The purpose of this paper is to present some recent advances in the study of water waves with vorticity and surface tension. These are periodic, two-dimensional waves over a flat bottom and the surface profiles are symmetric and monotone between crest and trough. The proofs rely on bifurcation theory.
It is a matter of common experience that many waves observed on the surface of a body of water can be roughly described by a periodic wave train, propagating steadily without change of form and with no variation along the crests. We consider two-dimensional wave trains propagating over water flowing on a current, over a flat, impermeable bed. A current here means a water flow with a flat surface and it is specified by vorticity. For example, a current which is uniform with depth is described by zero vorticity, while constant non-zero vorticity describes a linearly sheared current and so on. Ignoring viscosity, the restoring forces acting on these waves are due to gravity and surface tension.
Historically, mathematical studies of water waves have mainly been restricted to irrotational flows. Using bifurcation theory, a rich set of periodic wave trains has been found. When surface tension was absent, global continua were found, containing in their closure limiting waves with a sharp peak at the crests (Toland 1996). The crests are stagnation points, meaning that the vertical velocity component is zero while the horizontal velocity component equals the speed of the wave. At the other extreme, in the absence of gravity, explicit solution formulae were found for the case of infinite depth by Crapper (1957) and for the case of finite depth by Kinnersley (1976). For large amplitudes, these waves are overhanging, with highly rounded crests. As the amplitude increases, a limiting shape is reached, where the wave profile becomes self-intersecting, enclosing an air bubble at the trough. In the presence of both surface tension and gravity, research has mostly been confined to small-amplitude waves. It was formally observed by Wilton (1915) that the combination of gravity and surface tension causes a resonance to occur between two waves with a wavelength ratio 1 : 2, and that this creates mixed solutions, having two crests per period. The existence of such solutions, and other mixed solutions of different wavelength ratios, was rigorously confirmed later (e.g. Reeder & Shinbrot 1981a,b; Jones & Toland 1985, 1986; Jones 1989). The mathematical theory of large-amplitude waves remains largely unknown. Numerical studies indicate that there are limiting waves of two kinds: those with self-intersections, as in the pure capillary case, and those with contact points between the surface and the bottom (Okamoto & Shoji 2001).
While the irrotational setting is regarded as appropriate for waves travelling into still water (Johnson 1997), there are many situations in which it is necessary to take vorticity into account. For example, in any region where the wind is blowing, vorticity is generated (Da Silva & Peregrine 1988). Moreover, a realistic description of tidal flows is given by constant vorticity (Da Silva & Peregrine 1988), and the outflowing current at the mouth of an estuary generally exhibits a non-uniform vorticity distribution (Swan et al. 2001).
In the last few years, there has been a growing amount of interest in rotational waves. Constantin & Strauss (2004) proved the existence of global continua of finite-depth gravity waves with a general vorticity. These continua contain waves with horizontal velocity arbitrarily close to the propagation speed. Other areas of interest have been symmetry properties of these waves (Okamoto & Shoji 2001; Constantin & Escher 2004a,b), uniqueness issues (Kalisch 2004; Ehrnstöm 2005), as well as variational formulations (Constantin et al. 2006). These recent investigations have exclusively been restricted to pure gravity waves, ignoring the effects of surface tension.
Wahlén (2006b) recently considered the analogue of Kinnersley's capillary waves, and mathematically proved the existence of small-amplitude solutions for an arbitrary vorticity distribution using a method similar to Constantin & Strauss (2004). In Wahlén (2006a) the focus was shifted to capillary–gravity waves. Again, small-amplitude solutions for an arbitrary vorticity were found using bifurcation theory. The solutions found in both of these papers have one crest and one trough per period, and are monotonic between crest and trough. The existence of mixed waves remains an open question in the presence of vorticity, although Wahlén's (2006a) results indicate that such solutions could indeed exist. The aim of this paper is to present the results of Wahlén (2006a, b) to the scientific community in this field. Note that there is an earlier existence proof for capillary and capillary–gravity waves with vorticity using a different approach due to Zeidler (1973). It is the opinion of the author that the approach used by Wahlén (2006a,b) is somewhat simpler.
2. Mathematical formulation
Let us consider two-dimensional waves propagating over water with a flat bed. In its undisturbed state, the equation for the flat surface is y=0 and the flat bottom is given by y=−d for some d>0. The x-variable represents the direction of propagation and the wavelength is 2π/k, where k>0 is the wavenumber. In the presence of waves, the free surface is represented by y=η(t, x) and we choose a coordinate system so that the mean water level is zero, i.e. . Homogeneity (constant density) is a good approximation for water and yields the equation of mass conservation(2.1)The equation of motion is Euler's equation(2.2)where P(t, x, y) denotes the pressure; g is the gravitational constant; and ρ is the density. The boundary conditions for capillary–gravity waves are (cf. Johnson 1997) the dynamic boundary condition(2.3)where P0 is the constant atmospheric pressure and σ>0 is the coefficient of surface tension, as well as the kinematic boundary conditions(2.4)and(2.5)By making the change of variables and , we can assume that ρ=1.
For steady symmetric waves travelling at a speed c>0, i.e. the space–time dependence of the free surface, the pressure and the velocity field is of the form (x−ct). In a reference frame moving at a speed c, the wave becomes fixed and we can describe the problem in terms of the relative stream function ψ(x, y), defined by ψx=−v and ψy=u−c throughout the fluid and by ψ=0 on the free surface. On the flat bed, ψ is constant and its value is . Owing to the symmetry assumption, ψ is even in the x-variable. Previous study indicates that for waves not near the spilling or breaking state, the propagation speed c of the surface wave is considerably higher than the horizontal velocity u of each individual water particle (Lighthill 1978), so that u<c. Under this condition, the vorticity ω=vx−uy is a function of ψ, ω=γ(ψ). In other words, Δψ=−γ(ψ).
Introduce the functionand letwhere E is the total mechanical energy which is constant throughout the fluid according to Bernoulli's law. The dynamic boundary condition is equivalent to the conditionwhere Q=2(E−P0+gd).
The main difficulty in this problem is that η is not known a priori. For this purpose, we make a change of variables following Dubreil-Jacotin (1934). Since ψ is constant on the free surface and the bottom, and strictly decreasing as a function of y, we choose the new variables q=x and p=−ψ(x, y). Introducing the height function h(q, p)=y+d, we obtain the following formulation of the capillary–gravity problem:(2.6)where h is 2π/k-periodic and even in the q-variable.
3. The existence of steady waves
The construction of the solutions relies on the Crandall–Rabinowitz local bifurcation theorem (Crandall & Rabinowitz 1971). The first step is identifying the underlying current associated with the vorticity function γ. In terms of the formulation (2.6), this corresponds to a trivial solution, by which we mean a solution independent of q (in particular, the surface is flat). Fixing the propagation speed c, we find that there is a one-parameter family of trivial solutions, H(p; λ), parameterized by the square root of the relative horizontal velocity at the surface, , where U is the horizontal velocity component corresponding to the trivial solution H. The parameter λ ranges through the interval (−2Γmin,∞), where Γmin is the minimum value of the function Γ. The depth d and the parameter Q depend on λ.
(Pure capillary waves; Wahlén 2006b). Given the relative mass flux p0<0, the vorticity function γ∈Cα[0, |p0|] and the wave speed c>0, there exists for every sufficiently large wavenumber k>0 a curve of small-amplitude periodic waves with wavelength 2π/k.
The solutions are regular and symmetric, by which we mean (i) u and η are even, while v is odd and (ii) η has one maximum (crest) and one minimum (trough) per period and it is strictly monotone between the crest and the trough.
If γ≤0, then there exists such a curve for every positive wavenumber k.
(Capillary–gravity waves; Wahlén 2006a). Given the relative mass flux p0<0, the vorticity function γ∈Cα[0, |p0|] and the wave speed c>0, there exists for every sufficiently large wavenumber k>0 a curve of small-amplitude regular and symmetric periodic waves with wavelength 2π/k.
If σ≥σ0, where σ0 is given by equation (3.3), then there exists a solution curve for every positive wavenumber k.
Although σ is more or less constant for a water–air interface at a certain temperature, the value of σ0 depends upon the vorticity function and p0. The condition σ≥σ0 can therefore be seen as a condition on the vorticity and relative mass flux for water at a certain temperature.
When applying the Crandall–Rabinowitz theorem, one is led to study the linearization of equation (2.6) around a trivial solution. This takes the form(3.1)with w even and 2π/k-periodic. The function a is given explicitly by , and is related to the trivial solution H by a−1(p; λ)=Hp(p; λ). We remark that the second-order boundary condition on the top is rather unusual. Nevertheless, a theory for such problems does exist (Luo & Trudinger 1991) and was applied by Wahlén (2006a,b).
We are looking for points λ such that the linearized problem (3.1) has non-trivial solutions. Expanding in a cosine series leads to the spectral problem(3.2)If this problem has an eigenvalue μ=−k2n2, n∈ for some λ=λ*, then w(p, q)=u(p)cos(knq), where u is the corresponding eigenfunction, solves equation (3.1). Thus, we are particularly interested in the negative eigenvalues of equation (3.2). A description of the negative spectrum is given in lemmas 3.1 (figure 1) and 3.2 (figure 2). We begin with the pure capillary case.
(Wahlén 2006b). If g=0, then there exists for every λ>−2Γmin a unique non-positive eigenvalue μ0(λ). The function μ0(λ) is strictly decreasing with limλ→∞μ0(λ)=−∞ and .
If k>k0, we find that there is a unique λ* such that μ0(λ*)=−k2. Furthermore, for this value of λ, this is the unique negative eigenvalue of equation (3.2), and in particular the solution space of the linearized problem (3.1) is one-dimensional. It can be shown that all the conditions of the Crandall–Rabinowitz theorem are satisfied, and thus bifurcation occurs at λ*. In a neighbourhood of the bifurcation point, the non-trivial solutions inherit the nodal pattern of the linearized solution u(p)cos(kq). If γ≤0, then it can be shown that k0=0 so that there are waves for any wavenumber (Wahlén 2005).
In order to explain the situation in the capillary–gravity case, we introduce the parameters λ0 and σ0 determined by(3.3)
(Wahlén 2006a). If g>0 and σ≥σ0, then there are no non-positive eigenvalues for . For every λ≥λ0, there is a unique non-positive eigenvalue, μ0(λ), with , μ0(λ0)=0 and limλ→∞μ0(λ)=−∞.
If σ<σ0, then there is a λ1≥−2Γmin so that the following holds. For every λ<λ1 (this set can be empty), there exists no non-positive eigenvalues. For λ1<λ≤λ0, there exists two non-positive eigenvalues μ0(λ)<μ1(λ)≤0. For λ>λ0, there is only the non-positive eigenvalue μ0(λ). The function μ0 is strictly decreasing in λ>λ1 with limλ→∞μ0(λ)=−∞, while μ1 is strictly increasing in λ1<λ≤λ0 with μ1(λ0)=0. If λ1≥−2Γmin, then there is exactly one non-positive eigenvalue for λ=λ1.
When σ≥σ0, it is clear that for every k>0 there exists a λ*>λ0 such that μ0(λ*)=−k2 is the unique negative eigenvalue at λ*. Again, the Crandall–Rabinowitz theorem can be applied.
When σ<σ0, the situation is as before for λ>λ0, that is, for every k2>|μ0(λ0)|, there exists a unique λ*>λ0 such that equation (3.2) has the unique negative eigenvalue −k2 at λ*.
In the region λ<λ0, the situation is more complicated. For any λ* between λ1 and λ0, there are now two negative eigenvalues. Let and . If k0 and k1 are rationally dependent, say k0=nk and k1=mk for some k>0 and some n, m∈, then the solution space of equation (3.1) in a space of 2π/k-periodic functions is two-dimensional. If k0 is not an integer multiple of k1 (i.e. m does not divide n), then there are waves of wavenumbers k0 and k1 bifurcating at λ*. We simply apply the Crandall–Rabinowitz theorem after restricting the period to 2π/k0 and 2π/k1, respectively. However, if k0 is an integer multiple of k1, then the Crandall–Rabinowitz theorem only yields the existence of waves of the higher wavenumber k0. Restricting to 2π/k1-periodic functions does not reduce the dimension of the solution space.
It can be expected that the solution set close to these double bifurcation points is in fact much larger, containing ‘mixed’ solutions which are neither 2π/k0- nor 2π/k1-periodic (Jones & Toland 1985, 1986; Jones 1989). The existence of such solutions requires more sophisticated techniques.
Note that any sufficiently large ratio n/m can occur since μ0(λ)/μ1(λ)→∞ as λ→λ0. When λ1>−2Γmin, we can in fact get any ratio since μ0(λ)/μ1(λ)→1 as λ→λ0.
4. Some examples
(a) Irrotational flow
We now consider the special case of irrotational flow γ=0. The underlying current then has the form (U, 0), where U is a constant. The bifurcation points are solutions λ* of the equation(4.1)If g=0, then there exists a unique solution λ* for each k>0, and λ* is an increasing function of k. When g>0, the numbers λ0 and σ0 defined in equation (3.3) are given by λ0=(g|p0|)2/3 and σ0=1/3(g|p0|4)1/3. If σ≥σ0, then there is again a unique solution of equation (4.1), which is increasing in k. If σ<σ0, then there is still a unique solution for each k, but it may not be increasing in k. It is possible that for both k0>0 and k1>0, the same solution λ* can be obtained. If k0 and k1 are rationally dependent, then λ* is a double bifurcation point.
If we write equation (4.1) in terms of the depth d and the speed of the wave relative to the underlying uniform current U*, we obtainThis is the dispersion relation for linearized small-amplitude waves. Double bifurcation points occur if there are two rationally dependent wavenumbers k0 and k1 giving the same ‘eigenspeed’ c−U*. In principle, there exist periodic waves of minimal period 2π/k for every speed given by the above dispersion relation. The only restriction being that we have not proved the existence of waves of minimal period 2π/k1 when k0=nk1 for some n∈.
The dispersion relation shows that the relative wave speed is an increasing function of the depth. In the limiting cases g=0 and σ=0, two opposite phenomena occur. In the first case, the relative wave speed is an increasing function of the wavenumber, i.e. shorter waves travel faster. In the second case, the wave speed is a decreasing function of the wavenumber, so that longer waves travel faster. When both gravity and surface tension are present, the wave speed is an increasing function of k when σ/gd2≥1/3; otherwise, it is decreasing for small values of k, increasing for large values of k and has a unique minimum. In the latter case, the dispersion relation is qualitatively the same as for pure gravity waves for long waves and the same as for pure capillary waves for short waves. The minimum point of the wave speed as a function of k has a finite limit as d→∞, and under typical physical conditions it can be seen that this corresponds to a wavelength of approximately 1.7 cm (Crapper 1984). In this sense, it can be seen that if the water is shallow or if the waves are very short, surface tension is in some sense the dominating force, while in deeper water the force of gravity dominates for most wavelengths.
(b) Constant vorticity
In the case of a constant vorticity γ≠0, the underlying current has the form (U0−γy, 0), where U0 is the horizontal velocity component at the surface. We cannot give an explicit formula for λ0 and σ0. However, we can calculate the following dispersion relation in terms of the depth d and the speed of the trivial flow at the surface :As in the irrotational case, the relative wave speed is an increasing function of the depth. Furthermore, it can be seen that it is an increasing function of γ.
There are several unanswered questions about water waves with surface tension and vorticity. In this paper, we have considered only the waves on water of finite depth. However, in the case of gravity waves, there is an explicit family of deep-water solutions with a particular non-vanishing vorticity (cf. Gerstner 1809; see also Constantin 2001b), and recently Hur (2006) used global bifurcation theory to construct large-amplitude deep-water waves for a large class of vorticity functions. Using a combination of the methods of Wahlén (2006a,b) and Hur (2006), it should be possible to construct families of small-amplitude deep-water waves with vorticity and surface tension. Note that the existence proof of Zeidler (1973) also allows for infinite depth.
Another interesting direction is the study of solitary waves. In the irrotational case, there is a vast literature on capillary–gravity solitary waves (Kirchgässner 1988; Amick & Kirchgässner 1989; Iooss & Kirchgässner 1990, 1992; Sachs 1991; Iooss & Pérouème 1993; Buffoni et al. 1996; Buffoni & Groves 1999). On the other hand, Hur (in press) has recently extended the Beale (1977) construction of small-amplitude gravity solitary waves to include vorticity. Groves & Wahlén (2006) used the spatial dynamics methods (Kirchgässner 1982; Mielke 1991) to study the solitary capillary–gravity water waves with vorticity.
Finally, in order to fully understand the fluid motion, it is important to have a qualitative picture of the particle trajectories. The leading-order analysis of the linearized Stokes problem (periodic irrotational gravity waves) suggests that all particles move on closed orbits. However, in the recent paper by Constantin & Villari (in press), it was proved that there are no closed orbits for the linearized problem. Furthermore, a recent investigation by Constantin (2006) shows that this is also the case for the fully nonlinear problem. The particle trajectories for Crapper's and Kinnersley's waves were calculated by Hogan (1984, 1986); see also the numerical study of irrotational capillary–gravity waves by Hogan (1985). Again, the particle paths do not close up. On the other hand, the presence of vorticity can lead to closed orbits in some cases. In particular, the water waves discovered by Gerstner (1809) and some related edge waves (Constantin 2001a) have the property of all particles moving on circles. A natural direction would therefore be to make a rigorous study of the particle paths of irrotational capillary–gravity waves and of rotational waves, with and without surface tension.
The author is grateful to the organizers of the 2005 Wave Motion programme at the Mittag-Leffler Institute. The author would also like to thank the referee for several useful suggestions.
One contribution of 13 to a Theme Issue ‘Water waves’.
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