Geometrical description of non-linear electrostatic oscillations in relativistic thermal plasmas

We develop a method for investigating the relationship between the shape of a 1-particle distribution and non-linear electrostatic oscillations in a collisionless plasma, incorporating transverse thermal motion. A general expression is found for the maximum sustainable electric field, and is evaluated for a particular highly anisotropic distribution. Introduction High-power lasers and plasmas may be used to accelerate electrons by electric fields that are orders of magnitude greater than those achievable using conventional methods [1]. An intense laser pulse is used to drive a wave in an underdense plasma and, for sufficiently large fields, non-linearities lead to collapse of the wave structure (“wave-breaking”) due to sufficiently large numbers of electrons becoming trapped in the wave. Hydrodynamic investigations of wave-breaking were first undertaken for cold plasmas [2, 3] and thermal effects were later included in non-relativistic [4] and relativistic contexts [5–7] (see [8] for a discussion of the numerous approaches). However, it is clear that the value of the electric field at which the wave breaks (the electric field’s “wave-breaking limit”) is highly sensitive to the details of the hydrodynamic model. Plasmas dominated by collisions are described by a pressure tensor that does not deviate far from isotropy, whereas an intense and ultrashort laser pulse propagating through an underdense plasma will drive the plasma anisotropically over typical acceleration timescales. Thus, it is important to accommodate 3-dimensionality and allow for anisotropy when investigating wave-breaking limits. The sensitivity of the wave-breaking limit to the details of the plasma model suggests that it could depend on the anisotropy of the pressure tensor. One method for investigating the wave-breaking limit of a collisionless anisotropic plasma is to employ the warm plasma closure of velocity moments of the 1-particle distribution f satisfying the Vlasov-Maxwell equations [7]. Successive order moments of the Vlasov equation induce an infinite hierarchy of field equations for the velocity moments of f and at each finite order the number of unknowns is greater than the number of field equations. The warm plasma closure scheme sets the number of unknowns equal to the number of field equations by assuming that the terms containing the third order centred moment are negligible relative to those including second, first and zeroth order centred moments. Our aim is to uncover the relationship between wave-breaking and the shape of f . In general, the detailed structure of f cannot be reconstructed from a few low-order moments so we adopt a different approach based on a particular class of piecewise constant 1-particle distributions. Our choice of distribution, although somewhat artificial, reduces the Vlasov equation to that of a boundary in the unit hyperboloid bundle over spacetime. Combining the equation for the boundary with the Maxwell equations yields an integral for the wave-breaking limit in terms of the shape of the boundary. Our approach may be considered as a multi-dimensional generalization of the 1-dimensional relativistic “waterbag” model employed in [5]. 1 Vlasov-Maxwell equations The brief summary of the Vlasov-Maxwell equations given below establishes our conventions. Further discussion of relativistic kinetic theory may be found in, for example, [9,10]. We employ the Einstein summation convention