“…The homogenization theory of the Euler equations with oscillating data was first studied by McLaughlin-Papanicolaou-Pironneau (MPP for short) in 1985 [6]. To construct a·multiscale expansion for the Euler equations, they made an important assumption that the oscillation is convected by the mean flow: E E E where W(t,X,T,y), UI(t,x,T,y), q(t,X,T,y) and pI(t,x,T,y) are assumed to be periodic in y and T with zero mean, and the phase () is convected by the mean velocity field u:…”