  Context. An essential facet of turbulence is the space-time intermittency of
the cascade of energy that leads to coherent structures of high dissipation.
Aims. In this work, we attempt to investigate systematically the physical
nature of the intense dissipation regions in decaying isothermal
magnetohydrodynamical (MHD) turbulence. Methods. We probe the turbulent
dissipation with grid based simulations of compressible isothermal decaying MHD
turbulence. We take unprecedented care at resolving and controlling
dissipation: we design methods to locally recover the dissipation due to the
numerical scheme. We locally investigate the geometry of the gradients of the
fluid state variables. We develop a method to assess the physical nature of the
largest gradients in simulations and to estimate their travelling velocity.
Finally we investigate their statistics. Results. We find that intense
dissipation regions mainly correspond to sheets: locally, density, velocity and
magnetic fields vary primarily across one direction. We identify these highly
dissipative regions as fast/slow shocks or Alfv{\'e}n discontinuities (Parker
sheets or rotational discontinuities). On these structures, we find the main
deviation from 1D planar steady-state is mass loss in the plane of the
structure. We investigate the effect of initial conditions which yield
different imprints at early time on the relative distributions between these
four categories. However, these differences fade out after about one turnover
time, when they become dominated by weakly compressible Alfv{\'e}n
discontinuities. We show that the magnetic Prandtl number has little influence
on the statistics of these discontinuities, but it controls the Ohmic vs
viscous heating rates within them. Finally, we find the entrance
characteristics of the structures (such as entrance velocity and magnetic
pressure) are strongly correlated. Conclusions. These new methods allow to
consider developed compressible turbulence as a statistical collection of
intense dissipation structures. This can be used to post-process 3D turbulence
with detailed 1D models apt for comparison with observations. It could also
reveal useful as a framework to formulate new dynamical properties of
turbulence.
