Transformation of a shoaling undular bore

Abstract

We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variablecoefficient Korteweg -de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons -an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments.

Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation

“…An alternative method developed by Kamchatnov (2004) for a perturbed KdV equation requires a change of variable in (5), U ¼ bŨ andT ¼ R T b dT, to generate a KdV equation forŨ with a perturbation term of the form bTŨ=b. This approach was used by El et al (2007El et al ( , 2012 to study the evolution of solitary waves and undular bores over a slope, a study similar but complementary to that described here.…”

“…1 in the modulation equations (14, 15) by retaining the terms in 1/K(m), or more directly by averaging the wave action conservation law (8) for a solitary wave, see Whitham (1974) for the case of the constant-coefficient KdV equation, or Grimshaw (1979) and the discussion in El et al (2012) for the variable-coefficient KdV equation. The pair (13), (20) form a nonlinear hyperbolic system for a solitary wave train and can be solved explicitly.…”

“…Here, when the depth varies, we adapt and simplify the Gurevich-Pitaevskii approach, guided here by the detailed theory developed by El et al (2012) for the propagation of an undular bore up a slope. The main difference is that in El et al (2012) the undular bore was formed from a step initial condition (actually a box but much longer than (28) that we have used here).…”

“…The main difference is that in El et al (2012) the undular bore was formed from a step initial condition (actually a box but much longer than (28) that we have used here). Consequently, the undular bore in El et al (2012) was well formed when entering the slope, and in particular there was no sign in their simulations of the rarefaction forming at the rear of the evolving disturbance, see our simulations below. Thus, as described above by (25), the solution initially develops according to the Hopf equation and a shock forms at a time T 0 and place X 0 (26).…”

“…In turn the constant a 0 can be found using the theory of Gurevich and Pitaevskii (1974) for a constant depth, see El (2007), or the adaptation of that theory for a slope by El et al (2012). Since the shock solution of the Hopf equation develops a jump of U M , the outcome is that a 0 ¼ 2U M b 0 where b 0 ¼ bðT 0 Þ.…”