A 3D kinetic-fluid numerical code for stationary equilibrium states in magnetized plasmas

Abstract

In this paper we present a detailed description of a new stationary three-dimensional hybrid code with ions represented by particles and electrons by a massless fluid. We solve the ion equations of motion and ion distribution function moments are obtained. Conversely, we use a fluid approximation for the electrons. We determine the electron bulk velocity perpendicular to the magnetic field from the electron momentum equation with $m_e\to 0$ and without collisions. The parallel component of the electron bulk velocity is derived from the electron continuity equation. Once we have calculated the total current density, we determine the magnetic field from the Ampere's law, applying FFT algorithms, and from Faraday's law, we are able to calculate the components of the electric field. We use a recursive technique to obtain a self-consistent solution of our set of equations.

We use this code to study how electrons influence the structure of the magnetotail current sheet in many physical processes as, for example, the formation of a double peak in the current density, which is observed very often by the CLUSTER spacecraft crossing of the neutral sheet.

Introduction

There are three main techniques used to address issues concerning plasma physics: methods that examine the behavior of single charged particles; statistical methods, like kinetic theory; and fluid theory. Magnetohydrodynamics (MHD) methods have an important role in plasma research. This is attributable to the relative simplicity of their mathematical description (compared with the methods of kinetic theory) and to their ability to produce descriptions based on the use of averaged characteristics. Depending on the nature of the problem, the plasma can be treated either by the multiple fluid equations describing the different species present or the single fluid description can be used. The MHD codes are based on the fluid equations, that is plasma dynamics problem is approached from the point of view of a fluid.

Although the MHD approximation has been successfully used for several decades, the MHD theory is not suitable to describe some phenomena that are of considerable interest in physics. The MHD approximation breaks down for very weak magnetic fields (in this case the Larmor radius could be comparable or larger than the spacial scales of interest), in particular, in thin current sheets like those found in the Earth's magnetopause and magnetotail neutral sheets.

On the other hand, kinetic models, that treat the problem of plasma dynamics by addressing the motion of individual particles, produce a description of plasma behavior which is very detailed. Kinetic codes are the most accurate numerical simulation approach available for the simulation of plasma dynamics, but complete simulation, with all plasma components represented by a collection of discrete particles, is very demanding problem because of excessive computer resource requirements (primarily computer time). The electrons with relatively large ${\omega }_{pe}$ and ${\Omega }_{ce}$ have to be advanced at a relatively small $\mathrm{\Delta }{t}_e$, while the ions, with much smaller ${\omega }_{pi}$ and ${\Omega }_{ci}$, have to be advanced at a relatively large $\mathrm{\Delta }{t}_i$.

Summarizing, on one hand, MHD codes cannot capture all the physics in these plasma problems (e.g., finite Larmor radius effect). On the other hand, full particle codes are computationally demanding and it is not possible to simulate such large scale phenomena in three dimensions. To model phenomena that occur on scales which fall between longer scales obtained by magnetohydrodynamics simulations and shorter scales attainable by full particle simulations, it makes sense to use hybrid codes, that is codes which assume different models of the medium for different plasma components. Therefore hybrid codes are considered as a practical compromise. In hybrid algorithms one or more ion species are treated kinetically via standard PIC methods, Monte Carlo simulations or test particle methods used in particle codes, and the electrons are treated as a single charge neutralizing massless fluid.

Other types of hybrid models are possible (for a review see [27]), but hybrid codes with particle ions and massless electrons have become most common for simulating space plasma physics in the last decade. In such models, plasma is assumed quasi-neutral, which eliminates the need to solve the Poisson equation, and the displacement current is ignored in the Maxwell equations. In this approximation, the motion of plasma particles should be regarded as non-relativistic. To trace the evolution of the system on long time and large spatial regions, the fast electron dynamics must be completely eliminated. Electromagnetic hybrid codes (fluid electron, particle ions) have been used extensively in the past to study a variety of phenomenon occurring in space physics (see [28] and references therein). For example, a substantial number of large scale 1D, 2D and 2 and $\frac{1}{2}$ dimensional hybrid simulations have been involved to study magnetotail reconnection [4], [10], [12] and references therein, to investigate the macroscopic and microscopic structure of the dayside curved magnetopause [13] and to model the Earth's magnetosphere on global scales [24], [25]. These codes have proved to be quite good in reproducing the global configuration, comparing well with in situ observations; however, plasma dynamics evolve in a more complicated manner and 2D fields geometry is an obvious limitation.

This work addresses the development and use of a new stationary three-dimensional hybrid particle code to search for possible equilibrium magnetotail-like configurations. The interest for a new equilibrium structure stems from recent spacecraft observations in the earth's magnetotail which exhibit non conventional current profiles [16] and which have stimulated a number of new theoretical studies [20], [29].

In Section 2 we write the closed set of basic equations which describe the test particle simulation for ions and the fluid approximation for electrons and the numerical techniques we use to solve them. In Section 3 we specify the boundary conditions in order to apply the code for description of a magnetotail configuration. In Section 4 we present the numerical checks of the code and the simulation results. The conclusions are drawn in Section 5.