14 April 2026 · 9 min read

What the Lattice Does and Does Not Do

The integer lattice fixes which triads exist and at what angles. That is a fact about the interaction set, not about any solution.

On the torus, the 3D Navier–Stokes equations live on the integer lattice Z³ in Fourier space. The nonlinear term couples modes in triads k = p + q, and which triads exist, and at what angles, is fixed by the lattice before any initial datum is chosen. It is natural to ask what that geometry does for you. Here is what I can state about it, and where its reach ends.

Two Facts About the Lattice

The angular variance is positive

For the triad interactions on Z³, the variance of cos²α_T across the triads feeding a shell is positive at every shell computed and tends to 4/45 as the shell radius grows. The triads never all sit at one angle.

A shared mode cannot serve every triad

A mode at wavenumber k belongs to many triads. Maximising the stretching in one of them fixes the mode’s orientation; the other triads through the same mode then see it at a different angle. No mode can maximise the stretching of all its triads at once. This is qualitative: a statement about the interaction set, not a bound.

Both facts are kinematic. They describe the set of interactions available to any field on the torus, and they hold for every field, including one that is about to form a singularity. That is the “does” part, and it is all of it.

Phase Cancellation Is a Property of Data

A tempting next step is to measure, on a running solution, how much the triad contributions to the stretching at shell K cancel: a resultant statistic that is near 0 when the contributions are spread in phase and equals 1 when they align. On generic initial data this statistic is small at every shell. It is easy to read that as a property of the operator, as if the −i factor and the Leray projection scatter the triads on their own.

They do not. Take a single triad: a trigonometric polynomial with one pair of modes p, q and their sum k, nothing else. The statistic equals 1 exactly. The lattice offers many triads at many angles, but the datum decides which of them carry energy. A field that puts its energy on one triad has no cancellation to speak of, and it is a perfectly smooth divergence-free field. Phase cancellation is a property of generic data, as explicit trigonometric polynomials show. It is not a property of the equations.

What an Inviscid Truncation Does

The second thing the lattice does not do is make a truncation behave like the PDE. Run the Galerkin truncation at ν = 0 on the modes |k| ≤ N, with the corrected complex-coefficient triad kernel, and enstrophy grows. That growth is not a cascade signature. The ratio Ω/E reaches the lattice mean ⟨k²⟩ over the retained modes to within 1% at every N from 3 to 8, and the value does not change when the time step is halved.

That is the Lee–Kraichnan absolute equilibrium. A finite set of modes conserving energy and interacting through a cascade goes to equipartition: every retained mode ends up holding the same energy, and then the enstrophy-to-energy ratio is the mean of |k|² over the lattice ball. Enstrophy growth in a truncated Euler run is thermalisation. The number it reaches is set by N, not by the flow.

What This Clarifies

The lattice constrains the interaction set. It does not bound any solution. The two facts above hold for a field on its way to a singularity as much as for one that is not. The cancellation one observes in a generic run is a property of the datum. And the finite-mode systems one can run on a computer are truncations whose long-time inviscid behaviour is equipartition, so their enstrophy is not a measurement of the PDE’s.

What would be needed is control of the covariance between strain and the vorticity direction at the flux scale. That is where the stretching lives: on direct numerical simulation at the enstrophy peak, the mean-strain part accounts for at most 6% of it, and the strain–direction covariance carries the rest. And no one-point statistic of the direction is a necessary condition for it. The Burgers vortex has exactly unidirectional vorticity and sustained stretching. Six Burgers-type patches can be arranged into a divergence-free field with an exactly isotropic direction second moment and enstrophy-weighted stretching 0.999. The lattice gives the stage. The covariance is the play.