## Abstract

A theory is presented which provides a model for the appearance of critical layers within the flow below a water wave. The wave propagates over constant depth, with constant (non-zero) vorticity. The mechanism described here involves adjusting the surface-pressure boundary condition; two models are discussed. In the first, the pressure at the surface is controlled (mimicking the movement of a low-pressure region associated with a storm) so that the speed and development of the pressure region ensure the appearance of a critical layer. In the second, the pressure boundary condition is allowed to accommodate the reduction of pressure with altitude, although the effects have to be greatly enhanced for this mechanism to produce a critical layer. These two problems are analysed using formal parameter asymptotics. In the second problem, this leads to a Korteweg–de Vries equation for the surface wave, and then the evolution of appropriate solutions of this equation gives rise to the appearance of a critical layer near the bottom; the corresponding problem at the surface can be formulated but not completely resolved. The appearance of a stagnation point and then a critical layer, either at the surface or the bottom, are discussed; the nature of the flow, and the corresponding streamlines are obtained and some typical flow fields are depicted.

## 1. Introduction

The existence of a critical layer below a surface wave that moves over a flow with non-zero vorticity—often referred to as a ‘shear’ flow—is well known. This layer is a region in the neighbourhood of a line (in a two-dimensional configuration) at which the wave speed is equal to the speed in the flow below the surface. The conditions appropriate for the existence of a critical layer, and the structure of this critical layer, have been understood for decades (see Benney & Bergeron [1], Davis [2] and Habermann [3] and references therein). Indeed, mechanisms for the generation of cats' eyes within an existing critical layer have also been proposed [4]. More recently, fundamental work on the nature and properties of steady water waves in the presence of vorticity has considerably advanced our understanding of water waves and the predictions of the Euler equation [5,6]. Further, careful numerical studies have produced detailed streamline patterns and surface profiles for arbitrary amplitude waves over (mainly) constant vorticity [7]. Under suitable conditions, such steady waves may contain a critical layer (with associated closed streamlines) [8–10]. (Although a possible description of the structure of the flow near a critical level involves the inclusion of a thin viscous layer [11,12], the work cited earlier makes use of nonlinearity to give the critical layer its structure.) However, all the aforementioned describe scenarios that either contain, or do not contain, a critical level: these exist, or do not, depending on the choice of vorticity and other parameters—mass flow and total energy—that prescribe the flow. There is no mechanism in any of these configurations that will allow the appearance of a critical layer in a flow that, initially, does not contain one. (Of course, one simple reason for this is that most of the analyses mentioned above are for steady flows.) It is the possibility of generating critical layers where none originally existed that we explore here.

The studies of steady, periodic water waves (of arbitrary amplitude), in the presence of vorticity—at least, on the basis of the classical Euler model—have shown that critical layers are possible for certain vorticity distributions, mass flows and total energy. The general background to these studies can be found in Constantin & Strauss [5], and the presence of a critical layer, and its properties, within this same scenario, are discussed in Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10]; see also some of the papers cited in these primary references. Most of this work has considered the case of constant vorticity (which, for linear, long waves does not predict the presence of a critical layer; e.g. [13]), and this is the choice that we make here. We suppose that, initially, the flow has no critical layer. For a critical level to appear, the flow necessarily must be unsteady, this unsteadiness being exhibited by allowing at least one of the flow parameters to vary. There is no natural mechanism for varying the vorticity—the flow is inviscid—and it would be an extreme choice to increase/decrease the mass flow rate (by allowing fluid to be pumped in/out in some fashion). Indeed, in the work of Constantin & Strauss [5] and Ko & Strauss [7], the mass flow rate is a fixed parameter, and the vorticity and energy are taken as variable parameters. The most obvious way to proceed, therefore, is to change the total energy (the total ‘head’) of the system by adjusting conditions on the surface, e.g. by imposing changes in the pressure (in some fashion) at the surface. This simple manoeuvre will allow a transfer of energy to/from the flow at the surface. So what mechanisms might we reasonably choose?

At the outset of this analysis, we must emphasize that the aim is to produce a (mathematical) model that enables a critical layer to appear in the flow. Of course, we shall endeavour to ensure that any mechanism that we invoke is based on sound physical principles, if that is possible, but ultimately we want to explore what is (reasonably) possible within the framework of the Euler equation. We also add that we aim to provide a simple, deterministic model, without recourse to some representation of collections of statistical data that describe the detailed, but complicated, nature of the motion of the oceans under wind action. Thus, with these restrictions in mind, our first task is to investigate what options are available to us; there would appear to be three obvious ones (in the absence of any viscous stresses).

The simplest option—and, in many ways, the most natural—is to impose pressure changes at the surface, so that the pressure changes in time and, possibly, also in space. (The classical theory of surface gravity waves takes the pressure on the surface to be constant, i.e. atmospheric pressure.) The variation of pressure is precisely how a storm manifests itself: typically, a low-pressure region moves over the surface of the ocean. Such a moving pressure distribution is used, for example, in Akers [14], which provides some analytical details, with numerical results. Thus our first model replaces constant pressure at the surface by the following: the pressure at the surface varies in space and time, and moves over the surface. One important consequence of this model is that the total energy (head), represented by Bernoulli's equation at the surface, is no longer conserved; it is this ingredient that opens the possibility to the appearance of critical levels in the flow.

A familiar property of pressure in hydrostatic equilibrium, in an incompressible fluid, is the linear variation in pressure with height or depth. Our second model uses this phenomenon; however, the pressure changes that we shall need are greatly in excess of those that arise by virtue of the existing density of the air above the water's surface. Nevertheless, the essential physical principle is sound. Thus, in this model, we adjust the pressure condition at the surface so that the pressure is reduced on higher waves, implying that energy is transferred to the system.

A third alternative is particularly relevant to wind-driven forcing, in which the pressure distribution is proportional to the wave slope (e.g. [15–17]). This model represents a positive pressure on one face—the wind blows on this face—and a negative pressure on the back face (corresponding, roughly, to flow separation and a reduced pressure); there is no wind load on a horizontal surface. Although this would seem to be a model worthy of detailed examination—and perhaps it should be in the future—it leads to a description of the wave development that is likely to obscure the features that are of paramount interest in this context. In particular, in the small amplitude, long-wave approximation (which is a reasonable starting point) we obtain an equation for the surface wave, *η*(*ξ*,*t*), which takes the form
written in suitable non-dimensional variables (and subscripts denote partial derivatives). Here, we are in the moving frame, *ξ*=*x*−*ct*, over a prescribed ‘shear’ flow *U*(*z*), with *β* the coefficient associated with the wind-driven forcing. If *β*=0, then we recover the familiar Burns condition defining the wave speed *c* (and hence describing the conditions under which a critical level might appear in this approximation) [13,18,19]. If *β* is non-zero, then this equation implies an exponential structure in *ξ*, describing an exponential increase in amplitude for one sign of *ξ*; for small *β*, this same result obtains, but now with . This effect is not what we seek; rather, we are interested in the situation where the variation occurs in time and applies, more-or-less equally, over a region of the surface and of the waves. The first two models (mentioned above) provide this type of mechanism; so we shall restrict our discussion to just these two cases.

In summary, we shall present a version of the classical theory of water waves, in the presence of an underlying vorticity (and we shall consider only constant vorticity), but with a new pressure boundary condition at the surface. Either the pressure will be prescribed at the surface, but allowed to vary on a suitable time scale, or the pressure provided by the atmosphere will include a reduction in pressure with wave height. In the former case, the pressure is independent of the surface wave (although this pressure will drive the surface wave); in the latter, the pressure at the surface is coupled to the amplitude of the wave. In each case, the intention is to seek, within the conventional framework of formal parameter asymptotics for small amplitude, long waves, conditions for the appearance of critical layers. In particular, we shall address the issue of whether the critical layer (and hence stagnation points in the appropriate frame) appears first at the surface or on the bottom; see the numerical work of Ko & Strauss [7], based on the analysis of Constantin & Strauss [5]. Further, we shall also be able to describe the shape of both the critical lines and the corresponding streamlines; some alternatives, that we can hope to recover, are discussed in the general considerations provided by Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10].

## 2. Governing equations

We consider a two-dimensional flow (**x**=(*x*,*z*)), such that (plane) waves propagate in the positive (or negative) *x*-direction, over a steady flow with constant vorticity; the fluid is both inviscid and incompressible. This problem is non-dimensionalized in the usual fashion: we introduce the undisturbed depth of water, *h*_{0}—the bottom is flat and horizontal—and a typical wavelength, *λ*, of the surface wave. Together with an initial, typical (or average) amplitude, *a*, of the wave, we have the two familiar parameters of classical water wave theory: *ε*=*a*/*h*_{0} and *δ*=*h*_{0}/*λ*. The velocity, **u**, is non-dimensionalized using (with , where → denotes ‘replace by’), and the pressure uses *ρgh*_{0}, where *ρ* is the constant density of the water and *g* is the acceleration of gravity. It is then convenient to scale in order to remove *δ* from the equations, by writing , and , so that the problem is now formulated for arbitrary *δ*. Thus, we obtain the set
2.1
with
2.2
and
2.3
where the surface is written as *z*=1+*εη*(*x*,*t*;*ε*) and *p* is the deviation of the pressure away from the hydrostatic pressure distribution. The term *P*(*x*,*t*;*ε*) represents the chosen pressure condition at the surface, relative to the constant atmospheric pressure. (Note that the condition *p*=*εη* on *z*=1+*εη*, i.e. *P*=0, is simply a result of measuring *p* relative to the hydrostatic pressure distribution, with pressure=atmospheric pressure=constant on the surface.)

The final form of the equations, as we shall work with them, is obtained by introducing a suitable frame associated with the wave, together with a suitable time scale:
2.4
respectively, where *c*(*τ*;*ε*) is a variable wave speed. Further, the problem of interest here requires the choice
2.5
and the underlying (Galilean) frame is chosen so that the undisturbed flow (*u*=−*γz*,*w*=0) is zero on the bottom. The vorticity in this background flow is simply
2.6
the sign being chosen to correspond to the convention used in much of the earlier work (which was based on (*x*,*y*) coordinates); we are not concerned, here, with the case of irrotational flow, *γ*=0. Now equations (2.1)–(2.3), with (2.4) and (2.5), become
2.7
2.8
and
2.9
with
2.10
2.11
and
2.12
The vorticity associated with the perturbation away from the background state is then
We seek a solution in 0≤*z*≤1+*εη*(*ξ*,*τ*;*ε*), for and *τ*≥0, which possesses an asymptotic structure based on *ε*→0. The two models that we now examine are

model 1: pressure

*P*(*ξ*,*τ*) is prescribed;model 2: set

*P*=−*αη*(*ξ*,*τ*;*ε*) for a given constant*α*>0.

## 3. Model 1: formulation

This choice involves a prescribed pressure on the surface that moves (at a given but, in general, variable speed) and which may also evolve on the slow time scale. An immediate issue is the relation between this speed and the speed of propagation of the surface waves in the absence of any forcing; this must be addressed before we can proceed. In order to do this, we briefly revert to the equations written in terms of the original (non-dimensional, scaled) *x* and *t*, with *ε*→0. From equations (2.7)–(2.12), we obtain the set
with
Directly, we find that the equation for *η*(*x*,*t*), at this leading order, is
and so we obtain
where
where *c*_{1}, *c*_{2} are the roots of *C*^{2}+*γC*−1=0.

Now with and *c*≠*c*_{1},*c*_{2}, the solution for *η* is
for arbitrary functions *F*(⋅) and *G*(⋅), given *P* and *c*(*τ*). On the other hand, if *c*=*c*_{1} (or, correspondingly, *c*_{2}), then the particular integral for *η* becomes
but, as we now discuss, this will not be relevant to the development that we present here.

We suppose, in the near-field (defined by *t*=O(1) and containing the time, e.g. *t*=0, at which initial data are prescribed), that we have suitable initial data on compact support (over a domain *x*=O(1)). In general, there will be three wave components: one moving upstream and one downstream (determined by the speeds *c*_{1}, *c*_{2}), and the one associated with *c*. Unless *c* is very close to either *c*_{1} or *c*_{2}, then when the three components have reached the far-field (*t*=O(*ε*^{−1}) or *τ*=O(1)) they will be distinct and separate waves, no longer overlapping and, indeed, moving further apart. Now, we want to vary *c*(*τ*) so that criticality occurs; this requires *c* to attain a value of 0 or −*γ*, and then to decrease/increase further to reach a value between 0 and −*γ*. In order to avoid the difficulties of *c* passing through the values *c*_{1} or *c*_{2}, we elect to discuss the problem in which *c*(*τ*) starts from a value that satisfies
3.1
and
3.2
and not close to either *c*_{1} or *c*_{2}. Thus, in the far-field, we have only one component to consider: the one moving at speed *c*, which is driven by the moving pressure distribution. (We should remember that the aim in this study is to develop a mathematical model that describes the appearance of a critical layer, not to analyse all aspects of a more general problem—which might be a worthy subject for a later study.) Consequently, we return to equations (2.7)–(2.12), which are valid in the appropriate far-field, and seek an asymptotic representation of the surface wave, *η*(*ξ*,*τ*;*ε*), and of the flow field.

To proceed, we assume that there exists a (formal) asymptotic solution of (2.7)–(2.12), which takes the form
3.3
and
3.4
for given *P*(*ξ*,*τ*) and *c*(*τ*). We shall assume throughout that the choice of *c*(*τ*), and particularly of *P*(*ξ*,*τ*), will allow the construction of uniform expansions. A cursory examination of the structure of this problem suggests that it is sufficient for *c*(*τ*) to be a *C*^{0} function, but we should allow *P*(*ξ*,*τ*) to be . (We are at liberty to choose both these functions, so we may be as prescriptive as necessary; we comment further on the uniform validity of this expansion later.) The solution, at leading order, follows directly as
but since *c*^{2}+*γc*−1<0 for our choice of *c*(*τ*), it is convenient to write
3.5
This is consistent with the scenario that we have constructed: a wave of elevation is produced by a reduced pressure (*P*<0) on the surface. (Similar issues, with *c* inside the domain (*c*_{1}, *c*_{2})—or (*c*_{2},*c*_{1}), depending on the sign of *γ*—are mentioned in Akers [14].) There is no need, at this stage of the analysis, to consider higher order terms; we shall do that later.

Thus, we have
3.6
with
3.7
and so stagnation points (*u*−*c*=0 in this frame), and then critical levels, can appear if *c*+*γ* or *c* are sufficiently small. Note that we use here the full, original (non-dimensional) expression for *u*: background flow+perturbation. The form of this asymptotic solution suggests that a critical size will be *c*+*γ*=O(*ε*) or *c*=O(*ε*)—and remember that, in this model, we may choose *c*(*τ*). Let us set
3.8
or
3.9
and so a stagnation point can occur first at *z*=0 (the bottom) for (3.8), and at *z*=1 (the surface, in this approximation) for (3.9). However, we require a choice that, initially, ensures that *u*−*c*<0 (which guarantees no stagnation within the flow field) but which allows *u*−*c* to become zero and then positive, as the pressure distribution moves and evolves. Of the various options available to us, we limit ourselves to waves of elevation (so −*P*>0; see (3.5)), and then for (3.8)
3.10
and for (3.9)
3.11
The possibility of stagnation points appearing first at the bottom or on the surface (and along the vertical line through the peak, as we shall also see) is already known [7,20–22]. Of course, our asymptotic approach—based on small amplitude, long waves—cannot predict a critical value of *γ* that separates these two regimes (as was found in the numerical work of Ko & Strauss [7,23], for steady, periodic waves).

Now that we have seen how our initial investigations have predicted the possibility of stagnation points and eventually (we expect) critical levels, we incorporate the choices (3.10) and then (3.11) into the full equations, and analyse more carefully.

## 4. Model 1 with *c*=*εC*(*τ*)

The procedure here is to write *c*=*εC*(*τ*) in equations (2.7)–(2.12) and, in addition, to observe that this choice implies *p*=O(*ε*) (see (3.7)) and so we also write *p*→*εp*. The governing equations in this case are therefore
4.1
4.2
and
4.3
with
4.4
4.5
and
4.6
Further, in order to take the asymptotic expansion to higher order, we employ the familiar technique in these problems that involves expanding the surface boundary conditions about *z*=1. (This is easily confirmed as a valid procedure here, in common with all long-wave approximations in water wave theory, because the solution is polynomial in *z*.) We seek an asymptotic solution of this set, (4.1)–(4.6), using the asymptotic expansion introduced earlier (see (3.3) and (3.4)); at leading order, this gives (for *γ*≠0)
with
The solution, for which any disturbance is driven solely by *P*, takes the form
The extension to the next order is routine, producing (for example)
where
corresponding results are available for *p*_{1} and *η*_{1}. It is clear that these terms, together with the higher order terms, constitute a representation that is uniformly valid, provided that *P* and all its derivatives—each term ‘*n*’ involves 2*n*+1 or 2*n* derivatives in *ξ* (depending on the function being expanded)—remain bounded. It is typical of far-field expansions for wave propagation, that validity can be demonstrated, for suitable initial data, for times *τ*∈[0,*τ*_{0}] (equivalently *t*∈[0,*τ*_{0}/*ε*]) for some fixed *τ*_{0} as *ε*→0.

Let us choose to consider the situation in which *P* moves according to *c*=*εC*(*τ*), but does not evolve in *τ*, i.e. *P*=*P*(*ξ*); then
On this basis, we construct an asymptotic representation of *u*−*c*:
The critical level (where *u*−*c*=0) is then determined by
4.7
to leading order, or
4.8
when the next order is included. If this prescription implies *z*<0, then there is no critical layer within the flow field; if *z*=0, then a stagnation point (or, possibly, stagnation line) appears on the bottom; if *z*>0, then a critical layer sits in the flow field. In order to describe, in simple terms, what this predicts for the flow, let us consider a function −*P*(*ξ*)>0 with a single peak at *ξ*=0, and which otherwise decays to zero as (or, essentially equivalently, is on compact support on [−*ξ*_{0},*ξ*_{0}] for some *ξ*_{0}>0).

For *γ*>0, then (to leading order) if there is no stagnation point, nor a critical layer within the flow; when *C*(*τ*) is reduced to , a stagnation point appears on the bottom, precisely below the peak of the surface wave (because *η*∼−*P*; figure 1*a*). Now, as *C* is reduced further, so that , there is a line given by (4.7)—the critical level for this flow—that follows the shape of the surface and extends between two stagnation points on the bottom. Once a critical line has formed inside the flow field, closed streamlines then arise. The streamlines, in this moving frame, are determined from
to this order. Thus, we obtain an expression for the streamlines:
4.9
which necessarily sit in an O(*ε*) neighbourhood of the bottom, by virtue of the approximations used here (although we can readily reposition this region by moving the critical line further into the flow). We comment that it is easily confirmed that there is not a separate asymptotic region (a boundary layer) near the bottom; the solution constructed here is uniformly valid from top to bottom, across the flow field. All this information is depicted in figure 1*b*. In these two cases, the wave is moving (slowly) to the right. If, eventually, *C*<0 (so the wave has now changed direction), a critical level extends to infinity in each direction (at least, for our chosen form of –*P*; figure 1*c*). (If we include the higher order contribution given in (4.8), then the values of *C* just described are adjusted by O(*ε*)—the changes depending on *γ*, and *P*^{2} and *P*_{ξξ} at the peak—but the overall picture is unaltered.) A similar sequence emerges in the case of *γ*<0 with *C*<0 initially: —no stagnation points; —there is a single stagnation point on *z*=0 directly below the peak; —a critical level exists within the flow field.

## 5. Model 1 with *c*=−*γ*+*εC*(*τ*)

The development here follows the previous one quite closely, although some of the details are, in important areas, different. First we note that the choice *c*+*γ*=*εC*(*τ*) implies that, not only is *p*=O(*ε*), but so are *u* and *w*. Thus, we perform the transformations
and then equations (2.7)–(2.12) become
with
The same asymptotic structure as used previously then gives
where
Again, let us allow *C*(*τ*) to evolve but choose *P*=*P*(*ξ*), then
5.1
thus, at this order, we obtain
5.2
Then, for *u*−*c*<0 near *z*=1 with *C*=O(1), we require *C*>0 and so *γ*<0 (as noted in (3.10)). When we include the detailed shape of the surface (*z*∼1+*εη*_{0}+*ε*^{2}*η*_{1}), we see that
5.3
At leading order, (*u*−*c*)|_{surface}<0 if ; a critical layer will appear (near the surface) if , and a stagnation point first arises at the peak when . The critical level (where *u*−*c*=0) is defined by the curve
which is independent of the shape of the free surface; cf. the critical level near *z*=0. Further, we can easily obtain the equation of the streamlines (in the moving frame, to this order) in the form
5.4
cf. the streamlines near the bottom, (4.9).

The picture here is therefore somewhat simpler than for that near the bottom: as the surface wave slows down (*C* decreases), so first a stagnation point appears at the peak and then a line (parallel to the bottom) becomes the critical level; figure 2. (If *C* decreases further, the critical line will eventually sit below the minimum of the surface and so will extend to infinity in each direction.) The higher order terms merely provide a small adjustment to the shape of the critical line, but this does now involve the shape of the surface. Thus, at leading order, we have predicted the appearance of a critical level near the surface but, importantly, the shape of this line does not follow that of the surface, although—of course—the streamlines within this region do (figure 2).

## 6. Model 2: formulation

In this model, we replace the arbitrarily assigned pressure (*P*) by −*αη*(*ξ*,*τ*;*ε*), where *α* is a positive constant. We could allow *α*=*α*(*ξ*,*τ*), but this would not constitute a physically reasonable option: the underlying model here makes use of the hydrostatic pressure distribution in which *α* is proportional to the density of an incompressible fluid. Here, the only relaxation of reality is to permit this constant to be as large as O(1), if this is required in order to accomplish our aim of generating a critical layer. As we found in model 1, we should first examine, in outline, the general effect of this new surface boundary condition; once we have some guidance, we can make specific choices and analyse the problem more carefully.

The governing equations, following directly from (2.7)–(2.12), with *P* replaced by −*αη* in equation (2.11), are
6.1
6.2
and
6.3
with
6.4
6.5
and
6.6
One further adjustment is that, here, we may take *ξ*=*x*−*ct*, where *c* is a constant to be determined (but which may depend on *ε*); *c* is the speed of linear, long waves in this flow. We therefore tackle the problem in a way similar to that which produces the classical Korteweg–de Vries (KdV) equation in water wave theory, although, as we shall see, the balance of terms is rather different here.

We invoke the same asymptotic structure as in our previous discussions (see (3.3) and (3.4)), together with ; at leading order we then obtain (from (6.1)–(6.6))
for arbitrary *η*_{0}(*ξ*,*τ*), this solution being defined for *z*∈[0,1]. In this derivation, we have assumed—as in our earlier work—that the only disturbance to the background state is that generated by the surface wave *η*∼*η*_{0}.

The possible speeds of propagation are given by
and these two roots give speeds that are either faster or slower, as appropriate, than the maximum and minimum values of the constant vorticity background flow (precisely in accordance with the Burns condition [13,19]). Nevertheless, we may get close to these maximum/minimum values (0,−*γ*) by allowing 1−*α*→0. Indeed, we see that, with 1−*α*=Δ→0, then
6.7
In this exercise, we seek a choice of (the size of) the constant Δ that will allow, as the wave evolves, the appearance of a critical layer. In the first option given for *u*−*c*, we see that a critical level might appear near *z*=0, as *η*_{0} increases, if Δ=O(*ε*) and *γ*>0. (We shall always ensure that 1−*α*=Δ>0 in order to maintain some contact with physical reality.) The second expression for *u*−*c* suggests that a critical layer, if it appears at all, will be near the surface. In this case, however, the situation is less clear: with *z*=1 (the surface, to leading order), the term associated with the surface wave (*ε*Δ*η*_{0}/*γ*) is always smaller than Δ/*γ*, and so it is not immediately obvious how a balance is possible that will change the sign of *u*−*c*. On the other hand, the surface is more accurately *z*∼1+*εη*_{0}, so a way forward may become evident; we shall show that the problem can be formulated, but it gives rise to a more general variant of the KdV equation (which we cannot solve).

Before we proceed, we should comment that the equations (6.1)–(6.6) possess a scaling property that leads to an alternative interpretation of our approximation. The scaling *p*→(1−*α*)*p*, , , (and then removes the factor 1−*α* from the problem; however, this is at the expense of changing the size of the vorticity as Δ→0—and we wish to retain *γ*=O(1)—and, more significantly, of changing the time scale. This latter result is not consistent with the underlying approach that mirrors the classical conditions and time scales that lead to the KdV equation, which is what we hope for. The scaling property does imply, however, that this same analysis can be recovered by a suitable limit of the constant vorticity, with an evolution of the flow on a suitable time scale. Here, we choose to retain, as a separate parameter to be adjusted, 1−*α*=Δ, with the time scale defined solely in terms of the initial amplitude of the wave, and the vorticity fixed as an O(1) constant.

In what follows, we set 1−*α*=Δ=*ελ* (*λ*=O(1)), and then consider the two available choices for *c*, the first being *c*=O(*ε*); so we write *c*=*εC*.

## 7. Model 2 with *c*=*εC*

The boundary condition (6.5), with 1−*α*=*ελ*, implies that *p*=O(*ε*), and so we transform according to *p*→*εp*; equations (6.1)–(6.6) then become
with
We seek a solution based on our familiar asymptotic expansion (see (3.3) and (3.4)) which gives, at leading order (valid for *z*∈[0,1])
for arbitrary *η*_{0}(*ξ*,*τ*) (and *C*_{0} is also undetermined at this order). At the next order, after employing—as before—Taylor expansions about *z*=1 in the surface boundary conditions, we find that
together with a corresponding expression for *p*_{1}, and with
7.1
and
7.2
We recognize equation (7.1) as the famous KdV equation, written in these far-field variables (*ξ*=*x*−*εCt*,*τ*=*εt*), here associated with a contribution from the constant vorticity (*γ*) of the background flow. This form of KdV equation is consistent with that obtained from the consideration of propagation over a flow with arbitrary vorticity [24]. As we have mentioned earlier, these expressions, and all those that follow this development, constitute terms in a uniformly valid asymptotic expansion (at least for *τ*∈[0,*τ*_{0}]), as *ε*→0, provided that *η* and all its derivatives remain bounded. This will certainly be the case if the initial data are suitably differentiable.

The conditions for the appearance of a critical layer require (at this order)
to be negative, then zero and eventually becoming positive. (We note that this expression for *u*−*c* corresponds precisely with that given in (6.7).) It is clear—again, as we have already noted—that any critical layer based on this expression will emanate from a stagnation point on the bottom (*z*=0). Now *u*−*c*<0 on *z*=0 if *γ*>0 (because *ελ*>0) and *γη*_{0} is small enough, i.e. (and we shall restrict ourselves to waves of elevation); thus the following description applies only to positive vorticity. The critical level is given (at this order) by
provided that this produces *Z*>0; a stagnation point appears when *η*_{0} attains the value *λ*/*γ*^{2}, and this will be a single point if *η*_{0} possesses an isolated maximum. When a critical level has appeared, the closed streamlines can be recovered from
7.3
at this order; cf. (4.9) and (5.4). So what mechanism will allow the appearance of a stagnation point and then a critical layer?

The simplest scenario is to consider a suitable initial-value problem for the KdV equation (which could be the corresponding initial data in the near-field, which then propagates, essentially unchanged, until it reaches the far-field). This is most easily accomplished by selecting an initial profile—such as a suitable sech^{2} function—that evolves into exactly two solitons. (This same device was used in Johnson [4] to describe the formation of a cat's eye within an existing critical layer.) Because the amplitudes of the two solitons necessarily satisfy the condition that one has an amplitude greater than the initial profile, and the other smaller [25] we can arrange the following. Let the initial amplitude be less than *λ*/*γ*^{2}: there is no stagnation point (or line) within the flow. As the leading—the larger—soliton develops, there will be a time at which the maximum value (the peak) is equal to *λ*/*γ*^{2}: a stagnation point appears (at this instant) on *z*=0, directly below the peak of the soliton. Thereafter, the soliton grows to its maximum amplitude and continues to propagate: there is a critical layer, with closed streamlines—a cat's eye—between points on *z*=0, which sit directly below those points on the surface at which *η*_{0}=*λ*/*γ*^{2}. For finite time, the surface shape in the neighbourhood of a soliton and the corresponding critical line are not quite symmetric about a vertical axis, but symmetry is approached as the wave moves off to infinity. The resulting flow configuration under this soliton is depicted in figure 3. We add that, for an initial profile that generates more solitons, it is possible to have more than one cat's eye, but these are separated from each other and move apart, because the amplitude between solitons tends to zero as they propagate forwards at different speeds.

## 8. Model 2 with *c*=−*γ*−*εC*

With 1−*α*=*ελ* and *c*=−*γ*−*εC*, we require (as we did in model 1 with the choice appropriate for a surface critical layer; see §5) the scaling
to give the equations
with
The vorticity associated with the perturbation of the background state is then *u*_{z}−*εw*_{ξ}=0. A routine asymptotic expansion, following exactly the pattern described earlier, yields (for example)
8.1
and
8.2
At leading order, we also find that *C*_{0}=*λ*/*γ* and
8.3
and this equation, which describes the surface to leading order, indicates the presence of a serious difficulty. In this approximation, any surface wave with a negative slope moving to the right (correspondingly: positive/left)—and a typical wave of elevation will certainly possess this property—will progressively steepen, tending towards the vertical at some point on the profile. Of course, we can expect this to be balanced by a contribution from dispersion in the neighbourhood of this steepening front. Indeed, the higher derivative terms in the expansions (8.1) and (8.2) are evident (and it is easy to obtain the pattern for all higher order terms in this respect), and these will become larger as the wave steepens. However, there is an additional complication: we expect such terms to play a role in the equation that describes the surface, and this requires evaluation of *w* on *z*=1 (being the surface after the use of Taylor expansions in the surface boundary conditions)—but then this higher derivative term vanishes in (8.1). These observations suggest that the structure of the problem is quite involved.

It is an instructive exercise to explore how the dispersive property depends on depth, driven by the relevant (variable) scaling:
8.4
where (*ξ*_{0},*τ*_{0}) is the position and time, respectively, at which the slope of the profile first becomes vertical (according to (8.3)). The neighbourhood of the surface (*n*=1), and the bottom (*n*=0, so *z*−1=O(1) contains *z*=0), must be treated separately, the expansions then being matched. We adopt a more direct route that picks out the surface wave (but gives less information about the behaviour-with-depth close to the steepening front); this approach makes full use of the matching principle. We present the main results in outline only because, as we shall see, there is a limit to what we can describe in any detail.

The first stage is to observe that the asymptotic expansion for *u*, (8.2), for any *z*∈[0,1+*εη*], is not uniformly valid where , this scaling corresponding to the general version in (8.4), evaluated for the region away from the surface, i.e. *n*=0. We introduce
With this scaling, we obtain the familiar Laplace's equation for , here unscaled:
and so the relevant solution (satisfying the bottom boundary condition) takes the form
8.5
where we have assumed that the appropriate Fourier transform exists; correspondingly,
8.6
both valid as *ε*→0. This solution, at leading order, incorporates full linear dispersion (in that there is no further approximation involving, for example, long waves). The kinematic condition at the surface requires that
to leading order, which then defines *A* in terms of *η*.

The solution just described is defined for all *z*∈[0,1+*εη*], and so it can be evaluated in the neighbourhood of the surface (with *z*=1+*εZ*). However, a careful examination of the form of this expansion—the higher order terms and the form of the governing equations make this clear—shows that it exhibits a further breakdown even closer to (*ξ*_{0},*τ*_{0}): this occurs for (*χ*,*σ*)=O(*ε*) where . Thus we now introduce
and then we recover the full, governing equations, bereft of any dependence on the parameter:
8.7
together with the surface boundary conditions
8.8
where we have written *η*(*ξ*,*τ*;*ε*)≡*H*(*X*,*T*;*ε*); the relevant vorticity is *U*_{Z}−*W*_{X}=0.

At first sight, it might be thought that we have a problem that cannot be usefully addressed. However, we have one significant advantage here: we know the asymptotic structure (and can find a solution to any desired order) for any *z*∈[0,1+*εη*], but for a slightly larger (*ξ*−*ξ*_{0},*τ*−*τ*_{0}); viz. (8.5) and (8.6). There is now a standard procedure available to us; we solve (8.7), with the boundary conditions (8.8), to produce a solution that is valid near the surface. Of course, this does not generate a solution away from the region of the surface, because it is valid only for *Z*=O(1). However, we have a solution for all *z* (and so for *Z*=O(1), in particular), but valid a little further away from the region of maximum steepness, i.e. (8.5) and (8.6). We match these two solutions, for *Z*=O(1), thereby ensuring that the solution very close to (*ξ*_{0},*τ*_{0}) contains information from a region further from (*ξ*_{0},*τ*_{0}) which does satisfy the bottom boundary condition; this is a familiar role of the matching principle.

From equations (8.7), with the irrotationality of the velocity field (*U*,*W*), we have
and so we write
8.9
where *β* is a constant chosen to ensure that the appropriate integrals exist (e.g. ). (These terms in *β* can be reabsorbed into *B* and *D*, if required.) Correspondingly, we then have
8.10
and we may obtain *p* directly; the kinematic condition at the surface becomes
There is a corresponding equation associated with the pressure condition at the surface (which, of course, we assume is consistent with the equation above—which is certainly the case if a solution exists). The procedure now involves matching the solution represented by (8.9) and (8.10), which is valid for *Z*=O(1) and , to the solution given by (8.5) and (8.6) (which is valid for , after evaluation on *Z*=O(1)).

The resulting calculation is altogether routine, but rather intricate; the complete solution in each region, to leading order, matches precisely with the choices
and
which shows the appropriate transformation between the integrals; the relabelling here, between *A* and *B*, is equivalent to that between *η* and *H* (used earlier). The essential argument that underpins this process is that we assume that the dominant contributions to the various Fourier transforms come, in all cases, from wavenumbers of O(1). This principle has been applied, most particularly, after employing the appropriate scalings (in terms of *ε*) between the integrals, followed by *ε*→0. Further, when we note, at leading order as *ε*→0, that
we can express the equation of the surface in the form
8.11
(The *H* associated with the hyperbolic functions is evaluated as *H*(*X*,*T*); the integrals in *κ* can be evaluated in terms of transcendental functions, but there is no virtue in pursuing this here.) This equation incorporates a contribution that is typically associated with linear dispersion—the term on the left—but here it involves full (linear) dispersion and is nonlinear! Correspondingly, the term on the extreme right is a contribution from dispersion to the familiar nonlinear dispersion (as appears, for example, in the Camassa–Holm equation).

It is clear that we cannot continue with our primary aim here: to describe the appearance of a critical layer, near the surface, as the surface wave evolves. We are unable to solve the governing equation, (8.11), for the surface wave, and a detailed discussion of this new equation—no doubt of some interest—is beyond the scope of this current investigation. We can reasonably suppose that solutions exist, analogous to those of the KdV equation, which describe an evolving wave; a suitable solution could then be used (as in §7) to provide a mechanism for the appearance of a critical layer at the surface. This is not a route that we choose to follow, at present.

## 9. Discussion

Our intention in this paper has been to present mechanisms for the appearance of critical levels in the flow (necessarily with vorticity) below a surface wave. We have seen that, even in the simplest case of constant vorticity, two possible mechanisms are available: one involves no more than the adjustment, in any desired fashion, of the pressure at the surface; the other allows the pressure in the atmosphere to reduce with height. The technique employed (familiar in classical water wave calculations) involves the construction of asymptotic expansions based on a small parameter. In this discussion, we have worked with just a small amplitude parameter—the (arbitrary) wavelength parameter has been scaled out—and the pressure boundary condition is then chosen appropriately.

The first model, which is the most natural one, allows the choice of surface pressure (in a manner that mimics the movement of a low-pressure region over the ocean). The form of this pressure distribution and, more particularly, the speed at which it moves enable a stagnation point, and then a critical level, to appear in the flow. This can be engineered to occur either at the surface or at the bottom, consistent with the general results obtained by Constantin & Strauss [5], Constantin & Escher [26] and Constantin *et al*. [27], and depicted in the numerical work of Ko & Strauss [7], albeit for steady, periodic waves. Our results produce flow fields of the type predicted by Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10]: a critical level, with associated streamlines (cats' eyes), either close to the surface or close to the bottom. In the case of a pressure (–*P*) with a single maximum, both these flows emanate from a stagnation point either at the peak of the wave, or at the point on the bottom directly below the peak, respectively. Of course, our approach—perforce—is based on a small amplitude approximation (which is the case discussed in Wahlén [9]), and so we are unable to apply our description to large amplitude waves (which are allowed in Constantin & Strauss [5] and Constantin & Escher [26]). However, we can permit the critical level to move away from the region of the corresponding boundary, and into the main body of the flow, by suitably adjusting the speed of the pressure disturbance. The critical layer structure, in this case, will still be constrained to an O(*ε*) thickness about the critical level, by virtue of the surface wave of amplitude O(*ε*).

One interesting feature of this description of the critical level, not apparent in the earlier, more general discussions cited above, is that the shape of the critical line is driven by the shape of the *other* boundary. That is, the critical layer near the bottom follows the shape of the surface wave, and correspondingly for the level near the surface. The shape of the streamlines that describe the critical layer (and represents a cat's eye), in both cases, follows the shape of the surface wave. Finally, we comment that our results predict the appearance of stagnation points, and then critical levels, at the bottom for both *γ*>0 and *γ*<0, but only for *γ*<0 at the surface; these results are in general agreement with those observed by Ko & Strauss [7], but these apply, as we have noted earlier, to steady, periodic waves of arbitrary amplitude.

The second model is, we submit, technically more interesting and, as we have seen, rather more challenging. We now allow the pressure at the surface to decrease linearly with height, so that higher waves are subjected to a lower external pressure. This is clearly what happens in the standard model of an incompressible gas (air) above the fluid interface, but the effects must be greatly enhanced to produce the development of the flow that we seek. Nevertheless, we have a viable mathematical model—our primary aim—that automatically accommodates the desired phenomenon within our governing equations (rather than being arbitrarily adjusted, as in model 1). In this case, the plan has been to develop a suitable KdV-type analysis and then to allow the critical level to appear as a specific surface wave naturally evolves.

We have demonstrated that this procedure works well for the description of a critical level near the bottom, but only for *γ*>0; no corresponding solution exists for *γ*<0. The appearance of a stagnation point, and then a critical level (and associated single cat's eye), can be obtained by invoking the two-soliton solution of the KdV equation. The resulting appearance and evolution of the critical level, and of the corresponding streamlines, follow closely that seen in model 1; indeed, the shape-following property described for model 1 is reproduced for model 2. However, the corresponding problem for the appearance of a stagnation point and critical level, at the surface, is far less straightforward.

The problem of a KdV-type description of the development of a critical layer at the surface has not been satisfactorily completed. The asymptotic structure first entails a description that involves the steepening of a wave front (represented by the solution of *η*_{τ}−*γηη*_{ξ}=0), which then leads to the need for a thin, scaled region close to this front. This in turn requires an analysis of the full, governing equations without the advantage of any small parameters. Fortunately, an asymptotic solution (valid throughout the depth) in an intermediate region *fairly* close to the front is available; the matching principle has been invoked to obtain a solution near the surface and close to the front. Sadly, as we have seen, the equation describing the surface wave is highly nonlinear—far more so that the KdV equation—incorporating very strong dispersive contributions. This equation cannot be solved; so our discussion of the appearance of a critical layer at the surface must end here. Suffice it to record that there may be some virtue in analysing this new equation—perhaps some additional modelling assumptions could be used (e.g. restricting the wavenumber, *κ*, in (8.11), or making choices for *β*)—but this is altogether outside the remit that we have set ourselves in this discussion.

In conclusion, we have introduced two models that admit the appearance, and development, of stagnation points and critical levels in a constant vorticity flow; no doubt the techniques could be applied to flows with more general vorticity distributions (but we should not expect any significantly different conclusions). Some details of the structure, albeit in the asymptotic limit *ε*→0, have been obtained and described. It has not been possible to complete the discussion of one of the four scenarios that we have introduced.

## Acknowledgements

The author is pleased to acknowledge that both referees indicated a few additional, relevant references that have now been included.

## Footnotes

One contribution of 13 to a Theme Issue ‘Nonlinear water waves’.

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