  The comprehension of stratified flows is important for geophysical and
astrophysical applications. The Weak Wave Turbulence theory aims to provide a
statistical description of internal gravity waves propagating in the bulk of
such flows. However, internal gravity waves are usually perturbed by other
structures present in stratified flow, namely the shear modes and the vortical
modes. In order to check whether a weak internal gravity wave turbulence regime
can occur, we perform direct numerical simulations of stratified turbulence
without shear modes, and with or without vortical modes at various Froude and
buoyancy Reynolds numbers. We observe that removing vortical modes naturally
helps to have a better overall balance between poloidal kinetic energy,
involved in internal gravity waves, and potential energy. However, conversion
between kinetic energy and potential energy does not necessarily show
fluctuations around zero in our simulations, as we would expect for a system of
weak waves. A spatiotemporal analysis reveals that removing vortical modes
helps to concentrate the energy around the wave frequency, but it is not enough
to observe a weak wave turbulence regime. Yet, we observe that internal gravity
waves whose frequency are large compared to the eddy turnover time are present,
and we also find evidences for slow internal gravity waves interacting by
Triadic Resonance Instabilities in our strongly stratified flows simulations.
Finally, we propose conditions that should be fulfilled in order to observe a
weak internal gravity waves turbulence regime in real flows.
