Shell-model experiment: 8-shell run
Overview
The system is a cubic shell ODE: independent velocity-shell variables \(u_k\) and vorticity-shell variables \(\omega_k\) at integer wavenumbers \(k\), coupled by hand-built cubic transfer terms, a stretching coupling \(\lambda\) and a feedback coupling \(\sigma\), with linear damping \(\nu k^2\). It is not a Galerkin truncation of Navier–Stokes; \(u_k\) and \(\omega_k\) are separate state variables and the coupling structure is chosen by hand. Every threshold reported on this page — the amplitude \(A^*\), times to threshold crossing and classification scores — is a finite-window excursion threshold: \(A^*\) is the smallest initial amplitude for which the shell enstrophy \(\sum_k \omega_k^2\) exceeds a fixed excursion level within the integration window, and its value depends on the window length, the excursion level and the solver.
Question
Does the H/H'/H'' diagnostic hierarchy developed on the 6-shell ODE apply unchanged to the 8-shell ODE (4 velocity + 4 vorticity variables, wavenumbers \(k=1,2,3\) with forward transfer)? Quantities recorded: the diagnostic loop, the fitted power law between amplitude and peak shell enstrophy, the doubling-time criterion and the threshold amplitude \(A^*\).
Method
- Implement the 8-shell ODE with coupled velocity and vorticity variables and forward transfer between wavenumbers.
- Run the full 26-level hierarchy with the doubling-time criterion \(P\) as the primary discriminator.
- Sweep amplitudes to find \(A^*(\text{truth})\) by direct integration and \(A^*(P)\) from the criterion. Evaluate classification across 16 test amplitudes.
- Check for the diagnostic loop \(L_1 \to L_4 \to L_2 \to L_1\) at every checkpoint. Fit the power-law exponent of peak shell enstrophy against amplitude.
- Vary the window from \(T=30{,}000\) to \(T=80{,}000\) and record the \(P/\text{truth}\) ratio.
Results
Threshold amplitude
| Quantity | Value | Note |
|---|---|---|
| \(A^*(\text{truth})\) | 0.290 | 1.136 in the 6-shell ODE; the two systems have different coupling structures |
| \(A^*(P)\) | 0.277 | Doubling-time criterion estimate |
| \(P/\text{truth}\) | 95.5% | 86.1% in the 6-shell run |
Classification
| Metric | Result |
|---|---|
| Score | 16/16 amplitudes classified |
| Diagnostic loop | \(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint |
| Fitted power-law exponent | 2.0 |
Window dependence
| Window | \(P/\text{truth}\) |
|---|---|
| \(T = 30{,}000\) | 93% |
| \(T = 50{,}000\) | 94% |
| \(T = 80{,}000\) | 95% |
The ratio moves from 93% to 95% as the window lengthens from 30,000 to 80,000 steps.
Analysis
- Classification. All 16 test amplitudes are placed on the correct side of the direct-integration threshold within the window.
- Criterion ratio. \(P/\text{truth} = 95.5\%\) for the 8-shell ODE against 86.1% for the 6-shell ODE.
- Diagnostics carried over. The diagnostic loop, the fitted power-law exponent (2.0) and a positive \(A^*\) are all present in the 8-shell run.
- Window dependence. The ratio rises from 93% to 95% as the window grows from \(T=30{,}000\) to \(T=80{,}000\).
Summary of the run
The diagnostic hierarchy runs unchanged on the 8-shell cubic ODE. The doubling-time criterion places 16/16 test amplitudes on the correct side of the direct-integration threshold, with \(P/\text{truth} = 95.5\%\), and the fitted power-law exponent is 2.0. These are finite-window excursion quantities of this ODE.
Reproducibility
../simplex/build/sxc exp_ns_8mode_solve.sx -o build/exp_ns_8mode_solve.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_8mode_solve.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_8mode_solve \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_8mode_solveRelated
- 6-shell run — criterion-by-criterion threshold estimates
- 10-shell run — 5 velocity + 5 vorticity shells
- Shell-count series — full table across 6–24 shells
- Microscope — side-by-side trajectory ratios at A=1.08 and A=1.15
- Threshold countdown — S score as an early indicator of threshold crossing