Kraichnan-Leith-Batchelor similarity theory and two-dimensional inverse cascades

B. H. Burgess, R. K. Scott, T.G. Shepherd

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16 Citations (Scopus)


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 81922. 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)\propto α k-2(α=1) and E(k)\propto α 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.

Original languageEnglish
Pages (from-to)467-496
Number of pages30
JournalJournal of Fluid Mechanics
Early online date18 Feb 2015
Publication statusPublished - Mar 2015


  • Intermittency
  • Isotropic turbulence
  • Turbulence simulation


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