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Collisional relaxation: Landau versus Dougherty operator

  • Oreste Pezzi (a1), F. Valentini (a1) and P. Veltri (a1)


A detailed comparison between the Landau and the Dougherty collision operators has been performed by means of Eulerian simulations, in the case of relaxation toward equilibrium of a spatially homogeneous field-free plasma in three-dimensional velocity space. Even though the form of the two collisional operators is evidently different, we found that the collisional evolution of the relevant moments of the particle distribution function (temperature and entropy) are similar in the two cases, once an ‘ad hoc’ time rescaling procedure has been performed. The Dougherty operator is a nonlinear differential operator of the Fokker-Planck type and requires a significantly lighter computational effort with respect to the complete Landau integral; this makes self-consistent simulations of plasmas in presence of collisions affordable, even in the multi-dimensional phase space geometry.


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Anderson, M. W. and O'Neil, T. M. 2007a Eigenfunctions and eigenvalues of the Dougherty collision operator. Phys. Plasmas 14, 052 103.
Anderson, M. W. and O'Neil, T. M. 2007b Collisional damping of plasma waves on a pure electron plasma column. Phys. Plasmas 14, 112 110.
Bhatnagar, P. L., Gross, E. P. and Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511525.
Dougherty, J. K. 1964 Model Fokker-Planck equation for a plasma and its solution. Phys. Fluids 7, 113133.
Dougherty, J. K. and Watson, S. R. 1967 Model Fokker-Planck equations: part 2. The equation for a multicomponent plasma. J. Plasma Phys. 1, 317326.
Driscoll, C. F., Anderegg, F., Dubin, D. H. E. and O'Neil, T. M. 2009 Trapping and frequency variability in electron acoustic waves. In: New Developments in Nonlinear Plasma Physics (AIP Conf. Proc., 1188), (eds.) Eliasson, B. and Shukla, P. K., pp. 272–279.
Filbet, F. and Pareschi, L. 2002 A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case. J. Comput. Phys. 179, 126.
Hinton, F. L. and Hazeltine, R. D. 1976 Theory of plasma transport in toroidal confinement systems. Rev. Mod. Phys. 48, 239303.
Kogan, V. I. 1961 The rate of equalization of the temperatures of charged particles in a plasma. Plasma Phys. Problem Control. Thermonuclear React. 1, 153.
Kumar, K., D. and Das, N. 2014 Electron-ion collisional effect on Weibel instability in a Kappa distributed unmagnetized plasma. Phys. Plasmas 21, 042 106.
Landau, L. D. 1936 The transport equation in the case of the Coulomb interaction. In: Collected Papers of L. D. Landau, Pergamon Press, pp. 163170.
Lenard, A. and Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112, 14561459.
Livi, S. and Marsch, E. 1986 Comparison of the Bhatnagar-Gross-Krook approximation with the exact Coulomb collision operator. Phys. Rev. A 34, 533540.
Marsch, E. 2006 Kinetic physics of the solar corona and solar wind. Living Rev. Sol. Phys. 3.1, 1100.
O'Neil, T. M. 1968 Effects of Coulomb collisions and microturbulence on plasma wave echo. Phys. Fluids 11, 24202425.
Pareschi, L., Russo, L. G. and Toscani, G. 2000 Fast spectral methods for the Fokker-Planck-Landau Collision operator. J. Comput. Phys. 165, 216236.
Peyret, R. and Taylor, T. D. 1983 Computational Methods for Fluid Flow, Springer.
Pezzi, O., Valentini, F., Perrone, D. and Veltri, P. 2013a Eulerian simulations of collisional effects on electrostatic plasma waves. Phys. Plasmas 20, 092 111.
Pezzi, O., Valentini, F., Perrone, D. and Veltri, P. 2013b Phys. Plasmas 20, 092 111.
Pezzi, O., Valentini, F., Perrone, D. and Veltri, P. 2014a Erratum: Eulerian simulations of collisional effects on electrostatic plasma waves. Phys. Plasmas 21, 019 901.
Pezzi, O., Valentini, F. and Veltri, P. 2014b Kinetic ion-acoustic solitary waves in collisional plasmas. Eur. Phys. J. D 68, 128.
Spitzer, L. Jr, 1956 Physics of Fully Ionized Gases, Interscience Publishers.
Valentini, F., O'Neil, T. M. and Dubin, D. H. E. 2006 Excitation of nonlinear electron acoustic waves. Phys. Plasmas 13, 052 303.
Vlasov, A. A. 1938 On vibration properties of electron gas. J. Exp. Theor. Phys. 8, 291.
Zakharov, V. E. and Karpman, V. I. 1963 On the nonlinear theory of the damping of plasma waves. Sov. Phys. JETP 16, 351357.
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