2.1 Governing equations for waves with constant vorticity

Let $\boldsymbol{U}=(u(x,y,t),v(x,y,t))$ denote the velocity field in a region bounded below by a horizontal bottom and above by the free surface $y=\unicode[STIX]{x1D702}(x,t)$ . The Euler equations for an ideal fluid are

where $p(x,y,t)$ is the pressure, $\unicode[STIX]{x1D70C}$ is the constant fluid density, $g$ is the acceleration due to gravity and $\boldsymbol{j}=(0,1)$ . We have the boundary conditions

where $p_{atm}$ is the atmospheric pressure. The vorticity is denoted by

Taking $\unicode[STIX]{x1D702}\equiv 0$ and a constant vorticity $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6FE}$ ,

is a solution of (2.1), satisfying (2.2)–(2.4). The background flow $\boldsymbol{U}_{\mathbf{0}}$ is a linear shear flow. Now, consider

where $\unicode[STIX]{x1D719}$ is the velocity potential of an irrotational perturbation of the shear flow. Since $\unicode[STIX]{x1D6E4}$ is constant, $\unicode[STIX]{x1D6E4}$ and $\boldsymbol{U}$ satisfy the vorticity equation $\unicode[STIX]{x1D6E4}_{t}+(\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D6E4}=0$ , and thus only the incompressibility condition $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}=\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=0$ remains to be satisfied in the fluid.

Equations (2.1)–(2.4) can written in dimensionless form as

with the pressure in the fluid body given by

Here, $\unicode[STIX]{x1D713}$ is the streamfunction associated with $\unicode[STIX]{x1D719}$ (i.e. its harmonic conjugate) and $B(t)$ is the Bernoulli constant.

The equations have been non-dimensionalised using $d$ and $\sqrt{d/g}$ (as the characteristic length and time) in order to obtain a problem with a channel of unit depth, and unit linear long-wave speed. The dimensional quantities are therefore

The dimensional pressure is given by $p_{atm}+(\unicode[STIX]{x1D70C}gd)p$ .

We now consider the case in which all functions are taken to be periodic in $x$ with wavelength $\unicode[STIX]{x1D706}$ , and travelling with speed $c$ . Thus, they can be written in terms of $X=(x-ct)$ and $Y=y$ . Without loss of generality, we assume that the mean of $\unicode[STIX]{x1D702}(x,t)$ is zero. The function $\unicode[STIX]{x1D702}$ is taken to be even and $\unicode[STIX]{x1D719}$ to be odd with respect to $x=0$ where the wave crest is positioned. In the frame moving with the wave, (2.8) becomes

and

2.2 The conformal map

Figure 1.

The conformal mapping from the canonical domain on the right onto the physical domain on the left.

The canonical strip is now conformally mapped onto the flow domain of the periodic steady wave. The coordinates in the canonical domain are $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})$ and the conformal map is defined as $\widetilde{Z}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})=\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})+\text{i}\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})$ , as indicated in figure 1. The real and imaginary parts of the conformal map are conjugate harmonic functions. The wave elevation $\unicode[STIX]{x1D702}(X)$ provides the Dirichlet boundary data for $\widetilde{Y}$ . Thus, defining $\widetilde{\boldsymbol{Y}}(\unicode[STIX]{x1D709}):=\unicode[STIX]{x1D702}(\widetilde{X}(\unicode[STIX]{x1D709},0))$ , we solve the problem

where the strip depth $D$ will be determined. The solution for $\widetilde{Y}$ is

where $k_{j}=(2\unicode[STIX]{x03C0}/L)j$ , for $j\in \mathbb{Z}$ , and

It should be noted that in (2.14) and elsewhere, unless indicated otherwise, the $j=0$ case is defined in the limit $k_{j}\rightarrow 0$ . Integrating the Cauchy–Riemann equation $\widetilde{X}_{\unicode[STIX]{x1D709}}=\widetilde{Y}_{\unicode[STIX]{x1D701}}$ , we obtain that

where

Defining $\widetilde{\boldsymbol{X}}(\unicode[STIX]{x1D709}):=\widetilde{X}(\unicode[STIX]{x1D709},0)$ and taking the mean of $\widetilde{\boldsymbol{X}}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709})$ yields

On the other hand, $\int _{-L/2}^{L/2}\widetilde{\boldsymbol{X}}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=\widetilde{\boldsymbol{X}}(L)-\widetilde{\boldsymbol{X}}(0)=\unicode[STIX]{x1D706}$ , and therefore

Henceforth, we choose the wavelengths in both domains to be equal ( $L=\unicode[STIX]{x1D706}$ ), and therefore

In the canonical domain, both $\widetilde{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}):=\unicode[STIX]{x1D719}(\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}),\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}))$ and its harmonic conjugate $\widetilde{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}):=\unicode[STIX]{x1D713}(\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}),\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}))$ satisfy the Laplace equation:

the solutions of which are given by

The arbitrary constant $A$ can be fixed by imposing that the mean horizontal velocity induced by the wave at the bottom is zero:

fixing the laboratory frame of reference. Making the change of variables $X=\widetilde{X}(\unicode[STIX]{x1D709},-D)$ , we have that

It is easy to show (as detailed in (2.33) and (2.34)) that

where $|\text{J}|(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}):=\widetilde{X}_{\unicode[STIX]{x1D709}}^{2}+\widetilde{Y}_{\unicode[STIX]{x1D709}}^{2}$ . Restricted to the bottom, this results in

and therefore

Imposing $\langle \widetilde{\unicode[STIX]{x1D713}}_{\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D709},-D)\rangle =0$ in (2.25), and taking the limit $k_{j}\rightarrow 0$ , yields $A=\langle \widetilde{\unicode[STIX]{x1D733}}\rangle$ .

Finally, we add that, using the Cauchy–Riemann equations $\widetilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D709}}=\widetilde{\unicode[STIX]{x1D713}}_{\unicode[STIX]{x1D701}}$ and restricting ourselves to the free surface, it follows that $\text{i}k_{j}\boldsymbol{F}_{k}[\widetilde{\unicode[STIX]{x1D731}}]=k_{j}\coth (k_{j}D)\boldsymbol{F}_{k}[\widetilde{\unicode[STIX]{x1D733}}]$ and therefore

with ${\mathcal{C}}[\cdot ]:=\boldsymbol{F}^{-\mathbf{1}}[\text{i}\coth (k_{j}D)]\boldsymbol{F}_{k}[\cdot ]$ . Similarly,

2.3 The wave elevation equation

We now proceed to translate from the boundary conditions at the free surface to ones in the canonical domain. Since $\widetilde{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}):=\unicode[STIX]{x1D719}(\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}),\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}))$ , the derivatives of $\widetilde{\unicode[STIX]{x1D719}}$ are

from which it follows that

where $|\text{J}|(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}):=\widetilde{X}_{\unicode[STIX]{x1D709}}^{2}+\widetilde{Y}_{\unicode[STIX]{x1D709}}^{2}$ is the Jacobian of the map.

It should be noted that

Substituting into the kinematic boundary condition (2.11) yields

Since

using the Cauchy–Riemann equations to eliminate $\unicode[STIX]{x1D719}$ for $\unicode[STIX]{x1D713}$ , we substitute into the boundary condition (2.36), resulting in

Integrating that equation, we obtain

We now proceed to the dynamic boundary condition. Using the equations (2.34) and the Cauchy–Riemann equations for ( $\widetilde{X}$ , $\widetilde{Y}$ ) and ( $\widetilde{\unicode[STIX]{x1D719}}$ , $\widetilde{\unicode[STIX]{x1D713}}$ ), we obtain that at the free surface ( $\unicode[STIX]{x1D701}=0$ ),

Substituting these equations and (2.35) into the dynamical condition in (2.11), we obtain

Equations (2.38) and (2.41) represent the free-surface boundary conditions in the canonical domain. Using (2.31) and (2.32), we combine them into a single equation for the free-surface elevation $\widetilde{\boldsymbol{Y}}=\unicode[STIX]{x1D702}(\widetilde{X}(\unicode[STIX]{x1D709},0))$ :

Hence, as unknowns, we have $\widetilde{\boldsymbol{Y}}(\unicode[STIX]{x1D709})$ , $c$ , $B$ and $D$ , and therefore (2.42) must be supplemented by three conditions: we fix the wave height $H$ through

and we impose a mean-zero wave profile in the physical space,

as well as the depth condition (2.21)

Equations (2.42)–(2.45) are solved using Newton’s method, as detailed in § 3.1.

Once a solution $\widetilde{\boldsymbol{Y}}$ is found, we use (2.39) to find $\widetilde{\unicode[STIX]{x1D733}}$ . The underlying velocity potential $\widetilde{\unicode[STIX]{x1D731}}$ is readily available from its harmonic conjugate $\widetilde{\unicode[STIX]{x1D733}}$ through the expression (2.31). Using the conformal mapping defined by (2.14), (2.17) and the harmonic function given by (2.25), the information needed for particle dynamics is complete. Namely, given a wave profile, we are in a position to compute values of $\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}),\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})$ and $\widetilde{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})$ anywhere in our canonical domain. The pressure is also readily available through expression (2.12).

2.4 Flow structure beneath the wave

In this section, we show how to use the formulation above in order to compute streamlines and stagnation points in a frame moving with the wave, as well as pressure and particle paths in arbitrary frames of reference.

In a frame moving with the speed $U$ , a fluid particle has the trajectory $(X(t),Y(t))$ satisfying

In the canonical domain, the corresponding trajectory $(\unicode[STIX]{x1D709}(t),\unicode[STIX]{x1D701}(t))$ , such that $(X(t),Y(t))=(\widetilde{X}(\unicode[STIX]{x1D709}(t),\unicode[STIX]{x1D701}(t)),\widetilde{Y}(\unicode[STIX]{x1D709}(t),\unicode[STIX]{x1D701}(t)))$ , is obtained from

Setting $U=c$ yields the paths in a frame moving with the wave.

The identification of stagnation points provides the structure of the phase portrait for particle trajectories beneath the travelling wave. In particular, for a periodic velocity field, at least two stagnation points are necessary for a closed recirculation zone to travel with the wave speed. The location of the stagnation points and their dependence on wave parameters for travelling nonlinear waves has been an issue of research interest (Ko & Strauss Reference Ko and Strauss2008a ,Reference Ko and Strauss b ), which we shall address in detail. In our periodic symmetric wave configuration, we shall search for a stagnation point $(X_{s},Y_{s})$ restricted to a subset of the fluid domain. Since the wave crest is at $(0,\unicode[STIX]{x1D702}(0))$ , by symmetry and periodicity, $\unicode[STIX]{x1D719}_{Y}(X,Y)$ must vanish at $X=-\unicode[STIX]{x1D706}/2,0,\unicode[STIX]{x1D706}/2$ . In addition, the impermeable bottom implies $\unicode[STIX]{x1D719}_{Y}(X,-1)=0$ . Therefore, the vertical velocity is zero in the set depicted by a thick dark line in figure 2. In fact, under reasonable assumptions, stagnation points must lie in this region. Consider the vertical velocity field which is itself a harmonic function and which vanishes on this set. If the free boundary is monotonic on each side and has no stagnation points away from the wave crest, then the vertical velocity has a definite sign on the free surface on each side of the crest, and, by the maximum principle, can vanish only on the indicated set.

Figure 2.

Locations for a vanishing vertical speed.

The search for stagnation points thus reduces to finding the zeros of

in the set shown in figure 2. With the present formulation, these zeros are found with great precision whatever the number of stagnation points. Since the horizontal velocity is symmetric with respect to $X=0$ , our search can be restricted to the right half of figure 2, namely within the viable set  $V$ :

The pre-image of this set under the conformal mapping $(\widetilde{X}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}),\widetilde{Y}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}))$ is

Stagnation points are first found in the canonical domain using (2.24) and (2.34), and then mapped to the physical domain through (2.14) and (2.17).

In order to find the streamlines, the shear flow is added to $\widetilde{\unicode[STIX]{x1D713}}$ (the harmonic conjugate of $\unicode[STIX]{x1D719}$ ), yielding the streamfunction  $\unicode[STIX]{x1D713}_{T}$ :

The streamfunction is evaluated on a uniform grid in the canonical plane. These values are assigned to the image grid in the physical domain and used to draw level curves of $\unicode[STIX]{x1D713}_{T}(X,Y)$ . A similar procedure is used for the pressure. From Bernoulli’s equation (2.12), we compute

on a uniform grid in the canonical domain, which is mapped to the physical domain having the corresponding pressure values assigned to its nodes.

3.1 Computing the waves

Recall that the full solution in the canonical domain can be computed in a straightforward fashion as soon as the $\unicode[STIX]{x1D706}$ -periodic wave elevation $\widetilde{\boldsymbol{Y}}$ has been found, by solving (2.42)–(2.45). Take an equally spaced set of grid points in $\unicode[STIX]{x1D709}$ at $\unicode[STIX]{x1D701}=0$ in the canonical domain:

With $N$ even, let $\widetilde{\boldsymbol{Y}}_{j}:=\widetilde{\boldsymbol{Y}}(\unicode[STIX]{x1D709}_{j})$ and $\widetilde{\boldsymbol{X}}_{j}:=\widetilde{\boldsymbol{X}}(\unicode[STIX]{x1D709}_{j})$ , and impose even symmetry about $\unicode[STIX]{x1D709}=0$ by taking $\widetilde{\boldsymbol{Y}}_{j}:=\widetilde{\boldsymbol{Y}}_{N-j+2}$ , $j=N/2+2,\ldots ,N$ .

Substituting these into a discretised version of (2.42) gives rise to a set of $N/2+1$ equations for $N/2+4$ unknowns $(\widetilde{\boldsymbol{Y}}_{1},\ldots ,\widetilde{\boldsymbol{Y}}_{N/2+1},c,B,D)$ . The discretisation we have chosen replaces all derivatives and non-local linear operators in (2.42) and (2.32) by their discretised Fourier symbol. Thus, their action is computed with spectral accuracy in Fourier space and all nonlinearities are computed in real space at the grid points. The $N/2+1$ equations are obtained by evaluating (2.42) at grid points $\widetilde{\boldsymbol{X}}_{j}$ , $j=1,\ldots ,N/2+1$ . These can be expressed as

The system is closed with the discretisation of the additional equations (2.43)–(2.45):

The system is solved by Newton’s method, with linear irrotational waves as initial guess:

where $H$ is small. The solution can then be continued in the parameters ( $H,\unicode[STIX]{x1D6FE},c$ ) by using the prior converged solution as the initial guess to a new solution. The Jacobian of the system for the Newton iteration is found by finite variations in the unknowns. The stopping criterion is given by

Due to Galilean invariance, the parameter $b$ affects neither the shape of the wave nor the streamlines in the frame of reference travelling with the wave. For simplicity, we always choose $b=0.5$ , corresponding to zero velocity at mid-depth for linear waves.

Particle trajectories are found by integrating in time the ordinary differential equations (2.47) in the canonical domain with a fourth-order Runge–Kutta method and mapping these trajectories on the moving frame $(X,Y)$ as described in § 2.4. Once we have computed $(X(t),Y(t))$ , the orbits in the laboratory frame of reference $(x,y)$ are obtained from $x(t)=X(t)+ct$ , $y(t)=Y(t)$ .

3.2 Streamlines and stagnation points

For surface waves in the absence of vorticity, it is known that there can be only one critical (stagnation) point, namely at the $120^{\circ }$ crest of the largest Stokes wave (Plotnikov Reference Plotnikov2002). For constant vorticity, various computations have provided evidence of either one or three stagnation points for certain values of $\unicode[STIX]{x1D6FE}$ (Teles da Silva & Peregrine Reference Teles da Silva and Peregrine1988; Ko & Strauss Reference Ko and Strauss2008b ; Vasan & Oliveras Reference Vasan and Oliveras2014). There is also an analysis by Constantin et al. (Reference Constantin, Strauss and Varvaruca2014) regarding the location of critical points under certain conditions. Here, we make a comprehensive study of the configurations and precise positions of stagnation points. We study their location dependence with respect to $\unicode[STIX]{x1D6FE}$ and wave amplitude. Having this in mind, we consider two types of parameter sweeps in system (2.46): (1) fixing a value of $\unicode[STIX]{x1D700}$ and varying $\unicode[STIX]{x1D6FE}$ ; (2) fixing a value of $\unicode[STIX]{x1D6FE}$ and varying $\unicode[STIX]{x1D700}$ . The search for the stagnation point position and the construction of the system phase portrait are performed as described in § 2.4.

Figure 3.

Phase portraits for waves with steepness $\unicode[STIX]{x1D700}=0.09$ for varying vorticity $\unicode[STIX]{x1D6FE}$ . The markers indicate the locations of the stagnation points (circles correspond to centres and squares to saddles).

We begin with a detailed study of an intermediate-depth regime where the wavelength is $2\unicode[STIX]{x03C0}$ (recall that the depth is always set to 1). In figure 3, a parameter sweep with $\unicode[STIX]{x1D700}=0.09$ is shown for values from $\unicode[STIX]{x1D6FE}=0$ up to the maximum vorticity value we can compute for this steepness. The stagnation points are shown with circles (centres) and squares (saddles). For large values of vorticity, there are two stagnation points inside the flow: a centre below the crest and a saddle below the trough, resulting in a cat’s eye region of closed streamlines (recirculation zone) centred below the crest. As $\unicode[STIX]{x1D6FE}$ decreases, the saddle point moves downward and reaches the lower boundary at $\unicode[STIX]{x1D6FE}\approx 6.8525$ , where the saddle splits into two. The recirculation zone is now attached to the lower boundary. As $\unicode[STIX]{x1D6FE}$ decreases further, this attached recirculation zone shrinks and disappears at $\unicode[STIX]{x1D6FE}^{\ast }\approx 1.0873$ when the three stagnation points coalesce at $x=0$ and disappear in a higher-order saddle–centre bifurcation. For smaller values of $\unicode[STIX]{x1D6FE}$ , the streamlines are qualitatively similar to Stokes waves. As expected, decreasing vorticity sharpens the crests.

We note that, as all of the bifurcations of the flow field occur at the lower boundary, they are simpler to understand if the flow is reflected symmetrically about this line. In this interpretation, the first bifurcation above is the result of the transcritical collision of two saddles, the one in the fluid and its image below the bottom boundary. After collision, they separate along the bottom boundary, creating the two stagnation points which are the leading and trailing edges of the recirculation zone. The second bifurcation corresponds to two saddles and two centres (one of which is below the bottom boundary) annihilating at the point $(0,-1)$ .

Figure 4.

Phase portraits for waves with $\unicode[STIX]{x1D6FE}=1$ at different values of the wave steepness $\unicode[STIX]{x1D700}$ . The markers indicate the location of the stagnation points (circles correspond to centres and squares to saddles).

In figure 4, a parameter sweep with $\unicode[STIX]{x1D6FE}=1$ is shown for values of $\unicode[STIX]{x1D700}=0.1$ up to $\unicode[STIX]{x1D700}=0.22$ . In this sweep, the flow never exhibits the cat’s eye flow, as the recirculation zone is always attached to the bottom boundary at large amplitudes. As $\unicode[STIX]{x1D700}$ decreases, the same behaviour occurs as observed in the second bifurcation described in the paragraph above: the three stagnation points disappear at the bottom (for $\unicode[STIX]{x1D700}^{\ast }\approx 0.1113$ ) under the crest, and the resulting flow resembles that of a small-amplitude periodic irrotational Stokes wave. In this regime, as the amplitude increases the waves become more peaked.

Figure 5.

Parameter dependence of flow fields as the wave steepness $\unicode[STIX]{x1D700}$ and vorticity $\unicode[STIX]{x1D6FE}$ are varied while the depth and wavelength are held fixed at $1$ and $2\unicode[STIX]{x03C0}$ respectively. The dashed lines A and B correspond to points at which bifurcations occur, changing the number of stagnation points in the fluid domain. The vertical bold dash–dot line corresponds to a linearised flow with a stagnation point. The vertical dashed asymptote corresponds to the large- $\unicode[STIX]{x1D6FE}$ limit of rigid-body rotation and the solid curve to the largest waves we have computed at each value of vorticity. The three regions correspond to two, three or zero stagnation points within one period of the wave. Streamline profiles corresponding to the parameters indicated in the figure are also shown illustrating each case. The curve beyond point 2ii, as it asymptotes to the value $\unicode[STIX]{x1D700}=\sqrt{2}/\unicode[STIX]{x03C0}$ , has been schematically extended by the authors.

The full $(\unicode[STIX]{x1D700},\unicode[STIX]{x1D6FE})$ parameter space for $\unicode[STIX]{x1D706}=2\unicode[STIX]{x03C0}$ is shown in figure 5, together with typical flow patterns. The curve with vertical asymptotes at $\unicode[STIX]{x1D700}=0$ and $\unicode[STIX]{x1D700}=\sqrt{2}/\unicode[STIX]{x03C0}$ corresponds to the locus of the largest-amplitude waves that we could compute by our method. The asymptote at $\unicode[STIX]{x1D700}=\sqrt{2}/\unicode[STIX]{x03C0}$ corresponds to the limit in which the full fluid domain is in rigid-body rotation. The point $(0,\unicode[STIX]{x1D6FE}_{l}),\unicode[STIX]{x1D6FE}_{l}=\tanh (1)/(1-\tanh (1))$ is found by considering whether the linearised flow admits stagnation points on the lower boundary. As $\unicode[STIX]{x1D6FE}$ increases on points $(0,\unicode[STIX]{x1D6FE}),\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{l}$ , the stagnation point rises into the fluid. The dash–dot bold line in the figure indicates the range of $\unicode[STIX]{x1D6FE}$ such that the linearised flow has a stagnation point inside the fluid domain. The dashed lines correspond to bifurcation points at which the number of stagnation points in the domain changes. The bold dashed line B separates region 3 of ‘irrotational-like’ behaviour, with no stagnation points in the fluid, and region 2, with three stagnation points: two saddles at the lower boundary and a centre under the crest. On B, there is one stagnation point at the lower boundary under the crest. The thin dashed line A separates region 2 from region 1, where there are two stagnation points. This loss of a saddle point occurs when the two saddles at the lower boundary are exactly one wavelength apart and merge under the trough of the wave. Within region 1, this saddle moves inside the fluid, under the trough.

Figure 6.

Phase portraits with $H=0.4$ , $\unicode[STIX]{x1D6FE}=10$ and varying the wavelength with $\unicode[STIX]{x1D706}=0.6$ , 1, 2, 6, 20, 50. The black markers depict the stagnation points as before. The thick black curve is at $\unicode[STIX]{x1D713}(X,Y)=0$ . It should be noted that these solutions have different wavelengths from those in figure 5 and therefore correspond to different wavelength regimes: from short waves to long waves, from (a–f).

We now turn our attention to the effect of the wavelength, focusing on the most interesting case of large positive vorticity. We fix $\unicode[STIX]{x1D6FE}=10$ , set the depth to unity and fix the amplitude as $H=0.4$ as we vary the wavelength $\unicode[STIX]{x1D706}$ from the short-wave (deep water) regime ( $\unicode[STIX]{x1D706}=0.6$ ) to the long-wave (shallow water) regime ( $\unicode[STIX]{x1D706}=50$ ). This parameter choice allows us to keep the solutions strongly nonlinear throughout the range. The results are shown in figure 6. In deep water, the wave affects only an upper layer of depth approximately equal to the wavelength, with a large recirculation zone near the crest. As the depth decreases, this recirculation zone descends and there are counter-propagating currents above and below the cat’s eye. As the relative depth further decreases, the recirculation zone attaches to the bottom and forms a stationary fluid region at the bottom with a free-surface streaming flow above it. The free-surface profile here is similar to a ‘surface shear wave’ discussed in Peregrine (Reference Peregrine1974) which is often seen in backwash on a beach (see figure 7).

Figure 7.

Nearly steady rounded wave profiles in the presence of a strongly shearing backwash flow. Photo by Roberto Ribeiro Jr.

3.3 Pressure

For irrotational flows, the following pressure properties are rigorously known (Constantin & Strauss Reference Constantin and Strauss2010, p. 539):

1. (i) the maximum pressure value is attained at the bottom, below the crest;

2. (ii) the minimum pressure value is attained at the free surface;

3. (iii) the pressure is strictly increasing with depth along a vertical line;

4. (iv) the pressure is strictly decreasing along horizontal lines starting below the crest and ending below the trough.

In the presence of constant vorticity, none of these facts hold in general. For example, recently Strauss & Wheeler (Reference Strauss and Wheeler2016, Proposition 16) proved that any wave that overturns has a pressure sink (minimum pressure) inside the fluid region, provided that it has no stagnation point on the free surface. By periodicity and symmetry about $x=0$ , the values of pressure below the crest or trough must be local extrema with respect to horizontal variations. Furthermore, the pressure at any saddle point on the bottom boundary must be a local maximum by Bernoulli’s theorem since the bottom is a streamline, and the pressure must also decrease along the other streamlines originating at that point and going upwards into the domain. This means that, referring to figure 5, we expect that for the pressure along the bottom, region 3 will behave like irrotational waves, while region 2 will have a local minimum under both crest and trough, and a maximum at the saddles. By a continuity argument, region 1 will have a maximum under the trough and a minimum under the crest.

Figure 8.

Pressure beneath a periodic wave of steepness $\unicode[STIX]{x1D700}=0.09$ , as the background vorticity is varied. Black dots and squares represent the locations of global minima and maxima of pressure respectively. The thick black curve indicates the contour for $p(X,Y)=0$ . Also shown is the pressure along the bottom boundary.

Figure 8 displays the contours of the computed pressure field and the pressure on the bottom boundary for wave steepness $\unicode[STIX]{x1D700}=0.09$ and the various values of $\unicode[STIX]{x1D6FE}$ used in figure 3. In the figure, the thick black curves correspond to $p(X,Y)=0$ , the black dot indicates the point of global minimum pressure (when it is below atmospheric pressure) and the black square indicates the point of global maximum of pressure. As the vorticity increases, a below-atmospheric minimum pressure appears inside the fluid domain, whereas the maximum pressure has moved from below the crest to below the trough. In general, neither the pressure along a vertical line below the crest or trough, nor along a horizontal line from under the crest is monotonic. Some of these properties had been reported by Teles da Silva & Peregrine (Reference Teles da Silva and Peregrine1988) and Vasan & Oliveras (Reference Vasan and Oliveras2014).

Figure 9.

Pressure beneath a wave, in the different regimes shown in figure 6. In the pressure contour graphs, the black dots and squares represent the locations of minimum and maximum pressure respectively. The thick black curve denotes $p(X,Y)=0$ .

Figure 9 shows the effect of varying the depth (for the same cases as in figure 6). Except in very shallow water, the global minimum of pressure lies inside the recirculation zone, near the stagnation point under the crest, and the global maximum in pressure lies near the stagnation point under the trough. In shallow water, when the recirculation zone is attached to the bottom, the minimum in pressure moves upwards to a point above the crest of this recirculation zone.