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Astron. Astrophys. 327, 662-670 (1997)

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5. The basic results of the numerical simulations

The two-step Lax-Wendroff differencing scheme was used for the numerical simulation (Press et al. 1988 ). It was performed in a quarter of the total box of simulation. It was assumed that the solution is symmetric in relation to the equator and to the axis of rotation. The dimensionless radius of the star was taken equal to 0.5. The outer boundary of the box of simulation was placed sufficiently far from the Fast Magnetosonic Surface (FMS). No signal can propagate from this boundary into the internal part of the box of simulation. We assumed continuous derivatives of all physical variables on the outer boundary. Boundary conditions on the axis of rotation and on the equator follow from the system of equations and from the symmetry of the problem in relation to the equator. On the stellar surface the boundary conditions were specified in accordance with our analysis (Bogovalov 1997 ). These boundary conditions are as follows

(i) The function [FORMULA] is specified on the stellar surface independently of time.

(ii) The tangential component of the electric field is continuous on the stellar surface.

(iii) The Lorentz - factor of the plasma on the stellar surface is equal [FORMULA].

(iv) The density of the matter flux on the stellar surface does not depend on time and is specified a priori.

(v) The temperature of plasma is equal to zero.
The number of the boundary conditions is equal to the number of MHD waves able to propagate from the stellar surface: one Alfven, one fast magnetosonic, two slow magnetosonic and one entropy wave. The toroidal magnetic field on the stellar surface was calculated from Eq. (29) under the assumption that the electric field in the star is known from the frozen in condition [FORMULA]. The simulation was performed on the latticies [FORMULA] and [FORMULA].

Two initial states were used for the simulation. In a first step of computations the magnetosphere of the nonrotating star was used as the initial state. These computations showed that after several periods of rotation the flow is transformed into the flow described in Sect. (3). This is why everywhere below we use the state of the magnetosphere obtained in Sect. (3) as the initial state of the flow.

The most important result from the numerical simulations is the discovery of an instability of the stationary relativistic plasma flow at the condition [FORMULA]. In the figures we use [FORMULA] instead [FORMULA]. They equal to each other at [FORMULA]. Our main purpose was to obtain reliable proves that this instability is not due to numerical effects.

First of all the dependence of the instability of the plasma flow on the ratio [FORMULA] was investigated. The simulation was performed for several values of this ratio. The results are presented in Fig. 1-4. The box of simulation is presented in these figures. The star is placed in the left lower corner of the simulation box. Dashed lines show the poloidal field lines. The solid lines show the lines of the poloidal electric currents. The Alfven (AS) and the fast magnetosonic surfaces (FMS) are presented in these figures. It follows from Fig. 1 that the plasma flow is stable when the kinetic energy of plasma dominates the Poynting flux. When these parameters become equal to each other on the equatorial field line the flow becomes nonstationary in a small region near the equator. The size of this region increases with the decrease of the ratio [FORMULA].

[FIGURE] Fig. 1. The result of numerical simulation of the relativistic plasma flow at low Poynting flux. The simulation was performed at the lattice dimension [FORMULA]. The flow is stable. There are sometimes small vorticities of the poloidal electric currents appear near the equator.

[FIGURE] Fig. 2. The result of numerical simulation of the relativistic plasma flow when the Poynting flux is equal to the flux of the kinetic energy of plasma on the equatorial field line. The simulation was performed at the lattice dimension [FORMULA]. The flow is stable. But the small vorticities of the poloidal electric currents appear more frequently.

[FIGURE] Fig. 3. The result of numerical simulation of the relativistic plasma flow when the Poynting flux slightly exceeds the flux of the kinetic energy of the plasma on the equatorial field line. The simulation was performed at the lattice dimension [FORMULA]. The flow becomes nonstationary. There is a region near the equator where the stationary flow is unstable.

[FIGURE] Fig. 4. The result of numerical simulation of the relativistic plasma flow when the Poynting flux remarkably exceeds the flux of the kinetic energy of plasma on the equatorial field line. The simulation was performed at the lattice dimension [FORMULA]. The flow is nonstationary.

Another way to demonstrate the physical reality of the instability of the stationary plasma flow is to investigate the dependence of this instability on the spatial resolution of the lattice used for the numerical simulation. Fig. 5 and 6 show the results of numerical simulation on a lattice twice as large as the lattice used for the numerical simulation presented in Fig. 4. Fig. 5 shows the results of the numerical simulation for the same box as in Fig. 4. It is seen that the instability of the stationary flow develops twice faster than in similar case shown in Fig. 4. This behavior is typical for real physical instability. This instability is supressed by positive numerical viscosity and conductivity. The improvement of the spatial resolution decreases the numerical dissipation. The growth rate of perturbations increases. It is this behavior that we observe in Fig.5. In the case of artificial numerical instability the dependence of the growth rate on the spatial resolution is opposite. The numerical instability is due to the negative numerical viscosity and conductivity. They decrease with an improvement of the spatial resolution. This is why for the numerical instability we have to expect another dependence of the growth rate of the instability on the spatial resolution. The numerical instability is supressed with improvement of the spatial resolution.

[FIGURE] Fig. 5. The result of numerical simulation of the relativistic plasma flow at the same parameters as in the previous figure. The simulation was performed at the lattice dimension [FORMULA]. The growth rate increased twice.

Fig. 6 shows the dependence of the flow on the position of the outer boundaries of the box of simulation. The simulation in this figure is performed on the lattice with dimension [FORMULA] but in the box twice more than in Fig. 4. No evidence of the dependence of the flow on the position of the outer boundaries was found.

[FIGURE] Fig. 6. The result of numerical simulation of the relativistic plasma flow at the same parameters as in the previous figure. The simulation was performed at the lattice dimension [FORMULA], but the size of the box of simulation is also increased twice. This figure shows that the instability does not depend on the position of the outer boundary.

So, the dependence of the character of the flow on the relationship between the Poynting flux and the flux of the kinetic energy, the dependence of the growth rate of perturbations on the spatial resolution of the lattice used for the numerical simulation and independence of the flow on the position of the outer boundaries prove that the instability of the stationary flow of relativistic plasma at high [FORMULA] is likely physically real. The determination and investigation of the mechanism of the instability is beyond the scope of this paper. It will be performed in future works. Here we stress some properties of the instability which can be useful for the analytical analysis. The simulation dealt with the time dependent axisymmetric problem. It means that the stationary plasma flow is unstable with respect to the axisymmetrical perturbations. The question about the role of nonaxisymmetrical perturbations is open. It is important to note that the nonstationarity of the plasma flow takes place only in the superfast magnetosonic region. We have found no cases where the relativistic plasma flow was nonstationary in the subsonic region.

Nonstationarity of the plasma flow leads to the acceleration of plasma. Fig. 7 shows the plot of the Lorentz - factor of the plasma with clear evidence of the plasma acceleration.

[FIGURE] Fig. 7. The plot of Lorentz - factor of the plasma at the nonstationary flow of the Poynting flux dominated plasma. The nonstationarity of the flow leads to the acceleration of the plasma.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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