Pinch (plasma physics)#Crushing cans with the pinch effect

File:Plasma Pinch Mechanism.png

File:plasma-filaments.jpg inside filaments of electrical discharge from a Tesla coil]]

File:MagLif Concept.png

Pinches exist in nature and in laboratories. Pinches differ in their geometry and operating forces.{{cite journal |last=Carlqvist |first=Per |date=May 1988 |title=Cosmic electric currents and the generalized Bennett relation |journal=Astrophysics and Space Science |volume=144 |number=1–2 |pages=73–84 |doi=10.1007/BF00793173 |bibcode=1988Ap&SS.144...73C |s2cid=119719745 |url=https://link.springer.com/article/10.1007/BF00793173}} These include:

• Uncontrolled – Any time an electric current moves in large amounts (e.g., lightning, arcs, sparks, discharges) a magnetic force can pull together plasma. This can be insufficient for fusion.
• Sheet pinch – An astrophysical effect, this arises from vast sheets of charged particles.{{cite book |last=Biskamp |first=Dieter |year=1997 |title=Nonlinear Magnetohydrodynamics |publisher=Cambridge University Press |place=Cambridge, England |page=130 |isbn=0-521-59918-0}}
• Z-pinch – The current runs down the axis, or walls, of a cylinder while the magnetic field is azimuthal
• Theta pinch – The magnetic field runs down the axis of a cylinder, while the electric field is in the azimuthal direction (also called a thetatron{{cite book |title=Dictionary of Material Science and High Energy Physics |url=https://books.google.com/books?id=vvJazZZpB1QC&dq=thetatron+pinch&pg=PA315 |page=315 |isbn=978-0-8493-2889-3 |last1=Basu |first1=Dipak K. |date=8 October 2018| publisher=CRC Press }})
• Screw pinch – A combination of a Z-pinch and theta pinch{{cite journal |last1=Srivastava |first1=K. M. |last2=Vyas |first2=D. N. |date=August 1982 |url=https://ui.adsabs.harvard.edu/abs/1982Ap%26SS..86...71S/abstract |title=Non-linear analysis of the stability of the screw pinch |journal=Astrophysics and Space Science |volume=86 |number=1 |pages=71–89 |doi=10.1007/BF00651831 |bibcode=1982Ap&SS..86...71S |s2cid=121575638}} (also called a stabilized Z-pinch, or θ-Z pinch)See "[http://silas.psfc.mit.edu/introplasma/chap4.html#tth_sEc4.7 MHD Equilibria" in Introduction to Plasma Physics by I.H.Hutchinson (2001)]{{cite journal |last1=Srivastava |first1=K. M. |last2=Waelbroeck |first2=F. |year=1976 |title=On the stability of the screw pinch in the CGL model |journal=Journal of Plasma Physics |volume=16 |issue=3 |page=261 |bibcode=1976JPlPh..16..261S |doi=10.1017/s0022377800020201|s2cid=123689314 }}
• Reversed field pinch or toroidal pinch – This is a Z-pinch arranged in the shape of a torus. The plasma has an internal magnetic field. As distance increases from the center of this ring, the magnetic field reverses direction.
• Inverse pinch – An early fusion concept, this device consisted of a rod surrounded by plasma. Current traveled through the plasma and returned along the center rod.{{cite journal |last1=Anderson |first1=O. A. |last2=Furth |first2=H. P. |last3=Stone |first3=J. M. |last4=Wright |first4=R. E. |date=November 1958 |title=Inverse Pinch Effect |journal=Physics of Fluids |volume=1 |issue=6 |pages=489–494 |doi=10.1063/1.1724372|bibcode=1958PhFl....1..489A }} This geometry was slightly different than a z-pinch in that the conductor was in the center, not the sides.
• Cylindrical pinch
• Orthogonal pinch effect
• Ware pinch – A pinch that occurs inside a Tokamak plasma, when particles inside the banana orbit condense together.{{cite journal |last1=Helander |first1=P. |last2=Akers |first2=R. J. |last3=Valovič |first3=M. |title=The effect of non-inductive current drive on tokamak transport |date=3 November 2005 |journal=Plasma Physics and Controlled Fusion |volume=47 |issue=12B |pages=B151–B163 |doi=10.1088/0741-3335/47/12b/s12 |bibcode=2005PPCF...47B.151H|s2cid=121961613 }}{{Cite book |last1=Nishikawa |first1=K. |url=https://books.google.com/books?id=4cHkd77TSHcC&dq=Ware+pinch&pg=PA266 |title=Plasma Physics: Third Edition |last2=Wakatani |first2=M. |date=2000-01-24 |publisher=Springer Science & Business Media |isbn=978-3-540-65285-4 |language=en}}
• Magnetized liner inertial fusion (MagLIF) – A Z-pinch of preheated, premagnetized fuel inside a metal liner, which could lead to ignition and practical fusion energy with a larger pulsed-power driver.{{cite journal |last1=Slutz |first1=Stephen |last2=Vesey |first2=Roger A. |year=2012 |title=High-Gain Magnetized Inertial Fusion |journal=Physical Review Letters |volume=108 |issue=2 |page=025003 |doi=10.1103/PhysRevLett.108.025003 |bibcode=2012PhRvL.108b5003S |pmid=22324693 |doi-access=free}}

Pinches may become unstable.{{cite journal |last=Hardee |first=P. E. |year=1982 |title=Helical and pinching instability of supersonic expanding jets in extragalactic radio sources |journal=Astrophysical Journal |volume=257 |pages=509–526 |doi=10.1086/160008 |bibcode=1982ApJ...257..509H|doi-access=free }} They radiate energy across the whole electromagnetic spectrum including radio waves, microwaves, infrared, x-rays,{{cite journal |last1=Pereira |first1=N. R. |display-authors=etal |year=1988 |title=X-rays from Z-pinches on relativistic electron-beam generators |journal=Journal of Applied Physics |volume=64 |issue=3 |pages=R1–R27 |doi=10.1063/1.341808 |bibcode=1988JAP....64....1P}} gamma rays,{{cite journal |last1=Wu |first1=Mei |last2=Chen |first2=Li |last3=Li |first3=Ti-Pei |title=Polarization in Gamma-Ray Bursts Produced by Pinch Discharge |year=2005 |bibcode=2005ChJAA...5...57W |journal=Chinese Journal of Astronomy & Astrophysics |volume=5 |issue=1 |pages=57–64 |doi=10.1088/1009-9271/5/1/007 |arxiv=astro-ph/0501334 |s2cid=121943}} synchrotron radiation,Peratt, A.L., "[https://books.google.com/books?id=ZSlJRAeL95sC&dq=synchrotron+pinch&pg=PA62 Synchrotron radiation from pinched particle beams]", (1998) Plasma Physics: VII Lawpp 97: Proceedings of the 1997 Latin American Workshop on Plasma Physics, Edited by Pablo Martin, Julio Puerta, Pablo Martmn, with reference to Meierovich, B. E., "[https://ui.adsabs.harvard.edu/abs/1984PhR...104..259M/abstract Electromagnetic collapse. Problems of stability, emission of radiation and evolution of a dense pinch]" (1984) Physics Reports, Volume 104, Issue 5, p. 259-346. and visible light. They also produce neutrons, as a product of fusion.{{cite journal |last1=Anderson |first1=Oscar A. |display-authors=etal |title=Neutron Production in Linear Deuterium Pinches |year=1958 |journal=Physical Review |volume=110 |issue=6 |pages=1375–1387 |doi=10.1103/physrev.110.1375 |bibcode=1958PhRv..110.1375A |url=https://escholarship.org/uc/item/0gj9x38q}}

File:Kink Modes Model.png

Pinches are used to generate X-rays and the intense magnetic fields generated are used in electromagnetic forming of metals. They also have applications in particle beams{{cite journal |last1=Ryutov |first1=D. D. |last2=Derzon |first2=M. S. |last3=Matzen |first3=M. K |title=The physics of fast Z pinches |year=2000 |journal=Reviews of Modern Physics |volume=72 |issue=1 |pages=167–223 |doi=10.1103/revmodphys.72.167 |bibcode=2000RvMP...72..167R |url=https://zenodo.org/record/1233969}} including particle beam weapons,Andre Gsponer, "[https://arxiv.org/abs/physics/0409157 Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas]" (2004) https://arxiv.org/abs/physics/0409157 astrophysics studiesPeratt, Anthony L., "[http://adsabs.harvard.edu/abs/1988LaPaB...6..471P The role of particle beams and electrical currents in the plasma universe]" (1988) Laser and Particle Beams (ISSN 0263-0346), vol. 6, Aug. 1988, p. 471-491. and it has been proposed to use them in space propulsion."Z-Pinch Pulsed Plasma Propulsion Technology Development" Final Report Advanced Concepts Office (ED04) Marshall Space Flight Center October 8, 2010, Tara Polsgrove, Et Al. A number of large pinch machines have been built to study fusion power; here are several:

• MAGPIE A Z-pinch at Imperial College. This dumps a large amount of current across a wire. Under these conditions, the wire becomes plasma and compresses to produce fusion.http://dorland.pp.ph.ic.ac.uk/magpie/?page_id=239 {{Webarchive|url=https://web.archive.org/web/20141105113922/http://dorland.pp.ph.ic.ac.uk/magpie/?page_id=239 |date=2014-11-05}} "Wire Arrays Z-Pinch" accessed: 3-27-2015
• Z Pulsed Power Facility at Sandia National Laboratories.
• ZETA device in Culham, England
• Madison Symmetric Torus at the University of Wisconsin, Madison
• Reversed-Field eXperiment in Italy.
• Dense plasma focus in New Jersey
• University of Nevada, Reno (USA)
• Cornell University (USA)
• University of Michigan (USA)
• University of California, San Diego (USA)
• University of Washington (USA)
• Ruhr University (Germany)
• École Polytechnique (France)
• Weizmann Institute of Science (Israel)
• Universidad Autónoma Metropolitana (Mexico).
• Zap Energy Inc. (USA)

File:Aluminium-can-white.jpg magnetic field created by rapidly discharging 2 kilojoules from a high voltage capacitor bank into a 3-turn coil of heavy gauge wire.]]

{{Main|Electromagnetic forming}}Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices.{{cite web |url=https://members.tm.net/lapointe/Main.html |title=High Voltage Devices and Experiments |access-date=February 21, 2013 |last=LaPointe |first=Robert}}{{cite web |url=https://members.tripod.com/extreme_skier/cancrusher/ |title=Electromagnetic Can Crusher |access-date=February 21, 2013 |author=Tristan}}{{cite web |url=http://www.powerlabs.org/pssecc.htm |title=Solid State Can Crusher |last=Borros |first=Sam |access-date=February 21, 2013}} They use pulsed power techniques to produce a theta pinch able to crush an aluminium soft drink can using the Lorentz forces created when large currents are induced in the can by the strong magnetic field of the primary coil.{{cite web |url=http://magnet-physik.de/st_magnetopuls.html#method |title=MagnetoPulS |website=Magnet-Physik, Dr. Steingroever GmbH |date=2002 |archive-date=2003-05-22 |access-date=February 21, 2013 |archive-url=https://web.archive.org/web/20030522114102/http://magnet-physik.de/st_magnetopuls.html#method}}

{{cite web |url=http://www.english.pstproducts.com/index_htm_files/English%20White%20Paper%20by%20PSTproducts.pdf |title=Industrial Application of the Electromagnetic Pulse Technology |publisher=PSTproducts GmbH |website=white paper |date=June 2009 |access-date=February 21, 2013 |url-status=dead |archive-url=https://web.archive.org/web/20110715125631/http://www.english.pstproducts.com/index_htm_files/English%20White%20Paper%20by%20PSTproducts.pdf |archive-date=July 15, 2011}}

An electromagnetic aluminium can crusher consists of four main components: a high-voltage DC power supply, which provides a source of electrical energy, a large energy discharge capacitor to accumulate the electrical energy, a high voltage switch or spark gap, and a robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below).

File:can-pincher.png

In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control the current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes.Examples of electromagnetic pinch can crushers can be found at (a) Bob LaPointe's site on [http://members.tm.net/lapointe/Main.html High Voltage Devices and Experiments] (b) Tristran's [https://members.tripod.com/extreme_skier/cancrusher/ Electromagnetic Can Crusher] (including schematic) (c) Sam Borros's [http://www.powerlabs.org/pssecc.htm Solid State Can Crusher]

File:ieee-emblem.jpg emblem shows the basic features of an azimuthal magnetic pinch.See also the IEEE History Center, "[https://web.archive.org/web/20020414172744/http://www.ieee.org/organizations/history_center/ieee_emblem.html Evolution of the IEEE Logo]" March 1963; see also the comments in "[http://public.lanl.gov/alp/plasma/lab_astro.html Laboratory Astrophysics]"]]

The first creation of a Z-pinch in the laboratory may have occurred in 1790 in Holland when Martinus van Marum created an explosion by discharging 100 Leyden jars into a wire.van Marum M 1790 Proc. 4th Int. Conf. on Dense Z-Pinches (Vancouver 1997) (Am. Inst. Phys. Woodbury, New York, 1997) Frontispiece and p ii The phenomenon was not understood until 1905, when Pollock and BarracloughPollock J A and Barraclough S (1905) Proc. R. Soc. New South Wales 39 131 investigated a compressed and distorted length of copper tube from a lightning rod after it had been struck by lightning. Their analysis showed that the forces due to the interaction of the large current flow with its own magnetic field could have caused the compression and distortion.R. S. Pease, "The Electromagnetic Pinch: From Pollock to the Joint European Torus", "[http://nsw.royalsoc.org.au/journal/118_12.html#pease Pollock Memorial Lecture for 1984 delivered at the University of Sydney, 28 November, 1984"] {{Webarchive|url=https://web.archive.org/web/20060529053852/http://nsw.royalsoc.org.au/journal/118_12.html#pease |date=2006-05-29}} A similar, and apparently independent, theoretical analysis of the pinch effect in liquid metals was published by Northrup in 1907.{{cite journal |last=Northrup |first=Edwin F. |title=Some Newly Observed Manifestations of Forces in the Interior of an Electric Conductor |journal=Physical Review |series=Series I |publisher=American Physical Society (APS) |volume=24 |issue=6 |year=1907 |issn=1536-6065 |doi=10.1103/physrevseriesi.24.474 |pages=474–497 |bibcode=1907PhRvI..24..474N |url=https://zenodo.org/record/1649961}} The next major development was the publication in 1934 of an analysis of the radial pressure balance in a static Z-pinch by Bennett{{cite journal |year=1934 |title=Magnetically Self-Focussing Streams |journal=Phys. Rev. |volume=45 |issue=12 |pages=890–897 |bibcode=1934PhRv...45..890B |doi=10.1103/physrev.45.890 |last1=Bennett |first1=Willard H.}} (see the following section for details).

Thereafter, the experimental and theoretical progress on pinches was driven by fusion power research. In their article on the "Wire-array Z-pinch: a powerful x-ray source for ICF", M G Haines et al., wrote on the "Early history of Z-pinches".{{cite journal |last1=Haines |first1=M G |last2=Sanford |first2=T W L |last3=Smirnov |first3=V P |year=2005 |title=Wire-array Z-pinch: a powerful x-ray source for ICF |journal=Plasma Phys. Control. Fusion |volume=47 |issue=12B |pages=B1–B11 |doi=10.1088/0741-3335/47/12b/s01 |bibcode=2005PPCF...47B...1H|s2cid=120320797 }}

:In 1946 Thompson and Blackman submitted a patent for a fusion reactor based on a toroidal Z-pinch{{cite journal |last1=Thompson |first1=G. P. |last2=Blackman |last3=Haines |first3=M. G. |year=1996 |title=Historical Perspective: Fifty years of controlled fusion research |journal=Plasma Physics and Controlled Fusion |volume=38 |issue=5 |pages=643–656 |bibcode=1996PPCF...38..643H |doi=10.1088/0741-3335/38/5/001|s2cid=250763028 }} with an additional vertical magnetic field. But in 1954 Kruskal and Schwarzschild{{cite journal |last1=Kruskal |first1=M D |last2=Schwarzschild |year=1954 |title=Some Instabilities of a Completely Ionized Plasma |bibcode=1954RSPSA.223..348K |journal=Proc. R. Soc. Lond. A |volume=223 |issue=1154 |pages=348–360 |doi=10.1098/rspa.1954.0120 |s2cid=121125652}} published their theory of MHD instabilities in a Z-pinch. In 1956, Kurchatov gave his famous Harwell lecture showing nonthermal neutrons and the presence of m = 0 and m = 1 instabilities in a deuterium pinch.Kurchatov I V (1957) J. Nucl. Energy 4 193 In 1957 Pease{{cite journal |last1=Pease |first1=R S |year=1957 |title=Equilibrium Characteristics of a Pinched Gas Discharge Cooled by Bremsstrahlung Radiation |bibcode=1957PPSB...70...11P |journal=Proc. Phys. Soc. Lond. |volume=70 |issue=1 |pages=11–23 |doi=10.1088/0370-1301/70/1/304}} and BraginskiiBraginskii S I 1957 Zh. Eksp. Teor. Fiz 33 645Braginskii S I 1958 Sov. Phys.—JETP 6 494 independently predicted radiative collapse in a Z-pinch under pressure balance when in hydrogen the current exceeds 1.4 MA. (The viscous rather than resistive dissipation of magnetic energy discussed above and inHaines M G et al. 2005 Phys. Rev. Lett.. submitted; see also EPS Conf. on Plasma Physics 2004 (London, UK) paper 73 would however prevent radiative collapse).

In 1958, the world's first controlled thermonuclear fusion experiment was accomplished using a theta-pinch machine named Scylla I at the Los Alamos National Laboratory. A cylinder full of deuterium was converted into a plasma and compressed to 15 million degrees Celsius under a theta-pinch effect. Lastly, at Imperial College in 1960, led by R Latham, the Plateau–Rayleigh instability was shown, and its growth rate measured in a dynamic Z-pinch.{{cite journal |last1=Curzon |first1=F. L. |display-authors=etal |year=1960 |title=Experiments on the Growth Rate of Surface Instabilities in a Linear Pinched Discharge |bibcode=1960RSPSA.257..386C |journal=Proc. R. Soc. Lond. A |volume=257 |issue=1290 |pages=386–401 |doi=10.1098/rspa.1960.0158 |s2cid=96283997}}

In plasma physics three pinch geometries are commonly studied: the θ-pinch, the Z-pinch, and the screw pinch. These are cylindrically shaped. The cylinder is symmetric in the axial (z) direction and the azimuthal (θ) directions. The one-dimensional pinches are named for the direction the current travels.

File:thet pinch.png

The θ-pinch has a magnetic field directed in the z direction and a large diamagnetic current directed in the θ direction. Using Ampère's circuital law (discarding the displacement term)

:$\backslash begin\{align\}$

\vec{B} &= B_z(r)\hat{z} \\

\mu_0 \vec{J} &= \nabla \times \vec{B} \\

&= \frac{1}{r} \frac{d}{d \theta}B_z(r) \hat{r} - \frac{d}{dr}B_z(r) \hat{\theta}

\end{align}

Since B is only a function of r we can simplify this to

:$\backslash mu\_0\; \backslash vec\{J\}\; =\; -\backslash frac\{d\}\{dr\}B\_z(r)\; \backslash hat\{\backslash theta\}$

So J points in the θ direction.

Thus, the equilibrium condition ($\backslash nabla\; p\; =\; \backslash mathbf\{j\}\backslash times\backslash mathbf\{B\}$) for the θ-pinch reads:

:$\backslash frac\{d\}\{d\; r\}\; \backslash left(\; p\; +\; \backslash frac\{B\_z^2\}\{2\backslash mu\_0\}\; \backslash right)\; =\; 0$

θ-pinches tend to be resistant to plasma instabilities; This is due in part to Alfvén's theorem (also known as the frozen-in flux theorem).

{{main|Z-pinch}}

File:z pinch.png

The Z-pinch has a magnetic field in the θ direction and a current J flowing in the z direction. Again, by electrostatic Ampère's law,

:$\backslash begin\{align\}$

\vec{B} &= B_{\theta}(r)\hat{\theta} \\

\mu_0 \vec{J} &= \nabla \times \vec{B} \\

&= \frac{1}{r}\frac{d}{dr}\left(r B_\theta(r)\right) \hat{z} - \frac{d}{dz}B_\theta(r) \hat{r} \\

&= \frac{1}{r}\frac{d}{dr}\left(r B_\theta(r)\right) \hat{z}

\end{align}

Thus, the equilibrium condition, $\backslash nabla\; p\; =\; \backslash mathbf\{j\}\backslash times\backslash mathbf\{B\}$, for the Z-pinch reads:

:$\backslash frac\{d\}\{d\; r\}\; \backslash left(\; p\; +\; \backslash frac\{B\_\backslash theta^2\}\{2\backslash mu\_0\}\; \backslash right)\; +\; \backslash frac\{B\_\backslash theta^2\}\{\backslash mu\_0\; r\}\; =\; 0$

Although Z-pinches satisfy the MHD equilibrium condition, it is important to note that this is an unstable equilibrium, resulting in various instabilies such as the m = 0 instability ('sausage'), m = 1 instability ('kink'), and various other higher order instabilities.Bellan, P. Fundamentals of Plasma Physics

The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampère's law,

:$\backslash nabla\; \backslash times\; \backslash vec\{B\}\; =\; \backslash mu\_0\; \backslash vec\{J\}$

But this time, the B field has a θ component and a z component

:$\backslash begin\{align\}$

\vec{B} &= B_\theta \hat{\theta} + B_z \hat{z} \\

\mu_0 \vec{J} &= \frac{1}{r}\frac{d}{dr}\left(r B_\theta\right) \hat{z} - \frac{d}{dr}B_z \hat{\theta}

\end{align}

So this time J has a component in the z direction and a component in the θ direction.

Finally, the equilibrium condition ($\backslash nabla\; p\; =\; \backslash mathbf\{j\}\backslash times\backslash mathbf\{B\}$) for the screw pinch reads:

:$\backslash frac\{d\}\{d\; r\}\; \backslash left(p\; +\; \backslash frac\{B\_z^2\; +\; B\_\backslash theta^2\}\{2\; \backslash mu\_0\}\; \backslash right)\; +\; \backslash frac\{B\_\backslash theta^2\}\{\backslash mu\_0\; r\}\; =\; 0$

The screw pinch might be produced in laser plasma by colliding optical vortices of ultrashort duration.

[https://www.sciencedirect.com/science/article/abs/pii/S0375960110011734 A.Yu.Okulov. "Laser singular Theta-pinch", Phys.Lett.A, v.374, 4523-4527, (2010)]

For this purpose optical vortices should be phase-conjugated.

Optical phase conjugation and electromagnetic momenta

The magnetic field distribution is given here again via Ampère's law:

:$\backslash nabla\; \backslash times\; \backslash vec\{B\}\; =\; \backslash mu\_0\; \backslash vec\{J\}$

[[File:toroidal coord.png|right|thumb|250px|A toroidal coordinate system in common use in plasma physics.

{{legend|red|The red arrow denotes the poloidal direction (θ)}}

{{legend|blue|The blue arrow denotes the toroidal direction (φ)}}

]]

A common problem with one-dimensional pinches is the end losses. Most of the motion of particles is along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles out of the end of the machine very quickly, leading to a loss of mass and energy. Along with this problem, the Z-pinch has major stability problems. Though particles can be reflected to some extent with magnetic mirrors, even these allow many particles to pass. A common method of beating these end losses, is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous Grad–Shafranov equation. Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of the reversed field pinch.

{{As of|2015}}, there is no coherent analytical theory for three-dimensional equilibria. The general approach to finding such equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for stellarators. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration, an equilibrium relation, similar to that of the 1-D configurations exists:Ideal Magnetohydrodynamics: Modern perspectives in energy. Jeffrey P. Freidberg. Massachusetts Institute of Technology. Cambridge, Massachusetts. Plenum Press - New York and London - 1987. (Pg. 86, 95)

:$\backslash nabla\_\backslash perp\; \backslash left(\; p\; +\; \backslash frac\{B^2\}\{2\; \backslash mu\_0\}\; \backslash right)\; -\; \backslash frac\{B^2\}\{\backslash mu\_0\; \}\backslash vec\{\backslash kappa\}\; =\; 0$

Where κ is the curvature vector defined as:

:$\backslash vec\{\backslash kappa\}\; =\; \backslash left(\backslash vec\{b\}\; \backslash cdot\; \backslash nabla\backslash right)\backslash vec\{b\}$

with b the unit vector tangent to B.

File:water-pinching.jpg breaks the narrowing water column into droplets (not shown, see Plateau–Rayleigh instability). This is analogous to the magnetic field suggested as the cause of pinching in bead lightning."The PLASMAK Configuration and Ball Lightning" ({{usurped|1=[https://web.archive.org/web/20060715074212/http://www.prometheus2.net/bl-tokyo.pdf PDF]}}) presented at the International Symposium on Ball Lightning; July 1988 The morphology (shape) is similar to the so-called sausage instability in plasma.]]

Consider a cylindrical column of fully ionized quasineutral plasma, with an axial electric field, producing an axial current density, j, and associated azimuthal magnetic field, B. As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of j x B. In a steady state with forces balancing:

:$\backslash nabla\; p\; =\; \backslash nabla(p\_e+p\_i)\; =\; \backslash mathbf\{j\}\backslash times\backslash mathbf\{B\}$

where ∇p is the magnetic pressure gradient, and p_e and p_i are the electron and ion pressures, respectively. Then using Maxwell's equation $\backslash nabla\backslash times\backslash mathbf\{B\}\; =\; \backslash mu\_0\backslash mathbf\{j\}$ and the ideal gas law $p=NkT$, we derive:

:$2\; N\; k(T\_e\; +\; T\_i)\; =\; \backslash frac\{\{\backslash mu\_0\}\}\; \{4\; \backslash pi\}\; I^2$ (the Bennett relation)

where N is the number of electrons per unit length along the axis, T_e and T_i are the electron and ion temperatures, I is the total beam current, and k is the Boltzmann constant.

File:Generalized Bennett Relation diagram.pngThe generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density j_z, resulting in an azimuthal magnetic field Β_φ. Originally derived by Witalis,Witalis, E. A. "[https://ui.adsabs.harvard.edu/abs/1981PhRvA..24.2758W/abstract Plasma-physical aspects of charged-particle beams]" (1981) Physical Review A - General Physics, 3rd Series, vol. 24, Nov. 1981, p. 2758–2764 the generalized Bennett relation results in:Anthony L . Peratt, "Physics of the Plasma Universe", 1992 Springer-Verlag, {{ISBN|0-387-97575-6}}

:$\backslash begin\{align\}$

\frac{1}{4} \frac{\partial^2 J_0}{\partial t^2} = {} &W_{\perp \text{kin}} + \Delta W_{E_z} + \Delta W_{B_z} + \Delta W_k - \frac{{\mu_0}} {8 \pi} I^2 (a) \\[8pt]

& {} - \frac{1}{2}G\overline{m}^2 N^2 (a) + \frac{1}{2}\pi a^2 \epsilon_0 \left(E_r^2 (a) - E_\phi^2 (a) \right)\\

\end{align}

• where a current-carrying, magnetic-field-aligned cylindrical plasma has a radius a,
• J₀ is the total moment of inertia with respect to the z axis,
• W_⊥kin is the kinetic energy per unit length due to beam motion transverse to the beam axis
• W_Bz is the self-consistent B_z energy per unit length
• W_Ez is the self-consistent E_z energy per unit length
• W_k is thermokinetic energy per unit length
• I(a) is the axial current inside the radius a (r in diagram)
• N(a) is the total number of particles per unit length
• E_r is the radial electric field
• E_φ is the rotational electric field

The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces.

The Carlqvist relation, published by Per Carlqvist in 1988, is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form

:$\backslash frac\{\{\backslash mu\_0\}\}\; \{8\; \backslash pi\}\; I^2\; (a)\; +\backslash frac\{1\}\{2\}G\backslash overline\{m\}^2\; N^2\; (a)\; =\; \backslash Delta\; W\_\{B\_z\}\; +\; \backslash Delta\; W\_k$

and is applicable to many space plasmas.

File:Bennett Pinch graph.png

The Carlqvist relation can be illustrated (see right), showing the total current (I) versus the number of particles per unit length (N) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (T_i = T_e = T_n = 20 K), containing mainly hydrogen with a mean particle mass 3×10⁻²⁷ kg. The thermokinetic energy W_k >> πa² p_k(a). The curves, ΔW_Bz show different amounts of excess magnetic energy per unit length due to the axial magnetic field B_z. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.

Chart regions: (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line ΔW_Bz = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve ΔW_Bz = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid.

Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the Jeans criterion (for gravitational instability,{{cite journal |last1=Jeans |first1=J. H. |title=The stability of a spherical nebula |journal=Phil. Trans. R. Soc. Lond. A |volume=199 |issue=312–320 |year=1902 |bibcode=1902RSPTA.199....1J |doi=10.1098/rsta.1902.0012 |pages=1–53 |doi-access=free}} in one and two dimensions), force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states.

A fictionalized pinch-generating device was used in Ocean's Eleven, where it was used to disrupt Las Vegas's power grid just long enough for the characters to begin their heist.{{cite news |publisher=American Physical Society |title=The Con-Artist Physics of 'Ocean's Eleven' |date=March 2002 |url=https://www.aps.org/publications/apsnews/200203/oceans-eleven.cfm}}

• Electromagnetic forming
• Explosively pumped flux compression generator
• Fusion power
• Madison Symmetric Torus (reversed field pinch)

{{Reflist|30em}}

• [http://www.capturedlightning.com/frames/shrinkergallery.html Examples of electromagnetically shrunken coins and crushed cans]
• [http://www.capturedlightning.com/frames/shrinker.html Theory of electromagnetic coin shrinking]
• [http://www.capturedlightning.com/frames/Shrinking_History.htm The Known History of "Quarter Shrinking"]
• [https://web.archive.org/web/20060910014555/http://tesladownunder.iinet.net.au/CanCrushing.htm Can crushing info using electromagnetism among other things]
• [https://web.archive.org/web/20100406085526/http://dorland.pp.ph.ic.ac.uk/magpie/ The MAGPIE project at Imperial College London] is used to study wire array Z-pinch implosions.

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Category:Fusion power

Category:Plasma phenomena

Category:Dutch inventions