Inverse energy cascade

The scaling laws in both cascades can be obtained from dimensional analysis of Navier-Stokes equation as well as in the three-dimensional case.

For the inverse energy cascade, the assumption of a constant flux of energy $\Pi(\ell) = - \epsilon$ toward large scales reproduces 3d-like scaling laws for velocities and characteristic times:

$$u_{\ell} \sim \epsilon^{1/3} \ell^{1/3}$$ (1.66)

$$\tau_{\ell} \sim \epsilon^{-1/3} \ell^{2/3}$$ (1.67)

This means that the velocity field in the inverse cascade is rough, with scaling exponent $h=1/3$, exactly as in the three-dimensional case. The prediction for the energy spectrum reads

$$E(k) = C {\epsilon}^{2/3} k^{-5/3}$$ (1.68)

Figure 1.4: Energy spectrum of the inverse energy cascade $E(k) \sim k^{-5/3}$. In the scaling range the energy flux (shown in the inset) is constant and negative.

The hypothesis of locality of triadic interactions in the inverse cascade is consistent with the $k^{-3/5}$ spectrum. The transfer is associated with the distortion of the velocity field by its own shear. The effective shear at given wavenumber $k$ is expected to be negligibly affected by wave-numbers $\ll k$ because the integral $\int_0^{\infty} k^2 E(k) dk$, which measure the mean-square shear, converges at $k = 0$. Also the contribution by high wavenumber $\gg k$ is negligible because vorticity associated with those wave-numbers fluctuates rapidly in space and times and gives a small effective shear across distances of order $k^{-1}$.

In absence of a large-scale sink of energy the inverse cascade can only be quasi-steady because the peak $k_E$ of the energy spectrum keeps moving down to ever-lower wave-numbers as

$$k_E(t) \sim {\epsilon}^{-1/2} t^{-3/2}$$ (1.69)

while the total energy grows linearly in time $E(t) = \epsilon t$. If the input of energy continues for a sufficiently long time, the cascade can eventually reach the integral scale and energy begins to accumulate in the lowest mode, a phenomenon which is the analogous of Bose-Einstein condensation [20] of a finite two-dimensional quantum gas. This pile up of energy can produce a large scale spectrum steeper than $k^{-3}$ which violates the hypothesis of locality of interactions.

The presence of friction stops the energy cascade at wavenumber

$$k_E \sim {\epsilon}^{-1/2} \alpha^{3/2}$$ (1.70)

where the energy dissipation balances the energy transfer $2 \alpha E_{eq} = \epsilon$.

In two-dimensional turbulence it is possible to demonstrate the analogous of the Kolmogorov's four-fifths law. In the limit of infinite Reynolds number the third order (longitudinal) structure function of two-dimensional homogeneous isotropic turbulence, evaluated for increments $\ell$ small compared to the integral scale, and larger than the forcing correlation length, is given in terms of the mean energy flux per unit mass $\epsilon$ by

$$S_3(\ell) \equiv \langle (\delta u_{\ell})^3 \rangle = {3 \over 2} \epsilon \ell$$ (1.71)

Together with the scaling hypothesis for the structure functions $S_P(\ell) \sim \ell^{\zeta_p}$ the three-half law allows to obtain the equivalent of $K41$ theory for the inverse energy cascade in two-dimensional turbulence.

At variance with the three-dimensional case, where the dimensional prediction for the scaling exponents is modified by the presence of small scale intermittency, the statistics of velocity fluctuations at different scale $\ell$ in the scaling range of the inverse energy cascade are found to be roughly self-similar [21], with small deviations from gaussianity.