Abbett, W. P. 2007. The magnetic connection between the convection zone and corona in the quiet Sun. Astrophys. J. 665, 1469–1488.Google Scholar

Acheson, D. J. 1979. Instability by magnetic buoyancy. Sol. Phys. 62, 23–50.Google Scholar

Acheson, D. J. 1990. Elementary Fluid Dynamics (Oxford: Clarendon Press).

Alfvén, H. 1942a. Existence of electromagnetic-hydrodynamic waves. Nature 150, 405–406.Google Scholar

Alfvén, H. 1942b. On the existence of electromagnetic-hydromagnetic waves. Ark. Mat. Astr. Fys. 29B, No. 2, 1–7.Google Scholar

Alfvén, H. 1950. Cosmical Electrodynamics (Oxford: Clarendon Press).

Andronov, A. A. 1929. Application of Poincaré's theorem on bifurcation points and change in stability to simple auto-oscillatory systems. C. R. Acad. Sci. 189(15), 559–561.Google Scholar

Andronov, A. A., Vitt, A. A. and Chaikin, S. E. 1966. Theory of Oscillators (trans. Immirzi; Oxford: Pergamon).

Antia, H. M. and Chitre, S. M. 1979. Waves in the sunspot umbra. Sol. Phys. 63, 67–78.Google Scholar

Antia, H. M., Chitre, S. M. and Kale, D. M. 1978. Overstabilization of acoustic modes in a polytropic atmosphere. Sol. Phys. 56, 275–292.Google Scholar

Arnéodo, A., Coullet, P. H. and Spiegel, E. A. 1985a. The dynamics of triple convection. Geophys. Astrophys. Fluid Dyn. 31, 1–48.Google Scholar

Arnéodo, A., Coullet, P. H., Spiegel, E. A. and Tresser, C. 1985b. Asymptotic chaos. Physica D 14, 327–347.Google Scholar

Arnéodo, A., Coullet, P. H. and Tresser, C. 1982. Oscillators with chaotic behavior: an illustration of a theorem by Shil'nikov. J. Stat. Phys. 27, 171–182.Google Scholar

Arnéodo, A. and Thual, O. 1985. Direct numerical simulations of a triple convection problem versus normal form predictions. Phys. Lett. A 109, 367–373.Google Scholar

Arnold, V. I. 1972. Lectures on bifurcations and versal systems. Russ. Math. Surveys 27, 54–123.Google Scholar

Arnold, V. I. 1977. Loss of stability of self oscillations close to resonances and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11(2), 1–10.Google Scholar

Arnold, V. I. 1983. Geometrical Methods in the Theory of Ordinary Differential Equations (New York: Springer).

Arter, W. 1983a. Nonlinear convection in an imposed horizontal magnetic field. Geophys. Astrophys. Fluid Dyn. 25, 259–292.Google Scholar

Arter, W. 1983b. Magnetic-flux transport by a convecting layer – topological geometrical and compressible phenomena. J. Fluid Mech. 132, 25–48.Google Scholar

Arter, W. 1985. Magnetic-flux transport by a convecting layer including dynamical effects. Geophys. Astrophys. Fluid Dyn. 31, 311–344.Google Scholar

Arter, W., Proctor, M. R. E. and Galloway, D. J. 1982. New results on the mechanism of magnetic flux pumping by three-dimensional convection. Mon. Not. Roy. Astron. Soc. 201, 57P–61P.Google Scholar

Bajer, K., Bassom, A. P. and Gilbert, A. D. 2001. Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411.Google Scholar

Balbus, S. A., Bonart, J., Latter, H. and Weiss, N. O. 2009. On differential rotation and convection in the Sun. Mon. Not. Roy. Astron. Soc. 400, 176–182.Google Scholar

Barker, A. J., Silvers, L. J., Proctor, M. R. E. and Weiss, N. O. 2012. Magnetic buoyancy instabilities in the presence of magnetic flux pumping at the base of the solar convection zone. Mon. Not. Roy. Astron. Soc. 424, 115–127.Google Scholar

Bassom, A. P. and Zhang, K. 1994. Strongly nonlinear convection cells in a rapidly rotating fluid layer. Geophys. Astrophys. Fluid Dyn. 76, 223–238.Google Scholar

Batchelor, G. K. 1953. The condition for dynamical similarity of a frictionless perfect gas atmosphere. Q. J. Roy. Met. Soc. 79, 224–235.Google Scholar

Batchelor, G. K. 1956. A proposal concerning laminar wakes behind bluff bodies at large Reynolds number. J. Fluid Mech. 1, 388–398.Google Scholar

Batiste, O., Knobloch, E., Alonso, A. and Mercader, E. 2006. Spatially localized binary-fluid convection. J. Fluid Mech. 560, 149–158.Google Scholar

Beaume, C., Bergeon, A. and Knobloch, E. 2011. Homoclinic snaking of localized states in doubly diffusive convection. Phys. Fluids 23, 094102.Google Scholar

Beer, J., Tobias, S. M. and Weiss, N. O. 1998. An active Sun throughout the Maunder Minimum. Sol. Phys. 181, 237–249.Google Scholar

Bekki, N. and Karakisawa, T. 1995. Bifurcations from periodic solution in a simplified model of two-dimensional magnetoconvection. Phys. Plasmas 2, 2945–2962.Google Scholar

Bekki, N. and Moriguchi, H. 2007. Temporal chaos in Boussinesq magnetoconvection. Phys. Plasmas 14, 012306.Google Scholar

Bénard, H. 1901. Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Ann. Chim. Phys. 23, 62–144.Google Scholar

Bercik, D. J., Fisher, G. H., Johns-Krull, C. M. and Abbett, W. P. 2005. Convective dynamos and the minimum X-ray flux in main-sequence stars. Astrophys. J. 631, 529–539.Google Scholar

Berdyugina, S. V. 2005. Starspots: a key to the stellar dynamo. Living Revs. Solar Phys. 2, 6 (livingreviews.org/lrsp-2005–6).Google Scholar

Berger, T. E., Rouppe van der Voort, L. and Löfdahl, M. G. 2007. Contrast analysis of solar faculae and magnetic bright points. Astrophys. J. 661, 1272–1288.Google Scholar

Berhanu, M., Verhille, G., Boisson, J., et al. 2010. Dynamo regimes and transitions in the VKS experiment. Eur. Phys. J. B 77, 459–468.Google Scholar

Bernoff, A. J. 1986. Transitions from Order in Convection (Ph.D. dissertation: University of Cambridge).

Bharti, L., Jain, R. and Jaaffrey, S. N. A. 2007. Evidence for magnetoconvection in sunspot umbral dots. Astrophys. J. 665, L79–L82.Google Scholar

Biermann, L. 1941. Der gegenwärtige Stand der Theorie konvectiver Sonnenmodelle. Vierteljahresschr. Astron. Ges. 76, 194–200.Google Scholar

Blanchflower, S. M. 1999a. Modelling Photospheric Magnetoconvection (Ph.D. dissertation: University of Cambridge).

Blanchflower, S. M. 1999b. Magnetohydrodynamic convectons. Phys. Lett. A 261, 74–81.Google Scholar

Blanchflower, S. and Weiss, N.O. 2002. Three-dimensional magnetohydrodynamic convectons. Phys. Lett. A 294, 297–303.Google Scholar

Blanchflower, S. M., Rucklidge, A. M. and Weiss, N. O. 1998. Modelling photospheric magnetoconvection. Mon. Not. Roy. Astron. Soc. 301, 593–608.Google Scholar

Bogdanov, R. I. 1975. Versal deformations of a singular point on the plane in the case of zero eigenvalues. Sel. Math. Sov. 1(4), 389–421.Google Scholar

Boldyrev, S. and Cattaneo, F. 2004. Magnetic-field generation in Kolmogorov turbulence. Phys. Rev. Lett. 92, 144501.Google Scholar

Borrero, J. M. and Ichimoto, K. 2011. Magnetic structure of sunspots. Living Revs. Solar Phys. 8, 4 (livingreviews.org/lrsp-2011–4).Google Scholar

Botha, G. J. J., Rucklidge, A. M. and Hurlburt, N. E. 2007. Nonaxisymmetric instabilities of convection around magnetic flux tubes. Astrophys. J. 662, L27–L30.Google Scholar

Botha, G. J. J., Rucklidge, A. M. and Hurlburt, N. E. 2012. Formation of magnetic flux tubes in cylindrical wedge geometry. Geophys. Astrophys. Fluid Dyn. 106, 701–709.Google Scholar

Boussinesq, J. 1903. Théorie Analytique de la Chaleur, vol. 2, 254–257. (Paris: Gauthier-Villars).

Braginsky, S. I. and Roberts, P. H. 1995. Equations governing convection in Earth's core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97.Google Scholar

Brandenburg, A. 2011. Nonlinear small-scale dynamos at low magnetic Prandtl numbers. Astrophys. J. 741, 92.Google Scholar

Brandenburg, A. and Subramanian, K. 2005. Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417, 1–209.Google Scholar

Bretherton, C. S. and Spiegel, E. A. 1983. Intermittency through modulational instability. Phys. Lett. A 96, 152–156.Google Scholar

Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S. and Toomre, J. 2010. Persistent magnetic wreaths in a rapidly rotating sun. Astrophys. J. 711, 424–438.Google Scholar

Brown, B. P., Miesch, M. S., Browning, M. K., Brun, A. S. and Toomre, J. 2011. Magnetic cycles in a convective dynamo simulation of a young solar-type star. Astrophys. J. 731, 69.Google Scholar

Browning, M. K. 2008. Simulations of dynamo action in fully convective stars. Astrophys. J. 676, 1262–1280.Google Scholar

Brownjohn, D. P., Hurlburt, N. E., Proctor, M. R. E. and Weiss, N. O. 1995. Nonlinear compressible magnetoconvection. Part 3. Travelling waves in a horizontal field. J. Fluid Mech. 300, 287–309.Google Scholar

Brummell, N. H., Tobias, S. M. and Cattaneo, F. 2010. Dynamo efficiency in compressible convective dynamos with and without penetration. Geophys. Astrophys. Fluid Dyn. 104, 565–576.Google Scholar

Brummell, N. H., Tobias, S. M., Thomas, J. H. and Weiss, N. O. 2008. Flux pumping and magnetic fields in the outer penumbra of a sunspot. Astrophys. J., 686, 1454–1465.Google Scholar

Bullard, E. C. 1949. Electromagnetic induction in a rotating sphere. Proc. Roy. Soc. Lond. A 199, 413–443.Google Scholar

Burke, J. and Knobloch, E. 2007a. Snakes and ladders: localized states in the Swift Hohenberg equation. Phys. Lett. A 360, 681–688.Google Scholar

Burke, J. and Knobloch, E. 2007b. Homoclinic snaking: structure and stability. Chaos 17, 037102.Google Scholar

Bushby, P. J. 2003. Modelling dynamos in rapidly rotating late-type stars. Mon. Not. Roy. Astron. Soc. 338, 655–664.Google Scholar

Bushby, P. J. 2006. Zonal flows and grand minima in a solar dynamo model. Mon. Not. Roy. Astron. Soc. 371, 772–780.Google Scholar

Bushby, P. J. and Houghton, S. M. 2005. Spatially intermittent fields in photo-spheric magnetoconvection. Mon. Not. Roy. Astron. Soc. 362, 313–320.Google Scholar

Bushby, P. J., Favier, B., Proctor, M. R. E. and Weiss, N. O. 2012. Convectively driven dynamo action in the quiet Sun. Geophys. Astrophys. Fluid Dyn. 106, 508–523.Google Scholar

Bushby, P. J., Houghton, S. M., Proctor, M. R. E. and Weiss, N. O. 2008. Convective intensification of magnetic fields in the quiet Sun. Mon. Not. Roy. Astron. Soc. 387, 698–706.Google Scholar

Bushby, P. J., Proctor, M. R. E. and Weiss, N. O. 2010. Small-scale dynamo action in compressible convection. In Numerical Modeling of Space Plasma Flows, eds. N. V. Pogorelov, E. Audit and G. P., Zauk (San Francisco: Astron. Soc. Pac. Conf. Ser. 429), 181–185.

Bushby, P. J., Proctor, M. R. E. and Weiss, N. O. 2011. The influence of stratification upon small-scale convectively-driven dynamos. In IAU Symp. 271, Astrophysical Dynamics: From Stars to Galaiies, eds. N.H. Brummell, A. S.Brun, M. S. Miesch and Y., Ponty (Cambridge: Cambridge University Press), 197–204.

Busse, F. H. 1970. Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.Google Scholar

Busse, F. H. 1975a. Non-linear interaction of magnetic field and convection. J. Fluid Mech. 71, 193–206.Google Scholar

Busse, F. H. 1975b. A model of the geodynamo. Geophys. J. Roy. Astron. Soc. 42, 437–450.Google Scholar

Busse, F. H. 1978. Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar

Busse, F. H. 1987. A new mechanism for the Evershed effect. In The Role of Fine-Scale Magnetic Fields on the Structure of the Solar Atmosphere, eds. E. H. Schröter, M. Vázquez and A. A., Wyller (Cambridge: Cambridge University Press), 187–195.

Busse, F. H. and Clever, R. M. 1982. Stability of convection rolls in the presence of a vertical magnetic field. Phys. Fluids 25, 931–935.Google Scholar

Busse, F. H. and Clever, R. M. 1983. Stability of convection rolls in the presence of a horizontal magnetic field. J. Mec. Theor. Appl. 2, 495–502.Google Scholar

Busse, F. H. and Frick, H. 1985. Square-pattern convection in fluids with strongly temperature-dependent viscosity. J. Fluid Mech. 150, 451–465.Google Scholar

Busse, F. H. and Heikes, K. E. 1980. Convection in a rotating layer: a simple case of turbulence. Science 208, 173–175.Google Scholar

Busse, F. H. and Simitev, R. 2005. Dynamos driven by convection in rotating spherical shells. Astron. Nachr. 326, 231–240.Google Scholar

Busse, F. H. and Simitev, R. 2007. Convection in rotating spherical fluid shells. In Mathematical Aspects of Natural Dynamos, eds. E., Dormy and A.M., Soward (Boca Raton, FL: Grenoble Sciences/CRC Press), 168–198.

Busse, F. H., Müller, U., Stieglitz, R. and Tilgner, A. 1996. A two-scale homogeneous dynamo: an extended analytical model and an experimental demonstration under development. Magnetohydrodynamics 32, 235–248.Google Scholar

Cameron, R. and Galloway, D. 2005. The structure of small-scale magnetic flux tubes. Mon. Not. Roy. Astron. Soc. 358, 1025–1035.Google Scholar

Cameron, R., Schüssler, M., Vöglar, A. and Zakharov, V. 2007. Radiative magnetohydrodynamic simulations of solar pores. Astron. Astrophys. 474, 261–272.Google Scholar

Cardin, P. and Brito, D. 2007. Survey of experimental results. In Mathematical Aspects of Natural Dynamos, eds. E., Dormy and A. M., Soward (Boca Raton, FL: Grenoble Sciences/CRC Press), 361–407.

Carlsson, M., Stein, R. F., Nordlund, Å and Scharmer, G. B. 2004. Observational manifestations of solar magneto convection: center-to-limb variation. Astrophys. J. 610(2), L137–L140.Google Scholar

Cattaneo, F. 1984a. Compressible Magnetoconvection (Ph.D. dissertation: University of Cambridge).

Cattaneo, F. 1984b. Oscillatory convection in sunspots. In The Hydromagnetics of the Sun, eds. T. D., Guyenne and J. J., Hunt (Paris: ESA SP-220), 47–50.

Cattaneo, F. 1999. On the origin of magnetic fields in the quiet photosphere. Astrophys. J. 515, L39–L42.Google Scholar

Cattaneo, F. and Hughes, D. W. 2006. Dynamo action in a rotating convective layer. J. Fluid Mech. 553, 401–418.Google Scholar

Cattaneo, F., Brummell, N. H., Toomre, J., Malagoli, A. and Hurlburt, N. E. 1991. Turbulent compressible convection. Astrophys. J. 370, 282–294.Google Scholar

Cattaneo, F., Emonet, T. and Weiss, N. O. 2003. On the interaction between convection and magnetic fields. Astrophys. J. 588, 1183–1198.Google Scholar

Cattaneo, F., Lenz, D. and Weiss, N. O. 2001. On the origin of the solar mesogranulation. Astrophys. J. 563, L91–L94.Google Scholar

Champneys, A. R. and Kuznetsov, Y. A. 1994. Numerical detection and continuation of codimension-two homoclinic bifurcations. Int. J. Bif. Chaos 4, 785–822.Google Scholar

Chan, S. K. 1974. Investigation of turbulent convection under a rotational constraint. J. Fluid Mech. 64, 477–506.Google Scholar

Chandrasekhar, S. 1952. On the inhibition of convection by a magnetic field. Phil. Mag. (7th Ser.) 43, 501–532.Google Scholar

Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability (Oxford: Clarendon Press). (Dover Edition, 1981.)

Charbonneau, P. 2010. Dynamo models of the solar cycle. Living Revs. Solar Phys. 2, 2 (livingreviews.org/lrsp-2005–2).Google Scholar

Cheung, M. C. M., Rempel, M., Title, A. M. and Schüssler, M. 2010. Simulation of the formation of a solar active region. Astrophys. J. 730, 233–244.Google Scholar

Childress, S. and Gilbert, A. D. 1995. Stretch, Twist, Fold: the Fast Dynamo (Berlin: Springer).

Childress, S. and Soward, A. M. 1972. Convection driven hydromagnetic dynamo. Phys. Rev. Lett. 29, 837–839.Google Scholar

Choudhuri, A. R. 1998. The Physics of Fluids and Plasmas (Cambridge: Cambridge University Press).

Choudhuri, A. R. 2010. Astrophysics for Physicists (Cambridge: Cambridge University Press).

Choudhuri, A. R., Schiissler, M. and Dikpati, M. 1995. The solar dynamo with meridional circulation. Astron. Astrophys. 303, L29–L32.Google Scholar

Christensen, U. R. 2010. Dynamo scaling laws and applications to the planets. Space Sci. Rev. 152, 565–590.Google Scholar

Christensen, U. R. and Aubert, J. 2006. Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields. Geophys. J. Int. 166, 97–114.Google Scholar

Christensen, U. R., Holzwarth, V. and Reiners, A. 2009. Energy flux determines magnetic field strength of planets and stars. Nature 457, 167–169.Google Scholar

Christensen-Dalsgaard, J. and Thompson, M. J. 2007. Observational results and issues concerning the tachocline. In The Solar Tachocline, eds. D., HughesR., Rosner and N., Weiss (Cambridge: Cambridge University Press), 53–85.

Christopherson, D. G. 1940. Note on the vibration of membranes. Quart. J. Math. 11, 63–65.Google Scholar

Clark, Jr., A. 1964. Production and dissipation of magnetic energy by differential fluid motions. Phys. Fluids 7, 1299–1305.Google Scholar

Clark, Jr., A. 1965. Some exact solutions in magnetohydrodynamics with astro-physical applications. Phys. Fluids 8, 644–649.Google Scholar

Clark, Jr., A. 1966. Some kinematical models for small-scale solar magnetic fields. Phys. Fluids 9, 485–492.Google Scholar

Clark, Jr., A. and Johnson, H. K. 1967. Magnetic-field accumulation in supergranules. Sol. Phys. 2, 433–440.Google Scholar

Clever, R. M. and Busse, F. H. 1989. Nonlinear oscillatory convection in the presence of a vertical magnetic field. J. Fluid Mech. 201, 507–523.Google Scholar

Clune, T. L. and Knobloch, E. 1994. Pattern selection in three-dimensional magnetoconvection. Physica D 74, 151–176.Google Scholar

Corfield, C. N. 1984. The magneto-Boussinesq approximation by scale analysis. Geophys. Astrophys. Fluid Dyn. 29, 19–28.Google Scholar

Coullet, P. H. and Iooss, G. 1990. Instabilities of one-dimensional cellular patterns. Phys. Rev. Lett. 64, 866–869.Google Scholar

Coullet, P. H. and Spiegel, E. A. 1983. Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43, 776–821.Google Scholar

Courvoisier, A., Hughes, D. W. and Proctor, M. R. E. 2010. A self-consistent treatment of the electromotive force in magnetohydrodynamics for large diffusivities. Astron. Nachr. 331, 667–670.Google Scholar

Cowling, T. G. 1953. Solar electrodynamics. In The Sun, ed. G. P., Kuiper (Chicago: University of Chicago Press), 532–591.

Cowling, T. G. 1957. Magnetohydrodynamics (New York: Interscience).

Cowling, T. G. 1976a. Magnetohydrodynamics, 2nd edition (Bristol: Adam Hilger).

Cowling, T. G. 1976b. On the thermal structure of sunspots. Mon. Not. Roy. Astron. Soc. 177, 409–414.Google Scholar

Cowling, T. G. 1985. Astronomer by accident. Ann. Rev. Astron. Astrophys. 23, 1–18.Google Scholar

Cox, S. M. and Matthews, P. C. 2000. Instability of rotating convection. J. Fluid Mech. 403, 153–172.Google Scholar

Cox, S. M. and Matthews, P. C. 2001. New instabilities in two-dimensional rotating convection and magnetoconvection. Physica D 149, 210–229.Google Scholar

Cox, S. M., Matthews, P. C. and Pollicott, S. L. 2004. Swift-Hohenberg model for magnetoconvection. Phys. Rev. E 69, 066314.Google Scholar

Crawford, J. D. and Knobloch, E. 1991. Symmetry and symmetry-breaking bifurcations in fluid dynamics. Ann. Rev. Fluid Mech. 23, 341–387.Google Scholar

Curry, J. H. 1978. A generalized Lorenz system. Commun. Math. Phys. 60, 193–204.Google Scholar

Da Costa, L. N., Knobloch, E. and Weiss, N. O. 1981. Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar

Dangelmayr, G. 1996. Ginzburg-Landau description of waves in extended systems. In Pitman Res. Notes in Math. Ser. 352, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, eds. G., Dangelmayr, B., Fiedler, K., Kirchgiässner and A., Mielker (Harlow: Longmans), 5–78.

Dangelmayr, G. and Knobloch, E. 1986. Interaction between standing and travelling waves and steady states in magnetoconvection. Phys. Lett. A 117, 394–398.Google Scholar

Dangelmayr, G. and Knobloch, E. 1987. The Takens-Bogdanov bifurcation with O(>2) symmetry. Phil. Trans. R. Soc. A 322, 243–279.Google Scholar

Danielson, R. E. 1961. The structure of sunspot penumbras. II. Theoretical. Astrophys. J. 134, 289–311.Google Scholar

Danielson, R. E. 1965. Sunspots: theory. In IAU Symp. 29Stellar and Solar Magnetic Fields, ed. R., Lüst (Amsterdam: North-Holland), 314–329.

Dawes, J. H. P. 2007. Localized convection cells in the presence of a vertical magnetic field. J. Fluid Mech. 570, 385–406.Google Scholar

Dawes, J. H. P. 2008. Localized pattern formation with a large-scale mode. SIAM J. Appl. Dyn. Syst. 7, 186–206.Google Scholar

Dawes, J. H. P. 2010. The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Phil. Trans. Roy. Soc. Lond. A 368, 3519–3534.Google Scholar

Deane, A. E., Toomre, J. and Knobloch, E. 1987. Traveling waves and chaos in thermosolutal convection. Phys. Rev. A 36, 2862–2869.Google Scholar

Dikpati, M. and Gilman, P. A. 2009. Flux-transport solar dynamos. Space Sci. Rev. 144, 67–75.Google Scholar

Doedel, E. and Kernévez, J. 1986. AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations (Pasadena, CA: CalTech Press).

Donati, J.-F. 2011. Large -scale magnetic fields of low-mass dwarfs: the many faces of dynamo. In IAU Symp. 271Astrophysical Dynamics: from Stars to Galaxies, eds. N. H., Brummell, A. S., Brun, M. S., Miesch and Y., Ponty (Cambridge: Cambridge University Press), 23–31.

Dorch, S. B. F. and Nordlund, A. 2001. On the transport of magnetic fields by solar-like stratified convection. Astron. Astrophys. 365, 562–570.Google Scholar

Dormy, E. and Soward, A. M. (eds.) 2007. Mathematical Aspects of Natural Dynamos (Boca Raton, FL: Grenoble Sciences, CRC Press).

Dormy, E., Soward, A. M., Jones, C. A., Jault, D. and Cardin, P. 2004. The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 501, 43–70.

Drazin, P. G. 1992. Nonlinear Systems (Cambridge: Cambridge University Press).

Drobyshevski, E. M. and Yuferev, V. S. 1974. Topological pumping of magnetic flux by three-dimensional convection. J. Fluid Mech. 65, 33.Google Scholar

Dudley, M. L. and James, R. W. 1989. Time-dependent kinematic dynamos with stationary flows. Proc. Roy. Soc. Lond. A 425, 407–429.Google Scholar

Eddington, A. S. 1926. The Internal Constitution of the Stars (Cambridge: Cambridge University Press).

Elgin, J. N. and Molina Garza, J. B. 1989. On the travelling wave solutions of the Maxwell-Bloch equations. In Structure, Coherence and Chaos in Dynamical Systems, eds. P. L., Christiansen and R. D., Parmentier (Manchester: Manchester University Press), 553–561.

Eltayeb, I. A. 1972. Hydromagnetic convection in a rapidly rotating fluid layer. Proc. Roy. Soc. Lond. A 326, 229–254.Google Scholar

Eltayeb, I. A. 1975. Overstable hydromagnetic convection in a rotating fluid layer. J. Fluid Mech. 71, 161–179.Google Scholar

Fan, Y. 2001. Nonlinear growth of the three-dimensional undular instability of a horizontal magnetic layer and the formation of arching flux tubes. Astrophys. J. 546, 509–527.Google Scholar

Fan, Y. 2009. Magnetic fields in the solar convection zone. Living Revs. Solar Phys. 6, 4 (livingreviews.org/lrsp-2009–4).Google Scholar

Fautrelle, Y. and Childress, S. 1982. Convective dynamos with intermediate and strong fields. Geophys. Astrophys. Fluid Dyn. 22, 235–279.Google Scholar

Favier, B. and Bushby, P. J. 2012. Small-scale dynamo action in rotating compressible convection. J. Fluid Mech. 690, 262–287.Google Scholar

Favier, B. and Proctor, M. R. E. 2013. Kinematic dynamo action in square and hexagonal patterns. Phys. Rev. E 88(5).Google Scholar

Favier, B. F. N., Jouve, L., Edmunds, W., Silvers, L. J. and Proctor, M. R. E. 2012. How can twisted magnetic structures naturally emerge from buoyancy instabilities?Mon. Not. Roy. Astron. Soc. 426, 3349–3359.Google Scholar

Fearn, D. R. 1979. Thermal and magnetic instabilities in a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 14, 103–126.Google Scholar

Fearn, D. R. and Ogden, R. R. 2000. Magnetostrophic magnetoconvection. Phys. Earth Planet. Inter. 117, 273–294.Google Scholar

Ferraro, V. C. A. 1937. The non-uniform rotation of the Sun and its magnetic field. Mon. Not. Roy. Astron. Soc. 97, 458–472.Google Scholar

Ferraro, V. C. A. and Plumpton, C. 1961. Magneto-fluid Mechanics (Oxford: Clarendon Press).

Forest, C. B., O'Connell, R., Kendrick, R., Spence, E. and Nornberg, M. D. 2002. Hydrodynamic and numerical modeling of a spherical homogeneous dynamo experiment. Magnetohydrodynamics 38, 107–120.Google Scholar

Foukal, P. V. 2004. Solar Astrophysics, 2nd edition (Weinheim: Wiley-VCH).

Foukal, P. V. and Jokipii, J. R. 1975. On the rotation of gas and magnetic fields at the solar photosphere. Astrophys. J. 199, L71–L73.Google Scholar

Gailitis, A., Lielausis, S., Dementev, S., et al. 2000. Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Phys. Rev. Lett. 84, 4265–4368.Google Scholar

Gailitis, A., Lielausis, S., Dementev, S., et al. 2001. Magnetic field saturation in the Riga dynamo experiment. Phys. Rev. Lett. 86, 3024–3027.Google Scholar

Galloway, D. J. 1975. Fine structure and Evershed motions in the sunspot penumbra. Sol. Phys. 44, 409–415.Google Scholar

Galloway, D. J. and Moore, D. R. 1979. Axisymmetric convection in the presence of a magnetic field. Geophys. Astrophys. Fluid Dyn. 12, 73–105.Google Scholar

Galloway, D. J. and Proctor, M. R. E. 1983. The kinematics of hexagonal magnetoconvection. Geophys. Astrophys. Fluid Dyn. 24, 109–136.Google Scholar

Galloway, D. J. and Proctor, M. R. E. 1992. Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion. Nature 356, 691–693.Google Scholar

Galloway, D. J. and Weiss, N. O. 1981. Convection and magnetic fields in stars. Astrophys. J. 243, 945–953.Google Scholar

Galloway, D. J., Proctor, M. R. E. and Weiss, N. O. 1978. Magnetic flux ropes and convection. J. Fluid Mech. 87, 243–261.Google Scholar

Gaspard, P., Kapral, R. and Nicolis, G. 1984. Bifurcation phenomena near homoclinic systems: a two parameter analysis. J. Stat. Phys. 35, 697–727.Google Scholar

Ghizaru, M., Charbonneau, P. and Smolarkiewicz, P. K. 2010. Magnetic cycles in global large-eddy simulations of solar convection. Astrophys. J. 715, L133–L137.Google Scholar

Gibson, R. D. 1966. Overstability in the magnetohydrodynamic Bénard problem at large Hartmann numbers. Proc. Camb. Phil. Soc. 62, 287–299.Google Scholar

Gilman, P. A. 1983. Dynamically consistent nonlinear dynamos driven by convection in a rotating shell. II. Dynamos with cycles and strong feedbacks. Astrophys. J. Suppl. 53, 243–268.Google Scholar

Gilman, P. A. and Glatzmaier, G. A. 1981. Compressible convection in a rotating spherical shell. I – Anelastic equations. Astrophys. J. Suppl. 45, 335–349.Google Scholar

Gizon, L. and Birch, A. C. 2005. Local helioseismology. Living Revs. Solar Phys. 2, 6 (livingreviews.org/lrsp-2005–6).Google Scholar

Glatzmaier, G. A. 1984. Numerical simulations of stellar convective dynamos. I – The model and method. J. Comp. Phys. 55, 461–484.Google Scholar

Glatzmaier, G. A. 2013. Introduction to Modeling Convection in Planets and Stars (Princeton: Princeton University Press).

Glatzmaier, G. A. and Roberts, P. H. 1995. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377, 203–209.Google Scholar

Glatzmaier, G. A. and Roberts, P. H. 1996. Rotation and magnetism of the Earth's inner core. Science 274, 1887–1891.Google Scholar

Glendinning, P. 1994. Stability, Instability and Chaos (Cambridge: Cambridge University Press).

Glendinning, P. and Sparrow, C. 1984. Local and global behavior near homoclinic orbits. J. Stat. Phys. 35, 645–696.Google Scholar

Goldhirsch, I., Pelz, R. B. and Orszag, S. A. 1989. Numerical simulation of thermal convection in a two-dimensional finite box. J. Fluid Mech. 199, 1–28.Google Scholar

Golubitsky, M. and Stewart, I. N. 1985. Hopf bifurcation in the presence of symmetry. Arch. Rat. Mech. Anal. 87, 107–165.Google Scholar

Golubitsky, M., Stewart, I. and Schaeffer, D. G. 1998. Singularities and Groups in Bifurcation Theory, Volume II (New York: Springer).

Gorkov, L. P. 1958. Stationary convection in a plane liquid layer near the critical heat transfer point. JETP 6, 311.Google Scholar

Gough, D. O. 1969. The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456.Google Scholar

Gough, D. O. and Tayler, R. J. 1966. The influence of a magnetic field on Schwarzschild's criterion for convective instability in an ideally conducting fluid. Mon. Not. Roy. Astron. Soc. 133, 85–98.Google Scholar

Gough, D. O., Moore, D. R., Spiegel, E. A. and Weiss, N. O. 1976. Convective instability in a compressible atmosphere. II. Astrophys. J. 206, 536–542.Google Scholar

Grad, H. and Rubin, H. 1958. Hydrodynamic equilibria and force-free fields. In Proc. 2nd U.N. Conf. on Peaceful Uses of Atomic Energy, vol. 31 (Geneva: United Nations), 190–197.

Graham, E. 1975. Numerical simulation of two-dimensional compressible convection. J. Fluid Mech. 70, 689–703.Google Scholar

Graham, E. 1977. Compressible convection. In Problems of Stellar Convection, eds. E. A., Spiegel and J. P., Zahn (Berlin: Springer), 151–155.

Graham, E. and Moore, D. R. 1978. The onset of compressible convection. Mon. Not. Roy. Astron. Soc. 183, 617–632.Google Scholar

Guckenheimer, J. 1984. Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15, 1–49.Google Scholar

Guckenheimer, J. and Holmes, P. 1986. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 2nd Printing (New York: Springer).

Guckenheimer, J. and Knobloch, E. 1983. Nonlinear convection in a rotating layer – amplitude expansions and normal forms. Geophys. Astrophys. Fluid Dyn. 23, 247–272.Google Scholar

Hagenaar, H. J., Schrijver, C. J. and Title, A. M. 2003. The properties of small magnetic regions on the solar surface and the implications for the solar dynamo(s). Astrophys. J. 584, 1107–1119.Google Scholar

Haken, H. 1975. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53(1), 77–78.Google Scholar

Hale, G. E. 1908. On the probable existence of a magnetic field in sun-spots. Astrophys. J. 28, 315–343.Google Scholar

Halford, A. R. and Proctor, M. R. E. 2002. An oscillatory secondary bifurcation for magnetoconvection and rotating convection at small aspect ratio. J. Fluid Mech. 467, 241–257.Google Scholar

von Hardenberg, J., Parodi, A., Passoni, G., Provenzale, A. and Spiegel, E. A. 2008. Large-scale patterns in Rayleigh-Bénard convection. Phys. Lett. A 372, 2223–2229.Google Scholar

Hart, A. B. 1954. Motions in the Sun at the photospheric level. IV. The equatorial rotation and possible velocity fields in the photosphere. Mon. Not. Roy. Astron. Soc. 114, 19–38.Google Scholar

Hart, A. B. 1956. Motions in the Sun at the photospheric level. VI. Large-scale motions in the equatorial region. Mon. Not. Roy. Astron. Soc. 116, 38–55.Google Scholar

Harvey-Angle, K. L. 1993. Magnetic Bipoles on the Sun (Ph.D. dissertation: University of Utrecht).

Hasan, S. S. 1983. Time-dependent convective collapse of flux tubes. In IAU Symp. 102, Solar and Stellar Magnetic Fields: Origins and Coronal Effects, ed. J. O., Stenflo (Dordrecht: Reidel), 73–77.

Hasan, S. S. 1985. Convective instability in a solar flux tube. II. Nonlinear calculations with horizontal radiative heat transport and finite viscosity. Astrophys. J. 143, 39–45.Google Scholar

Hasan, S. S. 1986. Oscillatory motions in intense flux tubes. Mon. Not. Roy. Astron. Soc. 219, 357–372.Google Scholar

Hathaway, D. H. 1982. Nonlinear simulations of solar rotation effects in supergran ules. Solar Phys. 77, 341–356.Google Scholar

Hathaway, D. H. 2012a. Supergranules as probes of solar convection zone dynamics. Astrophys. J. 749, L13.Google Scholar

Hathaway, D. H. 2012b. Supergranules as probes of the Sun's meridional circulation. Astrophys. J. 760, 84.Google Scholar

Heinemann, T., Nordlund, å., Scharmer, G. B. and Spruit, H. C. 2007. MHD simulations of penumbra fine structure. Astrophys. J. 669, 1390–1394.Google Scholar

Hollerbach, R. and Jones, C. A. 1995. On the magnetically stabilizing role of the Earth's inner core. Phys. Earth. Planet. Inter. 87, 171–181.Google Scholar

Hood, A. W., Archontis, V., Galsgaard, K. and Moreno-Insertis, F. 2009. The emergence of toroidal flux tubes from beneath the solar photosphere. Astron. Astrophys. 503, 999–1011.Google Scholar

Hood, A. W., Archontis, V. and MacTaggart, D. 2012. 3D MHD flux emergence experiments: idealised models and coronal interactions. Sol. Phys. 278, 3–31.Google Scholar

Hopf, E. 1942. Abzweigung einer periodischen Löisung von einer stationiären Lösung eines Differential-systems. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. 94, 1–22.Google Scholar

Houghton, S. M. and Bushby, P. J. 2011. Localized plumes in three-dimensional compressible magnetoconvection. Mon. Not. Roy. Astron. Soc. 412, 555–560.Google Scholar

Howard, L. N. and Krishnamurti, R. 1986. Large-scale flow in turbulent convection: a mathematical model. J. Fluid Mech. 170, 385–410.Google Scholar

Howe, R. 2009. Solar interior rotation and its variation. Living Revs. Solar Phys. 6, 1 (livingreviews.org/lrsp-2009–1).Google Scholar

Hoyle, R. 2006. Pattern Formation (Cambridge: Cambridge University Press).

Hughes, D. W. 1985. Magnetic buoyancy instabilities for a static plane layer. Geophys. Astrophys. Fluid Dyn. 32, 273–316.Google Scholar

Hughes, D. W. 2007. Magnetic buoyancy instabilities in the tachocline. In The Solar Tachocline, eds. D. W., Hughes, R., Rosner and N. O., Weiss (Cambridge: Cambridge University Press), 275–298.

Hughes, D. W. and Cattaneo, F. 2008. The alpha-effect in rotating convection: size matters. J. Fluid Mech. 594, 445–461.Google Scholar

Hughes, D. W. and Proctor, M. R. E. 1988. Magnetic fields in the solar convection zone: magnetoconvection and magnetic buoyancy. Ann. Rev. Fluid Mech. 20, 187–223.Google Scholar

Hughes, D. W. and Weiss, N. O. 1995. Double-diffusive convection with two stabilizing gradients: strange consequences of magnetic buoyancy. J. Fluid Mech. 301, 383–406.Google Scholar

Hughes, D. W., Cattaneo, F. and Kim, E.-J. 1996. Kinetic helicity, magnetic helicity and fast dynamo action. Phys. Lett. A 223, 167–172.Google Scholar

Hughes, D. W., Rosner, R. and Weiss, N. O. (eds.) 2007. The Solar Tachocline (Cambridge: Cambridge University Press).

Huppert, H. E. 1976. Transitions in double-diffusive convection. Nature 263, 20–22.Google Scholar

Huppert, H. E. and Moore, D. R. 1976. Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar

Hurlburt, N. E. and Rucklidge, A. M. 2000. Development of structure in pores and sunspots: flows around axisymmetric magnetic flux tubes. Mon. Not. Roy. Astron. Soc. 314, 793–806.Google Scholar

Hurlburt, N. E. and Toomre, J. 1988. Magnetic fields interacting with nonlinear compressible convection. Astrophys. J. 327, 920–932.Google Scholar

Hurlburt, N. E., Matthews, P. C. and Proctor, M. R. E. 1996. Nonlinear compressible convection in oblique magnetic fields. Astrophys. J. 457, 933.Google Scholar

Hurlburt, N. E., Matthews, P. C. and Rucklidge, A. M. 2000. Solar magnetocon-vection. Sol. Phys. 192, 109–118.Google Scholar

Hurlburt, N. E., Proctor, M. R. E., Weiss, N. O. and Brownjohn, D. P. 1989. Nonlinear compressible magnetoconvection. Part 1. Travelling waves and oscillations. J. Fluid Mech. 207, 587–628.Google Scholar

Hurlburt, N. E., Toomre, J. and Massaguer, J. M. 1984. Two-dimensional compressible convection extending over multiple scale heights. Astrophys. J. 282, 557–573.Google Scholar

Isobe, H., Proctor, M. R. E. and Weiss, N. O. 2008. Convection-driven emergence of granular scale magnetic field and its role in coronal heating and solar wind acceleration. Astrophys. J. 679, L57–L60.Google Scholar

Jeffreys, H. 1926. The stability of a layer of fluid heated below. Phil. Mag. (7th Ser.) 2, 833–844.Google Scholar

Jeffreys, H. 1930. The instability of a compressible fluid heated below. Proc. Camb. Phil. Soc. 26, 170–172.Google Scholar

Jones, C. A. 1976. Acoustic overstability in a polytropic atmosphere. Mon. Not. Roy. Astron. Soc. 176, 145–159.Google Scholar

Jones, C. A. 2011. Planetary magnetic fields and fluid dynamos. Ann. Rev. Fluid Mech. 43, 583–614.Google Scholar

Jones, C. A., Boronski, P., Brun, A. S., et al. 2011. Anelastic convection-driven dynamo benchmarks. Icarus 216, 120–135.Google Scholar

Jones, C. A. and Moore, D. R. 1979. The stability of axisymmetric convection. Geophys. Astrophys. Fluid Dyn. 11, 245–270.Google Scholar

Jones, C. A., Moore, D. R., and Weiss, N. O. 1976. Axisymmetric convection in a cylinder. J. Fluid Mech. 73, 353–388.Google Scholar

Jones, C. A., Moussa, A. I. and Worland, S. J. 2003. Magnetoconvection in a rapidly rotating sphere: the weak-field case. Proc. Roy. Soc. Lond. A 459, 773–797.Google Scholar

Jones, C. A. and Roberts, P. H. 1990. Magnetoconvection in rapidly rotating Boussinesq and compressible fluids. Geophys. Astrophys. Fluid Dyn. 55, 263–308.Google Scholar

Jones, C. A. and Roberts, P. H. 2000a. Convection driven dynamos in a rotating plane layer. J. Fluid Mech. 404, 311–343.Google Scholar

Jones, C. A. and Roberts, P. H. 2000b. The onset of magnetoconvection at large Prandtl number in a rotating layer. II. Small magnetic diffusion. Geophys. Astrophys. Fluid Dyn. 93, 173–226.Google Scholar

Jones, C. A., Soward, A. M. and Moussa, A. I. 2000. The onset of convection in a rapidly rotating sphere. J. Fluid Mech. 405, 157–179.Google Scholar

Jones, C. A., Thompson, M. J. and Tobias, S. M. 2010. The solar dynamo. Space Sci. Rev. 152, 591–616.Google Scholar

Julien, K., Knobloch, E. and Tobias, S. M. 1999. Strongly nonlinear magnetoconvection in three dimensions. Physica D 128, 105–129.Google Scholar

Julien, K., Knobloch, E. and Tobias, S. M. 2000. Nonlinear magnetoconvection in the presence of strong oblique fieldsJ. Fluid Mech. 410, 285.Google Scholar

Julien, K., Knobloch, E. and Tobias, S. 2003. Highly supercritical convection in strong magnetic fields. In Advances in Nonlinear Dynamos, eds. A., Ferriz-Mas and M., Núñez-Jimenez (Bristol: Taylor and Francis), 195–236.

Kato, S. 1966. The effect of compressibility upon oscillatory convection. Pub. Astron. Soc. Japan 18, 201–208.Google Scholar

Kerswell, R. and Childress, S. 1992. Equilibrium of a magnetic flux tube in a compressible flow. Astrophys. J. 385, 746–757.Google Scholar

Kitiashvili, I. N., Kosovichev, A. G., Wray, A. A. and Mansour, N. N. 2010. Mechanism of spontaneous formation of stable magnetic structures on the Sun. Astrophys. J. 719, 307–312.Google Scholar

Kloosterziel, R. C. and Carnevale, G. F. 2003a. Closed-form linear stability conditions for rotating Rayleigh-Bénard convection. J. Fluid Mech. 480, 25–42.Google Scholar

Kloosterziel, R. C. and Carnevale, G. F. 2003b. Closed-form linear stability conditions for magneto-convection. J. Fluid Mech. 490, 333–344.Google Scholar

Knobloch, E. 1986. On convection in a horizontal magnetic field with periodic boundary conditions. Geophys. Astrophys. Fluid Dyn. 36, 161–177.Google Scholar

Knobloch, E. 2008. Spatially localized structures in dissipative systems: open problems. Nonlinearity 21, T45–T60.Google Scholar

Knobloch, E. 2014. Spatial localization in dissipative systems. Ann. Rev. Condensed Matter Phys. 6, in press.Google Scholar

Knobloch, E. and Proctor, M. R. E. 1981. Nonlinear periodic convection in double diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar

Knobloch, E. and Weiss, N. O. 1983. Bifurcations in a model of magnetoconvection. Physica D 9, 379–407.Google Scholar

Knobloch, E., Moore, D. R., Toomre, J. and Weiss, N. O. 1986. Transitions to chaos in two-dimensional double-diffusive convection. J. Fluid Mech. 166, 409–448.Google Scholar

Knobloch, E., Proctor, M. R. E., Rucklidge, A. M. and Weiss, N. O. 1996. Comment on ‘Bifurcations from periodic solution in a simplified model of two-dimensional magnetoconvection’ by N., Bekki and T., Karakisawa. Phys. Plasmas 3, 2475–2476.Google Scholar

Knobloch, E., Proctor, M. R. E. and Weiss, N. O. 1992. Heteroclinic bifurcations in a simple model of double-diffusive convection. J. Fluid Mech. 239, 273–292.Google Scholar

Knobloch, E., Tobias, S. M. and Weiss, N. O. 1998. Modulation and symmetry changes in stellar dynamos. Mon. Not. Roy. Astron. Soc. 297, 1123–1138.Google Scholar

Knobloch, E., Weiss, N. O. and Da Costa, L. N. 1981. Oscillatory and steady convection in a magnetic field. J. Fluid Mech. 113, 153–186.Google Scholar

Komarova, N. L. and Newell, A. C. 2000. Nonlinear dynamics of sand banks and sand waves. J. Fluid Mech. 415, 285–321.Google Scholar

Kovshov, V. I. 1978. Three approximations for free convection. Astron. Zh. 55, 501–515; Sov. Astron. 22, 288–296.Google Scholar

Krause, F. and Radler, K.-H. 1980. Mean-field Magnetohydrodynamics and Dynamo Theory (Berlin: Akademie-Verlag).

Krishnamurti, R. and Howard, L. N. 1986. Large-scale flow generation in turbulent convection. Proc. Nat. Acad. Sci. 78, 1981–1985.Google Scholar

Kuang, W. and Bloxham, J. 1997. An Earth-like numerical dynamo model. Nature 389, 371–374.Google Scholar

Kuppers, G. and Lortz, D. 1969. Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609–620.Google Scholar

Kuznetsov, Y. A. 1998. Elements of Applied Bifurcation Theory, 2nd edition (New York: Springer).

Landsberg, A. S. and Knobloch, E. 1991. Direction-reversing traveling waves. Phys. Lett. A 159, 17–20.Google Scholar

Lantz, S. R. 1992. Dynamical Behavior of Magnetic Fields in a Stratified, Convect ing Fluid Layer (Ph.D. dissertation: Cornell University).

Lantz, S. R. 1995. Magnetoconvection dynamics in a stratified layer. 2: A low-order model of the tilting instability. Astrophys. J. 441, 925–941.Google Scholar

Lantz, S. R. and Fan, Y. 1999. Anelastic magnetohydrodynamic equations for modeling solar and stellar convection zones. Astrophys. J. Suppl. 121, 247–264.Google Scholar

Lantz, S. R. and Sudan, R. N. 1995. Magnetoconvection dynamics in a stratified layer. I. 2D simulations and visualization. Astrophys. J. 441, 903–924.Google Scholar

Lehnert, B. 1954. Magneto-hydrodynamic waves in liquid sodium. Phys. Rev. 94, 815–824.Google Scholar

Leighton, R. B. 1969. A magneto-kinematic model of the solar cycle. Astro phys. J. 156, 1–20.Google Scholar

Leighton, R. B., Noyes, R. W. and Simon, G. W. 1962. Velocity fields in the solar atmosphere. I. Preliminary report. Astrophys. J. 135, 474–499.Google Scholar

Leka, K. D. and Skumanich, A. 1998. The evolution of pores and the development of penumbrae. Astrophys. J. 507(1), 454–469.Google Scholar

Liang, S. F., Vidal, A. and Acrivos, A. 1969. Buoyancy-driven convection in cylindrical geometries. J. Fluid Mech. 36, 239–256.Google Scholar

Lo Jacono, D., Bergeon, A. and Knobloch, E. 2011. Magnetohydrodynamic convectons. J. Fluid Mech. 687, 595–605.Google Scholar

Lo Jacono, D., Bergeon, A. and Knobloch, E. 2012. Spatially localized magneto convection. Fluid Dyn. Res. 44, 031411.Google Scholar

Lorenz, E. N. 1963. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar

Lorenz, E. N. 1980. Noisy periodicity and reverse bifurcations. Ann. N. Y. Acad. Sci. 357, 282–291.Google Scholar

Lundquist, S. 1952. Studies in magneto-hydrodynamics. Ark. Fys. 5, 297–347.Google Scholar

Malkus, W. V. R. 1959. Magnetoconvection in a viscous fluid of infinite electrical conductivity. Astrophys. J. 130, 259–275.Google Scholar

Malkus, W. V. R. 1964. Boussinesq equations and convection energetics. WHOI GFD Notes, 64–46.Google Scholar

Malkus, W. V. R. and Veronis, G. 1958. Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar

Marcus, P. S. 1981. Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241–255.Google Scholar

Massaglia, S., Rossi, P. and Bodo, G. 1989. Overstability of magnetic flux tubes in the Eddington approximation. Astron. Astrophys. 209, 399–405.Google Scholar

Matloch, L., Cameron, R., Shelyag, S., Schmitt, D. and Schüssler, M. 2010. Mesogranular structure in a hydrodynamical simulation. Astron. Astrophys. 519, A52.Google Scholar

Matthews, P. C. 1998. Hexagonal patterns in finite domains. Physica D 116, 81–94.Google Scholar

Matthews, P. C. 1999. Asymptotic solutions for nonlinear magnetoconvection. J. Fluid Mech. 387, 397–409.Google Scholar

Matthews, P. C. and Cox, S. M. 2000. Pattern formation with a conservation law. Nonlinearity 13, 1293–1320.Google Scholar

Matthews, P. C. and Rucklidge, A. M. 1993. Travelling and standing waves in magnetoconvection. Proc. Roy. Soc. Lond. A 441, 649–658.Google Scholar

Matthews, P. C., Hughes, D. W. and Proctor, M. R. E. 1995. Magnetic buoyancy, vorticity, and three-dimensional flux-tube formation. Astrophys. J. 448, 938–941.Google Scholar

Matthews, P. C., Hurlburt, N. E., Proctor, M. R. E. and Brownjohn, D. P. 1992. Compressible magnetoconvection in oblique fields: linearized theory and simple nonlinear models. J. Fluid Mech. 240, 559–569.Google Scholar

Matthews, P. C., Proctor, M. R. E., Rucklidge, A. M. and Weiss, N. O. 1993. Pulsating waves in nonlinear magnetoconvection. Phys. Lett. A 183, 69–73.Google Scholar

Matthews, P. C., Proctor, M. R. E. and Weiss, N. O. 1995. Compressible magnetoconvection in three dimensions: planform selection and weakly nonlinear behaviour. J. Fluid Mech. 305, 281–305.Google Scholar

Matthews, P. C., Rucklidge, A. M., Weiss, N. O. and Proctor, M. R. E. 1996. The three-dimensional development of the shearing instability of convection. Phys. Fluids 8, 1350–1352.Google Scholar

Maneguzzi, M. and Pouquet, A. 1989. Turbulent dynamos driven by convection. J. Fluid Mech. 205, 297–318.Google Scholar

Meneguzzi, M., Sulem, C., Sulem, P. L. and Thual, O. 1987. Three-dimensional numerical simulation of convection in low-Prandtl-number fluids. J. Fluid Mech. 182, 169–191.Google Scholar

Mercader, I., Batiste, O., Alonso, A. and Knobloch, E. 2011. Convectons, anticonvectons and multiconvectons in binary fluid convection. J. Fluid Mech. 667, 586–606.Google Scholar

Mestel, L. 2012. Stellar Magnetism, 2nd edition (Oxford: Clarendon Press).

Meyer, F., Schmidt, H. U., Weiss, N. O. and Wilson, P. R. 1974. The growth and decay of sunspots. Mon. Not. Roy. Astron. Soc. 169, 35–57.Google Scholar

Miesch, M. S. 2005. Large-scale dynamics of the convection zone and tachocline. Living Revs. Solar Phys. 2, 1 (livingreviews.org/lrsp-2005-1).Google Scholar

Miesch, M. S. and Hindman, B. W. 2011. Gyroscopic pumping in the solar near surface shear layer. Astrophys. J. 743(1), 79.Google Scholar

Miesch, M. S. and Toomre, J. 2009. Turbulence, magnetism and shear in stellar interiors. Ann. Rev. Astron. Astrophys. 41, 317–345.Google Scholar

Miesch, M. S., Brown, B. P., Browning, M. R., Brun, A. S. and Toomre, J. 2011. Magnetic cycles and meridional circulation in global models of solar convection. In IAU Symp. 271, Astrophysical Dynamics: from Stars to Galaxies, eds. N. H., Brummell, A. S., Brun, M. S., Miesch and Y., Ponty (Cambridge: Cambridge University Press), 261–269.

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

Mihaljan, J. M. 1962. A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J. 136, 1126–1133.Google Scholar

Moffatt, H. K. 1978. Magnetic Field Generation in Electrically Conducting Fluids (Cambridge: Cambridge University Press).

Moffatt, H. K. and Kamkar, H. 1983. The time-scale associated with flux expulsion. In Stellar and Planetary Magnetism, ed. A. M., Soward (New York: Gordon and Breach), 91–97.

Moffatt, H. K. and Proctor, M. R. E. 1985. Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493–507.Google Scholar

Moll, R., Cameron, R. H. and Schüssler, M. 2011. Vortices in simulations of solar surface convection. Astron. Astrophys. 533, A126.Google Scholar

Moll, R., Cameron, R. H. and Schüssler, M. 2012. Vortices, shocks and heating in the solar photosphere: effect of a magnetic field. Astron. Astrophys. 541, A68.Google Scholar

Monchaux, R., Berhanu, M., Bourgoin, M., et al. 2007. Generation of magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98, 044502.Google Scholar

Moore, D. W. and Spiegel, E. A. 1966. A thermally excited non-linear oscillator. Astrophys. J. 143, 871–887.Google Scholar

Moore, D. R. and Weiss, N. O. 1973. Two-dimensional Rayleigh-Bénard convection. J. Fluid Mech. 58, 289–312.Google Scholar

Moore, D. R. and Weiss, N. O. 1990. Dynamics of double convection. Phil. Trans. Roy. Soc. A 332, 121–134.Google Scholar

Moore, D. R. and Weiss, N. O. 2000. Resonant interactions in thermosolutal convection. Proc. Roy. Soc. Lond. A 456, 39–62.Google Scholar

Moore, D. R., Peckover, R. S. and Weiss, N. O. 1973. Difference methods for time dependent two-dimensional convection. Comp. Phys. Comm. 6, 198–220.Google Scholar

Moore, D. R., Toomre, J., Knobloch, E. and Weiss, N. O. 1983. Period doubling and chaos in partial differential equations for thermosolutal convection. Nature 303, 663–667.Google Scholar

Moore, D. R., Weiss, N. O. and Wilkins, J. M. 1990a. Symmetry-breaking in thermosolutal convection. Phys. Lett. A 147, 209–214.Google Scholar

Moore, D. R., Weiss, N. O. and Wilkins, J. M. 1990b. The reliability of numerical experiments: transitions to chaos in thermosolutal convection. Nonlinearity 3, 997–1014.Google Scholar

Moore, D. R., Weiss, N. O. and Wilkins, J. M. 1991. Asymmetric oscillations in thermosolutal convection. J. Fluid Mech. 233, 561–585.Google Scholar

Morin, J., Donati, J.-F., Petite, P., et al. 2010. Large-scale magnetic topologies of late M dwarfs. Mon. Not. Roy. Astron. Soc. 407, 2269–2286.Google Scholar

Morin, V. and Dormy, E. 2009. The dynamo bifurcation in rotating spherical shells. Int. J. Mod. Phys. B 23, 5467–5482.Google Scholar

Müller, U., Stieglitz, R. and Horanyi, S. 2004. A two-scale hydromagnetic dynamo experiment. J. Fluid Mech. 498, 31–71.Google Scholar

Musman, S. 1967. Alfvén waves in sunspots. Astrophys. J. 149, 201–209.Google Scholar

Nagata, M., Proctor, M. R. E. and Weiss, N. O. 1990. Transitions to asymmetry in magnetoconvection. Geophys. Astrophys. Fluid Dyn. 51, 211–241.Google Scholar

Nakagawa, Y. 1955. An experiment on the inhibition of thermal convection by a magnetic field. Nature 175, 417–419.Google Scholar

Nakagawa, Y. 1957. Experiments on the inhibition of thermal convection by a magnetic field. Proc. Roy. Soc. Lond. A 240, 108–113.Google Scholar

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S. and Toomre, J. 2011. Buoyant magnetic loops in a global dynamo simulation of a young Sun. Astrophys. J. 739, L38.Google Scholar

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S. and Toomre, J. 2012. Magnetic wreaths and cycles in convective dynamos. Astrophys. J. 762, 73.Google Scholar

Newcomb, W. A. 1961. Convective instability induced by gravity in a plasma with a frozen-in magnetic field. Phys. Fluids 4, 391–396.Google Scholar

Nordlund, Å. 1984. Magnetoconvection: the interaction of convection and small scale magnetic fields. In The Hydromagnetics of the Sun, eds. T. D., Guyenne and J. J., Hunt (Paris: ESA SP-220), 37–46.Google Scholar

Nordlund, Å., Stein, R. F. and Asplund, M. 2009. Solar surface convection. Living Revs. Sol. Phys. 6, 2 (livingreviews.org/lrsp-2009–2).Google Scholar

Oberbeck, A. 1879. Über die Wärmeleitung der Flüissigkeiten bei Berüicksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. 243, 271–292.Google Scholar

Ogura, Y. and Phillips, N. A. 1962. Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179.Google Scholar

Ossendrijver, M., Stix, M., Brandenburg, A. and Rüdiger, G. 2002. Magnetoconvection and dynamo coefficients II. Field-direction dependent pumping of magnetic field. Astron. Astrophys. 394, 735–745.Google Scholar

Ott, E. 1993. Chaos in Dynamical Systems (Cambridge: Cambridge University Press).

Parker, E. N. 1955. Hydromagnetic dynamo models. Astrophys. J. 122, 293–314.Google Scholar

Parker, E. N. 1963. Kinematical hydromagnetic theory and its application to the low solar photosphere. Astrophys. J. 138, 552–575.Google Scholar

Parker, E. N. 1974. The nature of the sunspot phenomenon I: solutions of the heat transport equation. Sol. Phys. 36, 249–274.Google Scholar

Parker, E. N. 1978. Hydraulic concentration of magnetic fields in the solar photosphere. VI – Adiabatic cooling and concentration in downdrafts. Astro phys. J. 221, 368–377.Google Scholar

Parker, E. N. 1979. Cosmical Magnetic Fields: their Origin and their Activity (Oxford: Clarendon Press).

Parker, E. N. 1993. A solar dynamo surface-wave at the interface between convection and non-uniform rotation. Astrophys. J. 408, 707–719.Google Scholar

Parker, R. L. 1966. Reconnection of lines of force in rotating spheres and cylinders. Proc. R. Soc. Lond. A 291, 60–72.Google Scholar

Peckover, R. S. and Weiss, N. O. 1978. On the dynamic interaction between magnetic fields and convection. Mon. Not. Roy. Astron. Soc. 182, 189–208.Google Scholar

Pietarila Graham, J., Cameron, R. and Schüssler, M. 2010. Turbulent small-scale dynamo action in solar surface simulations. Astrophys. J. 714, 1606–1616.Google Scholar

Pikelner, S. B. 1961. Osnovy Kosmicheskoy Elektrodinamiki (Moscow: Nauka).

Poincaré, H. 1885. Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation. Acta Math. 7, 259–380.Google Scholar

Pomeau, Y. 1986. Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11.Google Scholar

Ponomarenko, Y. B. 1973. On the theory of the hydromagnetic dynamo. J. Appl. Mech. Tech. Phys. 14, 775–778.Google Scholar

Prandtl, L. 1904. Über Flüissigkeitsbewegung bei sehr kleiner Reibung. Proc. Internat. Math. Cong., Heidelberg (see Gesammelte Abhandlungen, 575–584, 1981).Google Scholar

Priest, E. R. 2014. Magnetohydrodynamics of the Sun (Cambridge: Cambridge University Press).

Proctor, M. R. E. 1983. Amplification of fields by compressible convection. In IAU Symp. 102, Solar and Stellar Magnetic Fields: Origins and Coronal Effects, ed. J. O., Stenflo (Dordrecht: Reidel), 301–305.

Proctor, M. R. E. 1986. Columnar convection in double-diffusive systems. Contemp. Math. 56, 267–276.Google Scholar

Proctor, M. R. E. 1992. Magnetoconvection. In Theory and Observation of Sunspots, eds. J. H., Thomas and N. O., Weiss (Dordrecht: Kluwer), 221–241.

Proctor, M. R. E. 2001. Finite amplitude behaviour of the Matthews-Cox instability. Phys. Lett. A 292, 181–187.Google Scholar

Proctor, M. R. E. 2005. Magnetoconvection. In Fluid Dynamics and Dynamos in Astrophysics and Geophysics, eds. A. M., Soward, C. A., Jones, D. W., Hughes and N. O., Weiss (Boca Raton, FL: CRC Press), 235–276.

Proctor, M. R. E. and Galloway, D. J. 1979. The dynamic effect of flux ropes on Rayleigh-Buenard convection. J. Fluid Mech. 90, 273–287.Google Scholar

Proctor, M. R. E. and Matthews, P. C. 1996. 2: 1 resonance in non-Boussinesq convection. Physica D 97, 229–241.Google Scholar

Proctor, M. R. E. and Weiss, N. O. 1982. Magnetoconvection. Rep. Prog. Phys. 45, 1317–1349.Google Scholar

Proctor, M. R. E. and Weiss, N. O. 1984. Amplification and maintenance of thin magnetic flux tubes by compressible convection. In The Hydromagnetics of the Sun, eds. T. D., Guyenne and J. J., Hunt (Paris: ESA SP-220), 77–80.Google Scholar

Proctor, M. R. E. and Weiss, N. O. 1990. Normal forms and chaos in thermosolutal convection. Nonlinearity 3, 619–637.Google Scholar

Proctor, M. R. E. and Weiss, N. O. 1993. Symmetries of time-dependent magneto convection. Geophys. Astrophys. Fluid Dyn. 70, 137–160.Google Scholar

Proctor, M. R. E., Weiss, N. O., Brownjohn, D. P. and Hurlburt, N. E. 1994. Nonlinear compressible magnetoconvection. Part 2. Streaming instabilities in two dimensions. J. Fluid Mech. 280, 227–253.Google Scholar

Proctor, M. R. E., Weiss, N. O., Thompson, S. D. and Roxburgh, N. T. 2011. Effects of boundary conditions on the onset of convection with tilted magnetic fields. Geophys. Astrophys. Fluid Dyn. 105, 82–89.Google Scholar

Racine, E., Charbonneau, P., Ghizaru, M., Bouchat, A. and Smolarkiewicz, P. K. 2011. On the mode of dynamo action in a global large-eddy simulation of solar convection. Astrophys. J. 735, 46.Google Scholar

Radko, T. 2013. Double-Diffusive Convection (Cambridge: Cambridge University Press).

Rand, D. 1982. Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1–37.Google Scholar

Rayleigh, Lord (J. W., Strutt). 1896. Theory of Sound, vol. II, 232. (London: Macmillan).

Rayleigh, Lord (J. W., Strutt). 1916. On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 329–346.Google Scholar

Rempel, M. 2010. 3D numerical MHD modeling of sunspots with radiative transport. In IAU Symp 273, Physics of Sun and Starspots, eds. D. P., Choudhary and K. G., Strassmeier (Cambridge: Cambridge University Press), 8–14.

Rempel, M. 2011a. Penumbral fine structure and driving mechanisms of large-scale flows in simulated sunspots. Astrophys. J. 729, 5.Google Scholar

Rempel, M. 2011b. Subsurface magnetic field and flow structure of simulated sunspots. Astrophys. J. 740, 15.Google Scholar

Rempel, M. 2012a. Numerical sunspot models: robustness of photospheric velocity and magnetic field structure. Astrophys. J. 750, 62.Google Scholar

Rempel, M. 2012b. Numerical models of sunspot formation and fine structure. Phil. Trans. Roy. Soc. Lond. A 370, 3114–3128.Google Scholar

Rempel, M. and Schlichenmaier, R. 2011. Sunspot modeling: from simplified models to radiative MHD simulations. Living Revs. Solar Phys. 8, 3 (livingreviews.org/lrsp-2011-3).Google Scholar

Rempel, M., Schüssler, M., Cameron, R. H. and Knölker, M. 2009a. Penumbral structure and outflows in simulated sunspots. Science 325, 171–174.Google Scholar

Rempel, M., Schüssler, M. and Knölker, M. 2009b. Radiative magnetohydrody namic simulation of sunspot structure. Astrophys. J. 691, 640–649.Google Scholar

Rhines, P. B. and Young, W. R. 1983. How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.Google Scholar

Rieutord, M. and Rincon, F. 2010. The Sun's supergranulation. Living Revs. Solar Phys. 7, 2 (livingreviews.org/lrsp-2010-2).Google Scholar

Rimmele, T. R. 2008. On the relation between umbral dots, dark-cored filaments, and light bridges. Astrophys. J. 672, 684–695.Google Scholar

Rincon, F., Lignières, F. and Rieutord, M. 2005. Mesoscale flows in large aspect ratio simulations of turbulent compressible convection. Astron. Astrophys. 430, L57–L60.Google Scholar

Roberts, B. and Webb, A. R. 1978. Vertical motions in an intense magnetic flux tube. Sol. Phys. 56, 5.Google Scholar

Roberts, G. O. 1970. Spatially periodic dynamos. Phil. Trans. Roy. Soc. Lond. A 266, 535–558.Google Scholar

Roberts, G. O. 1972. Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. Roy. Soc. Lond. A 271, 411–454.Google Scholar

Roberts, P. H. 1967. An Introduction to Magnetohydrodynamics (London: Long-mans).

Roberts, P. H. 1994. Fundamentals of dynamo theory. In Lectures on Solar and Planetary Dynamos, eds. M. R. E., Proctor and A. D., Gilbert (Cambridge: Cambridge University Press), 1–58.

Roberts, P. H. and Jones, C. A. 2000. The onset of magnetoconvection at large Prandtl number in a rotating layer. I. Finite magnetic diffusion. Geophys. Astrophys. Fluid Dyn. 92, 289–325.Google Scholar

Roberts, P. H. and Zhang, K. 2000. Thermal generation of Alfvén waves in oscillatory magnetoconvection. J. Fluid Mech. 420, 201–223.Google Scholar

Rosenblum, E., Garaud, P., Traxler, A. and Stellmach, S. 2011. Turbulent mixing and layer formation in double-diffusive convection: three-dimensional numerical simulations and theory. Astrophys. J. 731, 66.Google Scholar

Roxburgh, N. T. 2007. Anelastic Convection in an Inclined Magnetic Field (Ph.D. dissertation: University of Cambridge).

Rucklidge, A. M. 1991. Chaos in Models of Double Convection (Ph.D. dissertation: University of Cambridge).

Rucklidge, A. M. 1992. Chaos in models of double convection. J. Fluid Mech. 237, 209–229.Google Scholar

Rucklidge, A. M. 1993. Chaos in a low-order model of magnetoconvection. Physica D 62, 323–337.Google Scholar

Rucklidge, A. M. 1994. Chaos in magnetoconvection. Nonlinearity 7, 1565–1591.Google Scholar

Rucklidge, A. M. and Matthews, P. C. 1993. Shearing instabilities in magnetocon vection. In Theory of Solar and Planetary Dynamos, eds. M. R. E., Proctor, P. C., Matthews and A. M., Rucklidge (Cambridge: Cambridge University Press), 257–264.

Rucklidge, A. M. and Matthews, P. C. 1995. The shearing instability in magne-toconvection. In Double-Diffusive Convection, Geophys. Monogr. Ser. 94, eds. A., Brandt and H. J. S., Fernando (Washington, DC: AGU), 171–184.

Rucklidge, A. M. and Matthews, P. C. 1996. Analysis of the shearing instability in nonlinear convection and magnetoconvection. Nonlinearity 9, 311–351.Google Scholar

Rucklidge, A. M., Weiss, N. O., Brownjohn, D. P., Matthews, P. C. and Proctor, M. R. E. 2000. Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer. J. Fluid Mech. 419, 283–323.Google Scholar

Rucklidge, A. M., Weiss, N. O., Brownjohn, D. P. and Proctor, M. R. E. 1993. Oscillations and secondary bifurcations in nonlinear magnetoconvection. Geophys. Astrophys. Fluid Dyn. 68, 133–150.Google Scholar

Rüdiger, G. and Hollerbach, R. 2004. The Magnetic Universe (Weinheim: Wiley VCH).

Ruelle, D. 1973. Bifurcations in the presence of a symmetry group. Arch. Rat. Mech. Anal. 51, 136–152.Google Scholar

St. Pierre, M. G. 1993. The stability of the magnetostrophic approximation. I: Taylor state solutions. Geophys. Astrophys. Fluid Dyn. 67, 99–115.Google Scholar

Saltzman, B. 1962. Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19, 329–341.Google Scholar

Savage, B. D. 1969. Thermal generation of hydromagnetic waves in sunspots. Astrophys. J. 156, 707–729.Google Scholar

Scharmer, G. B. and Henriques, V. M. J. 2012. SST/CRISP observations of convective flows in a sunspot penumbra. Astron. Astrophys. 540, A19.Google Scholar

Scharmer, G. B., de la Cruz Rodriguez, J., Süitterlin, P. and Henriques, V. M. J. 2013. Opposite polarity field with convective downflow and its relation to magnetic spines in a sunspot penumbra. Astron. Astrophys. 553, A63.Google Scholar

Scharmer, G. B., Gudiksen, B. V., Kiselman, D., Löfdahl, M. G. and Rouppe van der Voort, L. H. M. 2002. Dark cores in sunspot penumbral filaments. Nature 420, 151–153.Google Scholar

Scharmer, G. B., Henriques, V. M. J., Kiselman, D. and de la Cruz Rodriguez, J. 2011. Detection of convective downflows in a sunspot penumbra. Science 333, 316–319.Google Scholar

Schekochihin, A. A., Iskakov, A. B., Cowley, S. C., et al. 2007. Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys. 9, 300.Google Scholar

Schlüter, A., Lortz, D. and Busse, F. 1965. On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar

Schou, J., Antia, H. M., Basu, S., et al. 1998. Helioseismic studies of differential rotation in the solar envelope by the Solar Oscillations Investigation using the Michelson Doppler Imager. Astrophys. J. 505, 390–417.Google Scholar