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Shell-model experiment: shell-count series

Domain: Shell-model ODE dynamics  |  Systems: 6, 8, 10, 12, 16, 20, 24 shells

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. This page collects the runs at seven shell counts, from 6 to 24, each integrated with the H/H'/H'' diagnostic hierarchy and the doubling-time criterion \(P\). The 6-shell system uses a different coupling structure from the 8- to 24-shell systems.

The series table

RunShells\(A^*(\text{truth})\)\(A^*(P)\)\(P/\text{truth}\)ScorePower-law exponent
6-shell61.1361.02086.1%17/202.0
8-shell80.2900.27795.5%16/162.0
10-shell100.3020.29096.1%13/142.0
12-shell120.3280.30793.8%14/142.0
16-shell160.3470.32894.6%14/142.0
20-shell200.3470.32894.6%14/142.0
24-shell240.3470.32894.6%14/142.0

Threshold values by shell count

The direct-integration threshold \(A^*\) at the fixed window and excursion level, by shell count:

A* values:  0.290  →  0.302  →  0.328  →  0.347  →  0.347  →  0.347
              8       10       12       16       20       24   shells

In these runs \(A^*\) increases from 8 to 16 shells and takes the same value, 0.347, at 16, 20 and 24 shells. The 6-shell value (1.136) belongs to a system with a different coupling structure. All values are specific to the window length, excursion level and solver used; the series does not measure how they change under other settings.

Recorded observations

Quantities recorded at every shell count:
  1. \(A^*\) is positive at every shell count — i.e. some initial amplitudes stay below the excursion level within the window.
  2. \(A^*\) increases from 8 to 16 shells.
  3. \(A^*\) is 0.347 at 16, 20 and 24 shells.
  4. The fitted power-law exponent of peak shell enstrophy against amplitude is 2.0 in every run.
  5. \(P/\text{truth}\) lies in the 93–96% band for the 8- to 24-shell runs.
  6. The doubling-time criterion classifies 14/14 test amplitudes at 12, 16, 20 and 24 shells.

1. Positive thresholds

\(A^*(\text{truth}) > 0\) at every shell count: at each shell count there is a finite initial amplitude below which the shell enstrophy stays under the excursion level for the length of the window.

2. Threshold values across shell counts

\(A^*\) rises from 0.290 (8 shells) to 0.347 (16 shells). Additional shells add cubic transfer channels among which the initial amplitude is distributed. At 20 and 24 shells the recorded value is again 0.347.

3. Fitted power-law exponent

The exponent of the fitted power law between initial amplitude and peak shell enstrophy is 2.0 at every shell count run: 6, 8, 10, 12, 16, 20 and 24.

4. Criterion ratio

The \(P/\text{truth}\) ratio lies in a 93–96% band across the 8- to 24-shell runs (86.1% for the 6-shell system with its different coupling structure). The doubling-time criterion is used with the same settings at every shell count.

5. Classification scores

The 12-, 16-, 20- and 24-shell runs each place 14/14 test amplitudes on the correct side of the direct-integration threshold; the 8-shell run places 16/16.

What the series measures

The series records how the H/H'/H'' diagnostic quantities of this cubic shell ODE — threshold amplitude, criterion ratio, classification score, fitted exponent and the diagnostic loop — vary with the number of shells at one fixed window, excursion level and solver. It is a study of the hand-built ODE, not of Navier–Stokes, and its numbers are finite-window excursion quantities of that ODE.

Individual run pages