Reply to “Comments on ‘A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism’”

Abstract

AbstractFujiwara et al. explicitly simulated Langmuir circulations using a wave-resolving simulation (WRS) technique and found that the residual wave effect on vorticity was well represented by the vortex force of the Craik–Leibovich (CL) equation, at least in the simulated situation. In response to the simulation results, Mellor has proposed a view that ubiquitous applicability of the CL formulation is still questionable and that the three-dimensional radiation stress (3DRS) formulation that he has derived encompasses both of the vortex force effect and an effect that is lower order in terms of wave steepness. Here, these opinions are discussed in terms of the approximations used in the wave-averaged formulations. The asymptotic expansion of the Eulerian-averaged momentum equation allows the separate discussion of two different wave effects: pressure correction and torque. It is argued that the approximation adopted in Mellor’s 3DRS formulation is presumably not accurate enough to properly parameterize the wave torque effect, and possible approaches to examine its performance are proposed. We agree with the view that the applicability of the CL formulation needs further investigation. WRS will be a helpful tool for this purpose.