Volume 110A, number 4 PHYSICS LETTERS 22 July 1985 NONLINEAR DYNAMICS IN EXPANDING PLASMAS Ch. SACK and H. SCHAMEL Institut fftr Theoretische Physik, Ruhr-Universiti~t Bochum, D-4630 Bochum 1, Fed. Rep. Germany Received 5 October 1984; revised manuscript received 25 January 1985; accepted for publication 15 May 1985 The expansion of a plasma occupying initially a half-space is investigated numerically and, by means of a novel description of the ion fluid, also analytically. A simple wave structure is found in the collisionless approximation. Stabilized by dissipation, the associated ion bunching gives rise to a fast ion component, similar to the ion blow-off in laser fusion. Three non-stationary regimes of this strongest nonlinear and inhomogeneous dynamical system are distinguished and discussed. For large t the ion front propagates with a speed proportional to (t - t 1 )1/2 where t 1 is a reference time. A simple picture emerges, explaining the diverse experimental data. Plasma expansion is believed to play an important role in areas such as laser-matter interaction, pellet ablation as means of refuelling, plasma scrape-off in divertor or limiter tokamaks, arcing on surfaces, polar or stellar winds in astrophysics, etc. Belonging to one of the oldest disciplines in plasma physics, extensive numerical, analytical and experimental investigations have been carried out. In this letter we shall show first of all that an expanding plasma develops dynam- ical structures that are yet unexplored and not yet described in the plasma literature. In an earlier paper [1 ] we have described our hydro- dynamic lagrangian expansion code and have inter- preted a singularity occurring in the ion flow as being due to a numerical instability. Here, we stress on the mathematical and physical side of the expansion prob- lem and offer a new interpretation of this singular plasma behaviour, part of which has been published elsewhere [2]. We assume that the plasma occupies initially a half- space, with some smooth, narrow transition, and that the ions are cold and without drift, initially. Normaliz- ed by the plasma quantities in the unperturbed region, the equations which govern the evolution, are the cold ion equations which are supplemented by Poisson's equation and by a Boltzmann law for the electrons: atn + ax(nU ) = O, ~t v + OaxO = -ax~ , ~2~b = n e - n, ne(~ ) = exp(~b). (1) 206 Fig. 1 is taken from ref. [1] and shows the spatial dependence of the ion density at different time steps; fig. la presents a global view whereas in fig. lb the leading front is drawn on a larger scale. We notice the appearance of a peaked ion front and the abovemen- tioned singularity. A collapse of the ion density takes place at x = 4.11 and t = 17.94, which is called hence- forth the critical point. Fig. 2 represents a local view of the front showing the final stage of the collapse in more detail. The ion velocity, plotted in fig. 2a, is seen to steepen progressively, and its negative gradient diverges at the critical point. The electric field, shown in fig. 2b behaves similarly with a reversed sign of the gradient. Interesting details of the ion density n, respec- tively of its inverse, are found in fig. 2c where V = 1/n, the specific ion volume is displayed. It is V-shaped with two individual prongs, and its minimum is found to decrease linearly, reaching zero at the critical point. The electric potential and the electron density, not shown here, remain well behaved and smooth. Our first aim will be to show that this collapse reflects an intrinsic nonlinear process of the system, with other words, it is not of a numerical origin as believed earlier [1 ]. We do this by establishing for the first time a single scalar wave equation represent- ing the entire system (1), and by solving this equa- tion perturbatively near the critical point. We introduce the lagrangian mass variable 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)