Bifurcated Tokamak equilibria with vanishing edge current density
Enzo Lazzaro · Sergey V. Shchepetov
Abstract and summary · read the original at the source · none found
In one page
Ask what shape a tokamak plasma settles into and the textbook answer is that there is one answer: minimise the plasma’s energy at a fixed total current and read off the profile. Enzo Lazzaro and Sergey Shchepetov show the question has more than one answer. Doing the minimisation properly, they find that when there is a vacuum gap between a free plasma boundary and the conducting wall, the number of allowed equilibria is not one but infinite; pin the boundary in place instead and the count drops to a finite handful. They then work the same calculation for the awkward case that matters at the edge of a real machine — an average current density that falls to zero at the boundary — and get canonical profiles there too. The interesting part is what the multiplicity implies: a plasma can jump from one branch to another without anything new being applied to it, and the authors note that such a bifurcation looks like the low-to-high confinement transition machines actually show.
Why it matters hereChapter 9 needs the physics of how a confined plasma chooses its own configuration, and this is a clean statement that the choice is not unique — the same current in the same vessel admits more than one stable arrangement. That is the mathematical shape of a self-organising plasma, and it is the same shape the chapter’s larger structures are argued to have.
What it claims
01Tokamak equilibria with canonical profiles of current density and pressure are obtained by minimisation of the total plasma energy under the constraint of a constant total plasma current.Abstract, first sentence
Published and peer-reviewed02There are an infinite number of such solutions when there is a vacuum region between a free plasma boundary and a conducting wall.Abstract, second sentence
Published and peer-reviewed03For the case with a fixed boundary, a finite multiplicity of solutions exists — so the number of allowed equilibria is decided by how the plasma is bounded, not only by the current it carries.Abstract, third sentence
Published and peer-reviewed04Canonical profiles are obtained also in the case of an average current density vanishing at the boundary, which is the awkward edge condition of a real machine rather than an idealised one.Abstract, fourth sentence, read with the paper’s own title
Published and peer-reviewed05The bifurcation between these solution branches resembles a low-to-high confinement transition, so a change of confinement mode can be read as the plasma moving between equilibria that the energy minimum already permits, rather than as the arrival of a new force.Abstract, fifth sentence
Published and peer-reviewed06What would settle the reading is a machine measurement that catches the plasma on each branch under the same total current and edge condition, and shows the jump between them following the calculated bifurcation rather than an externally imposed change.Abstract, read as a whole; the body of the Letter was not available to the editorial rail
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Abstract
Tokamak equilibria with canonical profiles of current density and pressure are obtained by minimisation of the total plasma energy under the constraint of a constant total plasma current. It is shown that there are an infinite number of solutions when there is a vacuum region between a free plasma boundary and a conducting wall. For the case with fixed boundary a finite multiplicity of solutions exists. Canonical profiles are obtained also in the case of an average current density vanishing at the boundary. The bifurcation of solutions, similar to an L to H transition, is briefly discussed.
Enzo Lazzaro and Sergey V. Shchepetov. Physics Letters A 143, issue 8, pages 393 to 399, January 1990.
(Abstract only — no open full text of this Letter was reachable; see the rights note above for what was and was not read. On this site the companion tokamak sheets are /library/stm-100dc9cb9f, /library/stm-c728aa7225 and /library/stm-1990916bc3.)
The way in
https://doi.org/10.1016/0375-9601(90)90379-3PUBLICATION. Physics Letters A, volume 143, issue 8, pages 393 to 399, January 1990, published by Elsevier; International Standard Serial Number 0375-9601. The only licence recorded in the Crossref deposit is Elsevier’s text-and-data-mining user licence, which is not an open licence, and OpenAIRE marks the record Closed Access, so this page reproduces the published abstract only. WHAT WAS READ, 2026-09-08. No open full text was reachable — the article predates Elsevier’s open-access programmes and no green copy is indexed by OpenAIRE, OpenAlex or Unpaywall. The abstract below is the abstract Elsevier deposited for the article, retrieved from the OpenAIRE record for this digital object identifier; the Crossref and OpenAlex abstract fields for this identifier are empty, which is why OpenAIRE was used. The summary and the claims were written from that abstract together with the article’s bibliographic record, which shows a seven-page Letter with fourteen references; nothing here is drawn from the body of the paper, which the editorial rail did not see, and every locator says so. AUTHOR NAMES. The journal prints initials only, E. Lazzaro and S. V. Shchepetov. The given names are expanded here from the authors’ own records: Enzo Lazzaro of the Istituto di Fisica del Plasma of the Consiglio Nazionale delle Ricerche in Milan, and Sergey Shchepetov, whose given name is taken from his own public researcher record attached to this article. RELATED PAGES on this site: Lazzaro’s later work on how fast a heat pulse crosses a tokamak plasma at /library/stm-100dc9cb9f; rotating nonlinear magnetic islands at /library/stm-c728aa7225; and the multi-institution review of macroscopic magnetic islands in tokamaks at /library/stm-1990916bc3.
How to cite it
Enzo Lazzaro, Sergey V. Shchepetov (1990) Bifurcated Tokamak equilibria with vanishing edge current density. doi:10.1016/0375-9601(90)90379-3
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