  Via the process of accretion, magnetorotational turbulence removes energy
from a disk's orbital motion and transforms it into heat. Turbulent heating is
far from uniform and is usually concentrated in small regions of intense
dissipation, characterised by abrupt magnetic reconnection and higher
temperatures. These regions are of interest because they might generate
non-thermal emission, in the form of flares and energetic particles, or
thermally process solids in protoplanetary disks. Moreover, the nature of the
dissipation bears on the fundamental dynamics of the magnetorotational
instability (MRI) itself: local simulations indicate that the large-scale
properties of the turbulence (e.g. saturation levels, the stress-pressure
relationship) depend on the short dissipative scales. In this paper we
undertake a numerical study of how the MRI dissipates and the small-scale
dissipative structures it employs to do so. We use the Godunov code RAMSES and
unstratified compressible shearing boxes. Our simulations reveal that
dissipation is concentrated in ribbons of strong magnetic reconnection that are
significantly elongated in azimuth, up to a scale height. Dissipative
structures are hence meso-scale objects, and potentially provide a route by
which large scales and small scales interact. We go on to show how these
ribbons evolve over time --- forming, merging, breaking apart, and
disappearing. Finally, we reveal important couplings between the large-scale
density waves generated by the MRI and the small-scale structures, which may
illuminate the stress-pressure relationship in MRI turbulence.
