3. Electric field generation by a neutral beam in a realistic plasma
3.1. Background plasma response - analytic description
Assuming again that protons are infinitely heavy (so ), then the motions of the beam and background electrons in the mean particle description can be expressed as:
where is the plasma density, while and are beam and plasma electron velocities. (Note that here this set of mean particle equations is used solely for the study of collisional effects and cannot describe strong collective effects such as the two-stream instability).
First, let us consider times sufficiently short so that - and so , and 0.
Then the previous equations can be rewritten:
Elimination of from Eqs. (16-18) gives:
the solution of which is
Now, using Eq. (23), the equation for the second time derivative of the relative beam electron velocity is
where is the combined plasma frequency of beam + plasma electrons.
The solution of Eq. (24) is
Now using Eq. (23) again and the definition of u we can express the velocity of the background plasma as
From these equations it can be seen that, as the beam electrons decelerate, the background plasma electrons react to the electric field (generated by the beam separation), increasing their velocities in the beam direction. Oscillatory terms appear in both equations (for and ) and have the same amplitude whereas the secular terms differ in size by a factor , as should be the case for a return current (see below).
The above describes the early evolution on plasma timescales, driven by the (`infinitely' massive) protons. The electric field created again oscillates, as for an unresponsive plasma background, but it is reduced compared to (8) by the factor since the free plasma electrons now provide part of the total zero current condition (). The beam electrons are thus enabled to separate from the protons with a differential speed growing with time at a rate depending on . In the limit the previous neutral gas solution is recovered.
One must next ask how the system evolves on longer timescales and, in particular, whether it approaches a steady state after, and averaged over, times long compared to . If we set all time derivatives in Eqs. (20) and (21) to zero we get
Since is monotonic (above thermal speeds) implies so that the beam and plasma electrons become indistinguishable, the motion of both being described by Ohm's Law and the total current being zero i.e. (the ambient protons remaining stationary). Whether this steady state is approached in practice depends on whether it is established on a timescale short compared to the beam lifetime as set by the actual stopping time of finite mass protons (during which the ambient protons will also react to E and make a small correction to j). By Eq. (24) this will occur on a time over which or
Thus provided (ie the beam is not much denser than the plasma) we expect the beam electrons to be essentially decoupled from the beam protons by the action of the plasma electron drift current to reduce E, with negligible runaway. In solar flare beam and plasma conditions this condition is very amply met.
3.2. Numerical simulation
To check these conclusions we simulated the above processes by a numerical electrostatic particle code (see Birdsall and Langdon, 1985; Peratt, 1992). Firstly, we consider the case of a neutral beam in an unresponsive plasma and then proceed to consider a realistic background plasma. A homogeneous infinite neutral beam, as studied in Sect. 2.1 cannot be simulated in a finite space code, even with periodic boundary conditions. The reason is that in the code there can be no charges at infinity and it is such charges which can be thought of as the source of E in this problem. To get around 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 L =20 ( = 256 grid points) with the beam initially between and , i.e. occupying half of the system length. We assume the protons to be infinitely heavy. As convenient numerical values, the beam and background densities were taken to be 1 and 100 respectively, the ratio of the total numbers of plasma and beam electrons in the system then being 200, because the beam occupies only one half of the system.
For the case of a neutral beam in an unresponsive plasma we considered the problem in the beam frame and took the initial velocities of both electron and proton components to be zero. Then at every time step =0.2 (31.4 steps = plasma period of background plasma = 2 ), to simulate the effect of collisions, we reduce the velocity of beam electrons by (where the velocity is expressed in the code velocity unit (). The result is shown in Fig. 1, where the oscillatory motion of the electron component of the neutral beam with the beam plasma period can be seen. This result confirms our previous conclusions of Sect. 2.1.
Now we investigate the case where the background plasma can respond. In this case, however, as shown in the following, there is a strong collisionless effect - the two-stream instability - not reproducible in a mean particle approach. In reality, though, a number of factors serve to reduce the effect of the two stream instability, eg a distribution of beam particle velocities. However, to examine the collisional effect numerically, we need to perform our simulation in an artificial reference frame where the neutral beam, which has zero velocity relative to the background plasma, is nevertheless forced to experience a collisional drag. If the angular plasma frequency of the background plasma is 1 and that of the beam is 0.1 then the beam/plasma density ratio is the same as the beam/gas density ratio in the previous case. Because all parameters of this simulation are the same as in the unresponsive plasma case, these situations can be simply compared. Results are shown in Fig. 2, where the mean velocities of the proton beam component (full line), electron beam component (dashed line), and background plasma electrons (dotted line - values are multiplied by 10 for better visualization) are depicted. Weak plasma oscillations of the background plasma were recognized in the evolution of the electrostatic field energy.
This simulation, despite being artificial so as to suppress collective effects, confirms clearly the very important effect of the previous analysis. That is, in the presence of a background plasma, electron and proton components of a neutral beam become spatially separated due to collisions. This is because the growth of E arising from the beam electron-proton separation current is neutralized by a background plasma electron current - i.e. instead of just .
3.3. The electric field at the beam head
It was shown in Sect. 3.1that collisional separation of electron and proton components of a neutral beam is possible only in the presence of a sufficiently dense background plasma (). Here we follow in detail the process of charge separation at the electron and proton head fronts. We used the following numerical simulation parameters: numerical system length , where is the Debye length, is the thermal plasma electron velocity; 10000 and 10000 numerical electrons and protons representing background plasma; time step ; proton-electron mass ratio is 100.
At position 50 along the beam we mimic/exaggerate the collisional separation process in the neutral beam as follows: one electron is stopped there (to confine the process under study within our limited numerical space we adopt an abrupt stop to zero velocity); a corresponding proton is allowed to propagate with the velocity in code unit () to the right up to the position 156 , where it is suddenly stopped. The evolution of the system is shown in Figs. 3a and 3b, for the velocities and positions of electrons and protons of the neutral beam, the potential, together with the charge density and electric field, for two times very early in the process. To see the neutralizing effect of the background plasma we add Figs. 4a and 4b where the same variables at the same times are shown for the beam evolution, but without the background plasma effects. We see that a double layer is formed with an electric field, where the electrons stop, of and where the protons stop of where is the particle flux.
Comparing Figs. 3 and 4 we see the plasma neutralizing effect. Near the position x=50 the electric field starts to be screened - see sharp peaks of the electric field at this position (Figs. 3a and 3b).
Simultaneously, the plasma between the two stopping places starts to oscillate with the plasma frequency. This oscillation process can be seen also in Fig. 5, where the plasma and beam currents are depicted. The beam current then increases up to a constant value over the finite propagation time of protons from where the electrons are stopped to where the protons are stopped. On the other hand, the plasma (return) current oscillates from zero to a value of twice the beam current so that the total mean charge current is zero and the instantaneous total current including the displacement current is zero - i.e. , as discussed previously.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998