Numerical Solution for 1.5 Dimensional, Time Dependent MagnetoHydrodynamic Solutions

Klaus Hain

NASA Goddard Space Flight Center

1967

In some important problems such as θ-Pinches with parallel fields (after the first phases of implosion or solar flares), the overall structure is governed by a strong magnetic field. The Alfen speed V_A across the field lines is so high that the transversal time t = R/V_A, where the radius R of the plasma is small compared with a characteristic time t_C of the system, so that a quasiequilibrium can be reached in the time interval t_C The dimension perpendicular to the field is therefore counted as 0.5 dimension. In contrast, the full magnetohydrodynamic equations apply parallel to the field lines. In particular the behavior of shockwaves parallel to the magnetic field is studied here under various conditions.

The geometry is as follows: axial symmetry perpendicular to the field-lines with a boundary for t=0 at R₀. The initial conditions for t=0 define an equilibrium. The time dependence is brought in by varying the magnetic field at the plasma boundary with time and as a function of z.

Two cases are studied here:

θ Pinch. A conducting wall is assumed at R = R_C. The magnetic is then compressed at one end with strength δB, and a frequency ω. Also a damping time τ is specified such that B ~ ē^t/τ
Simulation of the behavior of solar flares, which are assumed to have the structure of arcs consisting of magnetic fieldlines, produced by ring currents at the bottom. The following approximations are made: initially a constant magnetic field is assumed. The time dependent behavior is simulated by compressing the magnetic field. The field-lines become weaker if one assumes that the flux containing walls to recede further as they are further away from the current carrying coils.

To give a good indication of how the plasma behaves under those conditions the results will be shown in movies.

The computations for the θ pinch show that no strong shock waves (Mach number < 2) are produced by compressing the field, unless the field becomes so strong as to confine the plasma very strongly or if the distance to the wall becomes very small.

As for the second case, the solar flares, it shows that a strong and long lasting field is required to achieve an appreciable amount of change in the geometry.

In a forthcoming paper the properties of injection of plasma and the eventual reaching of a steady state including gravity will be studied.

INTRODUCTION

In many problems the magnetic field determines the main structure of the plasma. Dynamics in these cases can be studied by assuming quasistationarity perpendicular to the magnetic field, or expressed in another way: the time t_A=R/v_A,where R is the extent of the plasma perpendicular to the field lines, and v_A, the Alfen velocity, is small compared to the transversal time t_C (or another characteristic time); i.e. t_C=L/C≫t_A where L is the length of the cylinder and C the velocity of sound parallel to the magnetic field.

Here for the numerical computation axial symmetry is assumed. The magnetic field is initially parallel to the axis. The main advantage of the assumption of quasistationarity and axial symmetry is that one needs only relatively few space points in r-direction, so that the computation is reasonably fast, but the computer program can be extended to take into account any shape perpendicular to the field lines.

As the magnetic field is very strong, the field lines are taken as one set of (moving) coordinates. This excludes antiparallel field lines with points of zero magnetic fields.

In the following the magnetohydrodynamic equations are set up in a moving general coordinate system. Then the numeric scheme is discussed briefly. The results show how the strength of shock waves parallel to the magnetic field depends on the mirror ratio and the compression ratio. The results of the numerical computations are also represented in a movie to give an indication of the dynamical processes which may occur in a plasma.

The following assumptions are made here:

Assumption of quasistationarity across the field lines.
```
  t_A = R/σ_A ≪ t_C = L/σ_C     (1)
```
Infinite electrical conductivity is assumed, heat conductivity is neglected. (But it would not be diffucult to introduce heat conductivity parallel to the magnetic field).
Shocks parallel to the magnetic field are treated by the method of von Neumann.
Axial symmetry perpendicular to the magnetic field lines, finite cylinder of length L.
T_i=T_e The electron temperature is set equal to the ion temperature. This implies that the plasma is so dense that the temperature equalization time is short compared to t_C.

At the first maximum compression(t = 1.5 ≈π/2ω), the radius at the com-pression is roughly 0.3, and a weak SHOCK is developing at z=5.0 in the density and temperature distribution. The higher compression for the cases (2) and (3) compared to (1) are largely compensated by higher stiffness of the field originating from the nearby coil.

At the time t=3.0≈π/ω), as the compression fields move through zero there is a shock moving toward z=0 (Mπ1.3), and the density has a minimum at z=7. The Mach number is higher for case (3) where compression occurs at z = 0.0.

The last two graphs of cases (2) and (3) sho the distribution at the time t=4.5≈3π/2ω) and t=6.0≈2π/ω. Whereas the Mach number in case (3) remains below 2, in case (4) the Mach number gets to be of the order of M≈4. For laboratory plasma the last case is not interesting, but in the dynamics at the sun surface these results may have some bearing.

The shape of the plasma surface as shown in Fig. 1d shortly after the reflection of the shock at z=0 is quite typical; the shock wave is now approximately at z=3, and at this higher pressure the plasma is pushed aside.

Other computer runs have shown that it has been not possible to get higher Mach numbers than approximately 1.5. The temperature increases approximately 2-3 times . The conclusion is that in medium ≈ range (≈ ≈0.1) one can not produce strong shock waves (in z-direction). With magnetic field changes whose frequency is comparable to transversal time of a sound wave along the field lines, one can increase the pressure by a factor of about 4. The next three graphs show the change in the geometry with the second set of boundary conditions.

For ω=10.0, as the frequency is very low, the changes are very slow. Nevertheless, the differences in pressure may be a factor, too. As it was proposed that the breaking up of solar flares may be due to resistive instabilities, it is planned to extend the calculation so to take into account the asymmetry around the axis, in order to see if a geometry develops which allows an instability.

Figs. 5a, 5b show an equilibrium situation, where a source is on the bottom, which ejects plasma. As a boundary condition at z = z_e, v1=V_C (sound speed) was assumed. These kinds of situations could occur in sun spots, if the field lines which form the boundary no longer come back to the sun. The pressure and density then follow approximately the adiabatic law, which means that the entropy is approximately a constant along the field lines. Different runs were made for varying r_a. The results are almost identical, except for a greater radius r_a, magnetic field strength. as indicated in Fig. 5a, there appears a hump in the magnetic field strength. In Fig. 5b, the deviations indicated (near z=8.0) show the influence of the boundary condition for z=10.0. Before reaching this point the distributions are identical. At z=15.0 the ρ_H max is a bit higher than thermal speed.

ACKNOWLEDGEMENTS

The author would like to thank A. Jaggi for suggesting the equilibrium problem. Also he would like to thank the Culham Laboratories (UKAEA), England, where part of the work was performed. Special thanks go to E. Monasterski of the Theoretical Division, GSFC, who did most of the programming.