2.1 The variational principle for water waves

The governing equation for irrotational inviscid two-dimensional surface gravity waves is (see, for example, Phillips Reference Phillips1977)

together with boundary conditions

and the condition at the bottom $\unicode[STIX]{x1D719}_{z^{\prime }}^{\prime }\rightarrow 0$ as $z^{\prime }\rightarrow -\infty$ , where $\unicode[STIX]{x1D719}^{\prime }$ is the velocity potential, $\unicode[STIX]{x1D702}^{\prime }$ is the free-surface displacement, $\unicode[STIX]{x1D735}^{\prime }=(\unicode[STIX]{x2202}_{x^{\prime }},\unicode[STIX]{x2202}_{z^{\prime }})$ , and $g$ is the acceleration due to gravity. Although the governing equation is linear, the boundary conditions are nonlinear and, more severely, are evaluated at an (unknown) dependent variable of the system.

To make analytic progress, asymptotic approximations are employed. The wave slope $\unicode[STIX]{x1D716}\equiv a_{0}k_{0}$ is a measure of the nonlinearity of the system, and is assumed to be small. Furthermore, if $\unicode[STIX]{x0394}k$ is a perturbation to the characteristic wavenumber $k_{0}$ , then we assume $|\unicode[STIX]{x0394}k|/k_{0}=O(\unicode[STIX]{x1D716})$ , which corresponds to a narrow-banded (and hence slowly varying) wave packet. The characteristic angular frequency, $\unicode[STIX]{x1D714}_{0}$ , is related to the wavenumber by the linear dispersion relation, i.e.  $\unicode[STIX]{x1D714}_{0}^{2}=gk_{0}$ .

We non-dimensionalize our system as follows:

Expanding the velocity potential and free-surface displacement as a series, we have (Chu & Mei Reference Chu and Mei1970)

where $X=\unicode[STIX]{x1D716}x$ , $T=\unicode[STIX]{x1D716}t$ , $\unicode[STIX]{x1D703}=x-t$ , and c.c. means complex conjugate. $\overline{\unicode[STIX]{x1D719}}$ and $\overline{\unicode[STIX]{x1D702}}$ are the mean (phase-averaged) velocity potential and free-surface displacement, respectively, with their scale to be determined. Finally, we take the complex-valued coefficient $\unicode[STIX]{x1D719}_{11}\equiv B$ as our dependent variable.

The expansions of the dependent variables to second order in $\unicode[STIX]{x1D716}$ , appropriate for the MNLSE, are (Trulsen & Dysthe Reference Trulsen and Dysthe1997; Trulsen Reference Trulsen2006)

Equations (2.1), (2.2) can be reformulated in terms of a variational principle through the action (Luke Reference Luke1967; Zakharov Reference Zakharov1968; Miles Reference Miles1977)

where $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D719}(x,z=\unicode[STIX]{x1D716}\unicode[STIX]{x1D702},t)$ is the velocity potential evaluated at the free surface.

Next, we substitute the form of the velocity potential and free-surface displacement into (2.8). Furthermore, following Lo & Mei (Reference Lo and Mei1985), we map into a spatial reference frame by introducing the following transformation

This reference frame is travelling at the linear group velocity and, for a fixed position $X$ , $\unicode[STIX]{x1D70F}$ is proportional to the negative of the elapsed time, while $\unicode[STIX]{x1D712}$ is related to the fetch (or stretched distance). Finally, for clarity of presentation we transform from the dependent variable $B$ to the dependent variable in the mapped frame  $A$ .

Employing the mapping given in (2.9), the averaged Lagrangian (Whitham Reference Whitham1965, Reference Whitham1974) then becomes

where $\mathscr{H}$ is defined as (cf. Gramstad & Trulsen Reference Gramstad and Trulsen2011, appendix C)

and ^* denotes the complex conjugate. The Hilbert transform $\mathbb{H}$ is defined as

with $\mathit{P.V.}$ meaning we are to take the principal value of the integral (Titchmarsh Reference Titchmarsh1948). Following Janssen (Reference Janssen1983, see also Akylas Reference Akylas1989) we have rewritten our action in terms of one variable by recognizing that $\overline{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D70F}}$ and $\overline{\unicode[STIX]{x1D711}}_{Z}$ are harmonic conjugates, where $\unicode[STIX]{x1D711}$ is the velocity potential in this spatial reference frame. That is, these variables are real and imaginary parts of an analytic function, and hence we can connect the two variables, evaluated at $Z=0$ , via the Hilbert transform  $\mathbb{H}$ .

Note, as the linear dispersion is exactly represented in this spatial reference frame (Trulsen et al. Reference Trulsen, Kliakhandler, Dysthe and Velarde2000; Kit & Shemer Reference Kit and Shemer2002), only one term is needed to completely describe it. This offers numerical advantages (Lo & Mei Reference Lo and Mei1985), and simplifies the equations algebraically.

The action associated with the Lagrangian density in the spatial reference frame, i.e. (2.10), is defined as

which is a functional over all admissible functions satisfying the prescribed conditions at $A(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D712}_{0})$ and $A(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D712}_{f})$ . Hamilton’s principle states that the governing equations of the system are found by requiring that $\mathscr{S}$ be stationary. That is, we seek solutions $(A,A^{\ast })$ such that

for infinitesimal $(\unicode[STIX]{x1D6FF}A,\unicode[STIX]{x1D6FF}A^{\ast })$ . Substituting the Lagrangian density into the action, then applying Hamilton’s principle, and recalling the anti-self-adjoint nature of the Hilbert transform (Titchmarsh Reference Titchmarsh1948), gives us (Lo & Mei Reference Lo and Mei1985; Kit & Shemer Reference Kit and Shemer2002)

where $\unicode[STIX]{x1D6FC}_{0}=2\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6FD}_{0}=8\unicode[STIX]{x1D716}$ . The constants $(\unicode[STIX]{x1D6FC}_{0},\unicode[STIX]{x1D6FD}_{0})$ also make it easy to track the influence of the induced mean flow (which is related to the term with coefficient  $\unicode[STIX]{x1D6FC}_{0}$ ), and the asymmetric self-steepening term (which is related to the term with coefficient  $\unicode[STIX]{x1D6FD}_{0}$ ) in the ensuing calculations.

This equation is usually written as a set of coupled PDEs in two dependent variables, $(A,\overline{\unicode[STIX]{x1D711}}|_{Z=0})$ . To second order in $\unicode[STIX]{x1D716}$ , the two variables are connected by the relation

which implies that gradients in the radiation stress lead to the generation of a mean flow (Dysthe Reference Dysthe1979; McIntyre Reference McIntyre1981).

4.1 Evolution of the centroid of the linear energy density

We begin by considering the motion of the centroid of the action density of the wave packet, $M$ , defined as

so that, from (3.4), we have

Unlike the lower-order NLSE, this quantity is in general not a constant, and depends on the behaviour of $\int |A|^{4}\,\text{d}\unicode[STIX]{x1D70F}$ . Whereas $P/E$ arises due to linear dispersion, the second term is exclusively due to amplitude dependent nonlinear effects.

We define the integral

which serves as another measure of the distribution of the action density of the wave packet.

Next, we define the characteristic frequency $\unicode[STIX]{x1D714}_{0}$ so that the integral $P$ is 0, which means that the evolution of $M$ can be written as

As $K$ is positive definite, this shows that the centroid velocity is always positive, and is greater than the predictions of the lower-order theory (i.e. when $\unicode[STIX]{x1D6FD}_{0}=0$ ). This is consistent with the numerical results of Lo & Mei (Reference Lo and Mei1985, see also Trulsen Reference Trulsen1998, Chereskin & Mollo-Christensen Reference Chereskin and Mollo-Christensen1985).

In appendix A, we map (4.4) back into the laboratory reference frame, for ease of use in laboratory studies.

4.2 Evolution of the centroid of the spectrally weighted frequency

We next derive an evolution equation for the centroid of the spectrally weighted frequency $\mathscr{P}$ (Trulsen & Dysthe Reference Trulsen and Dysthe1997, see also § 3), related to the chirp of the signal (Gordon Reference Gordon1986) which we define as $C$ . That is,

so that from (3.10) its evolution is given by

with $\mathscr{G}$ given by (3.8).

Note, we can relate $C$ and $M$ :

The quantity $H$ is constant, so that to first-order deviations to the speed of the centroid are a result of the evolution of the chirp of $C$ , with focusing packets leading to larger velocities than defocusing packets.

4.3 Evolution of the variance of the spatial MNLSE

Finally, we derive an evolution equation for the variance of the MNLSE. Following the literature in plasma physics (Goldman & Nicholson Reference Goldman and Nicholson1978; Sulem & Sulem Reference Sulem and Sulem1999), the variance of the wave packet is defined as

with the nomenclature based on similar results from the $N$ -body problem (Robinson Reference Robinson1997). The variance is sensitive to the point about which it is computed, so that the centroid $M$ serves as a natural choice.

Differentiating $I$ with respect to $\unicode[STIX]{x1D712}$ , making use of (3.3), and integrating by parts, we have

which connects the previous two moments, $C$ and $M$ with the variance of the packet. The second term in (4.9) is negative definite, and hence always enhances packet focusing, while the last term is positive definite, which serves to retard focusing.

This relationship leads to a simple criterion for packet focusing or defocusing, which is defined as $I_{\unicode[STIX]{x1D712}}<0$ , $I_{\unicode[STIX]{x1D712}}>0$ , respectively. In particular, when

$I_{\unicode[STIX]{x1D712}}<0$ and we expect packet focusing. To first order in $\unicode[STIX]{x1D716}$ , this corresponds to $C>0$ , which simply states linear dispersion leads to focusing when shorter slower waves precede longer faster waves. Note, this dispersive focusing technique is commonly employed in laboratory studies of wave breaking (Longuet-Higgins Reference Longuet-Higgins1974; Rapp & Melville Reference Rapp and Melville1990). Furthermore, the minimum of $I$ , corresponding to the maximum focusing, occurs when

We have found that the evolution of the moments depends on the behaviour of $A$ , which we do not know a priori. To better constrain the behaviour of $A$ , we derive exact solutions for the MNLSE in the limit of vanishing linear dispersion.

7.1 Numerical scheme

The numerical scheme used to integrate the spatial MNLSE is that of Lo (Reference Lo1985, appendix D) and Lo & Mei (Reference Lo and Mei1985, § 3). This is a split-step spectral method, based on Fornberg & Whitham (Reference Fornberg and Whitham1978). That is, the linear part of (2.15) is mapped into spectral space, where differentiation becomes multiplication and the subsequent ordinary differential equation is solved, and then transformed back into physical space. The nonlinear terms are more conveniently computed via point-wise multiplication (as opposed to double convolution sums in spectral space) at half time steps, where a midpoint finite-difference approximation is then used to solve for  $A$ . The integration scheme is accurate to second order in integration step size, $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}$ , and we have found that no numerical instability occurs if $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}$ is kept sufficiently small. Note, to have the motion be $2\unicode[STIX]{x03C0}$ periodic in $\unicode[STIX]{x1D70F}$ , for ease of computation, we must introduce a scaling constant into the numerics, which we take to be 0.1 for all of these numerical experiments (Lo & Mei Reference Lo and Mei1985, see also Lo Reference Lo1985, Trulsen & Dysthe Reference Trulsen and Dysthe1997).

The numerical model was validated by Lo and Mei in several ways, which we have reproduced. For the examples given below, we use $N=2^{11}$ modes, with a time step of the order of $10^{-4}$ , and take $\unicode[STIX]{x1D70F}\in (0,2\unicode[STIX]{x03C0})$ , which we ensure at the very least satisfies the Courant–Friedrichs–Lewy (CFL) condition for waves governed by linear theory, that is $N\unicode[STIX]{x0394}\unicode[STIX]{x1D712}/2\leqslant 1$ . The resolution in $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D712}$ were both doubled for the simulations presented below, and suitable convergence of the model was corroborated. Finally, we numerically confirmed the first integral of the spatial MNLSE, $E$ , is conserved to one part in $O(10^{8})$ while the second integral $P$ is conserved to one part in $O(10^{5})$ for the examples considered in this work.

7.2 Forward leaning of wave packet; $\unicode[STIX]{x1D716}=0.4$ , $\mathscr{C}_{0}=0$

Our first example takes $\unicode[STIX]{x1D716}=0.4$ and $\mathscr{C}_{0}=0$ . That is, this packet has large nonlinearity, while the initial frequency chirp is taken to be zero. The value of $\unicode[STIX]{x1D716}$ , although potentially outside of the strict region of validity of the MNLSE, is chosen to illustrate the effects that nonlinearity has on the evolution of the wave packet.

Figure 3 shows the evolution of $|A|$ as predicted by the MNLSE (shown in blue), as well as the prediction of the lower-order NLSE (shown by the grey dashed curve), while the black line shows the solution of the IBE. We see that this packet is characterized by an asymmetric leaning on the forward face. At $\unicode[STIX]{x1D712}=0.15$ , the slope of $|A|$ increases by 45 %, corresponding to local slopes on the free surface that are greater than 0.5. This exceeds the values at which highly nonlinear dissipative processes occur, such as the formation of parasitic capillary waves (Longuet-Higgins Reference Longuet-Higgins1995; Fedorov & Melville Reference Fedorov and Melville1998; Melville & Fedorov Reference Melville and Fedorov2015) and wave breaking (Melville Reference Melville1982; Su et al. Reference Su, Bergin, Marler and Myrick1982b ; Melville Reference Melville1983).

Note, the initial condition is a soliton for the NLSE, and hence does not evolve with increasing fetch. Furthermore, we see that the solution to the IBE (which is known a priori) agrees well with the evolution of $|A|$ for small values of $\unicode[STIX]{x1D712}$ . For larger values, the forward leaning is arrested by dispersion; hence, the difference between the predictions of the MNLSE and the IBE become manifest. Finally, this self-steepening on the forward face of the group has been repeatedly observed in laboratory studies of wave groups (see, for example, Su et al. Reference Su, Bergin, Marler and Myrick1982a ; Melville Reference Melville1983; Chereskin & Mollo-Christensen Reference Chereskin and Mollo-Christensen1985; Pierson, Donelan & Hui Reference Pierson, Donelan and Hui1992).

Figure 3.

The evolution of the wave envelope with initial conditions given in (6.1) with $\unicode[STIX]{x1D716}=0.4$ , $\mathscr{C}_{0}=0$ , as predicted by the MNLSE (blue curve), the inviscid Burgers’ equation (IBE) (black curve) and the NLSE (dashed grey curve). The packet evolution is characterized by a forward leaning due to nonlinearity. We see that the solutions to the MNLSE and the IBE agree well for small values of  $\unicode[STIX]{x1D712}$ .

7.3 Focusing chirped packet; $\unicode[STIX]{x1D716}=0.1$ , $\mathscr{C}_{0}=1$

Next, we consider a second example with $\unicode[STIX]{x1D716}=0.1$ and $\mathscr{C}_{0}=1$ to induce packet focusing. In this example, from the moment evolution equations derived in § 6, we expect the packet to focus, and hence the amplitude will increase in order to conserve action (cf. (6.6)). This implies the IBE will do a poor job of predicting the evolution of this packet, as the amplitude of solutions to this equation stay nearly constant (see figure 3).

Therefore, in this section we analyse the closed set of moment evolution equations derived in § 6 to both predict the early evolution of the packet, as well as to interpret the mechanisms responsible for modulating the moment equations.

The evolution of $|A|$ as predicted by the MNLSE, and the theory presented in § 6, is shown in figure 4. The packet focuses and steepens on the forward face. For small values of $\unicode[STIX]{x1D712}$ the agreement between (6.5) and the MNLSE is good. It must be emphasized that (6.5) does not rely on full solutions to the MNLSE, rather we just solve the simple ODE given in (6.14). For larger values of $\unicode[STIX]{x1D712}$ , the two solutions deviate, as the asymptotic assumptions used in § 6 break down. This manifests itself in an over prediction of the packet amplitude.

Figure 4.

The evolution of $|A|$ , based on integrating the full MNLSE with initial conditions given in (6.1) with $\unicode[STIX]{x1D716}=0.1$ , $\mathscr{C}_{0}=1$ . The packet evolution is characterized by a large reduction in the packet variance, i.e. a focusing, which together with the self-steepening mechanism leads to large slopes on the forward face of the wave group. The dashed line shows the initial condition.

At $\unicode[STIX]{x1D712}=0.12$ the slope is more than three times its initial value, and exceeds 0.4. Like the example considered in § 7.2, this exceeds the slope at which highly dissipative processes are observed. Furthermore, this example makes it clear that chirped signals are effective for generating large slopes in the free-surface displacement, even for waves that are initially only weakly nonlinear. This has been exploited in laboratory studies on wave breaking (Longuet-Higgins Reference Longuet-Higgins1974; Rapp & Melville Reference Rapp and Melville1990).

The theoretical predictions of $w$ , $S$ , $C$ and $\text{d}M/\text{d}\unicode[STIX]{x1D712}$ are shown in figure 5. The initial conditions imply $S_{0}=0$ , while $w_{0}=1$ . In general for small values of $\unicode[STIX]{x1D712}$ there is good agreement between the theoretical predictions and the full MNLSE.

Figure 5.

A comparison between the predictions of the moments based on the theory developed in § 6 (shown in red) and the full solutions to the MNLSE (shown in blue). (a) Evolution of the packet width  $w$ . The packet focuses, corresponding to a decrease in its variance. (b) Evolution of the chirp, which decreases monotonically as the packet focuses. (c) Evolution of the asymmetric contribution to  $A$ . (d) Evolution of the centroid, which speeds up as the packet focuses. For small values of $\unicode[STIX]{x1D712}$ , there is good agreement between the theory developed in § 6 and the full solutions to the MNLSE.

Figure 5(a) shows the packet variance, which decreases during the focusing event due to the initial positive chirp of the signal (cf. (6.13)). We see good agreement between the theoretical predictions and the full numerical solutions for small $\unicode[STIX]{x1D712}$ , and larger discrepancies between the two for larger  $\unicode[STIX]{x1D712}$ . Next, the chirp is shown in figure 5(b). It is seen that the initially positive chirp decreases monotonically with $\unicode[STIX]{x1D712}$ , corresponding to the longer faster waves catching the shorter slower waves due to linear dispersion.

Next, the evolution of $S(\unicode[STIX]{x1D712})$ is shown in figure 5(c). This corresponds to the growth of the asymmetric wave in (6.5). Finally, figure 5(d) shows the speed of the centroid, which accelerates as the packet focuses. Again, there is good agreement between the theoretical prediction and the full MNLSE over the range of $\unicode[STIX]{x1D712}$ shown here.