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This article surveys the development of observational understanding of the interior rotation of the Sun and its temporal variation over approximately forty years, starting with the 1960s attempts to determine the solar core rotation from oblateness and proceeding through the development of helioseismology to the detailed modern picture of the internal rotation deduced from continuous helioseismic observations during solar cycle 23. After introducing some basic helioseismic concepts, it covers, in turn, the rotation of the core and radiative interior, the “tachocline” shear layer at the base of the convection zone, the differential rotation in the convection zone, the near-surface shear, the pattern of migrating zonal flows known as the torsional oscillation, and the possible temporal variations at the bottom of the convection zone. For each area, the article also briefly explores the relationship between observations and models.

We describe the defining observations of the solar cycle that provide constraints for the dynamo processes operating within the Sun. Specifically, we report on the following topics: historical sunspot numbers and revisions; active region (AR) flux ranges and lifetimes; bipolar magnetic region tilt angles; Hale and Joy’s law; the impact of rogue ARs on cycle progression and the amplitude of the following cycle; the spatio-temporal emergence of ARs that creates the butterfly diagram; polar fields; large-scale flows including zonal, meridional, and AR in-flows; short-term cycle variability; and helioseismic results including mode parameter changes.

We present evidence of hitherto undiscovered global-scale oscillations in the near-surface shear layer of the Sun. These oscillations are seen as large-scale variations of radial shear in both the zonal and meridional flows relative to their mean values. The variations cover all or most of a visible hemisphere, and reverse with a timescale on the order of a solar rotation. A large annual variation in the meridional shear anomaly is understandable in terms of the tilt of the rotation axis, but the rapid oscillations of the shear anomalies in both zonal and the meridional directions appear to be modulated in a more complex, not-quite-annual way, although the latter are also strongly modulated by the projected rotational axis angle. Small-scale anomalies in the neighborhood of active regions lend support to their solar origin and physical interpretation. These results were obtained by analyzing ring-diagram fits of low-order modes in high-resolution Doppler data from the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory.

Near-surface flows measured by the ring-diagram technique of local helioseismology show structures that persist over multiple rotations. We examine these phenomena using data from the Global Oscillation Network Group (GONG) and the Helioseismic and Magnetic Imager (HMI) and show that a correlation analysis of the structures can be used to estimate the rotation rate as a function of latitude, giving a result consistent with the near-surface rate from global helioseismology and slightly slower than that obtained from a similar analysis of the surface magnetic field strength. At latitudes of 60 ∘ and above, the HMI flow data reveal a strong signature of a two-sided zonal flow structure. This signature may be related to recent reports of “giant cells” in solar convection.

We studied the presence and spatiotemporal characteristics and evolution of the variations in the differential rotation rates and radial magnetic fields in the Schwabe and quasi-biennial-oscillation (QBO) timescales. To achieve these objectives, we used rotation rate residuals and radial magnetic field data from the Michelson Doppler Imager on the Solar and Heliospheric Observatory and the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory, extending from 1996 May to 2020 August, covering solar cycles 23 and 24, respectively. Under the assumption that the radial surface magnetic field is nonlocal and the differential rotation is symmetric around the equator, our results suggest that the source region of the Schwabe cycle is confined between ∼30° N and S throughout the convection zone. As for the source region of the QBO, our results suggest that it is below 0.78 R ⊙.

High-resolution helioseismology observations with the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) provide a unique three-dimensional view of the solar interior structure and dynamics, revealing a tremendous complexity of the physical processes inside the Sun. We present an overview of the results of the HMI helioseismology program and discuss their implications for modern theoretical models and simulations of the solar interior.

This chapter introduces the reader to the solar interior, in particular the convection zone. The first section explores the solar cycle and the dynamo models that have been studied to explain this cycle. The second section explores helioseismology observations and analysis of the solar interior, and reviews the fundamental knowledge that has been gained from these studies. The third and final section reviews observations and theory of magnetic fields emerging from the convection zone into the solar corona.

While the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) provides Doppler-velocity [ V ], continuum-intensity [ I C ], and line-depth [Ld] observations, each of which is sensitive to the five-minute acoustic spectrum, the Atmospheric Imaging Array (AIA) also observes at wavelengths – specifically the 1600 and 1700 Å bands – that are partly formed in the upper photosphere and have good sensitivity to acoustic modes. In this article we consider the characteristics of the spatio–temporal Fourier spectra in AIA and HMI observables for a 15-degree region around NOAA Active Region 11072. We map the spatio–temporal-power distribution for the different observables and the HMI Line Core [ I L ], or Continuum minus Line Depth, and the phase and coherence functions for selected observable pairs, as a function of position and frequency. Five-minute oscillation power in all observables is suppressed in the sunspot and also in plage areas. Above the acoustic cut-off frequency, the behaviour is more complicated: power in HMI I C is still suppressed in the presence of surface magnetic fields, while power in HMI I L and the AIA bands is suppressed in areas of surface field but enhanced in an extended area around the active region, and power in HMI  V is enhanced in a narrow zone around strong-field concentrations and suppressed in a wider surrounding area. The relative phases of the observables and their cross-coherence functions are also altered around the active region. These effects may help us to understand the interaction of waves and magnetic fields in the different layers of the photosphere and will need to be taken into account in multi-wavelength local-helioseismic analysis of active regions.

We study temporal variations in the amplitudes and widths of high-degree acoustic modes by applying the ring-diagram technique to the GONG+, MDI and HMI Dopplergrams during the declining phase of cycle 23 and rising phase of cycle 24. The mode parameters from all three instruments respond similarly to the varying magnetic activity. The mode amplitudes and widths show consistently lower variation due to smaller magnetic activity in cycle 24 as compared to the previous solar cycle.