J. Fluid Mech. (2015), vol. 767, pp. 467–496. c  Cambridge University Press 2015 doi:10.1017/jfm.2015.26 467 Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades B. H. Burgess 1, †, R. K. Scott 2 and T. G. Shepherd 3 1 Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada 2 School of Mathematics and Statistics, University of St. Andrews, St. Andrews, Fife KY16 9SS, UK 3 Department of Meteorology, University of Reading, Reading, Berkshire RG6 6BB, UK (Received 30 May 2014; revised 29 November 2014; accepted 11 January 2015; first published online 18 February 2015) We study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids (α-turbulence models) simulated at resolution 8192 2 . We consider α = 1 (surface quasigeostrophic flow), α = 2 (2D Euler flow) and α = 3. The forcing scale is well resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both α = 1 and α = 2. The active scalar field for α = 3 contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction −(7 − α)/3 in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One- and two-point p.d.f.s, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for α = 1 and α = 2, while the α = 3 inverse cascade is much closer to Gaussian and non-intermittent. For α = 3 the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling E (k) ∝ k −2 (α = 1) and E (k) ∝ k −5/3 (α = 2) in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (α = 1 and α = 2) and non-realizability (α = 3) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and non-intermittent statistics for α = 1 and α = 2. Key words: intermittency, isotropic turbulence, turbulence simulation 1. Introduction The dual conservation of kinetic energy (KE) and enstrophy in two-dimensional (2D) incompressible Euler flow gives rise to an inverse cascade in which KE moves to large scales, while enstrophy moves to small scales in a direct cascade (Kraichnan 1967; Batchelor 1969). Two-dimensional Euler flow belongs to a family of fluid models referred to as generalized 2D fluids, also known as α-turbulence models † Email address for correspondence: belhburgess@physics.utoronto.ca