Back to Experiments

Shell-model experiment: H diagnostic and threshold

Experiment: exp_ns_H_regularity.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.

This run tracks the composite diagnostic \(H\) instead of the shell enstrophy \(\Omega = \sum_k \omega_k^2\) alone. \(H\) combines three diagnostic levels, is monotonic over the window once the \(L_1 \to L_4 \to L_2 \to L_1\) feedback pattern is active (the enstrophy itself oscillates), and has a computable derivative \(dH/dt\). The run records the sign of \(dH/dt\) on either side of \(A^*\) for four values of \(\lambda_2\).

Key Results

H Compared With Enstrophy

PropertyEnstrophy \(\Omega\)Composite \(H\)
Reflects the feedback pattern?No (output only)Yes (built from three coupled levels)
Monotonic over the window once the pattern is active?No (oscillates)Yes
Computable derivative?Involves the stretching term\(dH/dt = 0.3 \cdot dL_{1,n}/dt - 0.4 \cdot dL_2/dt + 0.3 \cdot dL_{4,n}/dt\)
Sign within the windowNot directly usable\(dH/dt < 0\) on trajectories that stay below threshold

Threshold Behaviour

The \(L_1 \to L_4 \to L_2 \to L_1\) pattern becomes active over a narrow amplitude range around \(A^*\). Below \(A^*\) the viscous term dominates and \(dH/dt < 0\); above \(A^*\) the stretching term dominates and \(dH/dt > 0\), within the window.

\(A^*\) Across Stretching Strengths

\(\lambda_2\)\(A^*\)\(dH/dt\) below \(A^*\)\(dH/dt\) above \(A^*\)
10.301NegativePositive
100.289NegativePositive
500.275NegativePositive
1000.270NegativePositive
Observations. The threshold \(A^*\) moves from 0.301 to 0.270 as \(\lambda_2\) increases 100-fold, an 11% variation. At every tested \(\lambda_2\) it is positive: there is an amplitude range in which \(dH/dt < 0\) throughout the window. These values are for this ODE, this window and this solver.

Analysis

  • \(A^*\) from 0.270 to 0.301. An 11% variation across a 100× change in \(\lambda_2\).
  • Sensitivity at small amplitude. \(|dH/dA|\) grows without bound as \(A \to 0\) in this ODE, the same pattern seen in the time-to-threshold-crossing curve.
  • Diagnostic scope. \(H\) and \(dH/dt\) are computed on the shell ODE and describe its trajectories within the window.

Reproducibility

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

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

./build/exp_ns_H_regularity

Related