Abstract
We integrate Fourier–Galerkin truncations of the 3D incompressible Euler equations (\(N \le 8\)), the 3D incompressible MHD equations (\(N \le 6\)) and the 2D surface quasi-geostrophic equation (\(N \le 15\), critical and inviscid) with a first-order splitting scheme and record energy, enstrophy and per-shell diagnostics across nested truncation levels. At \(\nu = 0\) the truncated systems relax to the Lee–Kraichnan absolute equilibrium: the ratio \(\Omega/E\) reaches the lattice mean \(\langle k^2 \rangle\) to within \(1\%\) at every \(N\), which fully accounts for the observed enstrophy growth (\(\approx 22\%\), stable across \(N\)) and is unchanged under time-step halving.
The cross-truncation ratios between nested levels are ratios of initial-energy increments between levels initialised with different fields (the phase assignment depends on the mode index, which depends on \(N\)), not a property of the dynamics. A finite Galerkin system cannot blow up, so bounded enstrophy in these runs is not evidence about any of the three PDEs; critical SQG is globally regular by theorem (Kiselev–Nazarov–Volberg 2007; Caffarelli–Vasseur 2010), inviscid SQG and 3D Euler blow-up for smooth periodic data are open, and this paper bears on neither.
What stands: the corrected \(-i\) nonlinearity of the kernel conserves energy up to first-order integrator error (drift halves with \(dt\)), the kernel-computed energy and enstrophy trajectories, and the thermalisation check.
Configurations Run
| Equation | Truncation | Dissipation | Initial data | Window |
|---|---|---|---|---|
| 3D Euler | \(N = 2\)–\(8\) | \(\nu = 0\) | distributed, Taylor–Green, concentrated, random | \(T = 2\) |
| 2D SQG (critical) | \(N \le 15\) | \(\kappa = 0.01\), \(\alpha = 1/2\) | distributed | \(T = 2\) |
| 2D SQG (inviscid) | \(N \le 15\) | \(\kappa = 0\) | distributed | \(T = 2\) |
| 3D MHD | \(N \le 6\) | \(\nu = \eta = 0.01\); \(\nu = \eta = 0\); \(\nu = 0.01,\ \eta = 0.001\) | distributed velocity and field | \(T = 2\) |
All runs use the corrected complex-coefficient triad kernel with the \(-i\) factor in the Fourier nonlinearity and a first-order splitting integrator. The nonlinear term conserves energy up to integrator error; the drift halves when \(dt\) is halved.
3D Euler Trajectories (\(\nu = 0\))
Enstrophy Across \(N\)
| \(N\) | \(\Omega(0)\) | \(\Omega(T=2)\) | Growth |
|---|---|---|---|
| 2 | 1.88 | 1.93 | +2.5% |
| 4 | 28.4 | 33.3 | +17% |
| 6 | 151.5 | 190.3 | +26% |
| 8 | 469.7 | 574.7 | +22% |
Thermalisation Check
The enstrophy growth in the inviscid truncations is the approach to the Lee–Kraichnan absolute equilibrium of the truncated system. In equilibrium the energy is equidistributed over the retained modes, so \(\Omega/E\) tends to the lattice mean \(\langle k^2 \rangle\) over the truncation. The measured \(\Omega/E\) at \(T = 2\) reaches \(\langle k^2 \rangle\) to within \(1\%\) at every \(N\) from 3 to 8; halving the time step changes \(\Omega(2)\) by \(0.13\%\). This accounts for the \(\approx 22\%\) growth seen in the distributed and random runs and the larger growth from concentrated data (whose initial \(\Omega/E\) sits further below \(\langle k^2 \rangle\)). A finite Galerkin system cannot blow up, so these trajectories carry no information about singularity formation in the Euler equations.
2D SQG Trajectories
Critical (\(\kappa = 0.01\), \(\alpha = 1/2\))
- Enstrophy decreases by 30% over the window; the fractional diffusion \(\kappa|k|\) removes energy at every shell.
- Kernel energy balance closes to \(10^{-16}\)–\(10^{-19}\) per step on the retained modes.
Inviscid (\(\kappa = 0\))
- Enstrophy grows by 2.5% over the window at the truncations run.
Critical SQG is globally regular by theorem (Kiselev–Nazarov–Volberg 2007; Caffarelli–Vasseur 2010). Blow-up of inviscid SQG and of 3D Euler for smooth periodic data are open problems. Bounded enstrophy in a finite truncation bears on neither.
3D MHD Trajectories
MHD carries a second field (magnetic \(B\)) coupled to the velocity; the kernel records the kinetic and magnetic energies separately.
| Config | \(E_{\text{kin}}\) change | \(E_{\text{mag}}\) change | Total energy |
|---|---|---|---|
| \(\nu = \eta = 0.01\) | −47% | +267% | decreasing |
| \(\nu = \eta = 0\) (ideal) | −46% | +754% | conserved to 0.5% (integrator error) |
| \(\nu = 0.01\), \(\eta = 0.001\) | −71% | +437% | decreasing |
In every configuration kinetic energy is transferred to the magnetic field over the window; the transfer is larger at the smaller magnetic diffusivity. These are trajectories of a six-mode-per-axis truncation and are reported as such.
Method Note
The instrument that carries over from these runs is the thermalisation check: for an inviscid truncation, compare \(\Omega/E\) at late time with the lattice mean \(\langle k^2 \rangle\) of the retained modes, and confirm the value is unchanged under time-step halving. Together with the energy balance of the corrected \(-i\) kernel, it separates integrator error and equilibrium relaxation from any dynamical effect before a truncated run is read.
References
- Higgins, R. (2026). Energy Conservation, Cascade Stabilisation, and the Global Regularity of the 3D Navier–Stokes Equations, v19. Zenodo. 10.5281/zenodo.22710239.
- Beale, J. T., Kato, T., & Majda, A. (1984). Breakdown of smooth solutions for 3-D Euler. Comm. Math. Phys., 94(1), 61–66.
- Luo, G., & Hou, T. Y. (2014). Potentially singular solutions of the 3D axisymmetric Euler equations. PNAS, 111(36), 12968–12973.
- Constantin, P., Majda, A. J., & Tabak, E. (1994). Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 7(6), 1495–1533.
- Biskamp, D. (2003). Magnetohydrodynamic Turbulence. Cambridge University Press.
- Kiriukhin, O. (2026). Orbit-Level Stretching in Cubic Fourier–Galerkin NS. arXiv: 2603.23293.
- Higgins, R. (2026). Unified Adaptation Theorem, v3. Zenodo. 10.5281/zenodo.22714409.
- Majda, A. J., & Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge University Press.