Interplay of kinematic and magnetic forcing in a model of a conducting fluid with randomly driven magnetohydrodynamic equations has been studied in space dimensions d > or =2 by means of the renormalization group. A perturbative expansion scheme, parameters of which are the deviation of the spatial dimension from two and the deviation of the exponent of the powerlike correlation function of random forcing from its critical value, has been used in one-loop approximation. Additional divergences have been taken into account that arise at two dimensions and have been inconsistently treated in earlier investigations of the model. It is shown that in spite of the additional divergences, the kinetic fixed point associated with the Kolmogorov scaling regime remains stable for all space dimensions d > or =2 for rapidly enough falling off correlations of the magnetic forcing. A scaling regime driven by thermal fluctuations of the velocity field has been identified and analyzed. The absence of a scaling regime near two dimensions driven by the fluctuations of the magnetic field has been confirmed. A renormalization scheme has been put forward and numerically investigated to interpolate between the epsilon expansion and the double expansion.
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