Note on ‘Multi-frequency Craik–Criminale solutions of the Navier–Stokes equations’ by B. R. Fabijonas and D. D. Holm

Abstract

In a recent paper (J. Fluid Mech. vol. 506, 2004, p. 207), B. R. Fabijonas and D. D. Holm claim that they have found a general method to construct new solutions of the Navier–Stokes equations from any known base flow solution. In this note, we argue that Fabijonas & Holm's solutions are very special in character. Although they can be defined in all space, they satisfy the Navier–Stokes equations on a single fixed trajectory of the chosen base flow. We show that this limits the usefulness of the solution and the applicability of the method. In particular, it is demonstrated that, in general, the iterative ‘WKB-bootstrapping’ algorithm designed by the authors cannot be applied after the first iteration. We also show that a second iteration is possible only if the base flow satisfies strong constraints. The consequence of these constraints is that no extension of the Craik–Criminale solutions to multiple frequencies is found to be possible. By applying Fabijonas & Holm's construction to a simple model equation, we demonstrate that their solution can also predict (unphysical) behaviours which cannot be reproduced by any global solution.