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Shell-model experiment: Six-shell system with a stretching term
Experiment: exp_ns_3d_vortex.sx | Domain: Shell-model ODE dynamics
Overview
The system studied here is a cubic shell ODE with independent velocity-shell and vorticity-shell variables \(u_k\) and \(\omega_k\): three velocity variables at wavenumber \(k_1 = 1\) and three vorticity variables at \(k_2 = 2\), coupled by a linear stretching coefficient \(\lambda\), a quadratic self-amplification coefficient \(\lambda_2\), and a vorticity-to-velocity feedback coefficient \(\sigma\), with viscous damping \(\nu k^2\) on each shell. It is not a Galerkin truncation of Navier–Stokes. The time to threshold crossing \(T\) and every threshold derived from it are finite-window excursion quantities: they depend on the integration window, the excursion threshold and the solver. This run adds the stretching terms to the six-shell system and records which of the diagnostics developed on the 1D Burgers and four-mode 2D runs still register on it.
Run Description
Model Parameters
| Parameter | Role | Effect |
|---|
| \(\sigma\) | Vorticity → velocity feedback | Drives the velocity shells via a \(\omega \times u\)-type product |
| \(\lambda\) | Linear stretching coefficient | \(\lambda = 0\): no stretching term; \(\lambda > 0\): stretching term active |
| \(\lambda_2\) | Quadratic self-amplification | \(\lambda_2 |\omega|^2 \omega\): cubic growth term on the vorticity shells |
| \(\nu\) | Viscosity | Damping proportional to \(k^2\) on each shell |
Diagnostics Recorded
| Diagnostic | Four-mode 2D run | Six-shell run with stretching | Registers? |
|---|
| \(|\partial T / \partial A|\) growing as \(A \to 0\) | Yes | Yes | Yes |
| I-ratio \(= -0.5\) at force balance | Yes | Yes (damping vs stretching terms) | Yes |
| \(S\)-score early warning | 37× lead | Reduced but present | Yes |
| Stability margin \(M(t)\) | Positive | Positive (reduced by stretching) | Yes |
Observation. All four diagnostics computed on the Burgers and four-mode runs also register on the six-shell system with the stretching terms switched on. The divergence of \(\partial T / \partial A\) as \(A \to 0\) (with \(T\) the time to threshold crossing within the window) is the clearest of the four at the stretching strengths used here. The stability margin is reduced by stretching but remains positive at these parameters, i.e. the damping terms dominate within the window.
Notes
- What the stretching term is. The \(\lambda\) and \(\lambda_2\) terms are sign-indefinite products of shell variables; they can increase or decrease \(|\omega_k|\) depending on the sign of the product, which is the property the run is designed to exercise. They are hand-built couplings between scalar shell variables, not projections of \((\omega \cdot \nabla)u\).
- I-ratio on this system. With stretching active, the I-ratio compares the damping and stretching terms rather than damping and advection; the value \(-0.5\) marks the point where the summed terms cancel, which is the balance identity, not a statement about the flow.
- Stretching reduces but does not remove the stability margin. Damping at \(k_2^2 = 4\) on the vorticity shells is sufficient at the parameters used here. The dependence on larger \(\lambda\) is recorded in the parameter-sweep run linked below.
Reproducibility
../simplex/build/sxc exp_ns_3d_vortex.sx -o build/exp_ns_3d_vortex.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_3d_vortex.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_3d_vortex \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_3d_vortex
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