Effects of differential rotation and meridional circulation in solar oscillations of high degree l

S. V. Vorontsov 0 1 0 Institute of Physics of the Earth , B. Gruzinskaya 10, Moscow 123810 , Russia 1 Astronomy Unit, Queen Mary, University of London , Mile End Road, London E1 4NS A B S T R A C T Helioseismic measurements at high degree l are sensitive to mode couplings induced by large-scale flows, with the largest contribution coming from the solar differential rotation. The coupling becomes strong at high degree l, when frequency spacings between interacting modes become small, calling for a more accurate theoretical prediction of the mode-coupling effects in helioseismic data analysis. In this paper, the currently available description based on the quasi-degenerate perturbation theory is developed to a higher order, and extended to include the possible contributing effects of solar meridional circulation. A semi-analytic asymptotic description is developed for effects of the centrifugal forces on the oscillation frequencies of high degree l. Numerical estimates of the corresponding effects in the observational data are discussed. 1 I N T R O D U C T I O N Sounding solar subsurface layers with p modes of high degree l is the most challenging area in global solar seismology. The diagnostic potential of the high-degree modes is exceptionally important. When the horizontally averaged component of the solar stratification is addressed by helioseismic inversions (measurement of the solar He abundance, calibration of the equation of state), a very high accuracy can potentially be achieved by properly averaging the huge amount of data (2l + 1 modes of different azimuthal order m at each l and radial order n). In helioseismic inversions with modes of lower degree l, targeted at the diagnostic of the deep solar interior, the properties of the outer layers restored with high-degree modes provide invaluable outer boundary conditions. When horizontal inhomogeneities are addressed (active regions, fluid flows on a supergranular scale or below), modes of high degree l are the only modes that allow proper spatial resolution in the horizontal dimensions. Since modes of the high-degree domain can be interpreted in terms of surface waves (e.g. Vorontsov 2006), it is the domain where we have an important link between methods of global and local helioseismology, a link which still needs a deeper theoretical investigation. A large volume of high-resolution Doppler velocity measurements has been accumulated with the Solar and Heliospheric Observatory (SOHO) MDI instrument in observations covering more than a solar cycle, and new data of better spatial resolution are now coming from the Solar Dynamics Observatory (SDO) HMI project. The major difficulties are the accurate measurements of the oscillation frequencies (Korzennik, Rabello-Soares & Shou 2008; Larson & Schou 2008; Rabello-Soares, Korzennik & Shou 2008). Principally, the difficulties come from spatial leakage in the sphericalharmonic decomposition of the Doppler-velocity images, which arises because spherical harmonics are not orthogonal over a visible hemisphere. At high degree, modes of the same radial order become closely separated in frequency, resonant line profiles become wider and individual spatial leaks blend into a continuous ridge in the observational power spectra. To recover the underlying mode parameters, we need an accurate response function (leakage matrix; see e.g. Vorontsov & Jefferies 2005), an accurate model of the acoustic line profiles (e.g. Jefferies, Vorontsov & Giebink 2006) and an accurate physical description of the mode-coupling effects induced by internal velocity fields, effects which become strong at high degree l. This study is focused on the effects of mode coupling. The mode coupling induced by advection effects caused by solar differential rotation was first addressed by Woodard (1989); a more accurate description, convenient for helioseismic data analysis, was developed by Vorontsov (2007). When taken into account in the observational data analysis, this effect allows us to eliminate the apparent systematic errors in frequency measurements in the degree range up to somewhere between l = 150 and 200 (Vorontsov et al. 2009). Further (unpublished) attempts to extend the degree range to higher l reveal that the systematic errors reappear again, accompanied with rapid degradation in the quality of the theoretical fits to the observational power spectra. Since mode coupling becomes strong, and its theoretical description is based on a perturbational analysis, we have to address the accuracy of the theoretical description, as well as s=1,2,... vmer = s=1,2,... which is equivalent to The fluid flow v is assumed to be stationary, and hence vmer satisfies the mass-conservation equation smaller contributions from sources other than differential rotation, first of all the effects of solar meridional circulation. In Section 2, we extend the perturbational analysis of the effects of differential rotation to higher order by lifting the simplifying assumptions adopted in the previous work (Vorontsov 2007). The effects of the meridional circulation are addressed in parallel, within a single analysis. Together with higher order effects in the mode-coupling coefficients, we address the corresponding effects induced in the oscillation frequencies. In Section 3, we develop a semi-analytical description of frequency corrections produced by centrifugal effects. Our numerical estimates are described in Section 4, and Section 5 contains a short summary. 2 D I F F E R E N T I A L R O TAT I O N A N D M E R I D I O N A L C I R C U L AT I O N Throughout this paper, we use a tilde to designate the eigenfrequencies and eigenfunctions u of linear adiabatic oscillations of a non-rotating Sun with no meridional flows. In the operator form with the equilibrium density 0 = 0(r), self-adjoint operator H and displacement field where 1 = / + sin1 / is the angular part of the gradient operator and the time dependence is separated as exp(it ). As in this paper we only address the axisymmetric components of the fluid flows, mode interaction is limited to modes of the same azimuthal order m. At high degree l, the interaction is also limited by modes of the same radial order n (we have small frequency separations along the p- and f-mode ridges). Therefore, we drop n and m from indexing the solutions for brevity, when this does not lead to confusion. In the perturbational analysis which follows, we assume the unperturbed eigenfunctions to be normalized as with the scalar product defined as (u1, u2) = u1 u2 dv, v = vrot + vmer, vrot = ws (r) = s=1,2,... d (0r2us) = 0rs(s + 1)vs, s = 1, 2, . . . , (13) dr with the surface boundary conditions us(R) = 0. We represent the self-adjoint operator H by the sum of two operators induced by the toroidal (differential rotation) and poloidal (meridional circulation) flows, H = Hrot + Hmer. The general expressions for the matrix elements of H can be found in Lavely & Ritzwoller ( (...truncated)