1953-41 International Workshop on the Frontiers of Modern Plasma Physics 14-5 July 008 Patterns of sound radiation behind point-like charged obstacles in plasma flows. H. Pecseli University of Oslo Institute for Physics Norway
1 Patterns of sound radiation behind point-like charged obstacles in plasma flows P. Guio Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom W.J. Miloch University of Oslo, Institute of Theoretical Astrophysics, Box 109 Blindern, N-0315 Oslo, Norway and School of Physics, The University of Sydney, Sydney, NSW 006, Australia H.L. Pécseli University of Oslo, Department of Physics, Box 1048 Blindern, N-0316 Oslo, Norway J. Trulsen University of Oslo, Institute of Theoretical Astrophysics, Box 109 Blindern, N-0315 Oslo, Norway
Basic form of the dielectric function with isothermal electrons and cold ions: ε ( 1 Ω pi, ) 1 ( ) ω k = + kλd ω Dispersion relation, ε(k,ω) = 0: ω( k) = C 1+ s k ( kλ De )
3 Basic assumtions: 1. All the relevant waves can be described by the (normalized) dispersion relation ω(k) = k /(1+k ), as found before.. The wavepattern has reached a stationary stage. 3. We consider only the far field, which can be described by geometrical optics.
4 Definitions of angles and directions:
5 In a moving frame: Ω( k) = ω(k) U k The group velocity for the isotropic case ω = ω( k ): k k Ω( k) = ug U = u g - U k k k Ω component parallel Ω component perpendicular to AC = to AC = u g cos( ψ ) u g U sin( ψ )
6 The direction of the group velocity must be parallel to AP. We then find: k Ω(k) tan( θ ) = U u g u sin( ψ ) g cos( ψ ) We have not yet used the form of ω = ω(k), only its isotropy.
7 Use u g = 1/(1+k ) 3/. Also, for given U we have the relation between k and ψ: Ω( k, ψ ) cos( ψ ) We find: = = ω( k) U tan( U (1 + k ) θ ) = U (1 + k ) 1 1+ k Uk cos( ψ ) = 0 1 1 gives
8 For U < 1 (i.e. Mach number < 1) we have a limiting minimum wavenumber (by cos(ψ) = 1/U (1+k ) < 0) k 0 = 1 U U. We have no such wavenumber for the supersonic case, U > 1
9 U = 0.85
10 U = 1.75
11 The maximum opening angle θ max in the Mach cone, for varying U:
Phase-fronts determined by: 1 dr k θ rdθ dr = giving = r k dθ r tan( θ + ψ ) NB. No distinction here between two or three spatial dimensions! Also, the same results are found for moving single charges or dipoles, for instance.
13 U = 1.33, 0.75, 0.4
14 Importance of ion Landau damping, here with T i = T e, assuming quasi-neutrality in the analysis:
15 Limiting case of fluid vs. kinetic model? Here T e = 5 T i.
16 Ion focusing:
17 T e = 80 T i. Insulating and conducting materials, here with U =.5 C s.
18 References: J. R. Sanmartin and S. H. Lam, Phys. Fluids 14, 6 (1971) K. R. Svenes and J. Trøim, Planet. Space Sci. 4, 81 (1994) P. Guio, W. J. Miloch, H. L. Pécseli, and J. Trulsen, Phys. Rev. E 78, 016401 (008) P. Guio and H. L. Pécseli, Phys. Plasmas 10, 667 (003) W. J. Miloch, H. L. Pécseli, and J. Trulsen, Nonlinear Processes Geophys. 14, 575 (007) W. J. Miloch, H. L. Pécseli, and J. Trulsen, Phys. Rev. E 77, 056408 (008)
19 Thank you for your attention!