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Hamiltonian long-wave approximations of water waves with constant vorticity
Physics Letters A 372 (2008) 2597–2602 www.elsevier.com/locate/pla
Hamiltonian long-wave approximations of water waves with constant vorticity Erik Wahlén Department of Mathematics, Lund University, PO Box 118, 22100 Lund, Sweden Received 6 September 2007; received in revised form 30 November 2007; accepted 5 December 2007 Available online 15 December 2007 Communicated by A.P. Fordy
Abstract Starting from a Hamiltonian formulation of water waves with constant vorticity we derive several long-wave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory. © 2007 Elsevier B.V. All rights reserved. MSC: 76B15; 35Q35; 37K05 Keywords: Water waves; Hamiltonian systems; Perturbation theory
1. Introduction An elegant way of deriving the classical model equations for long irrotational water waves is to expand the Hamiltonian in a power series in small parameters and truncate at the desired order. By keeping track of how the symplectic structure is transformed when the small parameters are introduced, the connection between the Hamiltonian formulation of the original problem and the model equations becomes apparent (see [1,2]). The Hamiltonian for irrotational water waves involves the Dirichlet–Neumann operator, which maps Dirichlet boundary data for a harmonic function to Neumann data. Although the operator is non-local, it can be approximated by differential operators in the long-wave regime. The aim of this Letter is to carry out the same procedure in the case of a underlying shear flow with constant vorticity. A Hamiltonian formulation for water waves with constant vorticity, which also includes the Dirichlet–Neumann operator, was recently derived in [3] (see also [4]). While approximate models for water waves on shear flows, both with constant and more general vorticities, have been derived previously [5–8,32], we are not aware of any derivations within the Hamiltonian E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.12.018
framework. We will work formally, leaving aside questions of function spaces etc. Lately several results concerning the rigorous justification of model equations for irrotational water waves have appeared—see, e.g., [9–13,33,35]. It would be interesting to see if the same methods can be applied in the case of constant vorticity. Note however that there is to this date no local wellposedness result for the constant vorticity water wave problem with decay at spatial infinity. While the justification of the full dynamics of the model equations is an open problem, there are several papers treating solitary travelling waves with constant vorticity. Long-wave approximations as well as numerical results can be found in [14–18,34], while existence results for the full water wave problem are given in [19–21] (see also [22,23] for particle trajectories and [24] for solitary capillary-gravity waves). To the lowest order these waves are described by a stationary KdV equation. 2. Preliminaries Let us briefly recall the governing equations for twodimensional water waves with constant vorticity. The fluid domain Ωη = {(x, y) ∈ R2 : −h < y < η(t, x)} is bounded from below by a flat rigid bottom B = {(x, y) ∈ R2 : y = −h} and above by a free surface Sη = {(x, y) ∈ R2 : y = η(t, x}), which
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we assume to be the graph of a function. We will concentrate on waves with decay and so we assume that η(t, x) → 0 as |x| → ∞. In the case of periodic waves η is assumed to be periodic and the fluid domain is restricted to a periodic cell see [25]. The wave is described by η and a velocity field (u, v) defined in Ω which satisfies Euler’s equations ut + uux + vuy = −Px , in Ωη vt + uvx + vvy = −Py − g, and the equation of mass conservation ux + vy = 0. We only deal with gravity driven waves, so that the dynamic boundary condition takes the form P = Patm on Sη , where P is the pressure and Patm is the constant atmospheric pressure. In addition, the kinematic boundary conditions are v = 0 on B and v = ηt + uηx on Sη . The vorticity of the flow is defined by ω := vx − uy . Irrotational waves correspond to ω = 0. Here we treat waves with constant ω. A shear flow (flat surface) with constant vorticity is described by a velocity field (u, v) = (−ωy + u0 , 0) for some u0 ∈ R. We seek waves which are perturbations of shear flows. In particular, at infinity the velocity field tends to that of the shear flow. For simplicity we will assume that u0 = 0, so that the asymptotic horizontal velocity at the surface is 0. Some authors advocate different choices of u0 , e.g., u0 = −ωh (meaning that u = 0 on the bottom). Choosing a different value of u0 will simply result in changes in some formulae (e.g., the formulae for wave speeds will still be correct if c is replaced by c − u0 ). Following [3,4] we introduce a generalized velocity potential ϕ. ϕ is a harmonic function satisfying ϕx = u + ωy,
ϕy = v,
H (η, ξ ) = R
ωξx η2 ω2 η3 gη2 ξ G(η)ξ − + + dx. 2 2 6 2
(2.2)
The system (2.1) can be brought into canonical form by introducing the new coordinates (η, ζ ), where ω −1 ∂ η 2 x (see [3]), but we prefer to work with (2.1)–(2.2) directly. The operator G appearing in (2.2) is the Dirichlet–Neumann operator for the fluid domain. For a given η and ξ , let ϕ be the unique solution of the problem ⎧ ⎨ ϕ = 0 in Ωη , ϕ=ξ on Sη , ⎩ ϕy = 0 on B,
ζ =ξ −
G(η) =
∞
Gj (η),
j =0
where each Gj (η) is a linear operator which is homogeneous of order j in η [26]. The terms Gj (η) may be computed by the use of a recursion formula [27]. The first few terms are given by G0 (η) = D tanh(hD), G1 (η) = DηD − D tanh(hD)ηD tanh(hD), 1 G2 (η) = − D 2 η2 D tanh(hD) + D tanh(hD)η2 D 2 2
− 2D tanh(hD)ηD tanh(hD)ηD tanh(hD) , where D = −i∂/∂x. Another important aspect of G is that it can be written as G(η) = DK(η)D, where K is positive, bounded and self-adjoint on L2 (R). K can itself be expanded in a power series ∞
Kj (η),
j =0
δξ
It is known that G is analytic in η ∈ C ∞ in a ball of radius O(h) about η = 0, with a Taylor expansion of the form
K(η) =
and ϕy = 0 on the bottom y = −h. We let ξ = ϕ|y=η be the restriction of ϕ to the free surface. As shown in [3], the water wave problem with constant vorticity ω can be formulated as the Hamiltonian system δH 0 1 ηt δη = , (2.1) δH ξt −1 ω∂x−1 where
together with either periodic or asymptotic conditions on ϕ. We then define
1/2 ∂ϕ G(η)ξ = 1 + ηx2 = (−ϕx ηx + ϕy )|y=η . ∂n y=η
where Kj (η) = D −1 Gj (η)D −1 . 3. Hamiltonian transformation and perturbation theory We will derive approximations of (2.1)–(2.2) in various parameter regimes. In order to describe these regimes, we introduce two dimensionless scaling parameters α = a/ h and β = (h/ l)2 , where a is a typical amplitude and l is a measure of the width of the wave. Different parameter regimes are introduced by scaling both independent and dependent variables in the governing equations. Although this will not change the Hamiltonian nature of the equations, the Hamiltonian function and the structure map will change. We therefore need to investigate how the structure map is affected by a change of variables (see [1,2] for a more detailed discussion). We consider a Hamiltonian system defined on a phase space X, which is assumed to be function space endowed with an inner product (·,·). The Hamiltonian is a function H : X → R with gradient δH (w) defined by dH |w (q) = (δH (w), q), for w, q ∈ X. The Hamiltonian system has the form wt = J δH (w), where the structure map J is skew-symmetric. In general, a change of variables w˜ = f (w) transforms the Hamiltonian ˜ where H˜ (w) ˜ = system wt = J δH (w) into w˜ t = J˜δ H˜ (w), −1 ∗ ˜ ˜ and J = ∂w f (w)J (∂w f (w)) . We now specialise H (f (w))
E. Wahlén / Physics Letters A 372 (2008) 2597–2602
to the structure map 0 1 J= −1 ω∂x−1 above. Suppose that f is a scaling: f (η, ξ ) = (κη, λξ ), κ, λ ∈ R. Then J is transformed into 0 1 ˜ . J = κλ (3.1) −1 κλ ω∂x−1 Spatial scaling x1 = εx, ε ∈ R, on the other hand, transforms J into 0 1 J˜ = ε . −1 1ε ω∂x−1 Since our Hamiltonian functions will be integrals over R, note that the relation dx = ε −1 dx1 introduces a factor ε −1 in the Hamiltonian. Finally, it is sometimes useful to write the Hamiltonian systems in terms of u = ∂x ξ . This corresponds to a transformation f (η, ξ ) = (η, ∂x ξ ) = (η, u) under which J changes to 0 1 J˜ = −∂x . 1 ω We will also expand the Hamiltonian function in a series H (u, μ) = H (0) (u) + μH (1) (u) + · · · + μn H (n) (u) + O(μn+1 ), where H (k) is homogeneous of order k and μ ∈ R is a small parameter. The approximate models are derived by systematically truncating the Taylor expansion of H , so that the new Hamiltonian function is Hn = H (0) (u) + μH (1) (u) + · · · + μn H (n) (u). Unlike the situation in [1,2] our original structure map is not canonical. Although some of the above transformations introduce coefficients in front of the lower right entry of the structure map matrix, we will always use combinations of transformations resulting in a cancellation between these coefficients. E.g., the map f (η, ξ ) = (μ−1 η, μ−1 ξ ) has the effect of multiplying J by μ−2 , since κ and λ are equal in (3.1). 4. The linearised equations Before proceeding to the long-wave expansions, it is instructive to look at the equations of motion linearised around a steady state. The steady state is in our case the underlying shear flow, represented by η = 0 and ξ = 0. In terms of the parameters described in the previous section, the small-amplitude regime occurs when α = μ is small while β = O(1). It is introduced by scaling the dependent variables ξ = μξ1 .
η = μη1 ,
H (η, ξ ) =
μj H (j ) (η1 , ξ1 ),
H
(2)
and
H (2+j ) (η1 , ξ1 ) = R
ξ1 Gj (η1 )ξ1 dx, 2
for j 2. On the other hand J is transformed into μ−2 J . We can now write down a sequence of approximate equations, where we retain terms up to order j and then reintroduce the unscaled variables by letting μ = 1. The linearisation is the first equation: ηt = G0 ξ, (4.1) ξt = −gη + ω∂x−1 G0 ξ (note that G0 is independent of η). Writing this in terms of u = ξx , we instead obtain ηt = −∂x K0 u, (4.2) ut = −gηx − ω∂x K0 u. This is a dispersive wave equation with dispersion relation
g tanh(hk) ω2 tanh2 (hk) ω tanh(hk) , ± + c±,ω = (4.3) 2k k 4k 2 where c±,ω is the phase speed and k is the wave number. Here ±c±,ω > 0. The presence of an underlying shear flow with positive vorticity has the effect of increasing the velocity in the positive x-direction, so that c+,ω > c+,0 and c−,ω > c−,0 . Similarly, negative vorticity leads to c+,ω < c+,0 and c−,ω < c−,0 . Note that for ω > 0 the positive phase speed equals the horizontal velocity component somewhere in the fluid if k is sufficiently large (the fluid is stagnant in a reference frame moving at the phase speed). Many existence proofs for periodic water waves exclude this situation since it leads to mathematical problems [25]. The long-wave limit, which is studied in Sections 6, 7, is formally obtained by letting k → ∞. 5. Shallow water approximation The shallow water regime, α = O(1) and β = μ2 , where μ is a small parameter, is introduced by the transformation to stretched variables x1 = μx,
t1 = μt
and scaling of ξ ξ1 = μξ.
D = μD1 . The terms in the expansion of the Dirichlet–Neumann operator become
j =2
where
R
ξ1 G1 (η1 )ξ1 ωξ1x η12 ω2 η13 − + dx 2 2 6
This induces the transformation
We then obtain ∞
H (3) (η1 , ξ1 ) =
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(η1 , ξ1 ) = R
ξ1 G0 (η1 )ξ1 gη12 + dx, 2 2
G0 = μD1 tanh(μhD1 ),
G1 (η) = μ2 D1 ηD1 − D1 tanh(μhD1 )ηD1 tanh(μhD1 ) ,
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etc. These expression can then be expanded in the small parameter μ, using 1 2 tanh(μhD1 ) = μhD1 − (μh)3 D13 + (μh)5 D15 + · · · . 3 15 We thus obtain 2 2 gη ωξ1x1 η2 ω2 η3 (η + h)ξ1x1 dx1 − + + H (η, ξ ) = 2 2 6 2 μ R
+ O(μ), and J is transformed into Jˆ = μ2 J . Alternatively, since ηt is replaced by μηt1 we can incorporate the change in t by replacing μ2 J with μJ . Returning to unscaled variables we obtain the approximate Hamiltonian 2 gη ωξx η2 ω2 η3 (η + h)ξx2 dx. − + + HSW (η, ξ ) = 2 2 6 2 R
The corresponding equations are ηt = −((η + h)ξx )x + ωηηx , ξt = −gη − 12 ξx2 − ωhξx , which upon introducing u = ξx , may be written as ηt = −((η + h)u − ω2 η2 )x , ut = −(gη + 12 u2 + ωhu)x .
(5.1)
(5.2)
6. The Boussinesq regime In this regime one studies long, small-amplitude waves. In other words, α = β = μ2 is a small parameter. The regime is induced by introducing stretched variables t1 = μt,
and scaled dependent variables 1 η1 = 2 η, μ
1 ξ1 = ξ. μ
The corresponding structure map is μ−3 J (if we incorporate the stretching of t). In the first approximation of the Hamiltonian, terms up to order O(μ4 ) are retained. All but the first term in the Taylor series for G(η) may therefore be neglected. Expanding G0 in μ and keeping terms of order at most O(μ4 ) leads to the approximate Hamiltonian 4 2 hμ4 ξ 2 gμ η1 1x1 dx1 + . HW = (6.1) 2 2 μ R
The approximate equations of motion are, in unscaled variables, ηt = −hξxx , (6.2) ξt = −gη − ωhξx .
(6.3)
This wave equation has solutions propagating at speeds
ωh ω2 h2 ± gh + , 2 4 which agrees with the long-wave limit of the dispersion relation. Note that in contrast to the periodic case stagnation points are excluded, since, e.g., the positive phase speed is always greater than the maximum horizontal velocity ωh. In the next approximation terms of order up to O(μ6 ) are retained. With the next section in mind, we write the approximate Hamiltonian in the scaled variables 4 2 gμ η1 ωμ6 ξ1x1 η12 ω2 μ6 η13 − + HB = 2 2 6 R
+
Higher order equations may be obtained analogously.
x1 = μx,
Introducing u = ξx , this may be rewritten as ηt = −hux , ut = −gηx − ωhux .
2 (h + μ2 η1 )μ4 ξ1x 1
2
−
2 μ6 h3 ξ1x 1 x1
6
dx1 . μ
(6.4)
The corresponding equations of motion are ⎧ 2 ⎪ ⎨ η1t1 = −((h +3 μ η1 )ξ1x1 )x1 h 2 − μ 3 ξ1x1 x1 x1 x1 + μ2 ωη1 η1x1 , ⎪ ⎩ 2 − ωhξ 2 ωh3 ξ1t1 = −gη1 − μ2 12 ξ1x 1x1 − μ 3 ξ1x1 x1 x1 . 1 If we let u = ξx we have ⎧ 2 ⎪ ⎨ η1t1 = −((h +3 μ η1 )u1 )x1 h − μ2 3 u1x1 x1 x1 + μ2 ωη1 η1x1 , ⎪ 3 ⎩ u1t1 = −(gη1 + μ2 12 u21 + ωhu1 + μ2 ωh3 u1x1 x1 )x1 .
(6.5)
(6.6)
The next approximation is obtained by retaining terms of order up to O(μ8 ). Written in the variables (η1 , u1 ), the corresponding Hamiltonian is 4 2 gμ η1 μ6 ωu1 η12 μ6 ω2 η13 (h + μ2 η1 )μ4 u21 − + + H= 2 2 6 2 R
−
μ6 h3 u21x1 6
−
μ8 h2 u21x1 η1 2
+
μ8 h5 u21x1 x1 15
dx1 . μ
In unscaled variables the corresponding equations are ⎧ 3 ⎪ ηt = −((h + η)u)x − h3 uxxx + ωηηx ⎪ ⎪ ⎪ 2 5 ⎨ − h2 (ηux )xx − 15 h uxxxxx , 3 1 2 ⎪ ⎪ ut = −(gη + 2 u + ωhu + ωh3 uxx − 12 h2 u2x ⎪ ⎪ ⎩ 2 + ωh2 (ηux )x + 15 ωh5 uxxxx )x .
(6.7)
7. Unidirectional waves of KdV type We now concentrate on unidirectional waves. Let us first nondimensionalise the equations by setting 1/2 g 1 u (x, y) = (x , y ), t = t , u= √ , h h gh
E. Wahlén / Physics Letters A 372 (2008) 2597–2602
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(1 + c02 )(r12 + s12 ) μ4 2
where a prime denotes a dimensional variable. This is equivalent to setting g = h = 1 and introducing the nondimensional vorticity ω˜ = √ωh . gh In the lowest order the Boussinesq equation is the linear system ηt = −ux , (7.1) ut = −ηx − ωu ˆ x,
H=
with Hamiltonian
1 2 η + u2 dx H= 2
We can write the equations in a frame moving at the speed of the right-moving wave front associated with the longest wave. This is achieved by introducing new variables x2 = x1 − c0 t1 , t2 = μ2 t1 . The change of reference frame is equivalent to subtracting c0 I from the Hamiltonian [1], where I is the conserved quantity impulse (referred to as momentum in [3]) ωη ˆ 2 I= dx. uη − 2
R
and structure map 0 1 . J = −∂x 1 ωˆ The system can be diagonalised by introducing characteristic coordinates. Introduce new variables r and s defined by (r, s) = T (η, u) where 1 1 c0 , T= 1 + c2 c0 −1 0
with c0 =
ωˆ + 2
1+
ωˆ 2 . 4
(7.2)
A simple calculation shows that J is then transformed into 0 − c0 2 1+c0 ˜ , J = ∂x 0 − c˜0 2 1+c0 2 where c˜0 = −1/c0 = ω2ˆ − 1 + ωˆ4 is the conjugate of c0 , and that H is transformed into
1 + c02 2 r + s 2 dx. H˜ = 2 R
The new equations are rt = −c0 rx , st = −c˜0 sx . Writing (6.5) in characteristic coordinates we obtain ⎧ 2 c03 ⎪ ⎪ r1t1 = −c0 r1x1 + μ2 (− c0 (3+ω2ˆ ) r1 r1x1 − ⎪ 2 r1x1 x1 x1 1+c 3(1+c ⎪ 0 0) ⎪ ⎪ 3 2 2 ⎪ c c0 c0 ⎪ ⎨ + 0 2 s1 s1x1 + 2 s1x1 x1 x1 + 2 (r1 s1 )x1 ), 1+c0
3(1+c0 )
1+c0
1+c0
3(1+c0 )
1+c0
⎪ c4 +c2 +1 ⎪ ⎪ s1t1 = −c˜0 s1x1 + μ2 ( 0 0 2 s1 s1x1 − c˜0 2 s1x1 x1 x1 ⎪ ⎪ 1+c0 3(1+c0 ) ⎪ ⎪ c0 ⎪ 1 1 ⎩ − 2 r1 r1x1 − 2 r1x1 x1 x1 − 2 (r1 s1 )x1 ),
R
2 c2 r1 s 2 c0 (r1 s1 )x1 (3 + ωˆ 2 )r13 c02 r1x 1 − − 0 1 + 6 6 2 3 2 3 4 2 2 s1x c0 r1 s1 c0 (c0 + c0 + 1)s1 dx1 − (7.4) + − 1 . 2 6 6 μ
+μ
6
R
We concentrate on the region corresponding to predominantly right-moving waves by assuming that s1 = O(μ2 ). Under this assumption the first equation in (7.3) reduces to r1t2 +
c03 c0 (3 + ωˆ 2 ) r r + r1x2 x2 x2 = O μ2 . 1 1x 2 2 2 1 + c0 3(1 + c0 )
Thus we see that, to this order of approximation, the KdV equation also provides a model for water waves with constant vorticity. Ignoring terms of order O(μ2 ), Eq. (7.5) constitutes a Hamiltonian system with 3 + ωˆ 2 3 c02 2 r − rx dx HKdV = 6 6 R
and c0 ∂x , Jˆ = − 1 + c02 which is formally obtained by setting s1 = 0 in (7.4). One may rightfully wonder about the validity of the assumption s1 = O(μ2 ). Note that under this assumption the O(μ2 ) part of the second equation in (7.3) is a linear transport equation for s1 with a forcing term which depends on r1 . Thus we may formally argue as follows: We first solve the KdV equation for r1 and then solve the resulting equation for s1 with a forcing term depending on the KdV solution r1 . If the initial data of s1 is chosen in a particular way, then s1 will remain O(μ2 ) on some time-scale. The necessary condition is that (c0 − c˜0 )s1 − μ2 c˜0 2 Es1 (Hˆ 5 )|s1 =0 = O(μ2 ) at time 0, where 1+c0
(7.3)
where we have used the relation between c0 and ωˆ to simplify the coefficients. The Hamiltonian is given explicitly by the formula
(7.5)
Hˆ 5 denotes the order O(μ5 ) terms of the Hamiltonian density in (7.4), and Es1 is the variational derivative with respect to s1 . This is of course equivalent to requiring that s1 = O(μ2 ) initially. In order to obtain a KdV equation for η1 , note that r1 = η1 − s1 . Since
2 2 c˜0 ˆ μ s1t2 = ∂x2 (c0 − c˜0 )s1 − μ Es1 (H5 )|s1 =0 + O μ4 , 2 1 + c0
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E. Wahlén / Physics Letters A 372 (2008) 2597–2602
we have to require that (c0 − c˜0 )s1 − μ2
c˜0 E (Hˆ 5 )|s1 =0 1+c02 s1
=
O(μ4 ) initially, in order to replace r1 in (7.5) by η1 . For further details we refer to [28,29]. Let us briefly discuss a different way of obtaining the model (7.5). In [7] (see also [8]) a KdV equation for waves over general shear flows was derived. If the background shear flow in nondimensional variables is given by u = U (y), v = 0, the equation reads −2I3 ηt + 3I4 ηηx + J1 ηxxx = 0,
(7.6) Acknowledgements
where 0 In = −1
The author is grateful for constructive suggestions made by the referees.
dy , (U (y) − c)n
References
and 0 0 z1 J1 = −1 z −1
(U (z1 ) − c)2 dz2 dz1 dz. (U (z) − c)2 (U (z2 ) − c)2
In our situation U (y) = −ωy ˆ and c = c0 . Straightforward calculations yield ωˆ − 2c0 , 2(ωˆ − c0 )2 c02 1 J1 = . 3(ωˆ − c0 )2
I3 =
Finally, we remark that while it is possible to derive fifth order unidirectional models from the higher order Boussinesq equation (6.7), the status of these equations is unclear. Several different models have been derived in the irrotational case [1,30,31], some of which are Hamiltonian and some of which are not. We therefore believe that a proper derivation of a higher order model within the Hamiltonian framework would need a separate discussion. A non-Hamiltonian fifth order model for water waves with constant vorticity was however derived in [5].
I4 = −
ˆ 0 + 3c02 ωˆ 2 − 3ωc 3(ωˆ − c0 )3 c03
,
Dividing (7.6) by −2I3 and simplifying the coefficients, we arrive at (7.5). The solitary waves predicted by the approximation (7.5) are waves of elevation with profile √ 3 + ωˆ 2 ˆ 2) 2 2 2 c0 (3 + ω t , x − c0 t − μ η(t, x) = μ sech μ 2c0 3(1 + c02 ) in unscaled, but nondimensional variables, where we have taken the amplitude equal to μ2 (this can always be achieved by redefining μ). Let us make some comparisons with the irrotational solitary wave (ωˆ = 0). We choose to compare wave profiles with the same amplitude. First note that for small μ, the speed of the rotational wave is greater than in the irrotational case for positive vorticity, while it is smaller for negative vorticity. The O(μ2 ) correction of the speed is an even positive function which is strictly increasing for positive ω. In other words a large value of |ω| increases the wave speed. Next we compare the width of the waves. It is an easy exercise to show that the expression √ 3 + ωˆ 2 , 2c0 where c0 depends on ωˆ through (7.2), is a decreasing function of ω. ˆ In other words the width of the wave (defined, say, as the distance between to points with η = μ2 /2) is an increasing function of the vorticity.
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Hamiltonian long-wave approximations of water waves with constant vorticity Erik Wahlén Department of Mathematics, Lund University, PO Box 118, 22100 Lund, Sweden Received 6 September 2007; received in revised form 30 November 2007; accepted 5 December 2007 Available online 15 December 2007 Communicated by A.P. Fordy
Abstract Starting from a Hamiltonian formulation of water waves with constant vorticity we derive several long-wave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory. © 2007 Elsevier B.V. All rights reserved. MSC: 76B15; 35Q35; 37K05 Keywords: Water waves; Hamiltonian systems; Perturbation theory
1. Introduction An elegant way of deriving the classical model equations for long irrotational water waves is to expand the Hamiltonian in a power series in small parameters and truncate at the desired order. By keeping track of how the symplectic structure is transformed when the small parameters are introduced, the connection between the Hamiltonian formulation of the original problem and the model equations becomes apparent (see [1,2]). The Hamiltonian for irrotational water waves involves the Dirichlet–Neumann operator, which maps Dirichlet boundary data for a harmonic function to Neumann data. Although the operator is non-local, it can be approximated by differential operators in the long-wave regime. The aim of this Letter is to carry out the same procedure in the case of a underlying shear flow with constant vorticity. A Hamiltonian formulation for water waves with constant vorticity, which also includes the Dirichlet–Neumann operator, was recently derived in [3] (see also [4]). While approximate models for water waves on shear flows, both with constant and more general vorticities, have been derived previously [5–8,32], we are not aware of any derivations within the Hamiltonian E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.12.018
framework. We will work formally, leaving aside questions of function spaces etc. Lately several results concerning the rigorous justification of model equations for irrotational water waves have appeared—see, e.g., [9–13,33,35]. It would be interesting to see if the same methods can be applied in the case of constant vorticity. Note however that there is to this date no local wellposedness result for the constant vorticity water wave problem with decay at spatial infinity. While the justification of the full dynamics of the model equations is an open problem, there are several papers treating solitary travelling waves with constant vorticity. Long-wave approximations as well as numerical results can be found in [14–18,34], while existence results for the full water wave problem are given in [19–21] (see also [22,23] for particle trajectories and [24] for solitary capillary-gravity waves). To the lowest order these waves are described by a stationary KdV equation. 2. Preliminaries Let us briefly recall the governing equations for twodimensional water waves with constant vorticity. The fluid domain Ωη = {(x, y) ∈ R2 : −h < y < η(t, x)} is bounded from below by a flat rigid bottom B = {(x, y) ∈ R2 : y = −h} and above by a free surface Sη = {(x, y) ∈ R2 : y = η(t, x}), which
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we assume to be the graph of a function. We will concentrate on waves with decay and so we assume that η(t, x) → 0 as |x| → ∞. In the case of periodic waves η is assumed to be periodic and the fluid domain is restricted to a periodic cell see [25]. The wave is described by η and a velocity field (u, v) defined in Ω which satisfies Euler’s equations ut + uux + vuy = −Px , in Ωη vt + uvx + vvy = −Py − g, and the equation of mass conservation ux + vy = 0. We only deal with gravity driven waves, so that the dynamic boundary condition takes the form P = Patm on Sη , where P is the pressure and Patm is the constant atmospheric pressure. In addition, the kinematic boundary conditions are v = 0 on B and v = ηt + uηx on Sη . The vorticity of the flow is defined by ω := vx − uy . Irrotational waves correspond to ω = 0. Here we treat waves with constant ω. A shear flow (flat surface) with constant vorticity is described by a velocity field (u, v) = (−ωy + u0 , 0) for some u0 ∈ R. We seek waves which are perturbations of shear flows. In particular, at infinity the velocity field tends to that of the shear flow. For simplicity we will assume that u0 = 0, so that the asymptotic horizontal velocity at the surface is 0. Some authors advocate different choices of u0 , e.g., u0 = −ωh (meaning that u = 0 on the bottom). Choosing a different value of u0 will simply result in changes in some formulae (e.g., the formulae for wave speeds will still be correct if c is replaced by c − u0 ). Following [3,4] we introduce a generalized velocity potential ϕ. ϕ is a harmonic function satisfying ϕx = u + ωy,
ϕy = v,
H (η, ξ ) = R
ωξx η2 ω2 η3 gη2 ξ G(η)ξ − + + dx. 2 2 6 2
(2.2)
The system (2.1) can be brought into canonical form by introducing the new coordinates (η, ζ ), where ω −1 ∂ η 2 x (see [3]), but we prefer to work with (2.1)–(2.2) directly. The operator G appearing in (2.2) is the Dirichlet–Neumann operator for the fluid domain. For a given η and ξ , let ϕ be the unique solution of the problem ⎧ ⎨ ϕ = 0 in Ωη , ϕ=ξ on Sη , ⎩ ϕy = 0 on B,
ζ =ξ −
G(η) =
∞
Gj (η),
j =0
where each Gj (η) is a linear operator which is homogeneous of order j in η [26]. The terms Gj (η) may be computed by the use of a recursion formula [27]. The first few terms are given by G0 (η) = D tanh(hD), G1 (η) = DηD − D tanh(hD)ηD tanh(hD), 1 G2 (η) = − D 2 η2 D tanh(hD) + D tanh(hD)η2 D 2 2
− 2D tanh(hD)ηD tanh(hD)ηD tanh(hD) , where D = −i∂/∂x. Another important aspect of G is that it can be written as G(η) = DK(η)D, where K is positive, bounded and self-adjoint on L2 (R). K can itself be expanded in a power series ∞
Kj (η),
j =0
δξ
It is known that G is analytic in η ∈ C ∞ in a ball of radius O(h) about η = 0, with a Taylor expansion of the form
K(η) =
and ϕy = 0 on the bottom y = −h. We let ξ = ϕ|y=η be the restriction of ϕ to the free surface. As shown in [3], the water wave problem with constant vorticity ω can be formulated as the Hamiltonian system δH 0 1 ηt δη = , (2.1) δH ξt −1 ω∂x−1 where
together with either periodic or asymptotic conditions on ϕ. We then define
1/2 ∂ϕ G(η)ξ = 1 + ηx2 = (−ϕx ηx + ϕy )|y=η . ∂n y=η
where Kj (η) = D −1 Gj (η)D −1 . 3. Hamiltonian transformation and perturbation theory We will derive approximations of (2.1)–(2.2) in various parameter regimes. In order to describe these regimes, we introduce two dimensionless scaling parameters α = a/ h and β = (h/ l)2 , where a is a typical amplitude and l is a measure of the width of the wave. Different parameter regimes are introduced by scaling both independent and dependent variables in the governing equations. Although this will not change the Hamiltonian nature of the equations, the Hamiltonian function and the structure map will change. We therefore need to investigate how the structure map is affected by a change of variables (see [1,2] for a more detailed discussion). We consider a Hamiltonian system defined on a phase space X, which is assumed to be function space endowed with an inner product (·,·). The Hamiltonian is a function H : X → R with gradient δH (w) defined by dH |w (q) = (δH (w), q), for w, q ∈ X. The Hamiltonian system has the form wt = J δH (w), where the structure map J is skew-symmetric. In general, a change of variables w˜ = f (w) transforms the Hamiltonian ˜ where H˜ (w) ˜ = system wt = J δH (w) into w˜ t = J˜δ H˜ (w), −1 ∗ ˜ ˜ and J = ∂w f (w)J (∂w f (w)) . We now specialise H (f (w))
E. Wahlén / Physics Letters A 372 (2008) 2597–2602
to the structure map 0 1 J= −1 ω∂x−1 above. Suppose that f is a scaling: f (η, ξ ) = (κη, λξ ), κ, λ ∈ R. Then J is transformed into 0 1 ˜ . J = κλ (3.1) −1 κλ ω∂x−1 Spatial scaling x1 = εx, ε ∈ R, on the other hand, transforms J into 0 1 J˜ = ε . −1 1ε ω∂x−1 Since our Hamiltonian functions will be integrals over R, note that the relation dx = ε −1 dx1 introduces a factor ε −1 in the Hamiltonian. Finally, it is sometimes useful to write the Hamiltonian systems in terms of u = ∂x ξ . This corresponds to a transformation f (η, ξ ) = (η, ∂x ξ ) = (η, u) under which J changes to 0 1 J˜ = −∂x . 1 ω We will also expand the Hamiltonian function in a series H (u, μ) = H (0) (u) + μH (1) (u) + · · · + μn H (n) (u) + O(μn+1 ), where H (k) is homogeneous of order k and μ ∈ R is a small parameter. The approximate models are derived by systematically truncating the Taylor expansion of H , so that the new Hamiltonian function is Hn = H (0) (u) + μH (1) (u) + · · · + μn H (n) (u). Unlike the situation in [1,2] our original structure map is not canonical. Although some of the above transformations introduce coefficients in front of the lower right entry of the structure map matrix, we will always use combinations of transformations resulting in a cancellation between these coefficients. E.g., the map f (η, ξ ) = (μ−1 η, μ−1 ξ ) has the effect of multiplying J by μ−2 , since κ and λ are equal in (3.1). 4. The linearised equations Before proceeding to the long-wave expansions, it is instructive to look at the equations of motion linearised around a steady state. The steady state is in our case the underlying shear flow, represented by η = 0 and ξ = 0. In terms of the parameters described in the previous section, the small-amplitude regime occurs when α = μ is small while β = O(1). It is introduced by scaling the dependent variables ξ = μξ1 .
η = μη1 ,
H (η, ξ ) =
μj H (j ) (η1 , ξ1 ),
H
(2)
and
H (2+j ) (η1 , ξ1 ) = R
ξ1 Gj (η1 )ξ1 dx, 2
for j 2. On the other hand J is transformed into μ−2 J . We can now write down a sequence of approximate equations, where we retain terms up to order j and then reintroduce the unscaled variables by letting μ = 1. The linearisation is the first equation: ηt = G0 ξ, (4.1) ξt = −gη + ω∂x−1 G0 ξ (note that G0 is independent of η). Writing this in terms of u = ξx , we instead obtain ηt = −∂x K0 u, (4.2) ut = −gηx − ω∂x K0 u. This is a dispersive wave equation with dispersion relation
g tanh(hk) ω2 tanh2 (hk) ω tanh(hk) , ± + c±,ω = (4.3) 2k k 4k 2 where c±,ω is the phase speed and k is the wave number. Here ±c±,ω > 0. The presence of an underlying shear flow with positive vorticity has the effect of increasing the velocity in the positive x-direction, so that c+,ω > c+,0 and c−,ω > c−,0 . Similarly, negative vorticity leads to c+,ω < c+,0 and c−,ω < c−,0 . Note that for ω > 0 the positive phase speed equals the horizontal velocity component somewhere in the fluid if k is sufficiently large (the fluid is stagnant in a reference frame moving at the phase speed). Many existence proofs for periodic water waves exclude this situation since it leads to mathematical problems [25]. The long-wave limit, which is studied in Sections 6, 7, is formally obtained by letting k → ∞. 5. Shallow water approximation The shallow water regime, α = O(1) and β = μ2 , where μ is a small parameter, is introduced by the transformation to stretched variables x1 = μx,
t1 = μt
and scaling of ξ ξ1 = μξ.
D = μD1 . The terms in the expansion of the Dirichlet–Neumann operator become
j =2
where
R
ξ1 G1 (η1 )ξ1 ωξ1x η12 ω2 η13 − + dx 2 2 6
This induces the transformation
We then obtain ∞
H (3) (η1 , ξ1 ) =
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(η1 , ξ1 ) = R
ξ1 G0 (η1 )ξ1 gη12 + dx, 2 2
G0 = μD1 tanh(μhD1 ),
G1 (η) = μ2 D1 ηD1 − D1 tanh(μhD1 )ηD1 tanh(μhD1 ) ,
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etc. These expression can then be expanded in the small parameter μ, using 1 2 tanh(μhD1 ) = μhD1 − (μh)3 D13 + (μh)5 D15 + · · · . 3 15 We thus obtain 2 2 gη ωξ1x1 η2 ω2 η3 (η + h)ξ1x1 dx1 − + + H (η, ξ ) = 2 2 6 2 μ R
+ O(μ), and J is transformed into Jˆ = μ2 J . Alternatively, since ηt is replaced by μηt1 we can incorporate the change in t by replacing μ2 J with μJ . Returning to unscaled variables we obtain the approximate Hamiltonian 2 gη ωξx η2 ω2 η3 (η + h)ξx2 dx. − + + HSW (η, ξ ) = 2 2 6 2 R
The corresponding equations are ηt = −((η + h)ξx )x + ωηηx , ξt = −gη − 12 ξx2 − ωhξx , which upon introducing u = ξx , may be written as ηt = −((η + h)u − ω2 η2 )x , ut = −(gη + 12 u2 + ωhu)x .
(5.1)
(5.2)
6. The Boussinesq regime In this regime one studies long, small-amplitude waves. In other words, α = β = μ2 is a small parameter. The regime is induced by introducing stretched variables t1 = μt,
and scaled dependent variables 1 η1 = 2 η, μ
1 ξ1 = ξ. μ
The corresponding structure map is μ−3 J (if we incorporate the stretching of t). In the first approximation of the Hamiltonian, terms up to order O(μ4 ) are retained. All but the first term in the Taylor series for G(η) may therefore be neglected. Expanding G0 in μ and keeping terms of order at most O(μ4 ) leads to the approximate Hamiltonian 4 2 hμ4 ξ 2 gμ η1 1x1 dx1 + . HW = (6.1) 2 2 μ R
The approximate equations of motion are, in unscaled variables, ηt = −hξxx , (6.2) ξt = −gη − ωhξx .
(6.3)
This wave equation has solutions propagating at speeds
ωh ω2 h2 ± gh + , 2 4 which agrees with the long-wave limit of the dispersion relation. Note that in contrast to the periodic case stagnation points are excluded, since, e.g., the positive phase speed is always greater than the maximum horizontal velocity ωh. In the next approximation terms of order up to O(μ6 ) are retained. With the next section in mind, we write the approximate Hamiltonian in the scaled variables 4 2 gμ η1 ωμ6 ξ1x1 η12 ω2 μ6 η13 − + HB = 2 2 6 R
+
Higher order equations may be obtained analogously.
x1 = μx,
Introducing u = ξx , this may be rewritten as ηt = −hux , ut = −gηx − ωhux .
2 (h + μ2 η1 )μ4 ξ1x 1
2
−
2 μ6 h3 ξ1x 1 x1
6
dx1 . μ
(6.4)
The corresponding equations of motion are ⎧ 2 ⎪ ⎨ η1t1 = −((h +3 μ η1 )ξ1x1 )x1 h 2 − μ 3 ξ1x1 x1 x1 x1 + μ2 ωη1 η1x1 , ⎪ ⎩ 2 − ωhξ 2 ωh3 ξ1t1 = −gη1 − μ2 12 ξ1x 1x1 − μ 3 ξ1x1 x1 x1 . 1 If we let u = ξx we have ⎧ 2 ⎪ ⎨ η1t1 = −((h +3 μ η1 )u1 )x1 h − μ2 3 u1x1 x1 x1 + μ2 ωη1 η1x1 , ⎪ 3 ⎩ u1t1 = −(gη1 + μ2 12 u21 + ωhu1 + μ2 ωh3 u1x1 x1 )x1 .
(6.5)
(6.6)
The next approximation is obtained by retaining terms of order up to O(μ8 ). Written in the variables (η1 , u1 ), the corresponding Hamiltonian is 4 2 gμ η1 μ6 ωu1 η12 μ6 ω2 η13 (h + μ2 η1 )μ4 u21 − + + H= 2 2 6 2 R
−
μ6 h3 u21x1 6
−
μ8 h2 u21x1 η1 2
+
μ8 h5 u21x1 x1 15
dx1 . μ
In unscaled variables the corresponding equations are ⎧ 3 ⎪ ηt = −((h + η)u)x − h3 uxxx + ωηηx ⎪ ⎪ ⎪ 2 5 ⎨ − h2 (ηux )xx − 15 h uxxxxx , 3 1 2 ⎪ ⎪ ut = −(gη + 2 u + ωhu + ωh3 uxx − 12 h2 u2x ⎪ ⎪ ⎩ 2 + ωh2 (ηux )x + 15 ωh5 uxxxx )x .
(6.7)
7. Unidirectional waves of KdV type We now concentrate on unidirectional waves. Let us first nondimensionalise the equations by setting 1/2 g 1 u (x, y) = (x , y ), t = t , u= √ , h h gh
E. Wahlén / Physics Letters A 372 (2008) 2597–2602
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(1 + c02 )(r12 + s12 ) μ4 2
where a prime denotes a dimensional variable. This is equivalent to setting g = h = 1 and introducing the nondimensional vorticity ω˜ = √ωh . gh In the lowest order the Boussinesq equation is the linear system ηt = −ux , (7.1) ut = −ηx − ωu ˆ x,
H=
with Hamiltonian
1 2 η + u2 dx H= 2
We can write the equations in a frame moving at the speed of the right-moving wave front associated with the longest wave. This is achieved by introducing new variables x2 = x1 − c0 t1 , t2 = μ2 t1 . The change of reference frame is equivalent to subtracting c0 I from the Hamiltonian [1], where I is the conserved quantity impulse (referred to as momentum in [3]) ωη ˆ 2 I= dx. uη − 2
R
and structure map 0 1 . J = −∂x 1 ωˆ The system can be diagonalised by introducing characteristic coordinates. Introduce new variables r and s defined by (r, s) = T (η, u) where 1 1 c0 , T= 1 + c2 c0 −1 0
with c0 =
ωˆ + 2
1+
ωˆ 2 . 4
(7.2)
A simple calculation shows that J is then transformed into 0 − c0 2 1+c0 ˜ , J = ∂x 0 − c˜0 2 1+c0 2 where c˜0 = −1/c0 = ω2ˆ − 1 + ωˆ4 is the conjugate of c0 , and that H is transformed into
1 + c02 2 r + s 2 dx. H˜ = 2 R
The new equations are rt = −c0 rx , st = −c˜0 sx . Writing (6.5) in characteristic coordinates we obtain ⎧ 2 c03 ⎪ ⎪ r1t1 = −c0 r1x1 + μ2 (− c0 (3+ω2ˆ ) r1 r1x1 − ⎪ 2 r1x1 x1 x1 1+c 3(1+c ⎪ 0 0) ⎪ ⎪ 3 2 2 ⎪ c c0 c0 ⎪ ⎨ + 0 2 s1 s1x1 + 2 s1x1 x1 x1 + 2 (r1 s1 )x1 ), 1+c0
3(1+c0 )
1+c0
1+c0
3(1+c0 )
1+c0
⎪ c4 +c2 +1 ⎪ ⎪ s1t1 = −c˜0 s1x1 + μ2 ( 0 0 2 s1 s1x1 − c˜0 2 s1x1 x1 x1 ⎪ ⎪ 1+c0 3(1+c0 ) ⎪ ⎪ c0 ⎪ 1 1 ⎩ − 2 r1 r1x1 − 2 r1x1 x1 x1 − 2 (r1 s1 )x1 ),
R
2 c2 r1 s 2 c0 (r1 s1 )x1 (3 + ωˆ 2 )r13 c02 r1x 1 − − 0 1 + 6 6 2 3 2 3 4 2 2 s1x c0 r1 s1 c0 (c0 + c0 + 1)s1 dx1 − (7.4) + − 1 . 2 6 6 μ
+μ
6
R
We concentrate on the region corresponding to predominantly right-moving waves by assuming that s1 = O(μ2 ). Under this assumption the first equation in (7.3) reduces to r1t2 +
c03 c0 (3 + ωˆ 2 ) r r + r1x2 x2 x2 = O μ2 . 1 1x 2 2 2 1 + c0 3(1 + c0 )
Thus we see that, to this order of approximation, the KdV equation also provides a model for water waves with constant vorticity. Ignoring terms of order O(μ2 ), Eq. (7.5) constitutes a Hamiltonian system with 3 + ωˆ 2 3 c02 2 r − rx dx HKdV = 6 6 R
and c0 ∂x , Jˆ = − 1 + c02 which is formally obtained by setting s1 = 0 in (7.4). One may rightfully wonder about the validity of the assumption s1 = O(μ2 ). Note that under this assumption the O(μ2 ) part of the second equation in (7.3) is a linear transport equation for s1 with a forcing term which depends on r1 . Thus we may formally argue as follows: We first solve the KdV equation for r1 and then solve the resulting equation for s1 with a forcing term depending on the KdV solution r1 . If the initial data of s1 is chosen in a particular way, then s1 will remain O(μ2 ) on some time-scale. The necessary condition is that (c0 − c˜0 )s1 − μ2 c˜0 2 Es1 (Hˆ 5 )|s1 =0 = O(μ2 ) at time 0, where 1+c0
(7.3)
where we have used the relation between c0 and ωˆ to simplify the coefficients. The Hamiltonian is given explicitly by the formula
(7.5)
Hˆ 5 denotes the order O(μ5 ) terms of the Hamiltonian density in (7.4), and Es1 is the variational derivative with respect to s1 . This is of course equivalent to requiring that s1 = O(μ2 ) initially. In order to obtain a KdV equation for η1 , note that r1 = η1 − s1 . Since
2 2 c˜0 ˆ μ s1t2 = ∂x2 (c0 − c˜0 )s1 − μ Es1 (H5 )|s1 =0 + O μ4 , 2 1 + c0
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we have to require that (c0 − c˜0 )s1 − μ2
c˜0 E (Hˆ 5 )|s1 =0 1+c02 s1
=
O(μ4 ) initially, in order to replace r1 in (7.5) by η1 . For further details we refer to [28,29]. Let us briefly discuss a different way of obtaining the model (7.5). In [7] (see also [8]) a KdV equation for waves over general shear flows was derived. If the background shear flow in nondimensional variables is given by u = U (y), v = 0, the equation reads −2I3 ηt + 3I4 ηηx + J1 ηxxx = 0,
(7.6) Acknowledgements
where 0 In = −1
The author is grateful for constructive suggestions made by the referees.
dy , (U (y) − c)n
References
and 0 0 z1 J1 = −1 z −1
(U (z1 ) − c)2 dz2 dz1 dz. (U (z) − c)2 (U (z2 ) − c)2
In our situation U (y) = −ωy ˆ and c = c0 . Straightforward calculations yield ωˆ − 2c0 , 2(ωˆ − c0 )2 c02 1 J1 = . 3(ωˆ − c0 )2
I3 =
Finally, we remark that while it is possible to derive fifth order unidirectional models from the higher order Boussinesq equation (6.7), the status of these equations is unclear. Several different models have been derived in the irrotational case [1,30,31], some of which are Hamiltonian and some of which are not. We therefore believe that a proper derivation of a higher order model within the Hamiltonian framework would need a separate discussion. A non-Hamiltonian fifth order model for water waves with constant vorticity was however derived in [5].
I4 = −
ˆ 0 + 3c02 ωˆ 2 − 3ωc 3(ωˆ − c0 )3 c03
,
Dividing (7.6) by −2I3 and simplifying the coefficients, we arrive at (7.5). The solitary waves predicted by the approximation (7.5) are waves of elevation with profile √ 3 + ωˆ 2 ˆ 2) 2 2 2 c0 (3 + ω t , x − c0 t − μ η(t, x) = μ sech μ 2c0 3(1 + c02 ) in unscaled, but nondimensional variables, where we have taken the amplitude equal to μ2 (this can always be achieved by redefining μ). Let us make some comparisons with the irrotational solitary wave (ωˆ = 0). We choose to compare wave profiles with the same amplitude. First note that for small μ, the speed of the rotational wave is greater than in the irrotational case for positive vorticity, while it is smaller for negative vorticity. The O(μ2 ) correction of the speed is an even positive function which is strictly increasing for positive ω. In other words a large value of |ω| increases the wave speed. Next we compare the width of the waves. It is an easy exercise to show that the expression √ 3 + ωˆ 2 , 2c0 where c0 depends on ωˆ through (7.2), is a decreasing function of ω. ˆ In other words the width of the wave (defined, say, as the distance between to points with η = μ2 /2) is an increasing function of the vorticity.
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