  We study the two-dimensional (2D) shear flow of amorphous solids within
variants of an elastoplastic model, paying particular attention to spatial
correlations and time fluctuations of, e.g., local stresses. The model is based
on the local alternation between an elastic regime and plastic events during
which the local stress is redistributed. The importance of a fully tensorial
description of the stress and of the inclusion of (coarse-grained) convection
in the model is investigated; scalar and tensorial models yield very similar
results, while convection enhances fluctuations and breaks the spurious
symmetry between the flow and velocity gradient directions, for instance when
shear localisation is observed. Besides, correlation lengths measured with
diverse protocols are discussed. One class of such correlation lengths simply
scale with the spacing between homogeneously distributed, simultaneous plastic
events. This leads to a scaling of the correlation length with the shear rate
as $\dot{\gamma}^{\frac{-1}{2}}$ in 2D in the athermal regime, regardless of
the details of the model. The radius of the cooperative disk, defined as the
near-field region in which plastic events induce a stress redistribution that
is not amenable to a mean-field treatment, notably follows this scaling. On the
other hand, the cooperative volume measured from the four-point stress
susceptibility and its dependence on the system size and the shear rate are
model-dependent.
