Longitudinal moment hierarchy for shear transport (${\partial }_t{m}_j=\kappa {\partial }_y^2{m}_j+ju{m}_{j-1}+\kappa j(j-1)m_{j-2}$)

For a passive scalar in a planar shear flow $u(y,t)$, let $m_j(y,t)={\int }_{\mathbb{R}}{x}^j\chi (x,y,t)dx$. Integrating the advection-diffusion equation by parts in $x$ gives the displayed hierarchy for $j\ge 0$, with negative-index terms omitted. Sufficient decay and finite moments justify the integrations. Insulating walls give Neumann boundary conditions for every moment. Solving the first two equations then determines the long-time growth of the second moment without solving the full scalar field.

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.