Exploring the latitude and depth dependence of solar Rossby waves using ring-diagram analysis

Context. Global-scale equatorial Rossby waves have recently been unambiguously identified on the Sun. Like solar acoustic modes, Rossby waves are probes of the solar interior.
Aims: We study the latitude and depth dependence of the Rossby wave eigenfunctions.
Methods: By applying helioseismic ring-diagram analysis and granulation tracking to observations by HMI aboard SDO, we computed maps of the radial vorticity of flows in the upper solar convection zone (down to depths of more than 16 Mm). The horizontal sampling of the ring-diagram maps is approximately 90 Mm (∼7.5°) and the temporal sampling is roughly 27 hr. We used a Fourier transform in longitude to separate the different azimuthal orders m in the range 3 ≤ m ≤ 15. At each m we obtained the phase and amplitude of the Rossby waves as functions of depth using the helioseismic data. At each m we also measured the latitude dependence of the eigenfunctions by calculating the covariance between the equator and other latitudes.
Results: We conducted a study of the horizontal and radial dependences of the radial vorticity eigenfunctions. The horizontal eigenfunctions are complex. As observed previously, the real part peaks at the equator and switches sign near ±30°, thus the eigenfunctions show significant non-sectoral contributions. The imaginary part is smaller than the real part. The phase of the radial eigenfunctions varies by only ±5° over the top 15 Mm. The amplitude of the radial eigenfunctions decreases by about 10% from the surface down to 8 Mm (the region in which ring-diagram analysis is most reliable, as seen by comparing with the rotation rate measured by global-mode seismology).
Conclusions: The radial dependence of the radial vorticity eigenfunctions deduced from ring-diagram analysis is consistent with a power law down to 8 Mm and is unreliable at larger depths. However, the observations provide only weak constraints on the power-law exponents. For the real part, the latitude dependence of the eigenfunctions is consistent with previous work (using granulation tracking). The imaginary part is smaller than the real part but significantly nonzero.