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The unbounded two-dimensional guiding-centre plasma

Published online by Cambridge University Press:  13 March 2009

Michael K. -H. Kiessling
Michael K. -H. Kiessling
Affiliation:
Institut für Theoretische Weltraumn und Astrophysik, Ruhr-Universität, D-44780 Bochum, Germany
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Abstract

The thermal mean-field equilibrium of a translation-invariant, unbounded one- component guiding-centre plasma is studied by analytical techniques. A variational principle is constructed. It is shown that only radial symmetric, decreasing density profiles occur. Prescribing the total number of gyro centres N ∈ (0, ∞), the energy E ∈ (E0, ∞) and the canonical angular momentum M ∈ (0, ∞]) uniquely determines a profile. Metastable or unstable profiles do not exist. A simple algebraic relation between N, M, the guiding-centre temperature β−1 and the characteristic Larmor frequency ω is derived. This explains Williamson's computer-based observations.

Type
Research Article
Information
Journal of Plasma Physics , Volume 54 , Issue 1 , August 1995 , pp. 11 - 29
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Aly, J. J. 1994 Some properties of rotating coherent structures in a non-neutral Plasma column. Phys. Plasmas 1, 2401–2403.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover, New York.Google Scholar
Bennett, W. H. 1934 Magnetically self-focussing streams. Phys. Rev. 45, 890–897.CrossRefGoogle Scholar
Brezis, H. & Merle, F. 1991 Uniform estimates and blow-up behavior of solutions of —Δu = V (x)eu in two dimensions. Commun. PDE 16, 1223–1253.Google Scholar
Caglioti, E., Lions, P.-L., Marchioro, C. & Pulvirenti, M. 1992 A special class of stationary flows for two-dimensional Euler equations: a statistical mechanical description. Commun. Math. Phys. 143, 501–525.CrossRefGoogle Scholar
Caglioti, E., Lions, P.-L., Marchioro, C. & Pulvirenti, M. 1995 A special class of stationary flows for two-dimensional Euler equations: a statistical mechanical description. Part II. Commun. Math. Phys. (to appear).CrossRefGoogle Scholar
Carlen, E. A. & Loss, M. 1992 Competing symmetries, the logarithmic HLS inequality, and Onofri's inequality on Sn. Geom. Funct. Anal. 2, 90–104.Google Scholar
Chanillo, S. & Kiessling, M. K.-H. 1994 Rotational symmetry of solutions to some nonlinear problems in statistical mechanics and geometry. Commun. Math. Phys. 160, 217–238.CrossRefGoogle Scholar
Chanillo, S. & Kiessling, M. K.-H. 1995 Conformally invariant systems of nonlinear PDEs of Liouville type. Geom. Funct. Anal. (to appear).Google Scholar
Chen, W. & Li, C. 1991 Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622.Google Scholar
Corngold, N. R. 1993 Virial equation for the two-dimensional pure electron plasma. Phys. Fluids B 5, 3847–3851.CrossRefGoogle Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics II. Wiley, New York.Google Scholar
Eyink, G. & Spohn, H. 1993 Negative temperature states and large-scale long-lived vortices in two-dimensional turbulence. J. Stat. Phys. 70, 833–886.CrossRefGoogle Scholar
Fine, K. S., Driscoll, C. F. & Malmberg, J. H. 1989 Measurement of a nonlinear diocotron mode in pure electron plasmas. Phys. Rev. Lett. 63, 2232–2235.CrossRefGoogle ScholarPubMed
Gidas, B., Ni, W. M. & Nirenberg, L. 1979 Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243.Google Scholar
Gidas, B., Ni, W. N. & Nirenberg, L. 1981 Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical Analysis and Applications, Part A, Advances in Mathematics Supplementary Studies, Vol. 7A (ed. Nachbin, L.), pp. 369–402. Academic Press, New York.Google Scholar
Gilbarg, D. & Trudinger, N. 1983 Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin.Google Scholar
Hardy, G., Littlewood, J. E. & Polya, G. 1934 Inequalities. Cambridge University Press.Google Scholar
Joyce, G. & Montgomery, D. C. 1973 Negative temperature states for the two-dimensional guiding-centre plasma. J. Plasma Phys. 10, 107–121.CrossRefGoogle Scholar
Kiessling, M. K.-H. 1988 Existence and uniqueness of the pressure profile p(A) for a current-carrying quasi-neutral equilibrium plasma. J. Math. Phys. 29, 1258–1263.Google Scholar
Kiessling, M. K.-H. 1993 Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Maths. 46, 27–56.Google Scholar
Kiessling, M. K.-H. & Lebowitz, J. L. 1994 Dissipative stationary plasmas: kinetic modeling, Bennett's pinch and generalizations. Phys. Plasmas 1, 1841–1849.Google Scholar
Levy, R. H. 1965 Diocotron instability in a cylindrical geometry. Phys. Fluids 8, 1288.Google Scholar
Li, C.-M. 1991 Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Commun. PDE 16, 585–615.CrossRefGoogle Scholar
Liouville, J. 1853 Sur l'équation aux différences Partielles ∂2 log λ/∂u ∂u ± λ /2a2 = 0. J. Math. Pures Appl. 18, 71–72.Google Scholar
Lund, S. M., Ramos, J. J. & Davidson, R. C. 1993 Coherent structures in rotating non-neutral plasma. Phys. Fluids B 5, 19–41.Google Scholar
Lundgren, T. S. & Pointin, Y. B. 1977 Statistical mechanics of two-dimensional vortices. J. Stat. Phys. 17, 323–355.Google Scholar
Marchioro, C. & Pulvirenti, M. 1994 Mathematical Theory of Incompressible Non Viscous Fluids. Springer-Verlag, Berlin.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Suppl. Nuovo Cim. 6, 279–287.CrossRefGoogle Scholar
Peurrung, A. J. & Fajans, J. 1993. A limitation to the analogy between pure electron plasmas end two-dimensional inviscid fluids. Phys. Fluids B 5, 4295–4298.Google Scholar
Reed, M. & Simon, B. 1980 Functional Analysis. Academic Press, New York.Google Scholar
Smith, R. A. & O'neil, T. M. 1990 Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding center plasma or discrete vortex system. Phys. Fluids B 2, 2961–2975.Google Scholar
Smith, R. A., O'neil, T. M., Lund, S. M., Ramos, J. J. & Davidson, R. C. 1992 Comment on the stability theorem of Davidson and Lund. Phys. Fluids B 4, 1373.CrossRefGoogle Scholar
Taylor, J. B, 1993 Filamentation, current profiles, and transport in a tokamak. Phys. Fluids B 5, 4378–4383.Google Scholar
Williamson, J. H. 1977 Statistical mechanics of a guiding center plasma. J. Plasma Phys. 17, 85–92.CrossRefGoogle Scholar