Energy spectrum is the energy distributed in the wavenumber space $E(k)=\frac{dE}{dk}$.
$dE=E(k)dk$ is the energy of a particular wavenumber and $\int E(k)dk$ is the total energy.
Energy self-similarly cascades through the series of scales known as the inertial range.
A turbulent flow is composed by "eddies" of different sizes. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy
Cascade means that the energy is being transferred from one scale to another without dissipation.
The characteristic velocity on scale $l$ is $u_l$.
Assuming the energy cascade rate $\varepsilon=\frac{u_l^2}{t_c}$ and the cascading timescale $t_c$ is a dynamic time $l/u_l$, we get
\[ \varepsilon \sim u_l^3/l , \\ u_l \sim (\varepsilon l)^{1/3} \sim \varepsilon^{1/3}k^{-1/3} .\]
Since $E(k)k \sim u_l^2$, we have
\[ E(k) \sim \varepsilon^{2/3}k^{-5/3} .\]
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