Shell model of turbulence

A shell model of turbulence replaces a spatially resolved fluid by a sequence of amplitudes associated with progressively smaller length scales. A typical choice assigns a complex velocity amplitude $u_n$ to the wave-number shell $k_n=k_0{\lambda }^n$, with $\lambda >1$. Neighboring shells exchange energy through nonlinear couplings, while forcing supplies energy and viscosity damps high wave numbers. The model retains a scale cascade at far lower cost than direct numerical simulation, while discarding most spatial geometry.^[ref]

The basic energy bookkeeping is transparent. WriteIf the nonlinear coupling and end conditions satisfy ${\sum }_n\mathrm{Re}(u_n^{\ast }{N}_n)=0$, differentiating givesThe nonlinearity transfers energy between shells without creating its total. Viscous damping becomes faster at larger $k_n$: an isolated unforced linear mode decays as $e^{-\nu k_n^{2}t}$.

For an illustrative constant-flux estimate, a transfer time ${\tau }_n\sim (k_n|u_n|{)}^{-1}$ gives $\epsilon \sim |u_n{|}^2/{\tau }_n\sim k_n|u_n{|}^3$, hence $|u_n|\sim (\epsilon /k_n{)}^{1/3}$. Deviations of higher moments from a single scaling rule can then diagnose intermittency. The estimate is dimensional reasoning, not an exact solution. The published improved coupling was designed to reduce spurious shell correlations while preserving useful conservation and scaling properties.^[ref]