Two-dimensional turbulence has both kinetic energy and mean-square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k)∼ε2/3k−5/3 and E(k)∼η2/3k−3, where ε is the rate of cascade of kinetic energy per unit mass, η is the rate of cascade of mean-square vorticity, and the kinetic energy per unit mass is ∫0∞E(k) dk. The −53 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero-vorticity flow. The −3 range gives an upward vorticity flow and zero-energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber interactions. The formal −3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers ∼ki and the Reynolds number is large, it is conjectured that a quasi-steady-state results with a −53 range for k « ki and a −3 range for k » ki, up to the viscous cutoff. The total kinetic energy increases steadily with time as the −53 range pushes to ever-lower k, until scales the size of the entire fluid are strongly excited. The rate of energy dissipation by viscosity decreases to zero if kinematic viscosity is decreased to zero with other parameters unchanged.

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Batchelor's theory of the turbulent straining of small-spatial-scale amplitude variations of a convected scalar field is re-examined to see the effects of fluctuation of the rates of strain in space and time. The k−1 viscous-convective-range spectrum is unaltered, except for the constant of proportionality, but spectrum level in the viscous-diffusive range displays a sensitivity to fluctuations which increases with wavenumber. The Gaussian cutoff found by Batchelor is replaced by more gently decreasing dependences of spectrum level on wavenumber. The scalar spectrum is also treated by the Lagrangian-history direct-interaction approximation. The k−5/3 inertial-convective range of Obukhov and Corrsin, the k−1 viscous-convective range, and the k−17/3 inertial-diffusive range of Batchelor, Howells, and Townsend all are recovered. For a given rate of spectral transport of scalar variance, the predicted spectrum levels in the k−5/3 and k−1 ranges are too small, in comparison with experiment, by numerical factors in the neighborhood of one to three. The largest error is for the k−1 range at large Reynolds number.

In physical systems, a reduction in dimensionality often leads to exciting new phenomena. Here we discuss the novel effects arising from the consideration of fluid turbulence confined to two spatial dimensions. The additional conservation constraint on squared vorticity relative to three-dimensional (3D) turbulence leads to the dual-cascade scenario of Kraichnan and Batchelor with an inverse energy cascade to larger scales and a direct enstrophy cascade to smaller scales. Specific theoretical predictions of spectra, structure functions, probability distributions, and mechanisms are presented, and major experimental and numerical comparisons are reviewed. The introduction of 3D perturbations does not destroy the main features of the cascade picture, implying that 2D turbulence phenomenology establishes the general picture of turbulent fluid flows when one spatial direction is heavily constrained by geometry or by applied body forces. Such flows are common in geophysical and planetary contexts, are beautiful to observe, and reflect the impact of dimensionality on fluid turbulence.