The Modified Cumulant Expansion For Two-Dimensional Isotropic Turbulence

Tomomasa Tatsumi, Shinichiro Yanase

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k-3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

Original languageEnglish
Pages (from-to)475-496
Number of pages22
JournalJournal of Fluid Mechanics
Publication statusPublished - Sept 1981
Externally publishedYes

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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