Three-dimensional Bernstein-Greene-Kruskal (BGK) Modes in a Finite Magnetic Field

There has been renewed interest in the theory of Bernstein-Greene-Kruskal (BGK) modes, motivated by recent identifications of electrostatic solitary waves in space plasmas from spacecraft such as Geotail, Fast, Polar, Cassini, and Cluster. Some of these observations show 3D features, and thus require 3D solutions. While 1D BGK theory is quite mature, there appears to be no exact 3D solution in the literature, except for the limiting case when the magnetic field is infinitely strong [Chen & Parks, Geophys. Res. Lett., 29, 1331 (2002); Chen et al., Phys. Rev. E, 69, 055401(R) (2004)], as well as 3D BGK modes in zero magnetic field [Ng & Bhattacharjee, Phys. Rev. Lett., 95, 245004, 2005], and 2D BGK modes in finite magnetic field [Ng, Bhattacharjee & Skiff, Phys. Plasmas 13, 055903 (2006)]. Here we demonstrate that approximate 3D BGK modes can be constructed such that they tend to Chen-Parks solutions in the strong magnetic field limit, while the electron distributions are generalized to functions in full 3D velocity space. We also show that there exists a large portion of observed electrostatic structures for which the background magnetic field is strong enough that the approximation for such solutions is well satisfied. This work is supported by a National Science Foundation grant PHY-1004357.