The Properties of G-modes in Layered Semi-Convection

We study low frequency waves that propagate in a region of layered semi-convection. Layered semi-convection is predicted to be present in stellar and planetary interiors and can significantly modify the rate of thermal and compositional mixing. We derive a series of analytical dispersion relations for plane-parallel layered semi-convection in the Boussinesq approximation using a matrix transfer formalism. We find that like a continuously stratified medium, a semi-convective staircase -- in which small convective regions are separated by sharp density jumps -- supports internal gravity waves (g-modes). When the wavelength is much longer than the distance between semi-convective steps, these behave nearly like g-modes in a continuously stratified medium. However, the g-mode period spacing in a semi-convective region is systematically {\em smaller} than in a continuously stratified medium, and it decreases with decreasing mode frequency. When the g-mode wavelength becomes comparable to the distance between semi-convective steps, the g-mode frequencies deviate significantly from those of a continuously stratified medium (the frequencies are higher). G-modes with vertical wavelengths smaller than the distance between semi-convective steps are evanescent and do not propagate in the staircase. Thus, there is a lower cutoff frequency for a given horizontal wavenumber. We generalize our results to gravito-inertial waves relevant for rapidly rotating stars and planets. Finally, we assess the prospects for detecting layered semi-convection using astero/planetary seismology.