Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows

Peter Maxwell,Simen Å Ellingsen

doi:10.1007/s42286-020-00030-0

Abstract

The path-following scheme in Loisel and Maxwell (SIAM J Matrix Anal Appl 39(4):1726–1749, 2018) is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current. This is equivalent to solving the Rayleigh stability equation with linearized free-surface boundary condition for each sought point on the curve. Taking advantage of the analyticity of the dispersion relation, a path-following or continuation approach is adopted. The problem is discretized using a collocation scheme, parametrized along either a radial or angular path in the wave vector plane, and differentiated to yield a system of ODEs. After an initial eigenproblem solve using QZ decomposition, numerical integration proceeds along the curve using linear solves as the Runge–Kutta F(cdot ) function; thus, many QZ decompositions on a size 2N companion matrix are exchanged for one QZ decomposition and a small number of linear solves on a size N matrix. A piecewise interpolant provides dense output. The integration represents a nominal setup cost whereafter very many points can be computed at negligible cost whilst preserving high accuracy. Furthermore, a two-dimensional interpolant suitable for scattered data query points in the wave vector plane is described. Finally, a comparison is made with existing numerical methods for this problem, revealing that the path-following scheme is the most competitive algorithm for this problem whenever calculating more than circa 1,000 data points or relative normwise accuracy better than 10^{-4} is sought.

Highlights

We develop a path-following method to numerically calculate the dispersion relation curve for linear surface waves atop a horizontal current of arbitrary depth dependence, an adaptation of the scheme developed by Loisel and Maxwell [29]
When a vertically varying shear current is present beneath a water surface, the dispersion of water waves riding atop it is affected in a complicated way
The high-precision initial calculation for the PFmp algorithm avoids this roundoff error and it can be seen that the path-following method a b itself retains this improved accuracy even in double precision

Summary

Introduction

We develop a path-following method to numerically calculate the dispersion relation curve for linear surface waves atop a horizontal current of arbitrary depth dependence, an adaptation of the scheme developed by Loisel and Maxwell [29]. Waves in this regime are dispersive with their behaviour being entirely characterized by the dispersion relation, i.e. the dependence of phase velocity c(k) on the wave vector k whose modulus k is the wavenumber. When a vertically varying shear current is present beneath a water surface, the dispersion of water waves riding atop it is affected in a complicated way. The well-known expression for the phase velocity of a line√ar surface wave in inviscid, initially quiescent water of depth h, c0(k) = c0(k) = (g/k) tanh(kh), where g is gravitational acceleration, does not hold when considering general horizontal depth-dependent currents of the form U(z) where z is the depth. In the cases of quiescent water, uniform current, and strictly linearly varying current is a closed-form expression for c(k) known for all permissible values of k (and vice versa).

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