Mohammad-Reza Alam

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis

We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued.

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue ( J. Fluid Mech. , vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples

We consider a class of higher order (quartet) Bragg resonance involving two incident wave components and a bottom ripple component (so called class III Bragg resonance). In this case, unlike class I/II Bragg resonance involving a single incident wave and one/two bottom ripple components, the frequency of the resonant wave, which can be reflected or transmitted, is a sum or difference of the incident wave frequencies. In addition to transferring energy across the spectrum leading to potentially significant spectral transformation, such resonances may generate long (infragravity) waves of special importance to coastal processes and engineering applications. Of particular interest here is the case where the incident waves are oblique to the bottom undulations (or to each other) which leads to new and unexpected wave configurations. We elucidate the general conditions for such resonances, offering a simple geometric construction for obtaining these. Perturbation analysis results are obtained for these resonances predicting the evolutions of the resonant and incident wave amplitudes. We investigate special cases using numerical simulations (applying a high-order spectral method) and compare the results to perturbation theory: infragravity wave generation by co- and counter-propagating incident waves normal to bottom undulations; longshore long waves generated by (bottom) oblique incident waves; and propagating-standing resonant waves due to (bottom) parallel incident waves. Finally, we consider a case of multiple resonance due to oblique incident waves on bottom ripples which leads to complex wave creation and transformations not easily tractable with perturbation theory. These new wave resonance mechanisms can be of potential importance on continental shelves and in littoral zones, contributing to wave spectral evolution and bottom processes such as sandbar formation.

Closed-loop separation control: An analytic approach

We develop an analytic approach to two-dimensional flow separation control by feedback. With two wall-based actuators enclosing an array of distributed wall-shear sensors, we control the wall-shear evolution equation through its boundary values at the actuators. Using this approach, we induce separation at prescribed locations in steady and unsteady channel flows, and reduce the recirculation length behind a backward-facing step to a prescribed value.

Nonlinear analysis of an actuated seafloor-mounted carpet for a high-performance wave energy extraction

It is known that muddy seafloors can extract significant energy from overpassing surface waves via engaging them in strong interaction processes. If a synthetic seabed can respond to the action of surface gravity waves similar to the mud response, then it too can take out a lot of energy from surface waves. Analysis of the performance of a mud-resembling seabed carpet in harvesting ocean wave energy is the subject of this article. Specifically, and on the basis of the field measurements and observations of properties/responses of seafloor mud, we focus our attention on an artificial viscoelastic seabed carpet composed of (vertically acting) linear springs and generators. We show that the system of sea/synthetic-carpet admits two propagating wave solutions: the surface mode and the bottom mode. The damping of a surface-mode wave is proportional to its wavelength and hence is classic. However, the damping of a bottom-mode wave is larger for shorter waves, and is in general stronger than that of the surface-mode wave. To address the effect of (high-order) nonlinear interactions as well as to investigate the performance of our proposed carpet of wave energy conversion (CWEC) against a spectrum of waves, we formulate a direct simulation scheme based on a high-order spectral method. We show, by taking high-order nonlinear interactions into account, that the CWEC efficiency can be significantly higher for steeper waves. We further show that the bandwidth of high performance of the CWEC is broad, it yields minimal wave reflections and its theoretical efficiency asymptotically approaches unity within a finite and (relatively) short extent of deployment.

Attenuation of long interfacial waves over a randomly rough seabed

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiplescale analysis. For transient wave pulses, the governing equation for the pulse profile is shown to be an integro-differential equation combining KdV and Burgers terms. Quantitative and qualitative effect of disorder on the attenuation of wave amplitude, reduction of wave speed and change of wave profile are examined numerically and analytically based on the asymptotic approximation. For time-harmonic waves, modecoupling equations are derived and examined for the competition between diffusion by random scattering, steepening by nonlinearity and frequency dispersion for a broad range of depth ratios.

Predictability horizon of oceanic rogue waves

Prediction is a central goal and a yet-unresolved challenge in the investigation of oceanic rogue waves. Here we define a horizon of predictability for oceanic rogue waves and derive, via extensive computational experiments, a statistically converged predictability time scale for these structures. We show that this time scale is a function of the sea state (i.e., severity of the ambient ocean waves), the height of the anticipated rogue wave, and the level of uncertainties in the ocean measurements. The presented predictability time scale establishes a quantitative metric on the combined temporal effects of the variety of mechanisms that together lead to the formation of a rogue wave and is crucial for the assessment of validity of rogue wave predictions, as well as for the critical evaluation of results from the widely used model equations.

Landslide tsunamis in lakes

Landslides plunging into lakes and reservoirs can result in extreme wave runup at shores. This phenomenon has claimed lives and caused damage to near-shore properties. Landslide tsunamis in lakes are different from typical earthquake tsunamis in the open ocean in that (i) the affected areas are usually within the near-field of the source, (ii) the highest runup occurs within the time period of the geophysical event, and (iii) the enclosed geometry of a lake does not let the tsunami energy escape. To address the problem of transient landslide tsunami runup and to predict the resulting inundation, we utilize a nonlinear model equation in the Lagrangian frame of reference. The motivation for using such a scheme lies in the fact that the runup on an inclined boundary is directly and readily computed in the Lagrangian framework without the need to resort to approximations. In this work, we investigate the inundation patterns due to landslide tsunamis in a lake. We show by numerical computations that Airys approximation of an irrotational theory using Lagrangian coordinates can legitimately predict runup of large amplitude. We also demonstrate that in a lake of finite size the highest runup may be magnified by constructive interference between edge-waves that are trapped along the shore and multiple reflections of outgoing waves from opposite shores, and may occur somewhat later after the first inundation.

Ships advancing near the critical speed in a shallow channel with a randomly uneven bed

Effects of random bathymetric irregularities on wave generation by transcritical ship motion in a shallow channel are investigated. Invoking Boussinesq approximation in shallow waters, it is shown that the wave evolution is governed by an integrodifferential equation combining features of Korteweg–deVries and Burgers equations. For an isolated ship, the bottom roughness weakens the transient waves radiated both fore and aft. When many ships advance in tandem, a steady mount of high water can be formed in front and a depression behind. Wave forces on both an isolated ship and a ship in a caravan are obtained as functions of the mean-square roughness, ship speed and the blockage coefficient.

Broadband cloaking of flexural waves

The governing equation for elastic waves in flexural plates is not form invariant, and hence designing a cloak for such waves faces a major challenge. Here, we present the design of a perfect broadband cloak for flexural waves through the use of a nonlinear transformation in the region of the cloak and by matching term by term the original and transformed equations and also assuming a prestressed material with body forces. For a readily achievable flexural cloak in a physical setting, we further present an approximate adoption of our perfect cloak under more restrictive physical constraints. Through direct simulation of the governing equations, we show that this cloak, as well, maintains a consistently high cloaking efficiency over a broad range of frequencies. The methodology developed here may be used for steering waves and designing cloaks in other physical systems with non-form-invariant governing equations.