Shell-model experiment: Threshold across parameter combinations
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 finite-window threshold \(A^*\) strictly positive for every parameter combination and initial-condition type tested? A threshold of zero would mean that every trajectory crosses the excursion threshold within the window. The run covers 12 parameter combinations and 3 initial-condition families.
Method
- Sweep 12 parameter combinations: \(\nu \in \{0.001, 0.005, 0.01, 0.02\}\), \(\lambda_2 \in \{1.0, 5.0, 10.0\}\).
- For each combination, find \(A^*\) via bisection to 3 decimal places.
- Additionally run 3 initial-condition types at a representative parameter set: sinusoidal, constant, and concentrated (energy in a single shell).
- Record whether \(A^* > 0\) in every case.
Results
Parameter Combinations
| \(\nu\) | \(\lambda_2 = 1.0\) | \(\lambda_2 = 5.0\) | \(\lambda_2 = 10.0\) |
|---|---|---|---|
| 0.001 | 0.312 | 0.289 | 0.247 |
| 0.005 | 0.421 | 0.378 | 0.319 |
| 0.01 | 0.498 | 0.441 | 0.372 |
| 0.02 | 0.606 | 0.523 | 0.431 |
Initial Condition Types
| IC Type | \(A^*\) | Positive? |
|---|---|---|
| Sinusoidal (equal energy per shell) | 0.347 | Yes |
| Constant (flat spectrum) | 0.352 | Yes |
| Concentrated (single shell dominant) | 0.331 | Yes |
Summary
| Metric | Result |
|---|---|
| Parameter combinations | 12/12 have \(A^* > 0\) |
| Initial condition types | 3/3 have \(A^* > 0\) |
| Minimum \(A^*\) | 0.247 (\(\nu = 0.001\), \(\lambda_2 = 10\)) |
| Maximum \(A^*\) | 0.606 (\(\nu = 0.02\), \(\lambda_2 = 1\)) |
Summary
\(A^*\) positive in all 15 tested configurations. The finite-window threshold ranges from 0.247 to 0.606 across the parameter grid and from 0.331 to 0.352 across initial-condition types.
Related
- Feedback pattern across parameters — the \(L_1 \to L_4 \to L_2\) pattern on a 3×3 grid
- Trajectory classification checks — four checks, 5 trials each
- Doubling-time criterion — classification of six amplitudes
- Shell-count summary — threshold values from 6 to 24 shells