Steepening of the energy spectrum

In absence of friction the requirement of a constant enstrophy flux in the direct cascade allows to obtain a dimensional prediction for the energy spectrum $E(k) = C' \epsilon_{\omega}^{2/3} k^{-3}$ (see Chapter 1). As already shown by Bernard [33] and Nam et al [30], a non-vanishing friction regularizes the flow depleting the formation of small-size structures and results in a steeper spectrum [30]

$$E(k) \sim k^{-3-\xi}$$ (2.5)

In the range $0 < \xi < 2$ the exponent $\xi$ coincides with the scaling exponent $\zeta_2$ of the second-order moment of vorticity fluctuations $S^\omega_2(r)=\langle (\delta_r \omega)^2 \rangle \sim r^{\zeta_2}$. An explicit expression for the correction $\xi$ to the spectral slope will be given in Section. 2.4.

The physical origin of the steepening of the spectrum is clear: part of the enstrophy which is transported to small scales is removed by the friction during the cascade process, thus the amplitude of fluctuations which reach the small scales is reduced.

Since the energy spectrum is steeper than $E(k) \sim k^{-3}$, the second-order velocity structure function $S_2(\ell) = \langle \delta u_{\ell}^2 \rangle$ is dominated by the IR contribution of the spectrum and trivially displays smooth scaling independently of the value of $\xi$. Thus the presence of a non-vanishing friction term ensures that the velocity fields in the direct enstrophy cascade is smooth.