Shell Models on Recurrent Sequences: Fibonacci, Padovan and Other Series

TL;DR

The paper addresses how to formulate shell models on non-self-similar, recurrent wave-number sequences while preserving key invariants. It develops a GOY/Sabra-like framework on sequences such as Fibonacci, Lucas, Narayana, Padovan, Perrin, and a square-root Fibonacci variant, ensuring conservation of energy and helicity and comparing to conventional self-similar shells. The results show that these recurrent sequences reproduce standard spectral scaling, constant energy flux, and intermittency patterns, with inter-shell spacing primarily controlling intermittency; a compound Fibonacci–Lucas sequence and a helically decomposed version demonstrate robustness and the emergence of inverse cascades. The findings suggest that shell-model dynamics are robust to discretization choices and can inform diagnostic tools, subgrid models, and connections to log-lattice and nested polyhedra frameworks for turbulence modeling.

Abstract

A new class of shell models is proposed, where the shell variables are defined on a recurrent sequence of integer wave-numbers such as the Fibonacci or the Padovan series, or their variations including a sequence made of square roots of Fibonacci numbers rounded to the nearest integer. Considering the simplest model, which involves only local interactions, the interaction coefficients can be generalized in such a way that the inviscid invariants, such as energy and helicity, can be conserved even though there is no exact self-similarity. It is shown that these models basically have identical features with standard shell models, and produce the same power law spectra, similar spectral fluxes and analogous deviation from self-similar scaling of the structure functions implying comparable levels of turbulent intermittency. Such a formulation potentially opens up the possibility of using shell models, or their generalizations along with discretized regular grids, such as those found in direct numerical simulations, either as diagnostic tools, or subgrid models. It also allows to develop models where the wave-number shells can be interpreted as sparsely decimated sets of wave-numbers over an initially regular grid. In addition to conventional shell models with local interactions that result in forward cascade, a particular helical shell model with long range interactions is considered on a similarly recurrent sequence of wave numbers, corresponding to the Fibonacci series, and found to result in the usual inverse cascade.

Figures (8)

• Figure 1: Log-log plot of the spectral energy for each shell $\langle|u_{n}|^{2}\rangle$ as a function of the wave-number $k_{n},$ with the wave-numbers corresponding to the Fibonacci numbers, Lucas numbers and the respective self-similar spacing $k_{n}\propto\varphi^{n}.$ In the inset, a semi-log plot shows the spectral energy flux $\langle\Pi_{n}^{E}\rangle$, defined in Eq. (\ref{['eq:goy-flux-definition']}), as a function of the wavenumber $k_{n}$ (log. scale).
• Figure 2: Log-log plot of the spectral energy for each shell $\langle|u_{n}|^{2}\rangle$ as a function of the wave-number $k_{n},$ for a wave-number sequences corresponding to the Padovan numbers, Perrin numbers and the respective self-similar spacing $k_{n}\propto\rho^{n}.$ The inset semi-log plot displays the spectral energy flux as a function of the wave-number $k_{n}$ (log. scale).
• Figure 3: Log-log plot of the energy spectra for the square root of the Fibonacci sequence and the corresponding self-similar sequence. The inset semi-log plot displays the spectral energy flux as a function of the wave-number $k_{n}$ (log. scale).
• Figure 4: Log-log plot of the structure functions of order $p$ (from two to eight, top to bottom), defined as $S_{p}(k_{n})=\langle|u(k_{n})|^{p}\rangle$ as a function of the wave-number $k_{n}$, for both the self similar spaced model and the asymptotically self similar one in two different spacing considered, the golden ratio and the square root of the golden ratio. The inset plot indicates the structure function after eliminating the periodic three oscillations, showing the scaling of the quantity $\Sigma_{p}$.
• Figure 5: In the figure, for the Fibonacci sequence case, we illustrate the procedure for estimating the scaling exponents $\xi(p)$ following the ESS method. The scalings of the quantity $\Sigma_{p}$ are plotted as a function of $\Sigma_{3}.$ The structure function $p=1,2,\dots,10$ are displayed from top to bottom. The dots represent the set of points used for the fitting procedure, corresponding to approximately four decades in the inertial range, which aligns with the constant flux window.