2026/08/05 by C. M. Schober, A. Islas
Physics and Astronomy · #physics.flu-dyn #nlin.PS
20 pages, no figures
arxiv created 2026/08/05 · arxiv updated 2026/08/06
We extend the Carter-Govan multiple-scales analysis of weakly viscous, narrowband deep-water wave packets beyond Dysthe order within the potential-flow reduction of Dias, Dyachenko, and Zakharov (DDZ). We seek to determine whether this framework generates the complex multiplier (1 + iβ) used phenomenologically to modify the nonlocal Dysthe mean-flow interaction. Although order counting places a direct viscous carrier-mean interaction at sixth order, it does not exclude an indirect fifth-order contribution arising from viscosity dependent lower-order harmonics and nonlinear interactions. We therefore derive the first correction to the induced mean flow and the complete fifth-order first-harmonic solvability condition. The resulting nonlocal terms are derivative--dependent and contain no explicit viscosity, excluding the proposed indirect mechanism within the DDZ framework. At sixth order, a restricted calculation of the nonlinear viscous block isolates a direct carrier-mean contribution with the same operator structure as the imaginary component of the prescribed mean-flow correction. Independently, a finite-adjustment-time model yields an exact, frequency dependent mean-flow response. Its low-frequency expansion produces the multiplier 1 + iβeff(Ω), with βeff(Ω) =Ωτ. When Ωτ= \mathcal O(ε), the resulting phase-lag correction enters at fifth order, one order beyond the leading Dysthe mean-flow interaction.