Turbulent backscatter ($\Pi =-{\tau }_{ij}{\bar{S}}_{ij}<0$)

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.^[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 ${\bar{u}}_i$ givesThe 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 conventionmakes $\Pi <0$ a gain of resolved kinetic energy.

For a trace-free resolved strain $\bar{S}=\mathrm{diag}(\gamma ,-\gamma ,0)$ and stress $\tau =\mathrm{diag}(a,b,0)$, with nonnegative $a,b$,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 $\Pi =2{\nu }_t{\bar{S}}_{ij}{\bar{S}}_{ij}\ge 0$. 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.