Kolmogorov-Kraichnan Theory

Both laws describe how energy moves (cascades) between different scales in turbulent fluids.

isotropic: the fluids properties are uniform in all directions homogenous: the fluids properties are uniform at each point of it

The Kolmogorov Law

Applies to an incompressible fluid in three dimensions.

$$E(k)\sim {ϵ}^{2/3}{k}^{-5/3}$$

• As $k$ increases (smaller eddies), energy decreases as $k^{-5/3}$

Assumptions

1. Turbulence is statistically homogenous and isotropic
2. Energy is injected at large scales and dissipated at small scales
3. There exists an intermediate region the inertial range where the rate of energy transfer per unit mass, $ϵ$ is constant

The Kraichnan Law

Applies to two-dimensional systems and shows, that two-dimensional turbulence produces two cascades simultaneously.

Inverse energy cascade: (energy flows to larger scales → structures grow)

$$E(k)\sim C\_E{ϵ}^{2/3}{k}^{-5/3}$$

Direct enstrophy cascade: (enstrophy = vorticity²/2, flows to smaller scales)

$$E(k)\sim C\_\Omega {\mu }^{2/3}{k}^{-3}$$

• $\mu$ is the enstrophy flux (rate of its transfer)