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We show that an infinite number of non-unitary minimal models may describe two
dimensional turbulent magnetohydrodynamics (MHD), both in the presence and
absence of the Alf'ven effect. We argue that the existence of a critical
dynamical index results in the Alf'ven effect or equivalently the
equipartition of energy. We show that there are an infinite number of conserved
quantities in $2D-MHD$ turbulent systems both in the limit of vanishing the
viscocities and in force free case. In the force free case,
using the non-unitary minimal model $M_{2,7}$ we derive the correlation
functions for the velocity stream function and magnetic flux function.
Generalising this simple model we find the exponents of the energy spectrum in
the inertial range for a class of conformal field theories.
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