Rossby waves are a pervasive feature of the large-scale motions of the Earth’s atmosphere and oceans. These waves (also known as planetary waves and r-modes) also play an important role in the large-scale dynamics of different astrophysical objects such as the solar atmosphere and interior, astrophysical discs, rapidly rotating stars, planetary and exoplanetary atmospheres. This paper provides a review of theoretical and observational aspects of Rossby waves on different spatial and temporal scales in various astrophysical settings. The physical role played by Rossby-type waves and associated instabilities is discussed in the context of solar and stellar magnetic activity, angular momentum transport in astrophysical discs, planet formation, and other astrophysical processes. Possible directions of future research in theoretical and observational aspects of astrophysical Rossby waves are outlined.

We present a nonlinear magnetohydrodynamic shallow-water model for the solar tachocline (MHD-SWT) that generates quasi-periodic tachocline nonlinear oscillations (TNOs) that can be identified with the recently discovered solar “seasons.” We discuss the properties of the hydrodynamic and magnetohydrodynamic Rossby waves that interact with the differential rotation and toroidal fields to sustain these oscillations, which occur due to back-and-forth energy exchanges among potential, kinetic, and magnetic energies. We perform model simulations for a few years, for selected example cases, in both hydrodynamic and magnetohydrodynamic regimes and show that the TNOs are robust features of the MHD-SWT model, occurring with periods of 2–20 months. We find that in certain cases multiple unstable shallow-water modes govern the dynamics, and TNO periods vary with time. In hydrodynamically governed TNOs, the energy exchange mechanism is simple, occurring between the Rossby waves and differential rotation. But in MHD cases, energy exchange becomes much more complex, involving energy flow among six energy reservoirs by means of eight different energy conversion processes. For toroidal magnetic bands of 5 and 35 kG peak amplitudes, both placed at 45° latitude and oppositely directed in north and south hemispheres, we show that the energy transfers responsible for TNO, as well as westward phase propagation, are evident in synoptic maps of the flow, magnetic field, and tachocline top-surface deformations. Nonlinear mode–mode interaction is particularly dramatic in the strong-field case. We also find that the TNO period increases with a decrease in rotation rate, implying that the younger Sun had more frequent seasons.

Powerful ‘space weather’ events caused by solar activity pose serious risks to human health, safety, economic activity and national security. Spikes in deaths due to heart attacks, strokes and other diseases occurred during prolonged power outages. Currently it is hard to prepare for and mitigate the impact of space weather because it is impossible to forecast the solar eruptions that can cause these terrestrial events until they are seen on the Sun. However, as recently reported in Nature, eruptive events like coronal mass ejections and solar flares, are organized into quasi-periodic “seasons”, which include enhanced bursts of eruptions for several months, followed by quiet periods. We explored the dynamics of sunspot-producing magnetic fields and discovered for the first time that bursty and quiet seasons, manifested in surface magnetic structures, can be caused by quasi-periodic energy-exchange among magnetic fields, Rossby waves and differential rotation of the solar interior shear-layer (called tachocline). Our results for the first time provide a quantitative physical mechanism for forecasting the strength and duration of bursty seasons several months in advance, which can greatly enhance our ability to warn humans about dangerous solar bursts and prevent damage to satellites and power stations from space weather events.