## Abstract

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.

This article is part of the theme issue ‘Nonlinear water waves’.

## 1. Introduction

The analysis of the water wave problem dates back to the studies of Newton, with Gerstner [1,2] being the first to consider nonlinear waves. A detailed review on the origins of water wave theory is given in [3], where the main contributions until the great work of Stokes [4] are presented. Since then, an abundance of work has appeared, using a variety of mathematical formulations of the physical problem, and studying analytical, numerical and experimental aspects of the problem. Not in a few cases, numerical investigations have guided subsequent rigorous mathematical analysis. Furthermore, a detailed insight may be extracted through the combined application of analytical and numerical approaches, which it is not otherwise possible to obtain from either approach in isolation.

In the present work, we start from Euler's equations, and by reviewing the methodology of Constantin & Strauss in [5], we derive a mathematical formulation for two-dimensional, periodic, travelling water waves with variable vorticity. It is worth pointing out that, while a recent alternative approach for constant vorticity was devised in [6], the methodology in [5] is more adequate for the considerations that are pursued in this paper.

Firstly, by introducing a streamfunction we formulate the free boundary value problem (BVP) (2.15), which was rigorously derived in [7]. Consecutively, a partial hodograph transformation yields the nonlinear BVP (3.3); the equivalence of the latter with the Euler equations is proved in [5]. A key role in the analysis of this problem is the occurrence of a bifurcation parameter which is indicative of the total energy of the wave; this parameter, denoted by *Q*, is called the hydraulic head of the flow. In [5], it was proved that for a specific value *Q*=*Q**, there exists a bifurcation point on the diagram of the solution. In particular, in the neighbourhood of *Q** and for the same value *Q*>*Q**, the problem (3.3) admits two qualitatively different solutions: the first one corresponds to a laminar flow, and the second one to a water wave with non-vanishing amplitude, a so-called genuine wave. Moreover, the latter waves come as a family, formulating a bifurcating branch.

In the present work, we review one analytical and two numerical methods for approximating solutions which belong to the interesting branch of the bifurcation diagram, i.e. solutions that correspond to genuine waves. Furthermore, through a systematic analysis, we reconstruct the whole branch of the solution until a limiting solution which was predicted by the analysis in [5], and corresponds to waves of maximal amplitude. It is important to mention that some results obtained from the analytical approach provide the background of the development of these numerical techniques.

In [8], high-order approximations to periodic travelling wave profiles are derived, through a novel expansion. This expansion incorporates the variation of the total mechanical energy of the water wave and yields the extension of these approximations to any finite order. The basic ideas and results of this work are reviewed in §4, and these expansions are given in equations (4.2) and (4.1). In the absence of flow reversal, the analysis in [9,10] ensures the convergence of the infinite Taylor expansions, due to the analyticity of the streamlines, whereas the available results on regularity for flow reversal are ; see the results in [11]. For the first rigorous constructions of travelling waves occurring via power series approximations, we refer to the works of Nekrasov [12], Levi-Civita [13] and Struik [14].

The first numerical approach, which is reviewed here, computes large-amplitude travelling water waves in flows with constant vorticity and is based on a penalization method. The algorithm is realized as a partial differential equation (PDE) constrained optimization formulation, maximizing the wave amplitude subject to the PDE constraint induced by (3.3). This algorithm is quite general, displaying results for some specific cases of non-constant vorticity in [7]. The second computational approach is based on numerical continuation techniques, which are suitably adapted to the water wave problem for obtaining non-laminar flows of constant vorticity and particular cases of vorticity which varies linearly and quadratically with the streamfunction; see [15]. Through this, the bifurcation curve, which starts from *Q** and contains non-laminar flows, is reconstructed. This procedure is limited by waves that include a stagnation point, i.e. a point where the horizontal velocity of the flow becomes equal to the propagation speed of the wave. Water waves with maximal amplitude are connected with the existence of points of stagnation in their flow. Both these approaches are reviewed in §5.

Computations which are based on the numerical continuation approach have been performed earlier in [16,17], with several interesting results, which agree with the relevant analytical predictions. Among the most important results of those works, we point out the following. Stagnation can occur not only at the crest but also at the point on the bottom directly below the crest. Along the bifurcation curve, for fixed relative mass flux^{1} *p*_{0}, the amplitude of the wave is increasing and the depth *d* varies only slightly. Furthermore, waves of maximal amplitude are obtained at the end of the bifurcation curve, and the maximal amplitude is an increasing function of |*p*_{0}|, in the case of constant vorticity. Finally, the shapes of the streamlines of the extreme waves depend on the vorticity, which is a result observed also in several numerical works and indicates the importance of the effect that the vorticity has on the features of water waves.

One important aspect of the formulation (3.3) is that, once a solution is found, then one can readily obtain other important features of the flow, such as the velocity vector and the pressure on the flat bottom. In several instances of this work, we illustrate these flow characteristics, derived from either analytical or numerical approaches, for several different types of wave–current interactions. Laboratory experiments and numerical simulations for irrotational waves are discussed in [18–21], while these types of studies for wave–current interactions in flows of constant vorticity were pursued in [16,17,22–25]. In a number of recent works, waves with critical layers are studied numerically; see for example [26,27].

## 2. Mathematical formulation

### (a) Mathematical modelling

We discuss two-dimensional, periodic waves with general vorticity and large amplitude. We assume that the water is incompressible and inviscid without surface tension, lies over a flat bottom and is acted upon by gravity *g*. Moreover, we study waves travelling at constant speed and without change of shape at the surface of a layer of water. In this formulation, we make no shallowness or small-amplitude approximation.

In particular, the water waves have the following properties:

—

*Two-dimensional*. The motion is identical in any direction parallel to the crest line. Thus, we analyse a cross section of the flow that is perpendicular to the wave crests, choosing appropriate Cartesian coordinates (*X*,*Y*). Let the cross section of the fluid domain be of the form where*d*>0 is the average depth and*ξ*is the free surface.—

*Incompresible*. Let (*U*(*t*,*X*,*Y*),*V*(*t*,*X*,*Y*)) be the velocity field. The constant density flow implies the equation of mass conservation 2.1—

*Inviscid*. Let*P*(*X*,*Y*,*t*) be the fluid pressure. The Euler equations take the form 2.2 where*P*is the pressure and*g*is the gravitational constant.—

*Flat bottom*. In its undisturbed state (no waves), the equation of the flat surface is*Y*=0, and the flat bottom is given by*Y*=−*d*for some*d*>0, i.e.*d*represents the average depth. Moreover, the flow does not penetrate the horizontal bottom, which is given by the boundary condition 2.3—

*Free water surface*. Let*Y*=*ξ*(*t*,*X*) be the free surface; the same particles always form this surface and this is expressed by 2.4—

*No surface tension*. The motion of the water and the air is decoupled; this is translated to the condition that the pressure*P*is equal to the atmospheric pressure*P*_{atm}on the free surface.

Moreover, we discuss water waves that are *travelling* at constant speed *c*>0. This has a twofold meaning. First, in this regime, the free surface is a graph; for irrotational travelling waves, see the considerations in [28]. Second, the space–time dependence of the free surface, of the pressure and of the velocity field has the form (*X*−*ct*), i.e. for (*x*,*y*)=(*X*−*ct*,*Y*) we have that
Obviously, this change of variables transforms accordingly the Euler equations and the relevant boundary conditions.

Restricting our attention to waves which are *periodic* in *x*, and taking, for convenience, the length scale to be 2*π*, we obtain that the velocity field (*u*,*v*), the pressure *P* and the free boundary *η* are 2*π*-periodic functions.

Consequently, in a frame moving at the (constant) wave speed, the BVP is defined in the two-dimensional bounded domain bounded above by the free surface profile and below by the flat bed

As *d* represents the average depth, the waves oscillate around the flat free surface *y*=0, that is
2.5

One may think that the above-mentioned assumptions will significantly restrict the amount of water waves that can be described by this model. However, this is a realistic model for large families of waves which display a plethora of different characteristics; see the relevant discussion in [5,29].

An essential flow characteristic is the vorticity 2.6 which is indicative of underlying currents. With respect to the flow beneath the waves, we restrict our attention to flows for which 2.7 This condition prevents the appearance of stagnation points in the flow and the occurrence of flow reversals.

### (b) Free boundary value problem

A flow characteristic that is necessary for the definition of the free BVP is the *relative mass flux*
2.8
which is independent of *x*, see [7]. We introduce the streamfunction *ψ*(*x*,*y*) as the unique solution of the differential equations
2.9
subject to
2.10
Note that *ψ*(*x*,*y*) is periodic in the *x*-variable, and that the constraint (2.10) is consistent with the flat bottom boundary condition (2.3). Moreover, (2.9) and the definition of vorticity yield
2.11
The free boundary condition (2.4) is equivalent to *ψ* being constant on *S*, while (2.8) and (2.10) ensure that this constant must vanish, i.e.
2.12
On the other hand, due to (2.9), we see that we can re-express, through integration, the mass conservation (2.1) and the Euler equations in (2.2) by the expression
2.13
where *Q* is a constant called the hydraulic head, which is given by
2.14
with *E* being a constant which represents the total mechanical energy throughout the flow.

### Remark 2.1

For two-dimensional travelling water waves, if condition (2.7) holds, then vorticity is necessarily a function of the streamfunction, that is, *γ*=*γ*(*ψ*)=*γ*(−*p*), cf. [22].

Following this concept, in the instances where vorticity is non-constant in this work, we refer to its particular dependence on the streamfunction; see for example the term ‘linear vorticity’ in §5.

The previous considerations show that the governing equations can be formulated in terms of the streamfunction as the free boundary problem:

### Definition 2.2

The constants *g* (gravitational constant), *p*_{0} (relative mass flux), *Q* (hydraulic head) and the function (vorticity) are given.

Moreover, for given *η*, which we assume to be normalized by (2.5), let *ψ*=*ψ*[*η*] be the even and 2*π*-periodic (in the *x*-variable) solution of the linear equation (2.11) with boundary conditions (2.10) and (2.12).

For given *η*, this *linear* PDE is overdetermined by imposing the nonlinear boundary condition (2.13) on *S*.

The free boundary problem consists in using the overdeterminacy to determine *η*.

The free BVP can also be viewed as solving an operator equation 2.15 where .

Evenness of *ψ* reflects the requirement that *u* and *η* are symmetric, while *v* is antisymmetric about the crest line *x*=0; here, we shift the moving frame to ensure that the wave crest is located at *x*=0. Symmetric waves present these features and it is known that a solution with a free surface *S* that is monotonic between crest and trough has to be symmetric; see [30,31].

## 3. A nonlinear boundary value problem

Starting from the free boundary formulation of travelling water waves, which was derived by the Euler equations in the above section, we apply the Dubreil-Jacotin transformation, introduced in [32]. This is a partial hodograph transformation, which is described as (figure 1)
and transforms the unknown domain to the rectangle
3.1
Let
3.2
be the height above the flat bottom *B*. As *ψ* is a strictly decreasing function of *y*, for every fixed *x* the height *h* above the flat bottom is a single-valued function of *ψ* (equivalently, of *p*).

Consequently, through the methodology in [5,7], the free boundary problem defined by (2.15) takes the form of the following second-order elliptic nonlinear fixed BVP for the even (in *q*) function *h*(*q*,*p*):
3.3
where *R* is given by (3.1), *γ* is the given vorticity, *g* is the gravitational constant and *Q* is given in (2.14), being indicative of the total mechanical energy of the wave. Moreover, the free boundary *η*(*x*) is given by *h*(*q*,0)=*η*(*x*)+*d*, for an average depth *d*.

### Remark 3.1

The average depth of the fluid can be recovered only *a posteriori* from the height function by taking the mean integral of *h*(*q*,0) over a period and using relation (2.5).

A novel approach to overcome this issue was initiated in [33,34], where an alternative bifurcation approach which fixes the mean depth *a priori* is presented; see also [35]. Based on this formulation, one may proceed in a similar way in both the analytical and numerical aspects in order to compute families of non-laminar water waves of fixed depth, where the relative mass flux *p*_{0} will be varying. We consider this a very fruitful approach in order to see, among other interesting features, how the variation of the quantity *p*_{0} will affect important characteristics of waves. For a brief overview on the influence of *p*_{0} on the wave characteristics, we refer to [15].

In this formulation, the parameter *Q* is treated as a bifurcation parameter. To make clear this statement, we first have to discuss some specific solutions of the problem (3.3).

### (a) Laminar solutions

The laminar flows are readily obtained as the *q*-independent solutions of (3.3) by the following formula:
3.4
provided that the parameter λ>0 satisfies the equation
3.5
where .

### (b) Bifurcation and linearized solution

Initially, we restrict our attention to the case of constant vorticity, i.e. *γ*=constant. For *γ*=0, we obtain irrotational waves, which have as a limiting case the so-called Stokes’ wave of greatest height. Moreover, the case *γ*≠0 gives rise to a large number of rotational waves, which display a variety of different characteristics. This mechanism describes waves where wave–current interactions cause significant increase of the wave amplitude in little time; such a case appears at the Columbia River entrance, where tidal currents cause a doubling in the wave height; see [36,37].

In the case of constant vorticity, i.e. *γ*=constant, apart from the laminar flows one can obtain the solution of the linearized problem. In this case, the BVP (3.3) is linearized around the laminar flow *H*(*p*;λ) and gives a linear separable PDE and a Robin boundary condition. For some specific value λ* (equivalently *Q**), an existence result for the solution of this BVP is presented in [5]. Moreover, the linearized solution is given by
3.6
where λ_{*}>0 is the solution of the dispersion relation
3.7

Equations (3.4) and (3.5) are now simplified to
3.8
and
3.9
respectively, with λ_{*} satisfying the dispersion equation (3.7).

It was shown in [5] that the point (*Q**,*H**) is a bifurcation point in the parameter space (figure 2). In brief, near the laminar flows (3.4), as the parameter λ varies, there are generally no genuine waves, except at critical value λ=λ* which is determined by the dispersion relation (3.7). Near this bifurcating laminar flow *H**, we have two solution curves: one laminar solution curve λ↦*H*(*p*;λ), where λ and *Q* are related by (3.5), and one non-laminar solution curve *Q*↦*h*(*q*,*p*;*Q*) such that *h*_{q}≢0 unless *h*=*H** (figure 2). In [5], it was shown that the curve containing the non-laminar solutions can be extended to a global continuum that contains solutions of (3.3) with 1/*h*_{p}(*q*_{s},*p*_{s})→0 at some (*q*_{s},*p*_{s}). This condition is characteristic of flows whose horizontal velocity *u* is arbitrarily close to the speed *c* of the reference frame, at some point in the fluid, the limiting configuration being a flow with stagnation points.

In what follows, we start from the results produced by this analysis and we obtain general methods for approximating solutions of (3.3) that correspond to flows of large amplitude.

Firstly, we interpret the function 3.10 as a perturbation of a laminar solution of the system (3.3), in the sense that 3.11 and that the amplitude of the water wave is of order . Having in mind that the wave amplitude vanishes for laminar flows, we can see the expression (3.10) as an approximation of small-amplitude water waves. Thus, the expression given in (3.10) may serve as an initial step in iterative numerical procedures for computing waves of large amplitude. Two different such approaches are described in §5.

Second, we can see this expression as the first-order asymptotic expansion of a general solution of the BVP (3.3) which represents a large-amplitude water wave. This point of view on this expression is triggered by the bifurcation argument, which was described above and was proved in [5]. A rigorous construction of the full asymptotic expansion is presented in [8]. Higher-order expansions result in waves of large amplitude; this analytical approximation of genuine waves is presented in §4.

## 4. Analytical approximation

The analytical approach starts from the perturbation of a laminar solution around the bifurcation point described above. The location of the bifurcation point and the direction of this perturbation are given by (3.8) and (3.9) and determined as the solution of an eigenvalue problem described in [5]. Then, asymptotic techniques are applied in order to approximate genuine water waves, i.e. solutions of (3.3) that correspond to non-laminar flows. This idea is introduced in [38] and analysed in [8], in order to approximate the bifurcation branches in the solution diagram, figure 2. Developing this idea, in [8] we are able to derive explicit formulae for families of water waves contained in the bifurcation branch of non-laminar flows. Some of the results related to important characteristics of the waves are depicted in figure 3. In particular, the free surface and the pressure of the water at the bottom are depicted for different values of constant vorticity; the latter characteristic is particularly important, because in practice the state of the sea surface is often gathered from knowledge of subsurface pressure [33,39–43]. Knowledge of these flow characteristics is very useful in qualitative studies; see [44,45]. The latter works are particularly noteworthy because the qualitative studies contained therein relate to the fully nonlinear exact governing equations; in this context, we also refer to [46–49]

Using the above ‘bifurcation argument’, we observe that the variation of the parameter *Q* (the hydraulic head of the flow) about the uniquely determined value *Q** induces an approximation of non-laminar flows, in the following sense.

### Definition 4.1

Define the approximation for the hydraulic head of the flow, 4.1

### Definition 4.2

Define the approximation for the height function *h*(*q*,*p*;*Q*),
4.2
with
4.3
and
4.4
where .

Our goal is to *simultaneously* determine the functions and the constants so that the system (3.3) is satisfied up to order *b*^{2N+2}, i.e.
4.5

The fact that the asymptotic expansion *h*^{(2N+1)} is an approximation of a non-laminar wave is established by the following theorem, which has a central role in the current analysis of the water wave problem, and is stated and proved in [8].

### Theorem

*Let g, p*_{0} *and γ be fixed, λ*_{*} *be defined as the solution of (3.7) and Q** *be given by (3.5).*

*There exist specific sets of functions* *and constants* *, such that the function h*^{(2N+1)}*(q,p;b) defined in (4.2) satisfies the system (4.5), under the constraint that the hydraulic head Q is given by (4.1).*

Furthermore, the details of the proof presented in [8] suggest that the theorem is true for a larger class of vorticities. This class is described through an appropriate change of variables, which reduces the complexity of the proof to the one for the constant-vorticity case; for candidate members of this class, we refer to [50–52].

The above theorem implies that the dominant term of the wave height of this approximation of the water wave is of order *b*, i.e.
4.6
This condition indicates the accuracy of approximation of genuine waves in the following sense: equations (4.5) show that *h*^{(2N+1)} is an approximation of a solution up to order *b*^{2N+2}, whereas the relation (4.6) shows that *h*^{(2N+1)} differs from the laminar solution at order *b*. Consequently, *h*^{(2N+1)} is ‘closer’ to a non-laminar solution. An illustration of this is given in figure 4, for a fixed relative error.

For the explicit formulae of higher-order expansions, we refer to [8]; fixing the values of the parameters 4.7 we depict these expansions for the irrotational case, up to the fifth order, in figure 4. More complicated formulae for the case of constant vorticity are provided in [8], which we avoid presenting here for matters of brevity; the illustration of some examples is given in figure 3.

Finally, the proof of the theorem in [8], being constructive, provides a deterministic algorithm for computing water waves of maximal amplitude. The realization of such an algorithm for the computation of water waves in this class of vorticities is a work in progress [53].

## 5. Computational approaches

In what follows, we discuss two iterative methods for computing water waves of large amplitude, following the numerical treatment performed in [7,15], obtaining several interesting results which agree with the relevant analytical predictions. We point out that the initial guess for these iterative schemes is of great importance, so that our algorithms select the branch of the bifurcation diagram containing non-laminar flows. Low-order expansions arising from (4.2) are suitable for providing this initial guess.

### (a) A penalization method

In [7], for the selection of waves of large amplitude, we impose the constraint of maximization of the norm of the slope of the wave profile, over one wavelength. A penalization method to solve numerically this constrained optimization problem in the fixed rectangular domain *R*, given in (3.1), is proposed. The main point is the minimization of the quantity
subject to the PDE constraint that *h* satisfies (3.3).

The energy function is chosen in such a way that it vanishes for laminar flows (in which *h*_{q}≡0), thus selecting genuine waves. This permits us to provide accurate simulations of the surface water wave and of the main flow characteristics (fluid velocity components, pressure) beneath it.

A brief description of the algorithm, which implements the penalization method, is given as follows:

(i)

*Initialize*. Choose a constant*k*=0*ν*_{0}>0 (typically small). Use an initial guess*h*^{(0)}of the solution of (3.3), by selecting from (3.10). These forms guarantee that (3.11) holds.(ii)

*k*→*k*+1. Given*h*^{(k)}we solve the linear equation for*h*, obtained by (3.3) when freezing the coefficients of lower order from the previous iteration step. The solution is denoted by*h*^{(k+1)}.(iii)

*Compute*. Because we work with a semi-implicit scheme, we have to use a relatively small step size, which is determined here.— If then put

*ν*_{k+1}=*ν*_{k}and update We emphasize that is the steepest descent energy of the quadratic functional . From this perspective, we might call this algorithm a*steepest descent*algorithm.— Else put

*ν*_{k+1}=0 and update The function*F*is given by*F*(*p*)≃2*H*(*p*,λ*).

(iv) Among the stopping criteria of the algorithm is the satisfaction of the system of equations (3.3) up to small error. If the algorithm is not terminated, then move to the second step.

The different branches of the third step guarantee that the residuals of the boundary conditions and the differential equations are decreasing. More details on the realization of the algorithm, as well as the justification of this realization, are presented in [7]. In figure 5, we present the basic results for the irrotational case, considering the wave profile and the velocity field. This algorithm is general in the sense that we obtain results for other cases of vorticity, with some results for constant and linear cases being presented in [7]. Here, we will illustrate results of this more general case through the more sophisticated approach which is presented in the following subsection.

### (b) A numerical continuation technique

In this section, we describe a numerical continuation approach for computing water waves of large amplitude through the mathematical formulation that is given by (3.3). Appropriate techniques are applied in order to overcome some particular obstacles of the problem related to the appearance of turning points, bifurcating points and stagnation points. As a result, the induced algorithm in [15] is efficient and not expensive. Additionally, it uncovers new parts of the interesting branch of the bifurcation diagram, i.e. families of waves with novel characteristics. This algorithm, at its limiting point, computes water waves near stagnation; a family of such waves is depicted in figure 6. Furthermore, these numerical results agree with the ones in the literature (see [16,17]), and have a somewhat improved accuracy, which, in some cases, is important enough to give an additional understanding of the solutions of the problem. Finally, the generality of this algorithm is established by the computation of some cases of continuous and non-constant vorticity.

#### (i) Predictor–corrector method

Discretization of (3.3) leads to a system of nonlinear equations 5.1 with a given mapping and unknowns and . To overcome the two major problems in solving (5.1), the underdetermined system and the need for a good initial guess, we employ a numerical continuation strategy. We consider the case that at least one solution is known and the goal is to compute additional solutions. Parametrization of the solution curve and fixing the step size results in a well-defined problem and the known solutions are used to predict an initial guess. As corrector we use a Newton algorithm to solve the system of nonlinear equations. In what follows, we give a short introduction to this topic; for a more elaborated study, we refer to [54,55].

Let be the set of all pairs satisfying (5.1) and a known solution; then the problem is to find a new solution
Since this is an underdetermined system, we first have to find an additional equation. Assume that a parametrization of the solution curve in a neighbourhood of (*θ*_{k},**h**_{k}) is given by the equation *p*(*θ*,**h**,*s*)=0. Then for a fixed step size *s*, we obtain the following system:
5.2
The existence and uniqueness of a solution to (5.2) depends on the choice of *p* and the solution set . In [15], we make the following two choices for *p*:

— The natural parametrization which corresponds to fixing the value of

*θ*and is given by 5.3— The local parametrization, where instead of

*θ*we can choose any entry of**h**as parameter. Let*i*be an index of**h**; this parametrization is given by

Figure 7 shows both these parametrizations on a bifurcation diagram that plots (*θ*,**h**[*i*]) for all solutions in . In addition to , the line of all pairs (*θ*,**h**) satisfying *p*(*θ*,**h**,*s*)=0 is also plotted; the solution we seek is the intersection of these two lines. The existence and uniqueness of such an intersection depends on and the choice of *s*. In the example, has a turning point in *θ*, so no solution pair exists that satisfies the natural parametrization if *s* is too large. Indeed, if the current solution is exactly on the turning point, no *s*_{0}>0 exists such that (5.3) describes a parametrization of . In such a case, one way to proceed with the numerical continuation is by parametrizing with respect to another characteristic such that the bifurcation curve does not have a turning point; in the given example, this is **h**[*i*].

For our model problem (3.3), it proved most effective to parametrize by the entry of **h** corresponding to the highest point of the wave.

The predictor is the initial guess for the Newton scheme and will be denoted as . Of the many predictors described in the literature, we briefly introduce only two, the trivial and secant predictors:
The value *s*^{(0)} is chosen such that the parametrization equation is satisfied.

#### (ii) Constant vorticity

Let *b* be fixed, then , which is given in (3.10), is a predictor for a solution on the bifurcation branch. Note that a too small *b* will urge the corrector to converge to a laminar wave, while a too big *b* can result in a diverging corrector step. In our experience, the choice *b*=*s* with a small enough *s* leads to convergence to the non-laminar branch. Efficient predictors are also higher-order asymptotic expansions given by (4.2).

The natural choice for the bifurcation parameter *θ* is *Q*, which is indicative of the total mechanical energy of the wave, and we assume all other parameters to be fixed; see for example (4.7). Although for most of our computations, we used *Q* as the continuation parameter, we note that for waves of constant vorticity we can alternatively fix *Q* and choose *γ* to be the continuation parameter. This choice can be beneficial for overcoming some limitations of this approach and computing waves which belong to new parts of the bifurcation curve.

Considering the validation of our algorithm, in [15] we have illustrated the performance of the corrector step to solve (5.2), for which we employ a Newton algorithm. Furthermore, we have performed the same error tests on examples for which we have *a priori* knowledge of the analytical expression of the solution.

The computed waves in this section demonstrate different characteristics below and above a critical vorticity, which is *γ*_{crit}≈−2.971, for the choice of parameters given in (4.7). We emphasize the fact that we obtain, also, some new qualitative results, compared with [17].

The appearance of two turning points in *Q* and a monotonically increasing wave height along the solution curve is observed for all constant vorticities that are larger than *γ*_{crit}. The value of *γ* does have a significant influence on the stagnating waves. Indeed, the results presented in figure 6 display the following characteristics for the wave profile:

— The wave height decreases, as the vorticity

*γ*gets larger.— The shape of the wave changes with the vorticity; the crest becomes sharper and the trough wider and flatter as the vorticity increases.

The above procedure does not compute a maximal wave with a sharp angle at the top for all constant vorticities. In particular, if the vorticity is below *γ*_{crit}, it breaks down earlier; this behaviour was observed in [16,17] and attributed to a stagnation point at the bottom of the wave. In [15], we modify the above approach. We consider some *γ*_{0} above the critical value and compute the solution with maximum value of *Q* along the bifurcation curve . Then we fix *Q* and instead consider *γ* as the bifurcation parameter, looking for a solution for *γ*_{1}<*γ*_{crit}<*γ*_{0}. Now we switch back to *Q* as the bifurcation parameter and compute the upper section of , bounded above by the wave of maximal wave height and below by the wave with stagnation point at the bottom. The last wave we could compute was at *γ*=*γ*_{ℓ}=−3.062. Numerical experiments show that this approach works and that waves with stagnation point at the top exist even for vorticities below the critical value (figure 8).

In summary, the *γ*-continuation technique in the interval (*γ*_{ℓ},*γ*_{crit}) indicates the existence of waves that have a stagnation point on the free surface, for vorticities *below* the critical value, and conjectures the existence of waves with stagnation points both at the crest and at the bottom, right below the crest. We emphasize the fact that the pressure of the fluid displays a qualitative change for values of vorticity smaller than the critical value. Indeed, in figure 9*a*, where we depict the pressure of the fluid at the bottom, we see that when *γ*<*γ*_{crit} the pressure at the point right below the crest displays a ‘plateau’ behaviour instead of a local maximum, which characterizes flows with *γ*>*γ*_{crit}.

The former behaviour is more transparent in figure 9*b*. Considering the point of the bifurcation curve for which this ‘plateau’ behaviour of the pressure is more distinct, we make the following three observations. First, for *γ*=*γ*_{crit}, this takes place in the neighbourhood of the point on the bifurcation curve where the gap will appear, namely *Q*_{G}. Second, if *γ*_{ℓ}<*γ*<*γ*_{crit}, then this behaviour occurs for the two substantially different waves, which correspond to the endpoints of the gap in the bifurcation curve. Third, if *γ*<*γ*_{ℓ}, then this behaviour occurs at the last computable wave.

For a more detailed analysis of the characteristics (including pressure) of the waves corresponding to points around the gap on the bifurcation curve, we refer to [15].

### Remark 5.1

We find it worth noting that the distribution of the pressure in the bulk of the fluid could take substantially different form, depending on the coordinate system used for studying the problem. An illustrative example is presented in figure 10, for *γ*=*γ*_{crit} and *Q*=*Q*_{G}. In figure 10*a*, the level curves of the pressure are drawn in the physical domain, i.e. for the (*x*,*y*)-variables, whereas in figure 10*b* this depiction is given for the rectangle *R*, which is obtained through the partial hodograph transformation, i.e. for the (*q*,*p*)-variables. Figure 10 suggests that, for particular waves, along any streamline, the pressure displays a local minimum at *q*=0 – equivalently *x*=0.

Gathering the results of the above procedure, for different values of constant vorticity, we display the wave height and average depth for maximal waves in figure 11.

#### (iii) Linear vorticity

As a last example, we consider the case that vorticity depends linearly on the streamfunction,
Then the laminar solution is given by
The dispersion relation as given in [50] reads
with
In [15], we consider *γ*_{α}(−*p*) such that *γ*_{α}(−*p*_{0})=0 and *γ*_{α}(0)=*α* for *α*=±1, where we observe that the considered waves display similarities with waves of constant vorticity. The analysis of this particular class of examples was motivated by the discussion in [17], where it is mentioned that, from experiments, wind typically has the effect of producing vorticity in the water near the surface. Thus, we discuss a case of non-constant, continuous distribution of vorticity, which vanishes at the bottom but not at the surface of the fluid. A more elaborate study on other cases of non-constant vorticity—apart from linear—is performed in [15].

Finally, we have computed a wave for which the linear vorticity distribution takes the value *α*=−10 at the free surface and vanishes at the bottom; see figure 12. For this wave, we observe a different behaviour from the waves of negative constant vorticity *γ*<*γ*_{ℓ}=−3.062, with the most striking being the significantly larger wave height and the existence of a stagnation point at the free surface.

## Data accessibility

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## Competing interests

I declare I have no competing interests.

## Funding

The author was supported by the project *Computation of large amplitude water waves* (P 27755-N25), funded by the Austrian Science Fund (FWF).

## Acknowledgements

The author thanks A. Constantin and O. Scherzer for discussion of and useful comments on this work. Furthermore, he is grateful to the reviewers for useful comments on the presentation of this work.

## Footnotes

One contribution of 19 to a theme issue ‘Nonlinear water waves’.

- Accepted August 11, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.