4.1 Experimental and numerical results

Six different values of the slope steepness were considered. In the laboratory experiments, the base length of all slopes was kept fixed at $1.5~\text{m}$ and the vertical height $H_{s}$ varied from $0.05~\text{m}$ to $0.60~\text{m}$ . This corresponds to a non-dimensional slope steepness of $s=H_{s}/1.5\in [0.03,0.40]$ and a slope angle (as defined in figure 1) of $\unicode[STIX]{x1D703}\in [1.9^{\circ },21.8^{\circ }]$ . Some reference cases for a vertical wall ( $\unicode[STIX]{x1D703}=90^{\circ }$ ) were also run. In the numerical simulations, the laboratory slopes were replicated in the first instance and in some cases, extended in length to allow further analysis of the fluid dynamics. In this section, results are presented for four of the six slopes considered, spanning the range of slopes investigated. The experiments that have been chosen are presented for waves at a fixed non-dimensional incident wave amplitude of $a/H=0.11$ . The wave shoaling characteristics did not vary (in a qualitative sense) on a given slope over the range of wave amplitudes considered.

In figure 3, observations of wave propagation over the shallowest slope, $s=0.03$ , are presented. Figure 3(a–e) images are taken from the experimental movie, using a false-colour scheme to visualise optimally the flow field in the illuminated section. Figure 3(f–j) represents the corresponding vorticity fields as computed by PIV. The images are ordered sequentially in time from top to bottom at fixed time intervals of $\unicode[STIX]{x0394}t$ . Here, $\unicode[STIX]{x0394}t=5~\text{s}$ ; in all subsequent figures $\unicode[STIX]{x0394}t=3~\text{s}$ unless stated otherwise. In all experiments, the incident mode-2 wave did not have a tail associated with it as it propagated over the flat bed where the upper and lower layer depths were approximately equal and the pycnocline was centred on the mid-depth of the water column. Figure 3(d,i,e,j) shows that, as the wave propagates over the slope, and as the ratio of the upper and lower layer depths varies, an oscillatory tail develops behind the mode-2 wave. Moreover, the amplitude of the wave tail becomes more pronounced (figure 3 e,j) as the mode-2 wave continues to propagate further along the slope and (i) the difference in the layer depths increases and (ii) the stratification profile becomes increasingly asymmetrical across the mid-depth.

Figure 3.

Sequence of raw images (a–e) and corresponding vorticity fields (f–j) from experiment s1a2 ( $s=0.03,a/H=0.11$ ). The time interval between panels is $\unicode[STIX]{x0394}t=5~\text{s}$ . Note that images are shown from two different cameras arranged horizontally to cover an integrated field of view; (a,f,b,g) camera 1, (c,h,d,i,e,j) camera 2.

Figure 4.

Sequence of raw images (a–e) from experiment s2a2 ( $s=0.07,a/H=0.11$ ) and corresponding vorticity field (f–j). The time interval between panels is $\unicode[STIX]{x0394}t=3~\text{s}$ .

In figure 4, images are presented from the case of $s=0.07$ , a steeper slope than for the data shown in figure 3. In this case, the oscillatory tail is more pronounced than in the shallowest case ( $s=0.03$ , figure 3). This is due to the ratio of the fluid layer depths changing at a quicker rate than in the previous case (figure 3) as the wave propagates up the slope. This is in keeping with the simulations of Olsthoorn et al. (Reference Olsthoorn, Baglaenko and Stastna2013) who investigated mode-2 waves propagating over a flat bed but for cases in which the ratio of the layer depths was varied parametrically. They showed that the efficiency of energy transfer from the mode-2 wave body to the oscillatory tail is enhanced with increasing asymmetry over the layer depths. The proximity to the slope of the lower interface of the wave causes the wave shape to be modified and the bottom half of the mode-2 wave to be flattened (figure 4 b,g,c,h,d,i,e,j). As the lower half of the wave is modulated by the bottom boundary in this way, asymmetry is introduced to the wave form. The asymmetry is similar to that documented in Cheng et al. (Reference Cheng, Hsieh, Hsu and Hwang2017) for mode-2 ISW propagation over slope/shelf topography (see their figure 6 a). This flow development, coupled with the presence of the oscillatory tail, causes the mode-2 wave to change form; the top interface of the wave has two peaks and the lower interface has one trough (figure 4 d,i,e,j). A similar change of form was observed by Terletska et al. (Reference Terletska, Jung, Talipova, Maderich, Brovchenko and Grimshaw2016) for mode-2 waves propagating over a step. In addition, the proximity of the mode-2 wave to the bottom boundary causes a build up of vorticity in the bottom boundary layer directly below the mode-2 wave (black signal at the lower boundary under the mode-2 wave).

Figure 5.

Numerical simulation of case s2a2. Panels (a–e) show the density field with a colour map created to approximate the experimental images. The densities near the centre of the undisturbed pycnocline centre have been set to black to visually accentuate this region. Panels (f–j) show the vorticity capped at $\unicode[STIX]{x1D714}=\pm 3~\text{s}^{-1}$ and the slope boundary is marked by a dashed grey line. The time interval between images in the sequence is $\unicode[STIX]{x0394}t=3~\text{s}$ . The location in the numerical domain is chosen to approximate figure 4.

Figure 6.

Numerical simulation of case s2a2. The density field with a colour map created to approximate the experimental images. The densities near the centre of the undisturbed pycnocline centre have been set to black to visually accentuate this region. The time interval between images in the sequence is $\unicode[STIX]{x0394}t=3~\text{s}$ and are chosen during the terminal shoaling portion of evolution.

In figure 5, numerical simulations that can be directly compared with figure 4 are shown. The horizontal location of the numerical simulations has been chosen to approximate that of figure 4. These two reference locations differ slightly from each other due to the manner in which the numerical model specifies the bottom topography (to maintain spectral accuracy the piecewise linear slope is smoothed). In the density plots of figure 5(a–e), fluid with a density within $2\,\%$ of the density at the mid-point of the undisturbed pycnocline, is darkened to illustrate the overturning core region of the leading mode-2 wave. A clear bias in fluid at this density toward the bottom half of the mode-2 wave is evident in the core. The ‘black’ interfacial fluid, is split apart asymmetrically by the propagating wave and stretched along the wave boundary. It subsequently forms cusp-like structures that rotate inward due to the vorticity (f–j) associated with the main wave body. This motion is attributed by Schmidt (Reference Schmidt1998) to an asymmetric Holmboe instability as documented for mode-2 ISWs propagating over a flat bed. In the vorticity evolution sequence (f–j), the leading mode-2 wave is clearly delineated by the spatial distribution of baroclinic vorticity. The core region exhibits considerable small scale vorticity variation. Ejection of material out of the core is evident in the density and vorticity (figures 5 d,i and 5 e,j) plots. This feature is associated with short length scale variability in the vorticity field extending behind the leading mode-2 wave body and is again in agreement with Schmidt (Reference Schmidt1998). Like the laboratory observations (figure 4), once shoaling begins, the early evolution of the mode-2 wave is marked by the formation of a clear trailing oscillatory tail. Agreement between the numerical simulation and the laboratory observations is excellent. No significant short length scale activity is observed in the bottom boundary layer.

Figure 7.

Sequence of raw images (a–e) from experiment s4a2 ( $s=0.20$ , $a/H=0.11$ ) and corresponding vorticity field (f–j). The time interval between panels is $\unicode[STIX]{x0394}t=3~\text{s}$ .

Figure 8.

Sequence of raw images (a–e) from experiment s4a2 ( $s=0.20$ , $a/H=0.11$ ) and corresponding vorticity field (f–j). The time interval between panels is $\unicode[STIX]{x0394}t=3~\text{s}$ . The first panel (a,f) is displayed at the corresponding time to figure 7(e,j).

The terminal phase of shoaling and, in particular, the subsequent flow development once the pycnocline intersects the slope is shown in figure 6. (Note similar observations were not possible in the laboratory due to the limited length of the wave flume). Figure 6 has the same aspect ratio as figure 5, but has been shifted downstream. Only the density field is shown. It can be seen that the mode-2 wave degenerates into several mode-1 waves of elevation, a behaviour resembling in some respects the fission process described by Aghsaee et al. (Reference Aghsaee, Boegman and Lamb2010) for shoaling of mode-1 waves on shallow slopes. This breakdown lowers the mechanical energy density of the flow, so that the wave train effectively ‘runs out of steam’ as it moves up the slope. In contrast to the shoaling flow behaviour observed for steeper slopes (see below), little boundary layer activity is observed and there is no evidence of any reflected wave signal.

In figures 7–9 the slope steepness is increased to $s=0.2$ . Figures 7 and 8 are laboratory observations while figure 9 is numerical simulation. In this case, the undisturbed pycnocline intersects the slope in the laboratory observations. An oscillatory tail appears to form (see figure 7 c,h,d,i) but the time scale over which it has time to develop before the wave collides with the slope is much less than in the shallower slope cases (figures 3–5). As the wave shoals, the bottom of the mode-2 wave flattens (figure 7 c,h), vorticity is intensified at the bottom boundary, the boundary layer separates (figure 7 d,i; see figure 10 below for validation) and a vortex is generated at the bed leading to re-suspension of tracer particles from the bed (figure 7 e,j). A second mode-2 ISW (of smaller amplitude and slower speed) appears as part of the experimentally generated rank-ordered wave train and can be seen in figures 7(d,i) and 7(e,j). In figure 8 the dynamics further upstream and at later times can be seen. Panel 8(a,f) is displayed at the same time as panel 7(e,j). As the wave collides with the slope there is run up (figure 8 a,f,b,g,c,h), run down (figure 8 d,i,e,j), re-suspension and the mode-2 form of the wave is destroyed as the wave evolves into small amplitude mode-1 waves of elevation.

Figure 9 shows the numerical evolution of density and vorticity for run s4a2 and can be compared directly with the laboratory observations of figure 7. Agreement with the experimental results is again excellent. During shoaling the lower half of the mode-2 wave broadens (figure 9 c,h) and begins to lag (figure 9 d,i) the main wave. Interaction with the boundary layer occurs behind the wave as evidenced by the near-bottom swirl of density in figure 9(e). In the vorticity evolution, the main wave body is clearly marked by the distribution of baroclinic vorticity it induces. The core region exhibits considerable small scale vorticity variation. Near-bottom vorticity is largest beneath the lagging bottom half of the wave (white in vorticity panel of figure 9 i) and in the forming separation bubble (black in vorticity panel of figure 9 i) and the resulting roll up (figure 9 j).

Figure 9.

Numerical simulation of case s4a2. Panels (a–e) show the density field with a colour map created to approximate the experimental images. The densities near the centre of the undisturbed pycnocline centre have been set to black to visually accentuate this region. Panels (f–j) show the vorticity capped at $\unicode[STIX]{x1D714}=\pm 3~\text{s}^{-1}$ and the slope boundary is marked by a dashed grey line. The time interval between images in the sequences is $\unicode[STIX]{x0394}t=3~\text{s}$ . Location in the numerical domain is chosen to approximate figure 7.

Figure 10 shows the development and bursting of the separation bubble below the shoaling wave. The figure shows the shaded horizontal velocity, as opposed to vorticity, in order to highlight the flow features. The horizontal velocity is saturated at $u=\pm 0.04~\text{m}~\text{s}^{-1}$ . Panels are taken every 2 s, and the initial time is chosen so that figure 10(b) corresponds to figure 9(d,i). In figure 10(a), it can be seen that the down slope flow (black) rides up and over the separation bubble (bottom white) near $x=5.52~\text{m}$ . The separation region subsequently grows in vertical extent, leading to vortex roll up and the generation of short length scale variations. The bottom boundary layer dynamics is similar to that observed under mode-1 ISWs of depression and are attributed to an adverse pressure gradient at the trailing edge of the wave and ‘global instability’ in the boundary layer (Bogucki, Dickey & Redekopp Reference Bogucki, Dickey and Redekopp1997; Hammond & Redekopp Reference Hammond and Redekopp1998; Stastna & Lamb Reference Stastna and Lamb2002; Diamessis & Redekopp Reference Diamessis and Redekopp2006; Carr, Davies & Shivaram Reference Carr, Davies and Shivaram2008; Boegman & Ivey Reference Boegman and Ivey2009; Boegman & Stastna Reference Boegman and Stastna2019).

Figure 10.

Detail of the separation bubble below the shoaling wave in the numerical simulation of case s4a2. Shaded horizontal velocity saturated at $u\pm 0.04~\text{m}~\text{s}^{-1}$ . The time interval between panels is $\unicode[STIX]{x0394}t=2~\text{s}$ . Panel (b,g) corresponds to panel (d,i) in figure 9, although note the different horizontal and vertical extent of the two figures.

The numerical results corroborate the experimental observation that the shoaling mode-2 wave can induce boundary layer separation before the terminal phase of shoaling. For completeness, the terminal phase itself is shown in figure 11. Figure 11(a) is at the same time as figure 9(e,j) but the domain has been shifted in the horizontal direction to allow visualisation of where the pycnocline intersects the slope. It can be seen that the mode-2 wave breaks down into two mode-1 waves of elevation. In figure 11(c) the leading wave has broken down into a thin jet that runs up slope (cf. figures 8(c,h) and 8(d,i) from the experimental results). As the fluid in the jet begins to run down the slope, a breaking internal wave is evident (figure 11 d cf. figure 8 d,i). In MacIntyre et al. (Reference MacIntyre, Flynn, Jellison and Romero1999), the authors noted that internal waves of different modes were generated upon the cessation of winds over Mono Lake, California. These waves were observed to steepen and enhance turbulence (greatly for slopes on the order of $10\,\%$ ) in shoaling regions. It is possible that the mechanisms observed here in the experiments and simulations could be the source of the turbulence observed in the field.

Figure 11.

Numerical simulation of case s4a2. The density field with a colour map created to approximate the experimental images. The densities near the centre of the undisturbed pycnocline centre have been set to black to visually accentuate this region. Outputs every 3 s during the terminal shoaling portion of evolution. Panel (a) corresponds to figure 9(e,j).

Figure 12.

Sequence of raw images (a–e) from experiment s6a3 ( $s=0.40$ , $a/H=0.11$ ) and corresponding vorticity field (f–j). The time interval between panels is $\unicode[STIX]{x0394}t=3~\text{s}$ .

Table 2.

Shoaling behaviour observed for different slopes.

In figure 12 the case for a slope of steepness $s=0.4$ is presented. In this case, as the slope is steep, there is no time for an oscillatory tail to develop before the wave collides with the slope. Like the previous case, the collision of the wave with the slope causes run up (figure 12 c,h,d,i), run down (figure 12 d,i,e,j), boundary layer separation (figure 12 b,g), vortex generation (figure 12 c,h) and re-suspension at the bed (figure 12 c,h,d,i). In this case, however, the wave does not degenerate into mode-1 waves of elevation and a reflected mode-2 wave signal is seen at later time, propagating back upstream. Note that the boundary layer vortex travels up slope but does not surpass the region of mixed fluid in the pycnocline above.

In table 2, the flow characteristics associated with shoaling for different slope steepness are summarised. It can be seen that for slope steepness, $s\geqslant 0.13$ , a reflected signal and significant near-bed dynamics (run up, run down, boundary layer separation and vortex formation) occur. An oscillatory tail only forms for slopes of steepness $s\leqslant 0.27$ ; for steeper slopes there is not enough time for the tail to form before the wave collides with the slope. Moreover, the smaller the slope steepness the greater the degeneration of the wave form.

4.2 Run up

Figure 13.

Schematic diagram to illustrate the definition of run up length $L_{max}$ . The dashed lines indicate the undisturbed pycnocline location. $(x_{p},y_{p})$ are the coordinates of the intersection of the top of the undisturbed pycnocline with the slope and $(x_{m},y_{m})$ are the coordinates of the highest up slope point that the run up reaches.

In cases where run up was observed in the laboratory, it was possible to measure the run up length and speed. Measurements were not possible for cases in which $s\leqslant 0.13$ as the maximum attained by the run up was downstream of the field of view of the cameras. Run up length was measured by tracing the top of the pycnocline and identifying the undisturbed intersection point of the top of the pycnocline with the slope (or wall in the case $s=\infty$ ) $(x_{p},y_{p})$ and the location at which the run up reached a maximum $(x_{m},y_{m})$ (see figure 13). The run up length was defined as

(4.1) $$\begin{eqnarray}\displaystyle L_{max}=\sqrt{(x_{m}-x_{p})^{2}+(y_{m}-y_{p})^{2}}. & & \displaystyle\end{eqnarray}$$

Figure 14 shows how the run up length, $L_{max}$ , varies with slope steepness, $s$ , and non-dimensional wave amplitude, $a/H$ . For a given wave amplitude, the run up length is reduced as the slope steepness increases. This is due to an increase in the gravitational force opposing the flow up the slope as the slope steepness increases. For a given slope steepness the run up length increases with wave amplitude. This is due to an increase in the momentum associated with an increase in wave amplitude.

Figure 14.

Composite plot of run up length $L_{max}$ (m) versus slope $s$ for different ranges of amplitude $a/H$ . Different shaped symbols correspond to different values of non-dimensional wave amplitude $a/H$ . $s=1$ represents collision with a vertical wall ( $s=\infty$ ). Error bars associated with the data points are not included as they are smaller than the marker size.

Run up speed was measured from along slope time series, constructed by recording pixel intensity values along diagonal slices through images for a reference plane $0.002~\text{m}$ above the slope and parallel to it (Sutherland et al. Reference Sutherland, Barrett and Ivey2013). The slices were then arranged sequentially from left to right as shown for example in figure 15. The along slope coordinate is denoted $x_{s}$ . In figure 15, the pycnocline (bright green patch to the left hand side) is seen to run up the slope (vertical/diagonal advancement of thin green line) before reaching a maximum height and running back down. The run up speed, $c_{r}$ , was computed by fitting a straight line to the portion of the time series where run up occurred and measuring the slope of the line. The run up illustrated in figure 15 was typical for all observations made i.e. the run up attained a constant up slope speed before instantaneously stopping and returning back down the slope. The bright white patch in the lower portion of the time series corresponds to resuspended particles associated with the near-bed vortex. It can be seen that this patch travels a short distance up slope but it does not pass the upper interface of the pycnocline.

Figure 15.

Diagonal time series for case s4a2 taken along a slice parallel to the slope and $0.002~\text{m}$ above it. The along slope coordinate is denoted $x_{s}$ . The extent of the horizontal and vertical axes are $\unicode[STIX]{x0394}t=25.1$ s and $\unicode[STIX]{x0394}x_{s}=0.69~\text{m}$ respectively.

Figure 16 shows how the run up speed, $c_{r}$ , varies with slope steepness, $s$ , and non-dimensional wave amplitude, $a/H$ . For a given wave amplitude, the run up speed increases as the slope steepness increases. This is due to a reduction in collision time with increasing slope steepness. For a given slope steepness the run up speed increases with wave amplitude. This is due to an increase in the momentum associated with an increase in wave amplitude and is in keeping with the trend for run up length in figure 14.

Figure 16.

Plot of run up speed (see text) $c_{r}$ ( $\text{m}~\text{s}^{-1}$ ) versus boundary slope $s$ . Different shaped symbols correspond to different values of non-dimensional amplitude $a/H$ . $s=1$ represents $s=\infty$ (collision with a vertical wall). Error bars associated with the data points are not included as they are smaller than the marker size.

A scaling for the maximum run up length $L_{max}$ of the shoaling ISW can be obtained by assuming that, at the instant of maximum run up, (i) fluid within the portion of the wave above the slope is momentarily at rest and (ii) the down slope component of the excess weight of this fluid is proportional to the incident momentum flux of the incident waves generating the run up. Such an approach has been utilised successfully by Archetti & Brocchini (Reference Archetti and Brocchini2002) and Hughes (Reference Hughes2004a ,Reference Hughes b ) for cases of near-shore surface waves (including solitary waves) impinging on inclined beaches. For the mode-2 ISWs under consideration here, the incident momentum flux can be estimated as $M_{x}\approx \unicode[STIX]{x1D70C}_{2}ac^{2}W$ , of where $W$ is the width the channel. A proportion $K_{1}$ of this momentum flux is directed up the slope during shoaling such that the up slope momentum flux $(M_{x})_{up}\approx K_{1}\unicode[STIX]{x1D70C}_{2}ac^{2}W$ , where $K_{1}$ may be assumed a priori to be a function of the slope angle $\unicode[STIX]{x1D703}$ and the wave parameters $a$ and $c$ . When the run up length reaches its maximum value $L_{max}$ , the down slope component of the excess weight of the fluid in the run up current is $mg\sin \unicode[STIX]{x1D703}\approx K_{2}(\unicode[STIX]{x0394}\unicode[STIX]{x1D70C})(L_{max}h_{2}W)g\sin \unicode[STIX]{x1D703}$ , where the volume of run up fluid is assumed on trigonometric grounds to be proportional to $(L_{max}h_{2}W)$ and $K_{2}$ is the proportionality coefficient. Equating the net up slope momentum flux in the up slope direction with the down slope component of the excess weight (and assuming no frictional forces are significant) allows the length scale $L_{max}$ to be estimated from above as

(4.2) $$\begin{eqnarray}\displaystyle L_{max}\approx \frac{Kac^{2}}{g^{\prime }h_{2}\sin \unicode[STIX]{x1D703}},\end{eqnarray}$$

where $K(\unicode[STIX]{x1D703},a,c)=K_{1}/K_{2}$ is a consolidated coefficient. In order to evaluate this scaling, the experimental run up length data are plotted in figure 17. The degree of agreement between the measured and predicted values of $L_{max}$ indicates that the above scaling provides a satisfactory indicator for the run up behaviour of shoaling mode-2 waves.

Figure 17.

Run up length $L_{max}$ (m) versus scaling parameter $ac^{2}/g^{\prime }\sin \unicode[STIX]{x1D703}h_{2}$ . The solid line is the linear best fit. Error bars associated with the data points are not included as they are smaller than the marker size.