Issue 
A&A
Volume 626, June 2019



Article Number  A3  
Number of page(s)  11  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201834849  
Published online  30 May 2019 
Timedistance helioseismology of solar Rossby waves^{⋆}
^{1}
MaxPlanckInstitut für Sonnensystemforschung, JustusvonLiebigWeg 3, 37077 Göttingen, Germany
email: zhichao@mps.mpg.de
^{2}
Institut für Astrophysik, GeorgAugustUniversität Göttingen, FriedrichHundPlatz 1, 37077 Göttingen, Germany
^{3}
Center for Space Science, NYUAD Institute, New York University Abu Dhabi, PO Box 129188 Abu Dhabi, UAE
Received:
13
December
2018
Accepted:
17
April
2019
Context. Solar Rossby waves (r modes) have recently been discovered in the nearsurface horizontal flow field using the techniques of granulationtracking and ringdiagram analysis applied to six years of SDO/HMI data.
Aims. Here we apply timedistance helioseismology to the combined SOHO/MDI and SDO/HMI data sets, which cover 21 years of observations from May 1996 to April 2017. The goal of this study is to provide an independent confirmation over two solar cycles and in deeper layers of the Sun.
Methods. We have measured southnorth helioseismic travel times along the equator, which are sensitive to subsurface northsouth flows. To reduce noise, the travel times were averaged over travel distances from 6° to 30°; the mean distance corresponds to a pmode lower turning point of 0.91 R_{⊙}. The 21year time series of traveltime measurements was split into three sevenyear subsets and transformed to obtain power spectra in a corotating frame.
Results. The power spectra all show peaks near the frequencies of the classical sectoral Rossby waves for azimuthal wavenumbers in the range 3 ≤ m ≤ 15. The mode frequencies and linewidths of the modes with m ≤ 9 are consistent with a previous study whereas modes with m ≥ 10 are shifted toward less negative frequencies by 10–20 nHz. While most of these modes have efolding lifetimes on the order of a few months, the longest lived mode, m = 3, has an efolding lifetime of more than one year. For each mode, the rms velocity at the equator is in the range of 1–3 m s^{−1}, with the largest values for m ∼ 10. No evidence for the m = 2 sectoral mode is found in the power spectrum, implying that the rms velocity of this mode is below ∼0.5 m s^{−1}.
Conclusions. This work confirms the existence of equatorial global Rossby waves in the solar interior over the past two solar cycles and shows that timedistance helioseismology is a promising technique to study them deep in the convection zone.
Key words: Sun: helioseismology / Sun: oscillations / Sun: interior / waves
The movie associated to Fig. 1 is available at https://www.aanda.org
© Z.C. Liang et al. 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open Access funding provided by Max Planck Society.
1. Introduction
Spheroidal oscillations associated with solar f, p, and g modes have been studied intensively in the past few decades (see, e.g., Basu 2016, for a recent review). Toroidal oscillations, including r modes, have been discussed in the literature as well (e.g., Papaloizou & Pringle 1978; Unno et al. 1989). For these modes the Coriolis force is the dominant restoring force; they are similar to the Rossby waves observed in the Earth’s atmosphere and oceans. Provost et al. (1981), Smeyers et al. (1981), and Saio (1982) derived linearized equations for the analysis of r modes in uniformly and slowly rotating stars. Wolff & Blizard (1986) applied the analysis of r modes to the solar case and discussed the possibility of different radial orders. Later, Wolff (1998) studied the effect of differential rotation on r modes and suggested that nonsectoral modes might be suppressed while sectoral modes are the least affected because their motions are concentrated in the equatorial regions.
In the corotating frame and to lowest order in the mean solar rotation rate Ω, the frequencies of r modes are given by
where ℓ is the harmonic degree, and m is the azimuthal order. A consequence of this is that the horizontal flow field arising from the rmode oscillations characterized by (ℓ,m) drifts at a phase velocity of ω/m = −2Ω/[ℓ(ℓ+1)] < 0. The negative phase velocity means the drift direction is retrograde.
Löptien et al. (2018) used six years of observations from the Helioseismic and Magnetic Imager on board the Solar Dynamical Observatory (SDO/HMI: Scherrer et al. 2012; Schou et al. 2012) to provide a direct and unambiguous detection of Rossby waves at the surface and in the outer 20 Mm of the Sun. Using both granulationtracking (e.g., Löptien et al. 2016, 2017) and ringdiagram analysis (e.g., Bogart et al. 2015), Löptien et al. (2018) find radial vorticity patterns along the equator which propagate retrograde in the corotating frame with a dispersion relation that is consistent with Eq. (1) for the m = ℓ case (sectoral modes). Löptien et al. (2018) find no evidence for the existence of the nonsectoral modes.
The rmode velocity for sectoral modes at the equator is solely in the northsouth direction (see Saio 1982, for sketches of the flow field of r modes). Figure 1 shows the flow field and vorticity of classical rmode oscillations for the case of m = ℓ = 5. For this paper we have used timedistance helioseismology to measure subsurface flows in the meridional direction along the equator. The timedistance helioseismology pipeline of Liang et al. (2018) applies not only to the recent SDO/HMI data but also to SOHO/MDI data (Scherrer et al. 1995). The use of timedistance technique and the SOHO/MDI data would provide an independent confirmation of the findings of Löptien et al. (2018).
Fig. 1.
Classical sectoral r mode with m = 5, seen from the equatorial plane of a uniformly rotating solar model. The three panels show the southward flow v_{θ}, the prograde flow v_{ϕ}, and the radial vorticity ζ_{r} (from left to right) in the corotating frame. The color scale is the same for v_{θ} and v_{ϕ} with red positive and blue negative while the colors for the ζ_{r} indicate radially outward (red) or inward (blue) directions. The black dotted lines represent constant longitudes fixed in the corotating frame. The green rectangle marks the equatorial area (±15°) in which we measure v_{θ} in this paper. A movie showing the patterns propagating in the retrograde direction in the corotating frame is available online. 
2. Data analysis and results
2.1. Timedistance analysis
We use mediumℓ Dopplergrams taken by SOHO/MDI and SDO/HMI covering the period from May 1996 to April 2017. Background Doppler signals, such as solar rotation, are removed by subtracting a onehour running mean for each pixel. A bandpass frequency filter is further applied to isolate the p modes within 2–5 mHz. Every 24h time series of Dopplergrams are then projected into heliographic coordinates with a map scale of 0.6° pixel^{−1} and tracked at the Carrington rotation rate. An improvement to the mapping procedure used in Liang et al. (2018) is made here by taking an error in the inclination angle of the solar rotation axis into account (see Appendix A.1).
We compute the crosscovariance function (CCF) between pairs of points arranged in the arctoarc geometry used by Liang et al. (2018) where two 30°wide concentric arcs, separated by an angular distance Δ, are aligned in the northsouth direction. The CCFs between pairs of points on the opposing arcs are averaged and associated with the central point of the two arcs. The procedure is repeated for different central points located within ±15° latitude at intervals of 0.6° in longitude and latitude, and for the distance range Δ = 6°–30° in steps of 0.6°. Unlike the averaging scheme in Liang et al. (2018) where the CCFs were averaged over longitude and over days in each month, the daily CCFs are Gaussian smoothed with FWHM = 12° in longitude and latitude, and subsampled along the equator at intervals of 10° in longitude.
The southnorth traveltime shift in this work is defined as δτ = τ_{s} − τ_{n}, where τ_{n} and τ_{s} are the northward and southward travel times of the firstskip wavelets in the CCF, respectively. We measure the southnorth traveltime shifts from the spatially smoothed CCFs using the linearized oneparameter fitting method (Gizon & Birch 2002) as this algorithm is more robust to noise (Gizon & Birch 2004). More precisely, a 20min interval around the firstskip wavelet in the CCF for positive time lag (i.e., the wavelet traveling in the northward direction) is selected as the reference function to compute the weight function derived by Gizon & Birch (2002). The weighted sum of the differences between the southward and northward wavelets in the CCF gives an estimate of the southnorth traveltime shift. The measured southnorth traveltime shifts for different travel distances are sensitive to subsurface northsouth flows around the equator as depicted in Fig. 2. Data points within active regions are included in the averaging of CCFs in order to reduce the noise level. Also, the periods when the SOHO spacecraft was rotated by 180°, which were not used in Liang et al. (2018), are used here for these alternate threemonth gaps would result in strong leakage sidelobes in the Fourier domain.
Fig. 2.
Schematic plot of acoustic ray paths (red lines) that connect pairs of points across the equator in a meridional plane. The traveltime difference between the southward and northward propagating acoustic waves is sensitive to the northsouth flow along the ray path. The larger the angular distance between the observation points at the surface, the deeper the lower turning point of the rays. 
To enhance the signaltonoise ratio, we average the traveltime shifts over all travel distances with a weighting function that takes the noise correlation between different distances into account (see Appendix A.2). The weighted mean distance is about 14.6°, for which the corresponding depth of the lower turning point from the ray approximation is about 63 Mm. The weighted average of the traveltime shifts suffers from a strong annual variation, mostly due to the centertolimb effects (Zhao et al. 2012)^{1}. To remove this timevarying background, we fit and subtract a periodic function from the measured traveltime shifts (see Appendix A.3).
The Carrington coordinate system rotates at Ω_{cr}/2π = 456.03 nHz in a sidereal frame. As viewed from the Earth, the Carrington coordinate system rotates at Ω_{cr} − Ω_{⊕}, where Ω_{⊕}/2π = 31.69 nHz is the Earth’s mean orbital frequency. The central times of the daily measurements are denoted by t_{j} = t_{0} + jΔt, where Δt = 24 h and j ∈ {0, 1, …, 7669}, covering 21 years of data. At each time t_{j} the Carrington longitudes of the traveltime measurements are given by
where i ∈ { − 5, −4, …, 5} (11 measurements along the equator each day) and Δψ = 10° is the spatial sampling rate. The longitude ψ_{00} is the Carrington longitude of the central meridian as seen by the observer at time t_{0}, which is the time when we start using MDI observations:
To compare with the results of Löptien et al. (2018), we transform to a frame that rotates at Ω_{eq}/2π = 453.1 nHz, that is the solar surface equatorial rotation rate in the sidereal frame measured by globalmode helioseismology. The transformation between the two coordinate systems is given by
where ϕ_{ij} is the longitude measured in the new frame. Figure 3 is a schematic plot of the corresponding window function W(ϕ_{ij}, t_{j}), which is equal to 1 for i ∈ { − 4, …, 4}, tapered to 1/2 at the boundaries i = ±5 where noise is slightly higher, and zero elsewhere. The window function is also zero for missing data.
Fig. 3.
Section of the window function W(ϕ_{ij}, t_{j}) in the frame that rotates at Ω_{eq}/2π = 453.1 nHz. At time t_{j} the longitudes of the travel times are denoted by ϕ_{ij} (see main text). The window function is equal to one for i ∈ { − 4, …4}, one half at the boundaries i = ±5, and zero elsewhere. The window function is also zero for missing data. The temporal periodicity of the window is 2π/(Ω_{eq} − Ω_{⊕}) = 27.46 days. 
We denote by δτ(ϕ_{ij}, t_{j}) the travel times interpolated at the longitudes ϕ_{ij} in the new frame. In practice, we implement the interpolation in the spatial Fourier domain by multiplying the spatial Fourier transform of the travel times by the phase factor exp[−im(Ω_{cr} − Ω_{eq})(t_{j} − t_{0})], where m is the azimuthal wavenumber.
2.2. Power spectra
The full data set δτ(ϕ_{ij}, t_{j}) is divided into three consecutive periods of seven years. The starting times of these three periods, denoted by t^{(k)} with k = {1, 2, 3}, are given by
The window functions for each 7yr period is
where C_{T}(t_{j} − t^{(k)}) is a taper time window that selects an observation period of T = NΔt ≈ 7 yr where N = 2556, starting from time t^{(k)} (see Fig. 4).
Fig. 4.
Window function C_{T}(t) (thick solid line) that selects an observation period of T = 2556Δt ≈ 7 yr, tapered with oneyear raised cosines at both ends. 
The power spectrum of each 7yr period is
where the azimuthal order m is in the range m ≤ 18 (spatial Nyquist frequency). For each , the temporal Nyquist frequency is 5787 nHz and the frequency resolution is 4.5 nHz.
We denote the 2D Fourier transform of by . Because the resulting involves the convolution of , it is necessary to investigate the power spectrum of the window function, , from which the spectral leakage arises. We note that since the window function is real.
The mean power spectrum of the window functions, , consists of welldefined peaks, as can be seen in Fig. 5a (plot restricted to m ≥ 0). The maximum values of the peaks at m = 1 and m = 2 are 76% and 30% of the peak at the origin (m, ω) = (0, 0). The maximum power of the other peaks drops rapidly, 3% for m = 3 and 1% for m = 4. The decrease in the peak power with increasing m (as shown in Fig. 5c) corresponds to that of a square window covering 100° longitudes. In Fig. 5a, adjacent peaks are shifted in frequency by (Ω_{eq} − Ω_{⊕})/2π = 421.41 nHz due to the choice of coordinate frame. Since the power spectrum involves a convolution in Fourier space of the solar data with , the power spectrum of the data at (m, ω) leaks to multiple sidelobes at (m + δm, ω − δm (Ω_{eq} − Ω_{⊕})), where δm = 1, 2, …
Figure 5b shows a cut at m = 0 through the power spectrum of the window function. The width of the main lobe along the ωaxis is just one frequency bin and the frequency leaks to the neighboring frequency bins are nearly zero (two order of magnitude smaller than the central peak). Since we observe only a fraction of the Sun (see Fig. 3), there is spatial leakage to the neighboring m at frequency separations that are integer multiples of 421.41 nHz. Figure 5c shows these leaks, which implies a resolution in m of about 4.
Fig. 5.
Panel a: mean power spectrum of the window functions . The power spectrum is normalized to unity at (m, ω) = (0, 0). The maximum value of each peak is written right next to the peak. The frequency offset between the peaks with adjacent m is (Ω_{eq} − Ω_{⊕})/2π = 421.41 nHz. Panel b: cut at m = 0 through the power spectrum from panel a. Panel c: maximum power of the peaks as a function of m, depicting the spatial leakage. 
Figure 6 shows the for the three different periods, as well as the mean spectrum
Fig. 6.
Power spectra of southnorth traveltime shifts measured in the frame rotating at Ω_{eq}/2π = 453.1 nHz. Panels a–c: from three time periods. Panel d: mean spectrum from an average of the three (seven years each). The blue lines highlight the dispersion relation of the classical Rossby waves described in Eq. (1) with ℓ = m and Ω = Ω_{eq}. The orange ellipse marks the excess lowfrequency power at low m and the purple ellipse marks the spectral leakage from the lowfrequency power. The gray scale is the same for the four panels and is shown in the color bar on the right. For clarity, the spectra are rebinned in frequency by a factor of three, such that the frequency resolution is 3/T = 13.6 nHz. 
The power distribution in all three peaks around the eigenfrequencies of r modes as described in Eq. (1) for the case of ℓ = m ≥ 3. However, there is no clear sign of ℓ = m ≤ 2. It is interesting to note that the mode amplitudes vary among the three , suggesting a possible temporal evolution of Rossby waves over the solar cycles. After the averaging, the ridge structure in the along ω = −2Ω_{eq}/(m + 1) becomes more prominent. The other two ridges of power seen in Fig. 6, which are separated from the central ridge by (δm, δω) = ( ± 1, ∓(Ω_{eq} − Ω_{⊕})) with a reduced amplitude (60 ∼ 80%), are due to the spectral leakage discussed previously. Excess power at low frequencies is present at low m, which also leaks and modulates the spectrum at frequencies between −300 and −500 nHz. It might be caused by largescale convection or local flows around active regions that corotate with the Sun, and should not be mistaken for oscillation power. While the annual variation (31.7 nHz) in the background has been removed in the analysis, harmonics at ±63.4 nHz remain for m = 0 as the background fitting in Appendix A.3 only accounts for the first order term of B_{0}angle variation. We do not expect any aliasing at high m since we applied a Gaussian smoothing with FWHM = 12° in longitude as mentioned in Sect. 2.1.
Figure 7 shows the in closeup around the frequency range of interest. The distribution of rmode power seems to shift toward less negative frequencies by 10–20 nHz for m ≥ 10 compared to the mode frequencies measured by Löptien et al. (2018).
Fig. 7.
Enlargement of Fig. 6d around the frequency range of interest (see Fig. 8 for the line profiles of individual modes in the range 3 ≤ m ≤ 14). The red circles show the mode frequencies estimated from Lorentzian fits (see Sect. 2.3). The errors in the mode frequencies are given roughly by the size of the red circles (see errors in Table 1). For comparison, the mode frequencies measured by Löptien et al. (2018) are also indicated by green circles. 
We note that our analysis is independent of the choice of reference frame. To illustrate this point, we present the power spectrum of data computed in the frame of the observer (as seen from Earth) in Appendix A.4. Compared to the power spectrum computed in the corotating frame, this power spectrum is shifted by m × 421.41 nHz for each m value. Figure A.4 displays a set of horizontal segments of size set by the resolution in m (≈4, see Fig. 5c). While the resolution in m is not one, one can easily retrieve the mode frequencies since the sectoral Rossby modes are well separated in frequency due to their long lifetimes.
2.3. Mode frequencies and linewidths of r modes
To quantify the line profiles of r modes, a model consisting of a Lorentzian function plus a constant background,
is fit to the at individual m. Here A is the maximum height of the Lorentzian function, ω_{m} the mode frequency, Γ_{m} the full width at half maximum, and B the constant background power. To ensure that the Lorentzian fits are not affected by the spatial leaks from neighboring m (see Fig. 8, at negative frequencies, on the left) nor by the lowfrequency power from active regions or convection (near zero frequency, on the right), we chose a fitting range [ν_{start}, ν_{start} + 300 nHz], with ν_{start} given in Table 1.
Fig. 8.
Power spectra of southnorth traveltime shifts for modes in the range 3 ≤ m ≤ 14 (black curves, with the frequency resolution 1/T = 4.5 nHz). The travel times are measured in the frame rotating at equatorial rotation rate Ω_{eq}/2π = 453.1 nHz. The red lines are the fits F_{m}(ω) given by Eq. (8). The green vertical lines indicate the mode frequencies from Löptien et al. (2018). The orange arrows mark the excess lowfrequency power at low m that might be caused by active regions or largescale convection. The purple arrows mark the leaks from the m − 1 rmode power and the lowfrequency power at m − 1. Each power spectrum is accompanied by a plot of the ratio in gray in the lower panel, which is expected to have a mean of unity (dashed line) and a constant variance if the fit is not biased. 
Measured characteristics of solar sectoral r modes.
The best fit is obtained by minimizing the sum
with respect to the model parameters A, B, ω_{m}, Γ_{m} (Duvall & Harvey 1986). Although Eq. (9) was derived assuming that the quantity has a chisquared distribution with two degrees of freedom for a single realization, we can minimize the same function for the case of multiple realizations (Anderson et al. 1990) as the is an average over three power spectra. The minimization algorithm used is the downhill simplex method (e.g., Press et al. 1992, Sect. 10.4). Table 1 lists the resulting best fit parameters for 3 ≤ m ≤ 15.
Figure 8 shows the fitted F_{m}(ω) on top of the , along with the results of Löptien et al. (2018) for comparison. We have also computed the spectrum without splitting the 21yrlong time series, the results of which appear to be similar to Fig. 8 but with better frequency resolution and higher noise level. The fitted mode frequencies and linewidths are remarkably consistent with that of Löptien et al. (2018) for 3 ≤ m ≤ 9. However, excess power on the right of the main peak is present for m ≥ 8, the extent of which for m ≥ 10 becomes so large that the overall profiles shift toward less negative frequencies.
2.4. Estimates of rmode velocity
To obtain a rough estimate of the flow speed associated with a mode, first we compute in the ray approximation the traveltime shifts due to a prescribed toroidal flow field (see, e.g., Saio 1982). For the forward calculation, we choose a maximum horizontal flow speed at the surface of 2 m s^{−1}. We considered two toroidal flows: the first one is independent of depth and the second one decreases linearly with depth to vanish at 0.9 R_{⊙}. The forwardmodeled traveltime shifts in the northsouth direction are Gaussian smoothed in longitude and latitude and averaged over travel distances in the same way as the measurements. The resulting traveltime shifts decrease with increasing m due to the smoothing in longitude. The maximum traveltime shifts from the first flow model range from 0.13 s to 0.05 s depending on m, from which we derive conversion constants of 15.6–41.7 m s^{−2} to convert from traveltime shifts to the surface flow speed. The conversion constants from the second model are in general larger by a factor of ∼1.6 for each mode.
Next, we obtain the rms traveltime shifts δτ_{rms} from the fitted Lorentzian profile F_{m}(ω)−B for each mode using Parseval’s theorem. The effect of the incomplete data coverage and the spectral leakage is estimated by applying the same window functions and analysis to synthetic data, and is taken into account when computing the δτ_{rms}.
Last, we use the above conversion constants to convert from δτ_{rms} to the surface rms flow speed v_{rms}. The conversion constants adopted are from the first model in which the flow is independent of depth, and thus the estimated v_{rms} are conservative. The results for modes with 3 ≤ m ≤ 15 are listed in Table 1. For these modes, the surface velocity is on the order of 1–3 m s^{−1}, with larger values for 7 ≤ m ≤ 13. The modes with the lowest m values have velocities of order 1–2 m s^{−1}, as assumed in the RV study by Lanza et al. (2019).
3. Discussion
Using an independent helioseismic method and a different data set, we have confirmed the existence of the equatorial global Rossby waves reported by Löptien et al. (2018). We have extended the observations to deeper layers (down to ∼63 Mm) and to a total period of 21 years by combining SOHO and SDO observations. The power spectra obtained from three sevenyear periods covering cycles 23 and 24 all show signatures of r modes for 3 ≤ m = ℓ ≤ 15. The measured mode frequencies and linewidths are generally consistent with the granulationtracking results of Löptien et al. (2018). However, in our data, excess power is observed on the lowfrequency side of the line profiles for m ≥ 10 which leads to a systematic shift of the fitted mode frequencies with respect to Löptien et al. (2018).
In order to check if the spectral leakage comes into play, we applied the same window functions and Fourier analysis to synthetic data. The resulting spectra do not show apparent frequency shifts, implying that the excess power is not due to the leakage of neighboring modes. Because rmode frequencies may vary over the solar cycle, we also examined separately the spectrum shown in Fig. 6c computed over a similar period as that used by Löptien et al. (2018). The abovementioned excess power remains present, which means that this difference with Löptien et al. (2018) is not due to the use of MDI data in earlier periods.
Systematic effects associated with surface magnetic field cause solar cycle variations in the traveltime measurements (Liang & Chou 2015a). We tried excluding the data points inside the active regions from the averaging of CCFs; however, the resulting power spectrum was dominated by noise as the masking procedure removed a considerable amount of pixels. Even if we could resolve the systematic effects of the surface magnetic field, the local flows surrounding the active regions may still enter the traveltime measurements (e.g., Gizon et al. 2001; Gizon 2004). We suspect that the excess lowfrequency power at low m is caused by active regions and associated local flows since the rotation rate of the active regions is rather close to zero in our chosen rotation frame. Also the lowfrequency power is stronger during the first (May 1996–April 2003) and third (May 2010–April 2017) periods when the solar activity is relatively higher. However, we cannot strictly exclude the possibility that the lowfrequency power results from largescale convection. Examination of the power spectra of traveltime measurements at different latitudes may help clarify this issue.
The errors in mode frequency measurements listed in Table 1 were estimated from Monte Carlo simulations. These errors can also be estimated using Eq. (2) from Libbrecht (1992) which relates the errors to the fitted parameters A, B, Γ_{m} and the total observation time. The estimation of the errors on mode frequencies from the two methods are consistent. Our signaltonoise ratio A/B is smaller than the granulationtracking observations of Löptien et al. (2018). Twentyone years of data analyzed using timedistance helioseismology give similar frequency error estimates as six years analyzed using granulation tracking.
It is interesting to compare the mode amplitudes measured here with those reported by Löptien et al. (2018). To this end, we first computed the radial vorticity of a prescribed rmode toroidal flow field as used in the forward modeling, from which we obtained the ratio of rms velocity to rms vorticity in the latitude range between ±20° for each mode. We then used the ratio to convert the rms radial vorticity reported by Löptien et al. (2018) to the rms velocity. The rms velocity estimated from their rms vorticity is on the order of 1–2 m s^{−1} and is consistent with the v_{rms} in Table 1.
We do not see any evidence for the m = 2 mode (see Fig. 9). The traveltime measurements suffer from both timeindependent and annual variations for m ≤ 2 due to centertolimb effects. The m = 2 mode is, however, expected to have a period of about 3/2 of the rotation period and should be cleanly separated from the centertolimb systematics in the Fourier domain. To place an upper limit on the velocity of a possible m = 2 mode, we generated synthetic m = 2 sinusoidal traveltime shifts with frequencies in the range between −350 and −150 nHz. Three different amplitudes that correspond to rms velocity of 0.4, 0.5, and 0.6 m s^{−1} were implemented. These synthetic data were added into the measurements and Fourier analysis was applied with the window functions being taken into account. Figure 9 shows the resulting power levels. We find that an m = 2 sectoral mode with v_{rms} < 0.5 m s^{−1} would be difficult to identify in the power spectrum.
Fig. 9.
Power spectrum of southnorth traveltime shifts for m = 2 (black solid line). The blue vertical line indicates the frequency of the classical m = 2 sectoral r mode. The red solid line is the background B estimated by a fit to the power in the frequency range between −350 and −150 nHz. The red dashed line is the threshold for 95% confidence level; that is, the noise in the background only has a 5% chance of being higher than this threshold for at least one frequency bin. We note that the spike around −295 nHz (on the right side of the blue line) is above the background but much lower than the 95% confidence level. The three green lines indicate the power that would correspond to a m = 2 sectoral mode with rms velocity of 0.4, 0.5, or 0.6 m s^{−1}. 
The efolding lifetimes of r modes, 2/Γ_{m}, are on the order of a few months for most of the modes observed here. The longest lived modes (m = 3 and m = 5) have lifetimes of more than one year. Figure 10 shows a schematic excitation event (an exponentially decaying cosine) for the m = 5 r mode as seen in the corotating frame. The physical mechanisms responsible for the excitation, or damping, of the r modes are not yet known.
Fig. 10.
Schematic damped oscillation of a sectoral r mode with m = 5, seen at a fixed longitude in the Sun’s corotating frame. The value of the efolding lifetime (vertical line) is as observed (see Table 1). We note that if the oscillation were seen in the Earth’s frame, the observed oscillation period would be about 6 days for m = 5. 
While the rmode oscillations in the nearsurface layers have been remarkably determined using the granulationtracking and ringdiagram analysis, the depth dependence of solar r modes throughout the convection zone is largely unknown. Timedistance helioseismology (Duvall et al. 1993) as used in this work has shown to be an important tool to study the r modes deep inside the Sun. More measurements of deep rmode oscillations such as the one presented here will provide some insight into the nature of r modes and, hopefully, further constrain theories of solar r modes.
Acknowledgments
We thank B. Löptien, B. Proxauf and J. Schou for useful discussions. We also thank the referee for comments that helped improve the manuscript. 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. The data were processed at the German Data Center for SDO (GDCSDO), funded by the German Aerospace Center (DLR). L.G. acknowledges partial support from the NYU Abu Dhabi Center for Space Science under grant G1502. We used the workflow management system Pegasus funded by The National Science Foundation under OCI SI2SSI program grant #1148515 and the OCI SDCI program grant #0722019.
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Appendix A: Supplementary m aterial
A.1. Inclination of the rotation axis
Because the inclination angle of the Sun’s rotation axis determined by Carrington (1863) has been found to be slightly in error (Beck & Giles 2005; Hathaway & Rightmire 2010), corrections to both the instrument roll angle and the solar tilt angle B_{0} are needed, otherwise a strong annual variation appears in the measured traveltime shifts and modulate the resulting spectrum. Accordingly, an improvement to the mapping procedure used in Liang et al. (2018) is made by including corrections to the values of the two keywords CRLT_OBS (B_{0} angle) and CROTA2 (instrument roll angle). Following Larson & Schou (2015, Eqs. (3) and (4)), the first order corrections are implemented as
where the primed keywords denote the updated values, δI is the error in the inclination angle, δP is an instrumentspecific correction for CCD misalignment, t and t_{ref} are the observation time and the time when CRLT_OBS is close to zero, both expressed in years (365.25 days). The t_{ref} is determined by fitting a sinusoid with a fixed period of 365.25 days to a time series of keyword CRLT_OBS, and is 7 June 1996 01:19:34_TAI for MDI data and 7 June 2010 14:17:20_TAI for HMI data. The adopted value of δP for MDI data is 0.2° (Liang et al. 2017, 2018), while that for HMI data is zero since the HMI team had calibrated and updated the value of CROTA2 (Couvidat et al. 2016; Hoeksema et al. 2018). As for δI, a value of −0.08° from Hathaway & Rightmire (2010) is adopted. The effect of this correction is quantified by fitting a periodic function to the southnorth traveltime shifts (see Appendix A.3).
A.2. Weighting function
Here we use the notation introduced in Sect. 2.1 and denote the measured southnorth traveltime shifts by δτ^{(0)}(ψ, t, Δ), where ψ is the Carrington longitude, t the observation time, and Δ the travel distance. We have dropped the indices i and j in the subscripts of ψ and t for simplicity. To combine the traveltime shifts for different travel distances, we follow the procedure in Jackiewicz et al. (2008, Sect. 5.1). Given two random variables X and Y, the covariance is defined by Cov[X, Y]=⟨XY⟩−⟨X⟩⟨Y⟩, where the angle brackets denote the expectation value (ensemble average). The noise covariance matrix of the traveltime shifts between different distances is given by Λ(Δ, Δ′) = Cov[δτ^{(0)}(ψ, t, Δ),δτ^{(0)}(ψ, t, Δ′)]. The weighting for each distance is then given by
and the weighted average of the measured traveltime shifts is
The square root of Λ(Δ, Δ) is the standard deviation of the traveltime shifts, denoted by σ(Δ), which gives an estimate of the noise level at each distance. Figure A.1 shows the σ(Δ) and w(Δ) as a function of distance. The trend of σ is consistent with Fig. 2 of Beck & Giles (2005) except that the noise level in their result is much lower probably owing to the use of phasespeed filter and different longitude and latitude ranges in the averaging. For Δ < 10°, the σ increases rapidly because of the low spatial resolution of mediumℓ Dopplergrams and thus lack of information about the acoustic waves residing in the nearsurface layers. Consequently, the weights for short distances are small. For Δ > 10°, the σ increases with increasing distance because of geometrical spreading and damping. Also, the noise correlation between different distances (offdiagonal elements of Λ) becomes greater when both Δ and Δ′ are large (Gizon & Birch 2004), which further reduces the weights for largedistance cases.
Fig. A.1.
Top: standard deviation of the measured traveltime shifts (square root of the diagonal elements of Λ) as a function of distance. The zigzag might be due to the discreteness of the windows that isolate the singleskip wavelet in the crosscovariance function when fitting the traveltime shifts, particularly for shortdistance cases in which the slope of the singleskip ridge in the timedistance diagram is steepest. Bottom: weighting function, w(Δ), as a function of distance. The corresponding radii of the lower turning points from the ray approximation are indicated at the top. The weighted mean distance (vertical dashed line) is about 14.6°, which corresponds to a lower turning point of ∼0.91 R_{⊙}. 
A.3. Removal of the background variation
Following the notation in Sect. 2.1, at time t_{j}, the Carrington longitudes of the traveltime measurements are denoted by ψ_{ij}, and the central meridian as seen by the observer is denoted by ψ_{0, j}. Because the discussion here involves the centertolimb effects, it is convenient to define the separation from the central meridian by ψ′_{ij} ≡ ψ_{ij} − ψ_{0, j}. The weighted average of the traveltime shifts resulting from Appendix A.2 is expressed in this coordinate system and denoted by δτ^{(1)}(ψ′_{ij}, t_{j}).
At each longitude ψ′_{ij}, a function representing the temporal variation of the background
is fitted to and subtracted from the δτ^{(1)}(ψ′_{ij}, t_{j}) for MDI and HMI data separately. Here t_{ref} is taken from Appendix A.1. The a_{0}(ψ′_{ij}) represents the timeindependent background, and a_{1}(ψ′_{ij}) represents the linear component, if any, of the slowly varying background. The a_{2}(ψ′_{ij}) accounts for an annual variation due to systematic errors such as the centertolimb effects that vary with the B_{0} angle (CRLT_OBS); in addition, the a_{3}(ψ′_{ij}) is added since the magnitude of centertolimb effects was found to be time dependent (Liang & Chou 2015b; Liang et al. 2018). The a_{4}(ψ′_{ij}) accounts for another annual variation caused by the error in the inclination of the rotation axis, if not completely removed in Appendix A.1; the resulting error in the roll angle (i.e., Eq. (A.2)) may introduce a leakage of the solar rotation signal into the southnorth traveltime shifts (Beck & Giles 2005).
Figure A.2 shows the δτ^{(1)}(ψ′_{ij}, t_{j}), the fitted background S(ψ′_{ij}, t_{j}), and the residuals
Fig. A.2.
Left: traveltime measurements δτ^{(1)}(ψ′_{ij}, t_{j}), Gaussian smoothed with FWHM = 90 days in time for better visualization, where ψ′_{ij} ≡ ψ_{ij} − ψ_{0, j} is the separation from the central meridian. Middle: fitted background S(ψ′_{ij}, t_{j}). Right: difference between the two panels on the left. The horizontal dashed lines in all panels indicate the division between the results from MDI and HMI data; the MDI data used in this work span from May 1996 to April 2010 while HMI data span from May 2010 to April 2017. 
The annual variation is clearly seen in the δτ^{(1)}(ψ′_{ij}, t_{j}), suggesting the necessity of dealing with this systematic effect. Also, the magnitude of δτ^{(1)}(ψ′_{ij}, t_{j}) from HMI data at larger ψ′_{ij} is systematically greater than the rest. The fitted S(ψ′_{ij}, t_{j}) accounts for a large amount of the periodic variation of the background. After removing the fitted background, the resulting δτ^{(2)}(ψ′_{ij}, t_{j}) from the two data sets are more consistent with each other, and are ready for the Fourier analysis.
Figure A.3 shows the S(ψ′_{ij}, t_{j}) at selected longitudes as well as the fitted values of the parameters as a function of longitude for the two data sets. It is apparent that a_{0}(ψ′_{ij}) and a_{2}(ψ′_{ij}) (i.e., the centertolimb effects) dominate the background variation while a_{4}(ψ′_{ij}) is nearly zero (0.008 ± 0.021 s for MDI and −0.015 ± 0.038 s for HMI within the longitude range ψ′_{ij}≤30°) since we have applied a correction δI = −0.08° to the inclination angle of solar rotation axis as discussed in Appendix A.1. Without this correction, the value of a_{4}(ψ′_{ij}) is around 0.1–0.2 s. We also tried δI = −0.1° used by Larson & Schou (2015) who rounded the value −0.095 ° ±0.002° determined by Beck & Giles (2005) up to −0.1°; however, the a_{4}(ψ′_{ij}) becomes negative in this case (−0.035 ± 0.019 s for MDI and −0.058 ± 0.038 s for HMI).
Fig. A.3.
Panel a: fitted S(ψ′_{ij}, t_{j}) at selected longitudes, where ψ′_{ij} ≡ ψ_{ij} − ψ_{0, j} is the separation from the central meridian. The vertical dashed line indicates the division between the results from MDI and HMI data. Panels b–e: fitted values of parameters as a function of longitude for the MDI and HMI data sets, respectively. The error bars are estimated from the covariance matrix of the fitting parameters. 
The background variation in the measurements from HMI is larger than that from MDI since the magnitude of centertolimb effects for HMI is greater (Liang et al. 2017). The value of a_{2}(ψ′_{ij}) is expected to be negative because when the solar north pole is tilted toward the observer (CRLT_OBS > 0) the solar equator is on the southward side of the images where the centertolimb variation in the southnorth traveltime shifts is negative.
Furthermore, the background variation shows an eastwest asymmetry which changes over time, especially in the measurements from MDI data. The eastwest asymmetry in the traveltime measurements from MDI was reported by Giles (2000, Sect. 4.7.2). He attributed the asymmetry to the nonuniform focus of the MDI camera, which changed at times during the mission. We note that the granulationtracking measurements also suffer from systematic errors similar to the centertolimb effects, termed the shrinkingSun effect (Lisle et al. 2004). The shrinkingSun effect not only depends on the distance to the disk center, but also exhibits an asymmetry in the eastwest direction (Löptien et al. 2016, 2017). Löptien et al. (2016) suggested that the eastwest asymmetry in the shrinkingSun effect could be caused by the influence of the solar rotation on the observing height. If so, it would have a similar effect on the traveltime measurements since the magnitude of centertolimb effects is strongly dependent upon the lineformation height of the observables (Zhao et al. 2012; Baldner & Schou 2012).
A.4. Power spectrum as seen from Earth
Figure A.4 shows the power spectrum computed in the frame of the observer (Earth’s frame).
Fig. A.4.
Power spectrum of southnorth traveltime shifts computed in the Earth’s frame. The blue line highlights the dispersion relation of sectoral modes of classical Rossby waves in this frame. The mode frequencies (red circles) are shifted by m × 421.41 nHz when measured in the Earth’s frame. The lowfrequency power from active regions or convection (near zero frequency at low m in Figs. 6 and 7) is also shifted to m × 421.41 nHz (green crosses) when measured in the Earth’s frame. 
Movie
Movie of Fig. 1 (Access here)
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All Figures
Fig. 1.
Classical sectoral r mode with m = 5, seen from the equatorial plane of a uniformly rotating solar model. The three panels show the southward flow v_{θ}, the prograde flow v_{ϕ}, and the radial vorticity ζ_{r} (from left to right) in the corotating frame. The color scale is the same for v_{θ} and v_{ϕ} with red positive and blue negative while the colors for the ζ_{r} indicate radially outward (red) or inward (blue) directions. The black dotted lines represent constant longitudes fixed in the corotating frame. The green rectangle marks the equatorial area (±15°) in which we measure v_{θ} in this paper. A movie showing the patterns propagating in the retrograde direction in the corotating frame is available online. 

In the text 
Fig. 2.
Schematic plot of acoustic ray paths (red lines) that connect pairs of points across the equator in a meridional plane. The traveltime difference between the southward and northward propagating acoustic waves is sensitive to the northsouth flow along the ray path. The larger the angular distance between the observation points at the surface, the deeper the lower turning point of the rays. 

In the text 
Fig. 3.
Section of the window function W(ϕ_{ij}, t_{j}) in the frame that rotates at Ω_{eq}/2π = 453.1 nHz. At time t_{j} the longitudes of the travel times are denoted by ϕ_{ij} (see main text). The window function is equal to one for i ∈ { − 4, …4}, one half at the boundaries i = ±5, and zero elsewhere. The window function is also zero for missing data. The temporal periodicity of the window is 2π/(Ω_{eq} − Ω_{⊕}) = 27.46 days. 

In the text 
Fig. 4.
Window function C_{T}(t) (thick solid line) that selects an observation period of T = 2556Δt ≈ 7 yr, tapered with oneyear raised cosines at both ends. 

In the text 
Fig. 5.
Panel a: mean power spectrum of the window functions . The power spectrum is normalized to unity at (m, ω) = (0, 0). The maximum value of each peak is written right next to the peak. The frequency offset between the peaks with adjacent m is (Ω_{eq} − Ω_{⊕})/2π = 421.41 nHz. Panel b: cut at m = 0 through the power spectrum from panel a. Panel c: maximum power of the peaks as a function of m, depicting the spatial leakage. 

In the text 
Fig. 6.
Power spectra of southnorth traveltime shifts measured in the frame rotating at Ω_{eq}/2π = 453.1 nHz. Panels a–c: from three time periods. Panel d: mean spectrum from an average of the three (seven years each). The blue lines highlight the dispersion relation of the classical Rossby waves described in Eq. (1) with ℓ = m and Ω = Ω_{eq}. The orange ellipse marks the excess lowfrequency power at low m and the purple ellipse marks the spectral leakage from the lowfrequency power. The gray scale is the same for the four panels and is shown in the color bar on the right. For clarity, the spectra are rebinned in frequency by a factor of three, such that the frequency resolution is 3/T = 13.6 nHz. 

In the text 
Fig. 7.
Enlargement of Fig. 6d around the frequency range of interest (see Fig. 8 for the line profiles of individual modes in the range 3 ≤ m ≤ 14). The red circles show the mode frequencies estimated from Lorentzian fits (see Sect. 2.3). The errors in the mode frequencies are given roughly by the size of the red circles (see errors in Table 1). For comparison, the mode frequencies measured by Löptien et al. (2018) are also indicated by green circles. 

In the text 
Fig. 8.
Power spectra of southnorth traveltime shifts for modes in the range 3 ≤ m ≤ 14 (black curves, with the frequency resolution 1/T = 4.5 nHz). The travel times are measured in the frame rotating at equatorial rotation rate Ω_{eq}/2π = 453.1 nHz. The red lines are the fits F_{m}(ω) given by Eq. (8). The green vertical lines indicate the mode frequencies from Löptien et al. (2018). The orange arrows mark the excess lowfrequency power at low m that might be caused by active regions or largescale convection. The purple arrows mark the leaks from the m − 1 rmode power and the lowfrequency power at m − 1. Each power spectrum is accompanied by a plot of the ratio in gray in the lower panel, which is expected to have a mean of unity (dashed line) and a constant variance if the fit is not biased. 

In the text 
Fig. 9.
Power spectrum of southnorth traveltime shifts for m = 2 (black solid line). The blue vertical line indicates the frequency of the classical m = 2 sectoral r mode. The red solid line is the background B estimated by a fit to the power in the frequency range between −350 and −150 nHz. The red dashed line is the threshold for 95% confidence level; that is, the noise in the background only has a 5% chance of being higher than this threshold for at least one frequency bin. We note that the spike around −295 nHz (on the right side of the blue line) is above the background but much lower than the 95% confidence level. The three green lines indicate the power that would correspond to a m = 2 sectoral mode with rms velocity of 0.4, 0.5, or 0.6 m s^{−1}. 

In the text 
Fig. 10.
Schematic damped oscillation of a sectoral r mode with m = 5, seen at a fixed longitude in the Sun’s corotating frame. The value of the efolding lifetime (vertical line) is as observed (see Table 1). We note that if the oscillation were seen in the Earth’s frame, the observed oscillation period would be about 6 days for m = 5. 

In the text 
Fig. A.1.
Top: standard deviation of the measured traveltime shifts (square root of the diagonal elements of Λ) as a function of distance. The zigzag might be due to the discreteness of the windows that isolate the singleskip wavelet in the crosscovariance function when fitting the traveltime shifts, particularly for shortdistance cases in which the slope of the singleskip ridge in the timedistance diagram is steepest. Bottom: weighting function, w(Δ), as a function of distance. The corresponding radii of the lower turning points from the ray approximation are indicated at the top. The weighted mean distance (vertical dashed line) is about 14.6°, which corresponds to a lower turning point of ∼0.91 R_{⊙}. 

In the text 
Fig. A.2.
Left: traveltime measurements δτ^{(1)}(ψ′_{ij}, t_{j}), Gaussian smoothed with FWHM = 90 days in time for better visualization, where ψ′_{ij} ≡ ψ_{ij} − ψ_{0, j} is the separation from the central meridian. Middle: fitted background S(ψ′_{ij}, t_{j}). Right: difference between the two panels on the left. The horizontal dashed lines in all panels indicate the division between the results from MDI and HMI data; the MDI data used in this work span from May 1996 to April 2010 while HMI data span from May 2010 to April 2017. 

In the text 
Fig. A.3.
Panel a: fitted S(ψ′_{ij}, t_{j}) at selected longitudes, where ψ′_{ij} ≡ ψ_{ij} − ψ_{0, j} is the separation from the central meridian. The vertical dashed line indicates the division between the results from MDI and HMI data. Panels b–e: fitted values of parameters as a function of longitude for the MDI and HMI data sets, respectively. The error bars are estimated from the covariance matrix of the fitting parameters. 

In the text 
Fig. A.4.
Power spectrum of southnorth traveltime shifts computed in the Earth’s frame. The blue line highlights the dispersion relation of sectoral modes of classical Rossby waves in this frame. The mode frequencies (red circles) are shifted by m × 421.41 nHz when measured in the Earth’s frame. The lowfrequency power from active regions or convection (near zero frequency at low m in Figs. 6 and 7) is also shifted to m × 421.41 nHz (green crosses) when measured in the Earth’s frame. 

In the text 
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