The properties of g-modes in layered semiconvection

We study low-frequency waves that propagate in a region of layered semiconvection. Layered semiconvection 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 semiconvection in the Boussinesq approximation using a matrix transfer formalism. We find that like a continuously stratified medium, a semiconvective 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 semiconvective steps, these behave nearly like g-modes in a continuously stratified medium. However, the g-mode period spacing in a semiconvective region is systematically 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 semiconvective 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 semiconvective steps are evanescent and do not propagate in the staircase. Thus, there is a lower cut-off frequency for a given horizontal wavenumber. We generalize our results to gravitoinertial waves relevant for rapidly rotating stars and planets. Finally, we assess the prospects for detecting layered semiconvection using astero/planetary seismology.