  We evaluate the suppression of $J/\psi$ production in an equilibrating quark
gluon plasma for two competing mechanisms: Debye screening of colour
interaction and dissociation due to energetic gluons. Results are obtained for
$S+S$ and $Au+Au$ collisions at RHIC and LHC energies. At RHIC energies the
gluonic dissociation of the charmonium is found to be equally important for
both the systems while the screening of the interaction plays a significant
role only for the larger systems.
  At LHC energies the Debye mechanism is found to dominate for both the
systems. While considering the suppression of directly produced $\Upsilon$ at
LHC energies, we find that only the gluonic dissociation mechanism comes into
play for the initial conditions taken from the self screened parton cascade
model in these studies. Thus we find that a systematic study of quarkonium
suppression for systems of varying dimensions can help identify the source and
the extent of the suppression.
