= Shell model of turbulence
= Shell models of turbulence
{synonym}
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.<Improved shell model of turbulence by Victor S. L'vov and others>[Author preprint, section I and equations 1–5: shell variables and conservation constraints]{ref}
The basic energy bookkeeping is transparent. Write
$$
\dot u_n=N_n(u)-\nu k_n^2u_n+f_n,\qquad E=\frac12\sum_n|u_n|^2.
$$
If the nonlinear coupling and end conditions satisfy $\sum_n\operatorname{Re}(u_n^*N_n)=0$, differentiating gives
$$
\frac{dE}{dt}=\sum_n\operatorname{Re}(u_n^*f_n)-\nu\sum_n k_n^2|u_n|^2.
$$
The nonlinearity transfers energy between shells without creating its total. Viscous damping becomes faster at larger $k_n$: an isolated unforced linear mode decays as $e^{-\nu k_n^2t}$.
For an illustrative constant-flux estimate, a transfer time $\tau_n\sim(k_n|u_n|)^{-1}$ gives $\varepsilon\sim|u_n|^2/\tau_n\sim k_n|u_n|^3$, hence $|u_n|\sim(\varepsilon/k_n)^{1/3}$. Deviations of higher moments from a single scaling rule can then diagnose <intermittency (turbulence)>. 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.<Improved shell model of turbulence by Victor S. L'vov and others>[Abstract and sections I–II, model motivation and scale correlations]{ref}
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