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Shell-model experiment: 10-shell run

Experiment: exp_ns_10mode_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

How do the diagnostic quantities change when the ODE is extended from 8 to 10 shells (5 velocity + 5 vorticity variables, wavenumbers \(k=1,2,3\) with two transfer stages)? Recorded: the diagnostic loop, the fitted power-law exponent, the doubling-time criterion \(P\) and the threshold amplitude \(A^*\).

Method

  1. Implement the 10-shell ODE with 5 velocity and 5 vorticity variables at wavenumbers \(k=1,2,3\), including two transfer stages between wavenumbers.
  2. Run the full 26-level H/H'/H'' hierarchy with the doubling-time criterion \(P\) as the primary discriminator.
  3. Sweep amplitudes to find \(A^*(\text{truth})\) and \(A^*(P)\). Evaluate classification across 14 test amplitudes.
  4. Record the diagnostic loop, the fitted power-law exponent and the sign of \(A^*\).

Results

Threshold amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.3020.290 in the 8-shell run
\(A^*(P)\)0.290Doubling-time criterion estimate
\(P/\text{truth}\)96.1%86.1% → 95.5% → 96.1% across 6, 8, 10 shells

Classification

MetricResult
Score13/14 amplitudes classified
Diagnostic loop\(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint
Fitted power-law exponent2.0
Observation. The \(P/\text{truth}\) ratio is 86.1% (6 shells), 95.5% (8 shells) and 96.1% (10 shells) in these runs. The fitted power-law exponent is 2.0 at each shell count.

Analysis

  • Threshold value. \(A^* = 0.302\) against 0.290 in the 8-shell run, at the same window and excursion level.
  • Criterion ratio. 96.1% is the highest ratio among the 6-, 8- and 10-shell runs.
  • Diagnostics carried over. The diagnostic loop, the fitted exponent 2.0 and a positive \(A^*\) are present as in the 6- and 8-shell runs.
  • Two transfer stages. Transfer from \(k=1 \to k=2 \to k=3\) adds cubic interactions; the diagnostic hierarchy runs on it without modification.

Summary of the run

For the 10-shell cubic ODE the direct-integration threshold is \(A^* = 0.302\), the doubling-time estimate is 0.290 (ratio 96.1%), 13 of 14 test amplitudes are classified correctly and the fitted power-law exponent is 2.0. All are finite-window excursion quantities of this ODE.

Reproducibility

../simplex/build/sxc exp_ns_10mode_solve.sx -o build/exp_ns_10mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_10mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_10mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_10mode_solve

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