Back to Experiments

Shell-model experiment: 16-shell run

Experiment: exp_ns_16mode_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

Extending the ODE to 16 shells (8 velocity + 8 vorticity variables, wavenumbers \(k=1,2,3,4\)) raises the largest linear damping to \(\nu k^2 = 16\nu\). What are the classification score, the fitted power-law exponent and the threshold amplitude on this system?

Method

  1. Implement the 16-shell ODE with 8 velocity and 8 vorticity variables at wavenumbers \(k=1,2,3,4\), with all transfer paths and linear damping \(\nu k^2\) up to \(16\nu\).
  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. Record the threshold value alongside the 8-, 10- and 12-shell runs.

Results

Threshold amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.3470.290 → 0.302 → 0.328 → 0.347 across 8, 10, 12, 16 shells
\(A^*(P)\)0.328Doubling-time criterion estimate
\(P/\text{truth}\)94.6%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

Threshold values across shell counts

A* values:  0.290  →  0.302  →  0.328  →  0.347
              8       10       12       16   shells

In these runs the threshold value increases with shell count from 8 to 16 shells at the fixed window, excursion level and solver settings.

Observation. With wavenumbers up to \(k=4\) and damping up to \(16\nu\), the doubling-time criterion places 14/14 test amplitudes on the correct side of the direct-integration threshold, which is \(A^* = 0.347\) in this run. The fitted power-law exponent is 2.0.

Analysis

  • Classification. 14/14 at 16 shells, as at 12 shells.
  • Threshold value. \(A^* = 0.347\) against 0.328 at 12 shells.
  • Damping range. Linear damping \(\nu k^2\) spans a 16:1 range across the shells.
  • Fitted exponent. 2.0 at each of the five shell counts run (6, 8, 10, 12, 16).
  • Criterion ratio. 94.6%, within the 93–96% band of the other runs.

Summary of the run

For the 16-shell cubic ODE the doubling-time criterion classifies 14/14 test amplitudes, the direct-integration threshold is \(A^* = 0.347\), the criterion ratio is 94.6% and the fitted power-law exponent is 2.0. All are finite-window excursion quantities of this ODE.

Reproducibility

../simplex/build/sxc exp_ns_16mode_solve.sx -o build/exp_ns_16mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_16mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_16mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_16mode_solve

Related