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Astron. Astrophys. 353, 729-740 (2000)

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2. Numerical simulations

We investigate the propagation of an initially monoenergetic neutral beam. For simplicity particles of the background plasma are assumed `immobile' so that they provide only beam electron scattering and collisional drag on the beam particles, i.e. as already stated, no beam current neutralization by the background plasma flow is considered.

If we assume a neutral beam with an infinite cross-section or a beam propagating along straight magnetic field lines, the electron beam heating can be considered as 1-D in the electric field and 2-D in electron beam velocities. Thus, to demonstrate the heating, a modified 1-D electrostatic particle numerical code can be used.

Even in the simplest case of an infinite homogeneous neutral beam, this purely temporal problem is complicated by the large range of time-scales involved - from the electron plasma period up to the proton collision time. Since the proton collision time is long compared to the electron collision time and extremely long compared to the electron plasma period, in this Sect. we simplify the problem by considering collisional effects on electrons only - i.e. we approximate beam protons as infinitely heavy. Moreover, to shorten computation times the energy losses and scattering of electrons are artificially increased. For this reason we are speaking only about a scaled demonstration of the initial stages of the heating process.

In the standard numerical electrostatic particle code (Birdsall & Langdon 1985) a homogeneous infinite neutral beam cannot be simulated, even with periodic boundary conditions. The reason is that in the code there can be no charge at infinity and it is such charges which can be thought of as the source of the charge separation electric field. To overcome this, we made a simulation using a code with periodic boundary conditions, but with the neutral beam shorter than the length of the system so that beam-end effects are present. For the length of the system we took [FORMULA] ([FORMULA] = 256 grid points) with the neutral beam initially between [FORMULA] and [FORMULA], i.e. occupying half of the system length. 10000 numerical electrons and 10000 numerical protons were considered. It is useful to make particle code computations in the frame of the beam protons, where the initial velocities of both electrons and protons are zero. Nevertheless, for scattering computations the initial beam speed in the background plasma frame of [FORMULA] = 10 code velocity units ([FORMULA]) needs to be defined (scaled code velocities are denoted by upper case `V').

Our numerical scheme then proceeds as follows. At each time step [FORMULA] units (1 unit [FORMULA] 0.01 beam plasma periods), we simulate the effect of collisions by reducing the total speed [FORMULA] of each electron by [FORMULA]. Here [FORMULA] is a scaled up collisional constant, [FORMULA] is the speed across the longitudinal (beam) direction and [FORMULA] is the longitudinal speed relative to the protons. This deterministic reduction in [FORMULA] has the effect of reducing [FORMULA] for each electron, by an amount determined by its pitch angle [FORMULA] which is also altered by collisions but in a stochastic fashion which we treat fully in accordance with the approach of Bai (1982). In subsequent time steps the electrons fall behind the protons creating an electric field (determined in the code by Poisson's equation) which accelerates them, bringing them back to the proton speed. The net result is a transfer of longitudinal proton energy into random perpendicular and longitudinal velocity components of the electrons.

The simulation described above was used in two regimes: a) with and b) without electron pitch angle scattering, energy losses due to collisions being considered in both cases. First, let us consider the case with scattering (ignored in Brown et al. 1998a). The computations were made up to 230 plasma periods of the beam electrons. Oscillations of the beam electron component, caused by collisional drag and the resulting electric field, are seen in Fig. 1, where the mean parallel component velocities (in the proton beam frame) are shown for both electrons (dashed line) and protons (thick line). In the bottom part of this figure there is an enlargement of the electron oscillations. The state of the electron velocity distribution at t = 230 plasma periods are shown in Fig. 2, as velocity components and the cosine [FORMULA] of electron pitch angles. The velocities of 10000 electrons are shown in units where the initial proton speed is 10. All initial perpendicular velocities were taken to be zero. The protons having constant speed stay inside the 64-128 grid, whereas electrons are kept there by the charge separation electric field. A small number of electrons escape for numerical reasons, but this effect is not important here. The most important result is that the electrons are heated, in that energy is accumulated in their perpendicular velocity distribution - see also Fig. 3, where the evolution of the electron distribution function in electron energy ([FORMULA]) is shown.

[FIGURE] Fig. 1. Time evolution of mean parallel-component velocities of electrons (dashed line) and protons (thick line). Velocities are expressed in the the proton beam frame. In the bottom part of the figure a detail of the electron component oscillation for 10 plasma periods is shown.

[FIGURE] Fig. 2. Electron distribution at t = 230 plasma periods for the case with scattering. VTOT is the total electron velocity in the background plasma frame, VPER is the perpendicular velocity, VX is the parallel velocity in the proton beam frame, MU is the cosine of the electron pitch angle. The initial beam velocity in the background plasma frame is 10 units.

[FIGURE] Fig. 3. Evolution of the electron distribution function during collisional deceleration and scattering. Its state at 45, 90, 135, 180, and 225 plasma periods is shown by the solid, short dashed, dot-dashed, dot-dot-dot-dashed, and long dashed lines, respectively. The electron energy equals VTOT*VTOT expressed in the background plasma frame.

To illustrate the key role of electron pitch angle scattering, computation of the case without scattering was also carried out (Fig. 4) and we see that the gain in energy is small, consistent with previous results presented in Brown et al. (1998a).

[FIGURE] Fig. 4. The same as Fig. 2 but without scattering.

From these scaled numerical simulations, we estimate that at the very beginning of this electron beam heating the heating rate is about the initial electron beam energy per electron collisional stopping time. To obtain the total heating gain of beam electrons the proton collisional deceleration needs to be included. If this heating rate continued the electrons would attain energies comparable to [FORMULA] as the protons stopped but in reality, discussed in the analytic treatment below, the heating rate of the electrons declines as they heat. However, such an extension of the computation is not possible in this type of numerical model because of beam end effects. To take the calculation further, having demonstrated the basic heating effect, we now revert to an approximate analytic treatment.

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999