Longitudinal wave-breaking limits in a unified geometric model of relativistic warm plasmas

The covariant Vlasov-Maxwell system is used to study breaking of relativistic warm plasma waves. The well-known theory of relativistic warm plasmas due to Katsouleas and Mori (KM) is subsumed within a unified geometric formulation of the ‘waterbag’ paradigm over spacetime. We calculate the maximum amplitude Emax of non-linear longitudinal electric waves for a particular class of waterbags whose geometry is a simple 3-dimensional generalization (in velocity) of the 1-dimensional KM waterbag (in velocity). It is well known that the value of limv→c Emax (with the effective temperature of the plasma electrons held fixed) diverges for the KM model; however, we show that a certain class of simple 3-dimensional waterbags yields a finite value for limv→c Emax, where v is the phase velocity of the wave and c is the speed of light. Introduction Considerable effort has been devoted to developing compact accelerators employing the enormous electric fields present in plasma wakes driven by intense lasers [1] or charged particle beams [2] (see [3,4] for recent discussions). Conventional accelerators operate by exciting RF microwave cavities with klystrons and use the longitudinal electric component of a cavity mode to accelerate bunches of charged particles for subsequent collision. However, it is anticipated that electric field strengths in the next generation of accelerators will be so high that the RF cavity walls may undergo electrical breakdown [5]. To address this issue, researchers have turned to plasmabased acceleration mechanisms whose field can be orders of magnitude beyond that of conventional accelerators. Recent years have seen the on-going development of compact sources of intense electromagnetic radiation in the X-ray to THz frequency range [6] that employ laser-driven plasma acceleration. Such sources promise a wide range of applications in medicine, material science and security. A sufficiently short and intense laser pulse propagating through a plasma may create a travelling longitudinal plasma wave whose velocity is approximately the same as the laser pulse’s group velocity. However, it is not possible to sustain arbitrarily large electric fields; substantial numbers of plasma electrons become trapped in the wave and are accelerated, which dampens the wave. Indeed, the trapping phenomenon in longitudinal plasma waves lies at the heart of the original laser wakefield accelerator concept [1]. Although the evolution of a plasma wave dynamically trapping particles is complex, over the years much effort has been devoted to analytically understanding the upper bound (‘wave-breaking limit’) on the amplitude of plasma waves. Wave-breaking limits were first calculated for cold plasmas [7, 8] undergoing non-linear longitudinal electrostatic oscillations, and thermal effects were later included in non-relativistic [9] and relativistic [10–12] contexts. The results for the cold plasma are uncontroversial, but recent discussion [13–15] has uncovered difficulties with establishing an agreed analytical description of longitudinal wave-breaking in warm plasmas; in particular, it has been noted that different plasma models based on different assumptions yield different results. Models of non-linear plasma waves near breaking are approaching the limits of their domain of applicability, and different models exhibit different wave-breaking limits. Although recent experiments [16–18] operate in the 3-dimensional ‘bubble’ (or ‘blow-out’) regime [19] and exploit transverse wave-breaking [20], recent work [13–15] has rekindled interest in the theory of longitudinal wave-breaking. Recent discussion [13–15] includes comparison of the behaviour of the relativistic ‘waterbag’ model [10, 21] due to Katsouleas and Mori (abbreviated as KM) and a warm plasma model [12] due to Schroeder, Esarey and