Perfectly conducting incompressible fluid model of a wire array implosion
Alexander L. Velikovich · Igor V. Sokolov · Andrey A. Esaulov
Abstract and summary · read the original at the source · none found
In one page
A wire-array Z-pinch begins as a cage of fine wires around a common axis. A colossal current explodes each wire into plasma, and the magnetic field then drives the whole cage inward. Whether that cage behaves as many separate columns or as one continuous shell decides how the implosion goes, and Velikovich, Sokolov and Esaulov answer it analytically. Treat each plasma column as an incompressible, perfectly conducting fluid, and the two-dimensional problem of columns that move and change shape collapses into one-dimensional equations written for the boundaries of those columns alone. Two pressures then compete. One drives the array toward the axis as a set of individual columns; the other, which the authors call a tidal pressure, pulls neighbours together into an annular conducting shell before anything converges. Which one wins is set by a single number, the gap between wires measured against their diameter. The model also names its own limit: it cannot produce the precursor plasma streams the experiments see.
Why it matters hereChapter 9 is about plasma that organises itself into structures that hold together and carry current, and this paper gives the arithmetic for when a ring of separate current-carrying columns stops being separate and becomes a single shell. Chapter 12 depends on the same machines, because wire arrays are the drivers behind the largest laboratory X-ray sources built for fusion, and the array’s geometry is the part the designer actually chooses.
What it claims
01An incompressible, perfectly conducting magnetohydrodynamic model can describe a multiwire array implosion in the plane transverse to the axis, using the theory of analytic functions: the plasma columns left by the electrical explosion of the individual wires move and change the shape of their cross section in the magnetic field produced by the currents flowing on the surfaces of the columns and closing through a cylindrical return-current can.Abstract, opening sentences
Published and peer-reviewed02The model solves the field geometry for an arbitrary array. The geometry of both the global and the private magnetic fields, and the self-consistent distributions of electric current on the conducting surfaces, are determined for any wire array configuration — including nested wire arrays and wires placed close to the return-current can.Abstract
Published and peer-reviewed03The problem reduces in dimension. The coupled equations of motion and magnetostatics for what is essentially a two-dimensional problem are reduced to one-dimensional parametric governing equations, written for the boundary of the fluid contours rather than for their interiors.Abstract
Published and peer-reviewed04The implosion is a competition between two pressures: an implosion pressure that makes the array converge to the axis as a set of individual plasma columns, and a tidal pressure that makes the wires merge and form an annular conducting shell.Abstract
Published and peer-reviewed05One ratio decides which pressure wins — the gap-to-diameter ratio, pi times the array radius divided by the number of wires times the wire radius, both taken as functions of time. If that ratio is large at early time the array implodes as a set of individual plasma columns; when it falls to about pi or less the tidal forces prevail and the columns tend to form a shell-like configuration before they start converging on the axis.Abstract
Published and peer-reviewed06The model states its own boundary. It does not allow the precursor plasma streams to be ejected from the wires toward the axis, which the authors read as an indication that this process is governed by the finite plasma conductivity and could only be described with a proper conductivity model.Abstract, closing sentences
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Read it · abstract
Abstract
An incompressible perfectly conducting magnetohydrodynamic model is applied to describe a multiwire array implosion on the (r,θ) plane using the theory of analytic functions. The plasma columns emerging from the electrical explosion of individual wires move and change the shape of their cross section in the magnetic field produced by the currents flowing on the surfaces of the columns and closing through a cylindrical return current can. Geometry of both the “global” and “private” magnetic fields and self-consistent distributions of the electric currents on the conducting surfaces are determined for any wire array configuration including nested wire arrays, wires close to the return current can, etc. The coupled equations of motion and magnetostatics for an essentially two-dimensional problem are reduced to one-dimensional parametric governing equations, written for the boundary of the fluid contours. The implosion dynamics is shown to be driven by a competition between the implosion pressure, making the array converge to the axis as a set of individual plasma columns, and the tidal pressure that makes the wires merge, forming an annular conducting shell. Their relative roles are determined by the gap-to-diameter ratio πRc(t)/NRw(t). If this ratio is large at early time, then the array implodes as a set of individual plasma columns. Otherwise, when the ratio is about π or less, the tidal forces prevail, and the plasma columns tend to form a shell-like configuration before they start converging to the axis of the array. The model does not allow the precursor plasma streams to be ejected from the wires to the axis, indicating that this process is governed by the finite plasma conductivity and could only be described with a proper conductivity model.
Alexander L. Velikovich, Igor V. Sokolov and Andrey A. Esaulov, Perfectly conducting incompressible fluid model of a wire array implosion, Physics of Plasmas 9, 1366 (2002), from the Plasma Physics Division of the US Naval Research Laboratory, Washington DC.
(Abstract only, and the full text was not reachable — see the rights note above. On this site, the Imperial College measurements this model is written against are at /library/stm-33d8266a67 for wire-array implosion dynamics, /library/stm-bb03ac17ef for what the discreteness of the wires does to an implosion, /library/stm-a2345abd6a for the steady-state ablation that feeds the flow, /library/stm-704d7be815 for the two-wire case reduced to its smallest form, /library/stm-02af410435 for how the wire material changes the answer, /library/stm-06c1a40035 for the prepulse trick on a fibre pinch, and /library/stm-69ab32821f for the fibre-ablation question in a solid-deuterium pinch. Sandia’s microfabricated arrays, built to hold exactly the geometry this model parameterises, are at /library/stm-98d874080d.)
The way in
https://doi.org/10.1063/1.1452104SOURCE NOT REACHED IN FULL. The article is held by AIP Publishing as Physics of Plasmas volume 9, issue 4, pages 1366 to 1380, and Crossref carries no licence statement for it. Unpaywall and OpenAlex both label it bronze open access, but the only address they give is the publisher’s own PDF at aip.scitation.org, which returns the publisher’s bot page rather than the file, and the AIP article-PDF path returns 403; an AIP bronze label is a journal-level guess and is not a Creative Commons licence in any case. On 2026-09-08 the paper was hunted through OpenAlex, Unpaywall, Crossref, OpenAIRE, Semantic Scholar, CORE and OSTI. OpenAIRE and OpenAlex both point at a repository copy in Deep Blue at the University of Michigan, handle 2027.42/70839, deposited because Igor Sokolov was then at Michigan; on the day of writing Deep Blue answered every request — landing page, bitstream and REST API alike — with a Cloudflare challenge, so that copy could not be read either. The work was done in the Plasma Physics Division of the US Naval Research Laboratory, whose reports are indexed by DTIC rather than by OSTI, and the DTIC search service is a client-side application with no reachable API, so no NRL report version could be retrieved. The summary and every claim below are therefore written from the authors’ own complete abstract together with the bibliographic record, and every locator says Abstract, because the abstract is what was read.
How to cite it
Alexander L. Velikovich, Igor V. Sokolov, Andrey A. Esaulov (2002) Perfectly conducting incompressible fluid model of a wire array implosion. doi:10.1063/1.1452104
Where it sits in the curriculum
Plasmoids, charge clusters and the orbsLattice confinement fusion