  The z-pinch is a classical steady state for the MHD model, where a confined
plasma fluid is separated by vacuum, in the presence of a magnetic field which
is generated by a prescribed current along the z direction. We develop a
variational framework to study its stability in the absence of viscosity
effect, and demonstrate for the first time that any such a z-pinch is always
unstable. Moreover, we discover a sufficient condition such that the
eigenvalues can be unbounded, which leads to ill-posedness of the linearized
MHD system.
