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Retrograde-propagating waves of vertical vorticity with longitudinalwavenumbers between 3 and 15 have been observed on the Sun with a dispersionrelation close to that of classical sectoral Rossby waves. The observedvorticity eigenfunctions are symmetric in latitude, peak at the equator, switchsign near $20^\circ$-$30^\circ$, and decrease at higher latitudes. We searchfor an explanation that takes into account solar latitudinal differentialrotation. In the equatorial $\beta$ plane, we study the propagation of linearRossby 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 oflatitude. In the inviscid case, the eigenvalue spectrum is real and continuousand 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-ordermodified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. Forreasonable values of the eddy viscosity corresponding to supergranular scalesand above (Reynolds number $100 \le Re \le 700$), all modes are stable. Atfixed longitudinal wavenumber, the least damped mode is a symmetric mode with areal frequency close to that of the classical Rossby mode, which we call the Rmode. For $Re \approx 300$, the attenuation and the real part of theeigenfunction is in qualitative agreement with the observations (unlike theimaginary part of the eigenfunction, which has a larger amplitude in the model.Conclusion: Each longitudinal wavenumber is associated with a latitudinallysymmetric R mode trapped at low latitudes by solar differential rotation. Inthe viscous model, R modes transport significant angular momentum from thedissipation layers towards the equator.

astronomy and astrophysics

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