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Shell-model experiment: Side-by-side amplitude comparison

Experiment: exp_ns_microscope.sx  |  Domain: Shell-model ODE dynamics

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.

At \(T = 50{,}000\) on the 6-shell ODE, the H' criterion classifies 91.5% of trajectories the same way as the full-window outcome. Is the remaining 8.5% visible in the raw trajectories when two nearby amplitudes are compared as ratios rather than as individual time series? The run compares \(A = 1.08\) (stays below threshold within the window) and \(A = 1.15\) (crosses the threshold) side by side.

Method

  1. Run two trajectories simultaneously: \(A = 1.08\) and \(A = 1.15\) with identical initial conditions except for amplitude.
  2. Compute the production ratio (ratio of the enstrophy production rates) at every time step.
  3. Track the enstrophy ratio \(E_b/E_a\) at key checkpoints.
  4. Record where and how the two trajectories separate.

Results

Production Ratio Difference

StepProduction Ratio DiffNote
03.1Separated from the first step
5,000wideningMonotonic increase begins
50,000largeClear separation

The production ratio differs by 3.1 at step 0. The separation is present from the first time step rather than emerging gradually.

Enstrophy Ratio Evolution

Checkpoint\(E_b / E_a\)
Early1.134
Mid1.598
Late24.856

The enstrophy ratio widens monotonically from step 5,000 onward, growing from 1.134 to 24.856 — a 22× increase.

Observations. The separation between the two trajectories is in the ratio rather than in either time series on its own. Both trajectories look qualitatively similar for thousands of steps; their ratio separates from step 0. The 8.5% of trajectories not resolved by H' at \(T = 50{,}000\) are ones for which the ratio has not yet grown large enough to be unambiguous within that window, which is what the H'' confidence tracker measures.

Analysis

  • Separation from step 0. The production-ratio difference of 3.1 at step 0 means the two trajectories differ in their initial dynamics, not only at late times.
  • Monotonic widening. The enstrophy ratio does not reverse in this run — once the trajectories start separating, they continue to separate.
  • 22× increase. The late ratio of 24.856 against an early 1.134 shows exponential separation.
  • Observation time. The 8.5% at \(T = 50{,}000\) corresponds to trajectories whose ratio is still growing at the end of that window; H'' tracks whether the ratio is still accelerating or has levelled off.

Summary

The side-by-side view shows that the 8.5% unresolved at \(T = 50{,}000\) is a matter of window length in this ODE. The production ratio separates from step 0 and the enstrophy ratio grows 22× over the trajectory; the signal used by the H'' tracker is present early and becomes unambiguous with more observation time.

Reproducibility

../simplex/build/sxc exp_ns_microscope.sx -o build/exp_ns_microscope.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_microscope.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_microscope \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_microscope

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