Applying static magnetic field can produce flows (thermoelectric magnetohydrodynamic flows, TEMHDF) in the melt by interacting with the thermoelectric currents (TEC) during solidification of metals. A physical model was proposed to interpret how these TEC appear at the solid-liquid interface and verified by a corresponding simulation. The influences of TEMHDF on solidification were investigated through both ex-situ experiments and n situ observations by means of synchrotron X-ray radiography. The 3D numerical simulations of TEMHDF were performed for these two cases, respectively, and suggested that both the change of interface shape with different transverse static magnetic fields demonstrated by the ex-situ experiments and the real time observed interface shape varying under a 0.08 T transverse static magnetic field could attribute to the TEMHDF advanced solid-liquid interface in the static magnetic field-assisted directional solidification. The TEMHDF produced by an axial static magnetic field were also computed along with the interface change predicted based on which is good in line with the published experimental results. This study of TEMHDF and their impacts on the solid-liquid interface shape provides a method to tailor the structure during directional solidification using static magnetic field.