An understanding of fully developed nonlinear MHD phenomena becomes increasingly essential for the interpretation of magnetospheric and interplanetary observations. In a series of four papers we shall study the nonlinear, self‐similar evolution of a cylindrical magnetic flux tube with two components of the magnetic field, axial (Bz) and azimuthal (Bϕ). In this first paper we restrict ourselves to the case of a plasma of low beta. (Subsequent papers deal with finite beta effects, with dissipation, and with a data example.) Introducing a special class of configurations we call “separable fields,” we reduce the problem to an ordinary differential equation. Two cases are to be distinguished: (1) when the total field minimizes on the symmetry axis, the magnetic configuration inexorably collapses, and (2) when, on the other hand, the total field maximizes on the symmetry axis, the magnetic configuration behaves analogously to a nonlinear oscillator. Here we focus on the latter case. The effective potential of the motion contains two terms: a strong, repulsive term associated with the gradient of Bz²/8π and a weak, restoring term associated with the pinch. We solve the nonlinear differential equation of motion numerically and find that the period of oscillations grows exponentially with the energy of the oscillator. Our treatment emphasizes the role of the force‐free configuration as the lowest potential energy state about which the system oscillates.
