Remarks on the KLB theory of two-dimensional turbulence

We study the inverse energy transfer in forced two-dimensional (2D) Navier--Stokes turbulence in a doubly periodic domain. It is shown that an inverse energy cascade that carries a nonzero fraction of the injected energy to the large scales via a power-law energy spectrum $\propto k^{-\alpha}$ requires that $\alpha\ge5/3$. This result is consistent with the classical theory of 2D turbulence that predicts a $k^{-5/3}$ inverse-cascading range, thus providing for the first time a rigorous basis for this important feature of the theory. We derive bounds for the Kolmogorov constant $C$ in the classical energy spectrum $E(k)=C\epsilon^{2/3}k^{-5/3}$, where $\epsilon$ is the energy injection rate. Issues related to Kraichnan's conjecture of energy condensation and to power-law spectra as the quasi-steady dynamics become steady are discussed.