2026/05/27 by Jeffrey R. Carpenter
Earth and Planetary Sciences · #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Coastal and Marine Dynamics
paper · doi:10.1175/jpo-d-25-0257.1
Abstract In the hydraulics of sheared, inviscid channel flows, where the horizontal velocity profile varies with height, the speed of long waves is governed by an integral equation with an apparent singularity that is present for wave speeds within the range of the profile. It is shown that simple linear profiles (or those composed of linear segments) do not have a singularity, but more complicated cases with curvature to the profile are singular. In singular cases, the singularity can be interpreted by analyzing the initial value problem, where it results in a so-called continuous spectrum solution with different space/time behavior than the classical wave mode solutions. This is relevant in the hydraulics of sheared flows since it affects the signal response carried by the upstream wave mode, with a rapidly decreasing flux found as the Froude number increases. Significance Statement Understanding the propagation characteristics of long waves on the ocean surface is crucial for predicting the flows in channels and constrictions. However, when the flow exhibits strong shear (i.e., large changes in current speed with depth), the equations predicting the speed of these long surface waves are no longer properly defined when the wave speed lies in the range of the current speeds. This note demonstrates that in certain simple conditions, the equations can be “corrected” to remove the mathematical problem, but in general, and more realistic, oceanic flows, this is not the case. However, the mathematical problem can be overcome and has a physical explanation revealing that the upstream propagation of wave signals in sheared channel flows is highly reduced.