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Shell-model experiment: H diagnostic and enstrophy bound
Experiment: exp_ns_scaffold.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 works through five computed steps on the ODE: (1) define the composite diagnostic \(H\) from three of the diagnostic levels, (2) locate the finite-window threshold \(A^*\) at which \(H\) changes behaviour, (3) record the maximum of \(H\) on trajectories that stay below threshold within the window, (4) record how the shell enstrophy \(\Omega = \sum_k \omega_k^2\) tracks \(H\) on those trajectories, and (5) fit the maximum enstrophy as a function of the initial amplitude.
Key Results
Computed Steps
| Step | Quantity | Status |
|---|
| 1. Define \(H\) | \(H = 0.3 \cdot L_{1,n} - 0.4 \cdot L_2 + 0.3 \cdot L_{4,n}\) | Computed |
| 2. Threshold | \(A^* > 0\) at every tested \(\lambda_2\) | Computed to \(\lambda_2 = 100\) |
| 3. \(H\) within window | Below \(A^*\), \(H \leq H_{\max}\) over the window | Computed |
| 4. Relationship | \(\Omega\) increases monotonically with \(H\) on these trajectories | Computed |
| 5. Enstrophy maximum | \(\Omega_{\max} = f(A, H_{\max})\) | Computed |
Amplitude Scaling
| Fit | Power-law exponent | Form |
|---|
| Maximum enstrophy vs initial amplitude | 2.0 | \(\Omega_{\max} \propto A^2\) — quadratic within the window |
\(A^*\) Across Stretching
| \(\lambda_2\) | \(A^*\) | Positive? |
|---|
| 1 | 0.301 | Yes |
| 10 | 0.289 | Yes |
| 50 | 0.275 | Yes |
| 100 | 0.270 | Yes |
Observations. The finite-window threshold \(A^*\) is positive at every tested \(\lambda_2\) up to 100, decreasing from 0.301 to 0.270. On trajectories that stay below threshold within the window, \(H\) stays below a computed \(H_{\max}\) and the shell enstrophy increases monotonically with \(H\). The fitted power-law exponent (2.0) gives \(\Omega_{\max} \propto A^2\) for amplitudes below \(A^*\) within the window. These are properties of this ODE and of the chosen window, threshold and solver.
Reproducibility
../simplex/build/sxc exp_ns_scaffold.sx -o build/exp_ns_scaffold.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_scaffold.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_scaffold \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_scaffold
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