# Simulation of Parameters of Plasma Dynamics of a Magneto Plasma Compressor
## 2 Model of the MPC Discharge and Numerical Solution Method  
The goal of this research is the development of a mathematical model as well suited as possible to the experimental conditions and the systematic numerical study of discharges in gases for a wide range of variation in basic parameters of the magneto plasma compressor and the ambient gas environment based on this model. The plasma formation processes of a magneto plasma compressor discharge in gas are generally three dimensional and some fluctuations are caused both by spatial and temporal inhomogeneity [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. The full model is presented in the literature [12; 14; 19]. In this paper, 3D effects are not taken into account and the mathematical model of the MPC discharge is presented on the basis of a 2D (axisymmetric) unsteady system of equations of viscous single-temperature radiation plasma dynamics. The term plasma dynamics used in the work implies a physical discipline that includes a mathematical description (in the form of a continuous medium) of plasma dynamics (taking into account the presence of electromagnetic fields and currents within them), radiation transfer processes in plasma (over a wide range of wavelengths), and its interaction with solids (electrodes, walls, etc.). The equation for the capacity is [26]:  
\[\frac{1}{\mathrm{c}^{2}}\frac{\mathrm{d}\mathrm{L}_{\mathrm{C}}\mathrm{J}}{ \mathrm{dt}}+\mathrm{R}_{\mathrm{C}}\mathrm{J}=\mathrm{U}_{\mathrm{k}},\ \frac{\mathrm{d}\mathrm{U}_{\mathrm{k}}}{\mathrm{dt}}=-\frac{\mathrm{J}}{ \mathrm{C}}\]  
where the conditions in the initial moment of time are \(\mathrm{t}=0\), \(\mathrm{J}=0\), \(\mathrm{U}_{\mathrm{k}}(\mathrm{t}=0)=\mathrm{U}_{0}\), \(\mathrm{J}\)--full current; \(\mathrm{U}_{0}=\mathrm{U}_{\mathrm{k}}(\mathrm{t}=0)\), \(\mathrm{U}_{\mathrm{k}}(\mathrm{t})\)--initial and current voltage on the C bank, respectively; \(\mathrm{R}_{\mathrm{C}}=\mathrm{R}_{0}+\mathrm{R}_{\mathrm{pl}}-\frac{1}{2 \mathrm{c}^{2}}\frac{\mathrm{d}\mathrm{L}_{\mathrm{pl}}}{\mathrm{dt}}\), \(\mathrm{L}_{\mathrm{C}}=\mathrm{L}_{0}+\mathrm{L}_{\mathrm{pl}}\); \(\mathrm{L}_{0},\ \mathrm{R}_{0}\)--inductance and resistance of the external circuit; \(\mathrm{L}_{\mathrm{pl}}=\frac{\mathrm{c}^{2}}{4\pi\mathrm{J}}\int\mathrm{H}^{ 2}\mathrm{d}\mathrm{V}\)--MPC plasma inductance; \(\mathrm{R}_{\mathrm{om}}=\frac{1}{\mathrm{J}}\int\limits_{\mathrm{V}}\frac{ \mathrm{i}^{2}}{\mathrm{\sigma}}\mathrm{dV}\), \(\mathrm{R}_{\mathrm{pl}}(\mathrm{t})=\frac{1}{\mathrm{J}}\int\limits_{0}^{ \mathrm{\ell}_{\mathrm{k}}}\int\limits_{0}^{\mathrm{\ell}_{\mathrm{m}}} \left(\overrightarrow{\mathrm{j}}\overrightarrow{\mathrm{E}}\right)\mathrm{dV}\), \(\mathrm{R}_{\mathrm{pl}}(\mathrm{t})=\frac{1}{\mathrm{d}^{2}(\mathrm{t})}\int \limits_{\mathrm{V}}\overrightarrow{\mathrm{V}}\left[\overrightarrow{ \mathrm{j}}\overrightarrow{\mathrm{H}}\right]\mathrm{dV}\)--respectively, ohmic, plasma, and plasma dynamic resistances of the discharge plasma. The radiation fields for seven spectral groups are calculated according to:  
\[\frac{1}{\mathrm{J}}\frac{\partial\left(\mathrm{J}\mathrm{q}_{\mathrm{i}\mathrm{ L}}\right)}{\partial\xi}+\frac{1}{\mathrm{J}}\frac{\partial\left(\mathrm{J} \mathrm{q}_{\mathrm{i}\mathrm{n}}\right)}{\partial\eta}+\chi_{\mathrm{i}} \mathrm{c}\mathrm{U}_{\mathrm{i}}=4\chi_{\mathrm{i}}\sigma_{\mathrm{i}}\tau^{ 4},\ \frac{\mathrm{c}}{3}\frac{\partial\mathrm{U}_{\mathrm{i}}}{\partial\xi}+\chi_{ \mathrm{i}}\mathrm{q}_{\mathrm{i}\mathrm{n}}=0,\ \frac{\mathrm{c}}{3}\frac{\partial\mathrm{U}_{\mathrm{i}}}{ \partial\eta}+\chi_{\mathrm{i}}\mathrm{q}_{\mathrm{i}\mathrm{n}}=0\]  
where \(\mathrm{U}_{\mathrm{i}}(\mathrm{y},\mathrm{z},\mathrm{t})\)--the energy density, \(\chi_{\mathrm{i}}\)--absorption coefficient. Functions of flow limiters are used in the calculations performed [27; 28]. The conductivity coefficient \(\chi_{\mathrm{E}}\) and viscosity coefficient \(\mu_{\mathrm{E}}\)[29], the ASTEROID computer system [30], and the Thomas-Fermi model [31] are used for calculations.