HELIOSEISMIC MEASUREMENTS OF THE SUBSURFACE MERIDIONAL FLOW D. C. Braun 1. and Y. Fan

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1 The Astrophysical Journal, 508:L105 L108, 1998 Noveber The Aerican Astronoical Society. All rights reserved. Printed in U.S.A. HELIOSEISMIC MEASUREMENTS OF THE SUBSURFACE MERIDIONAL FLOW D. C. Braun 1 Solar Physics Research Corporation, 4720 Calle Desecada, Tucson, AZ, and Y. Fan High Altitude Observatory, National Center for Atospheric Research, Mitchell Lane, Boulder, CO Received 1998 August 18; accepted 1998 Septeber 16; published 1998 Septeber 30 ABSTRACT We easure the ean frequencies of acoustic (p-ode) waves propagating toward and away fro the poles of the Sun fro observations ade with the Solar Oscillations Investigation Michelson Doppler Iager on board the Solar and Heliospheric Observatory and the ground-based Global Oscillations Network Group. We deonstrate that there is a significant frequency shift between poleward- and equatorward-traveling waves easured over solar latitudes 20 60, which is consistent with the Doppler effect of a poleward eridional flow on the order of 10 s 1. Fro the variation of the frequency shifts of p-odes with degree between 72 and 882 as a function of the lower turning point depth, we infer the speed of the eridional flow, averaged over these latitudes, over a range in depth extending over the top half of the solar convection zone. We find no evidence for a significant equatorward return flow within this depth range. Subject headings: Sun: interior Sun: oscillations 1. INTRODUCTION The solar eridional circulation transports angular oentu and agnetic flux across a wide range of latitudes within the convection zone and is a significant coponent of odels of the dynaics of rotating stellar convection zones, dynaos, and the solar cycle (Glatzaier & Gilan 1982; Gilan & Miller 1986; Choudhuri, Schüssler, & Dikpati 1995; Dikpati & Choudhuri 1995; Wang, Sheeley, & Nash 1991). Measureents of the surface anifestation of eridional circulation have typically indicated values of poleward flows between 10 and 20 s 1, although values vary significantly aong different groups and detection techniques. Hathaway (1996) and Hathaway et al. (1996) have presented evidence to suggest that soe of these discrepancies ay be due to episodic changes in the flow rates. Recently, several groups have eployed helioseisic techniques to easure the near-surface and subsurface properties of eridional flows. In particular, Giles et al. (1997) and Giles, Duvall, & Scherrer (1998) have used techniques based on the principles of tie-distance helioseisology (Duvall et al. 1993), while González Hernández et al. (1998) and Schou & Bogart (1998) have eployed ring diagra analysis (Hill 1988; Patrón et al. 1995). The analysis we utilize here for the study of the solar eridional flow is based upon a procedure previously eployed in the exploration of p-ode sunspot interactions. Its principal goal is a decoposition of the solar oscillation signal, observed in an annulus surrounding a selected point, into appropriate inward- and outward-propagating wave odes. This ethod has been tered Fourier-Hankel spectral decoposition, since in its application to active-region seisology the annulus is usually chosen to be sall enough so that the spatial for of the wave odes is described very closely by Hankel functions. The initial utility of this procedure was deonstrated by 1 Also at High Altitude Observatory, National Center for Atospheric Research. 2 The National Center for Atospheric Research is sponsored by the National Science Foundation. L105 the exploration of the p-ode absorption qualities of sunspots and other solar activity by looking at the difference in aplitudes between the inward and outward wave coponents (Braun, Duvall, & LaBonte 1988; Bogdan et al. 1993; Braun 1995). The ethod is readily extended to analysis of larger annular regions over the spherical solar surface by using the appropriate cobinations of Legendre functions of the first and second kind instead of Hankel functions (Duvall et al. 1988; Braun, Duvall, & Jefferies 1990; Bogdan et al. 1993; Chen et al. 1996). We note that this ethod provides an ideal diagnostic of the properties of the subsurface eridional circulation. For exaple, a net poleward or equatorward flow beneath the annular doain of the analysis, centered on the solar poles, will produce equal but opposite frequency shifts (at constant horizontal wavenuber) of zonal ( 0) or near-zonal p-odes (defined as those odes with FF K ), where is the aziuthal order and is the ode degree. This procedure has a well-known analogy in the deterination of internal solar rotation by the easureent of frequency differences between prograde- and retrograde-traveling sectoral p-odes. Just as in the case of solar rotation, the variation of the frequency shift with and teporal frequency n provides a powerful diagnostic of the subsurface variation of the eridional flow. The basic concept is siilar in nature to that eployed in ring diagra analysis, which easures subsurface flows by easuring the Doppler distortion in power spectra constructed assuing a local plane wave approxiation for p-odes. The technique discussed here is, however, optiized for the detection of eridional flows by properly accounting for spherical geoetry and spanning a uch larger range of latitude and longitude then typically utilized for ring diagras. The wider spatial coverage enables the detection and resolution of p- odes of significantly lower degree which penetrate deeper into the solar interior, while sacrificing the ability to easure sall-scale variations of the flow in latitude and longitude.

2 L106 BRAUN & FAN Vol DATA REDUCTION The data used in this analysis consists of concurrent tie series of full-disk solar Doppler velocity iages as obtained by the Solar Oscillations Investigation Michelson Doppler Iager (SOI-MDI), described by Scherrer et al. (1995), and the Global Oscillations Network Group (GONG), described by Harvey et al. (1996). We used 8 days of nearly continuous SOI-MDI data between 1997 May 7 and 14, obtained during one of the high teleetry rate Dynaics capaigns, which consist of 1024 # 1024 pixel 60 s averaged calibrated Doppler velocity iages sapled once per inute. Fro the GONG database, we selected 32 days of calibrated velocity iages for the overlapping period 1997 April 13 May 14, with the preference aong the six GONG sites given to iages obtained closest to the zenith at any given tie. In our analysis, the duty cycle of usable SOI-MDI and GONG iages over the selected tie periods was approxiately 83% and 75%, respectively. The Doppler velocity values are first interpolated onto a spherical polar coordinate syste (r, v, f) situated about either pole of the Sun and evaluated at the solar radius r R,, with the resolution of the grid chosen to slightly oversaple the pixel spacing at the largest value of v. To siplify the notation, we will define the polar angle (colatitude) to be zero at either the north or south pole, with v increasing away fro the pole, and let the aziuthal angle have the sae handedness with respect to either pole. The annular region is defined by a range of colatitudes between vin 30 and vax 70 (i.e., latitudes between 20 and 60). The coordinate grid rotates about the poles with the Carrington rate. To avoid data near the solar lib, we eliinate points with a heliocentric angle greater than 60 fro disk center and apodize the edges in aziuth f with a cosine bell having a width of 10 aziuthal eleents. Solar rotation and steady and lowteporal frequency coponents of the Doppler signal are reoved (see Braun 1997). The residual velocity signal dv in spherical coordinates and as a function of tie t is then expanded as a su of the for i(f2pnt) dv(r,,v,f,t) e [AnV (cos v) n B n(v ) (cos v)], (1) where V is a noralized function: [ ] 2i V (cos v) { N P (cos v) Q (cos v), (2) p (V ) is its coplex conjugate, P and Q are Legendre func- tions of the first and second kind, and N is a noralization constant such that, in the liit of sall v and large, V (cos v) approaches the value of the Hankel function (1) H (v). A n and B n represent the coplex aplitudes of poleward and equatorward waves, respectively. The nuerical transfors needed to copute the set of wave aplitudes A n and B n are analogous to those described in Braun, Duvall, & LaBonte (1988). The range of colatitudes spanned by the annulus gives a resolution D 2p/(vax v in) 9. Mode aplitudes and power spectra are only deter- ined for aziuthal orders and degree 1 25 in keeping with the desire to isolate near-zonal odes. The truncation of the observed signal near the lib results in leakage such that only about one-third of the 51 resulting power spectra for each are independent of one another. Fig. 1. Power spectra corresponding to poleward- and equatorward-propagating p-odes for a representative ode observed with SOI-MDI data. For the purpose of this illustration, both spectra have been soothed with a boxcar average over a width of five resolution eleents. The values of the centroid frequencies of each peak are indicated by the vertical lines with the appropriate line type. 3. RESULTS It is iediately apparent with only a visual exaination of the -averaged SOI-MDI power spectra at high that the ridges of the poleward-propagating p-ode power spectra are shifted to higher teporal frequencies relative to the equatorward-traveling spectra (Fig. 1). The large widths of the peaks are the result of a finite resolution in (and, for high degrees, the relatively short p-ode lifeties). In spite of this, frequency shifts on the order of 1 3 Hz are clearly seen between the poleward and equatorward profiles. To easure the shift, we deterine the centroid (power-weighted ean) frequency for both -averaged power spectra. We define Dn as the centroid frequency of the poleward-propagating ode inus the centroid frequency of the equatorward-propagating ode. The background power spectra are estiated fro a linear interpolation of the power averaged in sall bins at frequencies of 1 and 8 Hz and subtracted fro the p-ode ridges. The raw frequency shifts show a roughly linear increase with with considerable scatter for both the SOI-MDI and GONG data. Doppler frequency shifts of high degree p-odes produced by horizontal flows have been previously studied (e.g., Gough & Toore 1983; Hill 1988; Patrón et al. 1995). In spherical coordinates, the presence of a flow U(r, v, f) will produce a relative frequency shift between poleward- and equatorwardpropagating zonal p-odes of degree and radial order n: R, 0 ( AU v S/r) K n (r) dr pole eq Dn n { nn nn R,, (3) p K (r) dr where AU v S is the ean eridional flow over the colatitude range v in to v ax : 0 n v ax 1 v ax in U v in AU S { (v v ) 7 vˆ dv, (4) which is positive in the poleward direction for both heispheres, and the kernels K n are approxiately given by the ode kinetic energy in spherical shells of unit thickness. For near-zonal odes, it can be shown that the frequency shift is

3 No. 1, 1998 HELIOSEISMIC MEASUREMENTS L107 Fig. 2. Measureents of U, defined as the frequency shift ultiplied by pr, /, plotted as a function of W n/ for both the northern and southern heispheres in the SOI-MDI and GONG data sets. The vertical dotted lines indicate the p-ode lower turning point depths M fro left to right in increents of 10 M below the solar surface. slightly less than given by equation (3) by a factor that is essentially unity for the odes considered here. Equation (3) shows that frequency shifts due to flows are proportional to the horizontal wavenuber and that the quantity U { pdnn R,/ gives the weighted depth average of the angular ean flow AU v S/r ultiplied by the solar radius R,. If the angular ean eridional flow AU v S/r is constant over the range of depths spanned by a given p-ode, then it follows that U is a constant equal to the value of the ean eridional flow at the surface AU v S 0. To study the radial variation of the eridional flow, we have plotted in Figure 2 the observed U for both heispheres as deterined fro the SOI-MDI and GONG data versus W { n/, which is a function of the lower turning point depth of the acoustic odes. The vertical dotted lines indicate the values of W corresponding to depths of 10 M and deeper, increasing in steps of 10 M fro left to right, as deterined fro the standard solar odel of Christensen-Dalsgaard, Proffitt, & Thopson (1993). Individual easureents of U were sorted in order of increasing W and averaged in bins containing 10 points each. The error bars indicate the standard deviation of the ean for each average. These errors were found to be consistent with the uncertainties estiated for the centroid frequencies using standard error propagation techniques. Only odes with integrated ridge power greater than 10 ties the background level were used in this analysis, which results in a set of usable odes of and for the SOI-MDI and GONG data, respectively. All data show evidence for poleward-propagating eridional flows corresponding to surface values of about 10 s 1 over the entire range of ode depths studied. The SOI-MDI observations show evidence for an increase in U in depth above 10 M and a peak in U at W 0.01 Hz, especially in the northern heisphere. The SOI-MDI data also show a sall but significant difference in the ean values of U between the northern ( U ) and southern ( U ) heispheres. This difference persists in the average of U for saller subsets over the entire range in W and ay result fro leakage of the solar rotation signal caused by a position angle error of the MDI instruent of about 4, which is not unreasonable (Giles et al. 1997; T. L. Duvall, Jr. 1997, private counication). An error in the rotation rate applied to the coordinate syste, as well as the effects of differential rotation over the latitude range considered, should produce slight variations of the easured centroid frequencies, but not the poleward-equatorward frequency shifts, as a function of the aziuthal order of the odes. Nevertheless, as a test of the application of the ethod to p-odes of different aziuthal orders, we perfored the analysis for three subsets of power spectra sued over the ranges 25 9, 8 8, and 9 25 and found no appreciable systeatic differences aong the three sets fro the results shown in Figure 2. The data for both the northern and southern heispheres is averaged together and shown in Figure 3 (top). The results for the SOI-MDI and GONG data sets are shown as triangles and squares, respectively. To infer the depth variation of the flow, we copare the observations in Figure 3 with values of U coputed fro nuerical integrations of equation (3) assuing specific odels of the angular eridional velocity AU v SR, /r as a function of radius r. The significant scatter in the observed points warrants suitable caution in any odeling effort, and consequently we select a forward odeling approach over inversion techniques, which ay be overly sensitive to data errors. By trial and error we found that the data is well fit by a velocity profile given by the solid line in the lower panel of Figure 3. The calculated values of U for this profile are plotted over the observed heisphere-averaged points in the top panel. The scatter in the data allow soe variation in the iplied eridional velocities. Indeed, the data is not unreasonably fit with a constant value of AU v SR, /r of 10 s 1 (indicated by

4 L108 BRAUN & FAN Vol. 508 a longer tie series of SOI-MDI data between 1996 May 24 and July 24, find saller values of the eridional flow of between 10 and 20 s 1, which ay decrease slightly fro the surface. Both of these ring diagra analyses resolve only p-odes extending to depths of approxiately 0.97 R,. The easureents of Schou & Bogart (1998) give siilar results to those inferred fro tie-distance analysis ade with conteporary data (Giles et al. 1997). Giles, Duvall, & Scherrer (1998) have also recently reported results of inversions of considerably longer duration tie-distance easureents that indicate eridional flows at latitudes 45 decreasing fro 10 s 1 at the surface to zero at 0.94 R, and changing sign below this depth. We have copared our observed U with values calculated fro velocity profiles of siilar behavior and find that our data is not consistent with a reversal of the ean flow at or above 0.94 R,. Clearly, further work is necessary in order to understand these differences, which ay represent either real teporal variations of the flows or systeatic errors between differing techniques. In support of the results presented here, it is difficult, but perhaps not ipossible, to conceive of a situation in which systeatic effects should conspire to produce spurious frequency shifts of low- odes that are observed in our analysis to be quantitatively and qualitatively siilar to values obtained at higher, which are in good agreeent with expectations fro surface easureents of eridional flows. Fig. 3. Top: Values of U for each data set, averaged over both heispheres. The triangles indicate the SOI-MDI observations and the squares show the GONG observations. The solid and dash-dot curves show the predicted values of U coputed fro odel flows, with radial variations indicated in the botto panel by corresponding line types. the dash-dotted lines in both panels of Fig. 3) over the entire range of depth spanned by the observed odes. Our results ay be copared with recent results obtained fro ring diagra and tie-distance analyses. González Hernández et al. (1998) infer eridional flows of about 50 s 1 at idlatitudes, peaking at a depth of about 0.98 R,, fro SOI- MDI data obtained in 1996 June. Schou & Bogart (1998), using We thank To Duvall and Ellen Zweibel for useful discussions. We greatly appreciate the excellent support fro the SOHO SOI-MDI and GONG teas and the fine quality of the helioseisic data that they have given us. SOHO is a project of international cooperation between ESA and NASA. The GONG project is anaged by the National Solar Observatory, a Division of the National Optical Astronoy Observatories, which is operated by AURA, Inc., under a cooperative agreeent with the National Science Foundation. D. C. B., a longter visitor at the High Altitude Observatory, is grateful to Michael Knölker and the staff of HAO for their hospitality and support. The work of D. C. B. is supported by NSF grants AST and AST and NASA grants NAGW and NAG REFERENCES Bogdan, T. J., Brown, T. M., Lites, B. W., & Thoas, J. H. 1993, ApJ, 406, 723 Braun, D. C. 1995, ApJ, 451, , ApJ, 487, 447 Braun, D. C., Duvall, T. L., Jr., & Jefferies, S. M. 1990, in Progress of Seisology of the Sun and Stars, ed. Y. Osaki & H. Shibahashi (Berlin: Springer), 181 Braun, D. C., Duvall, T. L., Jr., & LaBonte, B. J. 1988, ApJ, 335, 1015 Chen, K.-R., Chou, D.-Y., & The TON Tea. 1996, ApJ, 465, 985 Choudhuri, A. R., Schüssler, M., & Dikpati, M. 1995, A&A, 303, L29 Christensen-Dalsgaard, J., Proffitt, C. R., & Thopson, M. J. 1993, ApJ, 403, L75 Dikpati, M., & Choudhuri, A. R. 1995, Sol. Phys., 161, 9 Duvall, T. L., Jr., Harvey, J. W., Braun, D. C., LaBonte, B. J., & Poerantz, M. A. 1988, BAAS, 20, 701 Duvall, T. L., Jr., Jefferies, S. M., Harvey, J. W., & Poerantz, M. A. 1993, Nature, 362, 430 Giles, P. M., Duvall, T. L., Jr., & Scherrer, P. H. 1998, in Structure and Dynaics of the Interior of the Sun and Sun-like Stars, ed. S. G. Korzennik & A. Wilson (ESA SP-418; Noordwijk: ESA), in press Giles, P. M., Duvall, T. L., Jr., Scherrer, P. H., & Bogart, R. S. 1997, Nature, 390, 52 Gilan, P. A., & Miller, J. 1986, ApJ, 61, 585 Glatzaier, G. A., & Gilan, P. A. 1982, ApJ, 256, 316 González Hernández, I., Patrón, J., Bogart, R. S., & The SOI Ring Diagras Tea. 1998, in Structure and Dynaics of the Interior of the Sun and Sunlike Stars, ed. S. G. Korzennik & A. Wilson (ESA SP-418; Noordwijk: ESA), in press Gough, D. O., & Toore, J. 1983, Sol. Phys., 82, 401 Harvey, J. W., et al. 1996, Science, 272, 1284 Hathaway, D. H. 1996, ApJ, 460, 1027 Hathaway, D. H., et al. 1996, Science, 272, 1306 Hill, F. 1988, ApJ, 333, 996 Patrón, J., Hill, F., Rhodes, E. J., Korzennik, S. G., & Cacciani, A. 1995, ApJ, 455, 746 Scherrer, P. H., et al. 1995, Sol. Phys., 162, 129 Schou, J., & Bogart, R. S. 1998, ApJ, 504, L131 Wang, Y.-M., Sheeley, N. R., Jr., & Nash, A. G. 1991, ApJ, 383, 431

5 The Astrophysical Journal, 510:L81, 1999 January The Aerican Astronoical Society. All rights reserved. Printed in U.S.A. ERRATUM In the Letter Helioseisic Measureents of the Subsurface Meridional Flow by D. C. Braun and Y. Fan (ApJ, 508, L105 [1998]), equation (4) was isprinted and should appear as follows: v ax 1 AU S { (v v ) U vˆ v ax in dv. (4) v in L81