities have length scales of ~1 cm.

A linear stability analysis (Gímez-Giraldo et al. Reference Gímez-Giraldo, Imberger, Antenucci and Yeates2008; Smyth, Moum & Nash Reference Smyth, Moum and Nash2011; Bouffard, Boegman & Rao Reference Bouffard, Boegman and Rao2012), using a tanh density profile and the mean observed horizontal velocity profile (figure 8a, trailing the bolus), showed peak growth of mode-one instabilities to be centred at a wavelength of 0.6 cm. This wavelength is consistent with the size of the small-scale overturns in figure 8(a) (figure S1).

A plan view (figure 8b, yellow arrow) shows the lobe-cleft instability of the leading face (Härtel, Carlsson & Thunblom Reference Härtel, Carlsson and Thunblom2000; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Xu, Subich & Stastna Reference Xu, Subich and Stastna2016; Jones et al. Reference Jones, Ivey, Rayson and Kelly2020). There is no return flow toward the leading bolus (figure 8b, yellow arrow), which is visually similar to a shoaling ISW of elevation or gravity current (e.g. Simpson Reference Simpson1972, their figure 11; Härtel et al. Reference Härtel, Carlsson and Thunblom2000, their figure 1; Xu et al. Reference Xu, Subich and Stastna2016, their figure 5). The irregular pattern of the leading edge (figure 8b; green arrow) shows that the lobe-cleft instability occurred during the presence of a return flow (before degeneration of the leading bolus), as simulated by Xu et al. (Reference Xu, Subich and Stastna2016). The three-dimensional structure of the lobe-cleft instability is evident in our observations, with finite-amplitude features that are long in the streamwise direction (e.g. velocity structure in figure 9b) and short in the spanwise direction (e.g. density field in figure 8b), as identified by Xu et al. (Reference Xu, Subich and Stastna2016; see their figure 7). Lobe-cleft instability has also been observed in oceanographic observations of shoaling NLIWs of elevation (Jones et al Reference Jones, Ivey, Rayson and Kelly2020).

Figure 9.

Hovmöller diagram of instantaneous velocity profiles for a bolus in (a) the streamwise direction (u component), (b) the vertical direction (w component) and (c) the spanwise direction (v component) for run 8 (table 2) as captured by the first ADVP. Measured data within 5 mm of the bed have been removed because of poor signal-to-noise ratio and correlation from acoustic reflection of the bottom. In all the experimental data, this is the largest bolus that is recorded with the strongest velocities.

A strong down-draft of lower layer fluid, near the bed, occurred during the progression of the bolus and diapycnal mixing across the pycnocline was weak, which suggested continued mass flux to the lower layer may be more important than diapycnal mixing across the pycnocline in regulating bolus progression and degeneration. This has implications for irreversible vertical flux of sediments, nutrients and buoyancy along coastal margins (Sandstrom & Elliott Reference Sandstrom and Elliott1984; Helfrich Reference Helfrich1992; Boegman & Stastna Reference Boegman and Stastna2019).

3.4. Velocity field

The velocity profiles from the three ADVPs, during the eight experiments, captured boluses with different sizes and velocity magnitudes, yet they were qualitatively similar. As an example, figure 9 shows time series of the ADVP-observed instantaneous velocity profiles (u,w,v) during the passage of the largest observed bolus (run 8; first ADVP). When the bolus formed, it was preceded by a thin downslope lower-layer flow (figure 9a; t = 0–9 s; also see figure 8a and supplementary movie S3), which formed either from the elongated leading face of the first incident NLIW of depression or the trailing edge of the preceding bolus. When the bolus passed the ADVP, there was an upslope flow of lower layer fluid inside the bolus and a downslope flow of the upper layer fluid above (figure 9a; t = 9–14 s). The flow along the leading face of the bolus was upward (figure 9b; t = 9–12 s) and was downward flow at the rear face (figure 9b; t = 12–15 s). Spanwise flow within the bolus core was evident, but was strongest through the pycnocline (figure 9c; t = 10–15 s); which indicated the presence of 3-D instability, particularly along the rear face (e.g. figure 8a).

After the bolus passed, a thin downslope return flow (e.g. figure 8 and supplementary movie S3) was also observed (figure 9a; t = 15–25 s). This drained the dense fluid inside of the bolus to the lower layer or toward the trailing bolus. These velocity fields (figure 9) are consistent with those observed by Richards et al. (Reference Richards, Bourgault, Galbraith, Hay and Kelley2013; their figure 9); although they were unable to capture velocity profiles close to the bottom (<1 m above the bed); potentially not resolving a return flow. The size and velocity magnitude, associated with the boluses, decreased as they propagated upslope (figure S2). The fluctuating velocity components were roughly one-tenth the magnitude of the instantaneous components and were elevated within the bolus and along its boundary, which indicated active turbulence (figure S3).

4. Discussion

4.1. Difference between a NLIW of elevation and a bolus

Here, we propose that a comparison of the velocity fields can be used to differentiate NLIWs of elevation and boluses. Shoaling NLIWs of elevation have been investigated in the literature (e.g. Stastna & Lamb Reference Stastna and Lamb2008; Carr & Davies Reference Carr and Davies2010; Xu et al. Reference Xu, Subich and Stastna2016), but their dynamics after generation through fission of an NLIW of depression and the difference between an NLIW of elevation and a bolus remain comparatively uninvestigated in the laboratory. Stastna & Lamb (Reference Stastna and Lamb2008) and Carr & Davies (Reference Carr and Davies2010) considered propagation of an ISW of elevation over a horizontal bed, while Wallace & Wilkinson (Reference Wallace and Wilkinson1988) and Xu et al. (Reference Xu, Subich and Stastna2016) considered shoaling over a planar slope and a small-amplitude shelf, respectively. Wallace & Wilkinson (Reference Wallace and Wilkinson1988) proposed that internal waves of elevation steepened over the slope until they became unstable and eventually overturned transforming into a bolus with instabilities at the crest. Each bolus moved upslope decreasing in size as it degenerated. Carr & Davies (Reference Carr and Davies2010) reported three wave types and observed shear instability through the pycnocline for one of them. Arthur & Fringer (Reference Arthur and Fringer2014) simulated the final stage of shoaling, for fission, to be a train of rank-ordered solitary waves of elevation (boluses). Xu et al. (Reference Xu, Subich and Stastna2016) simulated the start of lobe-cleft instability at the nose of the wave and shear instability in the form of Kelvin–Helmholtz billows through the rear-wave face during the shoaling over the shelf, but as with Carr & Davies (Reference Carr and Davies2010), they did not call these boluses. In the present study, our observations after the turning point were very similar to those of Wallace & Wilkinson (Reference Wallace and Wilkinson1988) (figure 6).

We compared the velocity fields for an NLIW of elevation and two boluses (figure 10) from run 1 (table 2). Both boluses formed from fission of one incident NLIW of depression; however, the second NLIW of elevation was not captured by the first and second ADVPs (figure 7). In this run, the boluses did not propagate to the third ADVP (figure 7i). Therefore, the NLIW of elevation (figure 10a,c,e) transformed into the first bolus (figure 10b,d,f; bolus on the left). The bolus velocity signatures (figure 10b,d,f) were qualitatively similar to the already investigated bolus from run 8 (figure 9a–c). However, the velocity signature associated with the NLIW of elevation (figure 10a,c,e) was distinct from those arising from the boluses. The u velocity (figure 10a,b) had a return flow in the lower layer behind the boluses (figure 10b, t = 8–13 s and t = 15.5–20 s) that was not present for the NLIW of elevation (figure 10a, t = 14–20 s). This is in agreement with the simulations of NLIWs of elevation by Xu & Stastna (Reference Xu and Stastna2020; their figure 2a). The v velocity (figure 10e,f) showed striking spanwise flow for the case with boluses (figure 10f, t = 5–7 s and t = 13.5–15 s), which was not observed in the NLIW of elevation (Hosegood et al. Reference Hosegood, Bonnin and Van Haren2004). These instantaneous velocities (figure S2) were maximum around the bolus edge, owing to the instabilities in the pycnocline.

Figure 10.

Hovmöller diagram showing contrasting instantaneous velocity profiles for an NLIW of elevation and two boluses. Panels (a,b), (c,d) and (e,f) show the streamwise (u component), vertical (w component) and spanwise (v component) velocity components, respectively, for the NLIW of elevation (left column)/boluses (right column). These data are for run 1 as captured by the first and second ADVPs. The first bolus forms from the shown NLIW of elevation, whereas the second bolus forms from the trailing NLIW of elevation (not shown). Both NLIWs of elevation occur through fission of the same incident NLIW of depression. See figure 7 and supplementary movie S1.

The thin region of three-dimensionalization < ~1 cm (figure 10f) is consistent with the wavelength of the most unstable mode (0.5 cm) from a linear stability analysis (figure S4). The size of our instabilities was likely limited by the large density difference across the pycnocline (20–40 kg m⁻³) in the present experiments, relative to the smaller density gradients in oceanic observations (Moum et al. Reference Moum, Farmer, Smyth, Armi and Vagle2003), which would allow for growth of larger instabilities relative to the wave/bolus height.

The current speed inside the first bolus (u and w; figure 10b,d) was greater than in the initial NLIW of elevation (u and w; figure 10a,c), even though the bolus was smaller and the data were captured 0.9 m further upslope. Considering the reduction in bolus size and velocity magnitude during propagation over the slope (figure S2), it can be surmised that the peak in u and w velocity components occurred at the bolus birth point.

4.2. Bolus formation

The bolus formation mechanism for shoaling internal waves on a slope depends on the wave form (depression versus elevation). From our results and Moore et al. (Reference Moore, Koseff and Hult2016), for an incident NLIW of depression, bolus formation depends on the breaker type (fission, collapsing, plunging and surging). However, for an ISW of elevation, Wallace & Wilkinson (Reference Wallace and Wilkinson1988) proposed that the structure of the bolus was independent of the breaking mechanism and only determined by the inclination of the slope.

The steep-slope boluses, observed by Moore et al. (Reference Moore, Koseff and Hult2016), consisted of a vortex with a mixed breaking region that was pushed shoreward under wave inertia. This has also been simulated (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010) and experimentally observed (Michallet & Ivey Reference Michallet and Ivey1999; Allshouse & Swinney Reference Allshouse and Swinney2020) by others. However, in the present mild-slope experiments (§ 4.1), all boluses were formed by transformation of an NLIW of elevation without intense mixing, as in the experiments by Wallace & Wilkinson (Reference Wallace and Wilkinson1988). Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) distinguished between ISWs of elevation with trapped cores and the turbulent boluses that emerged from collapsing breakings. However, their 2-D model was unable to resolve 3-D instability in the boluses after fission, which led to an overestimation of wave inertia during breaking and reduced mixing by secondary spanwise instabilities, relative to the 3-D case (Fringer & Street Reference Fringer and Street2003; Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Arthur & Fringer Reference Arthur and Fringer2016). Therefore, in general, a bolus formed by fission from an NLIW of depression (mild slope) will have less mixing than boluses generated on steep slopes, as it is initialized from a wave with minimal reflection.

Xu & Stastna (Reference Xu and Stastna2020) used a 3-D model (SPINS), restricted to two dimensions because of the computational cost, to simulate fission, where an ISW of elevation shoaled after the turning point. A separation bubble gradually developed beneath the wave and eventually broke down into two parts (their figures 3 and 5): one trapped inside the wave; and the other shed behind the leading wave to interact with the trailing waves. Their simulation of separation bubble breakdown provides insight into our bolus formation mechanism. We observed formation of a quasi-trapped core (figure 13c; see § 4.3.1) within the bolus and shear instability through the pycnocline, particularly along the rear face. This is where the velocities induced by the wave were maximum and they modelled strong cross-boundary-layer transport. Therefore, where the separation bubble broke down and shear instabilities formed through the wave boundary layer (Ma et al. Reference Ma, Yan, Hou, Lin and Zheng2016), the NLIW of elevation evolved into a bolus (figure 10).

Comparing the local bolus propagation speed ${C_b}$ (from the video) with the local fluid velocity u_max (from the ADVP) at the bolus birth point for runs 2, 5 and 7 (table 2), we evaluated the Lamb (Reference Lamb2002, Reference Lamb2003) criterion, where a trapped core will form when u_max > C_b. This criterion was valid for all boluses, with our observations showing the wave-induced current became unsteady at the bolus birth point and a quasi-trapped core formed inside of the NLIW of elevation (figure 13c; see § 4.3.1) and became a bolus (figure 10). These observations are consistent with the simulations by Xu et al. (Reference Xu, Subich and Stastna2016). At the bolus birth point, ${u_{max}} \approx 0.7{c_0}$, as observed for a NLIW degenerating through plunging breaking (Boegman & Ivey Reference Boegman and Ivey2009) following Sveen et al. (Reference Sveen, Guo, Davies and Grue2002).

The location of the bolus birth point, for all breaking mechanisms except fission, was near the breaking point and close to the ‘surging region’ (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; Moore et al. Reference Moore, Koseff and Hult2016; Allshouse & Swinney Reference Allshouse and Swinney2020). The simulations by Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010; their figure 18) introduced an equation for the breaking point and proposed that the breaking point for fission was where the first wave of elevation emerged. However, the present observations show bolus formation to begin at a point between the turning point and the interaction point. Here, shear instabilities form through the pycnocline, owing to separation bubble breakdown, as the NLIW of elevation emerges (figure 11). Regressing the bolus birth point against $Fr$ (2.11) gives an equation (R ² = 95 %, normalized root-mean-square error (NRMSE) = 12.9 %) to predict the location of bolus birth for fission on a mild slope (figure 12a):

(4.1)

\begin{equation}\frac{{{x_{bf}}}}{{{L_w}}} = 3.35F{r^{0.85}}.\end{equation}

Here, x_bf is the distance between the bolus birth point and the interaction point and ${L_w}$ is from (2.6). The bolus birth point was always between the turning and interaction points (figure 11). The location of the bolus birth point, occurring where ${u_{max}} \approx 0.7{c_0}$, depended on Fr (ratio of fluid velocity to wave celerity). The bolus birth point moved closer to the turning point as the incident wave energy increased (figure 12a) and local current velocities associated with larger incident NLIWs more rapidly approached the local phase speed. Wallace & Wilkinson (Reference Wallace and Wilkinson1988) also introduced a criterion for the bolus birth point but acknowledged that their results showed large variability and they did not report sufficient data to compare their result with our own.

Figure 11.

Schematic showing fission of a train of NLIWs of depression over a mild slope. Here, x_bf is the distance between the bolus birth point (star) and the interaction point and x ₀ is the bolus propagation distance.

Figure 12.

Parameterization of bolus characteristics. (a) Bolus birth point criterion: ratio of the distance between the bolus birth point and the interaction point (x_bf) normalized by L_w (2.6) versus Fr (2.11). (b) The total bolus propagation distance (x ₀) normalized by L_w versus Fr. (c) The initial height of the bolus (H ₀) normalized by L_w versus Fr. (d) Classification of number of boluses (Nb) birthed during shoaling of each NLIW of depression in relation to the initial wave forcing parameters Fr and Re (2.10). Data points are mean ± standard deviation in panels (a–c).

Figure 13.

Flow within a bolus. (a) Spatial distribution of the $\bar{U}$ velocity superimposed on the 2-D velocity vector field for a bolus from run 8 as captured by the first ADVP. (b) The 2-D velocity vector field superimposed on a video image of the bolus in panel (a). (c) Streamlines from a reference frame moving with the wave for the bolus in panel (a). In all panels, the bolus moves from left to right.

Moore et al. (Reference Moore, Koseff and Hult2016) classified different types of bolus formation according to Fr, Re and the wave steepness ka. They found that Fr ≥ 0.2 marked a transition from a coherent bolus to a turbulent surge/bore (i.e. a transition from a bolus owing to collapsing to a bolus as a result of either plunging or surging). The Re did not have a considerable effect on the bolus type, except for when Re < 500 (or ka < 0.4), the bolus would be coherent. While they did not consider changes in the bottom slope, our results using a mild slope show that the type of bolus depends on the wave-breaking mechanism. The present boluses (Fr ≤ 0.2 and ka < 0.4; table 2) were coherent and Re > 500 for all experiments, except runs 1 and 3. Inspection of the observed boluses and their mean velocity profiles (not shown) shows that v increases with Reynolds number, causing more shear instabilities and more mixing with associated energy loss.

4.3. Bolus characteristics

The boluses observed in this study were all similar in shape (figure 8), with shear instabilities through the pycnocline, especially along the rear face (e.g. Wallace & Wilkinson Reference Wallace and Wilkinson1988; Helfrich Reference Helfrich1992; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007; Arthur & Fringer Reference Arthur and Fringer2014). However, the boluses were not identical, therefore, we investigated what parameters led to changes in the physical characteristics of the boluses.

4.3.1. Bolus dynamics

Spatial velocity distributions, under a frozen-time assumption, were computed by multiplying the velocity profile time series (figure 9) by the local bolus propagation speed ${C_b}$. The spatial horizontal velocity distribution, for the bolus in figure 9, and the corresponding velocity vector field in the horizontal–vertical plane showed circulation around the bolus, with an upslope flow inside the bolus and downslope flow above (figure 13a). The centre of the circulation was considered to be the top of the bolus (x ≈ 41 cm and depth ≈ 8 cm) and the distance to the bottom was the bolus height ${H_b}$. The bolus length ${L_b}$ was equal to the distance along the bed between where $\bar{U} \approx 0\;\textrm{m}\;{\textrm{s}^{ - 1}}$ (x ≈ 20 and 58 cm).

Overlaying the vector field on the corresponding video (figure 13b) provided an image similar to field observations by Bourgault et al. (Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014; their figure 13) both in the St. Lawrence Estuary and in their Reynolds-averaged model. Venayagamoorthy & Fringer (Reference Venayagamoorthy and Fringer2007) and Vieira & Allshouse (Reference Vieira and Allshouse2020) also showed visually similar velocity vectors superimposed on the density contours for one of their simulated boluses (their figures 15a and 4d, respectively); although the formation mechanisms were different. Jones et al. (Reference Jones, Ivey, Rayson and Kelly2020) presented similar velocity vectors superimposed on temperature observations of a bolus on the Australian North West Shelf.

The top of the bolus found from the velocity vectors (centre of the circulation) was inside the bolus – as visualized from the video camera (figure 13b) – in agreement with field observations by Bourgault et al. (Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014; their figure 13). The red fluid above the centre of the circulation is the mixed-density downslope flow (Wallace & Wilkinson Reference Wallace and Wilkinson1988; their figure 9) created by shear instabilities through the pycnocline (Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007).

The bolus-induced currents may lead to sediment resuspension (Hosegood et al. Reference Hosegood, Bonnin and Van Haren2004; Boegman & Stastna Reference Boegman and Stastna2019). At the front face of the bolus, close to the bed (figure 13b; x = 58 cm), an upward flow may lift bed material, similar to the field-scale model results by Bourgault et al. (Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014; their figure 13b). However, the lack of near-bed velocity data makes interpretation of the resuspension mechanism difficult from backscatter observations both in the field (Richards et al. Reference Richards, Bourgault, Galbraith, Hay and Kelley2013, their figure 9; Klymak & Moum Reference Klymak and Moum2003, their figure 5; Jones et al. Reference Jones, Ivey, Rayson and Kelly2020, their figure 3e) and in the lab (Aghsaee & Boegman Reference Aghsaee and Boegman2015).

Streamlines were generated by subtracting ${C_b}$ from $\bar{U}$ (Wallace & Wilkinson Reference Wallace and Wilkinson1988; Venayagamoorthy & Fringer Reference Venayagamoorthy and Fringer2007) to achieve a reference frame moving with the wave (figure 13c). The streamlines show a quasi-trapped core (e.g. Lamb Reference Lamb2002, Reference Lamb2003; Klymak & Moum Reference Klymak and Moum2003), which is thought to carry sediments along the continental shelf (Richards et al. Reference Richards, Bourgault, Galbraith, Hay and Kelley2013; Bourgault et al. Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014). The size of the bolus decreased while moving upslope, owing to drainage of the core by the return flow and mixing between the core and the ambient fluid (Xu et al. (Reference Xu, Subich and Stastna2016), which confirmed the appropriateness of using ‘quasi-trapped core’ terminology to describe the boluses. The largest closed streamline passed over the top of the bolus, which showed that the boundary of the quasi-trapped core region was the same as what we considered to be the bolus. Venayagamoorthy & Fringer (Reference Venayagamoorthy and Fringer2007) identified two circulation regions within the bolus: one was larger and in the upper portion of the bolus, which is analogous to figure 13(c), and the other was close to the bottom with flow in the reverse direction that caused lower layer fluid to drain from the bolus through the rear face. In figure 13(c), there is flow separation beneath the circulating core, but owing to the lack of data close to the bottom, it is not clear if this is the secondary circulation cell modelled by Venayagamoorthy & Fringer (Reference Venayagamoorthy and Fringer2007). This requires further investigation using neutrally buoyant particles to visualize the near-bed flow. In their comprehensive field observations, Jones et al. (Reference Jones, Ivey, Rayson and Kelly2020) did not mention boluses with trapped cores, rather their most observed phenomenon was a solitary-like wave with a trapped core that had a recirculation region centred inside the wave (their figure 3a). We did not observe these waves in the present observations (e.g. figure 10).

To relate bolus propagation speed to the initial wave forcing, Moore et al. (Reference Moore, Koseff and Hult2016) plotted the ${C_b}/\omega a$ versus $Fr$ (figure 14a) and $ka$ (figure 14b). Despite the bolus generation mechanisms being different, our data agree with their trend, showing that by increasing the initial wave forcing, the normalized bolus speed decreased. Moore et al. (Reference Moore, Koseff and Hult2016) proposed that this was because some of the incident wave energy was lost to turbulence and when the bolus became unstable, more energy would be dissipated. Although they did not generate boluses by fission, our observations also show that for the runs with higher wave forcing (Re > 500), the edges of the boluses were more irregular with instability-driven mixing.

Figure 14.

Local bolus propagation speed $({C_b})$ (normalized by $\omega a$) versus (a) wave Froude number $(Fr)$ and (b) wave steepness $(ka)$. Average bolus height $({h_{ave}})$ (normalized by $a$) versus (c) wave Froude number $(Fr)$, (d) wave steepness $(ka)$.

4.3.2. Bolus propagation distance

It is important to know the total distance a bolus will propagate after formation, transporting cold dense fluid, sediments and nutrients. Figure 12(b) shows the bolus propagation distance ${x_0}$ normalized by ${L_w}$ (2.6), relative to $Fr$ (2.11). Regression data gives an equation for bolus propagation distance (R ² = 96 %, NRMSE = 8.2 %):

(4.2)

\begin{equation}\frac{{{x_0}}}{{{L_w}}} = 5.81F{r^{0.67}}.\end{equation}

Larger incident waves, with more fluid inertia and greater $Fr$, will form boluses that propagate further upslope (figure 12b).

4.3.3. Bolus size

The size of a bolus has been shown to consistently diminish as it propagates upslope in both process-based investigations (figures 5–8; Wallace & Wilkinson Reference Wallace and Wilkinson1988; Helfrich Reference Helfrich1992; Moore et al. Reference Moore, Koseff and Hult2016; Allshouse & Swinney Reference Allshouse and Swinney2020) and field observations (Sandstrom & Elliott Reference Sandstrom and Elliott1984; Moum et al. Reference Moum, Klymak, Nash, Perlin and Smyth2007; Bourgault et al. Reference Bourgault, Morsilli, Richards, Neumeier and Kelley2014). The bolus height-to-length ratio ${H_b}/{L_b} \approx 0.3$ (figure 15a) remained approximately constant (Wallace & Wilkinson Reference Wallace and Wilkinson1988). Figure 15(b) shows that the change in ${H_b}$ from the initial bolus height $({H_o})$ through bolus degeneration is linear (Wallace & Wilkinson Reference Wallace and Wilkinson1988; Helfrich Reference Helfrich1992, their (8); Bourgault et al. Reference Bourgault, Blokhina, Mirshak and Kelley2007). This is independent of whether the boluses were generated from NLIWs of elevation (Wallace & Wilkinson Reference Wallace and Wilkinson1988), collapsing NLIWs of depression (Helfrich Reference Helfrich1992) or fission (present study).

Figure 15.

Parameterizations for bolus aspect ratio. (a) The bolus height-to-length ratio as a function of the normalized distance from the bolus birth point. The ordinate scale is the distance of the bolus from the bolus birth point (x) normalized by the bolus propagation distance (x ₀). (b) Variation of normalized bolus height (H ₀ = initial bolus height) versus the distance to the bolus from the bolus birth point (x) normalized by the bolus propagation distance (x ₀). Wallace & Wilkinson (Reference Wallace and Wilkinson1988) is referred to as WW (Reference Wallace and Wilkinson1988).

To parameterize ${H_o}$, Helfrich (Reference Helfrich1992) proposed ${H_0}/a = 1.75 \pm 0.25$ for $0 < {\lambda _{KdV}}/{L_i} < 0.25$, which was shown by Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) to decrease with increasing ${{{\lambda _{KdV}}} / {{L_i}}}$ (figure S5). However, all of our NLIWs $(0.24 \le a/{h_1} \le 0.55)$, most of the NLIWs in Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) $(0.2 \le a/{h_1} \le 2.05)$ and some of the NLIWs in Helfrich (Reference Helfrich1992) $(0.07 \le a/{h_1} \le 3.4)$ were not of the weakly nonlinear KdV-type (i.e. $a/{h_1} > 0.22$, Cui et al. Reference Cui, Dong and Wang2021). Consequently, $\lambda > {\lambda _{KdV}}$, bringing into question the utility of this parameterization. To better classify surging and collapsing breakers, Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) introduced the parameter ${H_0}/{H_t}$, where ${H_t}$ is the depth where the separation bubble has reached the pycnocline. However, we found that this criterion was not applicable to breaking by fission, as the separation bubble split before reaching the pycnocline. Therefore, there was a need for a new relation to predict ${H_o}$ for fission.

Motivated by the results of Moore et al. (Reference Moore, Koseff and Hult2016) and given the data scatter (figure S5), we predicted the initial height of the bolus (normalized by ${L_w}$) relative to $Fr$ (R ² = 94 %, NRMSE = 9.5 %):

(4.3)

\begin{equation}\frac{{{H_0}}}{{{L_w}}} = 0.16F{r^{0.56}}.\end{equation}

The waves became taller and narrower as $Fr$ increased (figure 12c). This observation was surprising, given that field data shows higher frequency NLIWs to have taller (sech² versus sinusoidal) profiles and less energy (Boegman et al Reference Boegman, Ivey and Imberger2005; their figure 15). We surmise that $Fr$ is locally regulated by shoaling, enhancing ${u_{max}}$ relative to ${C_b}$.

The relation of the average bolus height $({h_{ave}} = {H_0}/2)$ normalized by the incident wave amplitude to wave inertia and steepness (figures 14c and 14d) were also investigated, as in Moore et al. (Reference Moore, Koseff and Hult2016). The average bolus height decreased with increasing wave inertia, as energy was lost to turbulence and drag. The discrepancy between our results and those of Moore et al. (Reference Moore, Koseff and Hult2016) results from the differences in incident wave types (periodic sinusoidal internal waves versus NLIWs of depression) and, as a result, different definitions of wave amplitude. We have also chosen a different criterion for bolus height (§ 4.2.1), using the top of the bolus to be the inflection point in the velocity field ($\bar{U} = 0$; figure 13a), whereas Moore et al. (Reference Moore, Koseff and Hult2016) chose the pycnocline (e.g. figure 13b). We believe that our definition is more conceptually sound, as it is based on the finding boundary between the upslope moving bolus and downslope return flow. Moreover, this will be easier to locate in profiles from moorings, considering that the height of the bolus is a function of pycnocline thickness (Vieira & Allshouse Reference Vieira and Allshouse2020).

Given the decrease in bolus size as it shoals, it remains important to quantify how much of the lower layer fluid contained in the bolus is lost to irreversible mixing with the upper layer, relative to what is lost to the near-bed return flow (Southard & Cacchione Reference Southard and Cacchione1972). We do not have sufficient data for a complete analysis; however, we were able to estimate the return flow from the mean velocity profiles for run 8 (figure 16a). The return flow thickness and velocity were observed to decrease as the boluses shoaled. Taking a control volume between ADV-1 and ADV-3 (figure 16b), the change in bolus volume was estimated from the digitized video images (not shown) as $\Delta V1 \approx {V_{\textrm{ADV-1}}}-{V_{\textrm{ADV-3}}} \approx 0.025 - 0.003 \approx 0.022 \pm 0.01\;{\textrm{m}^3}$. This was compared with the volume flux in the return flow at these two points (figure 16a) and the travel time of the bolus from ADV-1 to ADV-3 as ΔV2 ≈ Δt × ΔQ ≈ 19.8 × 0.0012 ≈ 0.023 ± 0.01 m³. The computed values for ΔV1 and ΔV2 were within the error bounds associated with estimating the density profiles of the return flow and the bolus. This back-of-the-envelope example suggests that, for this case, the bulk of the bolus core may be lost to the return flow, with irreversible mixing of bolus fluid to the upper fluid layer (e.g. as required for nutrient flux) being minimal. Here, we have neglected transport and mixing that may occur owing to other mechanisms and at different stages of bolus evolution. For example, Xu & Stastna (Reference Xu and Stastna2020) showed that when boluses started to form, significant cross-boundary-layer transport could occur. Given the implications for nutrient supply to the continental shelf, these back-of-the-envelope calculations should be investigated further.

Figure 16.

(a) Mean velocity profiles of the return flow at the ADVP for run 8. (b) Schematic showing boluses propagating over the boundary slope, with fluid draining to a return flow. In panel (a), the mean velocities for depth <0.5 cm are modelled by using the log-law.

4.3.4. Number of boluses

From the literature, it remains unclear how many boluses to expect from each incident NLIW of depression. For a lone NLIW of depression, a different number of boluses will be generated depending on the characteristics of the wave and the slope (Aghsaee et al. Reference Aghsaee, Boegman and Lamb2010; their figure 13). For a packet of two incident ISWs of depression, Helfrich (Reference Helfrich1992) observed that the first wave formed only one bolus, but for the second wave, the number of boluses was the same as the number of boluses for a similar lone wave (see their figure 18). For ISWs of elevation, Wallace & Wilkinson (Reference Wallace and Wilkinson1988) found one bolus for each wave, as did Moore et al. (Reference Moore, Koseff and Hult2016) for each wave in a train of sinusoidal internal waves.

We observed either one or two boluses to be generated from a single NLIW of depression; matching the number of NLIWs of elevation formed during fission after the turning point (Wallace & Wilkinson (Reference Wallace and Wilkinson1988). The number of NLIWs of elevation generated by fission depended on the characteristics of the incident NLIW of depression. Malomed & Shrira (Reference Malomed and Shrira1991) proposed a KdV-based equation to predict the number of secondary solitons (ISWs of elevation), but as with Helfrich (Reference Helfrich1992) and Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010), this relationship only worked when there was a lone ISW of depression; not a periodic train, as in the present study.

The number of boluses increased as both Froude number (2.11) and Reynolds number (2.10) decreased; with $Fr\; \approx 0.1$ separating the data (figure 12d). The decreasing number of boluses, with increasing wave inertia $(Fr)$, can be interpreted according to the generation time scale for each NLIW of elevation. Each NLIW of depression has a finite ‘duration of transition’ (Grimshaw et al. Reference Grimshaw, Pelinovsky and Talipova1998; transformation time scale from an NLIW of depression to a packet of NLIWs of elevation). Increasing $Fr$ causes the wave period to decrease and each NLIW of depression has a shorter duration of transition, before the arrival of the next NLIW in the packet, which prevents further boluses from forming (§ 3.2; figure 7).

5. Conclusions

The present study extends the prior work on NLIW fission and bolus formation on mild slopes as commonly observed in lakes and continental margins. As each NLIW of depression shoals, they evolve into a packet of NLIWs of elevation after passing the turning point; the number of waves of elevation depends on the Froude number of the incident wave. The waves of elevation transform into boluses where shear instability occurs through the pycnocline. At this ‘bolus birth point’, flow velocities are maximum. Each bolus propagates upslope as a quasi-trapped core, decreasing in size and velocity magnitude, through loss of volume to a near-bed return flow emanating from the rear face and diapycnal mixing from instability across the pycnocline. A simple calculation suggests that for some cases, the volume-flux to the return flow may predominate, thereby limiting the overall effectiveness of some boluses in driving biogeochemical fluxes in the coastal ocean. However, this topic requires further research as both shear and lobe-cleft instabilities were observed to drive mixing, which was not quantified experimentally.

The formation mechanism, birth point, propagation distance and initial height of the boluses generated from fission were distinct from those generated by other breaking mechanisms associated with NLIWs of depression or elevation, but the height to length ratio, change in size with propagation and the velocity field inside and around the boluses were all remarkably similar. The bolus birth point, initial height and dissipation length scale were parameterized in terms of variables that are measurable in the field by oceanographers. These bolus characteristics were related to $Fr$, with ${u_{max}} \approx 0.7{c_0}$ at the bolus birth point. Future work should investigate effects of changing the boundary slope (S ≤ 0.05) as well as the generality of the equations describing the bolus birth point, the initial bolus height and the total bolus propagation distance. We are presently extending this study to investigate sediment resuspension and transport by fission and bolus propagation over mild slopes.

Supplemental material and movies

Supplementary material and movies are available at https://doi.org/10.1017/jfm.2021.1033.

Acknowledgements

We thank R. Mulligan for lending us two Vectrino ADVPs for use in the lab. L.B. thanks J. Moum for pointing out that we experimentalists were using slopes that were different from what is actually in the coastal ocean. L.B. thanks N. Jones and G. Ivey for discussions on the challenges in measuring NLIW transformation between moorings. We thank P. Davies and J. Grue for an invitation to present our early findings from this study at the 6th Norway–Scotland Waves & Marine Hydrodynamics Symposium. Four anonymous reviewers are thanked for their constructive and valuable comments. Upon acceptance for publication, the laboratory data will be archived in the Dataverse repository at Queen's University (https://dataverse.scholarsportal.info/).

Funding

This research was funded by an NSERC Discovery Grant and Discovery Accelerator Supplement to L.B. and by Queen's University.

Declaration of interests

The authors report no conflict of interest.

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Figure 1.Schematic showing the propagation of an NLIW of depression in a two-layer stratification system toward a uniform boundary slope. Here the total depth (H) consists of a thicker (h2) and commonly denser (ρ2) lower layer which is overlaid by a thinner (h1) and lower denser (ρ1) layer. An NLIW of depression with amplitude a and wavelength $\lambda = 2{L_w}$ propagates towards a planer slope with steepness S. Here, Li is the pycnocline length above the slope.

Figure 2.Different breaking mechanism regimes for incident NLIWs, classified according to wave slope $({S_w})$ and bottom slope $(S)$ based on the results by Nakayama et al. (2019).

Table 1.Comparison between boundary slopes used in experimental studies of shoaling NLIWs and observed boundary slopes in the coastal ocean. Note that the experimental slopes are typically orders of magnitude greater than those observed in the ocean, which leads to differing breaking mechanisms in the laboratory compared with the ocean.

Figure 3.Schematic of experimental set-up. Periodic NLIWs of depression are generated within the two-layer fluid by flapping the wave paddle with an electric motor connected through an eccentric drive shaft. The upper fluid layer is fresh water and the lower fluid layer is saline. See table 2 for a list of the experimental parameters.

Table 2.Experimental parameters, where a is the wave amplitude (negative means wave of depression), ρ2 is the lower layer fluid density, h2 is the lower layer thickness, γ is non-dimensional nonlinearity parameter (2.6), T is the wave period (2.8), c is the nonlinear wave phase speed (2.4), Lw is the wave half-width (2.7), Sw is the wave slope, ka is the wave steepness (2.9), Re is the wave Reynolds number (2.10) and Fr is the wave Froude number (2.11) (the boundary slope S = 0.04 and the total height H = 0.4 cm are constant for all the tests).

Figure 4.Time series of the pycnocline displacement ($\eta$) for the run 6 recorded by the wave probe. The resultant wave amplitude (a) and wave period (T) are shown.

Figure 5.Snapshots in time for run 2 (each row represents a time step), which show fission for a typical NLIWs of depression over a mild slope and bolus formation when there is one bolus for each incident NLIW of depression. The blue arrows in b, d, g and k show the incident NLIW of depression, the evolving NLIW of elevation after the turning point, the bolus and the point where the bolus has completely dissipated, respectively. Time period between snapshots is Δt = 6 s and the locations of the ADVPs are shown with red dots. All three image panels are acquired simultaneously with the same camera.

Figure 6.Snapshots in time for the run 6 which show typical transformation of an NLIW of elevation to a bolus when there is one bolus for each incident NLIW of depression. The blue arrow shows the location where the rear face of the NLIW of elevation reaches maximum steepness, and shear instability in the form of Kelvin–Helmholtz billows forms through the rear face. We consider this to be the bolus birth point. Thereafter, the bolus progresses upslope, decreasing in size until it degenerates. The time interval between snapshots is Δt = 5 s and the locations of the ADVPs are shown with red dots. Each image panel is acquired with a different camera.

Figure 7.Snapshots in time for run 1 which show typical bolus formation mechanism when there are two boluses for each incident NLIW of depression. The blue arrow shows the tongue of the lower layer fluid attaching the first bolus to the pycnocline. The orange arrow shows the second NLIW of elevation and the green arrow shows the second bolus (see supplementary movie S1). The time interval between snapshots is Δt = 4 s and the locations of the ADVs are shown with red dots. The third ADVP is outside the field of view.

Figure 8.Images showing run-up of a bolus after its formation (run 4). The bolus is travelling from left to right in the upslope direction. (a) Side view (see supplementary movie S2), (b) top view of two boluses as indicated by arrows: the green arrow shows the top view of the bolus in panel (a); the yellow arrow shows the preceding bolus during its final stage of shoaling as it degenerates (see supplementary movie S3). In this run, there is one bolus for each NLIW of depression. The bolus in panel (a) has a height of ~4 cm and the smaller-scale shear instabilities have length scales of ~1 cm.

Figure 9.Hovmöller diagram of instantaneous velocity profiles for a bolus in (a) the streamwise direction (u component), (b) the vertical direction (w component) and (c) the spanwise direction (v component) for run 8 (table 2) as captured by the first ADVP. Measured data within 5 mm of the bed have been removed because of poor signal-to-noise ratio and correlation from acoustic reflection of the bottom. In all the experimental data, this is the largest bolus that is recorded with the strongest velocities.

Figure 10.Hovmöller diagram showing contrasting instantaneous velocity profiles for an NLIW of elevation and two boluses. Panels (a,b), (c,d) and (e,f) show the streamwise (u component), vertical (w component) and spanwise (v component) velocity components, respectively, for the NLIW of elevation (left column)/boluses (right column). These data are for run 1 as captured by the first and second ADVPs. The first bolus forms from the shown NLIW of elevation, whereas the second bolus forms from the trailing NLIW of elevation (not shown). Both NLIWs of elevation occur through fission of the same incident NLIW of depression. See figure 7 and supplementary movie S1.

Figure 11.Schematic showing fission of a train of NLIWs of depression over a mild slope. Here, xbf is the distance between the bolus birth point (star) and the interaction point and x0 is the bolus propagation distance.

Figure 12.Parameterization of bolus characteristics. (a) Bolus birth point criterion: ratio of the distance between the bolus birth point and the interaction point (xbf) normalized by Lw (2.6) versus Fr (2.11). (b) The total bolus propagation distance (x0) normalized by Lw versus Fr. (c) The initial height of the bolus (H0) normalized by Lw versus Fr. (d) Classification of number of boluses (Nb) birthed during shoaling of each NLIW of depression in relation to the initial wave forcing parameters Fr and Re (2.10). Data points are mean ± standard deviation in panels (a–c).

Figure 13.Flow within a bolus. (a) Spatial distribution of the $\bar{U}$ velocity superimposed on the 2-D velocity vector field for a bolus from run 8 as captured by the first ADVP. (b) The 2-D velocity vector field superimposed on a video image of the bolus in panel (a). (c) Streamlines from a reference frame moving with the wave for the bolus in panel (a). In all panels, the bolus moves from left to right.

Figure 14.Local bolus propagation speed $({C_b})$ (normalized by $\omega a$) versus (a) wave Froude number $(Fr)$ and (b) wave steepness $(ka)$. Average bolus height $({h_{ave}})$ (normalized by $a$) versus (c) wave Froude number $(Fr)$, (d) wave steepness $(ka)$.

Figure 15.Parameterizations for bolus aspect ratio. (a) The bolus height-to-length ratio as a function of the normalized distance from the bolus birth point. The ordinate scale is the distance of the bolus from the bolus birth point (x) normalized by the bolus propagation distance (x0). (b) Variation of normalized bolus height (H0 = initial bolus height) versus the distance to the bolus from the bolus birth point (x) normalized by the bolus propagation distance (x0). Wallace & Wilkinson (1988) is referred to as WW (1988).

Figure 16.(a) Mean velocity profiles of the return flow at the ADVP for run 8. (b) Schematic showing boluses propagating over the boundary slope, with fluid draining to a return flow. In panel (a), the mean velocities for depth <0.5 cm are modelled by using the log-law.

Ghassemi et al. supplementary movie 1

Bolus formation mechanism when there are two boluses for each incident ISW of depression.

Video 661.4 KB

Ghassemi et al. supplementary movie 2

The run-up of a bolus after its formation (Side view)

Video 3.8 MB

Ghassemi et al. supplementary movie 3

The run-up of a bolus after its formation (Top view)

Video 6.3 MB

Ghassemi et al. supplementary material

Supplementary figures

File 694.4 KB

Article contents

Bolus formation from fission of nonlinear internal waves over a mild slope

Abstract

JFM classification

Information

1. Introduction

2. Materials and methods

2.1. Theoretical background

2.2. Experimental set-up and measurement techniques

3. Results

3.1. Flow field

3.2. Wave shoaling and bolus formation

3.3. Bolus instability

3.4. Velocity field

4. Discussion

4.1. Difference between a NLIW of elevation and a bolus

4.2. Bolus formation

4.3. Bolus characteristics

4.3.1. Bolus dynamics

4.3.2. Bolus propagation distance

4.3.3. Bolus size

4.3.4. Number of boluses

5. Conclusions

Supplemental material and movies

Acknowledgements

Funding

Declaration of interests

References

Ghassemi et al. supplementary movie 1

Ghassemi et al. supplementary movie 2

Ghassemi et al. supplementary movie 3

Ghassemi et al. supplementary material

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