Mean and variance of the Eulerian and Lagrangian horizontal velocities induced by nonlinear multi-directional irregular water waves

Abstract

In this paper, we study the mean and variance of the Eulerian and Lagrangian fluid velocities as a function of depth below the surface of directionally spread irregular wave fields given by JONSWAP spectra in deep water. We focus on the behaviour of these quantities in the bulk of the water, and using second-order potential flow theory we derive new simple asymptotic approximations for their decay in the limit of large depth below the surface. Specifically, we show that when the depth is greater than about 1.5 peak wavelengths, the variance of the Eulerian velocity decays in proportion to exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) , and the mean Lagrangian velocity decays in proportion to 1 ( − k p z ) 1 / 6 exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) . Here, k p is the peak wave number and z is the vertical coordinate measured positively upwards from the still water level. We test the accuracy of the second-order formulation against new fully nonlinear simulations of both short crested and long crested irregular wave fields and find a good match, even when the simulations are known to be affected substantially by third-order effects. To our knowledge, this marks the first fully nonlinear investigation of the Eulerian and Lagrangian velocities below the surface in irregular wave fields.