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The goal of this paper is twofold. On the one hand, we introduce aquasi-homogeneous version of the classical ideal MHD system and study itswell-posedness in critical Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, $d\geq2$,with $1<p<+\infty$ and under the Lipschitz condition $s>1+d/p$ and$r\in[1,+\infty]$, or $s=1+d/p$ and $r=1$. A key ingredient is thereformulation of the system \textsl{via} the so-called Els\"asser variables. Onthe other hand, we give a rigorous justification of quasi-homogeneous MHDmodels, both in the ideal and in the dissipative cases: when $d=2$, we willderive them from a non-homogeneous incompressible MHD system with Coriolisforce, in the regime of low Rossby number and for small density variationsaround a constant state. Our method of proof relies on a relative entropyinequality for the primitive system, and yields precise rates of convergence,depending on the size of the initial data, on the order of the Rossby numberand on the regularity of the viscosity and resistivity coefficients.

Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system