A complex scalar form of the incompressible Navier–Stokes equations

Abstract

ABSTRACT It is demonstrated that in Fourier domain the Navier–Stokes equations for an incompressible fluid can be reduced to a single complex scalar equation. An advantage of this equation is that any solution of this equation (approximate or exact) automatically represents real incompressible velocity field. An attempt was undertaken to check that in the absence of viscosity this equation represents a Hamiltonian system expressed in non-canonical variables. However, only one of the two necessary conditions was shown to hold; the question of fulfillment of the second necessary condition (the Jacobi identity) remains open. Using the new representation, it was demonstrated that in the inviscous case there are only two translationally-invariant, second-order integrals of motion which correspond to conservation of energy and helicity.