  In this article, we aim to study the stability and dynamic transition of an
electrically conducting fluid in the presence of an external uniform horizontal
magnetic field and rotation based on a Boussinesq approximation model. By
analyzing the spectrum of the linear part of the model and verifying the
validity of the principle of exchange of stability, we take a hybrid approach
combining theoretical analysis with numerical computation to study the
transition from a simple real eigenvalue, a pair of complex conjugate
eigenvalues and a real eigenvalue of multiplicity two, respectively. The center
manifold reduction theory is applied to reduce the infinite dimensional system
to the corresponding finite dimensional one together with one or several
non-dimensional transition numbers that determine the dynamic transition types.
Careful numerical computations are performed to determine these transition
numbers as well as related temporal and flow patterns etc. Our results indicate
that both continuous and jump transitions can occur at certain parameter
region.
