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Shell-model experiment: 12-shell run

Experiment: exp_ns_12mode_solve.sx  |  Domain: Shell-model ODE dynamics

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

The 12-shell ODE carries 6 velocity + 6 vorticity variables (two components at each wavenumber \(k=1,2,3\)) and three cubic transfer paths: \(k_1 \to k_2\), \(k_2 \to k_3\) and the direct \(k_1 \to k_3\) term. What are the classification score, the fitted power-law exponent and the threshold amplitude on this system?

Method

  1. Implement the 12-shell ODE with two-component velocity and vorticity variables at each wavenumber, enabling all three transfer paths.
  2. Run the full 26-level H/H'/H'' hierarchy with the doubling-time criterion \(P\).
  3. Sweep amplitudes to find \(A^*(\text{truth})\) and \(A^*(P)\). Evaluate classification across 14 test amplitudes.
  4. Check that the three transfer paths \(k_1 \to k_2\), \(k_2 \to k_3\) and \(k_1 \to k_3\) are active.

Results

Threshold amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.3280.290 → 0.302 → 0.328 across 8, 10, 12 shells
\(A^*(P)\)0.307Doubling-time criterion estimate
\(P/\text{truth}\)93.8%Within the 93–96% band of the other runs

Classification

MetricResult
Score14/14 amplitudes classified
Diagnostic loop\(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint
Fitted power-law exponent2.0

Transfer paths

PathWavenumbersStatus
Forward 1\(k_1 \to k_2\)Active
Forward 2\(k_2 \to k_3\)Active
Direct\(k_1 \to k_3\)Active
Observation. With all three transfer paths active, the doubling-time criterion places 14/14 test amplitudes on the correct side of the direct-integration threshold. The threshold values in these runs are 0.290, 0.302 and 0.328 at 8, 10 and 12 shells.

Analysis

  • Classification. 14/14 test amplitudes placed on the correct side of the threshold within the window.
  • Threshold value. \(A^* = 0.328\) against 0.302 in the 10-shell run.
  • Direct transfer path. The \(k_1 \to k_3\) term adds further cubic interactions; the diagnostic hierarchy runs without modification.
  • Fitted exponent. The power-law exponent of peak shell enstrophy against amplitude is 2.0 at each shell count run so far (6, 8, 10, 12).

Summary of the run

For the 12-shell cubic ODE with three transfer paths, the doubling-time criterion classifies 14/14 test amplitudes, the direct-integration threshold is \(A^* = 0.328\) and the fitted power-law exponent is 2.0. All are finite-window excursion quantities of this ODE.

Reproducibility

../simplex/build/sxc exp_ns_12mode_solve.sx -o build/exp_ns_12mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_12mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_12mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_12mode_solve

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