Among the identified solar inertial modes the highlatitude mode with azimuthal order m1 HL1 has the largest amplitude and plays a role in shaping the Suns 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 lineofsight 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 modes 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 corr0.50 but not correlated with the polar field strength. Over the period 19832022 the mode amplitude is strongly anticorrelated with the rotation rate at latitude 60circ corr0.82 i.e. with the rotation rate near the modes critical latitude. The mode frequency variations are small and display no clear solar cycle periodicity above the noise level sim pm 3nHz. Since about 1990 the mode frequency follows an overall decrease of sim 0.25 nHzyear consistent with the longterm decrease of the angular velocity at 60circ 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 highlatitude modes rotation and the magnetic field.
['Astrophysics - Solar and Stellar Astrophysics']
Doppler velocity of m1 highlatitude inertial mode over the last five sunspot cycles
{'Doppler velocity of m = 1 high-latitude inertial mode over the last five sunspot cycles': 'Zhi-Chao Liang 1 and Laurent Gizon 1 , 2 , 3  \n- 1 Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany e-mail: zhichao@mps.mpg.de; gizon@mps.mpg.de\n- 2 Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen, 37077 Göttingen, Germany\n- 3 Center for Astrophysics and Space Science, NYUAD Institute, New York University Abu Dhabi, Abu Dhabi, UAE  \nReceived ⟨ date ⟩ / Accepted ⟨ date ⟩', 'ABSTRACT': "Context. Avariety of solar global modes of oscillation in the inertial frequency range have been identified in maps of horizontal flows derived from GONG and HMI data. Among these, the high-latitude mode with azimuthal order m = 1 ( HL 1) has the largest amplitude and plays a role in shaping the Sun's di ff erential rotation profile.  \nAims. We aim to study the evolution of the HL 1 mode parameters, utilizing Dopplergrams from the Mount Wilson Observatory (MWO), GONG, and HMI, covering together five solar cycles since 1967.  \nMethods. 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.  \nResults. The HL 1 mode is easily observed in the maps of zonal velocity at latitudes above 50 degrees, especially during solar minima. The mode parameters measured from the three independent data sets are consistent during their overlapping periods and agree with previous findings using HMI ring-diagram analysis. We find that the amplitude of the 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 · (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 ( ∼ ± 3 nHz). Since about 1990, the mode frequency follows an overall decrease of ∼ 0 . 25 nHz / year, consistent with the long-term decrease of the angular velocity at 60 · latitude.  \nConclusions. We have shown that the amplitude and frequency of the HL 1 mode can be measured over the last five solar cycles, together with the line-of-sight projection of the velocity eigenfunction. 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.  \nKey words. Sun: rotation - Sun: oscillations - Sun: interior - Sun: helioseismology - Sun: general", '1. Introduction': "Rossby and inertial modes of solar oscillation are observed in long-duration time series of the horizontal velocity field on / near the solar surface (Löptien et al. 2018; Gizon et al. 2021; Hanson et al. 2022). These horizontal flows may be inferred at a daily cadence using local helioseismology applied to the Global Oscillation Network Group (GONG: Harvey et al. 1996) and the Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory (SDO / HMI: Scherrer et al. 2012) data sets, or using local correlation tracking of granulation (seen in SDO / HMI intensity images). The observed modes have periods of order the solar rotation period and their surface eigenfunctions are quasitoroidal. Linear eigenvalue solvers enable us to identify the various modes that are clearly resolved in frequency space and to infer their radial eigenfunctions throughout the convection zone (Bekki et al. 2022). The equatorial Rossby modes have maximum colatitudinal velocity, vθ , at the equator, while the highlatitude modes have maximum longitudinal velocity, vϕ , in the polar regions. Nearly all modes have critical latitudes where their phase speeds equal the local di ff erential rotation velocity.  \nIn the observations, the inertial modes are unambiguously identified as global modes because each mode oscillates at the same frequency over a large range of latitudes (Gizon et al. 2021). Waidele & Zhao (2023) and Lekshmi et al. (2024) measured the temporal variations of the equatorial Rossby mode properties using the horizontal flows from helioseismic analysis of HMI and GONG data, and reported that the mode power (frequency shift) averaged over azimuthal wavenumber m is correlated (anticorrelated) with the sunspot cycle. We refer the reader to Gizon et al. (2024) for a review on the topic of solar inertial modes.  \nThe high-latitude inertial mode with m = 1 (hereafter HL 1 for short) is of particular interest. It has the largest amplitude among all modes in the inertial frequency range (time-averaged vϕ ∼ 10 m / s, see Gizon et al. 2021) and it is driven by a baroclinic instability (Bekki et al. 2022). This mode is very sensitive to the latitudinal entropy gradient and to other important properties of the solar convection zone, such as the superadiabatic temperature gradient (Gizon et al. 2021; Bekki et al. 2022, 2024). The mode is anti-symmetric across the equator in vϕ and symmetric in vθ . Its mean frequency over the period 2010 - 2020 is measured to be ν Carr HL 1 = -86 . 3 ± 1 . 6 nHz in the Carrington frame,  \nwhere the negative sign indicates retrograde propagation. In an inertial frame, the sidereal frequency is  \nν HL 1 = -86 . 3 + 456 . 0 nHz = 369 . 7 nHz (1)  \nand in the Earth frame the synodic frequency is  \nν synodic HL 1 = 369 . 7 -31 . 7 nHz = 338 . 0 nHz . (2)  \nAbove 50 · latitude, the mode has a characteristic spiral pattern in vϕ , which, according to models, depends sensitively on the properties of the convection zone (Gizon et al. 2021). The flow pattern associated with the m = 1 mode was first reported by Hathaway et al. (2013) using a supergranulation tracking method, although it was then misidentified as giant cell convection sheared by di ff erential rotation. Bogart et al. (2015) later confirmed the existence of the m = 1 polar flow pattern using ring-diagram analysis.  \nIt came to our attention only recently that the m = 1 high-latitude inertial mode of oscillation was clearly seen above 50 · latitude in the 'zonal velocity' time-series computed from Mt. Wilson Observatory (MWO) Dopplergrams by Ulrich (1993, his figure 5, top panel) and Ulrich (2001, his figure 1, top panel). Ulrich et al. (1988) defined the zonal velocity as  \nV zonal( θ ) = P | L | < 30 · (sin L )[ V los( θ, L ) -V background( θ, L )] P | L | < 30 · | sin L | , (3)  \nwhere V los is a Dopplergram and V background is the Doppler image consisting of the limb shift plus the line-of-sight (LOS) projection of the Sun's axisymmetric flows (rotation and meridional circulation). The co-latitude is denoted by θ and L = ϕ -ϕ c is the heliographic longitude measured with respect to the central meridian ϕ c. The zonal velocity informs us about the slowlyvarying large-scale components of the longitudinal velocity vϕ on the Sun. The velocity V zonal can be used to measure the 'torsional oscillation', and also to detect the inertial modes.  \nIn figure 4 of Ulrich (2001), the m = 1 high-latitude mode is visible in V zonal at latitudes ± 58 . 5 · as narrow peaks of power with very high signal-to-noise ratio at a sidereal frequency close to 0 . 8 × (Carrington Period) -1 = 0 . 8 × 456 nHz = 365 nHz. This frequency corresponds to ν HL 1 given in Eq. (1) and is slightly above the local rotation rate. At the lower latitudes (below 40 · ), other peaks of excess power are seen at frequencies that follow the local frequency of the surface di ff erential rotation (increases toward the equator); these peaks are unrelated to the HL 1 mode, but are likely due to velocity features that are associated with magnetic activity advected at the local rotation rate.  \nIn the present paper, we use all three available series of fulldisk Dopplergrams (MWO, GONG and HMI) to study the longterm evolution of the HL 1 mode. We find that the mode is visible in V zonal in the latitude range 60 · - 75 · throughout the entire period from 1967 to 2022. We characterize the evolution of the HL 1 mode parameters and study their correlations with the sunspot number, the polar field strength, and the solar rotation rate. We note that preliminary results using only GONG and HMI were presented by Gizon et al. (2024).", '2. Detection of HL 1 mode in MWO, GONG and HMI Dopplergrams': "The time periods used in the analysis are from 1967 to 2012 for MWO data, from 2001 to 2022 for GONG data ( tdvzi ), and from 2010 to 2022 for HMI data ( hmi.v\\_720s ). The original time cadences and spatial resolutions of the data sets are quite di ff erent. For the purpose of this work, we take one frame per day and reduce the resolution to 10 '' per pixel by binning.", '2.1. Correction of GONG and HMI Dopplergrams': 'The static limb shift, di ff erential rotation, and meridional circulation velocity signals were already subtracted from the MWO data by Ulrich (2001), thus no V background subtraction is necessary for this dataset. For the other two datasets, we perform low-order polynomial fits to V los to obtain V background. For the limb shift, we fit a model that consists of a polynomial of order five in µ , where µ is the cosine of the center-to-limb angle. For the LOS projection of the di ff erential rotation and the meridional flow, we fit the following models:  \nV (rot) los ( θ, L ) = cos B 0 sin θ sin L 4 X k = 0 bk (cos θ ) k , (4)  \nV (merid) los ( θ, L ) = (sin B 0 sin θ -cos B 0 cos θ cos L ) 4 X l = 1 clP 1 l (cos θ ) , (5)  \nwhere B 0 is the heliographic latitude at the disk center and the P 1 l are associated Legendre polynomials of degree l and order one. The model parameters are estimated using a least-squares method. We does not subtract the daily background fits as this would remove the low-degree inertial modes as well. Instead, we subtract a 35-day running median of the fits. We note that we apply the procedure to the unmerged GONG data (tdvzi), because the merged GONG data (mrvzi) are a ff ected by the removal of a surface fit to the data, which undesirably reduces the amplitude of the lowm modes. With this, we can correct the GONG and HMI Dopplergrams for V background.  \nThe HMI Doppler time series also contains a number of jumps, which coincide with the re-tuning of the optical filter system (Couvidat et al. 2016; Hoeksema et al. 2018). This affects the interference fringes. In the hmi.v\\_720s data, the fringes were largely removed from 1 October 2012 onward (see Couvidat et al. 2016, their section 2.8.2 and their figures 9 and 10). However the fine structure of the fringes still remains in this data set (see Fig. A.1). To remove the HMI fringes, we subtract the median values of the images computed over time intervals between adjacent re-tuning dates.', '2.2. Computation of zonal velocity': 'In order to compute the zonal velocity for MWO, GONG and HMI using Eq. (3), we remap the corrected Dopplergrams onto heliographic coordinates using the equidistant cylindrical projection with a map scale of 0 . 6 · per pixel, to obtain ( V los -V background) as a function θ , L , and time. We choose to sum over longitudes | L | < 30 · and average over nine consecutive days, in order to follow the procedure described by Ulrich et al. (1988). The resulting time series of zonal velocity, V zonal( θ, t ), are kept in the Earth frame.', '2.3. Visual detection of HL 1 mode in supersynoptic maps': "Figure A.2 shows all the supersynoptic maps of V zonal as functions of time and latitude for the three data sets (MWO, GONG, HMI). The black and white stripes at high latitudes correspond to the HL 1 inertial mode. Importantly, these stripes are consistent between the overlapping data sets (see arrows in Fig. A.2). Figure 1 shows selected time periods from Fig. A.2 showing only five years of data around each solar cycle minimum. The mode amplitude is strongest during these quiet Sun periods.  \nFig. 1. Example supersynoptic maps of V zonal in the Earth frame computed from the MWO (top three panels), GONG (fourth panel) and HMI data (bottom panel) near cycle minima. The full data set is shown in Fig. A.2). The HL 1 mode may be seen from time to time as stripes in V zonal above 50 · latitude in both hemispheres. For clarity, the maps were smoothed in latitude with a Gaussian kernel of width 10 · . A 35day running median was subtracted from each latitude to remove slow variations including the 'torsional oscillation'.  \n<!-- image -->", '2.4. Consistency between datasets during overlapping time periods': 'Since the global mode HL 1 is antisymmetric across the equator in V zonal, we compute the north-south antisymmetric zonal velocity V ( -) zonal ( θ, t ) = [ V zonal( θ, t ) -V zonal( π -θ, t )] / 2 to reduce noise. We then compute an average of V ( -) zonal over the latitudinal range 60 · - 75 · where the mode is easily detectable:  \nV (band) zonal ( t ) = D V ( -) zonal ( θ, t ) E | θ -22 . 5 · |≤ 7 . 5 · . (6)  \nFigure 2 shows extracts from V (band) zonal for overlapping time segments between MWO, GONG, and HMI. The consistency between the overlapping data sets is clear.  \nFig. 2. V (band) zonal ( t ) during di ff erent overlapping periods of the MWO, GONGandHMIdata. The long-term trend in each data set was removed by subtracting a 181-day running mean. The period of the predominant oscillations seen in V (band) zonal is about 34 days, which corresponds to the HL 1 mode frequency near ν synodic HL 1 = 338 nHz.  \n<!-- image -->', '3. Power spectra of zonal velocity': 'We then Fourier transform each of the MWO, GONG, and HMI V (band) zonal ( t ) to obtain three power spectra,  \nP ( ν ) = 1 ∆ ν N 2 t η GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> X t V (band) zonal ( t ) e -2 π i ν t GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 2 , (7)  \nwhere Nt = 16824 for MWO, Nt = 7823 for GONG, Nt = 4628 for HMI, with a common temporal sampling ∆ t = 1 day. The quantity ∆ ν = 1 / ( Nt ∆ t ) is the frequency resolution, equal to 0 . 7 nHz for MWO, 1 . 5 nHz for GONG, and 2 . 5 nHz for HMI. To correct for missing data, we divide the power by the duty cycle η = 0 . 96 for MWO, η = 0 . 96 for GONG, and η = 1 for HMI.  \nFigure 3 shows P ( ν ) for the full time series of the three data sets. In all cases, excess power is detected around the reference HL 1 mode frequency ν synodic HL 1 = 338 nHz. The mode power differs between the three data sets. This is not because of varying data quality, but because they do not cover the same time periods. Weshall confirm later that the mode power and frequencies measured from the three data sets are consistent during their overlap periods. The central frequency of the peak in the HMI P ( ν ) is 2 nHz o ff from the reference ν synodic HL 1 , although the reference frequency was obtained using the HMI data in a similar time period. One possible explanation is that Gizon et al. (2021) obtained the HL 1 mode frequency using the latitude range 37 . 5 · - 67 . 5 · where the excess power around the local di ff erential rotation rate (highlighted by the blue curves in their figure 1) remains significant and thus the measured frequency was slightly biased.  \nIn all three data sets the background noise level is extremely low. The full width of the peak is approximately w = 7 nHz for the MWO and GONG data sets and w = 4 nHz for HMI. This corresponds to e-folding lifetimes of about 1 / ( πw ) = 1 . 4 years and 2 . 5 years respectively. As will be seen later in the three-year  \nFig. 3. Power spectra of V (band) zonal in the frequency range 250-450 nHz (in the Earth frame), computed from the MWO, GONG and HMI Dopplergrams (from top to bottom respectively). The vertical red lines at 338 nHz mark the reference synodic mode frequency.  \n<!-- image -->  \npower spectra of Fig. 5 (middle column), the mode is continuously visible since 1969.', '4. Velocity pattern associated with the HL 1 mode': 'Let us infer the LOS velocity perturbation associated with the HL 1 mode from the series of Dopplergrams. We align the disk centers of the HMI V los images, rotate the images by the nominal roll angle provided by the HMI team ( -CROTA2 ), and re-scale the disk radius to a common value. Instead of taking one frame per day, we compute daily averages of the images to reduce the noise. At each pixel ( x , y ) on the disk, we apply a band-pass filter to the data in the Fourier domain centered around the synodic mode frequency to obtain  \nV HL 1 los ( x , y, t ) : = 1 Nt X filt ν -338 nHz 75 nHz ! ˆ V los( x , y, ν ) e 2 π i ν t , (8)  \nν  \nwhere filt( ξ ) is a symmetric function equal to 1 for | ξ | < 1 / 3, tapered to zero with raised cosines, and with a full width at half maximum of 1. In the above expression, ˆ V los denotes the temporal FFT of V los.  \nFigure 4 shows an example of a filtered Dopplergram during the solar minimum in 2018, at a time when the north pole is tilted toward the Earth. The image was smoothed using a 2D Gaussian kernel with a FWHM of 5 · to reduce noise on small spatial scales. The horizontal velocity of the HL 1 mode projected onto the line of sight is clearly seen above 50 · latitude in both hemispheres. Notice that there appears to be a polar-crossing flow with an amplitude over 10 m / s associated with the HL 1 mode (in agreement with an early statement by Ulrich 1993, page 35). In the left column of Fig. A.3, we can see the line-of-sight projection of the mode velocity at di ff erent times (di ff erent phases)  \nFig. 4. Line-of-sight Doppler velocity associated with the HL 1 mode as viewed from the Earth on 09 September 2018 when the solar tilt angle B 0 is 7 . 24 · , extracted from time series of daily HMI images by applying a band-pass filter around ν synodic HL 1 (see Eq. (8)). The LOS velocity pattern in the polar regions is associated with the HL 1 mode. The image was smoothed using a 2D Gaussian kernel with a FWHM of 5 · . A comparison with the reconstructed V HL 1 los using 5 · -tile ring-diagram flow maps is shown in Fig. A.3 and in the movie (Liang 2024).  \n<!-- image -->  \nwith di ff erent B 0 angles. The predominant pattern in V HL 1 los at high latitudes is mostly north-south antisymmetric.  \nTo compare with the Doppler data, we also reconstruct V HL 1 los using the 5 · -tile horizontal flow maps near the solar surface from the HMI ring-diagram pipeline (Bogart et al. 2011a,b). Because the ring-diagram flow maps ( V Carr) are given in the Carrington frame, it is necessary to apply at each position vector r (defined from the center of the Sun) the following velocity transformation from the Carrington frame to the Earth frame,  \nV ⊕ = V Carr + ( Ω Carr z ⊙ -Ω ⊕ z ) × r , (9)  \nwhere Ω Carr / 2 π = 456 nHz is the Carrington rotation rate, z ⊙ is the unit vector that points along the solar rotation axis, Ω ⊕ / 2 π = (1 yr) -1 = 31 . 7 nHz, and z is the unit vector that points along the normal to the ecliptic plane. The second term on the righthand side of Eq. (9) does not a ff ect the frequencies within the HL 1 band-pass filter. Thus the V HL 1 los reconstructed from the ringdiagram flow maps is essentially V Carr projected onto the line of sight, followed by the HL 1 mode filtering.  \nThe middle column of Fig. A.3 shows the reconstructed V HL 1 los from the ring-diagram flow maps, to be compared with V HL 1 los maps from the direct Doppler. The large-scale patterns and the velocity amplitudes are in very good agreement between the two data sets, especially at high latitudes. This suggests in particular that the HL 1 mode does not have any significant radial motion (since we did not consider this component in the reconstruction of the Doppler data). At lower latitudes, the large-scale features have much lower amplitudes but also agree. A movie showing the evolution V HL 1 los ( x , y, t ) and its ring-diagram reconstruction at a cadence of 1 day during 2010 - 2022 is available online (Liang 2024). At high latitudes, the agreement is good between the two data sets, although noise is higher in the ring-diagram data set. This is not too surprising as we expect helioseismic analysis to contain stochastic p-mode noise that is not present in Doppler data at a daily temporal cadence.  \nLet us study the random noise in V HL 1 los in more detail. We estimate the noise level by subtracting from V HL 1 los a spatial smoothing with a Gaussian FWHM of 15 · and computing the rms of the residuals in time for each pixel. As shown in Fig. A.3 (bottom  \nFig. 5. Visibility of HL 1 mode in consecutive three-year power spectra, using MWO (1969 - 2010) and HMI (2011 - 2022) data. Panel (a): Example MWO time series (1984-1986) of V ( -) zonal averaged over latitudes 60 · ≤ λ < 75 · (i.e., V (band) zonal ). Panel (b): Power spectra of V ( -) zonal for the same time period in various latitudinal bands of 15 · (labeled at the top right corner of each panel). The vertical red lines denote the reference HL 1 mode frequency ν synodic HL 1 = 338 nHz and the excess power around this frequency is highlighted in blue. The shaded pink areas indicate the di ff erential rotation rate in the outer 5% of the Sun from global helioseismology as viewed from the Earth. Panels (c) and (d): Same as above but for an example HMI three-year time series during 2017-2019. Panel (e): All available three-year power spectra of V (band) zonal stacked on top of each other. The crosses highlight the reference frequency of 338 nHz.  \n<!-- image -->  \nrow), the rms noise level in direct V HL 1 los is below 2 m / s, with a minimum of 0 . 2 m / s near disk center. The noise level increases with disk radius because an angular distance of 5 · corresponds to about 8 pixels near the disk center but only about one pixel near the limb. The noise level in the reconstructed data from ringdiagram flow maps is also around 0 . 2 m / s near disk center, but it increases much faster with disk radius than that in the direct Doppler data. We also note that the reconstructed images from helioseismology are only available up to 98% of the disk radius. On the other hand, the Dopplergrams cover the entire solar disk, which is much useful to observe the high-latitude HL 1 mode.  \nIn the right column of Fig. A.3 we also compare the observations with the line-of-sight projection of the theoretical velocity eigenfunction (Gizon et al. 2021; Bekki et al. 2022). We find a reasonable agreement, however there are also noticeable di ff erences between observations and theory. In particular the tilt of the velocity pattern changes faster with latitude in the observations than in theory.  \nWeremark that the tilt of the spiral pattern at high latitudes in Fig. 4 is in the opposite direction to that of Fig. 1 since in the supersynoptic maps the corresponding longitude decreases toward the right along the horizontal axis. We also note that we avoided performing spatial Fourier transform, since we do not see the entire solar surface and LOS projection e ff ects may complicate the interpretation of the spatial power spectra.', '5.1. Power spectra from three-year data segments': "We break down the time series into three-year segments to extract the HL 1 mode parameters as a function of time. We use the MWO data from 1969 to 2010 and the HMI data from 2011 to 2022, giving a total of 18 non-overlapping segments covering five solar cycles.  \nWe compute the power spectra for each of these three-year segments. Above 45 · latitude, the mode is detected in all data sets as narrow peaks of power near synodic frequency 338 nHz well above noise level. This can be seen for two example time segments in Fig. 5 (panels b and d). Below 45 · , the mode is in the noise in the MWO data (Fig. 5b), while it remains visible below in the HMI power spectra (Fig. 5d). We note that there is also some excess power at frequencies corresponding to the local di ff erential rotation (highlighted by the shaded pink areas), indicating that some velocity features are advected by the local rotation, such as flows around active regions (Gizon 2004), convective flows, or possibly critical-latitude modes (Gizon et al. 2021; Fournier et al. 2022). The background power gradually drops to zero below 60 nHz since we subtracted a 181-day running mean from the time series of V zonal to remove long-term trends including the 'torsional oscillation' and B 0-angle variation.  \nThe middle column of Fig. 5 shows the 3-year power spectra above 60 · latitude, stacked on top of each other in the (synodic) frequency range 250 - 450 nHz. The HL 1 mode power is always present, although it is smaller during the periods 1972 - 1974  \nand 1993 - 1995. We see that the noise level in the MWO data improved from 1982 when a new exit slit assembly was installed, and from 1986 when the fast scan program started, as reported by Ulrich et al. (1988). We find that it is possible to follow the evolution of the mode amplitude over the full extent of the data sets (MWO, GONG, HMI) with a time resolution of at least three years. It is remarkable that the MWO data enable us to add three decades of very useful observations.", '5.2. Estimation of mode parameters': 'To improve the temporal sampling, we now consider sliding time windows of three years with central times separated by one year, such that each time window has an overlap of two years with the neighboring windows. To extract the HL 1 mode parameters in each time window, we choose to fit a Lorentzian function to the corresponding power spectra,  \nL ( ν ) = A 1 + [( ν -ν HL 1 + 31 . 7nHz) / ( w/ 2)] 2 + B , (10)  \nwhere ν HL 1 is the sidereal mode frequency, A is the mode power, w is the full width at half maximum of the Lorentzian, and B the constant background over the fitting range 200 - 500 nHz. The total power of the HL 1 mode under the Lorentzian profile is π A w . The rms and maximum values of the mode velocity amplitude ( V (band) zonal ) are given by √ π A w and √ 2 π A w respectively.  \nWe note that we padded each 3-yr time segment of V (band) zonal with zeros to have a length of Nt = 8192 and a frequency resolution of ∆ ν = 1 . 4 nHz. The original frequency resolution from a 3-yr time segment (10 . 6 nHz) is coarser than the line width of the HL 1 mode (4 - 7 nHz; see § 3), making it tricky to fit. The mode power π A w is una ff ected by the zero padding as the duty cycle η has taken this into account in Eq. (7); however, the line profile is broadened by the padding so we do not consider w and A individually. We also note that the V zonal defined in Eq. (3) is smaller than the corresponding longitudinal velocity vϕ . Using synthetic data, we find that the ratio of vϕ to V zonal ranges from 2 . 1 to 3 . 3, depending on the latitude and the model used. We choose to scale the mode amplitude measured from V (band) zonal by a factor of 2 . 9 so that it has the same magnitude as the one measured from the vϕ component of HMI ring-diagram flow maps during solar minimum of cycle 24 / 25.', '5.3. Evolution of mode amplitude and frequency': 'Figure 6 shows the temporal variations of the measured amplitude and frequency of the HL 1 mode. The results from the three data sets during the overlapping periods are consistent to a large extent. The mode amplitude is mostly above 5 m / s with a median value of 9 . 2 m / s whereas the mode frequency has a median value of 370 . 0 nHz with a peak-to-peak variation of about 10 nHz. The mode amplitude reaches its overall maximum during the solar minimum of cycle 24 / 25 right after the weakest cycle 24 in the past five cycles. The 11-yr variation can be seen in the mode amplitude, especially during cycles 23 and 24. On the other hand, there is no clear 11-yr variation in the mode frequency.', '5.4. Evolution of mode velocity pattern': 'In addition to the mode amplitude and frequency, we also measure the slope of the striped pattern seen in the supersynoptic maps of V zonal at high latitudes. We define the slope of the striped pattern as the ratio of ∆ τ/ ∆ λ , where ∆ τ is the time lag between  \nFig. 6. Temporal variations of the amplitude ( top panel ) and frequency ( middle panel ) of the HL 1 mode, together with the sunspot number and the polar field strength ( bottom panel ). The orange, green, and red solid curves show the results from MWO, GONG and HMI data sets, respectively, with the shaded areas showing the 68% confidence interval estimated from 10 000 Monte Carlo simulations. The open circles indicate that the excess power around the mode frequency is below 90% confidence level. The horizontal dashed lines represent the median value over MWO (1968-2010) and HMI (2011 -2022 ) data sets.  \n<!-- image -->  \nthe time series of V zonal at di ff erent latitudes and ∆ λ is the latitudinal separation thereof, as depicted in the top panel of Fig. 7. In practice we determine the time lag that maximizes the correlation between V zonal at adjacent latitudes and average over latitude weighted by a raised cosine between latitudes 60 · and 75 · . The bottom panel of Fig. 7 shows its temporal variation. The measured ∆ τ/ ∆ λ varies mostly between 0 . 2 and 1 day / deg with a medium value of 0 . 54 day / deg. Although there are two points below zero (years 1969 and 1994), they fall within the range of uncertainty. It seems fair to say that ∆ τ/ ∆ λ does not change sign. The ∆ τ/ ∆ λ seems to have larger values during solar minima and exhibits the 11-yr variation to some extent, though not as clear as that in the mode amplitude. We note that the time lag ∆ τ has a dependence on the mode frequency. Nonetheless, the variation in the measured ν HL 1 (about ± 5 nHz) only a ff ects ∆ τ/ ∆ λ by ± 1 . 5% which is negligible.  \nFig. 7. Temporal variations of HL 1 mode velocity pattern. Top panel: HMI V ( -) zonal in the first half of 2021 applied with a band-pass filter around ν synodic HL 1 . The slope of the striped pattern in V ( -) zonal is highlighted by the tilted red lines. The ∆ τ (time lag) and ∆ λ (latitudinal separation) that determine the slope of the striped pattern ∆ τ/ ∆ λ are indicated in the plot (the two legs of the red right triangle). Bottom panel: ∆ τ/ ∆ λ as a function of time, using the same color conventions as in Fig. 6.  \n<!-- image -->', '5.5. Correlations between mode parameters, sunspot number, and polar field': 'In order to examine which periodicities that are present in the time series of the HL 1 mode parameters, we combine the measurements from MWO (1968 - 2010) and HMI (2011 - 2022) to form a 55-yr long data set, and we compute a temporal power spectrum for each of the parameters (velocity amplitude, frequency, and eigenfunction tilt). The right column of Fig. A.4 shows these three power spectra. We see that there is a clear 11yr periodicity in both the mode velocity amplitude and the tilt, while the power spectrum for the mode frequency does not show a distinct peak at (11 yr) -1 = 2 . 88 nHz. The second and third plots on the left column of Fig. A.4 highlight (in red) the 11yr component of the evolution of the velocity amplitude and the tilt. We see that they are anti-correlated with the sunspot number. The correlation coe ffi cients between the mode parameters and the sunspot number are -0 . 5 for the mode amplitude and -0 . 34 for the eigenfunction tilt. Since there are only 18 independent points in a period of 55 years, we find that coe ffi cients lower than 0 . 47 in absolute value are not significant ( p > 0 . 05). Thus only the anticorrelation between the mode amplitude and the sunspot number is significant.  \nWe also compute the correlation coe ffi cients between the mode parameters and the polar field strength over the period 1977-2022 shown in the bottom panel of Fig. 6. Surprisingly, these correlation coe ffi cients are all below 0 . 1 in absolute value, which means there is no correlation between the mode parameters and the polar field strength.', '5.6. Correlations between mode parameters and rotation rate': "We now compare the mode parameters with the solar surface rotation rate at 60 · latitude where the local rotation rate is close to  \nthe mode frequency. From 1996, we use the solar rotation rate obtained from global helioseismology (Larson & Schou 2015, 2018). The data series used are mdi.vw\\_V\\_sht\\_2drls from 1996 to 2010 and hmi.vw\\_V\\_sht\\_2drls from 2010 to 2022, at intervals of 72 days with NACOEFF = 6. The rotation rate is smoothed with a 360-day running window. We then extend the measurements back to 1983 using the MWO Doppler velocity data, as processed by Ulrich et al. (2023, their table 1 and figure 4). The MWO rotation rate is shifted by an o ff set of 16 nHz so it matches the surface rotation rate from global helioseismology during the overlapping time period. Also, because the global helioseismology data are north-south symmetric by construction, we symmetrize the MWO data. We remark that the surface rotation rate from the MWO Doppler velocity starts from 1983, despite the fact that the MWO Doppler data have been available since 1967. This is because there exists a systematic error that o ff sets the Doppler signals prior to 1983 (Ulrich et al. 2023, section 2). Since we only consider the signals in the inertial frequency range, this systematic error may not bias the measured mode parameters.  \nThe right panel of Fig. 8 compares the evolution of the mode frequency and the surface rotation rate at 60 · latitude for 1983 2022, together with the mode frequency (the shaded region covers the error range of ± 1 σ ). Both curves show the same longterm decrease, which amounts to about -7 nHz over 30 years. This decrease has been reported before at this latitude during cycles 23 and 24 throughout the convection zone (Basu & Antia 2019, their figure 2). In addition, the rotation rate also shows an 11-year component at the level of ± 2 . 5 nHz associated with the high-latitude branch of the so called 'torsional oscillation' (e.g., Howe 2009; Basu 2016). These small variations in the rotation rate are within the error bars of the inertial mode frequency ( ± 3 nHz or so, for a three year time window). The mode frequency measurements are too noisy to reveal any variation at this level of precision. Overall, the correlation coe ffi cient between the mode frequency and the rotation rate at 60 · is 0 . 55. This correlation coe ffi cient is significant ( p = 0 . 05).  \nThe left panel of Fig. 8 compares the evolution of the mode amplitude and the surface rotation rate at 60 · latitude. (Beware: note the reverse vertical axis for the rotation rate). We see that the relative mode amplitude variations are of order unity, while the relative change in rotation due to the 'torsional oscillation' is about 1%. The times of solar minima are indicated on the plot with vertical bars. We find that the times at which the mode amplitude reaches its maximum coincide with the beginnings of cycles 21, 22, and 25. But when the mode amplitude is smaller (solar minima of cycles 22 / 23 and 23 / 24), the mode has its maximum amplitude 1 - 3 years after the solar minima. Furthermore, cycle 21 is the strongest sunspot cycle in the past five decades and the polar field strength in the subsequent solar minimum is notably strong; however, the mode amplitude is not particularly small in that period. In contrast, the rotation period varies in step with the mode amplitude, including the long-term trend and the phase lags in cycles 23 and 24. We find a correlation coe ffi cient between the mode amplitude and the rotation rate at 60 · of -0 . 82. This anticorrelation is very significant ( p = 0 . 0006).", '6. Summary and discussion': "We have computed synoptic maps of zonal velocity V zonal using LOS Doppler velocity from MWO, GONG, and HMI, covering a total of 55 years. We have measured the HL 1 mode parameters from these synoptic maps at high latitudes. The time series of mode amplitude and eigenfunction tilt exhibit an 11-yr period-  \n<!-- image -->  \nFig. 8. Temporal variation of the mode amplitude ( left ) and frequency ( right ) taken from Fig. 6, together with surface rotation rates at 60 · latitude obtained from MWO Doppler velocity (1983 - 2013) in solid light blue, and from global helioseismology using MDI (1996 - 2010) and HMI (2010 - 2022) data in dashed blue. Note the reverse scaling of the rotation rate (indicated by the blue arrow) on the right vertical axis of the left panel. In the left panel, the beginnings of cycles 21 - 25 (solar minima) are marked with vertical lines and the cycle numbers are denoted at the top.  \n<!-- image -->  \nicity, whereas the 11-yr periodicity in mode frequency is below noise level. The time series of mode amplitude and eigenfunction tilt are weakly anticorrelated with the sunspot number. However, the anticorrelation with the high-latitude rotation rate is more pronounced, particularly for the mode amplitude. The mode amplitude peaks coincide with solar minima when the mode amplitude is large, but lag by 1 - 3 years when the mode amplitude is smaller. The long-term trend of the mode frequency follows the rotation rate near the critical latitude. The results presented in this work contain a wealth of information about the HL 1 mode. Improved linear and nonlinear models of inertial modes (with magnetic field) are needed to interpret the observations and further constrain the physics of the solar interior.  \nThe temporal changes in mode frequency are very small and can probably be explained using the linearized wave equation together with a prescription of the variations in solar rotation and solar structure. In particular, the solar-cycle variations of rotation ('torsional oscillation') are known from p-mode helioseismology throughout the convection zone, and their e ff ect on the HL 1 mode frequency can be inferred using a linear eigenvalue solver (Bekki et al. 2022) or a first-order perturbation method (see work by Goddard et al. 2020 for the equatorial Rossby modes). However, it remains to be seen if changes in the rotation alone will be su ffi cient to explain the data. The changes in the background entropy and the direct e ff ect of the magnetic field on the mode may have some importance as well. In turn, the measurements of mode frequency present an opportunity to constrain these changes in the convection zone (unlike p modes, inertial modes have large kinetic energy density deep in the convection zone).  \nThe much larger variations in mode amplitude and mode eigenfunctions will require more sophisticated modeling tools. Recent non-linear 3D hydrodynamic simulations carried out by  \nBekki et al. (2024) have revealed the complex relationship between the baroclinically-unstable high-latitude modes, the entropy gradient, and the rotation rate. It was shown that the modes may transport entropy toward the equator and thus back react on the latitudinal di ff erential rotation. Such work represents a first step in the understanding of the dynamics of the HL 1 mode, but will have to be extended to account for the e ff ects of the dynamo.  \nAcknowledgements. We are very grateful to Roger K. Ulrich for providing the data from the synoptic program at the 150-Foot Solar Tower of the Mt. Wilson Observatory. The Mt. Wilson 150-Foot Solar Tower is operated by UCLA, with funding from NASA, ONR and NSF, under agreement with the Mt. Wilson Institute. The HMI data used are courtesy of NASA / SDO and the HMI science team. SOHO is a project of international cooperation between ESA and NASA. This work utilizes GONG data from the National Solar Observatory (NSO), which is operated by AURA under a cooperative agreement with NSF and with additional financial support from NOAA, NASA, and USAF. The sunspot numbers are from the World Data Center SILSO, Royal Observatory of Belgium, Brussels. The polar magnetic field data are from the Wilcox Solar Observatory (http: // wso.stanford.edu / Polar.html), courtesy of J.T. Hoeksema. The Wilcox Solar Observatory is currently supported by NASA. The data were processed at the German Data Center for SDO (GDC-SDO), funded by the German Aerospace Center (DLR). L.G. acknowledges support from the NYU Abu Dhabi Center for Space Science.", 'References': 'Basu, S. 2016, Living Reviews in Solar Physics, 13, 2 Basu, S. & Antia, H. M. 2019, ApJ, 883, 93 Bekki, Y., Cameron, R. H., & Gizon, L. 2022, A&A, 662, A16 Bekki, Y., Cameron, R. H., & Gizon, L. 2024, Science Advances, 10, eadk5643 Bogart, R. S., Baldner, C., Basu, S., Haber, D. A., & Rabello-Soares, M. C. 2011a, in Journal of Physics Conference Series, Vol. 271, GONG-SoHO 24: A New Era of Seismology of the Sun and Solar-Like Stars (IOP), 012008 Bogart, R. S., Baldner, C., Basu, S., Haber, D. A., & Rabello-Soares, M. C. 2011b, in Journal of Physics Conference Series, Vol. 271, GONG-SoHO 24: A New Era of Seismology of the Sun and Solar-Like Stars (IOP), 012009 Bogart, R. S., Baldner, C. S., & Basu, S. 2015, ApJ, 807, 125  \nCouvidat, S., Schou, J., Hoeksema, J. T., et al. 2016, Sol. Phys., 291, 1887  \nFournier, D., Gizon, L., & Hyest, L. 2022, A&A, 664, A6 Gizon, L. 2004, Sol. Phys., 224, 217 Gizon, L., Bekki, Y., Birch, A. C., et al. 2024, in IAU Symposium, Vol. 365, Dynamics of Solar and Stellar Convection Zones and Atmospheres, ed. A. V. Getling & L. L. Kitchatinov Gizon, L., Cameron, R. H., Bekki, Y., et al. 2021, A&A, 652, L6 Goddard, C. R., Birch, A. C., Fournier, D., & Gizon, L. 2020, A&A, 640, L10 Hanson, C. S., Hanasoge, S., & Sreenivasan, K. R. 2022, Nature Astronomy, 6, 708 Harvey, J. W., Hill, F., Hubbard, R. P., et al. 1996, Science, 272, 1284 Hathaway, D. H., Teil, T., Norton, A. A., & Kitiashvili, I. 2015, ApJ, 811, 105 Hathaway, D. H., Upton, L., & Colegrove, O. 2013, Science, 342, 1217 Hoeksema, J. T., Baldner, C. S., Bush, R. I., Schou, J., & Scherrer, P. H. 2018, Sol. Phys., 293, 45 Howe, R. 2009, Living Reviews in Solar Physics, 6, 1 Larson, T. P. & Schou, J. 2015, Sol. Phys., 290, 3221 Larson, T. P. & Schou, J. 2018, Sol. Phys., 293, 29 Lekshmi, B., Gizon, L., Jain, K., Liang, Z.-C., & Philidet, J. 2024, in IAU Symposium, Vol. 365, Dynamics of Solar and Stellar Convection Zones and Atmospheres, ed. A. V. Getling & L. L. Kitchatinov Liang, Z.-C. 2024, Movie, https://www2.mps.mpg.de/projects/seismo/ zhichao/cf-vlos.mp4 Löptien, B., Gizon, L., Birch, A. C., et al. 2018, Nature Astronomy, 2, 568 Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207 Ulrich, R. K. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 40, IAU Colloq. 137: Inside the Stars, ed. W. W. Weiss & A. Baglin, 25-42 Ulrich, R. K. 2001, ApJ, 560, 466 Ulrich, R. K., Boyden, J. E., Webster, L., et al. 1988, Sol. Phys., 117, 291  \nUlrich, R. K., Tran, T., & Boyden, J. E. 2023, Sol. Phys., 298, 123 Waidele, M. & Zhao, J. 2023, Astrophys. J., 954, L26', 'Appendix A: Additional figures': "Fig. A.1. Median values over the time periods in between the HMI retuning dates noted at http: // jsoc.stanford.edu / doc / data / hmi / hmi\\_retuning.txt. The images have been remapped to have the same resolution (10 '' per pixel) and the disk center aligned with north up. The color scale is saturated at ± 50 m / s. Hathaway et al. (2015, their figure 2) shows a similar pattern for the HMI data during 2012 - 2013.  \n<!-- image -->  \nMWO  \nFig. A.2. Synoptic maps of V zonal as functions of latitude and time, obtained from direct V los of HMI, GONG and MWO observations. For clarity, the maps were Gaussian smoothed in latitude with a FWHM of 10 · . Also, a 35-day running median was subtracted from all latitudes to remove a variety of systematic trends including the 'torsional oscillation'. The arrows with the same color highlight the stripes for di ff erent data sets in the same time period.  \n<!-- image -->  \nFig. A.3. Comparison of the m = 1 high-latitude mode oscillations in the line-of-sight Doppler velocity as viewed from the Earth, obtained using various methods, when the solar tilt angle B 0 is 7 . 24 · (top), 0 · (middle), and -7 . 24 · (bottom). Left column : Observed Doppler velocity V los from HMI data band-pass filtered around ν synodic HL 1 ( V HL 1 los , see Eq. (8)). For a better comparison with the middle column, the images were Gaussian smoothed with a FWHM of 5 · . Middle column : reconstructed V los, obtained by projecting the 5 · -tile horizontal flow maps from the HMI ring-diagram analysis onto the line of sight and applying the same filter as in the first column. Right column : Line of sight projection of mode eigenfunction obtained using a linear eigenvalue solver (Bekki et al. 2022), scaled to an amplitude comparable to the observations. Bottom row : Maps of noise in V HL 1 los estimated from the temporal variance of the data at each spatial point.  \n<!-- image -->  \n)  \ng  \ne  \nd  \n/  \ny  \na  \nd  \n(  \n/  \nFig. A.4. Eleven-year periodic components and power spectra of HL 1 mode parameters. Left column, from top to bottom: sunspot number, HL 1 mode amplitude, eigenfunction tilt, and frequency. The 55-yr long time series combine the MWO (1968 - 2010) and HMI (2011 - 2022) measurements taken from Figs. 6 and 7. The gray curves are fitted parabolas representing the long-term trend. The pink curves show the 11-yr periodicity in the data added to the long-term trend. Right column: power spectra of the time series from the left column with the temporal mean subtracted. The vertical pink lines indicate the frequency of (11 yr) -1 = 2 . 88 nHz that corresponds to the 11-yr periodicity.  \n<!-- image -->"}