Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T13:16:32.873Z Has data issue: false hasContentIssue false
  • English
  • Français

Convection in a plasma with opposed temperature and density gradients

Published online by Cambridge University Press:  13 March 2009

Luigi Nocera
Luigi Nocera
Affiliation:
Istituto di Fisica Atomica e Molecolare, Consiglio Nazionale delle Richerche, Via Giardino 7, I-56127 Pisa, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a viscous and heat-conducting plasma is considered, with temperature and density gradients arranged so that the resulting pressure is homogeneous, as suggested by some experiments on laser-plasma interactions. Gravity and magnetic field effects are ignored throughout. A non-dissipative fluid is considered first. Under the assumption of an Epstein profile for the equilibrium temperature, the dispersion curves for the propagation of sound waves are determined using a relaxation technique applied to a second-order spheroidal boundary-value problem for mode numbers 1,…, 5: it is found that these modes may be destablized by inhomogeneity. The dynamics of vorticity is analysed in the framework of the WKB approximation, which allows reduction to a fourth-order boundary-value problem: this is tackled in the ‘quasi-classical’ limit by solution of the relevant Bohr-Sommerfeld eigenvalue problem. Convective flows arise in the classically accessible layers and evanesce outside. For the same Epstein temperature profile as above, two such layers are formed, in agreement with observation. Given the growth rate of the unstable flow, the modified Rayleigh number, the width of the convective layer and the temperature drop across it are calculated as functions of the transverse wavenumber and for mode numbers 1–5. Conversely, given the modified Rayleigh number, the growth rate is calculated. The modified Rayleigh number and the growth rate thus found have a minimum and a maximum respectively as functions of the wavenumber of the perturbation. These can be taken as the most favourable values for convection to develop. The value of the wavenumber at which these extrema are attained can be used to determine the aspect ratio of the convective cells.

Type
Research Article
Information
Journal of Plasma Physics , Volume 46 , Issue 3 , December 1991 , pp. 391 - 406
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bassett, D. & WILLI, O. 1991 Experimental evidence of convection in laser produced plasmas. To be submitted.Google Scholar
Bassett, D., Willi, O. & Evans, R. G. 1987 Rutherford-Appleton Laboratory Annual Report 1987. Key, M. H. ed., Chilton, Oxon p. 57.Google Scholar
Book, D. L. 1979 J. Fluid Mech. 95, 779.CrossRefGoogle Scholar
Book, D. L. 1987 NRL Plasma Formulary. NRL Publication 0084–4040, Washington D.C.Google Scholar
Bouwkamp, C. J. 1950 Philips Res. Rep. 5, 87.Google Scholar
Brueckner, K. A., Jorna, S. & Janda, R. 1974 Phys. Fluids 17, 1554.CrossRefGoogle Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chiu, H. Y. 1967 Stellar Physics. Ginn Blaisdell.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Field, G. B. 1965 Astrophys. J. 142, 531.CrossRefGoogle Scholar
Hughes, D. W. & Proctor, M. R. E. 1988 Ann. Rev. Fluid Mech. 20, 187.CrossRefGoogle Scholar
Kopochenova, N. V. & Maron, I. A. 1975 Computational Mathematics. Mir.Google Scholar
Kull, H. J. 1989 Phys. Fluids B 1, 170.CrossRefGoogle Scholar
Kull, H. J. & Anisomov, S. I. 1986 Phys. Fluids 29, 2067.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1958 Quantum Mechanics. Pergamon.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Lifshitz, E. M. & Pitaevskii, L. P. 1981 Physical Kinetics. Pergamon.Google Scholar
Meixner, J. & SCHÄFke, F. W. 1954 Mathieusche Funktionen und Sphäroidfunctionen. Springer.CrossRefGoogle Scholar
Phillips, O. M. 1980 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes. Cambridge University Press.Google Scholar
Proctor, M. R. E. & Weiss, N. O. 1982 Rep. Prog. Phys. 45, 1317.CrossRefGoogle Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar