Kraichnan's inverse cascade in two-dimensional turbulence

Builds on Turbulence Energy cascade Kolmogorov 1941 theory Euler equations

Statement

Consider statistically stationary, homogeneous and isotropic turbulence of an incompressible fluid in the plane. Assume that energy is injected by a forcing that acts in a narrow band of wavenumbers around $k_f$ . Assume that energy is removed only at scales much larger than $1/k_f$ , and enstrophy only at scales much smaller than $1/k_f$ . Let $E(k)$ be the energy spectrum, $\varepsilon$ the rate of energy injection and $\eta$ the rate of enstrophy injection. Kraichnan's theory predicts two inertial ranges.

For $k \ll k_f$ , energy flows at the constant rate $\varepsilon$ towards small wavenumbers, that is, towards large scales, and $E(k) = C \varepsilon^{2/3} k^{-5/3}$ .
For $k \gg k_f$ , enstrophy flows at the constant rate $\eta$ towards large wavenumbers, and $E(k) = C' \eta^{2/3} k^{-3}$ .

Here $C$ and $C'$ are dimensionless constants.

Remarks

The prediction rests on two invariants of the two-dimensional Euler equations: the energy $\int E(k)\, dk$ and the enstrophy $\int k^2 E(k)\, dk$ . An argument of Fjørtoft treats energy at $k_f$ that moves to two other wavenumbers while both invariants are conserved. It shows that most of the energy must go to the smaller wavenumber and most of the enstrophy to the larger one. The two spectra then follow from dimensional analysis, as in the Kolmogorov 1941 theory, once $\varepsilon$ or $\eta$ is the only relevant parameter in the range. The $k^{-3}$ law was later given a logarithmic correction. The energy cascade in three dimensions goes the other way, from large to small scales, because enstrophy is not conserved there. Numerical simulations and soap film experiments show the inverse cascade, with energy piling up at the largest scale of the domain unless friction removes it.

Going deeper

For a student

The theory is the two-dimensional analogue of Kolmogorov's theory, with one cascade for each inviscid invariant. The slopes $-5/3$ and $-3$ come from dimensional analysis once one knows which flux is constant in each range. The double cascade is often called the Kraichnan–Leith–Batchelor picture. It guides the reading of large-scale atmospheric and oceanic spectra and the design of two-dimensional simulations.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Energy cascade concept
Euler equations concept
Kolmogorov microscales concept
Turbulence concept
Viscosity concept
Kolmogorov 1941 theory statement
Kraichnan's inverse cascade in two-dimensional turbulence statement

Tag	`B32T5H7ZNP`
URI	`m2://B32T5H7ZNP`
Kind	statement, fact
Writing	W0 stub
MSC	76F05, 76F02
Status	This copy of the site has no build data. Run `m2 status B32T5H7ZNP`.
About	Energy cascade
Uses	Turbulence, Energy cascade, Kolmogorov 1941 theory, Euler equations
Changed	3 November 2026