The reflection–transmission problem for inertial waves on geostrophic shear layers

Inertial waves in fluid regions of planets and stars play an important role in their dynamics and evolution, through energy, heat and angular momentum transport and mixing of chemicals. While inertial wave propagation in flows prescribed by solid-body rotation is well understood, natural environments are often characterised by convection or zonal flows. In these more realistic configurations, we do not yet understand the propagation of inertial waves or their transport properties. In this work, we focus on the interaction between inertial waves and geostrophic currents, which has thus far only been investigated using ray theory, where the wavelength is assumed to be small relative to the length scale of the current, or averaging/statistical approaches. We develop a quasi-two-dimensional analytical model to investigate the reflection and transmission of inertial waves in the presence of a localised geostrophic shear layer of arbitrary width and compare our theoretical findings with a set of numerical simulations. We demonstrate that, in contrast to ray theory predictions, partial reflections occur even in subcritical shear layers and tunnelling with almost total transmission is possible in supercritical shear layers, if the layer is thin compared with the wavelength. That is, supercritical shear layers act as low-pass filters for inertial wave beams allowing the low-wavenumber waves to travel through. Thus, our analytical model allows us to predict interactions between inertial waves and geostrophic shear layers not addressed by ray-based or statistical theories and conceptually understand the behaviour of the full wave field around and inside such layers.