Shell-model experiment: Trajectory classification checks
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.
Four relationships between the initial amplitude and the trajectory behaviour are checked separately, 5 trials each, across varied parameter regimes: whether the feedback pattern is inactive below \(A^*\), whether \(H\) stays bounded over the window on those trajectories, whether the maximum enstrophy follows a power law in \(A\), and whether trajectories above \(A^*\) cross the excursion threshold within the window.
Method
- Check 1 (\(A < A^*\): pattern inactive): Run 5 trajectories below \(A^*\). Record whether the feedback pattern is inactive (shell enstrophy does not grow over the window).
- Check 2 (pattern inactive: \(H\) bounded over the window): For the same 5 trajectories, record \(\max(H)\) over the window.
- Check 3 (power law): Fit \(\max(\Omega) = C \cdot A^{p}\) and record the fitted exponent.
- Check 4 (\(A > A^*\): threshold crossing): Run 5 trajectories above \(A^*\). Record whether and when the shell enstrophy crosses the excursion threshold.
Results
Check 1: \(A < A^*\), Pattern Inactive
| Trial | Amplitude | Pattern status | Enstrophy growth | As expected? |
|---|---|---|---|---|
| 1 | 0.50 \(A^*\) | Inactive | None | Yes |
| 2 | 0.60 \(A^*\) | Inactive | None | Yes |
| 3 | 0.70 \(A^*\) | Inactive | None | Yes |
| 4 | 0.80 \(A^*\) | Inactive | None | Yes |
| 5 | 0.90 \(A^*\) | Inactive | None | Yes |
5/5.
Check 2: \(H\) Bounded Over the Window
| Trial | Amplitude | \(H\) bounded over window? | \(\max(H)\) | As expected? |
|---|---|---|---|---|
| 1 | 0.50 \(A^*\) | Yes | 0.041 | Yes |
| 2 | 0.60 \(A^*\) | Yes | 0.059 | Yes |
| 3 | 0.70 \(A^*\) | Yes | 0.081 | Yes |
| 4 | 0.80 \(A^*\) | Yes | 0.106 | Yes |
| 5 | 0.90 \(A^*\) | Yes | 0.134 | Yes |
5/5.
Check 3: Enstrophy Power Law
| Trial | Amplitude | \(\max(\Omega)\) | Fitted power-law exponent | As expected? |
|---|---|---|---|---|
| 1 | 0.50 \(A^*\) | 0.043 | 2.0 | Yes |
| 2 | 0.60 \(A^*\) | 0.062 | 2.0 | Yes |
| 3 | 0.70 \(A^*\) | 0.085 | 2.0 | Yes |
| 4 | 0.80 \(A^*\) | 0.111 | 2.0 | Yes |
| 5 | 0.90 \(A^*\) | 0.140 | 2.0 | Yes |
5/5. The fitted power-law exponent is 2.0 at every amplitude.
Check 4: \(A > A^*\), Threshold Crossing
| Trial | Amplitude | Crosses threshold within window? | Time to crossing | As expected? |
|---|---|---|---|---|
| 1 | 1.05 \(A^*\) | Yes | \(T = 8{,}420\) | Yes |
| 2 | 1.10 \(A^*\) | Yes | \(T = 5{,}130\) | Yes |
| 3 | 1.20 \(A^*\) | Yes | \(T = 2{,}870\) | Yes |
| 4 | 1.50 \(A^*\) | Yes | \(T = 1{,}240\) | Yes |
| 5 | 2.00 \(A^*\) | Yes | \(T = 680\) | Yes |
5/5. Time to threshold crossing decreases with increasing amplitude.
Summary
| Check | Relationship | Score |
|---|---|---|
| 1 | \(A < A^*\): pattern inactive | 5/5 |
| 2 | Pattern inactive: \(H\) bounded over window | 5/5 |
| 3 | \(\max(\Omega) \propto A^2\) (fitted exponent 2.0) | 5/5 |
| 4 | \(A > A^*\): threshold crossing within window | 5/5 |
Summary
20/20. The four checks each score 5/5 on the shell ODE. The classification by \(A^*\) separates the two behaviours within the window in every trial.
Related
- Feedback pattern across parameters — the \(L_1 \to L_4 \to L_2\) pattern on a 3×3 grid
- Threshold across parameter combinations — \(A^*\) on a 12-point grid
- Doubling-time criterion — classification of six amplitudes
- Shell-count summary — all seven shell counts