  The saturated state of turbulence driven by the ion-temperature-gradient
instability is investigated using a two-dimensional long-wavelength fluid model
that describes the perturbed electrostatic potential and perturbed ion
temperature in a magnetic field with constant curvature (a $Z$-pinch) and an
equilibrium temperature gradient. Numerical simulations reveal a well-defined
transition between a finite-amplitude saturated state dominated by strong
zonal-flow and zonal-temperature perturbations, and a blow-up state that fails
to saturate on a box-independent scale. We argue that this transition is
equivalent to the Dimits transition from a low-transport to a high-transport
state seen in gyrokinetic numerical simulations. A quasi-static staircase-like
structure of the temperature gradient intertwined with zonal flows, which have
patch-wise constant shear, emerges near the Dimits threshold. The turbulent
heat flux in the low-collisionality near-marginal state is dominated by
turbulent bursts, triggered by coherent long-lived structures closely
resembling those found in gyrokinetic simulations with imposed equilibrium flow
shear. The break up of the low-transport Dimits regime is linked to a
competition between the two different sources of poloidal momentum in the
system -- the Reynolds stress and the advection of the diamagnetic flow by the
$\boldsymbol{E}\times\boldsymbol{B}$ flow. By analysing the linear ITG modes,
we obtain a semi-analytic model for the Dimits threshold at large
collisionality.
