Ballooning instability

The ballooning instability (a.k.a. ballooning mode instability) is a type of internal pressure-driven plasma instability usually seen in tokamak fusion power reactors[1] or in space plasmas.[2] It is important in fusion research as it determines a set of criteria for the maximum achievable plasma beta.[3] The name refers to the shape and action of the instability, which acts like the elongations formed in a long balloon when it is squeezed. In literature, the structure of these elongations are commonly referred to as 'fingers'.[4][5][6]

The narrow fingers of plasma produced by the instability are capable of accelerating and pushing aside the surrounding magnetic field in order to cause a sudden, explosive release of energy. Thus, the instability is also known as the explosive instability.[7][8][9]

## Dispersion Relation

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The dispersion relation is

ω ( ω − ω ∗ p i ) = [ S k ∥ 2 − 2 μ 0 κ ∇ P / β 2 ] ( 1 + b i ) V A 2 {\displaystyle \omega (\omega -\omega _{*}pi)=[Sk_{\parallel }^{2}-2\mu _{0}\kappa \nabla P/\beta ^{2}](1+b_{i})V_{A}^{2}} where

S = 1 + n e δ / n e c {\displaystyle S=1+n_{e}\delta /n_{e}c} ,

δ = β e / ( ω ∗ p i − ω ∗ e p ) / 2 ( ω − q i T − ω ∗ p i ) b i ) / ( ω − ω ∗ e ) − 3 / 2 ( ω − ω ∗ p e ) b i / ( ω − ω − ω ∗ e ) ( ω B e + ω k e ) / 2 ω {\displaystyle \delta =\beta _{e}/(\omega _{*}pi-\omega _{*}ep)/2(\omega -q_{i}T_{-}\omega _{*}pi)b_{i})/(\omega -\omega _{*}e)-3/2(\omega -\omega _{*}pe)b_{i}/(\omega -\omega -\omega _{*}e)(\omega _{B}e+\omega _{k}e)/2\omega }

## Relation to interchange instability

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See also: Interchange instability

The interchange instability can be derived from the equations of the ballooning instability as a special case in which the ballooning mode does not perturb the equilibrium magnetic field.[2] This special limit is known as the Mercier criterion.[3]

## References

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1. ^ Dobrott, D.; Nelson, D. B.; Greene, J. M.; Glasser, A. H.; Chance, M. S.; Frieman, E. A. (1977-10-10). "Theory of Ballooning Modes in Tokamaks with Finite Shear". Physical Review Letters. 39 (15): 943–946\. doi:10.2172/5115796. OSTI 5115796.
2. ^ a b Hameiri, E.; Laurence, P.; Mond, M. (1991-02-01). "The ballooning instability in space plasmas". Journal of Geophysical Research: Space Physics. 96 (A2): 1513–1526\. Bibcode:1991JGR....96.1513H. doi:10.1029/90ja02100. ISSN 0148-0227.
3. ^ a b P., Freidberg, Jeffrey (1987). Ideal magnetohydrodynamics. New York: Plenum Press. ISBN 0306425122. OCLC 15428479.`{{cite book}}`: CS1 maint: multiple names: authors list (link)
4. ^ Kleva, Robert G.; Guzdar, Parvez N. (2001). "Fast disruptions by ballooning mode ridges and fingers in high temperature, low resistivity toroidal plasmas". Physics of Plasmas. 8 (1): 103–109\. Bibcode:2001PhPl....8..103K. doi:10.1063/1.1331098. ISSN 1070-664X.
5. ^ Cowley, Steven C.; Wilson, Howard; Hurricane, Omar; Fong, Bryan (2003). "Explosive instabilities: from solar flares to edge localized modes in tokamaks". Plasma Physics and Controlled Fusion. 45 (12A): A31. Bibcode:2003PPCF...45A..31C. doi:10.1088/0741-3335/45/12A/003. ISSN 0741-3335. S2CID 250824453.
6. ^ Panov, E. V.; Sergeev, V. A.; Pritchett, P. L.; Coroniti, F. V.; Nakamura, R.; Baumjohann, W.; Angelopoulos, V.; Auster, H. U.; McFadden, J. P. (2012). "Observations of kinetic ballooning/interchange instability signatures in the magnetotail". Geophysical Research Letters. 39 (8): n/a. Bibcode:2012GeoRL..39.8110P. doi:10.1029/2012gl051668. ISSN 0094-8276.
7. ^ Hamasaki, Seishi (1971). "Self-Consistent Calculation of an Explosive Instability". Physics of Fluids. 14 (7): 1441–1451\. Bibcode:1971PhFl...14.1441H. doi:10.1063/1.1693626. ISSN 0031-9171.
8. ^ Jones, Michael E.; Fukai, J. (1979). "Evolution of the explosive instability in a simulated beam plasma". Physics of Fluids. 22 (1): 132. Bibcode:1979PhFl...22..132J. doi:10.1063/1.862440. ISSN 0031-9171.
9. ^ Cowley, S. C.; Cowley, B.; Henneberg, S. A.; Wilson, H. R. (2015-08-08). "Explosive instability and erupting flux tubes in a magnetized plasma". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 471 (2180): 20140913. arXiv:1411.7797. Bibcode:2015RSPSA.47140913C. doi:10.1098/rspa.2014.0913. ISSN 1364-5021. PMC 4550006. PMID 26339193.

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