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Shell-model experiment: Long-window criteria
Experiment: exp_ns_close_gap.sx | Domain: Shell-model ODE dynamics
Overview
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.
Instantaneous criteria trigger on transient intervals in which the stretching term dominates diffusion, intervals from which the ODE trajectory then recovers. This run introduces five criteria based on long-window analysis (5,000–10,000 steps) that use the net effect over full cycles rather than instantaneous snapshots, and records how the threshold \(A^*\) changes as each criterion is added. With all five criteria the threshold moves from 0.298 to 1.020.
Criteria
| Criterion | Window | What It Measures |
|---|
| F: Net production integral | 5,000 steps | \(\int(\text{stretching} - \text{diffusion})\,dt > 0\) sustained |
| G: Enstrophy growth ratio | 5,000 steps | \(\Omega(t+N) / \Omega(t) > 1\) sustained |
| H: Recovery test | 5,000 steps | Does the production ratio return below 1 after exceedance? |
| I: Moving average trend | 10,000 steps | Is the trend in \(d\Omega/dt\) positive despite oscillations? |
| J: Ratchet detector | Consecutive windows | Does peak \(\Omega\) increase in successive windows? |
Key Results
Threshold by Criterion Set
| Criterion Set | \(A^*\) | Change |
|---|
| Instantaneous criteria only | 0.298 | — |
| + Net production (F) | 0.54 | +81% |
| + Growth ratio (G) | 0.72 | +141% |
| + Recovery test (H) | 0.88 | +195% |
| + Trend + Ratchet (I, J) | 1.020 | +242% |
Saturation and Doubling Metrics
| Metric | Value | Meaning |
|---|
| Saturation predictor | 54.6% | Fraction of instantaneous-criterion alarms that the long-window criteria do not raise |
| Doubling time | 86.1% | Increase in time to threshold crossing under the long-window criteria |
| Hierarchy depth | 15 levels | Total diagnostic levels in the combined set |
Observations. In this ODE, transient exceedance of the production ratio is not followed by threshold crossing in most cases. The instantaneous criteria give a threshold 3.4× lower than the long-window criteria. Requiring sustained net production over 5,000–10,000 steps moves \(A^*\) past 1.0 for the 6-shell system. The 54.6% saturation figure counts instantaneous alarms not raised by the long-window criteria; the 86.1% doubling-time figure is the increase in time to threshold crossing. All figures depend on the window lengths chosen.
Analysis
- From 0.298 to 1.020. A 3.4× increase in \(A^*\) under long-window analysis. The ODE recovers from most transient exceedances within the window.
- 15-level diagnostic set. The combined set has 15 levels: the original 10 plus the 5 long-window criteria.
- Window dependence. The threshold value is a function of the criterion set and window length; the table records that dependence.
Reproducibility
../simplex/build/sxc exp_ns_close_gap.sx -o build/exp_ns_close_gap.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_close_gap.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_close_gap \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_close_gap
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