Surface waves on currents with arbitrary vertical shear

Abstract

We study dispersion properties of linear surface gravity waves propagating in an arbitrary direction atop a current profile of depth-varying magnitude using a piecewise linear approximation, and develop a robust numerical framework for practical calculation. The method has been much used in the past for the case of waves propagating along the same axis as the background current, and we herein extend and apply it to problems with an arbitrary angle between the wave propagation and current directions. Being valid for all wavelengths without loss of accuracy, the scheme is particularly well suited to solve problems involving a broad range of wave vectors, such as ship waves and Cauchy-Poisson initial value problems for example. We examine the group and phase velocities over different wavelength regimes and current profiles, highlighting characteristics due to the depth-variable vorticity. We show an example application to ship waves on an arbitrary current profile, and demonstrate qualitative differences in the wake patterns between concave down and concave up profiles when compared to a constant shear profile with equal depth-averaged vorticity. We also discuss the nature of additional solutions to the dispersion relation when using the piecewise-linear model. These are vorticity waves, drifting vortical structures which are artifacts of the piecewise model. They are absent for a smooth profile and are spurious in the present context.

A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

“…It is very easy to follow all physical quantities explicitly throughout calculations. The rate of convergence of the PLA is typically similar to that of the iterative approach of the DIM employed here, ∼ N −2 (see Smeltzer & Ellingsen, ; Zhang, and Figure ). The DIM reaches a higher convergence rate if a numerical approach of higher accuracy (than the second‐order central difference approximation for ) is used.…”

“…The PLA initially produces N + 1 solutions to the dispersion relation, N − 1 of which are artifacts of the abrupt change of vorticity at layer interfaces. There are reliable ways to deal with this problem (see; Smeltzer & Ellingsen, ), but it remains a hurdle. Second, and for the same reason, computation is less costly, since detecting the correct eigenvalues of can be the most costly part of the PLA calculation.…”

Transient wave resistance upon a real shear current

“…The wave pattern is then calculated as the integral of emitted waves at all previous times. It is necessary to devise a scheme to obtain the dispersion relation numerically; in this paper we used the piecewise-linear approximation [20], but several other options are available.…”

“…A useful numerical scheme to this end is the piecewiselinear approximation (PLA), which was analysed in Refs. [20,22], and which we will use herein to obtain numerical results. As described herein the PLA is restricted to unidirectional U(z); extension to shear currents changing direction is relatively straightforward.…”

“…The next essential step is to obtain solutions for ω ± , exact or approximate, and to determine A 1 and B 1 via the PLA procedure [20]. The PLA is particularly suitable for problems which are solved in the Fourier plane since it provides a rapid and accurate solution to the dispersion relation ω(k) equally well for all wavelengths, converging to the exact value as N increases [20,22]. For our numerical demonstrations we find that 4-5 layers are typically enough at the 1% accuracy level.…”