Shell-model experiment: Doubling-time criterion
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.
The shell-enstrophy doubling time \(\tau_d\) is the criterion used to classify superexponential growth within the window: a shrinking \(\tau_d\) means the enstrophy is growing faster than any exponential over the window. Does the trend of \(\tau_d\) classify each of six amplitudes correctly against what the trajectory does within the window?
Method
- Select 6 amplitudes: 3 below \(A^*\) and 3 above \(A^*\).
- For each amplitude, compute the enstrophy doubling time \(\tau_d\) along the trajectory.
- Classify: if \(\tau_d\) is growing or stable, predict that the trajectory stays below threshold within the window. If \(\tau_d\) is shrinking, predict threshold crossing.
- Compare the prediction against the trajectory outcome within the window.
Results
Classification Table
| Amplitude | Relative to \(A^*\) | \(\tau_d\) trend | Prediction | Outcome in window | Correct? |
|---|---|---|---|---|---|
| 0.174 | 0.50 \(A^*\) | Growing | Below threshold | Below threshold | Yes |
| 0.243 | 0.70 \(A^*\) | Stable | Below threshold | Below threshold | Yes |
| 0.313 | 0.90 \(A^*\) | Stable | Below threshold | Below threshold | Yes |
| 0.382 | 1.10 \(A^*\) | Shrinking | Crosses threshold | Crosses threshold | Yes |
| 0.521 | 1.50 \(A^*\) | Shrinking | Crosses threshold | Crosses threshold | Yes |
| 0.694 | 2.00 \(A^*\) | Shrinking | Crosses threshold | Crosses threshold | Yes |
Doubling Time and Growth Type
| Doubling time behaviour | Growth type over the window | Classification |
|---|---|---|
| Growing \(\tau_d\) | Sub-exponential | Below threshold |
| Stable \(\tau_d\) | Exponential | Below threshold |
| Shrinking \(\tau_d\) | Super-exponential | Crosses threshold |
Summary
| Metric | Result |
|---|---|
| Amplitudes tested | 6 (3 below threshold, 3 crossing) |
| Correct classifications | 6/6 |
| False positives | 0 |
| False negatives | 0 |
Summary
6/6 correct classification. The doubling-time trend separates the three below-threshold trajectories from the three that cross the excursion threshold within the window.
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
- Trajectory classification checks — four checks, 5 trials each
- S-score lead time — S as an early indicator of threshold crossing