What is the Magnetic Greenhouse Effect?
We call magnetic greenhouse effect the trapping of gravity waves (waves for which the restoring force is buoyancy, akin to ocean waves) in the interior of a star due to the presence of a strong internal magnetic field. The trapping occurs because the (simple) geometry of an incoming wave is altered by the presence of a magnetic field, which generally has a complex geometry. As a result of this interaction, the incoming wave is “scattered” into a collection of waves with a lower degree of symmetry. These waves see a much larger evanescent region (a region they can hardly traverse, since there their amplitude decays exponentially) compared to the incoming wave and are prevented from escaping the core, where they remain trapped.

While the physics is somewhat different, the analogy is with the greenhouse effect in the Earth’s atmosphere. In that case part of the light coming from our Sun and hitting our planet’s surface is re-emitted as heat (longer wavelength, infrared radiation). This radiation is then trapped by greenhouse gases (like carbon dioxide), and prevented to leave the atmosphere.