 Oscillating shear dispersion in an insulating channel

For $u=U\cos (\omega t)\sin (\pi y/L)$ on $|y|<L/2$, positive diffusivity $\kappa$, nonzero frequency and insulating walls, a transversely uniform zeroth moment remains constant. Its first moment is proportional to $g(t)\sin (\pi y/L)$, where $g^{\prime }+ag=U\cos (\omega t)$. The periodic solution is $g=U[a\cos (\omega t)+\omega \sin (\omega t)]/(a^2+{\omega }^2)$. The longitudinal moment hierarchy for shear transport then gives $m_2=2Kt{m}_0+O(1)$ after both transverse and periodic transients. The spatial average of ${\sin }^2(\pi y/L)$ is $1/2$ and the time average of ${\cos }^2(\omega t)$ is $1/2$, producing the factor $1/4$. At exactly zero frequency the steady enhancement is instead $U^2/(2a)$; averaging indefinitely over slow cycles does not commute with setting the frequency to zero.