Doppler velocity of $m=1$ high-latitude inertial mode over the last five sunspot cycles

Among the identified solar inertial modes, the high-latitude mode with azimuthal order $m=1$ (HL1) has the largest amplitude and plays a role in shaping the Sun's differential rotation profile. We aim to study the evolution of the HL1 mode parameters, utilizing Dopplergrams from the Mount Wilson Observatory (MWO), GONG, and HMI, covering together five solar cycles since 1967. We calculated the averages of line-of-sight Doppler signals over longitude, weighted by the sine of longitude with respect to the central meridian, as a proxy for zonal velocity at the surface. We measured the mode's power and frequency from these zonal velocities at high latitudes in sliding time windows of three years. We find that the amplitude of the HL1 mode undergoes very large variations, taking maximum values at the start of solar cycles 21, 22 and 25, and during the rising phases of cycles 23 and 24. The mode amplitude is anticorrelated with the sunspot number (corr=$-0.50$) but not correlated with the polar field strength. Over the period 1983-2022 the mode amplitude is strongly anticorrelated with the rotation rate at latitude $60^\circ$ (corr=$-0.82$), i.e., with the rotation rate near the mode's critical latitude. The mode frequency variations are small and display no clear solar cycle periodicity above the noise level ($\sim \pm 3$~nHz). Since about 1990, the mode frequency follows an overall decrease of $\sim 0.25$ nHz/year, consistent with the long-term decrease of the angular velocity at $60^\circ$ latitude. We expect that these very long time series of the mode properties will be key to constrain models and reveal the dynamical interactions between the high-latitude modes, rotation, and the magnetic field.