Turbulence from First Principles

TL;DR

The paper tackles turbulence by deriving a first-principles Navier–Stokes–type framework from the electrodynamics of continuous media in the viscous limit, incorporating a spin-network of two orthogonal angular momenta and invariant toroidal de Sitter geometry to organize energy transfer across scales. The approach yields a momentum-dependent closure based on a degenerate spin–oscillator dispersion and invariant tori, culminating in a sixth-order dissipative closure that sets a dissipation-range spectrum $E(k)\propto T_b^{-1} k^{-6}$ and fixes anisotropy via the golden-ratio-derived factor $\varphi/2$. Key contributions include a closed viscous-limit NS-type model with a Laplacian smoothing operator $\Delta$, explicit ds_solutn constructions, and a diagnostic picture linking microscopic EM structure to macroscopic dissipation in neutral or near-neutral media. The work offers a tractable, geometry-informed route to understanding viscous turbulence and provides testable predictions for the dissipation range and spectral scaling in highly viscous flows.

Abstract

We provide a first-principles approach to turbulence by employing the electrodynamics of continuous media at the viscous limit to recover the Navier-Stokes equations. We treat oscillators with two orthogonal angular momenta as a spin network with properties applicable to the Kolmogorov-Arnold-Moser (KAM) theorem. The microscopic viscous limit has an irreducible representation that includes $O(3)$ expansion terms for a radiation-dominated fluid with a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, equivalent to an oriented toroidal de Sitter space. The turbulence solution in $\mathbb{R}^{3,1}$ lies on 6-choose-3 de Sitter intersections of three orthogonal $n$-tori.

Figures (5)

• Figure 1: Comparison of our analytic result (red line) to empirical energy cascade in turbulence. Power spectrum of a fluid exhibits Kolmogorov scaling (blue solid line) in the inertial subrange before viscous effects dominate at high wavenumber $k$. Our result converges on a half-golden-ratio inertial subrange exponent (dashed cyan), i.e. $(1+\sqrt{5})/4=\varphi/2$ with $\varphi=(1+\sqrt{5})/2$. $E^*_{11}(k)$ is the normalised longitudinal energy spectrum of the eddy domain ensemble with normalized wavevector $k^*$. The viscous scaling solution was evaluated using higher order terms of the velocity field to $O(3)$. One-dimensional turbulence spectrum data for various Taylor microscale Reynolds numbers with a fitted (dashed) curve adapted from chassaing_turbulence_2000antonia_collapse_2014.
• Figure 2: Left, unit cylinder with embedded torus showing longitudinal contact sites (blue). Right, fractally enveloping ring tori converge onto a dense $n$-dimensional horn torus filling the unit cylinder when $r_{n+1} = \frac{r_n(e-1)}{2}$, $n\rightarrow \infty$.
• Figure 3: For paths traced by the electric displacement vector $\vec{P}$ the maximum dissipation via viscous effects occurs in the limit $\Delta x \rightarrow r_{\text{bound}}$ and $r_0=r_1=k_{\max}^{-1}$. Toroidal moments arise from octupole ($O(3)$) irreducible terms and higher as in the configuration shown, having four dipoles centred about $\pm x_0$. They equally arise from the in-plane rotation of quadrupole moments, that is \ref{['eq:rofstrain']} in a rotating frame. Angular momentum is transported via \ref{['eq:rofstrain']} through either the $\pm y_0$ surface ($\parallel T^{(m)}$), or the $\pm x_0$ surface ($\perp T^{(m)}$).
• Figure 4: The $z$-oriented solutions of toroidal spin-oscillator spaces (red points) exist on hypersurfaces $dR_z e^{dR_z}$ that co-exist with 0, 1 or 2 independent time dimensions. We may interpret the opposing time directions associated with the $z$-hypersurface as an energy cascade balancing mechanism, or simply equal angular momentum dissipation rate into orthogonal directions, in this case $\hat{x}$ and $\hat{y}$. There are three positive time solutions and two time-independent solutions.
• Figure 5: Red: solution of this work given by eq (\ref{['eq:ds_solutn']}) and (\ref{['eq:roots']}). Blue: K41 inertial scaling regime. Dashed Blue: non-viscous limit of eq(\ref{['eq:ds_solutn']}). Normalised longitudinal spectra for various turbulence regimes and corresponding Taylor microscale Reynolds numbers $\mathrm{Re}_{\lambda}$. (x: $\mathrm{Re}_{\lambda}=40$, dots: 67, solid line: 65, circles: 89, triangles: 139, dash-dot line: 130). Fitting the dissipative curve to the turbulence data provides qualitative insight into the longitudinal energy gap at various frequencies. Adapted from antonia_collapse_2014.