  The aim of the present work is to derive rigorous estimates for turbulent MHD
flow quantities such as the size and anisotropy of the dissipative scales, as
well as the transition between 2D and 3D state. To this end, we calculate an
upper bound for the attractor dimension of the motion equations, which
indicates the number of modes present in the fully developed flow. This method
has already been used successfully to derive such estimates for 2D and 3D
hydrodynamic turbulence as a function of the $\mathcal L_\infty$ norm of the
dissipation, as in \cite{doering95}. We tackle here the problem of a flow
periodic in the 3 spatial directions (spatial period $2\pi L$), to which a
permanent magnetic field is applied. In addition, the detailed study of the
dissipation operator provides more indications about the structure of the flow.
In section 2, we review the tools of the dynamical system theory as well as the
results they have led to in the case of 3D hydrodynamic turbulence. Section 3
is devoted to the study of the set of modes which minimises the trace of the
operator associated to the total dissipation in MHD turbulence (viscous and
Joule). Eventually, the estimates for the attractor dimension and dissipative
scales in MHD turbulence under strong magnetic field are derived in section 4
and compared to results obtained from heuristic considerations.
