- Email: [email protected]
Variational derivation of KdV-type models for surface water waves
Physics Letters A 366 (2007) 195–201 www.elsevier.com/locate/pla
Variational derivation of KdV-type models for surface water waves E. van Groesen a,b,∗ , Andonowati a,b,c a Department of Applied Mathematics, University of Twente, The Netherlands b LabMath-Indonesia, Bandung, Indonesia c Department of Mathematics, Institut Teknologi Bandung, Indonesia
Received 27 September 2006; received in revised form 4 February 2007; accepted 9 February 2007 Available online 13 February 2007 Communicated by A.P. Fordy
Abstract Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order. © 2007 Elsevier B.V. All rights reserved. MSC: 35Q53; 35Q58; 49S05 Keywords: KdV equations; Hamiltonian formulation; Finite and infinite depth; Second-order nonlinear accurate
1. Introduction Surface water wave theory deals with the description of the surface elevation of an ideal fluid on a layer of finite or infinite depth. Water waves entering still water are described by irrotational flows (that is with zero vorticity), while the interaction of waves and currents is modeled by flows with non-zero vorticity, see [1,2] and the research literature survey in the introduction of [3]. This Letter deals exclusively with irrotational flow. To remove the depth dependence, it is custom to approximate the Laplace equation for the fluid potential in the interior with the aim to arrive at equations in the horizontal space variables only. We will refer to such resulting equations in one horizontal direction as Boussinesq equations. Typically such equations are described by two first-order dynamic equations for the surface elevation and a horizontal fluid velocity type of * Corresponding author at: Department of Applied Mathematics, University of Twente, The Netherlands. E-mail address: [email protected] (E. van Groesen).
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.031
variable. These dispersive, nonlinear equations describe waves in both directions, coupling left and right running waves. To arrive at unidirectional waves, the intention is to find one dynamic equation of first order for the surface elevation that mainly describes waves in one direction only. Although the notion of uni-directionality is not so obvious, Korteweg–de Vries (KdV)type of equations are supposed to have this property. Unidirectional motions can be interpreted as special solutions of the Boussinesq equation when the velocity variable is related to the surface elevation in a specific way so that the two dynamic equations become similar. In the literature many Boussinesq equations have been derived; see [4–23] for some of the most recent papers on the topic and the literature cited in these publications. To arrive at equations that to some order balance nonlinearity and dispersion, in many cases a perturbation series is used to solve up to the desired the Laplace equation in terms of the two surface variables. Recently variational Boussinesq models have been proposed that have an elliptic equation in addition to the dynamic equations that accounts for the dispersion [23,24].
196
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
To arrive at KdV-type of equations, a moving frame is commonly introduced, and attention is restricted to solutions that vary in time and space in a slow way compared to the moving frame. For waves above a finite bottom, various KdV-type of equations have been derived, which differ in the accuracy with which the dispersion and/or nonlinearity is taken into account. The celebrated work of Korteweg and de Vries [25] is the simplest example of an equation in which effects of dispersion and nonlinearity are balanced for certain wave fields. Since the exact dispersion properties of infinitesimal waves are given by a non-algebraic dispersion relation, the series expansion method leads to results in which the dispersion is necessarily approximated (in Boussinesq- and KdV-equations) by rational expressions of spatial differential operators. For deep water waves, the square root dispersion relation for unidirectional infinitesimal waves prevents the possibility to obtain a uniformly valid rational approximation, as a simple dimension analysis can show; this may be the reason that KdV type of equations seem not to have been considered in the literature for waves on infinite depth. In this Letter we will derive for waves above a flat bottom in a systematic way unidirectional wave equations by exploiting the variational formulation of surface water waves. This formulation is reflected in a canonical action principle in the surface elevation and surface potential in which the Hamiltonian is the total energy. By approximating the kinetic energy we will arrive naturally at a variational principle in the surface elevation and the potential at the still water level. Neglecting nonlinearity for infinitesimal waves, a complete splitting can be obtained between right and left traveling waves, for the exact dispersion as well as for approximate dispersions, both for finite and infinite depth. This uniquely defines a manifold in the elevationpotential space. Restricting the variational principle to this manifold we will derive new classes of unidirectional equations that are accurate in second order in the wave height, valid for any (order of approximation of the) dispersion relation. The obtained class of so-called AB equations can be interpreted as higher-order KdV equations for waves above finite depth; in lowest order it is the classical KdV equation. For waves above deep water, the AB equation seems to be new. The organization of the Letter is as follows. In Section 2 we describe the basic variational structure and the approximation of the action principle that will be used in the following. Section 3 deals with linear dispersion properties, and the linear uni-directionalization. The AB-equation is derived in Section 4, and in Section 5 various limiting cases and approximations are considered. Conclusions and some general remarks will finish the Letter. 2. Basic variational structure We consider the evolution of surface elevation of an ideal incompressible fluid in one horizontal direction x on a layer of finite (constant) depth h0 or of infinite depth. For irrotational fluid motion with fluid potential Φ(x, z, t), the dynamics is described by the surface elevation η(x, t) and the fluid potential at the free surface, φ(x, t) = Φ(x, z = η, t).
According to [26,27], the dynamic equations are given by a set of Hamilton equations for the canonical variables η, φ: ∂t η = δφ H (η, φ),
(1)
∂t φ = −δη H (η, φ)
(2)
which follow from variations with respect to φ and η from the action principle δA(η, φ) = 0, with A(η, φ) = φ . ∂t η dx − H (η, φ) dt. (3) The Hamiltonian is the total energy, the sum of potential and kinetic energy, expressed in the desired variables η, φ. The potential energy is calculated with respect to the undisturbed water level and the Hamiltonian is then given by 1 2 gη dx + K(φ, η). H (η, φ) = (4) 2 The kinetic energy is not easily expressed in the surface variables, but is formally given, for finite and infinite depth, by 1 K(η, φ) = |∇Φ|2 dz dx, 2 where Φ satisfies the Laplace equation in the fluid interior and the surface condition Φ = φ at z = η and the impermeable bottom boundary condition. To arrive at approximations for the kinetic energy, in the following we use besides the surface fluid potential φ = Φ(x, η, t), also the fluid potential and the vertical velocity at the still water level: φ0 = Φ(x, z = 0, t),
W0 = ∂z Φ(x, z = 0, t),
and will look for expression in η and φ0 . Remark 1. It should be noted that if the action functional is expressed in terms of (η, φ0 ) instead of in (η, φ), the correct surface wave equations are still obtained because of the one-toone correspondence in the pairs of variables. Hence, the use of φ0 instead of φ in the following is only for modeling purposes, but is not an approximation in itself. Splitting the kinetic energy expression in a term till the still water level and a term accounting for the actual surface elevation as z=0 K=
1 |∇Φ|2 dz dx + 2
z=η
1 |∇Φ|2 dz dx, 2
z=0
we first observe that by using the fluid potential φ0 and the vertical velocity W0 we can write z=0
1 1 2 |∇Φ| dz dx = φ0 W0 dx. 2 2 For the other term we will take in this Letter the approximation of lowest order in the wave height z=η z=0
1 1 |∇Φ|2 dz dx ≈ 2 2
η (∂x φ0 )2 + W02 dx.
(5)
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
Then we get the following approximation for the Hamiltonian 1 1 2 1 H (η, φ0 ) ≈ gη + φ0 W0 + η (∂x φ0 )2 + W02 dx. 2 2 2 (6) This will be the basic approximation we will use in the following. 3. Linear dispersive waves The Laplace problem with prescribed potential at the still water level can be solved explicitly. As a result we get 1 W0 = − C 2 ∂x2 φ0 . g Here C is the phase velocity operator, i.e. the symmetric ˆ pseudo-differential operator with symbol the phase velocity C, related to the dispersion relation ω = Ω(k) by Ω(k) ˆ , where C(k) = k Ω(k) = c0 k tanh(kh0 )/(kh0 ),
c0 =
gh0 .
The case of infinite depth is obtained by the limiting expression: Ω∞ (k) Cˆ ∞ (k) = = g/|k| with Ω∞ (k) = k g/|k|, k ˆ while the other limiting case for shallow water has C(k) = c0 . In the linear limit for infinitesimal waves, we approximate φ = φ0 and take z=0 K0 ≈
1 |∇Φ|2 dz dx ≈ 2
1 φ0 W0 dx 2
leading to the action principle A0 (η, φ0 ) = φ0 . ∂t η dx − H0 (η, φ0 ) dt, 1 2 1 gη + φ0 W0 dx. with H0 (η, φ0 ) = 2 2 The equations are explicitly given by
∂t φ0 = −gη. ∂t η = − C 2 /g ∂x2 φ0 ,
(7)
(8)
197
waves. This can be seen by writing the second-order equation with commuting operators like (∂t − C∂x )(∂t + C∂x )η = 0, and observing that (∂t + C∂x )η = 0 and (∂t − C∂x )η = 0 describe right and left traveling dispersive waves, respectively. Expressed for the (η, u0 ) dynamics, this corresponds to the observation that the linear manifolds in (η, u0 ) space given by the relation u0 = ±gC −1 η are invariant for the dynamics; that is to say that the two dynamic equations coincide on this manifold, and are then given by ∂t η = ∓C∂x η. In a different way, that is more suitable for generalization, we can obtain the uni-directionalization by restriction of the dynamics. Illustrated for the right traveling wave, we consider the restriction φ0 = g∂x−1 C −1 η in the action functional. This functional is then given by g∂x−1 C −1 η . ∂t η dx − H¯ 0 (η) dt, A¯ 0 (η) = where H¯ 0 is the restricted Hamiltonian H¯ 0 (η) = H (η, φ0 = g∂x−1 C −1 η), explicitly given by
−1 −1 ¯ H0 (η) = H η, φ0 = g∂x C η = gη2 . Observe that for linear unidirectional waves there is energy equipartition: the potential energy equals the kinetic energy at each instant. The dynamic equation for η then follows form the action principle δ A¯ 0 (η) = 0, and is given by 2g∂x−1 C −1 ∂t ηdx + δη H¯ 0 (η) = 0, i.e. ∂t η = −
C ∂x δη H¯ 0 (η). 2g
(12)
Observe that now −C∂x /2g is the anti-symmetric structure map of the Hamiltonian system. In the next sections we will generalize the above by including nonlinear terms. Then again we aim to arrive at a first order in time evolution equation for the elevation.
Introducing the velocity type variable u0 = ∂x φ0 , the equations are the usual linear dispersive wave equations
∂t u0 = −∂x [gη] ∂t η = −∂x C 2 /g u0 , (10)
Remark 2. In the restriction procedure, the translation invariance of the original system is retained. This means that the horizontal quadratic momentum functional M = ηu0 dx remains a constant of the motion for the unidirectional waves. This functional then becomes ¯ M(η) = g ηC −1 η. (13)
which can be written as a second-order equation in agreement with the dispersion relation: ∂t2 η = ∂x2 C 2 η . (11)
It is a simple verification that since δ M¯ = 2gC −1 η, the Hamiltonian flow of this functional for the restricted dynamics ∂t η = C ¯ ∂x δη M(η) − 2g is indeed a uniform translation.
3.1. Uni-directionalization
3.2. Approximate dispersion for finite and infinite depth
The above Eqs. (10) are the well-known formulations for bidirectional wave propagation. The bidirectionality is evident from the symmetry (η, u0 , x, t) → (η, −u0 , −x, t). These linear equations can be split exactly in left and right running
It is to be observed that the restricted dynamics for the unidirectional waves is valid for finite and for infinite depth by choosing (approximations of) the corresponding phase speed operator. These are linear versions of KdV type of equations.
(9)
198
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
To avoid that pseudo-differential operators appear in the equations, these operators are often replaced by approximations with differential operators, corresponding to rational approximations of the phase velocity. In that respect, the following arguments using simple dimensional analysis will show an essential difference between the case of finite and infinite depth. The vector field F in the evolution equation ∂t η = F (η) for unidirectional waves should have the dimension of a velocity. Since the gravitation is the only quantity that can account for the time dimension, the vector field will be of the √ form ∂t η = gF1 (η). Then the dimension of F1 should be the square root of length. For waves on a layer of finite depth h0 √ this can be achieved by an equation of the form ∂t η = gh0 F2 where F2 is a polynomial in the dimensionless surface height η/ h0 and its derivatives with respect to the dimensionless spatial variable x/ h0 . Hence, on finite depth, approximations of the dispersion with differential operators may be feasible. However, this is not the case for deep water, because then the only way to get the dimension of square root of length is by a pseudo differential operator like C∞ . We will see this in more detail in Section 5.3. 4. Nonlinear dispersive AB equation We will now incorporate nonlinear contributions that are consistent with the approximation of the kinetic energy given in (6). As stated before in Remark 1 we will use without restriction or approximation the variable φ0 instead of φ in the action functional. However, in the following, instead of first deriving Boussinesq type of equations for bidirectional waves, i.e. dynamic equations for φ and η, we will directly restrict to the unidirectional case by inserting the uni-directionalization in the action functional. This is achieved by taking the restriction to the manifold that produces the uni-directionalization in the linear limit, i.e. φ0 = g∂x−1 C −1 η. To simplify notation, we introduce a skew-symmetric operator A and a symmetric operator B as follows: √ √ B = gC −1 . A = C∂x / g, Then we have √ √ u0 = gBη, and φ0 = gA−1 η,
√ W0 = − gAη.
The restricted Hamiltonian is found by substituting these relations in the Hamiltonian (6); the result is explicitly given by 1 H¯ (η) = g (14) η2 + η (Aη)2 + (Bη)2 dx. 2 The same restriction in the action, using φ = φ0 + ηW0 results into √ φ . ∂t η dx dt = g [A−1 η − η . Aη] . ∂t η dx dt. ¯ Taken together, the action principle reads δ A(η) = 0, with √ ¯ A(η) = g[A−1 η − η . Aη] . ∂t η dx − H¯ (η) dt. (15)
Vanishing of the variational derivative of the action functional leads to the evolution equation √ g −2A−1 ∂t η + A(η . ∂t η) + η . A∂t η = δ H¯ (η) 1 1 2 2 = g 2η + (Aη) − A(ηAη) + (Bη) + B(ηBη) . 2 2 Although correct, the expression involving ∂t η is rather complicated. We will simplify it by observing that in lowest order of wave height the linear equation reads
√ −2 gA−1 ∂t η = 2gη + O η2 , i.e.
√ ∂t η = − gAη + O η2 . We substitute this approximation in the action to obtain √ −1 g A η − η . Aη . ∂t η dx √ −1 = gA η . ∂t η + gη . (Aη)2 dx. Then the total action functional is approximated correctly up to and including qubic terms in wave height. We write it as a modified action principle, reading explicitly √ −1 ¯ gA η . ∂t η dx − Hmod (η) dt, Amod (η) = (16) where the modified Hamiltonian contains a term from the original action and is given by H¯ mod (η) = H¯ (η) − g η . (Aη)2 dx 1 2 2 2 dx. =g (17) η + η (Bη) − (Aη) 2 The resulting equation δAmod (η) = 0: √ − g2A−1 ∂t η = δ H¯ mod (η) will be called the AB equation and can be written like 1 1 √ ∂t η = − gA η + (Bη)2 + B(ηBη) 4 2 1 1 − (Aη)2 + A(ηAη) . 4 2
(18)
(19)
Note that so far we have not specified the dispersive properties; this information is included in the operators A and B. Without specifying the details of the dispersion operator C the action principle can be rewritten with the phase velocity operator as follows A(η) = g C −1 ∂x−1 η . ∂t η dx −
2 1 g2
gη2 + η C −1 η − η(C∂x η)2 dx dt 2 2 (20)
and the AB equation becomes
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
199
g −1 2 g −1 −1 C η + C ηC η 4 2 1 1 2 − (C∂x η) + C∂x (ηC∂x η) . 4g 2g
∂t η = −C∂x η +
5.2. Higher-order KdV type of equations (21)
As a consequence, the AB equation is, within the approximation to be second-order accurate in the wave height, applicable for finite depth dispersion and for infinite depth dispersion; it also describes approximations when dispersive approximations are desired. In the following section we will consider some special cases and simplifications of this equation. 5. Limiting cases and approximations In the following we consider the case of shallow and of deep water as limiting cases, and for finite depth we will derive several dispersive approximations of the AB equation. As is custom, to be able to compare the orders of wave height and of dispersive effects, we introduce order quantities as measure of wave height and wavelength: ε = O(η),
μ = O(∂x ).
The AB equation is accurate up to and including quadratic terms (third order in the modified action principle) in wave height, without restriction in dispersive properties, which allows flexibility to approximate the dispersion depending on physical geometries or type of wavefields. 5.1. Shallow water equations For waves above shallow water, the lowest order approximation is obtained by completely neglecting √ dispersive√effects, i.e. √ by taking C = c0 = gh. Then A = h∂x , B = 1/ h, and the action functional (16) reads √ −1 gA η . ∂t η dx Amod (η) = 1 −g η2 + η −(Aη)2 + (Bη)2 dx dt 2 g/ h∂x−1 η . ∂t η dx = 1 3 h 2 2 −g η + η − η(∂x η) dx dt. 2h 2 The equation becomes a higher order shallow water equation h 2 3 2 h 2 ∂t η = −c0 ∂x η + η + (∂x η) + η∂x η . (22) 4h 4 2 Observe that the vectorfield has terms of order ε2 c0 μ ε, , hε2 μ2 . h Restricting to waves with wavelength long compared to bottom depth, h O(1/μ), leads to the usual unidirectional shallow water equation, i.e. Burgers equation without dissipation: 3 ∂t η = −c0 ∂x η + η2 . 4h
Higher order KdV equations for waves above a flat bottom can be obtained by taking approximations of the dispersion operator in the AB equation (21). We will keep the Hamiltonian structure, but simplify the effect of dispersion. As a reference we first formulate the standard Hamiltonian structure of the classical KdV equation, which is given by h2 3 ∂t η = −c0 ∂x 1 + ∂x2 η + η2 = −c0 ∂x δHKdV (23) 6 4h 1 3 1 2 h2 2 for HKdV = (24) η − (∂x η) + η dx. 2 12 4h Note that this classical KdV equation has terms of order c0 μ ε, εμ2 h2 , ε 2 / h ; the nonlinear quadratic term has no dispersive effects. This equation is used in cases that the Boussinesq assumption applies, i.e. that ε = O(μ2 ), ‘rather long, rather low waves’, so that then the terms εμ2 and ε 2 balance. Starting from the AB equation (21), there are many possibilities to get simplified equations which may be valid for specific classes of wave fields. Below we just give a few examples. First we neglect higher order dispersion in the qubic terms in the Hamiltonian (taking C = c0 in these terms), while keeping all two qubic terms. This leads to the approximate Hamiltonian 1 3 h 2 2 ¯ η + η − η(∂x η) dx H1 (η) = g 2h 2 C ∂x δH1 (η), explicitly and to the equation ∂t η = − 2g
∂t η = −C∂x
h 2 3 2 h 2 η + η + (∂x η) + η∂x η . 4h 4 2
(25)
This can be called a higher order KdV equation. Note that also with this simplified Hamiltonian, the Hamiltonian formulation implies that the dispersion influences the nonlinear terms by the action of the structure map. A major difference with the classical KdV equation are the additional nonlinear terms. The expression in brackets has terms [ε, ε 2 / h, hε 2 μ2 ], so that these additional two terms of order hε 2 μ2 will be relevant for shorter waves for which h ∼ O(1/μ). Concerning simplified dispersion, for the full dispersion operator C various approximations have been proposed in the literature. Most approximations take a polynomial or rational function, corresponding to combinations of differential operators and inverses of differential operators. One example is the approximation taken by Madsen and Sorensen [14], given by the symbol Cˆ 2 = c02
1+
1 2 2 15 k h 1 + 25 k 2 h2
see also [18]. However, in the equations they consider, this operator is not fully applied to the nonlinear terms, as is expressed in (25). This may cause that the Hamiltonian formulation is lost,
200
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
which may result in loss of energy and momentum conservation. To illustrate this, we consider a somewhat simpler approximation of C, namely the classical KdV choice C = c0 (1+ 16 ∂x2 ) obtained by approximation the symbol by a quadratic function in the rather long wave regime. Then (25) becomes h 2 h2 2 3 2 h 2 ∂t η = −c0 1 + ∂x ∂x η + η + (∂x η) + η∂x η . 6 4h 4 2 (26) The vectorfield of this equation contains terms of various orders
ε2 2 2 2 2 c0 1 + h μ μ ε, , hε μ . h When further simplifications are desired, for instance, in the Boussinesq regime, one may be inclined to consider the following equation in which terms of order ε2 μ5 are disregarded h2 2 3 2 ∂t η = −c0 ∂x 1 + ∂x η + η 6 4h h h + (∂x η)2 + η∂x2 η . 4 2 However, the term with ∂x2 (η2 ) is the only term that is not the variational derivative of a functional; hence this approximation in the equation does not lead to a Hamiltonian system with the obvious structure map −c0 ∂x . A simplified Hamiltonian model is still possible, albeit of a much lower order. In fact, omitting all terms of order higher than ε2 μ, there results the classical KdV equation (23). Hence, aiming at an Hamiltonian equation of higher order that may be more accurate for shorter waves, the additional terms of order ε 2 μ3 are then necessarily accompanied by terms of order ε2 μ5 . Remark 3. When changing the structure −C∂x map from to −c0 ∂x as has been done to arrive at the classical KdV equation, it should be noted that then the quadratic momentum func¯ tional has to be modified also, from (13) M(η) = g ηC −1 η ¯ to M(η) = g ηc0−1 η. 5.3. Deep water unidirectional equation √ ˆ ˆ On √ water we have √ C = g/|k| and A := √ infinitely deep ˆ ik/ |k| = i sign(k) |k| and B = |k|. Then the AB equation becomes 1 1 √ ∂t η = − gA η + (Bη)2 + B(ηBη) 4 2 1 1 − (Aη)2 + A(ηAη) (27) 4 2 with terms of order √ 2 gμ ε, ε μ . Since both operators A and B have the same order of differ√ entiation μ, all four quadratic terms in the equation are of the same order. These operators cannot be easily approximated with ordinary differential operators.
6. Conclusion and remarks In order to illuminate the approximative nature of the AB equation we briefly summarize the derivation. The basis is the use of the variational structure of the equations of surface waves. This requires the expression of the Hamiltonian (the total energy) in the basic variables of surface elevation and surface potential as canonical variables. The AB equation has been derived starting with a separation of the nonlinear and dispersive aspects. For given fluid potential, the kinetic energy has been approximated in firstorder accuracy in the waveheight. The surface potential has been approximated accurate in first order of the wave elevation in terms of the potential at the still water level. The unidirectionalization is then obtained by a restriction of the action functional to the manifold that relates the still water level potential to the elevation. From the quadratic dependence of the kinetic energy on the fluid potential, the resulting restricted action functional is third order accurate in the wave height. This approximate action functional is then further simplified by invoking an approximation using the linearized evolution that does not change the accuracy in third order. The above steps result into a modified action principle that is accurate in third order of the wave height, and where specification of (approximative) dispersive properties have not been invoked yet. Specifying the dispersion for infinite depth, a new evolution for unidirectional waves above deep water is obtained. In this case all quadratic terms in the governing equation are of the same spatial order. For waves on finite depth, the equation with second order accuracy in the wave height, the exact dispersion can be approximated in various ways. Taking the classical approximation for the lowest order approximation leads to higher order KdV equations. The higher-order terms in this equation account for additional effects for relatively short waves. We commented on loosing the Hamiltonian structure when just individual terms are deleted in the governing equations, since terms of different order may together form a variational structure; Hamiltonian consistent equations are easiest obtained by making approximations in the action functional. Using the variational approach has the advantage of maintaining a coupling between the velocity potential within the fluid and the free surface. In the derivation of KdV via series expansions in waveheight and dispersion, the vertical velocity component may vanish (see [28]). This means that within the framework provided by the series truncation viewpoint, the particles in the fluid below the surface may ignore vertical motion. This does not capture the particle motion below a surface water wave in irrotational flow, as described recently in [29]. Keeping a coupling between velocity potential and free surface, as is done in this Letter, offers the possibility of a more accurate approximation of the motion throughout the fluid. The above described summary shows the modifications that are necessary when one wants to arrive at equations that are accurate in higher than second order in the wave height. Then higher-order accurate approximations will be needed in the approximation of the kinetic energy for given fluid potential, in the approximation between surface potential and still-
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
water level potential, and in the uni-directionalization manifold. Acknowledgements
[12] [13] [14] [15] [16]
The authors gratefully acknowledge useful suggestions and comments made by the referee. This work is part of project TWI.5374 of the Netherlands Organization of Scientific Research NWO, subdivision Applied Sciences STW.
[17] [18] [19]
References [1] R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997. [2] J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978. [3] A. Constantin, W. Strauss, Comm. Pure Appl. Math. 57 (2004) 481. [4] M.B. Abbott, A.D. McCowan, I.R. Warren, J. Hydraulic Engrg. 110 (10) (1984) 1287. [5] Y. Agnon, P.A. Madsen, H.A. Schäffer, J. Fluid Mech. 399 (1999) 319. [6] H.B. Bingham, Y. Agnon, Eur. J. Mech. B: Fluids 24 (2) (2005) 255. [7] M.F. Gobbi, J.T. Kirby, Coastal Engrg. 37 (1999) 57. [8] M.F. Gobbi, J.T. Kirby, G. Wei, J. Fluid Mech. 405 (2000) 181. [9] A.B. Kennedy, J.T. Kirby, M.F. Gobbi, Coastal Engrg. 44 (2002) 205. [10] J.T. Kirby, Nonlinear, dispersive long waves in water of variable depth, in: J.N. Hunt (Ed.), Gravity Waves in Water of Finite Depth, in: Advances in Fluid Mechanics, vol. 10, Computational Mechanics Publications, 1997, pp. 55–125. [11] J.T. Kirby, Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents, in: V.C. Lakhan (Ed.), Advances in Coastal Modeling, Elsevier, 2003, pp. 1–41.
[20] [21] [22] [23]
[24]
[25] [26] [27] [28] [29]
201
C. Lee, Y.-S. Cho, S.B. Yoon, Ocean Engrg. 30 (2003) 1849. P.A. Madsen, R. Murray, O.R. Sørensen, Coastal Engrg. 15 (1991) 371. P.A. Madsen, O.R. Sørensen, Coastal Engrg. 18 (1992) 183. P.A. Madsen, H.A. Schäffer, Philos. Trans. Math. Phys. Engrg. Sci. Ser. A 1749 (1998) 3123. P.A. Madsen, H.A. Schäffer, A review of Boussinesq-type equations for surface gravity waves, in: P.L.-F. Liu (Ed.), Advances in Coastal and Ocean Engineering, vol. 5, World Scientific, 1999, pp. 1–95. P.A. Madsen, H.B. Bingham, H.A. Schäffer, Proc. R. Soc. London, Series A 459 (2003) 1075. P.A. Madsen, D.R. Fuhrman, B. Wang, Coastal Engrg. 52 (2006) 487. O. Nwogu, J. Waterway, Port, Coastal Ocean Engrg ASCE 119 (6) (1993) 618. H.A. Schäffer, P.A. Madsen, Coastal Engrg. 26 (1995) 1. O.R. Sørensen, H.A. Schäffer, L.A. Sørensen, Coastal Engrg. 50 (2004) 181. J.M. Witting, J. Comput. Phys. 56 (1984) 203. G. Klopman, M.W. Dingemans, B. van Groesen, A variational model for fully non-linear water waves of Boussinesq type, in: Proceedings IWWWFB, 2005. G. Klopman, E. van Groesen, M.W. Dingemans, Variational Boussinesq modelling of fully non-linear water waves, J. Fluid Mech. 2006, submitted for publication. D.G. Kortweg, G. de Vries, Philos. Mag. 39 (1895) 422. V.E. Zakharov, J. Appl. Mech. Techn. Phys. 9 (2) (1968) 190. L.J.F. Broer, Appl. Sci. Res. 29 (1974) 430. P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge Univ. Press, Cambridge, 1989. A. Constantin, Invent. Math. 166 (2006) 523.
Variational derivation of KdV-type models for surface water waves E. van Groesen a,b,∗ , Andonowati a,b,c a Department of Applied Mathematics, University of Twente, The Netherlands b LabMath-Indonesia, Bandung, Indonesia c Department of Mathematics, Institut Teknologi Bandung, Indonesia
Received 27 September 2006; received in revised form 4 February 2007; accepted 9 February 2007 Available online 13 February 2007 Communicated by A.P. Fordy
Abstract Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order. © 2007 Elsevier B.V. All rights reserved. MSC: 35Q53; 35Q58; 49S05 Keywords: KdV equations; Hamiltonian formulation; Finite and infinite depth; Second-order nonlinear accurate
1. Introduction Surface water wave theory deals with the description of the surface elevation of an ideal fluid on a layer of finite or infinite depth. Water waves entering still water are described by irrotational flows (that is with zero vorticity), while the interaction of waves and currents is modeled by flows with non-zero vorticity, see [1,2] and the research literature survey in the introduction of [3]. This Letter deals exclusively with irrotational flow. To remove the depth dependence, it is custom to approximate the Laplace equation for the fluid potential in the interior with the aim to arrive at equations in the horizontal space variables only. We will refer to such resulting equations in one horizontal direction as Boussinesq equations. Typically such equations are described by two first-order dynamic equations for the surface elevation and a horizontal fluid velocity type of * Corresponding author at: Department of Applied Mathematics, University of Twente, The Netherlands. E-mail address: [email protected] (E. van Groesen).
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.031
variable. These dispersive, nonlinear equations describe waves in both directions, coupling left and right running waves. To arrive at unidirectional waves, the intention is to find one dynamic equation of first order for the surface elevation that mainly describes waves in one direction only. Although the notion of uni-directionality is not so obvious, Korteweg–de Vries (KdV)type of equations are supposed to have this property. Unidirectional motions can be interpreted as special solutions of the Boussinesq equation when the velocity variable is related to the surface elevation in a specific way so that the two dynamic equations become similar. In the literature many Boussinesq equations have been derived; see [4–23] for some of the most recent papers on the topic and the literature cited in these publications. To arrive at equations that to some order balance nonlinearity and dispersion, in many cases a perturbation series is used to solve up to the desired the Laplace equation in terms of the two surface variables. Recently variational Boussinesq models have been proposed that have an elliptic equation in addition to the dynamic equations that accounts for the dispersion [23,24].
196
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
To arrive at KdV-type of equations, a moving frame is commonly introduced, and attention is restricted to solutions that vary in time and space in a slow way compared to the moving frame. For waves above a finite bottom, various KdV-type of equations have been derived, which differ in the accuracy with which the dispersion and/or nonlinearity is taken into account. The celebrated work of Korteweg and de Vries [25] is the simplest example of an equation in which effects of dispersion and nonlinearity are balanced for certain wave fields. Since the exact dispersion properties of infinitesimal waves are given by a non-algebraic dispersion relation, the series expansion method leads to results in which the dispersion is necessarily approximated (in Boussinesq- and KdV-equations) by rational expressions of spatial differential operators. For deep water waves, the square root dispersion relation for unidirectional infinitesimal waves prevents the possibility to obtain a uniformly valid rational approximation, as a simple dimension analysis can show; this may be the reason that KdV type of equations seem not to have been considered in the literature for waves on infinite depth. In this Letter we will derive for waves above a flat bottom in a systematic way unidirectional wave equations by exploiting the variational formulation of surface water waves. This formulation is reflected in a canonical action principle in the surface elevation and surface potential in which the Hamiltonian is the total energy. By approximating the kinetic energy we will arrive naturally at a variational principle in the surface elevation and the potential at the still water level. Neglecting nonlinearity for infinitesimal waves, a complete splitting can be obtained between right and left traveling waves, for the exact dispersion as well as for approximate dispersions, both for finite and infinite depth. This uniquely defines a manifold in the elevationpotential space. Restricting the variational principle to this manifold we will derive new classes of unidirectional equations that are accurate in second order in the wave height, valid for any (order of approximation of the) dispersion relation. The obtained class of so-called AB equations can be interpreted as higher-order KdV equations for waves above finite depth; in lowest order it is the classical KdV equation. For waves above deep water, the AB equation seems to be new. The organization of the Letter is as follows. In Section 2 we describe the basic variational structure and the approximation of the action principle that will be used in the following. Section 3 deals with linear dispersion properties, and the linear uni-directionalization. The AB-equation is derived in Section 4, and in Section 5 various limiting cases and approximations are considered. Conclusions and some general remarks will finish the Letter. 2. Basic variational structure We consider the evolution of surface elevation of an ideal incompressible fluid in one horizontal direction x on a layer of finite (constant) depth h0 or of infinite depth. For irrotational fluid motion with fluid potential Φ(x, z, t), the dynamics is described by the surface elevation η(x, t) and the fluid potential at the free surface, φ(x, t) = Φ(x, z = η, t).
According to [26,27], the dynamic equations are given by a set of Hamilton equations for the canonical variables η, φ: ∂t η = δφ H (η, φ),
(1)
∂t φ = −δη H (η, φ)
(2)
which follow from variations with respect to φ and η from the action principle δA(η, φ) = 0, with A(η, φ) = φ . ∂t η dx − H (η, φ) dt. (3) The Hamiltonian is the total energy, the sum of potential and kinetic energy, expressed in the desired variables η, φ. The potential energy is calculated with respect to the undisturbed water level and the Hamiltonian is then given by 1 2 gη dx + K(φ, η). H (η, φ) = (4) 2 The kinetic energy is not easily expressed in the surface variables, but is formally given, for finite and infinite depth, by 1 K(η, φ) = |∇Φ|2 dz dx, 2 where Φ satisfies the Laplace equation in the fluid interior and the surface condition Φ = φ at z = η and the impermeable bottom boundary condition. To arrive at approximations for the kinetic energy, in the following we use besides the surface fluid potential φ = Φ(x, η, t), also the fluid potential and the vertical velocity at the still water level: φ0 = Φ(x, z = 0, t),
W0 = ∂z Φ(x, z = 0, t),
and will look for expression in η and φ0 . Remark 1. It should be noted that if the action functional is expressed in terms of (η, φ0 ) instead of in (η, φ), the correct surface wave equations are still obtained because of the one-toone correspondence in the pairs of variables. Hence, the use of φ0 instead of φ in the following is only for modeling purposes, but is not an approximation in itself. Splitting the kinetic energy expression in a term till the still water level and a term accounting for the actual surface elevation as z=0 K=
1 |∇Φ|2 dz dx + 2
z=η
1 |∇Φ|2 dz dx, 2
z=0
we first observe that by using the fluid potential φ0 and the vertical velocity W0 we can write z=0
1 1 2 |∇Φ| dz dx = φ0 W0 dx. 2 2 For the other term we will take in this Letter the approximation of lowest order in the wave height z=η z=0
1 1 |∇Φ|2 dz dx ≈ 2 2
η (∂x φ0 )2 + W02 dx.
(5)
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
Then we get the following approximation for the Hamiltonian 1 1 2 1 H (η, φ0 ) ≈ gη + φ0 W0 + η (∂x φ0 )2 + W02 dx. 2 2 2 (6) This will be the basic approximation we will use in the following. 3. Linear dispersive waves The Laplace problem with prescribed potential at the still water level can be solved explicitly. As a result we get 1 W0 = − C 2 ∂x2 φ0 . g Here C is the phase velocity operator, i.e. the symmetric ˆ pseudo-differential operator with symbol the phase velocity C, related to the dispersion relation ω = Ω(k) by Ω(k) ˆ , where C(k) = k Ω(k) = c0 k tanh(kh0 )/(kh0 ),
c0 =
gh0 .
The case of infinite depth is obtained by the limiting expression: Ω∞ (k) Cˆ ∞ (k) = = g/|k| with Ω∞ (k) = k g/|k|, k ˆ while the other limiting case for shallow water has C(k) = c0 . In the linear limit for infinitesimal waves, we approximate φ = φ0 and take z=0 K0 ≈
1 |∇Φ|2 dz dx ≈ 2
1 φ0 W0 dx 2
leading to the action principle A0 (η, φ0 ) = φ0 . ∂t η dx − H0 (η, φ0 ) dt, 1 2 1 gη + φ0 W0 dx. with H0 (η, φ0 ) = 2 2 The equations are explicitly given by
∂t φ0 = −gη. ∂t η = − C 2 /g ∂x2 φ0 ,
(7)
(8)
197
waves. This can be seen by writing the second-order equation with commuting operators like (∂t − C∂x )(∂t + C∂x )η = 0, and observing that (∂t + C∂x )η = 0 and (∂t − C∂x )η = 0 describe right and left traveling dispersive waves, respectively. Expressed for the (η, u0 ) dynamics, this corresponds to the observation that the linear manifolds in (η, u0 ) space given by the relation u0 = ±gC −1 η are invariant for the dynamics; that is to say that the two dynamic equations coincide on this manifold, and are then given by ∂t η = ∓C∂x η. In a different way, that is more suitable for generalization, we can obtain the uni-directionalization by restriction of the dynamics. Illustrated for the right traveling wave, we consider the restriction φ0 = g∂x−1 C −1 η in the action functional. This functional is then given by g∂x−1 C −1 η . ∂t η dx − H¯ 0 (η) dt, A¯ 0 (η) = where H¯ 0 is the restricted Hamiltonian H¯ 0 (η) = H (η, φ0 = g∂x−1 C −1 η), explicitly given by
−1 −1 ¯ H0 (η) = H η, φ0 = g∂x C η = gη2 . Observe that for linear unidirectional waves there is energy equipartition: the potential energy equals the kinetic energy at each instant. The dynamic equation for η then follows form the action principle δ A¯ 0 (η) = 0, and is given by 2g∂x−1 C −1 ∂t ηdx + δη H¯ 0 (η) = 0, i.e. ∂t η = −
C ∂x δη H¯ 0 (η). 2g
(12)
Observe that now −C∂x /2g is the anti-symmetric structure map of the Hamiltonian system. In the next sections we will generalize the above by including nonlinear terms. Then again we aim to arrive at a first order in time evolution equation for the elevation.
Introducing the velocity type variable u0 = ∂x φ0 , the equations are the usual linear dispersive wave equations
∂t u0 = −∂x [gη] ∂t η = −∂x C 2 /g u0 , (10)
Remark 2. In the restriction procedure, the translation invariance of the original system is retained. This means that the horizontal quadratic momentum functional M = ηu0 dx remains a constant of the motion for the unidirectional waves. This functional then becomes ¯ M(η) = g ηC −1 η. (13)
which can be written as a second-order equation in agreement with the dispersion relation: ∂t2 η = ∂x2 C 2 η . (11)
It is a simple verification that since δ M¯ = 2gC −1 η, the Hamiltonian flow of this functional for the restricted dynamics ∂t η = C ¯ ∂x δη M(η) − 2g is indeed a uniform translation.
3.1. Uni-directionalization
3.2. Approximate dispersion for finite and infinite depth
The above Eqs. (10) are the well-known formulations for bidirectional wave propagation. The bidirectionality is evident from the symmetry (η, u0 , x, t) → (η, −u0 , −x, t). These linear equations can be split exactly in left and right running
It is to be observed that the restricted dynamics for the unidirectional waves is valid for finite and for infinite depth by choosing (approximations of) the corresponding phase speed operator. These are linear versions of KdV type of equations.
(9)
198
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
To avoid that pseudo-differential operators appear in the equations, these operators are often replaced by approximations with differential operators, corresponding to rational approximations of the phase velocity. In that respect, the following arguments using simple dimensional analysis will show an essential difference between the case of finite and infinite depth. The vector field F in the evolution equation ∂t η = F (η) for unidirectional waves should have the dimension of a velocity. Since the gravitation is the only quantity that can account for the time dimension, the vector field will be of the √ form ∂t η = gF1 (η). Then the dimension of F1 should be the square root of length. For waves on a layer of finite depth h0 √ this can be achieved by an equation of the form ∂t η = gh0 F2 where F2 is a polynomial in the dimensionless surface height η/ h0 and its derivatives with respect to the dimensionless spatial variable x/ h0 . Hence, on finite depth, approximations of the dispersion with differential operators may be feasible. However, this is not the case for deep water, because then the only way to get the dimension of square root of length is by a pseudo differential operator like C∞ . We will see this in more detail in Section 5.3. 4. Nonlinear dispersive AB equation We will now incorporate nonlinear contributions that are consistent with the approximation of the kinetic energy given in (6). As stated before in Remark 1 we will use without restriction or approximation the variable φ0 instead of φ in the action functional. However, in the following, instead of first deriving Boussinesq type of equations for bidirectional waves, i.e. dynamic equations for φ and η, we will directly restrict to the unidirectional case by inserting the uni-directionalization in the action functional. This is achieved by taking the restriction to the manifold that produces the uni-directionalization in the linear limit, i.e. φ0 = g∂x−1 C −1 η. To simplify notation, we introduce a skew-symmetric operator A and a symmetric operator B as follows: √ √ B = gC −1 . A = C∂x / g, Then we have √ √ u0 = gBη, and φ0 = gA−1 η,
√ W0 = − gAη.
The restricted Hamiltonian is found by substituting these relations in the Hamiltonian (6); the result is explicitly given by 1 H¯ (η) = g (14) η2 + η (Aη)2 + (Bη)2 dx. 2 The same restriction in the action, using φ = φ0 + ηW0 results into √ φ . ∂t η dx dt = g [A−1 η − η . Aη] . ∂t η dx dt. ¯ Taken together, the action principle reads δ A(η) = 0, with √ ¯ A(η) = g[A−1 η − η . Aη] . ∂t η dx − H¯ (η) dt. (15)
Vanishing of the variational derivative of the action functional leads to the evolution equation √ g −2A−1 ∂t η + A(η . ∂t η) + η . A∂t η = δ H¯ (η) 1 1 2 2 = g 2η + (Aη) − A(ηAη) + (Bη) + B(ηBη) . 2 2 Although correct, the expression involving ∂t η is rather complicated. We will simplify it by observing that in lowest order of wave height the linear equation reads
√ −2 gA−1 ∂t η = 2gη + O η2 , i.e.
√ ∂t η = − gAη + O η2 . We substitute this approximation in the action to obtain √ −1 g A η − η . Aη . ∂t η dx √ −1 = gA η . ∂t η + gη . (Aη)2 dx. Then the total action functional is approximated correctly up to and including qubic terms in wave height. We write it as a modified action principle, reading explicitly √ −1 ¯ gA η . ∂t η dx − Hmod (η) dt, Amod (η) = (16) where the modified Hamiltonian contains a term from the original action and is given by H¯ mod (η) = H¯ (η) − g η . (Aη)2 dx 1 2 2 2 dx. =g (17) η + η (Bη) − (Aη) 2 The resulting equation δAmod (η) = 0: √ − g2A−1 ∂t η = δ H¯ mod (η) will be called the AB equation and can be written like 1 1 √ ∂t η = − gA η + (Bη)2 + B(ηBη) 4 2 1 1 − (Aη)2 + A(ηAη) . 4 2
(18)
(19)
Note that so far we have not specified the dispersive properties; this information is included in the operators A and B. Without specifying the details of the dispersion operator C the action principle can be rewritten with the phase velocity operator as follows A(η) = g C −1 ∂x−1 η . ∂t η dx −
2 1 g2
gη2 + η C −1 η − η(C∂x η)2 dx dt 2 2 (20)
and the AB equation becomes
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
199
g −1 2 g −1 −1 C η + C ηC η 4 2 1 1 2 − (C∂x η) + C∂x (ηC∂x η) . 4g 2g
∂t η = −C∂x η +
5.2. Higher-order KdV type of equations (21)
As a consequence, the AB equation is, within the approximation to be second-order accurate in the wave height, applicable for finite depth dispersion and for infinite depth dispersion; it also describes approximations when dispersive approximations are desired. In the following section we will consider some special cases and simplifications of this equation. 5. Limiting cases and approximations In the following we consider the case of shallow and of deep water as limiting cases, and for finite depth we will derive several dispersive approximations of the AB equation. As is custom, to be able to compare the orders of wave height and of dispersive effects, we introduce order quantities as measure of wave height and wavelength: ε = O(η),
μ = O(∂x ).
The AB equation is accurate up to and including quadratic terms (third order in the modified action principle) in wave height, without restriction in dispersive properties, which allows flexibility to approximate the dispersion depending on physical geometries or type of wavefields. 5.1. Shallow water equations For waves above shallow water, the lowest order approximation is obtained by completely neglecting √ dispersive√effects, i.e. √ by taking C = c0 = gh. Then A = h∂x , B = 1/ h, and the action functional (16) reads √ −1 gA η . ∂t η dx Amod (η) = 1 −g η2 + η −(Aη)2 + (Bη)2 dx dt 2 g/ h∂x−1 η . ∂t η dx = 1 3 h 2 2 −g η + η − η(∂x η) dx dt. 2h 2 The equation becomes a higher order shallow water equation h 2 3 2 h 2 ∂t η = −c0 ∂x η + η + (∂x η) + η∂x η . (22) 4h 4 2 Observe that the vectorfield has terms of order ε2 c0 μ ε, , hε2 μ2 . h Restricting to waves with wavelength long compared to bottom depth, h O(1/μ), leads to the usual unidirectional shallow water equation, i.e. Burgers equation without dissipation: 3 ∂t η = −c0 ∂x η + η2 . 4h
Higher order KdV equations for waves above a flat bottom can be obtained by taking approximations of the dispersion operator in the AB equation (21). We will keep the Hamiltonian structure, but simplify the effect of dispersion. As a reference we first formulate the standard Hamiltonian structure of the classical KdV equation, which is given by h2 3 ∂t η = −c0 ∂x 1 + ∂x2 η + η2 = −c0 ∂x δHKdV (23) 6 4h 1 3 1 2 h2 2 for HKdV = (24) η − (∂x η) + η dx. 2 12 4h Note that this classical KdV equation has terms of order c0 μ ε, εμ2 h2 , ε 2 / h ; the nonlinear quadratic term has no dispersive effects. This equation is used in cases that the Boussinesq assumption applies, i.e. that ε = O(μ2 ), ‘rather long, rather low waves’, so that then the terms εμ2 and ε 2 balance. Starting from the AB equation (21), there are many possibilities to get simplified equations which may be valid for specific classes of wave fields. Below we just give a few examples. First we neglect higher order dispersion in the qubic terms in the Hamiltonian (taking C = c0 in these terms), while keeping all two qubic terms. This leads to the approximate Hamiltonian 1 3 h 2 2 ¯ η + η − η(∂x η) dx H1 (η) = g 2h 2 C ∂x δH1 (η), explicitly and to the equation ∂t η = − 2g
∂t η = −C∂x
h 2 3 2 h 2 η + η + (∂x η) + η∂x η . 4h 4 2
(25)
This can be called a higher order KdV equation. Note that also with this simplified Hamiltonian, the Hamiltonian formulation implies that the dispersion influences the nonlinear terms by the action of the structure map. A major difference with the classical KdV equation are the additional nonlinear terms. The expression in brackets has terms [ε, ε 2 / h, hε 2 μ2 ], so that these additional two terms of order hε 2 μ2 will be relevant for shorter waves for which h ∼ O(1/μ). Concerning simplified dispersion, for the full dispersion operator C various approximations have been proposed in the literature. Most approximations take a polynomial or rational function, corresponding to combinations of differential operators and inverses of differential operators. One example is the approximation taken by Madsen and Sorensen [14], given by the symbol Cˆ 2 = c02
1+
1 2 2 15 k h 1 + 25 k 2 h2
see also [18]. However, in the equations they consider, this operator is not fully applied to the nonlinear terms, as is expressed in (25). This may cause that the Hamiltonian formulation is lost,
200
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
which may result in loss of energy and momentum conservation. To illustrate this, we consider a somewhat simpler approximation of C, namely the classical KdV choice C = c0 (1+ 16 ∂x2 ) obtained by approximation the symbol by a quadratic function in the rather long wave regime. Then (25) becomes h 2 h2 2 3 2 h 2 ∂t η = −c0 1 + ∂x ∂x η + η + (∂x η) + η∂x η . 6 4h 4 2 (26) The vectorfield of this equation contains terms of various orders
ε2 2 2 2 2 c0 1 + h μ μ ε, , hε μ . h When further simplifications are desired, for instance, in the Boussinesq regime, one may be inclined to consider the following equation in which terms of order ε2 μ5 are disregarded h2 2 3 2 ∂t η = −c0 ∂x 1 + ∂x η + η 6 4h h h + (∂x η)2 + η∂x2 η . 4 2 However, the term with ∂x2 (η2 ) is the only term that is not the variational derivative of a functional; hence this approximation in the equation does not lead to a Hamiltonian system with the obvious structure map −c0 ∂x . A simplified Hamiltonian model is still possible, albeit of a much lower order. In fact, omitting all terms of order higher than ε2 μ, there results the classical KdV equation (23). Hence, aiming at an Hamiltonian equation of higher order that may be more accurate for shorter waves, the additional terms of order ε 2 μ3 are then necessarily accompanied by terms of order ε2 μ5 . Remark 3. When changing the structure −C∂x map from to −c0 ∂x as has been done to arrive at the classical KdV equation, it should be noted that then the quadratic momentum func¯ tional has to be modified also, from (13) M(η) = g ηC −1 η ¯ to M(η) = g ηc0−1 η. 5.3. Deep water unidirectional equation √ ˆ ˆ On √ water we have √ C = g/|k| and A := √ infinitely deep ˆ ik/ |k| = i sign(k) |k| and B = |k|. Then the AB equation becomes 1 1 √ ∂t η = − gA η + (Bη)2 + B(ηBη) 4 2 1 1 − (Aη)2 + A(ηAη) (27) 4 2 with terms of order √ 2 gμ ε, ε μ . Since both operators A and B have the same order of differ√ entiation μ, all four quadratic terms in the equation are of the same order. These operators cannot be easily approximated with ordinary differential operators.
6. Conclusion and remarks In order to illuminate the approximative nature of the AB equation we briefly summarize the derivation. The basis is the use of the variational structure of the equations of surface waves. This requires the expression of the Hamiltonian (the total energy) in the basic variables of surface elevation and surface potential as canonical variables. The AB equation has been derived starting with a separation of the nonlinear and dispersive aspects. For given fluid potential, the kinetic energy has been approximated in firstorder accuracy in the waveheight. The surface potential has been approximated accurate in first order of the wave elevation in terms of the potential at the still water level. The unidirectionalization is then obtained by a restriction of the action functional to the manifold that relates the still water level potential to the elevation. From the quadratic dependence of the kinetic energy on the fluid potential, the resulting restricted action functional is third order accurate in the wave height. This approximate action functional is then further simplified by invoking an approximation using the linearized evolution that does not change the accuracy in third order. The above steps result into a modified action principle that is accurate in third order of the wave height, and where specification of (approximative) dispersive properties have not been invoked yet. Specifying the dispersion for infinite depth, a new evolution for unidirectional waves above deep water is obtained. In this case all quadratic terms in the governing equation are of the same spatial order. For waves on finite depth, the equation with second order accuracy in the wave height, the exact dispersion can be approximated in various ways. Taking the classical approximation for the lowest order approximation leads to higher order KdV equations. The higher-order terms in this equation account for additional effects for relatively short waves. We commented on loosing the Hamiltonian structure when just individual terms are deleted in the governing equations, since terms of different order may together form a variational structure; Hamiltonian consistent equations are easiest obtained by making approximations in the action functional. Using the variational approach has the advantage of maintaining a coupling between the velocity potential within the fluid and the free surface. In the derivation of KdV via series expansions in waveheight and dispersion, the vertical velocity component may vanish (see [28]). This means that within the framework provided by the series truncation viewpoint, the particles in the fluid below the surface may ignore vertical motion. This does not capture the particle motion below a surface water wave in irrotational flow, as described recently in [29]. Keeping a coupling between velocity potential and free surface, as is done in this Letter, offers the possibility of a more accurate approximation of the motion throughout the fluid. The above described summary shows the modifications that are necessary when one wants to arrive at equations that are accurate in higher than second order in the wave height. Then higher-order accurate approximations will be needed in the approximation of the kinetic energy for given fluid potential, in the approximation between surface potential and still-
E. van Groesen, Andonowati / Physics Letters A 366 (2007) 195–201
water level potential, and in the uni-directionalization manifold. Acknowledgements
[12] [13] [14] [15] [16]
The authors gratefully acknowledge useful suggestions and comments made by the referee. This work is part of project TWI.5374 of the Netherlands Organization of Scientific Research NWO, subdivision Applied Sciences STW.
[17] [18] [19]
References [1] R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997. [2] J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978. [3] A. Constantin, W. Strauss, Comm. Pure Appl. Math. 57 (2004) 481. [4] M.B. Abbott, A.D. McCowan, I.R. Warren, J. Hydraulic Engrg. 110 (10) (1984) 1287. [5] Y. Agnon, P.A. Madsen, H.A. Schäffer, J. Fluid Mech. 399 (1999) 319. [6] H.B. Bingham, Y. Agnon, Eur. J. Mech. B: Fluids 24 (2) (2005) 255. [7] M.F. Gobbi, J.T. Kirby, Coastal Engrg. 37 (1999) 57. [8] M.F. Gobbi, J.T. Kirby, G. Wei, J. Fluid Mech. 405 (2000) 181. [9] A.B. Kennedy, J.T. Kirby, M.F. Gobbi, Coastal Engrg. 44 (2002) 205. [10] J.T. Kirby, Nonlinear, dispersive long waves in water of variable depth, in: J.N. Hunt (Ed.), Gravity Waves in Water of Finite Depth, in: Advances in Fluid Mechanics, vol. 10, Computational Mechanics Publications, 1997, pp. 55–125. [11] J.T. Kirby, Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents, in: V.C. Lakhan (Ed.), Advances in Coastal Modeling, Elsevier, 2003, pp. 1–41.
[20] [21] [22] [23]
[24]
[25] [26] [27] [28] [29]
201
C. Lee, Y.-S. Cho, S.B. Yoon, Ocean Engrg. 30 (2003) 1849. P.A. Madsen, R. Murray, O.R. Sørensen, Coastal Engrg. 15 (1991) 371. P.A. Madsen, O.R. Sørensen, Coastal Engrg. 18 (1992) 183. P.A. Madsen, H.A. Schäffer, Philos. Trans. Math. Phys. Engrg. Sci. Ser. A 1749 (1998) 3123. P.A. Madsen, H.A. Schäffer, A review of Boussinesq-type equations for surface gravity waves, in: P.L.-F. Liu (Ed.), Advances in Coastal and Ocean Engineering, vol. 5, World Scientific, 1999, pp. 1–95. P.A. Madsen, H.B. Bingham, H.A. Schäffer, Proc. R. Soc. London, Series A 459 (2003) 1075. P.A. Madsen, D.R. Fuhrman, B. Wang, Coastal Engrg. 52 (2006) 487. O. Nwogu, J. Waterway, Port, Coastal Ocean Engrg ASCE 119 (6) (1993) 618. H.A. Schäffer, P.A. Madsen, Coastal Engrg. 26 (1995) 1. O.R. Sørensen, H.A. Schäffer, L.A. Sørensen, Coastal Engrg. 50 (2004) 181. J.M. Witting, J. Comput. Phys. 56 (1984) 203. G. Klopman, M.W. Dingemans, B. van Groesen, A variational model for fully non-linear water waves of Boussinesq type, in: Proceedings IWWWFB, 2005. G. Klopman, E. van Groesen, M.W. Dingemans, Variational Boussinesq modelling of fully non-linear water waves, J. Fluid Mech. 2006, submitted for publication. D.G. Kortweg, G. de Vries, Philos. Mag. 39 (1895) 422. V.E. Zakharov, J. Appl. Mech. Techn. Phys. 9 (2) (1968) 190. L.J.F. Broer, Appl. Sci. Res. 29 (1974) 430. P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge Univ. Press, Cambridge, 1989. A. Constantin, Invent. Math. 166 (2006) 523.