Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorialβplane

Abstract

Context. 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°–30°, and decrease at higher latitudes. Aims. We search for an explanation that takes solar latitudinal differential rotation into account. Methods. In the equatorial β plane, we studied the propagation of linear Rossby waves (phase speed c < 0) in a parabolic zonal shear flow, U = − U̅ ξ2 < 0, where U̅ = 244 m s−1, and ξ is the sine of latitude. Results. 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 to the problem to account for wave attenuation. In the viscous case, the stream functions solve 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 ≤ Re ≤ 700), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode whose real frequency is close to that of the classical Rossby mode, which we call the R mode. For Re ≈ 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). Conclusions. 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 toward the equator.

Solar inertial modes: Observations, identification, and diagnostic promise

“…The high-latitude inertial modes are analogous to the "wall modes" in plane Poiseuille flows (Gizon et al 2020b). They are seen in both the 1D and 2D eigenvalue problems for m ≤ 5.…”

“…The power is strong above the critical latitude, but decreases toward the poles. The third example is the m = 3 equatorial Rossby modes (Löptien et al 2018) at a frequency of −269 nHz, for which the power is mostly confined to lie between the critical latitudes (±59 • for this mode's frequency) where the mode is trapped (Gizon et al 2020b).…”

“…Critical-latitude inertial modes are found for both the 1D and 2D models. Their amplitudes are maximum near their critical latitudes; they are known as "center modes" in 1D hydrodynamics (Gizon et al 2020b). The kinetic energy density of the m = 2 mode of the 2D model at −92 nHz (−73 nHz observed) is concentrated near the base of the convection zone near 45 • latitude (Fig.…”

“…In order to discretize the problem, we projected ψ onto a basis of associated Legendre polynomials. For the numerical value of the eddy viscosity at the surface, we used the value ν t = km 2 s −1 (Gizon et al 2020b), unless otherwise specified. The resulting eigenvalue problem was solved for each m using the eigenvalue solver scipy.linalg.eig.…”

“…Adding viscosity changes the eigenvalue problem from second order to fourth order (e.g., Baruteau & Rieutord 2013). The singularity disappears and new quasi-toroidal modes appear, which are analogous to those of the plane Poiseuille viscous flow in classical hydrodynamics (Gizon et al 2020b, and Send offprint requests to: L. Gizon, e-mail: gizon@mps.mpg.de references therein). In the following, we loosely refer to the modes with frequencies on the order of the rotational frequency as inertial modes.…”