  We present a 3D simulation of the dynamic emergence of a twisted magnetic
flux tube from the top layer of the solar convection zone into the solar
atmosphere and corona. It is found that after a brief initial stage of flux
emergence during which the two polarities of the bipolar region become
separated and the tubes intersecting the photosphere become vertical,
significant rotational motion sets in within each polarity. The rotational
motions of the two polarities are found to twist up the inner field lines of
the emerged fields such that they change their orientation into an inverse
configuration (i.e. pointing from the negative polarity to the positive
polarity over the neutral line). As a result, a flux rope with sigmoid-shaped,
dipped core fields form in the corona, and the center of the flux rope rises in
the corona with increasing velocity as the twisting of the flux rope footpoints
continues. The rotational motion in the two polarities is a result of
propagation of non-linear torsional Alfv\'en waves along the flux tube, which
transports significant twist from the tube's interior portion towards its
expanded coronal portion. This is a basic process whereby twisted flux ropes
are developed in the corona with increasing twist and magnetic energy, leading
up to solar eruptions.
