3.1. Overall flow development

The propagation of a gravity current in a homogeneous ambient, such as occurs in the upper layer in our experiments, has been studied extensively by, among others, Britter & Simpson (Reference Britter and Simpson1978), Britter & Linden (Reference Britter and Linden1980) and Simpson (Reference Simpson1997). The gravity current coming out of the gate quickly adjusts to balance buoyancy flux, entrainment and bottom stress in a constant velocity phase (Britter & Linden Reference Britter and Linden1980). Throughout the propagation in the upper layer, both entrainment of the ambient fluid and detrainment of the gravity current fluid is observed, similar to what was found by, for example, Odier, Chen & Ecke (Reference Odier, Chen and Ecke2012) and Hogg et al. (Reference Hogg, Dalziel, Huppert and Imberger2017).

Consistent with previous observations, when the gravity current reaches the pycnocline, part of it penetrates the pycnocline and continues to propagate along the slope as an underflow, and another part propagates into the pycnocline at a higher level of neutral buoyancy as an interflow (Monaghan et al. Reference Monaghan, Cas, Kos and Hallworth1999; Samothrakis & Cotel Reference Samothrakis and Cotel2006a,Reference Samothrakis and Cotelb; Monaghan Reference Monaghan2007; Cortés et al. Reference Cortés, Rueda and Wells2014,  Reference Cortés, Wells, Fringer, Arthur and Rueda2015; Tanimoto et al. Reference Tanimoto, Ouellette and Koseff2020,  Reference Tanimoto, Ouellette and Koseff2021). Figure 2 shows the progression in $20$ second increments beginning at $t = 13$ s for run 1-a, the lowest $Ri_{\rho }$ case with $Fr = 0$ (absence of waves). In figure 2(a) ($t =13$ s) the gravity current is seen just reaching the interface. In figure 2(b) ($t=33$ s) the pycnocline is raised above the head of the underflow, a phenomenon referred to as a locked wave shown in Tanimoto et al. (Reference Tanimoto, Ouellette and Koseff2020). In figure 2(c) ($t=53$ s) the interflow is seen propagating to the left, with the interflow seemingly symmetrically distributed about the original pycnocline location, except for the head which is slightly raised. Also of note when comparing figures 2(a) and 2(b) is the difference in both the horizontal and vertical extents of the lower layer. Initially, the lower layer (dyed in green) extends upslope to the position indicated by the white arrow in figure 2(a), but in the presence of the gravity current the initial shape is deformed and some of the lower layer fluid is displaced above the initial height of the pycnocline (shown as the dotted black line). Some of the lower layer fluid at this point is exiting the system through the main drain; however, the mass lost through the drain does not account for the observed reduction of the extent of the lower layer. The ‘generation’ site of the interflow is shown in figures 2(b) and 2(c) (indicated by the red arrows), where the interflow appears to detach from the gravity current underflow as it inserts itself along the pycnocline. In the last two panels (figure 2d and figure 2e), the interflow and underflow are clearly separate with a region of the ambient lower layer between them, in the region indicated by the green arrows. Although the experiment is designed to minimize reflections, after the locked wave reaches the downstream end of the tank reflections result in a low frequency upstream surge. The surge, which has a wavelength much larger than the wavelength of the oncoming waves, is evident in the slightly concave shape of the interface in figure 2(d). Further evidence of the presence of the surge is the horizontal extent of the lower layer in figure 2(e), which is similar to the initial conditions in figure 2(a), as is the vertical position of the interface. Finally, in figure 2(e) we observe a few higher frequency waves, as was the case in Tanimoto et al. (Reference Tanimoto, Ouellette and Koseff2021).

Figure 2.

Snapshots in 20 s intervals for run 1-a starting from $t=13$ s. The top layer is clear, the bottom layer is dyed green, and the gravity current is dyed blue and descending to the left along the slope. The dotted line is the initial location of the pycnocline. The white arrow in (a) marks the lower layer extending to this location; in (b,c) the red arrows mark the generation site of the interflow; and in (d,e) the green arrows mark where the interflow and underflow are separated by the lower layer fluid.

Figure 3 shows the equivalent development of the interflow but for conditions corresponding to run 1-d, with the lowest $Ri_{\rho }$ gravity current and the highest $Fr$ internal waves. In figure 3(a) ($t =12.5$ s) the first oncoming wave is observable spanning across the left support beams where the crest is marked by the red downward arrow. Subsequently, in figure 3(b) the oncoming wave interacts with the locked wave from the head of the gravity current (in the region indicated by the red upward arrow), and the amplitude of the resulting wave is much larger than the amplitude of the oncoming wave in figure 3(a). Starting in figure 3(c), the interflow front starts to exhibit a degree of sinuosity as it propagates along the pycnocline (above the white arrow), which is distorted by the oncoming internal waves. We observe, however, that during this time period the internal waves do not propagate upstream beyond the right-hand support beam. In figures 3(d) and 3(e), the interflow is seen travelling along the interface path set by the internal waves, but there appears to be significantly more interaction with the underflow evident in the areas closer to the interflow generation site, above the areas marked by the white arrows.

Figure 3.

Snapshots in 20 s intervals for run 1-d starting at $t=12.5$ s. See caption for figure 2 for other details. The red arrow in (a) marks the crest of the wave and in (b) the crest of the oncoming and locked wave superimposed. The white arrow in (c) marks where the interflow path is starting to get affected by the internal waves and in (d,e) the interaction between the interflow and underflow is seen in this region.

Figure 4 shows side-by-side comparisons of experimental snapshots 30 s apart for runs 4-a (figure 4a–e) and 4-d (figure 4f–j), the runs with the highest $Ri_{\rho }$ with $Fr=0$ and $Fr=0.14$, respectively. Compared with figures 2 and 3, the higher $Ri_{\rho }$ gravity current does not deform the lower layer as much, and the interflow generation site is much farther upstream, marked by the white arrows in figures 4(b) and 4(g). In figures 4(c) and 4(h) a weak lower-layer surge is observed, returning the pycnocline to its original height. The front for the surge is marked by the white arrows in 4(c) and 4(h), and the re-established pycnocline can be seen in figures 4(d) and 4(i). In the case of the $Fr=0.14$ wave, in figure 4(g) the structure of the internal wave is seen upslope of the right-hand support beam, a feature not observed in figure 2 in the lowest $Ri_{\rho }$ case. In figures 4(i) and 4(j) the waves (with crests marked by the red arrows) appear to be steepening and forming bolus-like structures similar to those in Moore et al. (Reference Moore, Koseff and Hult2016) but, owing to the much thicker interface from the interflow, the result is perhaps more similar to a turbulent surge rather than a well-defined breaking wave. Comparing the progression in figure 4 with those in figures 2 and 3, it is clear that although the interactions between the internal waves and dense gravity currents have similar qualities, these characteristics depend on the $Fr$ of the waves and $Ri_{\rho }$ of the gravity current.

Figure 4.

Snapshots at 30 s intervals starting at $t=17$ s, (a–e) are for run 4-a and (f–j) are for run 4-d. The white arrows in (b,g) mark the interflow generation site and in (c,h) the lower layer is returning toward its original position. The red arrows in (i,j) mark the crest of the oncoming internal waves.

3.2. Gravity current profile and interflow generation

Before looking in detail at the results from experiments 1-a to 4-d, we present results from run 5-a that illustrates the mechanism influencing the formation of the gravity current interflow. These experiments utilized a PLIF–PIV system placed along the slope near the top of the lower layer (see location of red box with the dashed outline in figure 1). This specific location was chosen so that simultaneous density and velocity profiles could be acquired both with and without the ambient pycnocline being located upstream of this point. This experiment was performed under the same conditions as run 1-a; the flow development for this case is shown in figure 2. A key observation from figures 2 and 3 is that the lower layer is originally displaced downslope (to the left), allowing the gravity current to flow freely as though it were propagating in a homogeneous ambient. At around $t=70$ s, the lower layer surges back and the pycnocline (and the top of the lower layer) re-establishes itself upstream of the measurement location.

Collocated temporal measurements of velocity and density in the $x$–$z$ plane were obtained from the combined PLIF–PIV measurements for run 5-a, made in the region where the initial pycnocline and bottom slope intersect (shown as the red box in figure 1). Figure 5 shows the vertical profiles of the density and the horizontal component of the velocity for the gravity current at three different times ($t=0$, 22 and 111 s). The profiles are averaged over 2 s to smooth the data and minimize the effect of turbulent fluctuations. The vertical coordinate has been normalized by the height of the gravity current defined by Ellison & Turner (Reference Ellison and Turner1959) as

(3.1)\begin{equation} h_u=\frac{\displaystyle\left(\int_0^{\infty} u\,\mathrm{d}z \right)^{2}}{\displaystyle\int_0^{\infty} u^{2} (z)\,\mathrm{d}z}. \end{equation}

Figure 5.

Velocity and density profiles of the initial conditions before the experiment ($t=0$ s), the gravity current in a free-flowing state ($t=22$ s) and the gravity current with the pycnocline upstream ($t=111$ s). The velocity axis is scaled by the buoyancy flux of the gravity current, and the density by the densities of the ambient stratification. The vertical axis is scaled by the height defined in (3.1). The coordinate system is the same as in figure 1.

The solid line is the initial density profile at $t=0$ of the ambient layers, which are separated by a thin interface with a hyperbolic tangent shape. The dashed line at $t=22$ s in the velocity profile (figure 5a) shows the propagation of the gravity current, which exhibits classic gravity current velocity characteristics with the highest shear at the bottom of the profile (Wells et al. Reference Wells, Cenedese and Caulfield2010; Wells & Dorrell Reference Wells and Dorrell2021). The density profile (figure 5b) at $t=22$ s shows the downslope displacement of the lower layer. After the lower layer has re-established itself upstream of this measurement location at $t=111$ s, the peak velocity is reduced and the gradient in the velocity profile appears also to be reduced. At $t=22$ s the high density of the gravity current is present from $z/h_u=0$ to $z/h_u = 0.5$ in the density profile, above which is a long tail where the density decreases to the density of the upper layer linearly above $z/h_u=0.5$. The density profile at $t=111$ s is different from at $t=22$ s in that the pycnocline is now upstream of the measurement location at $t=111$ s, and it appears that the flow region formerly forming the long tail now consists of fluid of intermediate densities due to mixing of the underflow with the lower ambient fluid. Furthermore, this fluid appears to have been guided into the interflow region of the gravity current, as seen by the increase of the fluid with non-dimensional densities between zero and one in the density profile. This is very similar to the phenomenon known as ‘peeling detrainment’, where partially mixed fluid from the gravity current will selectively separate from the gravity current and seek its own level of neutral buoyancy (Baines Reference Baines2005,  Reference Baines2008; Odier et al. Reference Odier, Chen and Ecke2012; Cortés et al. Reference Cortés, Rueda and Wells2014; Hogg et al. Reference Hogg, Dalziel, Huppert and Imberger2017). In the present case, the fluid that is being detrained is ‘guided’ to a region between the two ambient layers, thus forming the gravity current interflow. While the detrained fluid does have some excess horizontal momentum imparted from the underflow, the mixing with the slow-moving ambient fluid that occurs to form the intermediate-density fluid results in the interflow velocity being less than that of the underflow. Although there have been attempts to characterize the velocity of intrusive gravity currents in lock releases (see Cheong et al. (Reference Cheong, Kuenen and Linden2006), for example), there is no unified theory to predict the intrusion velocity of a detraining interflow intrusion forming from a gravity current on a slope, either in stratified environments or in the presence of internal waves.

Because of mixing and entrainment, the density of the detrained fluid is much lower than the density of the original underflow of the fluid. We therefore do not expect the gravity current underflow downstream of the stratified interface ($t=111$ s in figure 5) to have the same density-profile shape as upstream of that interface ($t=22$ s in figure 5). In some previously reported work, the shapes of the velocity and density profiles of the gravity current are found to be similar (or are assumed to be), and integral or bulk measures can be used to characterize the thickness or density of the gravity current using only the velocity profile (Wells et al. Reference Wells, Cenedese and Caulfield2010). However, as is evident here, such approaches may not always be suitable, especially in stratified environments where fluid can detrain and detach from the main gravity current, thus resulting in continually evolving shapes of the density and velocity profiles.

3.3. Fluxes

In this section we examine how the interflow is affected by the presence of oncoming internal waves. Using the velocity and density field measurements from PLIF–PIV for the 16 experiments in runs 1-a to 4-d, we computed the flux through a vertical interrogation plane at the centre of the blue box in figure 1. This location along the slope was chosen so that the upper and lower layer would both be within the imaging extent and the interflow could be thus characterized. We define the interflow as the mass of fluid with a density greater than that of the upper ambient layer and less than that of the lower layer. The mass flux per unit width across this plane can be calculated as

(3.2)\begin{equation} \dot{m}(t)=\int_{z_1}^{z_2} u(z,t)\rho(z,t)\,\mathrm{d}z, \end{equation}

where the $z_1$ and $z_2$ are the lower and upper bounds of the interflow, respectively. Both the velocity and density data are averaged horizontally across a length of 1 cm to calculate the flux.

The time series of the cumulative flux of the interflow $(\int _0^{t} \dot {m}(t')\,\mathrm {d}t')$, or equivalently the total mass per unit width of the interflow that has crossed the interrogation plane, for runs 2a–2d is shown in figure 6. The cumulative fluxes for runs 2-c ($Fr=0.09$) and 2-d ($Fr=0.14$) with higher $Fr$ are reduced following the initial formation of the interflow, as compared with runs 2-a ($Fr=0$) and 2-b ($Fr=0.05$). The cumulative flux of the $Fr=0$ and $Fr=0.05$ cases track each other to within the measurement uncertainty, which suggests that the effect of the waves for this $Fr$ are minimal. The $Fr=0.09$ and $Fr=0.14$ cases also have similar fluxes until $t=60$ s, when there is a slight reduction in the $Fr=0.14$ case. As the formation of the initial interflow occurs before the internal waves arrive at the slope, the effect of the waves appears to be to inhibit the formation of the subsequent interflow, especially at higher $Fr$. The reduction in the flux is clearly dependent on $Fr$ for this $Ri_{\rho }$, although the trends with $Fr$ are not gradual, but rather are sharp. The reduction is observed for $Fr=0.09$ and $Fr=0.14$, but for not $Fr=0.05$. It is possible that at some transitional $Fr$ between 0.05 and 0.09, the role of the waves changes to reduce the flux.

Figure 6.

Cumulative flux per unit width for $Ri_\rho =6.43$.

The cumulative flux at $t=150$ s for all 16 experiments (runs 1-a to 4-d) is shown in figure 7. The fluxes are normalized by the cumulative flux for the $Fr=0$ case for each corresponding $Ri_{\rho }$, which serves as a base case for evaluating the effects of the oncoming internal waves. Across all $Ri_{\rho }$, the effect of oncoming waves is to reduce the observed flux of the interflow. However, the complexity of the flows becomes evident when looking at the different wave cases: the degree of reduction in the interflow for each $Ri_{\rho }$ is not monotonic with increasing $Fr$. For the $Ri_\rho =5.15$, $6.43$ and $8.62$ cases, the effect of increasing $Fr$ is to monotonically reduce the cumulative flux, but for the $Ri_\rho =13.49$ case there is no trend with increasing $Fr$ other than to reduce the flux to approximately 85 % of that of the $Fr=0$ case. In the following sections, we evaluate possible causes for the reduced fluxes in the presence of oncoming internal waves.

Figure 7.

Cumulative interflow fluxes at $t=150$ s normalized by the $Fr=0$ case for each $Ri_{\rho }$.

3.4. Oncoming internal waves

The time series of the displacement for the $\rho =\bar {\rho }$ isopycnal (as defined in § 2.2 as $\bar {\rho }=(\rho _1+\rho _2)/2$) is shown in figure 8 for $Ri_\rho =8.62$ (runs from 2-a to 2-d). In the ${Fr=0}$ case, in the absence of oncoming waves, both the low frequency surge from $t=60$ to 90 s and higher frequency internal waves after $t=90$ s can be seen, consistent with what was observed in Tanimoto et al. (Reference Tanimoto, Ouellette and Koseff2021). In the case where $Fr>0$, the deflection seen between $t=20$ and 30 s is the initial internal wave, which is typically smaller in amplitude compared with the subsequent waves, as the amplitude on the wave maker is ramped up. The peak at around $t=35$ s for all cases (including the $Fr=0$ case) is the locked wave, or the upward displacement of the isopycnal owing to the passing of the underflow beneath the pycnocline (Tanimoto et al. Reference Tanimoto, Ouellette and Koseff2020). The crest at $t=47$ s is the sum of the third internal wave generated, as well as the small-amplitude launched wave generated by the interflow (Tanimoto et al. Reference Tanimoto, Ouellette and Koseff2020). One notable difference between the $Fr=0.09$ and $Fr=0.14$ cases, beginning with the crest at $t=52$ s, is that although the two waves rise in unison, the $Fr=0.09$ wave appears to fall faster than the $Fr=0.14$ wave. This is a common trend that continues for the following waves, and may be due to the increased mass in the interflow. With the increased mass, the energy in the $Fr=0.09$ wave would be insufficient to lift the pycnocline to the maximum wave amplitude as for the $Fr=0.14$ case.

Figure 8.

Observed displacement of the $\rho =\bar {\rho }$ isopycnal for $Ri_\rho =8.62$.

3.5. Observed velocities and isopycnal displacement

To examine the effects of the internal waves on the dynamics of the interflow, we phase average the flow with the period of the oncoming internal waves. For linear internal waves, the induced velocities in the ambient layers will change sign during a wave period, resulting in times when the induced velocities are aligned with the interflow and times when they are antialigned. Also, as seen in figure 8, the presence of the interflow may affect the waves differently at different phases of the wave, which can be illuminated by phase averaging. The phases of the internal waves are extracted based on the time series of the vertical displacement of the central isopycnal (where $\rho =\bar {\rho }$) in figure 8. Of the 10 waves generated for each experiment, only wave periods 3–8 were included in the phase averaging, as the first two waves coming off the wave maker were typically smaller in amplitude than the subsequent waves. Furthermore, for the lowest $Ri_{\rho }$, by the time the eighth wave is generated, the pycnocline is no longer at its original level and the wave maker is not as effective in generating sinusoidal waves. For all experiments with $Fr>0$, the phase ($\phi$) across the experiments was found to be very consistent, as the timing of the initiation of the waves and the generation of the gravity current was not altered between experiments.

For the $Fr=0$ case, because there are no oncoming internal waves, we do not have a wave period that we can use to phase-average the flow. However, given the following considerations, we choose to use the 10 s period of the oncoming waves to perform a phase average in order to compare the flow with the $Fr>0$ cases. Figure 8 shows that a gravity current released into a quiescent stratified environment will generate internal motions, the frequencies of which are somewhat constrained by the generation mechanism (namely the perturbation of the pycnocline by the gravity current) and the restoring force of the ambient stratification. Often the motions induced are not limited to a single frequency: the ‘locked’ waves produced by the underflow and the ‘launched’ waves produced by the interflow result in motions of different frequencies (see Tanimoto et al. Reference Tanimoto, Ouellette and Koseff2021). The upper limit of these frequencies is set by what the ambient stratification can support. The challenge when phase averaging using the period of the oncoming internal waves is that in addition to the frequency of the oncoming waves, we are likely to encounter very similar sets of frequencies generated by the gravity current itself. Here we are assuming that these frequencies encountered in the $Fr=0$ case also exist in the $Fr>0$ experiments and can be linearly superimposed. By averaging the $Fr=0$ case in the same way as the $Fr>0$ experiments, we are therefore able to compare any differences that may be caused by the oncoming internal waves. We note that we would not expect the phase-averaged quantities in the $Fr=0$ case to be independent of phase, because the frequencies of the launched and locked waves are close to the frequency of the oncoming internal waves, and averaging over our limited data set (due to experimental constraints such as thickening of the interface, among others) is not able to remove them completely. If, however, the time record were longer, with sufficient averaging the $Fr=0$ case would not vary with phase. With this caveat in mind, the phase-averaged quantities of $Fr=0$ provide us with a common basis for comparison between the wave cases for each $Ri_{\rho }$.

The phase-averaged vertical profiles of velocity, density and corresponding gradient Richardson number $Ri_{g}$ obtained with the PLIF–PIV for experiments 1a–d (the lowest $Ri_{\rho }$ case) are shown in figure 9. The velocities in figure 9 have been normalized by the overall maximum velocity of the corresponding $Fr=0$ case. The density measurements are normalized as

(3.3)\begin{equation} \tilde{\rho}(z,\phi)=\frac{\rho(z,\phi)-\rho_1}{\rho_2 - \rho_1} \end{equation}

so that the upper layer has a non-dimensional density of zero and the lower layer a density of one. The gradient Richardson number, defined as

(3.4)\begin{equation} Ri_g = \frac{N^{2}}{S^{2}}=\frac{\dfrac{-g}{\rho_1} \dfrac{\partial \rho}{\partial z}}{\left(\dfrac{\partial u}{\partial z} \right)^{2}} \end{equation}

where $N$ is the Brunt–Väisälä or buoyancy frequency and $S$ is the vertical gradient of the horizontal shear, is the ratio of the stabilizing effect of buoyancy to the destabilizing effect of shear in the velocity profiles. Its magnitude serves as an indicator of potential instabilities in the flow that may lead to stirring or irreversible mixing (Peltier & Caulfield Reference Peltier and Caulfield2003).

Figure 9.

Phase averaged normalized velocity, density and $Ri_{g}$ profiles for runs 1-a, 1-b, 1-c and 1-d. The red lines are the vertical location of the maximum velocity and the vertical location of $\tilde {\rho }=0.5$, and the dashed lines in the density are the 1 % and 99 % density values. See text for definitions of normalization.

The velocity profiles for all experiments typically show a core region with the highest observed velocities at the centre of the core similar to a jet, with the shear extending into the region above and below the core. For all cases, we found that the difference between the maximum velocity at the core of the interflow and the velocity of the fluid in either of the ambient layers surrounding it is smaller for the top layer than for the bottom layer. The asymmetry is most likely due to the presence of the gravity current underflow which would cause the lower layer fluid between the interflow and underflow to flow upstream faster (to conserve mass), thus increasing the shear between it and the core interflow region. With increasing $Fr$, we observe differences in the vertical structure in the velocity profile, not only in the vertical location of the interflow core following the sinusoidal pattern induced by the waves but also in the effects of the wave-induced horizontal velocities beneath the crest and above the trough.

For the $Fr=0.14$ case in figure 9(j–l) we see the most pronounced wave effects appearing as strong asymmetries in the vertical structure of velocities, which are also a strong function of the wave phase. For wave phases between 0 and ${\rm \pi}$, and most notably around $\phi ={\rm \pi} /2$, there is strong shear underneath the crest of the wave evidenced by closely grouped velocity contours; above the crest of the wave, the velocity contours are spaced farther apart and there is less apparent shear. The same, however, is not true at $\phi =3{\rm \pi} /2$, where the shear appears to be more uniformly distributed vertically. This is consistent with the velocity field induced by the internal waves: the vertical profile of the horizontal velocity induced by a mode 1 internal wave is given as

(3.5)\begin{equation} u(z,\phi)=a\omega \,\textrm{e}^{\textrm{i}\phi}(\textrm{sech}(lz))^{k/l} \tanh(lz), \end{equation}

where $l$ is related to the thickness of the density profile (Phillips Reference Phillips1977) and $z$ is the vertical coordinate (positive upwards). Beneath the wave crest the induced velocity is negative, leading to strong shear as the interflow and induced velocities have opposite signs. The opposite is true at the wave trough, where the velocities above the pycnocline are negative and those below the pycnocline are positive. While the shape of the velocity profile is clearly a function of phase, the interface thickness based on density does not vary across $Fr$, though the sinuosity of the interface position does become stronger (as expected) with increasing $Fr$.

The velocity and density profiles can be combined to create plots of the gradient Richardson number, $Ri_{g}$, as a function of $Fr$, which are also shown in figure 9. These plots also illuminate the changes in the buoyancy and shear balance in the presence of internal waves. The Richardson numbers in these plots are normalized by the critical value of 0.25 for stratified parallel shear flows (Howard Reference Howard1961; Miles Reference Miles1961), and are plotted using a logarithmic scale such that white regions indicate where $Ri_{g}$ is critical. In the $Fr=0$ case shown in figure 9, $Ri_{g}$ often approaches the critical value of 0.25, especially on the lower flank of the interflow. In the core of the velocity structure, higher values of $Ri_{g}$ are observed as the core has the strongest density gradients to counteract the destabilizing effects of the shear. Comparing the regions of lower $Ri_{g}$ with the velocity and density profiles, it is clear that the lower values of $Ri_{g}$ are observed at the edge of the shear envelope (the vertical extent of the shear layer) where the density gradients are the weakest. As $Fr$ increases, however, the regions of lower $Ri_{g}$ that are susceptible to instabilities shift to lie mainly below the crest, shown by the lighter shades of red. The region of lower values of $Ri_{g}$ at the edge of the shear envelope above the interflow is also further reduced for increasing $Fr$ (shown by the deeper red colours). This suggests that the effect of the waves could be to stabilize the water column above the interflow by reducing the susceptibility of the flow to instabilities for this $Ri_{\rho }$.

These dynamics are largely consistent across flows with increasing $Ri_{\rho }$, but the effects are most pronounced for $Ri_\rho =13.49$ (runs 4a–4b), the flow with the lowest gravity current flow rate. This low flow rate leads to the least amount of momentum imparted to the fluid forming the interflow, and the wave effects are more evident. Figure 10 shows the same set of plots as figure 9, but for the highest $Ri_{\rho }$ case of 13.49. The same increased shear under the crest of the wave is observed, and the effect of the internal waves on the velocity is consistent with the lower $Ri_{\rho }$ cases. Other general trends in the velocity and density profiles across the wave phases are consistent with the $Ri_\rho =5.15$ case, such as the increased wave effects with increasing $Fr$, though there are some important differences. Compared with $Ri_\rho =5.15$, the vertical extent of the shear envelope looks similar for all phases, but the same is not true for the density. In the $Ri_\rho =13.49$ case, the density envelope, the boundaries of which are indicated by the 99 % density contours, is much smaller compared with the $Ri_\rho =5.15$ case. The smaller density envelope leads to the simultaneous occurrence of elevated shear and low density gradients throughout the wave phase, resulting in lower values of $Ri_{g}$ especially on the upper flank of the velocity profile. While for $Ri_\rho =5.15$ the region above the crest for the $Fr=0.14$ case was stabilized (with higher values of $Ri_{g}$) compared with the $Fr=0$ case, in the case of $Ri_\rho =13.49$ even the largest waves of $Fr=0.14$ cannot stabilize and increase the $Ri_{g}$ values above the crest.

Figure 10.

Phase averaged normalized velocity, density and $Ri_{g}$ profiles for runs 4-a, 4-b, 4-c and 4-d. Normalizations and annotations are the same as in figure 9.

3.6. Structure of the velocity and density profile

Here we present more detailed vertical profiles of velocity and density taken in the interflow region to illustrate how the presence of oncoming internal waves reduces the interflow flux. The velocity and density profiles for select wave phases for the $Ri_\rho =5.15$, $Fr=0$ and $Fr=0.05$ flow cases are shown in figure 11 (which are snapshots from figure 9). The values have not been made non-dimensional as there is no inherent scaling for the velocity or density of the gravity current interflow. The $Fr=0$ case (solid black line) shows the presence of a jet-like velocity with the maximum velocity at the core of the interflow, but with the velocity on each flank reaching different values. The velocity at the bottom flank of the velocity profile is strongly in the upstream direction, whereas the velocity at the top flank is much smaller in magnitude, and generally points downslope. This difference in shear is due to the presence of the underflow of the gravity current along the slope, causing the lower layer to flow in the upstream direction. The velocity profiles for the $Fr=0.05$ flow case in figure 11 (dash–dotted red line) show how the vertical position of the velocity maxima in the interflow core is modulated by the oncoming internal waves. In addition, we observe that not only does the position of the peak velocity depend on wave phase but that there is also a reduction in the peak velocity at various wave phases. The density profiles follow a hyperbolic tangent shape, as expected. The density structure is maintained throughout the full wave period but exhibits a vertical shift induced by the sinusoidal pattern of the waves.

Figure 11.

Velocity and density profiles for different wave phases for $Ri_\rho =5.15$, $Fr=0$ (solid, black line) and $Fr=0.05$ (dash–dot, red line). The velocity and density for the full wave period are shown in figure 9.

To quantify the reduction in velocity, we calculate the average velocity of the interflow and examine how it varies for the different flow cases. To do this we need to define the boundaries of the interflow in the vertical velocity profile. This is done by calculating the height of the interflow on the upper and lower flank of the velocity profile, setting $z'=0$ as the vertical location where $u=\max {(u)}$, and carrying out the integration in either direction using (3.1). As the value of $h$ from (3.1) will not converge unless $u(|z'|=\infty )=0$, $h_u$ is calculated from the velocity profile with the velocity of the ambient layer on the corresponding flank removed such that

(3.6)\begin{equation} h_{u,j}=\frac{\displaystyle\left(\int_0^{\infty} u-u_{\infty,j}\,\mathrm{d}z \right)^{2}}{\displaystyle\int_0^{\infty} (u-u_{\infty,j})^{2}\,\mathrm{d}z} \end{equation}

and $u_{\infty,j}$ is obtained as the average velocity of layer $j$, where the density profile shows no significant gradients. The average velocity of the interflow is then given for each phase as

(3.7)\begin{equation} \bar{u}(\phi)=\left( h_{u,1}-h_{u,2} \right) ^{{-}1} \int_{h_{u,2}}^{h_{u,1}} u(z',\phi)\,\mathrm{d} z', \end{equation}

where $h_{u,j}$ refers to the length derived from the velocity profile, the subscript 1 denotes the upper flank of the velocity profile, and the subscript 2 the lower flank. Shown in figure 12 is $\bar {u}(\phi )$ for each $Ri_{\rho }$ wave flow case divided by $\bar {u}(\phi )$ for the $Fr=0$ case. Consistent with the reduced interflow fluxes observed in figure 7, we see that $\bar {u}(\phi )$ in cases with waves is reduced (normalized velocity value less than unity) compared with the $Fr=0$ case for most phases of the waves in all of the flow cases. The one notable exception is the $Fr = 0.05$ wave case for the gravity current of $Ri_{\rho }$ of 6.43. We checked to see whether these differences in $\bar {u}(\phi )$ between the $Fr>0$ and $Fr=0$ cases were in fact an artefact of the assumption of a simple linear superposition of the interflow velocity field and the wave induced velocity flow. To do this, we scaled the difference in velocities between the $Fr>0$ and $Fr=0$ case by the corresponding maximum wave-induced velocity, $a\omega$. Results were not bounded by values of $-1$ and 1, and therefore the reduction in velocity cannot be attributed solely to a linear superposition of the interflow velocity and the wave induced velocity field. For $Ri_{\rho }$ of 5.15 and 6.43, the normalized velocity is the smallest at $\phi = 0$, and peaks at approximately $\phi ={\rm \pi}$. The magnitude of the velocity reduction is also more pronounced than the slight increase in velocity observed at $\phi ={\rm \pi}$, such that the velocity is reduced overall. For $Ri_\rho =8.62$ and 13.49, the trends in the phase appear to follow the sinusoidal motion of the isopycnals, where a maximum is observed closer to $\phi ={\rm \pi} /2$ with a minimum value roughly half a wave period (${\rm \pi}$) later.

Figure 12.

Average interflow velocities normalized by the velocity of the $Fr=0$ case.

3.7. Arclength

For a given $Ri_{\rho }$, the interface undergoes a set of ever larger amplitude deflections as $Fr$ increases, which results in the interflow traversing a longer path to reach the same downstream location. Taking the time series of the vertical displacement of the central isopycnal where $\rho =\bar {\rho }$, we quantify the length of this path by calculating the arclength of the isopycnal. The arclength, however, is not a true measure of the distance travelled by a fluid parcel in the interflow, as such a calculation would require a measurement moving with the flow. For each $Ri_{\rho }$, the arclength for each case where $Fr>0$ (normalized by the arclength for $Fr=0$) linearly increases with $Fr$. On average across $Ri_{\rho }$, the arclengths were 1.3, 1.7 and 2.2 times longer than that of the $Fr=0$ case for flows with $Fr=0.05$, 0.09 and 0.14, respectively. The uncertainty in the ratios of the arclengths for all cases was $\pm$0.1. The increase in arclength in our experiments is driven by the internal waves, which add a vertical oscillation to the path of the interface. The increased arclength, along with the reduced velocities, will reduce the observed fluxes as the interflow travels along a longer path at a slower rate. Such a vertical oscillation will induce a mean vertical velocity ($w$) and also horizontal gradients of vertical velocity (${\partial w}/{\partial x}$) and density (${\partial \rho }/{\partial x}$), which could have implications for mixing.

3.8. Change in mass

As the gravity current splits and forms an interflow and underflow, the underflow is removed from the tank via the main drain. However, the mass in the interflow remains in the tank, and by comparing a vertical density profile with the CT probe before and after the experiment, we can assess the change in mass. The ambient upper and lower layers remain intact throughout the experiment (they are not ever fully mixed by the gravity current or internal waves), hence the boundaries between the ambient layers is clear, and by using the density profile we can quantify the change in mass in the pycnocline between the two layers. This change in mass, however, is not a pure measure of the interflow, as fluid of intermediate density can form as the internal waves break, generate turbulence, and lead to irreversible mixing.

Figure 13 shows the ratio of the final-to-initial mass of the interflow for all 16 experiments. The final profile is taken 40 min after the gravity current is released, and the error bars are derived from two successive density profiles taken 5 min apart. The change in mass is consistent for most cases compared with the final cumulative fluxes in figure 7, but the largest differences occur in the $Fr=0.14$ cases except for in the Ri$_\rho =8.62$ case. Whereas the $Fr=0.14$ case for each $Ri_{\rho }$ typically had the highest reduction in the final cumulative interflow flux compared with the corresponding $Fr=0$ case, the change in mass for the $Fr=0.14$ case does not show the same reduction (again, with the exception of Ri$_\rho =8.62$). There are two possible reasons for this discrepancy. First, the reduced velocity of the $Fr>0$ cases along with the increased arclength would lead to the cumulative observed fluxes being lower. Also, as the objective was to assess how internal waves could alter the flux of the gravity current, the integration of the cumulative flux was only carried out while the gravity current was being fed into the tank. Even after the gravity current feed valve is closed, the remaining flow in the apparatus continues to flow out, which would be captured in the final mass. Second, there are motions along the pycnocline after the experiment has ended, which could enhance mixing and lead to an increase in the change in mass, but not the cumulative fluxes. These effects will be most pronounced in the $Fr=0.14$ case.

Figure 13.

Relative change in the mass of the pycnocline for all experiments from CT probe measurements before and after the experiment.