4.1.1. Finite-amplitude periodic waves on arbitrarily sheared currents

Using a 2D stream function formulation, Chen & Basu (Reference Chen and Basu2021) obtained numerical solutions for large-amplitude periodic waves on sheared currents on the vertical plane. The present model is validated with a case of nonlinear Stokes waves on currents of constant horizontal vorticity. Following Chen & Basu (Reference Chen and Basu2021), the water depth is 200 m and the target wave height and wavelength are $H=10.08$ m and $L=155.91$ m, respectively. The current has a surface velocity of $1.31\,{\rm m}\,{\rm s}^{-1}$ and decreases to zero at the bottom. Three scenarios are considered, namely, (a) pure wave, (b) waves on following linearly sheared currents and (c) waves on opposing linearly sheared currents. To reach the same target wavelength under different current configurations, the wave period ($T$) is specified to be 9.79, 9.10 and 10.60 s, respectively, in Chen & Basu (Reference Chen and Basu2021). Finally, the dimensionless parameters are $kh\approx 8.06$ and $ka \approx 0.20$, which corresponds to a third-order Stokes wave in deep water.

A numerical wavemaker algorithm is employed to generate the desired waves. This algorithm was first proposed by Lee & Suh (Reference Lee and Suh1998); Lee, Cho & Yum (Reference Lee, Cho and Yum2001) and Hsiao et al. (Reference Hsiao, Lynett, Hwung and Liu2005) extended the original algorithm from a line source to a spatially distributed source. Based on the mild-slope equations, Schäffer & Sørensen (Reference Schäffer and Sørensen2006) provided the theoretical derivation of the wavemaker algorithm. A more detailed discussion on the algorithm can be found in Appendix B. The sponge layer treatment (Larsen & Dancy Reference Larsen and Dancy1983) is used at both ends of the numerical flume to absorb outgoing waves. In the numerical simulations, a horizontally uniform current with a certain vertical profile is specified by $u^*$. A transition region is introduced to control the current effects horizontally, changing from zero in the wave generation region to full strength away from the wave generation region (Nwogu Reference Nwogu2009). To accomplish this, a ramp-up function, changing from 0 to 1, is multiplied to the terms being associated with the prescribed current (those terms are shown in Appendix A) so as to allow waves to propagate gradually from calm water into the current region with full strength.

Because of relatively large $kh$ and $ka$ values, the $S4$ model is used to carry out all the simulations with spatial and temporal resolutions of $\Delta x=L/53$ and $\Delta t=T/80$. The numerical solutions quickly reach a quasi-steady and uniform state and figure 2 shows the comparisons of phase-averaged free surface elevations and horizontal velocity profiles under the wave crest for all three cases between the present numerical results and those of Chen & Basu (Reference Chen and Basu2021). It should be noted that the results for free surface elevation obtained from both approaches are almost the same for all three cases. Thus, only one set of results from each model is included in figure 2(a), for comparison. The present $S4$ model can well capture the wave free surface elevations with slight discrepancies appearing at the crest and trough. By using the wave periods specified in Chen & Basu (Reference Chen and Basu2021), the resulting waves in each scenario reach the same target wavelength, verifying the accurate frequency dispersion relation under the combined effects of third-order amplitude dispersion and wave–current interactions (Doppler-shift effects) embedded in the present model. On the other hand, the cubic function employed by the $S4$ model can also capture reasonably well the total horizontal velocity profiles in the entire water column as shown in figure 2(b). However, small undulations appear in the velocity profiles in the water column, and the surface velocity is underestimated slightly, whereas the bottom velocity is overestimated slightly.

Figure 2.

Comparisons of (a) free surface elevation and (b) horizontal velocity profiles under the wave crest between present numerical results and those of Chen & Basu (Reference Chen and Basu2021).

Swan & James (Reference Swan and James2000) derived a second-order Stokes wave-type analytical solutions with the consideration of vertically sheared currents. Here, we check the performance of present models by comparing our solutions with those of Swan & James (Reference Swan and James2000) for a weakly nonlinear wave ($T=10$ s, $h=200$ m, $ka=0.1$) on an exponentially sheared current of the form $U(z)=2\exp (0.02z)$. Note that Swan & James (Reference Swan and James2000) did not provide the explicit solutions for the free surface elevation. Therefore, in figure 3(a) only the present model's results of phase-averaged free surface elevations are shown. However, the wavelength can be compared directly between the present model results and Swan & James's (Reference Swan and James2000) analytical solutions; the difference is only less than 0.2 %. In Swan & James (Reference Swan and James2000) the wave-induced horizontal velocity was defined as the total velocity minus the prescribed current velocity. This wave-induced velocity under the wave crest is plotted in figure 3(b) and is compared with corresponding velocity $u'$ in the present model. Strictly speaking these two quantities are different in definition, but they should be close. The present solutions exhibit undulations because of higher degree of polynomial approximation employed in the $S4$ model. However, the overall agreement is quite good, especially close to the free surface where the actions are.

Figure 3.

(a) Numerical results ($S4$ model) of free surface elevations. (b) Comparisons of wave-induced horizontal velocity profiles under the wave crest between numerical results ($S4$ model) and Swan & James (Reference Swan and James2000).

In the previous example we have shown that the present model can perform well in the deep-water wave conditions, which is a severe test for any depth-integrated models. In the next example, the present models will be checked for their capabilities for simulating nonlinear waves in finite water depths interacting with currents with more complex profiles in the water column. We consider nonlinear periodic waves with two complex current profiles: the exponential profile, $u^c/\sqrt {gh}=0.13\exp [6(z/h)]$, and the sinusoidal profile, $u^c/\sqrt {gh}=0.13\sin [{\rm \pi} (z/h+1)]$, where $u^c$ and $\sqrt {gh}$ are the current velocity and shallow-water wave celerity, respectively. The normalized current velocity profiles in the water column are shown in figure 4. A distinguishing feature between these two current profiles is the pattern of their slopes (vertical shears). Although the slope (shear) of the exponential profile is always positive in the water column with its maximum value at the still water surface, the slope (shear) of the sinusoidal current profile changes sign from negative in the upper half of the water column to positive in the lower half water column. In figure 4, the current profiles are also approximated by quadratic (dashed lines) and cubic polynomials (dash-dotted lines). Although the quadratic approximation is reasonable for the sinusoidal profile, very large discrepancies appear throughout the water column for the exponential profile, suggesting that YL20's $S3$ model would not be able to model these current profiles accurately, especially for the exponential current profile. Overall the cubic approximation models both current profiles more closely, but it still cannot accurately capture the correct surface, bottom and maximum velocity for these two profiles. On the other hand, the present $S3$ model takes the current velocity profile into consideration without further approximation and, therefore, it is a more accurate approach. Finally, the incident wave conditions are prescribed as follows: wave amplitude, $a=0.1$ m; wave period, $T=2$ s; constant water depth, $h=1$ m. The corresponding wavelength is $L_w=5.306$ m, and the dimensionless wave parameters, $kh=1.18$ and $ka=0.12$ or $H/h=0.2$, are obtained based on the third-order Stokes wave theory.

Figure 4.

Normalized current velocity profiles. The black solid line denotes the exponential current profile $u^c/\sqrt {gh}=0.13\exp [6(z/h)]$ and the red solid line represents the sinusoidal current profile $u^c/\sqrt {gh}=0.13\sin [{\rm \pi} (z/h+1)]$. The dashed lines and dash-dotted lines are the quadratic and cubic polynomial fitted results, respectively.

In total four wave–current interactions cases are considered, i.e. waves on following (opposing) exponentially (sinusoidally) sheared current, and the numerical setup is the same as previous cases in this section. The resulting free surface elevations for four wave–current interaction scenarios and the wave-alone case are shown in figure 5. Overall, the wavelength is always lengthened when waves are propagating in the same direction as the current (see figure 5d,e). The opposite is true when waves propagate against the current (see figure 5a,b). The current with a sinusoidal profile in the water column has stronger effects on waves than the current with an exponential profile does. Quantitatively speaking, when waves are propagating against (following) the current with sinusoidal profile and exponential profile, wavelengths are 12 % and 8.0 %, respectively, shorter (longer) than that of the pure wave scenario.

Figure 5.

Comparisons of free surface elevations for (a) waves against current with sinusoidal profile in water column, (b) waves against current with exponential profile, (c) wave alone, (d) waves following current with exponential profile and (e) waves following current with sinusoidal profile. The horizontal axis is horizontal distance $(x)$ normalized by the wave-alone wavelength $L_w$.

The exponentially sheared current has a horizontal vorticity of $\varOmega _c=2.4\,{\rm s}^{-1}$ at the still water level. A snapshot of the calculated velocity and horizontal vorticity field for waves propagating against the exponentially sheared current (waves are propagating to the left) is shown in figure 6, where the vorticity field is normalized by $\varOmega _c$. As the current velocity and the wave particle velocity are in the same order of magnitude, the total velocity is almost zero at the wave crest. Moreover, the negative horizontal velocity under the wave trough is enhanced by the presence of the current. As for the vorticity field, when the wave is absent, the vorticity field is uniform in the horizontal direction, but has a maximum value at the still water surface ($z/h =0$) and decays into the water column. With the consideration of waves, the vertical stratification of the vorticity field is stretched or compressed as the wave comes by. This feature is further illustrated in figure 7, where the vertical profiles of the total horizontal velocity and vorticity at five phases, evenly spaced from wave crest to trough, are shown. The total horizontal velocity is normalized by the maximum horizontal velocity estimated from Stokes wave theory without current. In the same figure, the profiles of the horizontal velocity and vorticity for the pure current are also plotted. They match well with the corresponding profiles of wave–current interaction at the phase of $\eta =0$.

Figure 6.

The velocity field (arrows) and vorticity field (colour) for waves on opposing current with an exponential profile. Waves propagate to the left and currents move to the right.

Figure 7.

(a) The vertical profiles of horizontal velocity for waves on opposing current with an exponential profile. Red lines represent the total horizontal velocity at different phases from wave crest to wave trough (left to right) with equal increment; black dashed line denotes the pure current profile without waves. (b) The corresponding vertical profiles of the vorticity field. Black dashed line denotes the vorticity profile for current without waves.

Similar analyses are performed for waves on the sinusoidally sheared current. A snapshot of velocity and vorticity fields for waves propagating against the current is shown in figure 8. The horizontal velocity on the wave crest is negative (in the same direction as that of the wave propagation) and reverses its direction twice at approximately $z/h=-0.2$ and $z/h=-0.8$, which is strongly influenced by the sinusoidal current profile. Figure 9 shows the vertical structure of the horizontal velocity and vorticity at different phases from wave crest to trough. The variation of the vorticity structure again follows the free surface motions, and the influence of the wave motion on the vorticity field mainly takes effect for $z/h>-0.8$. For brevity, the discussions on the results for waves following currents are presented in supplementary materials.

Figure 8.

The velocity field (arrows) and vorticity field (colour) for waves on opposing current with a sinusoidal profile. Waves are propagating to the left and currents move to the right.

Figure 9.

(a) The vertical profile of horizontal velocity for waves on opposing sinusoidally sheared current. Red lines represent the total horizontal velocity at five phases from wave crest to trough (left to right) with equal increment. Black dashed line denotes the velocity profile for current without waves. (b) The corresponding vertical profiles of vorticity for waves on opposing current with a sinusoidal profile. Black dashed line denotes the vorticity profile for current without waves.

Finally, the vertical profiles of time-averaged horizontal velocity (averaged over five wave periods) are shown in figure 10(a,b). For both exponential and sinusoidal current profiles, comparing with the prescribed current profile below wave trough, the model result is weaker when waves follow the current and is stronger when waves are against the current. Furthermore, comparing the current profiles obtained by superimposing the time-averaged velocity associated with wave-alone and the prescribed steady-state currents with the numerical model results, it can be observed that wave–current interactions (numerical model results) induce stronger (weaker) time-averaged velocities if ${\partial u^c}/{\partial z}>0$ (${\partial u^c}/{\partial z}<0$), and is independent of the wave propagation direction. Between the wave trough and wave crest, the time-averaged horizontal velocity profiles are drastically different from those obtained from the combinations of the prescribed steady-state current and the time-averaged velocity induced by wave alone. It is not surprising because the model considers the full interactions between waves and currents. Lastly, the time-averaged and depth-integrated volume flux in the water column is calculated, and the difference between numerical results of wave–current interaction and linear superimposition of those obtained from wave-alone and current-alone simulations is less than $2\,\%$ for all cases considered here.

Figure 10.

Vertical profiles of the time-averaged horizontal velocity for waves on currents with (a) exponential profile and (b) sinusoidal profile in the water column, respectively. The black line represents the current-alone case; the black dashed line denotes the wave-alone case; the blue line shows the results of waves on following currents (numerical); the red line represents waves on opposing currents (numerical); the blue dash-dotted line denotes waves on following currents (linear superposition of wave-alone and current-alone); the red dash-dotted line represents waves on opposing currents (linear superposition of wave-alone and current-alone cases).

4.1.2. Solitary waves on arbitrarily sheared currents

In this section, the effects of arbitrarily sheared current on solitary waves are examined. As indicated in table 1 and figure 11 several current profiles are considered. The exponential velocity profile mimics the wind-induced current (Swan et al. Reference Swan, Cummins and James2001), whereas the 1/7 power profile represents a high-Reynolds-number open channel flow (Peregrine Reference Peregrine1976). The Froude number ($Fr$) is defined as the ratio of the current velocity at the still water level ($u^c_s$) and the linear shallow-water wave celerity ($c_0=\sqrt {gh}$). The sign of the Froude number indicates whether the wave is propagating following ($+$) or against ($-$) the current. The prescribed steady-state currents are uniform in the direction of solitary wave propagation and have a maximum velocity at the still water level ($z =0$) and decrease to zero at the bottom ($z=-h$).

Figure 11.

Sketch of current profiles listed in table 1 with $Fr=1$. 1.  Exponential: $u^c/c_0=Fr \exp (10z/h)$. 2.  Cubic: $u^c/c_0=Fr(1+z/h)^{3}$. 3.  Quadratic: $u^c/c_0=Fr(1+z/h)^{2}$. 4.  Linear: $u^c/c_0=Fr(1+z/h)$. 5.  Square root: $u^c/c_0=Fr(1+z/h)^{1/2}$. 6.  1/3 power $u^c/c_0=Fr(1+z/h)^{1/3}$. 7.  1/7 power $u^c/c_0=Fr(1+z/h)^{1/7}$, where $c_0=\sqrt {gh}$.

Table 1.

Test cases for solitary waves interacting with vertically sheared currents.

In the numerical simulations, the water depth is set to be $h= 1$ m. The leading order Boussinesq solitary wave solutions (Boussinesq Reference Boussinesq1872) are employed as the initial conditions. The spatial resolution of $\Delta x=0.3h, 0.2h, 0.1h$ is used for solitary wave of different nonlinearity, i.e. $A/h=0.1, 0.2, 0.5$, respectively. The corresponding temporal resolutions are 0.1, 0.06 and 0.03 s, respectively. As the Boussinesq solitary wave solutions are not the final solutions of the present wave–current system, small trailing waves are generated at the beginning of the simulations and are left behind the leading soliton due to the differences in propagation speeds. Because of these trailing waves, the amplitude of the initial input soliton form is determined by trial and error until a solitary wave with the desired amplitude is generated (Gobbi, Kirby & Wei Reference Gobbi, Kirby and Wei2000; Lynett & Liu Reference Lynett and Liu2004). In this study, for solitary waves of $A/h=0.1$ and 0.2, because of the relatively small amplitude of the solitary waves and the existence of vertically sheared currents, it takes a relatively long distance of propagation (roughly 800 water depths) for the leading soliton to be separated from the trailing waves. On the other hand, a distance of only 200 water depth is necessary for the solitary wave of $A/h=0.5$.

The present models are first validated using the perturbation solutions of Pak & Chow (Reference Pak and Chow2009). In figure 12(a) the free surface profiles for the solitary wave of $A/h=0.1$, following and opposing a current with linear profile ($|Fr|=0.6$), are compared with the third-order (in terms of $\epsilon =A/h$) perturbation solutions of Pak & Chow (Reference Pak and Chow2009). The free surface profile becomes narrower when the solitary wave follows the current, whereas the free surface profile is widened by the opposing current. Overall the agreement between the numerical results and the perturbation solutions is very good.

Figure 12.

Comparisons of the free surface profiles of solitary waves ($A/h=0.1$) on following and opposing currents between the present numerical results and those of Pak & Chow (Reference Pak and Chow2009). (a) Froude number $|Fr|=0.6$. Present model results: blue solid line, no current; red solid line, opposing linear current profile; green solid line, following linear current profile. Lines in black are the corresponding results from Pak & Chow (Reference Pak and Chow2009). (b) Froude number $Fr=-1.2$. Present model results: red line, opposing linear current profile ($S2$ model); dashed blue line, opposing square root current profile ($S3$ model); thick blue line, opposing square root current profile ($S5$ model). Numerical results from Pak & Chow (Reference Pak and Chow2009): dash-dotted black, opposing linear current profile; dotted black, opposing square root current profile.

It should be noted that the $S2$ model is used to produce the numerical results shown in figure 12(a). However, models of higher approximations may be required when the current has different current profiles and its intensity ($Fr$) becomes stronger. This feature is demonstrated in figure 12(b), in which numerical results for a stronger opposing current with $Fr=-1.2$ with both linear current profile and square root shear profile (profile number 5 in figure 11) are plotted. For the linear current profile, the red line in figure 12(b), which is the converged results from the $S2$ model, agrees well with Pak & Chow (Reference Pak and Chow2009). For the square root current profile, however, the $S5$ model is needed to produce the converged results, which agree well with the solution from Pak & Chow (Reference Pak and Chow2009). The higher approximations make the wave profile narrower and closer to the converged results (e.g. dashed blue line which is from the $S3$ model). This suggests that although a model of a lower approximation can consider the exact current profile, in the case of a strong current with a complex profile, models of higher approximations may necessary to obtain the converged solutions by providing more corrections through $u'$.

The free surface profiles of finite-amplitude solitary wave with $A/h=0.2$ and 0.5 on vertically sheared current are also studied here. Figure 13(a) shows the comparison of free surface profiles of the solitary wave ($A/h=0.2$) on a following current with a linear profile ($Fr=0.114$) between the numerical results and solutions from a weakly dispersive fully nonlinear model (Choi Reference Choi2003). The converged results from the $S2$ model predict a slightly narrower profile compared with the solutions from Choi (Reference Choi2003). For solitary waves with a larger amplitude of $A/h=0.5$, we first compare free surface profiles of the solitary wave without current obtained from the $S2$ model with Fenton's (Reference Fenton1972) ninth-order solution, which is shown in figure 13(b). The agreement is extremely good. In the presence of an opposing current, a solitary wave with the same amplitude is wider than that for the no current situation. The numerical results ($S2$ model) of the surface profile of solitary wave ($A/h=0.5$) on current with a linear profile and $Fr=-1.0$ is shown by the red line in figure 13(b), where the third-order solution from Pak & Chow (Reference Pak and Chow2009) is also included for comparison. The converged results from the present model predict a slightly wider profile than the weakly nonlinear and weakly dispersive solution. Lastly, to demonstrate the advantage of present models in taking into account arbitrarily sheared current, the free surface profile of solitary wave ($A/h=0.5$) on current with 1/7 power profile (no. 7 in figure 11) is considered, and the converged result from $S2$ model is shown by the green line in figure 13(b). Although the surface current velocity remains the same for different current profiles, the free surface profile of the solitary wave on a current with a 1/7 power profile is obviously narrower than that on a current with a linear profile, which may attribute to the weaker vertical shear near the surface for the 1/7 power current profile.

Figure 13.

(a) Comparisons of the free surface profiles of solitary waves ($A/h=0.2$) on following current with $Fr=0.114$ between numerical results (red line) and fully nonlinear solution from Choi (Reference Choi2003) (black dash-dotted line). (b) Comparisons of the free surface profiles of solitary waves ($A/h=0.5$). No current: numerical (blue line), ninth-order solution from Fenton (Reference Fenton1972) (black dashed line). Note these two lines are not distinguishable. Linearly sheared opposing current with $Fr=-1$: numerical (red line), third-order solution from Pak & Chow (Reference Pak and Chow2009) (black dotted line). 1/7 power sheared opposing current with $Fr=-1$: numerical (green line).

In addition to the linear and square root current profiles discussed above, the other current profiles sketched in figure 11 are also examined for the solitary wave of $A/h=0.1$ on opposing current of $Fr=-0.6$ using the $S2$ model. One can anticipate the $S2$ model developed in YL20, which assume the horizontal velocity varies linearly in the water column, is not able to approximate the current profiles studied here, especially for those current profiles with strong local shear requires (e.g. the fifth-degree polynomial is needed to approximate profiles of no. 1 and no. 6 in figure 11). By using the linearly sheared current as the dividing case, figure 14(a) shows the solitary wave free surface profiles for the group of current profiles whose volume flux is larger than that of the linear profile. It is observed that the linear current profile produces the widest solitary free surface profile followed by the square root shear, 1/3 power shear, and the 1/7 power shear, though the differences among them are not large. The above observations seem to suggest that for this group of opposing currents changes in the solitary wave profile are more significant for currents with stronger surface shear.

Figure 14.

Solitary wave free surface profiles on opposing currents ($Fr=-0.6$) with different vertical profiles.

For the group of current profiles that have less volume flux than the linear profile current, the behaviour of the solitary wave free surface profiles shows different trends. Figure 14(b) shows that both the quadratic profile and cubic profile produce very similar results to those from the linear profile. A closer inspection shows that whereas the quadratic profile induces a slightly wider solitary wave profile than the linear profile, the cubic profile produces a slightly narrower solitary wave surface profile than the linear profile. However, the modification is much more significant for the exponential profile, which induces a narrower profile than the other non-uniform shear profiles in the same figure. Finally, among all the opposing currents of $Fr=-0.6$ considered here, the solitary wave with $A/h=0.1$ has the widest free surface profile when the underlying current has a quadratic profile.

Similarly to § 4.1.1, the velocity and vorticity field of the solitary wave ($A/h=0.1$) on opposing current with exponential profile and $Fr=-0.6$ are shown in figure 15, where the vorticity has been normalized by the maximum vorticity of pure current (in magnitude). A vortex is formed under the wave crest at approximately $z/h=-0.17$. Although the solitary wave propagates to the right, the velocity near the free surface is negative because of the presence of the strong opposing current. For the vorticity field, the strong vorticity at the still water level, associated with an exponential current profile, is lifted up to the free surface by the presence of the solitary wave.

Figure 15.

The velocity field (arrows) and vorticity field (colour) for a solitary wave on opposing current with an exponential profile. Waves propagate to the right and currents move to the left.

Finally, figure 16 shows the wave celerity ($C$) normalized by shallow-water wave speed ($c_0$) of solitary wave ($A/h=0.1$) on currents ($Fr=-0.6$) with different vertical profiles, which are also summarized in table 1. It is not surprising that solitary waves travel at a slower speed on opposing currents compared with the pure wave scenario. However, because of the vertical shear that decreases the current intensity from the surface to the bottom, the celerity of solitary wave on opposing current is larger than the linear superposition of pure wave celerity ($C/c_0\approx 1.05$) and current surface velocity ($u^c_s/c_0=-0.6$). The solitary wave celerity can vary significantly for currents with the same surface velocity but different vertical profiles. For example, compared with the wave-alone scenario, whereas the current with an exponential profile reduces the wave celerity by 4.65 %, the 1/7 power profile causes a dramatic decrease of 49.8 %. For the group of opposing currents considered here, the wave celerity decreases monotonically with the increasing volume flux of the opposing current. However, this does not imply that the volume flux of the current is the only parameter that determines the wave celerity. The detailed current vertical profile still matters. Thus, additional analysis is required to identify the controlling current parameters that influence certain wave properties, e.g. the influences of currents with the same volume flux but different vertical profiles on solitary waves.

Figure 16.

Wave celerity of solitary wave on currents ($Fr=-0.6$) with different vertical profiles in the water column. The horizontal axis is the volume flux of pure current $F$ normalized by the depth-uniform current flux $F_0$. The symbols from left to right correspond to current profiles 1 to 7 in figure 11.

4.2.1. Wave propagation over a vortex-ring-like current

Vortex-ring-like currents can be generated by the Gulf Stream along eastern coastlines of the United States, which moves warm water from the Gulf of Mexico to the Atlantic Ocean (Mapp et al. Reference Mapp, Welch and Munday1985). The presence of this large-scale current may significantly modify incoming wave field, which are important for predicting both offshore and coastal wave climates. The effects of a vortex-ring-like current on linear shallow-water wave transformation have been studied using various numerical models, e.g. the parabolic approximation of Boussinesq model (Yoon & Liu Reference Yoon and Liu1989), the mild-slope equation model (Chen, Panchang & Demirbilek Reference Chen, Panchang and Demirbilek2005), the coupled-mode model (Belibassakis, Gerostathis & Athanassoulis Reference Belibassakis, Gerostathis and Athanassoulis2011) and the weakly nonlinear and weakly dispersive Boussinesq-type model (Zou et al. Reference Zou, Hu, Fang and Liu2013). In this section, in addition to the classic case of linear shallow-water incident waves, the present models are also used to examine weakly nonlinear intermediate-water-depth waves propagating over a vortex-ring-like current field, which is more realistic to the physical background of the vortex-ring-like current field. Differences in the resulting wave fields are discussed.

Following Mapp et al. (Reference Mapp, Welch and Munday1985), the depth-uniform current field ($U_r$, $U_\theta$, $U_z$) is expressed in the cylindrical coordinates ($r$, $\theta$, $z$) as

(4.1)\begin{gather} U_r=0, \end{gather}

(4.2)\begin{gather}U_\theta=\left\{ \begin{array}{@{}ll} C_5\left(\dfrac{r}{R_1}\right)^N, & r \leq R_1, \\ C_6 \exp{\left[-\left(\dfrac{R_2-r}{R_3}\right)^2\right]}, & r >R_1,\end{array} \right. \end{gather}

(4.3)\begin{gather}U_z=0, \end{gather}

(4.4)\begin{gather}\eta^c={-}\frac{1}{g}\int_r^\infty\frac{U^2_\theta}{r} \,\mathrm{d}r, \end{gather}

where $\eta ^c$ is the free surface set-down associated with the current field, which satisfies exactly the steady-state Euler equation. In (4.2), $R_3$ estimates the length scale of the current field and $C_6$ represents the maximum magnitude of current velocity. Adopting the parameters presented in previous studies (Yoon & Liu Reference Yoon and Liu1989; Zou et al. Reference Zou, Hu, Fang and Liu2013), the following values are used in the present numerical calculations, $N=2$, $R_1=343.706$ m, $R_2=384.881$ m, $R_3=126.830$ m, $C_5=0.9\,{\rm m}\,{\rm s}^{-1}$ and $C_6=1.0\,{\rm m}\,{\rm s}^{-1}$. The shapes of $U_\theta (r)$ and $\eta ^c (r)$ are shown in figure 17, in which the maximum value of $U_\theta$ is located at $R_2 =384.881$ m as denoted by a red dot.

Figure 17.

The variation of $U_\theta$ and $\eta ^c$ expressed in (4.2) and (4.4). Note that the red maker indicates the maximum velocity, $U_\theta = C_6$, located at $r =R_2$.

Two kinds of incident wave conditions are considered herein. The incident wave in the first case has an amplitude of $a_1=0.05$ m and a period of $T_1=19.43$ s, with $a_2=0.5$ m and $T_2=7.3$ s for the second case. The still water depth is fixed at $h=10$ m, yielding the following dimensionless parameters characterizing these two incident wave conditions: ($k_1h=0.33$, $k_1a_1=0.0017$) and ($k_2h=1.0$, $k_2a_2=0.05$). Thus, the first case represents a linear shallow-water wave and the second case is a second-order Stokes wave in intermediate water depth. The relative current strength ($C_6/\sqrt {gh}$) is fixed at 0.1. Accordingly, the wave ray patterns can be calculated from the linear wave ray theory (Arthur Reference Arthur1950; Kenyon Reference Kenyon1971) as shown in figure 18(a) for both the shallow-water wave (red lines) and the intermediate-water wave (blue lines) scenarios. In the figure, the vortex-ring-like current field, circulating in the clockwise direction, is symbolically represented by two concentric circles. The centre of the current field is located at $(x_0, y_0)=(1.0,0.0)$ km. The two concentric circles have a radius of $R_2$ and $R_3$, which correspond to the maximum current velocity of $C_6$ and the velocity of $C_6/e$, respectively. Without the current field, wave rays are straight lines parallel to the $x$-axis. In the presence of the current field, wave rays are refracted and wave ray crossings occur in a complex manner, suggesting that the linear wave ray theory is no longer valid and the wave diffraction needs to be considered.

Figure 18.

(a) The wave ray patterns over a vortex-ring-like current. Red lines are wave rays for shallow-water wave ($k_1h=0.33$); blue lines are for intermediate-water-depth wave ($k_2h=1.0$). (b) Contour plots of instantaneous free surface elevations at $t=28T_1$ for the shallow-water incident wave case.

As $kh$ values are relatively small, the present $S2$ model is used to simulate both scenarios. The lengths of numerical simulations are $28T_1$ for shallow-water-depth and $75T_2$ for intermediate-water-depth cases, respectively. The spatial and temporal resolutions are $\Delta x=\Delta y=L/18$ and $\Delta t=T/20$ for both cases, where $T$ and $L$ are the period and wavelength of the incident waves, respectively. Sponge layers are applied downstream for absorbing outgoing waves. The length of the computational domain in the $y$-direction is set to be sufficiently long enough to guarantee that the wall boundary conditions implemented at north and south boundaries do not influence solutions in the area of interest.

In figure 18(b), the contour lines of instantaneous free surface elevations at the end of the numerical simulation ($t=28T_1$) for the shallow-water incident wave scenario are plotted. The parallel contour lines of incident plane waves are significantly distorted by the presence of the vortex-ring-like current through wave refraction and diffraction. Referring to the wave ray patterns shown in the left panel, many surface undulations along the crest lines can be identified with the wave ray crossings.

In figure 19, spatial distributions of normalized wave heights for the shallow-water-depth ($k_1h=0.33$) case and the intermediate-water-depth ($k_2h=1.0$) case are displayed. Here, the wave heights are calculated by performing the wave-by-wave zero-crossing analysis on the last five simulated waves. Note that the wavelength of the waves in intermediate water depth is one-third of that of the shallow-water wave and the incident wave amplitude of the intermediate-water-depth case is 10 times larger than that of the shallow-water wave, resulting in stronger nonlinearity for the intermediate-water-depth case. Thus, the amplification of wave height by opposing current is more obvious in the intermediate-water-depth case; the maximum wave height is located at $(x,y)=(1.3\,{\rm km},\ 0.4\,{\rm km})$, which is behind the maximum opposing current (the inner circle). Moreover, the vortex-ring-like current has stronger refraction effects on shorter waves, resulting in different wave patterns for these two cases, especially on the lee side of the current.

Figure 19.

(a) Spatial distribution of the normalized wave heights for the case of shallow-water-depth wave ($k_1h=0.33$). (b) Spatial distribution of the normalized wave heights for the case of intermediate-water-depth wave ($k_2h=1.0$).

The normalized wave heights along six transects ($x=$ constants) for the case of shallow-water incident wave ($k_1h=0.33$) are shown in the left panels of figure 20. The results from a Boussinesq-type model (Zou et al. Reference Zou, Hu, Fang and Liu2013) are also shown (in black dots) in the same panel. Generally speaking, two models have produced similar results with slight discrepancies in the region of $y>0$, where waves follow the currents. Lastly, the wave height variations along the same transects for the case of intermediate water depth ($k_2h=1.0$) are also shown in the right panels of figure 20. At $x=1.2594$ km the maximum wave height is almost three times the incident wave height because of the enhanced nonlinearity. On the other hand, the diffracted waves behind the vortex-ring-like current show much stronger spatial variations for their shorter wavelengths.

Figure 20.

Normalized wave heights along six transects ($x=$ constants). Here $H_0$ is the incident wave height. (a,c,e,g) Comparisons between the present $S2$ model (red lines) and the Boussinesq model (Zou et al. Reference Zou, Hu, Fang and Liu2013) (black dots) for the shallow-water-depth case ($k_1h=0.33$). (b,d,f,h) The present $S2$ model results for the intermediate-water-depth case ($k_2h=1.0$).

For both cases, the steady current velocities from the present model are obtained by time-averaging the total horizontal velocities over the last five wave periods in the numerical simulations. The comparisons between the model results and the prescribed vortex-ring-like current (4.1)–(4.4) are made at several locations under the wave trough, showing almost identical results. For completeness, the comparisons for the intermediate-wave case are shown in the supplementary materials. The time-averaged velocities between the wave trough and wave crest can also be calculated from the present model solutions, which show very different patterns from the prescribed currents. To demonstrate this feature, in figure 21 the present model results for the time-averaged velocity at the elevation of $z/h=-0.02$ are compared with the prescribed current velocity along four transects $y = 0.4$ km, 0.3 km, $-0.3$ km and $-0.4$ km. The elevation $z/h=-0.02$ is above the wave trough but under the current-induced mean free surfaces. The time-averaged resulting velocities are generally weaker than the prescribed current because of the periodically fluctuating free surfaces induced by waves.

Figure 21.

Comparisons between the time-averaged resulting current field (red line) and the prescribed current field (black line) at four transects for $x$ component velocity, $u$ (a,c,e,g) and $y$ component velocity, $v$ (b,d,f,h) at $z/h=-0.02$.

4.2.2. Obliquely incident wave propagation over a 3D sheared current

Whereas the examples discussed in the previous sections all have a constant depth, in this section, the present models are employed to simulate wave propagation and transformation over a varying bathymetry and a current field that shears both horizontally and vertically.

The plan view of the numerical setup is sketched in figure 22(a). A numerical wavemaker is placed at $x=0$, generating obliquely incident waves with an incident angle of $15^\circ$ with respect to the $x$-axis, an amplitude of 0.016 m and a period of 1.2 s in a still water depth of $d= 0.35$ m. The corresponding dimensionless wave parameters are $kd=1.18$ and $ka=0.054$. A submerged ridge parallel to $y$-axis is installed inside the computational domain. The submerged ridge has the same fore-slope and back-slope of $1:20$ and the water depth on the crest of the ridge is 0.1 m with the corresponding ridge crest width of 1.5 m (see figure 22c). Sponge layers of 2.5 m wide are implemented on the western and eastern ($x$-direction) sides of the computational domain for absorbing outgoing waves, whereas wider sponge layers (8 m) are used on the northern and southern ($y$-direction) sides. Finally, the length of the computational domain in the $y$-direction is set to be $60$ m, which is verified to be sufficiently long to minimize influences from the sponge-layer boundaries to the central area of interest.

Figure 22.

(a) The plane view of the computational domain, where the dashed areas are the sponge layers, the thick black line is the location of the wavemaker and the coloured contours indicate the current region. (b) Vertical profiles of depth-uniform current (blue line) and vertically sheared current (red line) on a flat bottom of still water depth. (c) Bottom bathymetry in the $x$-direction.

A prescribed steady-state current field is installed inside the computational domain. The current field is unidirectional in the $y$-direction and has a Gaussian distribution in the $x$-direction, which is similar to the ‘steady shear current’ pattern proposed by Smith (Reference Smith2006). However, the present current field can have different velocity profiles in the water column. The current velocity components and the free surface elevation can be expressed as follows

(4.5a–c)\begin{gather} u^c = 0, \quad w^c = 0, \quad \eta^c = 0, \end{gather}

(4.6a,b)\begin{gather}v^c = v^c_s g(z)=Fr\sqrt{gd} \exp{[-((x-x_0)/r)^2]}g(z), \quad -h(x)\leq z \leq 0, \end{gather}

where $v^c_s$ is the $y$-component of the surface current velocity, and it has a Gaussian shape in the $x$-direction, being centred at $x_0=10$ m, which is aligned with the centre line of the ridge and has a length scale of $r=2$ m. The location of the current field is indicated by the contour lines in figure 22(a). The maximum intensity of the current is determined by the Froude number ($Fr$) and the velocity profile in the water column is defined by $g(z)$, where $-h \leq z \leq 0$. The current flows in the positive $y$-direction. Note that the prescribed current field over the submerged ridge satisfies the steady-state Euler equations.

In the present simulations, $Fr=0.25$ is used so that the current is in a weak current regime with a length scale comparable to the incident wavelength, which is essentially not slowing varying. Three different scenarios are investigated: (1) pure wave propagation (no current); (2) waves on depth-uniform currents ($g(z)=1$); and (3) waves on vertically sheared currents with $[g(z)=1.1(z/h)^2+2.1(z/h)+1, -h\leq z\leq 0]$. The vertical profile of the sheared current on a flat bottom is shown by the red line in figure 22(b). Note that the current profile in the water column does not change in the $y$-direction. However, the profile changes at different $x$-cross-section because the water depth changes.

Numerical simulations are conducted using the $S2$ model with spatial and temporal resolutions of $\Delta x=0.04$ m, $\Delta y=0.2$ m and $\Delta t=0.02$ s, and they are carried out for 25 wave periods, which is sufficiently long for reaching a quasi-steady state. Figure 23(a,b) show the snapshots of normalized (by incident wave amplitude) instantaneous free surface elevations for the pure wave case and depth-uniform current case in the area of interest ($4\,{\rm m} < x< 14\,{\rm m}$, $-6\,{\rm m}< y< 6\,{\rm m}$) at the end of the simulation, respectively. In figure 23(a), evidently, a train of uniform obliquely incident waves has been generated before the toe of the submerged ridge ($x=4.25$ m). Waves are refracted by the submerged ridge and shorter waves with larger wave amplitudes can be observed on the front face and on top of the ridge (around $x= 10$ m) because of shoaling effects. Compared with the pure wave case, the wavelengths for the depth-uniform current case are longer, which can be observed in figure 23(b). The differences of instantaneous free surface elevations between the pure wave case and the uniform current case at the same instant are also shown in figure 23(c). The differences mainly appear in the region $x>8.5$ m, which is primarily caused by the higher wave celerity in the case of wave–current interactions.

Figure 23.

The normalized instantaneous free surface elevations at the end of the numerical simulation for (a) the pure wave case and (b) depth-uniform current case, and (c) their differences.

As both waves and currents are uniform in $y$-direction, we focus our analysis only on the transect along $y=0$ m. The spatial variation of the prescribed depth-uniform current and vertically sheared current velocity at two water depths ($z=-0.01$ m and $z=-0.05$ m) are shown in figure 24(a,b) by lines in black, respectively. In the same figure, the corresponding numerical results from the present models are also shown for comparison. The numerical results are obtained by time-averaging the horizontal velocities of the last five waves in the numerical simulation (at the same transect and vertical elevation). For the case of uniform velocity profile, the differences between the prescribed current velocity and the model results are only noticeable at the water depth of $z=-0.01$ m, and the wave effects on the current field below the wave trough are very small. For the vertically sheared current case, whereas there is a significant difference at the water depth of $z= -0.01$ m, small differences can also be observed for $z= -0.05$ m, which may be attributed to the relatively abrupt changes of the bottom.

Figure 24.

(a) The spatial variation of the time-averaged velocities along $y=0$ m at $z=-0.01$ m (blue dash-dotted line) and $z=-0.05$ m (blue line) for uniform current case. The prescribed current at these two water depths are shown by lines in black. (b) The spatial variation of the time-averaged velocities along $y=0$ m at $z=-0.01$ m (red dash-dotted line) and $z=-0.05$ m (red line) for vertically sheared current case. The prescribed current at these two water depths are shown by lines in black. (c) The snapshot of free surface elevations at the end of simulations for pure wave (black), waves on depth-uniform current (blue), and waves on vertically sheared current (red) at $y=0$ m. (d) The spatial variation of wave heights for pure wave (black), waves on depth-uniform current (blue), and waves on vertically sheared current (red) at $y=0$ m. Here $\eta _0$ and $H_0$ are the incident wave amplitude and height, respectively.

Snapshots of the normalized instantaneous free surface elevations for all three cases are also shown in figure 24(c). The differences of free surface elevations in these three cases become significant starting from $x> 9.0$ m, which are caused by the combined effects of bathymetric variation and sheared currents. Both current fields make wave celerity faster and the depth-uniform current has more significant effects. Waves also become higher and higher harmonic waves are generated by shoaling effects, which can be clearly seen on top of the ridge. Wave heights are also calculated from the last five waves of each simulation and their spatial distributions are plotted in figure 24(d). On the front slope of the ridge, the shoaling effects tend to amplify the wave height, the following current act in an opposite way, which reduces the wave height. Compared with wave-alone scenario, the wave heights on the front slope of the ridge can be $5.5\,\%$ and $2.1\,\%$ smaller for depth-uniform current and vertically sheared current, respectively. On top of the ridge, the wave heights for the vertical sheared current case is even larger than the wave-alone scenario, followed by the depth-uniform current case. However, the effects of the vertical shear is less obvious on the back-slope of the ridge, where the current magnitude decreases, and the wave heights of wave–current interaction cases are approximately 5–$10\,\%$ larger than that of wave-alone situation. It is also interesting to note that the nonlinear features at downstream are also more significant for cases with currents, which can be observed from both figures 24(c) and 24(d).