Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

Nonlinear, dispersive waves appear in settings where nonlinear effects are relevant and waves of different wavelengths travel at different speeds. Examples abound in oceanic waves, atmospheric waves, including the morning glory of Australia [18], tidal bores in oceans and rivers [15] and pulses in viscoelastic vessels, [44]. In all these cases, traveling waves seem to play a fundamental role manifesting themselves via a wave front followed by undulations. The front is decreasing to rest, while the undulations appear as progressively decreasing in amplitude oscillations that expand backwards for long distances. Such waves are usually called undular bores, positive surges or Favre waves [30, 6, 31].

Undular bores have been studied using unidirectional nonlinear dispersive equations such as the Benjamin-Bona-Mahony (BBM) equation (also called Regularized Long Wave (RLW) equation) or the Korteweg-de-Vries (KdV) equation [46, 11, 28, 29], as well as bidirectional nonlinear dispersive equations such as the Serre equations [26, 27] or the Kaup-Boussinesq (linearly ill-posed [8]) system [25, 35, 32]. In this work we focus on the Peregrine-Boussinesq system [47, 12, 8] with flat bottom topography

where $x$, $t$ are the space and time variables, $\eta=\eta(x,t)$ denotes the free-surface elevation while $u=u(x,t)$ stands for the horizontal velocity of the fluid measured at some height above the flat bottom of depth $D$. The constant $g$ is the acceleration due to gravity. The particular system describes the propagation of long waves in both directions, [8], which provides a more accurate description of undular bores compared to its unidirectional counterparts.

Peregrine’s system (1) possesses solitary wave solutions that have smooth profiles and decay exponentially in space [16]. It is well-posed in the Hadamard sense: for initial data $(\eta(x,0),u(x,0))\in H^{s}\times H^{s+1}$, where $H^{s}=H^{s}(\mathbb{R})$ the usual Sobolev space with $s\geq 1$, and if $\inf_{x\in\mathbb{R}}\{1+\eta(x,0)\}>0$, then for any $T>0$ there is a unique solution $(\eta,u)\in C([0,T];H^{s}\times H^{s+1})$ of (1), [48, 3, 9], and the recent work [45] for $s>1/2$. It should be noted that the inviscid Peregrine system (1) does not possess undular bores as traveling wave solutions but instead possesses dispersive shock waves formed by series of solitary waves followed by dispersive tails, making the explanation of undular bores via this system (alone) questionable.

It has been experimentally confirmed that dissipation due to turbulence plays a role in the shape formation of undular bores [50], in addition to nonlinearity and dispersion, which are of course unavoidable components of water waves. Turbulence is expected to produce dissipation but its effect on the shape of undular bores is known only via experimental evidence [37, 38]. The objective of this work is to test the effect of dissipation on the Peregrine system and to show that it is a necessary ingredient for the existence of traveling waves. We consider the system

where $\varepsilon$ is a positive constant, in order to study the effect of viscosity on (1). The system (2) can be derived to describe free-surface water waves of long wavelength and small amplitude from dissipation approximations of potential flow [34] using asymptotic expansions as demonstrated in [21], see also Appendix A. Similar Boussinesq systems are derived in other settings for laminar flows with parabolic velocity profile, [21, 44], see also [41, 39, 22]. The selection of the parameter $\varepsilon$ relies on experimentation because the dissipative effects are known only via experimental evidence and are not well-formulated in mathematical terms [37, 38]. Note that the term $u_{xx}$ describes the bulk damping in fluid’s body. In more realistic situations, dissipation due to boundary layers and bottom friction must be included via the heuristic Manning’s frictional term $c_{m}u|u|/(D+\eta)^{1/3}$, [24]. Inclusion of such term will result eventually in the dissipation of any solution in time as it is demonstrated experimentally in [10].

We study traveling wave solutions (2) and their dependence as functions of the parameters $\varepsilon$ and $\delta=\tfrac{D^{2}}{3}$. Such traveling waves often approximate shocks and they are called diffusive-dispersive shock waves [29]. We show their shape depends on the balance among dispersion and dissipation: (i) When dispersion is weak, then the wave is monotone. (ii) When the dispersion is moderate then the traveling waves consist of undulations following the traveling front.

Our results are motivated and the techniques parallel the analysis of [11] where existence and the shape of traveling waves is studied for a KdV-Burgers equation. But there is a difference; as the first equation has no dissipation it leads to a singularity in the differential equations describing the traveling waves, that needs to be analyzed and causes the free surface elevation to have steep gradients for large speeds; this phenomenon is not present in the KdV-Burgers equation of [11]. It is convenient to consider a scaled variant of (1), with $g=1$ and $\delta=D^{2}/3$, where the speed $c$ is scaled by $\sqrt{gD}$ and leads to the Froude numbe. In the long wave limit we expect that traveling waves propagate with scaled speed $c>1$. We show that the system (2) has a unique up to horizontal translations traveling wave solution that resembles positive surge for each (scaled by $\sqrt{gD}$) phase speed $c>1$. These traveling waves are oscillatory shock waves when $\varepsilon^{2}<4\delta c\alpha(c)$ and regularized shock waves when $\varepsilon^{2}\geq 4\delta c\alpha(c)$ where

The scaled speed $c$ is referred in the literature as the Froude number. There are two asymptotic regimes as $\delta,\varepsilon\to 0$. The weak dispersion regime $\delta<\frac{\varepsilon^{2}}{4c\alpha(c)}$, characterized by the eigenvalues of a linearized operator being real, where the traveling wave converges monotonically to a shock of the associated shallow water system. Moreover, combined with the analysis in [7], there is a moderate dispersion regime $\frac{\varepsilon^{2}}{4c\alpha(c)}<\delta\leq o(\varepsilon)$, where the diffusive-dispersive traveling wave still converges to a classical isentropic shock but the tail exhibits an oscillatory wave-train that shrinks as $\delta,\varepsilon\to 0$.

From a physical point of view, fitting solutions of the dissipative Boussinesq system with laboratory data suggest the use of certain values for the parameter $\varepsilon$ depending on the Froude number $c$. In particular, when $c<1.3$ very little dissipation ($\varepsilon=0.05$) is adequate for the accurate description of undular bores. For larger values of $c$ when dissipation due to wave breaking becomes dominant, then values $\varepsilon=O(1)$ are necessary for a good approximation of breaking waves even for $c=1.45$, while for $c=1.6$ the current theory fails to predict accurately the fully-breaking bore.

Existence of smooth solutions has been proved for dissipative Boussinesq systems, [1]. In a final section we compare smooth solutions of the two systems (1) and (2) and show that solutions emanating from the same initial data remain close with error $O(\varepsilon t)$. We note that the Peregrine system does not possess traveling wave solutions of undular bore type, yet it is commonly used in practice for the description of undular bores. The error estimate explains the accuracy of the asymptotic models in describing dispersive shock waves for time scales of $O(1/\varepsilon)$. Due to the fact that the Peregrine system along with all the other Boussinesq systems of [8] are asymptotically equivalent with the same error estimates against the Euler equations, we expect that all these system have the same capabilities to approximate undular bores.

The structure of this paper is as follows: In Section 2 we prove the existence and uniqueness of the traveling wave solutions for the system (2). We also provide estimates for the amplitude, tail and phase speed of the waves. In Section 3 we describe two different types of traveling waves and we give the exact criterion for acquiring regularized shock waves to the non-dispersive shallow water equations by dispersive-dissipative regularization. Section 4 provides the error estimate between systems (1) and (2). The article closes with conclusions and an Appendix containing a brief review of the derivation of the dissipative Boussinesq system (2).

In this section we study the existence and uniqueness (up to horizontal translations) of traveling wave solutions to the following dissipative Peregrine-Boussinesq system

We will be considering traveling wave solutions to the Boussinesq system (3) of the form

where $\xi=x-ct$, for $c>1$.

The following limiting conditions are imposed on $\eta$ and $u$:

where $\eta^{-}>0$, $u^{-}>0$, and we also require

Substitution of (4) into the system (3), and integrating over $[\xi,\infty)$ we obtain the algebraic relation between $\eta$ and $u$

and the ordinary differential equation

where ^′ stands for the usual derivative $d/d\xi$. Substitution of (7) into (8) gives the second order equation

Assuming that the equation (9) has positive solutions that satisfy the conditions (5) and (6), we estimate the magnitude of the tail of the traveling wave (undular bore). In particular, we have the following lemma:

We first estimate the local extrema of the solution $u$.

We define a dependent variable $v=(\delta c)u^{\prime}$. Then (9) is equivalent to the first order system

which we will write in the compact form

where

The system (18) has three critical points: (0,0) and ($u_{\pm}$, 0).

When linearized about (0,0), system (18) has characteristic polynomial

with eigenvalues

Hence (0,0) is a saddle point. When the system is linearized about $\mathbf{u}_{-}=(u_{-},0)$, the Jacobian matrix is

where (recalling that $u_{0}=u_{-}$)

The characteristic polynomial of the Jacobian matrix is $\lambda^{2}-\varepsilon(\delta c)^{-1}\lambda+\alpha(c)~{}(\delta c)^{-1}=0$ with eigenvalues

These eigenvalues are complex when

in which case the critical point $(u_{0},0)$ is a spiral. The first derivative of the function

is greater than zero for $c>1$. Since $\alpha(c)$ is equal to zero at $c=1$, we conclude that the function is increasing for $c\geq 1$, and hence the right-hand side of (21) is positive for $c>1$. The real part of the eigenvalues $\Lambda_{\pm}$ is positive and the critical point is always unstable (node or spiral).

The third critical point $u_{+}>0$ satisfies $u_{+}>c$. As we see below, solutions to (18) must satisfy $u(\xi)<c$ for all $\xi\in\mathbb{R}$. Thus the critical point $u_{+}$ cannot be connected to $(0,0)$ at $+\infty$ and will be excluded from the possible asymptotic limits.

Now let $\mathcal{R}=\{(u(\xi),v(\xi)):\xi\in\mathbb{R}\}$ be any bounded orbit of the system (18). The following will be a study of the asymptotic states of $\mathcal{R}$ at $+\infty$ and $-\infty$. However first note that (18) has no non-constant periodic solutions. To see this, let ($u_{p},v_{p}$) be a periodic solution to (18) with period $p>0$. Then ($u_{p},v_{p}$) is also a solution to (9). Multiplying (9) by $u^{\prime}_{p}$ and integrating over the interval (0, $p$) yields the identity

since by periodicity, all terms integrate to zero except one. It follows from (23) that $u_{p}$ is constant. Alternatively, observe that $\nabla\!\cdot\!\mathbf{F}(\mathbf{u})=\varepsilon(\delta c)^{-1}>0$ for all $c>1$, and thus by Bendixson-Dulac Theorem, [53], system (18) has no closed orbits in its phase space. This excludes also the existence of classical solitary wave solutions, which are homoclinic orbits to the origin.

Lemma 2.3 shows that the set $\mathcal{M}=\{(u,v)\in\mathbb{R}^{2}:0<u<c\}$ is an invariant set. Since there are no non-trivial periodic solutions to (18) and we have that at infinity both $u^{\prime}$ and $v^{\prime}$ tend to zero, we conclude that both the $\alpha$-limit set and the $\omega$-limit set must contain critical points of the system. Since these limit sets are connected by the orbit $\mathcal{R}$, they must each contain exactly one critical point and hence the orbit must tend asymptotically to a critical point both at $+\infty$ and $-\infty$. At one side the orbit connects to $(0,0)$ due to (6) while at $-\infty$ the critical point must be different due to Lemma 2.1. Using a refined version Poincaré-Bendixson Theorem, [53], any bounded orbit of (18) connects the critical points $(0,0)$ and $(u_{0},0)$.

Changing the direction of the traveling wave using the variables $\tilde{\xi}=-\frac{\xi}{\delta c}$, $\tilde{v}=v$ and $\tilde{u}=u$, we write the system (17) as

where ${}^{\prime}=\frac{d}{d\tilde{\xi}}$ while $G(u)$ is the potential function

Observe that (27) is dissipative in the sense that its solutions $(\tilde{u}(\tilde{\xi}),\tilde{v}(\tilde{\xi}))$ satisfy

where $V$ is the Liapunov function

The potential is given within a normalization constant which is here selected so that $G$ is negative for the traveling waves under study.

By the LaSalle invariance principle, [53], every bounded orbit $\mathcal{R}$ of (17) converges to the equilibrium point $(u_{0},0)$ as $\xi\to-\infty$. Moreover, when $\varepsilon=0$ the systems (17) and (27) are Hamiltonian and $V$ coincides with their Hamiltonian. In such case, we know that for $c>1$ there is a unique (up to translations) traveling wave (homoclinic orbit to the origin) with infinite period and of maximal energy $G(\bar{u})=0$, where here $\bar{u}=\max_{\xi}(u)$ denotes the amplitude of the traveling wave [16, 20]. The particular equation $G(\bar{u})=0$ is known as the speed-amplitude relation for solitary waves ($\varepsilon=0$). Let us denote by $\bar{\eta}=\max_{\xi}(\eta)$ the amplitude of $\eta$. It satisfies the formula $\bar{\eta}=\bar{u}/(c-\bar{u})$. Solving the equation $G(\bar{u})=0$ we obtain the speed-amplidute relationship for the solitary wave

Note that this relationship does not depend neither on $\delta$ nor on $\varepsilon$. In the case of diffusive-dispersive shock waves, their amplitudes should vary between $\eta_{0}=u_{0}/(c-u_{0})$ and the value $\bar{\eta}$ of the solitary wave amplitude. As we shall see later, even in the case where the dissipation of $O(10^{-2})$, the experimental data confirm that the amplitude of the solitary wave is a very good approximation of the amplitude of undular bores.

$G(u)$ has an inflection point at $(u_{c},G(u_{c}))$ with $u_{c}=c-\sqrt[3]{c}$. It also has three extrema $(0,0)$ and $(u_{\pm},G(u_{\pm}))$, though we have excluded the case of $u_{+}$. Moreover, $G(u)$ has a singularity (vertical asymptote) at $u=c$ with $\lim_{u\to c}G(u)=\infty$. A sketch of the potential is presented in Figure 1. The traveling wave will take values in the region between the local maximum of $G$ and the upper bound of possible maximum amplitude of a traveling wave, depicted in the figure by $\bar{u}$. The local maxima of the traveling wave will be located on the right side of the minimum $u_{0}$ of $G$, while the local minima of $u$ on its left side. As we shall see in Proposition 2.1 the maximum amplitude of the traveling wave cannot exceed the point of intersection between the line $y=0$ and the potential function $G$. The line $y=0$ depicts also the barrier of the maximal energy a solution may have.

A consequence of the decreasing Liapunov function is the following Proposition on positive solutions.

Because $(0,0)$ is a saddle point, there are two semi-orbits that converge to $(0,0)$ as $\xi\to+\infty$ and approach the origin tangentially to the stable manifold determined by the slope

where $\lambda_{\pm}$ as defined in (19). Thus, one semi-orbit approaches the origin from the fourth quadrant $Q_{4}=\{(u,v):u>0,v<0\}$ and the other through the second quadrant $Q_{2}=\{(u,v):u<0,v>0\}$, [53]. The later case of negative $u$ is excluded though due to Lemma 2.3 and thus the only semi-orbit that can exist is the one with $u>0$.

We have seen that if equation (9), or the equivalent system (18), has a solution $u$ satisfying the asymptotic conditions (5), then $0<u<c$ ensuring that $\eta>0$, see Lemma 2.3 and (7). We next prove that the system (18) has at least one such solution $u$. The main result of this section is stated bellow.