Research

Nonlinear wave-wave interaction in deep water

For weakly nonlinear waves, nonlinear interaction appears with quadratic nonlinearities in shallow water, and cubic nonlinearities in deep-water. Such nonlinearities are responsible, for example, for the amplitude-dependent change in wave celerity found by Stokes in the mid 19th century. Since work by Phillips and others, the role of nonlinear resonant and near-resonant interactions has become appreciated as a major contributor to the dynamics of ocean surface waves.

In linear theory, waves pass through one another unchanged due to the principle of superposition. Once weak nonlinearities are taken into account, each wave is coupled with the other waves present in the sea. Thus, the frequency of a short wave may be appreciably changed by the presence of long waves (the short waves "surf on top of" the longer waves). The same principle can be shown for spectra of waves [2], where frequencies higher than the spectral peak frequency are most effected by a type of generalized Stokes' correction.

In deep water, the phenomena associated with cubic nonlinear interaction are captured by the Zakharov equation, from which the nonlinear Schrodinger equation can be derived in the limit of narrow-bandwidth. We have used this Zakharov equation formulation extensively in studying nonlinear waves, and likewise explore stochastic analogues which govern averaged properties, like the energy distribution of the sea-states. In particular, going beyond the paradigm of Hasselmann's kinetic equation, we are interested in the effect of statistical inhomogeneity [4], as well as fast evolution of homogeneous sea-states [2]. A review of some recent developments, in both deep and shallow-water recently appeared in [1].

  1. R. Stuhlmeier, T. Vrecica, and Y. Toledo, Nonlinear wave interaction in coastal and open seas - deterministic and stochastic theory, in Nonlinear Water Waves (D. Henry et al (Eds.), Springer, 2019
  2. D. Andrade, R. Stuhlmeier, and M. Stiassnie, On the generalized kinetic equation for surface gravity waves, blow-up and its restraint, Fluids, 4 (2019), 2.
  3. R. Stuhlmeier and M. Stiassnie, Nonlinear dispersion for ocean surface waves, Journal of Fluid Mechanics, 859 (2019), 49-58.
  4. R. Stuhlmeier and M. Stiassnie, Evolution of statistically inhomogeneous degenerate water wave quartets, Phil. Trans. R. Soc. A 376 (2018) 20170101.

Wave energy converters, survivability, and resource assessment

We developed a number of analytical models of wave energy converters, based on floating plates [4], floating cylinders [1], and coupled floating cylinders [2,3] in order to study device design, response, and survival in a range of sea states. While idealized, the linear theory for devices can be described in a mathematically simple form, and may be understood analytically without resorting to approximations.

The devices investigated were used to model arrays of converters, and to study the wave-energy potential in spectral sea-states, ultimately with a directional distribution. The potential for wind regrowth of waves between arrays was explored over an indicative ocean basin, and the effects of array spacing explored [3,4]. Using device responses, a refinement to the power matrix in terms of device survival was proposed in [1], which suggested a boolean survivability matrix be added to the usual calculation of mean annual energy production. This idea was explored further using data from the Israeli Mediterranean, resulting in recommendations for device sizing. While the Israeli Mediterranean is a high-variability but low mean power site, making it less than ideal for wave-energy extraction, the methodology used is easily applicable to any device design and location for which data is available.

  1. R. Stuhlmeier and D. Xu, WEC design based on refined mean annual energy production for the Israeli Mediterranean coast, Journal of Waterway, Port, Coastal, and Ocean Engineering, 144 (2018) 06018002.
  2. D. Xu, R. Stuhlmeier and M. Stiassnie, Assessing the size of a twin-cylinder wave energy converter designed for real sea-states, Ocean Engineering, 147 (2018), 243-255.
  3. D. Xu, R. Stuhlmeier and M. Stiassnie, Harnessing wave power in open seas II - Very large arrays of wave energy converters for 2D sea-states, J. Ocean Eng. Marine Energy, 3 (2017), 151-160.
  4. M. Stiassnie, U. Kadri and R. Stuhlmeier, Harnessing wave-power in open seas, J. Ocean Eng. Marine Energy 2 (2016), 47-57.

Acoustic-gravity waves

In the usual water wave problem, the small compressibility of water is neglected without great effect. In turn, studies of underwater acoustics, which focus on compressibility, usually neglect the influence of gravity and the presence of surface waves. Even in the linear problem, it is interesting to see the interplay between compressibility and the restoring force of gravity. By adapting Havelock's classical result for wavemakers in [1], we investigated how energy from a disturbance (like an ideal wavemaker) is transferred into the creation of surface and so-called acoustic-gravity wave modes. Formally these latter modes arise from the evanescent solutions to the linear water wave problem, some of which turn into propagating modes due to the effects of compressibility. With increasing depth of a disturbance, more and more of the energy of a disturbance goes to generating compressibility modes - a fact also observed by Nosov in connection with tsunami.

  1. R. Stuhlmeier and M. Stiassnie, Adapting Havelock's wave-maker theorem to acoustic-gravity waves, IMA J. Appl. Math , 81 (2016), 631-646.

Water waves in the Lagrangian framework, exact solutions

In 1804 Franz Josef Gerstner constructed a remarkable exact solution to the water wave problem in infinite depth, describing periodic traveling waves with trochoidal (or cycloidal) streamlines. This is all the more remarkable considering no nonlinear theory of water waves had been developed, and only the rudiments of the now-classical linear theory were being worked out. Gerstner's solution consisted of water particles moving in circular trajectories (some 30 years before George Green elucidated the particle paths of linear waves in deep water) with the free surface therefore forming a trochoid. The only caveat was that this solution was rotational, and written in Lagrangian (or particle following) coordinates.
Numerous extensions of Gerstner's wave to a variety of scenarios have been explored in recent years, and it remains fascinating to find exact solutions to the full, nonlinear equations in such a simple form. These solutions include edge waves on sloping beds [3,6], waves in stratified flow [4,5], geophysical waves (including Coriolis forces) [1], and others.

  1. M. Kluczek and R. Stuhlmeier, Mass transport for Pollard waves, Applicable Analysis , to appear.
  2. R. Stuhlmeier, On Gerstner's water wave and mass transport, J. Math. Fluid. Mech., 17 (2015), 761-767.
  3. R. Stuhlmeier, Internal Gerstner waves on a sloping bed, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 3183-3192.
  4. M. Stiassnie and R. Stuhlmeier, Progressive waves on a blunt interface, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 3171 - 3182.
  5. R. Stuhlmeier, Internal Gerstner waves: applications to dead water, Applicable Analysis, 93 (2014), 1451-1457.
  6. R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137.

Tsunami propagation and the KdV

The Korteweg de Vries equation (KdV), and particularly its sech2 soliton solutions, are commonly used to model tsunami. Since tsunami wavelengths are on the order of hundreds of kilometres, and the ocean depths are several kilometres at best, this shallow-water model seems appropriate. Two papers [1,2] more closely investigated the asymptotic regime associated with the KdV, to see whether it was strictly applicable to tsunami propagation. This so-called KdV balance - the balance between nonlinearity and dispersion needed for KdV to apply in space and time - is also affected by shear flow, which was investigated in [1] by means of a formulation developed by Freeman and Johnson.

  1. R. Stuhlmeier, Effects of shear flow on KdV balance - applications to tsunami, Commun. Pure Appl. Anal., 11 (2012), 1549 - 1561.
  2. R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Contin. Dyn. Syst. Ser. B 12, (2009), 623-632.

Edge waves

Edge waves are a curious wave pattern, which propagate in the longshore direction (i.e. along a beach, not towards it). Discovered as a mathematical curiosity by G. G. Stokes, they came into their own some 100 years later in discussions of nearshore wave dynamics. Difficult to detect experimentally, they have nevertheless been implicated in the formation of beach cusps, crescentic bars, and the spacing of rip currents (see P. D. Komar's engaging book Beach Processes and Sedimentation). Inspired by the commonalities between linear, deep water theory and the Gerstner wave, and building on a study of Gerstner-type edge waves [2], I also investigated the particle trajectories in linear and weakly nonlinear edge wave models [1]. Although the flow is three-dimensional, it is possible to show by a little manipulation that the particles in each plane parallel to the sea-bed stay within that plane. Subsequently the particle trajectories can be analysed using well-known techniques for 2D flows!

  1. R. Stuhlmeier, Particle paths in Stokes' edge wave, J. Nonlinear Math. Phys., 22 (2015), 507-515.
  2. R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137.