OBSERVATIONS of the solar atmosphere reveal a sharp increase of the temperature, from several thousands degrees in the photosphere (the solar surface) to
several millions degrees in the corona. This amazing increase of temperature is still a mystery. Observations at different wavelengths with space telescopes
like SoHO/ESA or SDO/NASA, show that the magnetic field plays a central role in the coronal heating but the exact mechanism at the origin of
this heating is still not understood. Many magnetic structures are present in the solar corona like coronal loops which may produce solar flares. They are
dynamic and intermittent both in space and time. Statistical studies (e.g. histograms of the total energy released by solar flares) reveal power laws over many
decades which means that the solar corona is structured over a large range of scales (see e.g. Aschwanden et al., ARAA, 2001). A possible explanation of this
absence of characteristic scale is that the corona is in a state of turbulence with (magnetic) Reynolds numbers as large as 1013. Then the coronal
heating would be the result of the energy dissipation by magnetic structures at very small scales, smaller than the smallest structures observed with the current
telescopes.
The energy produced by the sub-photospheric layer is large enough to compensate the radiative loss of the solar corona. Thus, the problem of the coronal heating
is to understand how the energy is transferred into the corona and how it is dissipated mainly at small scales. I tackled this problem during my PhD thesis by
developing a numerical model for coronal loops [3,10,12,18]. It is based on the nonlinear MHD equations which describe the evolution of the velocity and magnetic
field vectors along the loop. This approach allowed me to propose an intermediate description between pure statistical models (cellular automata models) and 3D MHD:
the former are limited by their simplicity (pure mechanical rules) whereas the latter find their limitation in the numerical resources available. In my model
the random motions of the magnetic foot-points are traduced by an external forcing in the MHD equations. These motions lead to the shearing and twisting of the
magnetic field and thus to an accumulation of energy in the loop. Following a nonlinear cascade the energy is eventually evacuated from the system by intermittent
ohmic dissipation at small scales. The ohmic dissipation is a fundamental observable in this type of models since it allows a comparison with emission lines in
particular in UV and X-rays. I showed that the global statistical behavior observed may be reproduced by the MHD model. Extensions to these works were realized
later [29,33,41,50,64].
