A Statistical Field Theory for Isotropic Turbulence
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913051182104576 |
|---|---|
| author | Farooq, Ahmed |
| author_facet | Farooq, Ahmed |
| contents | This article establishes a first-principles statistical field theory of fully developed isotropic turbulence. Applying an exact Helmholtz decomposition to the local angular momentum field ($\Lvec = \rvec \times \uvec$) reveals a segregation into two orthogonally distinct topological phases: a longitudinal condensate of macroscopic coherent structures ($\PhiL$) and a volume-filling, transverse thermal bath ($\AL$). Constructing a Hamiltonian and evaluating the partition function of these decoupled fields demonstrates that their ergodic exploration of phase space is topologically quantized, mandating a strict $1:2$ equipartition of degrees of freedom. Inverting this topological projection back to the velocity domain isolates the radial velocity field ($\uvec_r$) (which strictly resides in the null space of the $\Lvec$ framework) revealing a recursive partitioning scheme across the cascade into a precise $1/3 : 2/9 : 4/9$ fractional hierarchy. This geometric constraint forces the turbulent steady state into a rigorous canonical equilibrium governed by the equalization of phase chemical potentials ($μ_Φ= μ_A$). The radial component acts as a non-equilibrium mechanical piston, continuously injecting energy into the tangential modes to sustain the canonical equilibrium -- a mechanism that mathematically formalizes the classical phenomenology of vortex stretching. Spectral evaluations from direct numerical simulation strongly corroborate this thermodynamic framework, establishing the universality of the partition ratios $1:2$ and $1/3 : 2/9 : 4/9$ as a fundamental signature of three-dimensional isotropic turbulence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19458 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Statistical Field Theory for Isotropic Turbulence Farooq, Ahmed Fluid Dynamics This article establishes a first-principles statistical field theory of fully developed isotropic turbulence. Applying an exact Helmholtz decomposition to the local angular momentum field ($\Lvec = \rvec \times \uvec$) reveals a segregation into two orthogonally distinct topological phases: a longitudinal condensate of macroscopic coherent structures ($\PhiL$) and a volume-filling, transverse thermal bath ($\AL$). Constructing a Hamiltonian and evaluating the partition function of these decoupled fields demonstrates that their ergodic exploration of phase space is topologically quantized, mandating a strict $1:2$ equipartition of degrees of freedom. Inverting this topological projection back to the velocity domain isolates the radial velocity field ($\uvec_r$) (which strictly resides in the null space of the $\Lvec$ framework) revealing a recursive partitioning scheme across the cascade into a precise $1/3 : 2/9 : 4/9$ fractional hierarchy. This geometric constraint forces the turbulent steady state into a rigorous canonical equilibrium governed by the equalization of phase chemical potentials ($μ_Φ= μ_A$). The radial component acts as a non-equilibrium mechanical piston, continuously injecting energy into the tangential modes to sustain the canonical equilibrium -- a mechanism that mathematically formalizes the classical phenomenology of vortex stretching. Spectral evaluations from direct numerical simulation strongly corroborate this thermodynamic framework, establishing the universality of the partition ratios $1:2$ and $1/3 : 2/9 : 4/9$ as a fundamental signature of three-dimensional isotropic turbulence. |
| title | A Statistical Field Theory for Isotropic Turbulence |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2604.19458 |