Spectral gap of a shearing-periodic box ($⟨|\nabla f{|}^2⟩\ge (2\pi /L{)}^2⟨|f{|}^2⟩$)

For zero-mean fields in a shearing box, Fourier modes have wavevectors $(2\pi n/L_x+2At2\pi m/L_y,2\pi m/L_y,2\pi p/L_z)$. A nonzero azimuthal or vertical index supplies at least $2\pi /L$, where $L$ is the largest box side; otherwise the nonzero radial index does. Parseval identity gives the displayed Poincaré inequality uniformly in time.