Spherical harmonic analysis of steady photospheric flows

Steady photospheric flows can be represented by a spectrum of spherical harmonic modes. A technique is described in which full disc doppler velocity measurements are analysed using the spherical harmonic functions to determine the characteristics of this spectrum and the nature of these flows. Synthetic data is constructed for testing this technique. This data contains limb shift, rotation, differential rotation, meridional circulation, supergranules, giant cells and various levels of noise.The data is analysed in several steps. First, the limb shift is calculated by finding the average velocity in concentric rings about disc center. A polynomial representation of the limb shift is then removed from the data. Secondly, the rotation profile is calculated by finding an average slope in the velocity across the disc at each latitude position. This rotation profile is fit with Legendre polynomials and removed from the data. The third step is to find the meridional circulation by calculating the spherical harmonic transform for the axisymmetric poloidal modes and correcting for the effects of the limb shift analysis. The final step is to calculate the full spectrum of spherical harmonic components for the convective flows. Supergranules are separated from giant cells by spectral filtering for high (l >32) and low (l <32) wavenumbers, respectively.Some information about the spectrum is lost because only one hemisphere is seen, only the line-of-sight velocity is measured and the measurements contain noise. The lack of information about the motions on the backside of the Sun produces a broad smearing of the spectrum into nearby modes. The lack of information about the transverse velocity component produces a mixing between modes whose longitudinal wavenumbers differ by two and between the poloidal and toroidal components with the same wavenumber. In spite of this mode mixing much can be learned from this analysis. Solar rotation and differential rotation can be accurately measured and monitored for secular changes. Meridional circulations with small amplitudes can be measured and monitored and giant cells can be separated from supergranules.