Influence of numerical noise on surface quasi-geostrophic turbulence

In this paper, the surface quasi-geostrophic (SQG) turbulence with a spatially periodic initial condition is numerically simulated by means of two methods: one is the traditional direct numerical simulation (DNS) whose numerical noise quickly increases to the same order of magnitude as its physical solution, the other is the so-called clean numerical simulation (CNS) whose numerical noise is much smaller than the physical solution and thus is negligible in a finite but long enough interval of time. Influence of numerical noises is investigated in detail by comparing the two simulations of the SQG turbulence given, respectively, by the DNS and CNS. It is found that the artificial stochastic numerical noises would lead to huge derivations apart from CNS benchmark solution not only in the spatiotemporal trajectories but also in the spatial symmetry of turbulence and many statistical results. However, it is surprisingly found that, despite so many huge deviations mentioned earlier, the kinetic energy spectra given by the DNS agrees well with that given by the CNS benchmark solution: both of them satisfy the famous −5/3 law. It indicates that the kinetic energy spectra have stability, i.e., insensitive to stochastic disturbances. On one side, this is a good new. Unfortunately, on the contrast, it also provides us an evidence that correctly capturing the −5/3 law of kinetic energy spectra cannot guarantee the correction of statistical results of DNS. In other words, a correct kinetic energy spectra alone cannot serve as a reliable criterion for assessing the correctness of numerical solutions of turbulence.