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Solar differential rotation reproduced with high-resolution simulation


The Sun rotates differentially with a fast equator and slow pole1. Convection in the solar interior is thought to maintain the differential rotation. However, although many numerical simulations have been conducted to reproduce the solar differential rotation2,3,4,5,6,7, previous high-resolution calculations with solar parameters fall into the antisolar (fast-pole) differential rotation regime. Consequently, we still do not know the true reason why the Sun has a fast-rotating equator. While the construction of the fast equator requires a strong rotational influence on the convection, the previous calculations have not been able to achieve the situation without any manipulations. The problem is called the convective conundrum8. The convection and the differential rotation in numerical simulations were different from the observations. Here, we show that a high-resolution calculation succeeds in reproducing the solar-like differential rotation. Our calculations indicate that the strong magnetic field generated by a small-scale dynamo has a significant impact on thermal convection. The successful reproduction of the differential rotation, convection and magnetic field achieved in our calculation is an essential step to understanding the cause of the most basic nature of solar activity, specifically, the 11 yr cycle of sunspot activity.

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Fig. 1: Overall structure of convection and magnetic field.
Fig. 2: Dependence of differential rotation on the resolution.
Fig. 3: Dependence of the convection and magnetic field properties on the resolution.
Fig. 4: Dependence of kinetic and magnetic energy spectra on the resolution.

Data availability

The data generated, analysed and presented in this study are available at

Code availability

We have opted not to make R2D2 code publicly available. Running R2D2 code requires expert assistance and an appropriate computer system. The numerical method is explained in our previous publications in detail5,18.


  1. Schou, J. et al. Helioseismic studies of differential rotation in the solar envelope by the Solar Oscillations Investigation using the Michelson Doppler Imager. Astrophys. J. 505, 390–417 (1998).

    Article  ADS  Google Scholar 

  2. Miesch, M. S. et al. Three-dimensional spherical simulations of solar convection. I. Differential rotation and pattern evolution achieved with laminar and turbulent states. Astrophys. J. 532, 593–615 (2000).

    Article  ADS  Google Scholar 

  3. Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S. & Toomre, J. Rapidly rotating suns and active nests of convection. Astrophys. J. 689, 1354–1372 (2008).

    Article  ADS  Google Scholar 

  4. Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S. & Toomre, J. Magnetic wreaths and cycles in convective dynamos. Astrophys. J. 762, 73 (2013).

    Article  ADS  Google Scholar 

  5. Hotta, H., Rempel, M. & Yokoyama, T. High-resolution calculation of the solar global convection with the reduced speed of sound technique. II. Near surface shear layer with the rotation. Astrophys. J. 798, 51 (2015).

    Article  ADS  Google Scholar 

  6. Hotta, H., Rempel, M. & Yokoyama, T. Large-scale magnetic fields at high Reynolds numbers in magnetohydrodynamic simulations. Science 351, 1427–1430 (2016).

    Article  MathSciNet  ADS  Google Scholar 

  7. Hotta, H. Breaking Taylor–Proudman balance by magnetic fields in stellar convection zones. Astrophys. J. 860, L24 (2018).

    Article  ADS  Google Scholar 

  8. O’Mara, B., Miesch, M. S., Featherstone, N. A. & Augustson, K. C. Velocity amplitudes in global convection simulations: the role of the Prandtl number and near-surface driving. Adv. Space Res. 58, 1475–1489 (2016).

    Article  ADS  Google Scholar 

  9. Kageyama, A. & Sato, T. ‘Yin–Yang grid’: an overset grid in spherical geometry. Geochem. Geophys. Geosystems 5, Q09005 (2004).

    Article  ADS  Google Scholar 

  10. Christensen-Dalsgaard, J. et al. The current state of solar modeling. Science 272, 1286–1292 (1996).

    Article  ADS  Google Scholar 

  11. Miesch, M. S., Brun, A. S. & Toomre, J. Solar differential rotation influenced by latitudinal entropy variations in the tachocline. Astrophys. J. 641, 618–625 (2006).

    Article  ADS  Google Scholar 

  12. Fan, Y. & Fang, F. A simulation of convective dynamo in the solar convective envelope: maintenance of the solar-like differential rotation and emerging flux. Astrophys. J. 789, 35 (2014).

    Article  ADS  Google Scholar 

  13. Hotta, H., Rempel, M. & Yokoyama, T. Efficient small-scale dynamo in the solar convection zone. Astrophys. J. 803, 42 (2015).

    Article  ADS  Google Scholar 

  14. Hanasoge, S. M., Duvall, T. L. & Sreenivasan, K. R. Anomalously weak solar convection. Proc. Natl Acad. Sci. USA 109, 11928–11932 (2012).

    Article  ADS  Google Scholar 

  15. Hotta, H., Iijima, H. & Kusano, K. Weak influence of near-surface layer on solar deep convection zone revealed by comprehensive simulation from base to surface. Sci. Adv. 5, eaau2307 (2019).

    Article  ADS  Google Scholar 

  16. Hotta, H. & Iijima, H. On rising magnetic flux tube and formation of sunspots in a deep domain. Mon. Not. R. Astron. Soc. 494, 2523–2537 (2020).

    Article  ADS  Google Scholar 

  17. Hotta, H., Rempel, M., Yokoyama, T., Iida, Y. & Fan, Y. Numerical calculation of convection with reduced speed of sound technique. Astron. Astrophys. 539, A30 (2012).

    Article  ADS  Google Scholar 

  18. Hotta, H., Rempel, M. & Yokoyama, T. High-resolution calculations of the solar global convection with the reduced speed of sound technique. I. The structure of the convection and the magnetic field without the rotation. Astrophys. J. 786, 24 (2014).

    Article  ADS  Google Scholar 

  19. Rempel, M. Numerical simulations of quiet Sun magnetism: on the contribution from a small-scale dynamo. Astrophys. J. 789, 132 (2014).

    Article  ADS  Google Scholar 

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We thank T. Yokoyama, R. Shimada and T. Hanawa for comments on the manuscript. The results were obtained using the supercomputer Fugaku provided by the RIKEN Center for Computational Science, the supercomputer Flow at Nagoya University and the Cray XC50 provided by the Center for Computational Astrophysics, National Astronomical Observatory of Japan. This work was supported by MEXT/JSPS KAKENHI (grants JP20K14510—principal investigator (PI) H.H., JP21H04492—PI K.K., JP21H01124—PI T. Yokoyama—and JP21H04497—PI H. Miayahara) and MEXT as a Program for Promoting Research on the Supercomputer Fugaku (Toward a unified view of the universe: from large-scale structures to planets, grant 20351188—PI J. Makino).

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Authors and Affiliations



H.H. contributed to the design of the project, developed the numerical code, carried out simulations, performed analysis and wrote the first draft of the paper. K.K. contributed to the design of the project, interpretation of the result and writing of the final draft.

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Correspondence to H. Hotta.

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The authors declare no competing interests.

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Peer review information Nature Astronomy thanks Yuhong Fan, Juri Toomre and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–4 and Discussion.

Supplementary Video

Overall structure of convection and magnetic field. Left, right: the radial velocity and the radial magnetic field at r = 0.9R, respectively, where R is the solar radius.

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Hotta, H., Kusano, K. Solar differential rotation reproduced with high-resolution simulation. Nat Astron 5, 1100–1102 (2021).

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