= Turbulent backscatter
{title2=$\Pi=-\tau_{ij}\bar S_{ij}<0$}
<Turbulent backscatter> is transfer of <kinetic energy> from unresolved or smaller spatial scales into larger resolved scales. In a <large eddy simulation>, its sign is defined relative to the chosen <spatial averaging filter> and <subgrid-scale stress>. It can occur locally even when the mean <energy cascade> transfers energy toward small scales.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[Introduction, small-to-large-scale energy flow and inability of a fixed dissipative closure to represent it]{ref}
With $\tau_{ij}=\overline{u_i u_j}-\bar u_i\bar u_j$, the stress force in the filtered momentum equation is $-\partial_j\tau_{ij}$. Multiplying by $\bar u_i$ gives
$$
-\bar u_i\partial_j\tau_{ij}
=-\partial_j(\bar u_i\tau_{ij})+\tau_{ij}\partial_j\bar u_i.
$$
The first term transports <kinetic energy> through space. Because $\tau$ is symmetric, the second is $\tau_{ij}\bar S_{ij}$, the local exchange with unresolved motion. Thus the forward-transfer convention
$$
\Pi=-\tau_{ij}\bar S_{ij}
$$
makes $\Pi<0$ a gain of resolved <kinetic energy>.
For a trace-free resolved strain $\bar S=\operatorname{diag}(\gamma,-\gamma,0)$ and stress $\tau=\operatorname{diag}(a,b,0)$, with nonnegative $a,b$,
$$
\Pi=-\gamma(a-b).
$$
Taking $\gamma>0$ and $a>b$ produces backscatter even though the stress matrix is <positive semidefinite>. Positivity of a weighted <covariance> does not fix the sign of its contraction with an indefinite strain tensor.
The <Smagorinsky model> with nonnegative <eddy viscosity> instead gives $\Pi=2\nu_t\bar S_{ij}\bar S_{ij}\geq0$. A <dynamic subgrid-scale model> can allow the reverse sign. Backscatter describes exchange between scales, not negative molecular <viscous dissipation> or creation of total <energy>.
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