Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone

Abstract

Context. Several types of global-scale inertial modes of oscillation have been observed on the Sun. These include the equatorial Rossby modes, critical-latitude modes, and high-latitude modes. However, the columnar convective modes (predicted by simulations and also known as banana cells or thermal Rossby waves) remain elusive. Aims. We aim to investigate the influence of turbulent diffusivities, non-adiabatic stratification, differential rotation, and a latitudinal entropy gradient on the linear global modes of the rotating solar convection zone. Methods. We numerically solved for the eigenmodes of a rotating compressible fluid inside a spherical shell. The model takes into account the solar stratification, turbulent diffusivities, differential rotation (determined by helioseismology), and the latitudinal entropy gradient. As a starting point, we restricted ourselves to a superadiabaticity and turbulent diffusivities that are uniform in space. We identified modes in the inertial frequency range, including the columnar convective modes as well as modes of a mixed character. The corresponding mode dispersion relations and eigenfunctions are computed for azimuthal orders of m ≤ 16. Results. The three main results are as follows. Firstly, we find that, for m ≳ 5, the radial dependence of the equatorial Rossby modes with no radial node (n = 0) is radically changed from the traditional expectation (rm) for turbulent diffusivities ≳1012 cm2 s−1. Secondly, we find mixed modes, namely, modes that share properties of the equatorial Rossby modes with one radial node (n = 1) and the columnar convective modes, which are not substantially affected by turbulent diffusion. Thirdly, we show that the m = 1 high-latitude mode in the model is consistent with the solar observations when the latitudinal entropy gradient corresponding to a thermal wind balance is included (baroclinically unstable mode). Conclusions. To our knowledge, this work is the first realistic eigenvalue calculation of the global modes of the rotating solar convection zone. This calculation reveals a rich spectrum of modes in the inertial frequency range, which can be directly compared to the observations. In turn, the observed modes can inform us about the solar convection zone.

Identification of Inertial Modes in the Solar Convection Zone

“…If the radial displacements are small compared to the horizontal ones, then the difference would be mainly in the viscous term, which has in general only a weak impact on the modes' frequencies. In fact, even when compared with a fully compressible model that considers a realistic solar density profile (such as Bekki et al 2022, see their Figure 14), the results of our model differ at most by 15% for m = 2, and less than 2% for m = 16 Rossby-mode frequencies.…”

“…Such models, although numerically challenging, are in principle straightforward to develop. More refined inertial-wave models (e.g., Bekki et al 2022) Software: kore (Rekier et al 2019;Triana et al 2019), SHTns (Schaeffer 2013), PETSc/SLEPc (Balay et al 1997;Hernandez et al 2005;Dalcin et al 2011;Balay et al 2019;Roman et al 2022), MUMPS (Amestoy et al 2001(Amestoy et al , 2006, Matplotlib (Hunter 2007). We provide kore as an open source and freely available code via https://doi.org/10.5281/zenodo.…”

Radial Trapping of Thermal Rossby Waves within the Convection Zones of Low-mass Stars

“…Other examples can be viewed and downloaded at Hindman et al (2020b). The numerically computed eigenfunctions obtained by Bekki et al (2022) in spherical geometry also illustrate the cylindrical nature of the lower boundary of the wave cavity.…”

“…The unambiguous detection of Rossby waves in the Sun by Löptien et al (2018) has led to a flurry of observational efforts to characterize the waves and to search for other classes of inertial oscillations (e.g., Alshehhi et al 2019;Hanasoge & Mandal 2019;Liang et al 2019;Hanson et al 2020;Proxauf et al 2020;Gizon et al 2021;Hathaway & Upton 2021;Mandal et al 2021). As a result, there has been a resurrection of interest in inertial waves as they might apply to the Sun and solar-like stars (e.g., Lanza et al 2019;Damiani et al 2020;Gizon et al 2020;Cai et al 2021;Bekki et al 2022). In particular, Gizon et al (2021) have raised the intriguing possibility that inertial oscillations might be a sensitive seismic diagnostic for the properties of the deep convection zone, such as the radial profiles of the superadiabatic gradient and the turbulent viscosity.…”

Viscous inertial modes on a differentially rotating sphere: Comparison with solar observations

“…In this paper, we provide additional results based on the 1D solver. Additional results based on the 2D solver are provided in a companion paper (Bekki et al 2022).…”