# A Study of Current Controlled Discharge in a Nitrogen Filled Tube
## 3 Results
### Modelling of Laboratory Experiments  
Below we discuss results of simulations by two MHD codes, NPINCH and ZSTAR, which describe the behavior of plasma in magnetic field. These codes permit us to obtain all plasma parameters, magnetic and electric fields. We assume that the tube is pre-filled with non-ionized nitrogen gas of uniform density and temperature. The discharge is initiated by a pulse of current driven by an external circuit. After the breakdown, we can consider the total electric current to be a given function of time. We use the experimentally measured dependence of electric current (Figure 1b) on time to describe this function. Simulations have been performed for tube made from alumina with inner radius of \(4.6\,\mathrm{mm}\), filled by nitrogen at various pressures \((30,70,100,130,300)\,\mathrm{Pa}\).  
The initial breakdown of the discharge cannot be described by our models because the breakdown exhibits three-dimensional structure, and the effects of plasma non-quasi-neutrality are important. However, the initial stage is short compared with the full time of the discharge, and it does not affect the plasma parameters at later time. After the breakdown, the current pulse heats the plasma and creates an azimuthal component of the magnetic field. The plasma pressure is much higher than the magnetic field pressure. Simulations show that magnetic field pressure is more than ten times less than the plasma pressure. In the absence of the wall plasma would expand and not hold on in the initial radius.  
At first, as the electric current increases, the electron temperature \(T_{e}\) grows with time up to the order of \(1\,\mathrm{eV}\). It happens, approximately, when electric current reaches its maximum value. Due to energy exchange between ions and electrons, ion temperature is slowly growing and becomes equal to electron temperature at approximately \(3\,\mathrm{\SIUnitSymbolMicro s}\). Then, up to the end of discharge, both electron and ion temperatures grow very slowly. Almost all the energy of the discharge is spent on ionization of the tube filling gas. The electron density is  
Figure 4: Radial profiles of electron density averaged along Z axis measured for various time delays at filling pressure of \(100\,\mathrm{Pa}\).  
growing till the end of the current pulse. The characteristic time of the penetration of electric field in plasma, the skin time \(t_{\sigma}=4\pi\sigma R_{0}^{2}/c^{2}\), is much less than the hydrodynamic time \(t_{h}=R_{0}/c_{s}\), where \(c_{s}=\sqrt{(zT_{e}+T_{i})/m_{i}}\) is the ion sound velocity, \(m_{i}\) is the nitrogen ion mass, \(\sigma\) is plasma conductivity and \(c\) is velocity of light. As a result, the radial distribution of the electric field and electric current density are homogeneous and consequently, the plasma is heated and ionized locally. The radial profiles of plasma pressure, electron temperature and density are flat everywhere except near the wall. The plasma is cooled near the wall due to thermal conduction to the relatively cold tube wall. We see that due to the Ohmic heating, the plasma is ionized, and its electron density is growing, its radial profile is staying flat. The thermal conduction is not significant in the tube except region near the wall.