  We prove the incompressible limit of compressible ideal
magnetohydrodynamic(MHD) flows in a reference domain where the magnetic field
is tangential to the boundary. Unlike the case of transversal magnetic fields,
the linearized problem of our case is not well-posed in standard Sobolev space
$H^m~(m\geq 2)$, while the incompressible problem is still well-posed in $H^m$.
The key observation to overcome the difficulty is a hidden structure
contributed by Lorentz force in the vorticity analysis, which reveals that one
should trade one normal derivative for two tangential derivatives together with
a gain of Mach number weight $\varepsilon^2$. Thus, the energy functional
should be defined by using suitable anisotropic Sobolev spaces. The weights of
Mach number should be carefully chosen according to the number of tangential
derivatives, such that the energy estimates are uniform in Mach number.
Besides, part of the proof is similar to the study of compressible water waves,
so our result opens the possibility to study the incompressible limit of
free-boundary problems in ideal MHD.
