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Motivated by the possible utility of studying turbulent astrophysical flows as dynamical systems, solutions to the incompressible two‐dimensional Navier–Stokes equation in Fourier space are investigated employing a small subset of Fourier modes that span a range of nearly 3000:1 in size and that preserve the triadic nature of the nonlinear interactions. These solutions thus preserve the energy and enstrophy conservation properties of the full solutions. The statistical similarity of the reduced solutions to the full solutions is tested by comparing their respective energy spectra in the energy and enstrophy inertial ranges, as well as the energy and enstrophy cascade directions. The behavior of the reduced solutions is consistent with full solutions having similar physical conditions (rates of injection and dissipation of the energy and enstrophy), while providing a computational gain of about a factor ∼1000 for k max∼104 in two dimensions (2D) and ∼106 at k max∼103.5 in 3D. Also, the effect and importance of the forcing on the outcome of the calculations are discussed, namely the slope of the energy spectrum and the duration of the net energy and enstrophy transfer.