“…The physics of plasma expansion, which is one of the most interesting topics in plasma science, has attracted a great deal of attention during the last two decades (Gurevich, Pariiskaya & Pitaevskii 1966;Farnsworth et al 1979;Sack & Schamel 1987;Lonngren 1990;Singh & Narayan 1990;Anisimov, Luk'Yanchuk & Luches 1996;Baitin & Kuzanyan 1998;Dorozhkina & Semenov 1998;Kovalev & Bychenkov 2003;Laha et al 2007;Huang et al 2008;Djebli 2010;Doggett & Lunney 2011;Fermous, Bennaceur-Doumaz & Djebli 2012;Akbari-Moghanjoughi 2015;Bennaceur-Doumaz et al 2015;McQuillen et al 2015). Plasma expansion in a vacuum is one of the important dominating processes in different plasma situations, such as the dynamics of laser produced plasmas (Perry & Mourou 1994), inertial confinement fusion plasmas (Lindl 1995), plasma clusters (Symes et al 2007), plasma thin foils (Maksimchuk et al 2000) and laser accelerated plasmas (Dawson 1964;Ditmire et al 1996Ditmire et al , 1997Coppa, D'Angola & Mulas 2011;D'Angola, Boella & Coppa 2014;Boella et al 2016), etc.…”