The structure of the shock front for an incoming cylindrical shock wave in a plasma

Abstract

The question of incoming shock waves, in particular those which have cylindrical symmetry, is of great theoretical and practical interest. In the hydrodynamics of an ideal fluid there is a well-known similarity solution for this problem (see [1]–[2]) which is of the nature of an asymptotic solution in a sufficiently small region (for a cylinder). In this paper we shall consider a cylindrical incoming shock wave in a situation where it is impossible to use the approximation of ideal hydrodynamics. The need for stating the problem in this way is especially clear when the characteristic dimension of the problem is comparable with the width of the structure of the shock wave front. The latter, as we know, is determined by the dissipative processes in the given medium. Thus when solving the problem we must start from a system of equations which include dissipative processes and adequately describe the structure of the shock wave front. In formulating the more exact system of equations we have restricted ourselves below to what is clearly the most interesting case, a sufficiently high degree of ionisation takes place in the medium and all the dissipative processes are determined by the Coulomb interaction of the charged particles. The paper [3] gives a system of equations which describe the plasma if the characteristic scale R of the problem is greater than, or of the order of, the mean free path λ i of the ions of the plasma R ≳ λ i − (kt 2) Ne 4Z 4L where k is Boltzmann' s constant, T the temperature of the plasma, N the number of ions per unit volume, e the elementary charge, Ze the mean ion charge, L the Coulomb logarithm. This system of equations is, in brief, that of two-temperature hydrodynamics, with heat conduction and the viscosity of the plasma taken into account, as well as the energy exchange between the ion and electron components of the plasma. The part played by radiation was taken into account in the equations of [3]. We shall ignore this here; the conditions under which this can be done are formulated in [3], [4]. The theory of the structure of a plane stationary shock wave in a plasma on the basis of this system of equations is developed in [3]–[6]. Mathematically, the problem reduces to the integration of one ordinary non-linear differential equation [6]. It must be noted that in this case the coefficient of viscosity of the plasma is assumed to be zero and at the same time the concept of a viscous isoelectronic thermal front is introduced (the thermal conduction in the ion component of the plasma was also ignored in comparison with the electronic heat conduction). These additional approximations were based, on the one hand, on the fact that consideration of the viscosity (and ion heat conduction) gives blurring only (and not the detailed structure! ) of any front over a width ~ λ i and plays almost no part in other parts of the structure of the stationary shock wave, and on the other hand, greatly complicates the whole problem mathematically. In this paper we integrate the system of initial partial differential equations numerically and directly. We shall especially be interested in the state of the plasma in a narrow axial region of width ~ λ i when an essentially non-stationary process of reflection of the shock wave from the axis takes place. Therefore here, in contrast to the theory of a plane stationary shock wave, we need to describe the structure of a viscous front, even though very roughly. We can give an effective continuous description of the structure of the viscous front by taking the viscosity of the plasma into account. It must again be emphasised that the detailed structure of the viscous front and of the near-axial region obtained with this approximation cannot be thought of as satisfactory from the physical point of view. Therefore, to ease the choice of viscosity formulae we have made several simplifications and, in general, as before, take no account of viscosity in the equations. Mathematically it is even convenient not to take viscosity into account in the equations if the methods which have been evolved for the numerical solution with continuous computing across the fronts of the shock waves are used. Instead of introducing the usual artificial viscosity we naturally use the “physical” viscosity of the plasma. The accuracy of the numerical method will correspond to the limits of validity of the initial system of equations noted in (1), in particular the viscous front will be blurred over the width ~ λ i T, N). Finally we make one general comment. The problem can be considered as an example of the direct numerical integration of a system of gas dynamical equations with viscosity and heat conduction, usually nonlinear. To do this it is only necessary to make the temperature of the ion and electron components of the plasma equal to one another, to modify the actual expressions for the coefficients of viscosity and heat conduction and the equation of state.