Unimodal solutions of the generalized Constantin–Lax–Majda equation with viscosity

Abstract

Steady-states of the generalized Constantin–Lax–Majda equation with the viscosity and an external force are computed numerically by the spectral method. This equation is regarded as a model for two-dimensional turbulent motion of incompressible viscous fluid. We demonstrate numerically that the equation admits unimodal solutions—solutions with one and only one peak and bottom, if the Reynolds number is sufficiently large. We also report some interesting properties of the spectra of unimodal solutions.