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Shell-model experiment: Stretching-coefficient sweep

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

Overview

The system is the six-shell cubic ODE with independent velocity-shell and vorticity-shell variables \(u_k\) and \(\omega_k\), stretching coefficients \(\lambda\) and \(\lambda_2\), feedback coefficient \(\sigma\) and viscous damping \(\nu k^2\). It is not a Galerkin truncation of Navier–Stokes. The time to threshold crossing \(T\) is a finite-window excursion quantity that depends on the integration window, the excursion threshold and the solver. At moderate stretching (\(\lambda = 4\), \(\lambda_2 = 10\)) the derivative \(|\partial T / \partial A|\) grows as the amplitude \(A\) shrinks; this run sweeps \(\lambda\) and \(\lambda_2\) upward to record whether that growth changes within the tested range.

Method

At each \((\lambda, \lambda_2)\) pair, measure \(|\partial T / \partial A|\) by forward-mode dual numbers at amplitudes \(A = 1.0, 0.5, 0.25, 0.125\). Three outcomes are distinguished:

  • Growing: \(|\partial T / \partial A|\) increases as \(A\) shrinks
  • Flat: \(|\partial T / \partial A|\) roughly constant in \(A\)
  • Decreasing: \(|\partial T / \partial A|\) falls as \(A\) shrinks

Results

\(\lambda\)\(\lambda_2\)Growth Ratio \(|\partial T / \partial A|_{A=0.125} / |\partial T / \partial A|_{A=1}\)Outcome
4104.7×Growing
16504.7×Growing
321004.7×Growing
642504.7×Growing
1285004.7×Growing
Observation. At every tested \((\lambda, \lambda_2)\) pair, up to \(\lambda = 128\), \(\lambda_2 = 500\), the growth ratio is 4.7×. Neither the flat nor the decreasing outcome occurs within the tested range.

Notes

  • Constant growth ratio. The 4.7× ratio is unchanged under a 32× increase in \(\lambda\) and a 50× increase in \(\lambda_2\). In this ODE the ratio is set by the low-amplitude scaling of \(T\) with \(A\) and, within the window used, does not depend on the stretching coefficients.
  • No transition found. The three-outcome classification collapses to a single outcome across the range swept. A transition, if one exists, lies beyond \(\lambda = 128\) or beyond the integration window used.
  • Scope. The result is a property of this shell ODE and of the finite-window quantity \(T\); it says nothing about the behaviour of Navier–Stokes solutions.

Reproducibility

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

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

./build/exp_ns_breaking_point

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