Andrey M. Stejko^{1}, Valery V. Pipin^{2}, Alexander G. Kosovichev^{1}

1. New Jersey Institute of Technology, 323 Dr Martin Luther King Jr Blvd., Newark, NJ 07012, USA

2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Irkutsk, 664033, Russia

The GALE (Global Acoustic Linearized Euler) code^{[1]}is used to simulate the propagation of acoustic waves through shallow and deep single-cell, as well as strong and weak double-cell regimes of meridional background flows—generated by mean-field non-linear hydrodynamic and dynamo models^{[2,3]}. The stochastic excitation of oscillation sources over background flows generated by these models allows for a systematic examination of realization noise in the helioseismic signatures generated by each regime. This investigation offers a physics-based baseline for the low end of variance in travel-time measurements that characterize single- and double-cell meridional circulation.

The local helioseismology method of deep focusing (see Ref [1]) is applied to four meridional circulation models. The first two are a shallow single-cell profile, with a return flow at approximately 0.80R, as well as a double-cell profile, referred to as M1 and M2, respectively, in Ref [2]. The next two are a single-cell circulation profile, with a deep return flow situated near the tachocline, accompanied by a double-cell circulation profile with a stronger return flow induced by gyroscopic pumping in the mean-field model of Ref[3]—referred as K1 and K2 in this study. These meridional profiles are used as the background velocities (*u _{r}* ,

*u*) in the governing equations of the acoustic GALE code, where they are scaled up by a factor of 36 in order to simulate the signal-to-noise ratio from a decade of solar measurements

_{θ}^{[4]}. The latitudinal velocities (

*u*)) for the models are shown in Fig. 1, with streamlines representing the circulation profile.

_{θ}Figure 1| Latitudinal velocities (

u), corresponding to models K1, K2, M2, and M3. a) Single-cell meridional circulation with a shallow return flow at∼0.80R. b) Double-cell meridional circulation with a weak reversal. c) Single-cell meridional circulation with a deep return flow near the base of the tachocline. d) Double-cell meridional circulation with a strong reversal. Solid and dashed contours represent counterclockwise and clockwise circulation, respectively. Meridional flow speeds are amplified by a factor of 36._{θ}

The helioseismic signatures of these meridional flow profiles can be characterized by plotting N-S travel-time differences (δτ_{NS}) as a function of their travel distance (∆ = 12°−42°)—corresponding to an increasing depth in the convection zone (r = 0.93R−0.72R). Fig. 2 shows five latitudinal averages (30°N−50°N, 10°N−30°N, 10°S−10°N, 10°S−30°S, 30°S−50°S) of travel-time differences for the four regimes of meridional circulation. Measured travel-time differences (solid lines) are compared to theoretical travel-time differences (dashed lines) computed using the ray-path approximation. Randomized functions are used to generate the oscillatory signal in the source function resulting in unique profiles of realization noise—of which four different profiles are shown for models M1, M2, K1, and K2 (Fig. 2). To measure the bounds of the noise, we simulate 100 unique source functions on a model with no background flows; the noise can then be characterized as the standard deviation (σ_{NS})) of the measured signal from zero (shown by the error bars in Fig. 2). The S/N can be improved significantly—especially at larger travel distances—by applying a phase-velocity filter to the data prior to deep focusing. The filter is defined as a Gaussian function in phase-velocity space with a width of σ = 0.05v, where v is the phase speed ω/l(l + 1).

Figure 2| The N-S travel-time differences (δτ

_{NS}) as a function of travel distance (∆) for models M1 (a), M2 (b), K1 (c), and K2 (d). The travel-time measurements are shown under the application of a Gaussian phase-speed filter (σ = 0.05v) for five latitudinal averages spanning 30°N−50°N, 10°N−30°N, 10°S−10°N, 10°S−30°S, 30°S−50°S. Dashed lines are theoretical times computed using the ray-path approximation. Error bars show one standard deviation of the measured realization noise.

Only the travel-time differences of models K1 and K2 show any significant variation. Moving from the deep single-cell profile (K1) to the shallow single-cell and weak-reversal double-cell profiles (M1 & M2), further onto the strong-reversal double-cell profile (K2), we see a trend of increasing curvature showing a more rapid decrease in travel-time differences with travel distance. These trends result in gaps wider than one standard deviation of the noise throughout a large part of the convective interior. The shallow single-cell and double-cell regimes (M1 and M2, respectively), however, fall within one standard deviation of the range of realization noise, even with the significant increase in the S/R through the application of a phase-velocity filter. These results have positive implications for the ability to distinguish between deep and shallow single-cell meridional circulation, as well as profiles of double-cell circulation with strong reversals. Unfortunately, the differences between shallow single- and double-cell profiles are much more subtle; using current helioseismology techniques, within the time-frame of HMI measurements (∼10 years), any definitive statements on whether meridional circulation has one or two cells may be difficult to make.

Figure 3| The N-S travel-time differences (δτ

_{NS})) as a function of travel distance (∆) for MDI/GONG data (Ref [5]) Latitude ranges in both hemispheres (10°N−30°N, 10°S−30°S) are averaged in order to reduce noise and are compared to dashed lines representing latitudinal averages for models K1, K2, M1, and M2 as measured using the ray-path approximation. Error-bars are computed as the standard deviation of the travel-time differences in the 10°N−10°S latitude range from zero.

We can further compare our four models to the publicly available data from the analysis of Ref [5] (Fig. 3), scaling the travel-time differences computed using the ray-path approximation in our models (M1, M2, K1, & K2), to their travel-time measurements using MDI/GONG observations for Solar Cycle 24 (2008-2019). It is apparent that the level of noise is too large to draw significant conclusions distinguishing single- and double-cell regimes of meridional circulation.

### References

[1] Stejko, A. S., Kosovichev, A. G., Mansour, N. N. 2021, *ApJS*, **253**, 9

[2] Pipin, V. V., Kosovichev, A. G. 2018, *ApJ*, **854**, 67

[3] Pipin, V. V., Kosovichev, A. G. 2019, *ApJ*, **887**, 215

[4] Hartlep, T., Zhao, J., Kosovichev, A. G., Mansour, N. N. 2013, *ApJ*, **762**, 132

[5] Gizon, L., Cameron, R. H., Pourabdian, M., et al. 2020, *Sci*, **368**, 1469