Helioseismic Probing of Solar Variability: The Formalism and Simple Assessments

We derive formulae connecting the frequency variations in the spectrum of solar oscillations to the dynamical quantities that are expected to change over the solar activity cycle. This is done for both centroids and the asymmetric part of the fine structure (so-called even-a coefficients). We consider the near-surface, small-scale magnetic and turbulent velocity fields, as well as horizontal magnetic fields buried near the base of the convective zone. For the centroids we also discuss the effect of temperature variation. We demonstrate that there is a full, one-to-one correspondence between the expansion coefficients of the fine structure and those of both the averaged small-scale velocity and magnetic fields. Measured changes in the centroid frequencies and the even-a coefficients over the rising phase solar cycle may be accounted for by a decrease in the turbulent velocity of order 1%. We show that the mean temperature decrease associated with the net decrease in the efficiency of convective transport may also significantly contribute to the increase of the centroid frequencies. Alternatively, the increase may be accounted for by an increase of the small-scale magnetic field of order 100 G, if the growing field is predominantly radial. We also show that global seismology can be used to detect a field at the level of a few times 10⁵ G, if such a field were present and confined to a thin layer near the base of the convective envelope.