  The plasma edge of a tokamak configuration is characterized by turbulent
dynamics leading to enhanced transport. We construct a simplified 3D
Hasegawa--Wakatani model reducing to a single partial differential equation for
the turbulent electric potential dynamics. Simulations demonstrate how the 3D
turbulence relaxes on a 2D axisymmetric profile, corresponding to the so-called
interchange turbulence. The spectral features of this regime are found to be
strongly dependent on the initialization pattern. We outline that the emergence
of axisymmetric turbulence is also achieved when the corresponding mode
amplitude is not initialized. Then, we introduce the symmetries of the magnetic
X-point of a tokamak configuration. We linearize the governing equation by
treating the poloidal field as a small correction. We show that it is not
always possible to solve the electric potential dynamics following a
perturbative approach. This finding, which is due to resonance between the
modes of the background and the poloidal perturbation, confirms that the
X-point symmetries can alter the properties of turbulent transport in the edge
region.
