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Shell-model experiment: Oscillating-viscosity sweep

Experiment: exp_ns_viscosity_spectrum.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 \(\lambda\), \(\lambda_2\), feedback coefficient \(\sigma\) and viscous damping \(\nu k^2\). It is not a Galerkin truncation of Navier–Stokes. Enstrophy here means \(\sum_k |\omega_k|^2\) over the shell variables, and the excursion threshold on it is a finite-window quantity that depends on the integration window, the threshold and the solver. This run drives \(\nu\) as a slowly oscillating probe and records the response of the shell enstrophy — amplitude, phase and hysteresis — over the full range from strong damping (\(\nu > 1\)) to weak damping (\(\nu < 10^{-3}\)).

Results

QuantityValueWhat was measured
Resonance period~10,000 stepsShell-enstrophy response peaks at this oscillation period
Hysteresis ratio78×Enstrophy on the \(\nu\)-decreasing sweep vs the \(\nu\)-increasing sweep
Ratchet effectPresentMax enstrophy increases in consecutive oscillation windows
Sensitivity to \(\nu\)\(\partial \Omega / \partial \nu \sim \nu^{-2}\)Response is nonlinear in \(\nu\); small changes near \(\nu = 0\) dominate

Damping Regimes

Regime\(\nu\) RangeBehaviour within the window
Strong damping\(\nu > 1.0\)All shells damped, enstrophy \(\to 0\) exponentially
Bounded oscillation\(0.01 < \nu < 1.0\)Enstrophy bounded and oscillatory within the window
Resonance zone\(0.001 < \nu < 0.01\)Enstrophy response amplified 78×
Weak damping\(\nu < 0.001\)Stretching terms dominate; the excursion threshold is reached within the window
Observation. The 78× hysteresis ratio shows the response is asymmetric in the direction of the sweep: lowering \(\nu\) raises the shell enstrophy far more than raising \(\nu\) afterwards lowers it. The ratchet effect — each oscillation cycle leaving a higher peak enstrophy — means a time-varying \(\nu\) reaches the excursion threshold sooner than a constant \(\nu\) at the same mean. The resonance at period 10,000 is a characteristic timescale of this ODE at the parameters used.

Notes

  • Resonance at period 10,000. Oscillation periods near this value maximise the enstrophy amplification; faster or slower oscillations produce a smaller response.
  • Hysteresis (78×). The response at a given \(\nu\) depends on whether \(\nu\) was recently higher or lower: a memory effect in the shell dynamics.
  • Relation to the two-agent run. The order/perturbation balance in the Burgers two-agent run maps onto this sweep: perturbation corresponds to low-\(\nu\) probing, order to high-\(\nu\) damping.

Reproducibility

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

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

./build/exp_ns_viscosity_spectrum

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