2.1. Rays and their unit tangent vector

Wave rays are time dependent and described as position vectors in the horizontal plane, denoted by $\boldsymbol{r} = [x_r(t),y_r(t)]$ . Here, the subscript ‘r’ is used to distinguish it from the position vector ${\boldsymbol{x}}=(x,y)$ in the fixed Cartesian coordinate system. According to the definition (Mei et al. Reference Mei, Stiassnie and Yue2005,their § 3.6)

(2.4a,b) \begin{equation} \dfrac {\textrm{d} \boldsymbol{r}}{\textrm{d} t} = \boldsymbol{\nabla} _k\omega (\boldsymbol{k},{\boldsymbol{x}}) \equiv {\boldsymbol{U}} + \boldsymbol{c}_{g} \quad \to \quad \dfrac {\textrm{d} y_r}{\textrm{d} x_r} = \dfrac {\dot {y}_r}{\dot {x}_r}, \end{equation}

where the dot denotes the derivative with respect to $t$ . Here, (2.4a ) denotes the absolute group velocity vector and (2.4b ) leads to the general expression for the local slope of the rays which will be used for the ray curvature presented in § 2.2. The definition of the rays has the physical meaning of wave group trajectories, i.e. the rays are locally parallel to the absolute group velocity everywhere at all times, and denote the direction of wave action propagation (Whitham Reference Whitham1965).

The material derivative of the wave vector $\textrm{d} \boldsymbol{k}/\textrm{d} t$ is obtained by the substitution of (2.2) into (2.1c ), giving rise to

(2.5) \begin{equation} \partial _t\boldsymbol{k} + (\boldsymbol{\nabla} _k\omega \boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{k} + \boldsymbol{\nabla }\omega = 0, \end{equation}

where $\omega =\omega (\boldsymbol{k}({\boldsymbol{x}},t),{\boldsymbol{x}})$ was used. Hence, the material derivative of the wave vector is defined as

(2.6a,b) \begin{equation} \dfrac {\textrm{d} \boldsymbol{k}}{\textrm{d} t} \equiv \partial _t\boldsymbol{k} + (\boldsymbol{\nabla} _k\omega \boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{k} \textrm{, and thus}\ \dfrac {\textrm{d} \boldsymbol{k}}{\textrm{d} t} = - \boldsymbol{\nabla }\omega (\boldsymbol{k},{\boldsymbol{x}}), \end{equation}

according to (2.5). A different but equivalent definition for $\textrm{d} \boldsymbol{k}/\textrm{d} t$ was used in Landau & Lifshitz (Reference Landau and Lifshitz1987), which would also lead to (2.6b ).

The unit tangent vector of the rays $\boldsymbol{t}$ , is essential to the explicit expression of the ray curvature, which can be obtained by noting that $(\textrm{d} \boldsymbol{r}/\textrm{d} t) \mathbin {\!/\mkern -5mu/\!} \boldsymbol{t}$ and $|\boldsymbol{t}|=1$ , showing that the following identities hold:

(2.7a,b) \begin{equation} \boldsymbol{t} = \dfrac {{\boldsymbol{U}}+\boldsymbol{c}_{g}}{|{\boldsymbol{U}}+\boldsymbol{c}_{g}|} \quad\textrm{ and }\quad \boldsymbol{t} = \left [ \dfrac {\dot {x}_r}{\dot {s}}, \quad \dfrac {\dot {y}_r}{\dot {s}} \right]\!, \end{equation}

where $\dot {s}=|\dot {\boldsymbol{r}}| \equiv \sqrt {(\dot {x}_r)^2 + (\dot {y}_r)^2}$ , and $s(t)$ denotes the arc length. Here, both the expressions will be used for the derivation of the ray curvature. The assumption of weak current compared with the group velocity of the waves is translated to the definition of the dimensionless current velocity as follows

(2.8) \begin{equation} {\boldsymbol{u}}=[u,v] ={{\boldsymbol{U}}}/{c_{g}}, \end{equation}

such that ${\mathcal{O}}(|{\boldsymbol{u}}|)\sim \varepsilon$ , with $c_{g}=|\boldsymbol{c}_{g}|$ , $u$ and $v$ representing the component of $\boldsymbol{u}$ in the $x$ - and $y$ -direction, respectively, and $\varepsilon$ denoting a small dimensionless scaling parameter. Thus, the unit tangent vector given by (2.7a ) can be expressed as

(2.9) \begin{align} \boldsymbol{t} = \dfrac {{\boldsymbol{u}}+{\boldsymbol{e}}_k}{|{\boldsymbol{u}}+{\boldsymbol{e}}_k|} = {\boldsymbol{e}}_k + {\boldsymbol{u}} -({\boldsymbol{u}}\boldsymbol{\cdot }{\boldsymbol{e}}_k){\boldsymbol{e}}_k + {\mathcal{O}}(\varepsilon ^2), \end{align}

where ${\boldsymbol{e}}_k=\boldsymbol{k}/k$ denotes the unit vector in the same direction as the local wave vector. The approximation to the unit tangent vector given by (2.9) is identical to Dysthe (Reference Dysthe2001, expression (3)) for the limiting cases of deep water waves.

2.2. The ray curvature

In differential geometry, the curvature $\kappa$ is defined as the change in the tangent vector $\boldsymbol{t}$ , along the arc length

(2.10) \begin{equation} \kappa \boldsymbol{n} = \frac {\textrm d\boldsymbol{t}}{{\textrm d}s}, \end{equation}

where $\boldsymbol{n}$ is the ray normal vector, and thus the identity $\boldsymbol{t} \perp \boldsymbol{n}$ is admitted such that $\boldsymbol{n} = [-t_y, t_x]$ is defined. The expression (2.7b ) can therefore be re-written in a form as follows:

(2.11) \begin{equation} \dot {\boldsymbol{r}} = \dot {s}\boldsymbol{t}. \end{equation}

On the insertion of (2.10) into (2.11), the curvature $\kappa$ , of rays can be expressed in a parametric form as follows (e.g. Mathiesen Reference Mathiesen1987):

(2.12) \begin{align} \kappa = \dfrac {\dot {x}_r\ddot {y}_r - \dot {y}_r\ddot {x}_r } {\left [ (\dot {x}_r)^2 + (\dot {y}_r)^2 \right ]^{3/2}} \equiv \boldsymbol{t} \boldsymbol{\cdot }\dfrac {[\ddot {y}_r, - \ddot {x}_r ] }{(\dot {x}_r)^2 + (\dot {y}_r)^2} , \end{align}

where the double dots denotes the second-order derivative with respect to time. Here, we opted for the signed version of the curvature, in order to be compliant with previous works (e.g. Kenyon Reference Kenyon1971; Gallet & Young Reference Gallet and Young2014; Bôas & Young Reference Bôas and Young2020). It is now clearly seen in (2.12) that the curvature relies on an explicit expression for both the unit tangent vector of the rays and the second derivative of the ray trajectories with respect to time. The latter can be obtained by definition

(2.13a,b) \begin{equation} \dfrac {\textrm{d}^2 \boldsymbol{r}}{\textrm{d} t^2} = [\ddot {x}_r, \ddot {y}_r] \textrm{ or } \dfrac {\textrm{d}^2 \boldsymbol{r}}{\textrm{d} t^2} =\, \dfrac {\textrm{d}}{\textrm{d} t} ({\boldsymbol{U}} + \boldsymbol{c}_{g}), \end{equation}

the latter of which should be evaluated on the time-dependent rays, i.e. ${\boldsymbol{x}}=\boldsymbol{r}(t)$ should be noted for (2.13b ). Thereby, we arrive at

(2.14) \begin{align} \dfrac {\textrm{d}^2 \boldsymbol{r}}{\textrm{d} t^2} =\, \partial _t{\boldsymbol{U}} + \left(\dfrac {\textrm{d} {\boldsymbol{x}}}{\textrm{d} t}\boldsymbol{\cdot }\boldsymbol{\nabla}\right){\boldsymbol{U}} + \left (\dfrac {\textrm{d} {\boldsymbol{x}}}{\textrm{d} t}\boldsymbol{\cdot }\boldsymbol{\nabla }\right )\boldsymbol{c}_{g}(\boldsymbol{k},{\boldsymbol{x}}) + \left (\dfrac {\textrm{d} \boldsymbol{k}}{\textrm{d} t}\boldsymbol{\cdot }\boldsymbol{\nabla} _k\right )\boldsymbol{c}_{g}(\boldsymbol{k},{\boldsymbol{x}}), \end{align}

for ${\boldsymbol{x}}=\boldsymbol{r}(t)$ , where $\textrm{d}/\textrm{d} t$ denotes the material derivative with respective to time. Expression (2.14) can be simplified by applying the quasi-stationary background current assumption $\partial _t{\boldsymbol{U}}\simeq \boldsymbol{0}$ , meaning that the background current $\boldsymbol{U}$ is slowly varying in time compared with the rapidly varying phase and rays. The terms on the right-hand side of (2.14) can be otherwise readily evaluated by also applying (2.6b ).

Inserting (2.4a , b ), (2.9), (2.14) and (2.6b ) into (2.12) and keeping the terms to ${\mathcal{O}}(\varepsilon )$ eventually gives rise to the explicit expression of the ray curvature as follows:

(2.15) \begin{align} \kappa _\approx = \dfrac {\partial _xU{_2}-\partial _yU{_1}}{c_{g}} + \dfrac {{\boldsymbol{e}}_{k,\perp } \boldsymbol{\cdot }\boldsymbol{\nabla }c}{c_{g}}. \end{align}

Here, the assumption of $\partial _t{\boldsymbol{U}}\simeq \boldsymbol{0}$ was used, and the subscript ‘ $\approx$ ’ denotes an approximation to the curvature according to (2.12). We recall that the local curvature of the rays given by (2.15) was derived under the weak current and geometric optics approximations; the unit vector ${\boldsymbol{e}}_{k,\perp } = [-k_y,k_x]/k$ was defined, which obeys ${\boldsymbol{e}}_{k}\boldsymbol{\cdot }{\boldsymbol{e}}_{k,\perp }=0$ , suggesting that it is always orthogonal to the local wave vectors. We remark that the identities $\boldsymbol{\nabla }c=\boldsymbol{\nabla }h\partial _hc$ and $c(k,{\boldsymbol{x}})=\sqrt {g\tanh kh({\boldsymbol{x}})/k}$ are admitted in (2.15) where the terms are all evaluated on the time-dependent rays: ${\boldsymbol{x}}=\boldsymbol{r}(t)$ .

For deep water waves which admit $\boldsymbol{\nabla }c=\boldsymbol{0}$ , the local curvature of the rays by (2.15) recovers the expressions by Kenyon (Reference Kenyon1971) and Dysthe (Reference Dysthe2001) and, for the cases in the absence of current, it recovers Arthur et al. (Reference Arthur, Munk and Isaacs1952, their expression (1c)). We also note that the current-gradient term in (2.15) can be expressed by the vertical component of the vorticity vector $\zeta = \partial _xU{_2} - \partial _yU{_1}$ .

It shall be noted that the approximate curvature (2.15) can also be recovered from the exact form by Phillips (Reference Phillips1977, their (3.5.6)) using the same assumptions associated with the current and bathymetry. Compared with this exact but more complex form, the leading-order approximation $\kappa _\approx$ enables a direct evaluation of the individual effects of current- or bathymetry-gradient on the wave refraction.

3.1. Model validation

To validate the approximate curvature $\kappa _\approx$ denoted by (2.15), we compare its numerical predictions with those based on the direct numerical implementation of the definition according to (2.12). With the ray paths, the curvature based on (2.15) can be directly evaluated on the wave rays ${\boldsymbol{x}} = \boldsymbol{r}(t)$ for the time instants $t = n\Delta t$ with $n\in [0,1,2,\ldots ,N_t-1]$ , where $\Delta t$ and $N_t$ denote the time interval and total number of time steps used for the numerical results, and $T=\Delta t (N_t-1)$ . Similarly, the curvature based on (2.12) was readily evaluated applying a second-order accurate central difference scheme on $\boldsymbol{r}(t)$ .

Figure 1. Accuracy in (2.15) for a deep water $6$ s period wave on a shearing current starting at point ${\boldsymbol{x}}_r(0) =$ (0, 0). Panel (a) show the analytical ray path (black solid line) and modelled (orange dots) from numerical integration of the ray equations (2.4) and (2.6). The ray paths are normalised with the initial curvature $\kappa _0$ . Indeed, the shape of the normalised ray paths are independent of the shear magnitude and wave period. Panel (b) demonstrates the temporal evolution in percentage difference between $\kappa$ and $\kappa _\approx$ for the ray in panel (a). Vertical lines denote different values of $\varepsilon = U_1(y)/c_{g}$ .

We follow Kenyon (Reference Kenyon1971) as analytical solutions exist for both the ray paths and the material derivative of the wave vector expressed as (2.4) and (2.6), respectively (Longuet-Higgins & Stewart Reference Longuet-Higgins and Stewart1961). In particular, we consider a deep water wave train on a constant shear current

(3.1) \begin{equation} {\boldsymbol{U}}=[\tau y,0], \end{equation}

for $y\gt 0$ , where $\tau$ is the velocity shear. Inserting the current profile described by (3.1) into the approximate curvature leads to $\kappa _\approx = |\tau /c_{g}|$ , where it has been shown by Kenyon (Reference Kenyon1971) that this approximation leads to a difference with (2.12) by approximately 10 % for $\varepsilon \sim$ 0.05.

The finding reported by Kenyon (Reference Kenyon1971) is consistent with the comparison shown in figure 1. Here, the temporal evolution of wave rays with initial position ${\boldsymbol{x}}_r(0)=$ (0, 0) and initial propagation direction $\theta _0 = \pi /2$ were used, where ${\boldsymbol{e}}_k\perp {\boldsymbol{u}}$ is implied. Here, and throughout the text, subscript ‘ $0$ ’ refers to initial conditions, i.e. $t=0$ . Such rays admit the analytical form $x_r(t) = -y_r(t)^2 \kappa _0 /2$ (Kenyon Reference Kenyon1971), where $\kappa _0 = -2 \tau \varOmega _0 / \mathrm{g}$ denotes the initial curvature, and $\varOmega _0$ the initial intrinsic frequency. Here, the initial frequency corresponded to a 6 s period wave. Figure 1(a) demonstrates a good agreement between the ray path predictions by the ray tracing solver and the analytical approximation. As expected, the relative difference between $\kappa _\approx$ and $\kappa$ , being defined as $|(\kappa _\approx -\kappa )/\kappa |$ , grows in time due to the constant shear $\tau$ and the increase in current speed with $y$ (figure 1 b); it reaches approximately 6 % for $\varepsilon =0.2$ , which well falls within the regime as reported by Kenyon (Reference Kenyon1971).

3.2. Wave refraction by both current and bathymetry

Our derivations in § 2 have demonstrated that the presence of both current and a varying bathymetry can lead to complex interplay, and thereby play an important role in wave refraction. Roles due to current and a bathymetry can in particular be quantified in a separate manner by noting that the approximate curvature $\kappa _\approx$ expressed as (2.15) has the form of linear superposition of current- and bathymetry-altered curvature components denoted by $\kappa _c$ and $\kappa _d$ , respectively, where

(3.2a,b,c) \begin{equation} \kappa _c = \dfrac {\partial _xU_{2}-\partial _yU_{1}}{c_{g}} \quad \textrm{and}\quad \kappa _d = \dfrac {{\boldsymbol{e}}_{k,\perp } \boldsymbol{\cdot }\boldsymbol{\nabla }c}{c_{g}} \textrm{ such that } \kappa _\approx = \kappa _c+\kappa _d. \end{equation}

Here, (3.2c ) permits us to quantify the individual and combined contributions of current and depth to the wave refraction, as noted, where both the sign and magnitude of $\kappa _c$ and $\kappa _d$ determine the ultimate effect on the wave refraction. We will use a few limiting cases in the following subsections to explicitly explore the underlying physics. These cases include wave trapping – corresponding to a wave ray which cannot escape – and zero curvature when $\kappa _c=-\kappa _d$ . In particular, the wave trapping is typically manifested as total internal reflection, like waves on an opposing current jet or atop an elongated submarine shallow, or as a complete absorption, like waves propagating against a beach.

3.2.1. Wave trapping on jet-like currents and complex bathymetry

To demonstrate the wave trapping phenomena due to the joint influence of current and depth on the wave refraction, we use the bathymetry profile and jet-like current, being expressed as, respectively,

(3.3a,b) \begin{equation} h(x,y) = \frac {H}{2} \left (1 + \alpha \sin {\left (\pi \frac {x}{L_x} \right )} \cos {\left (\pi \frac {y}{L_y} \right )} \right ) \!\!\quad \textrm{and}\quad \!\! \boldsymbol{U} = \left[U_{*} \cos ^4{\left ( \pi \frac {y}{L_y} - \frac {\pi }{2}\right )} , 0 \right]\!. \end{equation}

Here, $L_x$ and $L_y$ are the characteristic length in the $x$ - and $y$ -directions, respectively; $H$ denotes the characteristic length in the depth direction; $\alpha \in [0,1]$ denotes the degree of varying bathymetry, leading to the measure of the bathymetry variation in the $x$ - and $y$ -directions being $\max |\boldsymbol{\nabla }h/h|$ . When ${\mathcal{O}} (\max |\boldsymbol{\nabla }h/h| / k) \ll 1$ it corresponds to a slowly varying bathymetry in the horizontal plane, as required in the assumption for the approximate curvature $\kappa _\approx$ . Likewise, $U_{*}$ denotes a characteristic current magnitude in the $x$ -direction. The corresponding slowly varying current assumption is fulfilled for ${\mathcal{O}} (\max |\boldsymbol{\nabla }\boldsymbol{U}/U| / k) \ll 1$ , where $U = |\boldsymbol{U}|$ (Peregrine Reference Peregrine1976). Here, and for the subsequent analysis, $L_x=20$ km, $L_y=10$ km and a wave period of 12 s has been used, unless otherwise stated. The remaining parameters chosen for the numerical predictions are listed in table 1.

Table 1. Values for the ambient conditions including VDs, DW, PC and NC current profiles used in the numerical ray tracing simulations in figures 2–4. Subscript ‘ $0$ ’ refers to initial conditions.

Figure 2. The influence by the variable depth (VD table 1) on the wave propagation. Panel (a) show a wave ray with initial period of $12$ s propagating from left to right, with initial propagation direction parallel to the $x$ -axis. The bathymetry-altered curvature $\kappa _d$ , normalised by its maximum value $\kappa _{d,m}=\max |\kappa _{d}|$ , is shown in panel (b). The shallowest region in the VD bathymetry simulates a seamount and is shown in panel (c), where the water depth $h$ is given by the background colour. White dashed line in panel a denote the $h=\lambda /2$ contour.

Figure 2 shows the role of depth-induced refraction on wave propagation, using a variable depth (VD) obtained by (3.3a ) by letting $\alpha =0.95$ . For this profile, $h\in [3.75,146.25]$ m and the shallowest region simulates a seamount. As expected, waves are refracted according to the water-depth gradients, and thus bend towards the shallower regions. The seamount attracts and traps a large portion of the wave rays which are initially located at $y\gt 0.5L_y$ . The bathymetry-altered curvature $\kappa _d$ obtains its largest magnitude for the rays located in the vicinity of the local seamount, but having slightly passed it (see figure 2 b around $(0.6L_x,0.8L_y)$ ).

Figure 3. Refraction of deep water (DW) 12 s period waves atop a negatively (NC) and positively oriented current (PC) are shown in panels (a) and (b), respectively. The waves initially propagate from left to right. Inset figures denote the current profiles and their orientation, with further details given in table 1. Panels (c) and (d) show the associated current-altered curvature $\kappa _c$ along each wave ray, which is normalised by its maximum value $\kappa _{c,m}=\max |\kappa _{c}|$ .

Refraction solely due to currents are shown by simulations in DW conditions, which include a positively (PC) and negatively oriented current jet (NC). The PC induces caustics at the edges of the jet, while the NC induces caustics at the centre of the jet (figure 3 a,b). If the domain for the NC case were extended in the $x$ -direction, we would have seen the characteristic wave trapping phenomenon at the centre of the jet (Peregrine Reference Peregrine1976). In this example, the current-induced curvature $\kappa _c$ is given by

(3.4) \begin{equation} \kappa _c = \dfrac {-2U_{*} \pi }{L_y c_{g}} \cos ^2\left ( \pi \frac {y}{L_y} - \frac {\pi }{2}\right )\sin \left ( \pi \frac {2y}{L_y} - \pi \right)\!. \end{equation}

Here, the sign of $\kappa _c$ is mirrored about the line $y=0.5L_y$ for the two cases (figure 3 c,d).

Figure 4. The joint influence by VD and jet-like currents (NC, PC) on wave propagation. Initial wave periods are 12 s. Upper panels show the difference in propagation between depth-only (VD, orange), current-only (yellow) and their joint influence (red), when starting at the same initial position; dashed and solid lines in panel b denote different initial positions. Lower panels show the refraction for several wave rays on the negative current (panel c) and positive current (panel d). The colour shading denote $kh/\pi$ , where yellow colour denote $kh/\pi \geqslant 1$ .

The joint influence by the bathymetry and current yield more complicated situations (figure 4). Combining the varying depth and negative current (VD + NC) causes two caustics; one is located at the local shallow and the other takes the form of a meandering tube steered by the bathymetry (figure 4 c). Conversely, when the current is positive, there is a stronger refraction against the local shallow since the following current and bathymetry work together (VD + PC, figure 4 d); this leads to a corresponding decrease of wave rays in the region where $x\gt 0.5L_X$ and $y\gt 0.5L_y$ , when compared with the VD-only case (see figure 2 a). However, when combining the variable currents and bathymetry, we do recognise the predominant wave ray propagation patterns from the isolated cases; the refraction against the local shallow (figure 2 a); the caustic at the centre of the opposing jet (figure 3 a); and the diverging wave ray pattern at the centre of the following jet (figure 3 b).

The combined depth- and current-induced refraction is further highlighted by comparing the ray paths in different cases. Figure 4(a,b) shows the ray paths which initiate at the same position of $(0,0.6L_y)$ but are affected by either the PC, NC, VD or the combined effects of a current and VD. We see that the effect by the NC is to extend the distance where the ray holds a substantial $k_x$ component compared with the VD solution. As a consequence, the wave ray exits the domain further away from the local shallow, which is illustrated by the red line of figure 4(a). The opposite is true for the PC case, where the current gradients refract the wave ray against the VD faster than the bathymetry alone, as is shown by the red dashed line in figure 4(b). Thus, the joint effect of the PC and VD is stronger than their individual contribution.

We also depict in figure 4(b) the rays with an initial position of $(0,0.44L_y)$ . Here, the gradients of the positive current PC almost resemble the gradient of VD, but with different signs, leading to a wave ray with a small curvature, as is illustrated by the solid red line in figure 4(b). Such a case will be investigated in more detail in the subsequent subsection.

We introduce the ratio

(3.5) \begin{equation} \gamma = {\kappa _d^2 }/{\big(\kappa _d^2 + \kappa _c^2\big)}, \end{equation}

such that $\gamma \in [0,1]$ , which is referred to as the refraction assessment metric because, particularly when $\kappa _c$ and $\kappa _d$ hold the same sign, it can be used to measure which of the two plays a more dominant role in the wave refraction. Specifically, $\kappa _d$ and $\kappa _c$ dominates for $\gamma \gt 0.5$ and $\gamma \lt 0.5$ , respectively.

Figure 5. The ray curvature ratio $\gamma$ computed locally along wave rays. Rows from top to bottom show ray tracing simulations for waves with initial periods of 14, 12, 10 and 8 s, respectively. Columns from left to right denote different initial propagation directions $\theta _0$ , as indicated by the arrows in the lower row plots. All model simulations have the conditions VD + NC (table 1).

The metric (3.5) is shown in figure 5 for an ensemble of ray tracing simulations, including four different initial wave periods and three initial directions, while propagating atop VD + NC. Besides mapping when and where depth- and current-induced refractions dominate, the results highlight some important physical aspects concerning the two refraction mechanisms. Firstly, figure 5 demonstrates that the horizontal extent of the bathymetry dominated refraction $\kappa _d$ increases with increasing wave period. This relation is due to the two following reasons: (i) $kh$ increases with wave period meaning that longer waves feel more the depth change than shorter ones (Kenyon Reference Kenyon1983), and thus scatter accordingly; (ii) both terms in $\kappa _\approx$ are inversely proportional to the group velocity $c_{g}$ , meaning that decreasing wave periods strengthen the contribution from $\kappa _c$ . Secondly, figure 5 highlights the directional dependence of $\kappa _d$ ; the regions dominated by depth refraction (redish colours) have a different horizontal extent when comparing the leftmost and rightmost columns in figure 5. The reason for this difference is that the depth gradients $\partial _x h \ne \partial _y h$ , and that the unit vector ${\boldsymbol{e}}_{k,\perp } = [-k_y,k_x]/k$ generally holds different values depending on the initial direction. As a consequence, $\kappa _d$ obtains different values. On the contrary, and as seen from (3.4), $\kappa _c$ does not have such directional dependence.

Figure 6. A special case where depth and current refraction equalise and cancel each other. Panel (b) show wave rays propagating from left to right atop a sloping beach in the $y$ -direction and a shear current of type (3.1) (see panel (a). The black solid ray, where $\kappa _c = -\kappa _d$ , is initially located at ${\boldsymbol{x}}_r(0) = (0.05L_x,y_0)$ , where $y_0 = 0.5L_y$ . Adjacent rays are perturbed $\pm \varDelta _0=\pm 0.005L_y$ in the $y$ -direction. Panel (c) show the evolution of $\kappa _c + \kappa _d$ along the wave propagation distance (Dist.). Dashed black line denotes a slightly modified twin experiment which adds a small bump in the bathymetry ( $h=ys+h_{bm}$ ) in a subset of the domain, with details outlined in the text.

3.2.2. Zero curvature

As highlighted in figure 4(b), zero curvature occurs when $\kappa _c$ and $\kappa _d$ have different signs but with a similar magnitude, i.e. $\kappa _c = -\kappa _d$ . Such situations thus prevent wave refraction. This is illustrated in figure 6 for a prescribed shear current described by (3.1) with $\tau = 0.0015$ s⁻¹ and a constant sloping beach in the $y$ -direction. Here, the depth

(3.6) \begin{equation} h(x,y) = ys, \end{equation}

where $s = 0.015$ . This example can be used to represent the circumstances where oblique waves enter a gently sloping beach which is subject to longshore currents. To construct a wave ray which recovers $\kappa _c = -\kappa _d$ in the $x$ -direction, we solve the implicit equation for wavenumber $k$

(3.7) \begin{equation} \boldsymbol{\nabla }c = \frac {1}{2} \sqrt {\frac {{\textrm{g}} k}{\tanh (kh)}} \big [1 - \tanh ^2(kh) \big ] \boldsymbol{\nabla }h. \end{equation}

We consider the initial position ${\boldsymbol{x}}_r(0) = (0.05L_x,y_0)$ , where $y_0 = 0.5L_y$ and the water depth $h = 75\,{\textrm m}$ . Then, (3.7) yields a wave period of 12.7 s, and $k h=1.95$ . The resulting wave ray becomes a straight line when the initial propagating direction is parallel to the $x$ -axis, and the wave propagates from left to right (figure 6 b). However, small changes in the initial position are decisive for the ray direction. We introduce the small perturbation in position $\varDelta _0 = 0.005L_y$ , which corresponds to 50 m. The wave ray becomes trapped by the bathymetry if initially located at $y_0 + \varDelta _0$ , i.e. 50 m closer to shore (red line figure 6). On the contrary, the ray escapes the bathymetry if initially located $\varDelta _0$ further away from shore ( $y_0 - \varDelta _0$ , green line).

In cases with zero refraction, the wave refraction is very sensitive to any modulation in wavenumber. This is illustrated in a slightly modified twin experiment where the sloping beach is subject to a small perturbation, in the form of a small bump in the bathymetry. Let

(3.8) \begin{equation} h_{{bm}} = 10 \sin { \left (\frac {\pi }{0.05 L_x}(x - 0.15 L_x)\right )} \cos {\left (\frac {\pi }{0.1 L_y} (0.5 + y - 0.45 L_y) \right )}, \end{equation}

represent a local shallow bank, such that $h = ys + h_{{bm}}$ in the region $x\in [0.15L_x,0.2L_x]$ and $y\in [0.45L_y,0.55L_y]$ . Such a model realisation is depiced by the dashed lines in figure 6(b,c); while $k$ starts to decrease on the hump, $\boldsymbol{\nabla }c$ starts increasing such that $\kappa _c$ starts dominating. Indeed, the effect of the hump is that the wave ray eventually gets trapped by the bathymetry.