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\large{\textbf{Investigation of charged gas instability in 2D model of lightning discharge}}
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%%%% AUTHORS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\underline{Oleg V. KRAVCHENKO}$^{1}$,Vladislav I. PUSTOVOIT$^{2}$
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$^{1}$ \textit{Ph.D. student, Bauman Moscow State Technical University, Moscow, Russia}\\
$^{2}$ \textit{Academician RAS, Scientific and Technological Center of Unique Instrumentation RAS, Moscow, Russia}\\
Corresponding author's e-mail: \href{mailto:olekravchenkol@gmail.com}{olekravchenkol@gmail.com}\\
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It was introduced earlier that a homogenous charged media could become
unstable under some initial conditions in \cite{Pst06}.
The corresponding initial--value problem in static case
was investigated in \cite{Kr12} also. In present paper a 2D case of the model is considered in cartesian axes.
Assuming the following dimensionless variables
\begin{eqnarray*}
z = \frac{n_e}{n_i},\,u=\frac{v_e}{v_{\star}},\,l=\frac{2kT}{l E_{\star}},\,
y = \frac{E}{E_{\star}},\,\xi=\frac{x}{l},\,v_{\star}=\frac{eE_{\star}}{m\nu},\,
\beta^2 = \frac{2P_0}{D_{\star}},\,D_{\star} = \frac{E^2}{8\pi},\,
\tau = \frac{\varepsilon_0}{4\pi e\nu n_i},
\end{eqnarray*}
the system of government equations in dimensionless
vector form becomes
\begin{eqnarray}
\nabla n_e &+& n_e(y + v_e) = 0, \label{eq01}\\
\partial_t n_e &+& \nabla(n_e v_e) = 0, \label{eq02}\\
\nabla \cdot y &=& \frac{\beta^2}{2}(1 - n_e). \label{eq03}
\end{eqnarray}
It could be shown that Eqs.\,\eqref{eq01}--\eqref{eq03} are transform into one equation
\begin{equation}\label{eq04}
\nabla \biggl({\partial}_{\tau} y -  \bigtriangleup y - y\nabla{y} + \frac{\beta^2}{2}y\biggr)=0.
\end{equation}
Eq.\,\eqref{eq04} becomes the (2+1)--Burgers equation with a linear elastic term ($-y\beta^2/2$)
and source $\nabla g~=~0$ after one integration:
\begin{equation}\label{eq05}
{\partial}_{\tau} y =  \bigtriangleup y + y\nabla{y} - \frac{\beta^2}{2}y + g.
\end{equation}
Here $y=y(x,y,t)$ is an electric field currency, $n_e=n_e(x,y,t)$ is
an electrons density, and $g=g(t)$ is a source term are unknown functions in general case.
Numerical solution of the boundary--value problem of Eq.\,\eqref{eq05} gives
the electric field currency $y(x,y,t)$ evolution in time.
The evolution becomes irregular under different values of the source term $g(t)$
and parameter $\beta$.

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%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{thebibliography}{00}
\bibitem{Pst06} Pustovoit V.I.,
Journal of Comm. Tech. and Electr. (2006), \textbf{8}, 937-943.
\vspace{-10pt}
\bibitem{Kr12} Kravchenko O.V., PIERS Proc. (2012), Moscow, Russia, 1228-1231.
\end{thebibliography}

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