Closing the gap to convergence of gravitoturbulence in local simulations
  Aims. Our goal is to find a converged cooling limit for fragmentation in
self-gravitating disks. This is especially interesting for the formation of
planets, brown dwarfs or stars and the growth of black holes. While
investigating the limit, we want to give a clear criterion for the state of
convergence.
  Methods. We run two-dimensional shearingsheet simulations with the
hydrodynamic package Fosite at high resolutions. Thereby resolution and
limiters are altered. Subsequently, we investigate the spectra of important
physical quantities at the length scales where fragmentation occurs. In order
to avoid prompt fragmentation at high resolutions we start these simulations
with a fully developed gravitoturbulent state obtained at a lower resolution.
  Results. We show nearly converged results for fragmentation with a critical
cooling timescale $t_{\mathrm{crit}} \sim 10\,\Omega^{-1}$ . We can backtrace
this claim by investigating the spectra of relevant physical variables at
length scales around and below the pressure scale height. We argue that well
behaved results cannot be expected if counteracting quantities are varying too
much on these critical length scales, either by change of resolution or
numerical method. A comparison of fragmentation behaviour with the related
spectra reveals that simulations behave similar, if the spectra are converged
to the length scales where self-gravity leads to instabilities. Observable
deviations in the results obtained with different numerical setup are confined
to scales below these critical length scales.
