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Abstract Cyclical variations of the solar magnetic fields, and hence the level of solar activity, are among the top interests of space weather research. Surface flows in global-scale, in particular differential rotation and meridional flows, play important roles in the solar dynamo that describes the origin and variation of solar magnetic fields. In principle, differential rotation is the fundamental cause of dipole field formation and emergence, and meridional flows are the surface component of a longitudinal circulation that brings decayed field from low latitudes to polar regions. Such flows are key inputs and constraints of observational and modeling studies of solar cycles. Here, we present two methods, local correlation tracking (LCT) and machine learning-based self-supervised optical flow methods, to measure differential rotation and meridional flows from full-disk magnetograms that probe the photosphere and $$\text{H}\alpha$$ H α images that probe the chromosphere, respectively. LCT is robust in deriving photospheric flows using magnetograms. However, we found that it failed to trace flows using time-sequence $$\text{H}\alpha $$ H α data because of the strong dynamics of traceable features. The optical flow methods handle $$\text{H}\alpha $$ H α data better to measure the chromospheric flow fields. We found that the differential rotation from photospheric and chromospheric measurements shows a strong correlation with a maximum of $$2.85~\upmu \text{rad}\,\text{s}^{-1}$$ 2.85 μrad s − 1 at the equator and the accuracy holds until $$60^{\circ }$$ 60 ∘ for the MDI and $$\text{H}\alpha$$ H α , $$75^{\circ }$$ 75 ∘ for the HMI dataset. On the other hand, the meridional flow deduced from the chromospheric measurement shows a similar trend as the concurrent photospheric measurement within $$60^{\circ }$$ 60 ∘ with a maximum of $$20~\text{m}\,\text{s}^{-1}$$ 20 m s − 1 at $$40^{\circ }$$ 40 ∘ in latitude. Furthermore, the measurement uncertainties are discussed. 

This white paper is on the HMCS Firefly mission concept study. Firefly focuses on the global structure and dynamics of the Sun's interior, the generation of solar magnetic fields, the deciphering of the solar cycle, the conditions leading to the explosive activity, and the structure and dynamics of the corona as it drives the heliosphere.