= Longitudinal moment hierarchy for shear transport
{title2=$\partial_t m_j=\kappa\partial_y^2m_j+ju m_{j-1}+\kappa j(j-1)m_{j-2}$}
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 $j\geq0$, 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.
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