Topic/Type: 2.1 MHD, EHD & other fluid methods, Poster

### Singularly Perturbed ODE Models in Viscous Magnetohydrodynamics

Wieslaw Marszalek

DeVry University, North Brunswick, NJ, USA

We consider singularly perturbed resistive-viscous MHD equations of the form $\small u'=f(u,v,\lambda)$, $\small \epsilon v'=g(u,v,\lambda)$, where $\small '$ stands for derivative with respect to $\small \theta =x-st$, $\small s$ is the wave speed, $\small 0<\epsilon \ll 1$ and $\small \lambda$ is a parameter. Such systems of singularly perturbed MHD equations include the MHD models of intermediate shocks [1], when the resistivity $\small \eta$ and viscosity $\small \mu$ and/or $\small \nu$ are present and one of the viscosity parameters plays the role of "small" $\small \epsilon$. The $\small u=[B^y,B^z]$, two components of the magnetic induction vector ($\small B^x=const$) and $\small v$ is the velocity. When $\small \epsilon \rightarrow 0$ we obtain a system of differential-algebraic equations [2] (or DAEs) rather than singularly perturbed ODEs. The former have singularities which typically behave as impasse points, singular pseudo nodes, saddles, foci points, or singularity induced bifurcation (SIB) points [3]. The pseudo equilibrium and SIB points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Riemann sheets with either one or two analytic trajectories crossing the singularity (sonic) curve and other trajectories of lower smoothness.

In the paper we analyze the singularly perturbed MHD equations with "small" $\small \epsilon$ and examine their qualitative properties for various sets of resistivity $\small \eta$ and viscosity coefficients $\small \mu$ and $\small \nu$.

[1] C. C. Wu, "Formation, structure, and stability of MHD intermediate
shocks", J. Geophys. Research, vol.95, no.A6, 1990, pp.8149-8175.

[2] W. Marszalek, "Fold points and singularities in Hall MHD differential-algebraic equations", IEEE Trans. on Plasma Sci., vol.37, no.1, 2009, pp.254-260.

[3] R. Beardmore, "The singularity-induced bifurcation and its Kronecker normal form", SIAM J. Matrix Analysis, vol.28, no.1, 2001, pp.126-137.