Constrained Minimization for Extracting Unstable Periodic Orbits from Shell-model Turbulence

Developed turbulence represents a canonical example of chaos, whose dynamics can be characterized in terms of invariant solutions embedded within the turbulent attractor. We propose an algorithm for examining such invariant solutions based on a constrained minimization problem. Small-scale motions of invariant solutions within the attractor are expected to satisfy the local energy-transfer balance arising from the scale-by-scale cascade. To incorporate this property, we impose equality constraints constructed by applying a high-pass filter to the governing equation. In contrast, to suppress large-scale fluctuations induced by external forcing, we introduce an L2-regularization term obtained through a low-pass filter. Applying the proposed framework to the shell model — which reproduces the universal scaling features of developed Navier–Stokes turbulence — we extract multiple steady and periodic solutions embedded in shell-model turbulence. These solutions exhibit energy spectra in close agreement with that of the turbulent attractor. The constrained minimization approach, formulated in accordance with turbulence theory, offers a new strategy for the analysis of invariant solutions embedded in developed turbulence.