The Kolmogorov 5/3 law says that in turbulent flow, the energy $E(k)$ contained in a certain wave number k is proportionate to $k^{-\frac{5}{3}}$. How should one interprete this result in physical terms?

One way I considered is this. Energy is proportionate to velocity squared, the velocity of the fluid flow in the mode with wave number $k$ is proportionate to $k^{-\frac{5}{6}}$. Hence I can hope to think of this exponent as a measure of the fractal dimension of the velocity profile for turbulent flow.

Does this interpretation make sense or am I misguided?

Edit: My question is not so much of how one can justify the physical mechanism that results in the law, as the answer to the other question addresses. I am more interested in what it means to design an experiment that verifies this law.