Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial $\beta$ plane

Abstract

Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near $20^{\circ }$-$30^{\circ }$, and decrease at higher latitudes. We search for an explanation that takes into account solar latitudinal differential rotation. In the equatorial $\beta$ plane, we study the propagation of linear Rossby waves (phase speed $c<0$) in a parabolic zonal shear flow, $U=-\overline{U}{\xi }^2<0$, where $\overline{U}=244$ m/s and $\xi$ is the sine of latitude. In the inviscid case, the eigenvalue spectrum is real and continuous and the velocity stream functions are singular at the critical latitudes where $U=c$. We add eddy viscosity in the problem to account for wave attenuation. In the viscous case, the stream functions are solution of a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number $100\le Re\le 700$), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode with a real frequency close to that of the classical Rossby mode, which we call the R mode. For $Re\approx 300$, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model. Conclusion: Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers towards the equator.

Submitted to Astron. Astrophys. on 29 May 2020

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• Exploring the latitude and depth dependence of solar Rossby waves using ring-diagram analysis
• Predicting frequency changes of global-scale solar Rossby modes due to solar cycle changes in internal rotation

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