  The vertical motion of a tokamak plasma is analytically modelled during its
non-linear phase by a free-moving current-carrying rod inductively coupled to a
set of fixed conducting wires or a cylindrical conducting shell. The solutions
capture the leading term in a Taylor expansion of the Green's function for the
interaction between the plasma column and the surrounding vacuum vessel. The
plasma shape and profiles are assumed not to vary during the vertical drifting
phase such that the plasma column behaves as a rigid body. In the limit of
perfectly conducting structures, the plasma is prevented to come in contact
with the wall due to steep effective potential barriers created by the induced
Eddy currents. Resistivity in the wall allows the equilibrium point to drift
towards the vessel on the slow timescale of flux penetration. The initial
exponential motion of the plasma, understood as a resistive vertical
instability, is succeeded by a non-linear "sinking" behaviour shown to be
algebraic and decelerating. The acceleration of the plasma column often
observed in experiments is thus concluded to originate from an early sharing of
toroidal current between the core, the halo plasma and the wall or from the
thermal quench dynamics precipitating loss of plasma current.
