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 ∝ k−α requires that α ≥ 5/3. 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) = Cǫ2/3k−5/3, where ǫ 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.