### 2.1 Global helioseismology

Granulation in the surface layers of the Sun is highly compressible (Mach numbers approaching or
exceeding unity) and is therefore a strong source of acoustic waves. These waves propagate
throughout the solar interior, reflecting off the surface and interfering with one another to form
global standing modes with characteristic periods of about five minutes. In this way the Sun
resonates with acoustic oscillations which can be used to probe its internal structure and
dynamics.
Helioseismic investigations typically begin with a stellar structure model. Resonant modes of oscillation
are then computed by considering linear, adiabatic perturbations about the spherically symmetric
background state obtained from these models. Perturbations are typically expressed in terms of spherical
harmonic basis functions in latitude and longitude, and in terms of eigenfunctions in radius
characterized by a radial order . The frequencies of these resonant oscillations depend on the spherical
harmonic degree, , the radial order, , and the properties of the background state, principally the
sound speed and the density (Christensen-Dalsgaard, 2002). The next step is to observe these oscillations
on the Sun and compare them to the theoretical predictions. Helioseismic measurements typically consist of
a time series of photospheric images in some dynamical variable such as the radial velocity as determined by
the Doppler shift of a spectral emission line. These observations are then subjected to spherical
harmonic transforms in space and Fourier transforms in time in order to determine oscillation
frequencies. The measured frequencies agree remarkably well with the theoretical predictions
(Gough, 1996; Christensen-Dalsgaard, 2002) and for low spherical harmonic degree ,
they are quantized, indicative of resonant oscillations. At higher spherical harmonic degree,
the frequencies are blurred in due to locally-excited traveling waves which have not yet
propagated around the solar sphere to interfere with other waves. These modes form the basis
of local-domain helioseismology which will be discussed in Section 2.2 (see also Gizon and
Birch, 2005).
Different oscillation modes are sensitive to different regions of the solar interior; for example, high-
modes sample only the near-surface layers whereas low- modes penetrate much deeper. The oscillation
frequencies are weighted integrals over the sampling region (loosely, the ray path) so some inversion
procedure is necessary to infer solar interior properties such as the variation of the sound speed with depth
(Christensen-Dalsgaard, 2002). The inversions are usually assumed to be linear so weighted summations
over different frequencies can be used to derive averaging kernals which are sensitive to localized regions of
the solar interior. Parametric representations may also be used, with minimization procedures to
determine the best fit to solar data. Global inversions generally become less reliable in the
polar regions and in the deep interior which are not well-sampled by observable oscillation
modes.

With regard to solar interior dynamics, the most important feature of global acoustic oscillations is their
so-called rotational splitting. In a non-rotating star, the frequencies of resonant acoustic oscillations are
independent of the spherical harmonic order (neglecting the asphericity caused by flows or magnetic
fields). This is no longer the case when the effects of rotation are included. The resulting frequency shifts
are small relative to the reference frequency so they can be reliably treated as perturbations. Helioseismic
inversions can then be used to infer the internal rotation profile as a function of latitude and radius as
shown in Figure 1.

A limitation of global helioseismology is that the inversions used to infer rotation profiles or structural
quantities such as sound speed are only sensitive to the component which is symmetric about the equator.
Furthermore, they are insensitive to meridional circulations and non-axisymmetric convective motions. In
order to probe such dynamics other techniques are necessary, the most promising being local
helioseismology.