---
slug: paper-kolmogorov-a-n-1941-dissipation-of-energy-in-the-locally-isotropic-turbulence
title: Kolmogorov 1941: Dissipation of Energy in Locally Isotropic Turbulence
canonical: https://miscsubjects.com/a/paper-kolmogorov-a-n-1941-dissipation-of-energy-in-the-locally-isotropic-turbulence
---

## What Kolmogorov saw

A. N. Kolmogorov examined incompressible fluid turbulence at very high Reynolds numbers. He isolated a regime of local isotropy inside small domains far from boundaries. In that regime the statistical laws of velocity differences depend only on distance r, the mean energy dissipation rate per unit mass ε, and viscosity ν.

## Core results

The paper derives an exact relation from the Navier-Stokes equations under local isotropy. For the third-order longitudinal structure function it obtains Bddd(r) = −(4/5)εr inside the inertial range where viscosity effects are negligible. This is the 4/5 law. It also recovers the earlier 2/3 law for second-order moments from similarity assumptions.

## Exact primary passages

From the 1941c paper (English translation, Proc. R. Soc. Lond. A 434, 1991, pp. 15–17):

“For the turbulence in an incompressible fluid we have the equation… 4dBddd/dr + 6ν d²Bdd/dr² = −(4/5)ε r” (equation 5, after integration and boundary conditions at r = 0).

“For large r … it is natural to assume that … Bddd(r) = −(4/5)ε r” (equation 7).

The constant C appears in the skewness: Bddd(r) = C [Bdd(r)]^{3/2} with C = (−4/5)^{1/3} under the constant-skewness assumption.

The companion 1941a paper (same volume, pp. 9–13) defines local isotropy and states the two similarity hypotheses that close the scaling.

## Convergence patterns touched

The 4/5 law and local isotropy supply a mechanistic account of bounded chaos and scale invariance. Energy injected at large scales cascades through an inertial range whose statistics are independent of viscosity yet terminate at the dissipative Kolmogorov scale η = (ν³/ε)^{1/4}. The structure functions exhibit power-law scaling, a concrete instance of scale invariance arising from energy flow.

## Distance from the full synthesis

The work reaches the flow-to-structure step of the Ladder and demonstrates how reliable energy dissipation produces universal statistical patterns. It stops short of memory, life, or mind. It does not address the Mirror Layer or the reader inside the system.

## Honest limits and disconfirming edges

The derivation assumes strict local isotropy and stationarity inside the chosen domain. Real flows show intermittency and deviations from constant skewness; Kolmogorov himself later modified the theory in 1962. The 4/5 law remains exact under its hypotheses, yet the universality of the similarity functions is an assumption, not a theorem. Reductionist objections note that the equations are deterministic; the statistical closure is an averaging step whose validity must be checked case by case.

## Mechanistic claims

The relation Bddd(r) = −(4/5)εr follows directly from integration of the Navier-Stokes equation under local isotropy (mechanistic, source: Kolmogorov 1941c eq. 7).

Second-order structure functions scale as Bdd(r) ∼ (ε r)^{2/3} inside the inertial range under the second similarity hypothesis (mechanistic under the stated assumptions).

Local isotropy decouples small-scale statistics from large-scale anisotropy when the Reynolds number is sufficiently high (mechanistic, Kolmogorov 1941a).

## Sources

Kolmogorov, A. N. (1941c). Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19–21. English translation in Proc. R. Soc. Lond. A 434 (1991): 15–17.

Kolmogorov, A. N. (1941a). The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303. English translation in Proc. R. Soc. Lond. A 434 (1991): 9–13.
