  This article presents the study of the effects of magnetic fields on
non-distributed nuclear burning fronts as a possible solution to a fundamental
problem for the thermonuclear explosion of a Chandrasekhar mass ($M_{Ch}$)
white dwarf (WD), the currently favored scenario for the majority of Type Ia
SNe (SNe~Ia). All existing 3D hydrodynamical simulations predict strong global
mixing of the burning products due to Rayleigh-Taylor (RT) instabilities, which
is in contradiction with observations. As a first step and to study the flame
physics we present a set of computational magneto-hydrodynamic (MHD) models in
rectangular flux tubes, resembling a small inner region of a WD. We consider
initial magnetic fields up to $10^{12}\,\,\mathrm{G}$ of various orientations.
We find an increasing suppression of RT instabilities starting at about
$10^9\,\,\mathrm{G}$. The front speed tends to decrease with increasing
magnitude up to about $10^{11}\,\,\mathrm{G}$. For even higher fields new small
scale finger-like structures develop, which increase the burning speed by a
factor of 3 to 4 above the field-free RT-dominated regime. We suggest that the
new instability may provide sufficiently accelerated energy production during
the distributed burning regime to go over the Chapman-Jougey limit and trigger
a detonation. Finally we discuss the possible origins of high magnetic fields
during the final stage of the progenitor evolution or the explosion.
