We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45 degrees. This "skew-Newtonian" stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.

Solving Navier-Stokes equations using monotone algorithms often produces results that are consistent with the dynamics of high Reynolds number turbulence despite formally insufficient numerical resolution to capture all physically relevant scales of motion. It is frequently argued that the truncation error of a numerical scheme provides a mechanism for the energy dissipation and serves as an implicit subgrid scale model. Such an approach to turbulence modeling, known loosely as the Monotonically Integrated LES (MILES), was originally proposed by Boris et al. (1992) and reviewed recently by Grinstein and Fureby (2002). The primary effect of a numerical discretization in MILES, which is similar to the effects of explicit SGS models, is the dissipation of turbulent kinetic energy. Despite its importance, very little is known about details of numerical dissipation in MILES methods and, more importantly, its relation (or lack of it) to the actual dissipative effects of turbulence. We describe a method for computing effective numerical eddy viscosity, useful for assessing numerical dissipation of such approaches. The method is evaluated on an example of a specific nonoscillatory finite volume scheme MPDATA developed for simulations of geophysical flows. A series of numerical simulations of homogeneous, isotropic turbulence are performed and the numerical eddy viscosities are determined. The detailed quantitative comparisons are made between the numerical eddy viscosities and the theoretical eddy viscosity as well as intrinsic eddy viscosities computed exactly from the velocity fields by introducing artificial wave number cutoffs. A typical example of such a comparison is shown in the enclosed figure.

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We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482 resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales as k−5/3, consistent with the predictions of Kraichnan (Phys. Fluids, vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525). The mean flux across a length scale ℓ was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink (J. Fluid Mech., vol. 549, 2006a, p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.