Shell-model experiment: Feedback pattern across parameters
Question
The system studied here is a cubic shell ODE with independent velocity-shell variables \(u_k\) and vorticity-shell variables \(\omega_k\), with stretching couplings \(\lambda\), \(\lambda_2\) and a feedback coupling \(\sigma\). It is not a Galerkin truncation of Navier–Stokes. The threshold \(A^*\) reported below is a finite-window excursion threshold: the initial amplitude at which a trajectory first crosses the excursion threshold within the integration window. Its value, the times to threshold crossing and the classification scores depend on the integration window, the excursion threshold and the solver.
Is the \(L_1 \to L_4 \to L_2 \to L_1\) pattern (shell enstrophy rising while the S-score falls) specific to one parameter choice, or does it appear at every tested combination of viscosity \(\nu\) and coupling \(\lambda_2\) in this ODE?
Method
- Select a 3×3 grid of parameter combinations: \(\nu \in \{0.001, 0.005, 0.02\}\), \(\lambda_2 \in \{1.0, 5.0, 20.0\}\).
- For each combination, find \(A^*\) via bisection.
- Run the ODE at 80% of \(A^*\) — below threshold within the window but close enough to it to be informative.
- Check whether \(L_1\) increases AND \(L_2\) decreases at every checkpoint (the signature used for an active pattern).
Results
Parameter Sweep
| \(\nu\) | \(\lambda_2\) | \(A^*\) | Pattern at 80% \(A^*\) | \(L_1\) trend | \(L_2\) trend |
|---|---|---|---|---|---|
| 0.001 | 1.0 | 0.312 | Active | Increasing | Decreasing |
| 0.001 | 5.0 | 0.289 | Active | Increasing | Decreasing |
| 0.001 | 20.0 | 0.261 | Active | Increasing | Decreasing |
| 0.005 | 1.0 | 0.421 | Active | Increasing | Decreasing |
| 0.005 | 5.0 | 0.378 | Active | Increasing | Decreasing |
| 0.005 | 20.0 | 0.334 | Active | Increasing | Decreasing |
| 0.02 | 1.0 | 0.606 | Active | Increasing | Decreasing |
| 0.02 | 5.0 | 0.523 | Active | Increasing | Decreasing |
| 0.02 | 20.0 | 0.447 | Active | Increasing | Decreasing |
Summary
| Metric | Result |
|---|---|
| Combinations tested | 9 |
| Pattern active (\(L_1 \uparrow\) AND \(L_2 \downarrow\)) | 9/9 |
| Depends on the parameter choice? | Not within this grid |
Summary
9/9. The \(L_1 \to L_4 \to L_2 \to L_1\) pattern is active at all nine parameter combinations in this shell ODE.
Related
- Threshold across parameter combinations — \(A^*\) on a 12-point grid and three initial-condition types
- Trajectory classification checks — four checks, 5 trials each
- Doubling-time criterion — classification of six amplitudes
- 8-shell run — the same diagnostics on the 8-shell ODE