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Shell-model experiment: Coupling matrix of diagnostic levels

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

Overview

The system studied here is a cubic shell ODE with independent velocity-shell variables \(u_k\) and vorticity-shell variables \(\omega_k\), with stretching couplings \(\lambda\), \(\lambda_2\) and a feedback coupling \(\sigma\). It is not a Galerkin truncation of Navier–Stokes. The threshold \(A^*\) reported below is a finite-window excursion threshold: the initial amplitude at which a trajectory first crosses the excursion threshold within the integration window. Its value, the times to threshold crossing and the classification scores depend on the integration window, the excursion threshold and the solver.

The 10 diagnostic levels computed on the ODE are treated as one coupled system. This run computes the coupling matrix \(J_{ij} = \partial L_i / \partial L_j\) between levels, recording which levels drive or suppress each other. The state vector is \(\mathbf{L} = (L_1, L_2, L_4, L_8, L_9, L_{10})\) where \(L_1\) = shell enstrophy, \(L_2\) = S-score, \(L_4\) = stability margin, \(L_8\) = \(\partial T / \partial A\), \(L_9\) = \(\partial T / \partial \lambda\), \(L_{10}\) = \(\partial T / \partial(\text{accel})\), with \(T\) the time to threshold crossing.

Key Results

Coupling Matrix Structure

\(J_{ij}\)\(L_1\) (enstrophy)\(L_2\) (S-score)\(L_4\) (margin)
\(L_1\)—Negative (S suppresses enstrophy growth)Negative (margin damps)
\(L_2\)Negative (enstrophy degrades S)—Positive (margin supports S)
\(L_4\)Negative (enstrophy erodes margin)Positive (S reinforces margin)—

Feedback Pattern

The dominant cycle in the coupling matrix is \(L_1 \to L_4 \to L_2 \to L_1\): shell enstrophy erodes the stability margin, which lowers the S-score, which allows the enstrophy to grow further. The viscous term acts on \(L_1\) directly.

Indirect Viscosity Sensitivity

MeasurementValue
Direct sensitivity of enstrophy to viscosity\(\partial L_1 / \partial \nu\): modest
Indirect sensitivity via the coupling matrix500,000× amplification through the \(L_1 \to L_4 \to L_2 \to L_1\) cycle
Composite diagnostic \(H\)\(H = 0.3 \cdot L_{1,n} - 0.4 \cdot L_2 + 0.3 \cdot L_{4,n}\)
Observations. Measured level by level, the sensitivity of the ODE to \(\nu\) is small; propagated through the coupling matrix along the \(L_1 \to L_4 \to L_2 \to L_1\) cycle, a small change in \(\nu\) is amplified 500,000× in this run. The composite diagnostic \(H\) collects the three levels of that cycle into one scalar computed on the ODE.

Analysis

  • One coupled set, not 10 separate diagnostics. The coupling matrix shows that changes in any level propagate through the others.
  • The dominant cycle. The \(L_1 \to L_4 \to L_2 \to L_1\) cycle is the strongest pattern in the matrix; the viscous term enters it at \(L_1\).
  • 500,000× indirect sensitivity. The direct \(\partial L_1 / \partial \nu\) understates the total sensitivity of the diagnostics to \(\nu\) by five orders of magnitude in this run.

Reproducibility

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

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

./build/exp_ns_holistic

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