Consider stationary and locally isotropic turbulence at high Renumber. A simple model for the energy spectrum can be constructed by defining a sequence of wavenumbers \(k_{n}\), such that

\[ k_{n+1}=2 k_{n} \]

Let the energy contained in the interval \(\left[k_{n}, k_{n+1}\right]\) be \(E(k)\) and the amount of energy transported during the time \(\tau(E, k)\) from \(k_{n}\) to \(k_{n+1}\) be \(\epsilon(k)\). Then compute 

18.1.1 \(\epsilon(k)\) as function of the spectrum \(E(k)\), the wavenumber \(k\) and the time interval \(\tau(E, k)\).

18.1.2 Determine the spectrally local time scale \(\tau(E, k)\) in the inertial subrange of the spectrum using dimensional analysis.

18.1.3 Assuming that the amount of energy fed into the inertial subrange is equal to the amount removed from it, compute the form of the energy spectrum \(E(k)\).