The Error in Predicted Phase Velocity of Surface Waves atop a Shear Current with Uncertainty

The effect of a depth-dependent shear current U ( z ) on surface wave dispersion is conventionally calculated by assuming U ( z ) to be an exactly known function, from which the resulting phase velocity c ( k ) is determined. This, however, is not the situation in reality. Field measurements of the current profile are performed at a finite number of discrete depths and with nonzero experimental uncertainty. Here we analyse how imperfect knowledge of U ( z ) affects estimates of c ( k ). We performed a numerical experiment simulating a large number of “measurements” of three different shear currents: an exponential profile, a 1 / 7-law profile, and a profile measured in the Columbia River delta. A number of measurement points were specified, the topmost of which at z=-h_s (permitting simulation of measurement points which do not fully extend to the surface at z = 0 ), and measurements taken from a normal distribution with standard deviation U . Four different methods of reconstructing a continuous U ( z ) from the measurements are compared with respect to mean value and variance of c ( k ). We find that an ordinary least-squares polynomial fit seems robust against mispredicting mean values at the expense of relatively high variance. Its performance is similar for all profiles, whereas a fit to an exponential form is excellent in one case and poor in another. A clear conclusion is the need for a measurement of the surface velocity U (0) when there is significant shear near the surface. For the exponential and Columbia profiles alike, errors due to extrapolation of U from z=-h_s to 0 dominate the resulting error of c , especially for shorter wavelengths. In contrast, the error in c ( k ) decreases slowly with a higher density of measurement points, indicating that better, not more, velocity measurements should be invested in. A pseudospectral analysis of the linear operator corresponding to the three velocity profiles was performed. In all cases, the pseudospectrum shows strong asymmetry around the eigenvalue for c , indicating that a perturbation in the underlying current is more likely to push c to higher, not lower, values. This is in tentative agreement with our observation that for sufficiently large U , c is found to have predominantly positive skewness, although the direct relationship between the two is not altogether obvious.