IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 5, OCTOBER 2004 1953 Asymmetry in Fast Z-Pinches With Thin Liners Don S. Lemons and Rickey J. Faehl, Member, IEEE Abstract—We use a well-known, two-dimensional solution to Laplace’s equation for the vector potential between two perfectly conducting, individually axisymmetric but mutually eccentric, current carrying cylinders to model the geometry and time evo- lution of an asymmetric Z-pinch. Cylinder eccentricity correlates with an azimuthal variation in the axial current, the magnetic field, and the force on the liner. The asymmetric force sums to a net force tending to restore the inner cylinder to concentricity. Complete pinch compression and concentricity are achieved simultaneously when the initial radius of the inner cylinder is about 2/3 the radius of the outer return current cylinder or, equiva- lently, when the initial liner inductance per unit length is about nH cm. Compressing the liner onto a finite-sized cylindrical target boosts this critical ratio only up to . Recent and planned liner compression experiments are evaluated according to these criteria. Index Terms—Magnetic compression, Z-pinch, Z-pinch liner. I. INTRODUCTION Z -pinches are designed to compress conducting material onto the pinch axis with an inwardly directed force. Fast Z-pinches [1] do so on short time scales compared to the time required to establish an equilibrium between the compres- sive magnetic pressure and a resisting thermal pressure. Indeed, fast Z-pinches may generate no effective resisting pressure be- cause the conducting material consists of wire arrays or thin- walled, metallic liners. For instance, the Atlas capacitor bank delivers mega-amps of current and terawatts of power in mi- croseconds in order to compress and place under extreme pres- sure centimeter-sized thin conducting liners and the material they surround [2]. Most pulsed power driven Z-pinch applications depend upon efficiently delivering high energy to the Z-pinch target. The most natural way of maximizing capacitor bank energy on target and minimizing the energy lost in creating useless magnetic field is to provide a cylindrically symmetric return current path around the Z-pinch liner that effectively confines the magnetic field to a relatively small annular region between the inner Z-pinch liner and the outer return current conducting cylinder. Efficiency in this regard, though, can, if the two cylinders are not perfectly concentric, lead to an unwanted interaction between the inner liner and the outer return current conductors. In this paper we analyze a simple, and, we believe, widely applicable, model of this liner—return current interaction. Our Manuscript received February 5, 2004. D. S. Lemons is with Department of Physics, Bethel College, North Newton, KS 67117 USA (e-mail: dlemons@lanl.gov). R. J. Faehl is with the Plasma Physics Group, Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545 USA. Digital Object Identifier 10.1109/TPS.2004.835965 model consists of two hollow, perfectly conducting, not nec- essarily concentric, current-carrying cylinders: an inner liner cylinder and an outer return current cylinder. The analysis re- veals the structure of the magnetostatic field between the liner and return current cylinder and a simple relation between the mutual cylinder eccentricity and the asymmetry in the current distributions. Within the context of perfectly conducting cylin- ders either kind of asymmetry, spatial and electrical, immedi- ately leads to the other. These relations can be used to initialize two-dimensional (2-D) codes that simulate liner physics. We find that cylinder eccentricity leads to a net force that pushes the liner back toward concentricity. Since the outer return current cylinder, in typical pulsed power applications, is larger, heavier, and experiences less electromagnetic pressure than the inner liner cylinder, we chose, in the next stage of our analysis, to consider the outer cylinder as fixed in place and allow the inner liner cylinder to move under the influence of an asymmetric force. In principle, the return current cylinder can also move and does so in isentropic, i.e., shock-free com- pression experiments [3], [4]. We investigate liner motion under the assumption that the liner remains cylindrical—an assumption most appropriate when the liner eccentricity, and consequent stresses, are small. In this case, this force causes the liner to oscillate around mutual cylinder concentricity with a frequency that depends only upon the ratio of the radii of the liner and return current cylinders, respectively and . We find that an initially stationary, slightly eccentric, liner can be made to oscillate back to concentricity in a time equal to the time required to pinch the liner onto its axis if the cylinders are initialized with a critical ratio of cylinder radii —a result that is independent of liner mass, current mag- nitude, and even whether or not the driving current evolves in time. Compressing the liner onto a finite-sized cylindrical target boosts this critical ratio only up to . Smaller than critical values of cause compression to occur before the liner returns to the common axis, while larger than critical values cause compression to occur after the liner has os- cillated past the common axis. These results suggest that the initial ratio of cylinder radii is an important design parameter for Z-pinch driven liner experiments. II. MAGNETOSTATIC FIELD Fig. 1 shows two individually axisymmetric, perfectly con- ducting, current-carrying, thin, hollow cylinders with axes par- allel to the axis and their centers displaced from one another in such a way that the outer cylinder still completely surrounds the inner cylinder. A direct current of magnitude flows out of the page (in the direction) on the inner cylinder and a re- turn current of the same magnitude flows into the page (in the 0093-3813/04$20.00 © 2004 IEEE