4.1 Turbulent convection and dynamo action

Meneguzzi et al. (1981) and Cattaneo (1999) were the first to demonstrate, via magneto-convection simulations, that dynamo action will occur in turbulent convection even in the absence of rotation. These calculations were for closed, Boussinesq systems. Questions were raised whether local dynamo action is possible in the highly stratified solar convection zone (Stein et al., 2003Jump To The Next Citation Point) because in a stratified atmosphere with much stronger downflows than upflows, magnetic flux is pumped down (Tobias et al., 2001Jump To The Next Citation Point; Dorch and Nordlund, 2001Jump To The Next Citation Point). Abbett (2007Jump To The Next Citation Point), Vögler and Schüssler (2007Jump To The Next Citation Point), and Pietarila Graham et al. (2010Jump To The Next Citation Point) showed that a local surface dynamo is indeed possible. Vögler and Schüssler (2007Jump To The Next Citation Point) and Pietarila Graham et al. (2010Jump To The Next Citation Point) used a shallow, very high resolution, magneto-convection simulation with no Poynting flux in or out of the domain, but with a high magnetic diffusivity in the bottom boundary layer to mimic the loss of magnetic flux to the deeper convection zone. Pietarila Graham et al. (2010Jump To The Next Citation Point) demonstrated that during the kinematic (linear) growth phase, the primary dynamo process (95%) was stretching of magnetic field lines against the magnetic tension component of the Lorentz force by convective motions at subgranule scales (0.1 – 1 Mm) in the turbulent downdrafts, which generates still smaller scale (20 – 200 km) magnetic field. The other 5% was from work against magnetic pressure by fluid motions at granule scales (Figures 6View Image and 7View Image). In addition, magnetic pressure also produces a cascade of magnetic energy from the dynamo generated scales to still smaller scales. In the saturated phase, generation by stretching is almost balanced by the compressive cascade with a little energy also going into MHD wave generation.
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Figure 6: Dynamo energy transfers in the kinematic phase. The dominant process is turbulent stretching of magnetic field lines against the magnetic tension component of the Lorentz force. Image reproduced by permission from Pietarila Graham et al. (2010Jump To The Next Citation Point), copyright by AAS.
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Figure 7: Dynamo net energy transfer rates as a function of horizontal spatial scale in the kinematic dynamo phase. Left: work against magnetic tension (pink dashed line) and work against magnetic pressure (green dotted line) as a function of the fluid motion spatial frequency. Right: dynamo stretching (blue dot-dash line), dynamo compression (black solid line), and magnetic energy removed by compression (red dotted line) as a function of the magnetic field spatial frequency. Image reproduced by permission from Pietarila Graham et al. (2010), copyright by AAS.

Abbett (2007Jump To The Next Citation Point) and Schüssler and Vögler (2008Jump To The Next Citation Point) showed that such small scale dynamo action produces many low-lying loops with large amounts of horizontal field overlying the granules (Figures 8View Image and 9View Image). Steiner (2010Jump To The Next Citation Point) argues that the preponderance of horizontal over vertical field is an inherent consequence of the fact that granules are wider than a scale height. Consider an area of length L and height h. The horizontal (ΦH) and vertical (ΦV) fluxes for a loop are the same, so that 2 ΦH = ⟨BH ⟩Lh = ΦV = ⟨BV ⟩L, where BH is the horizontal and BV is the vertical field and L and h are the horizontal and vertical extents of the field. Hence ⟨BH ⟩∕⟨BV ⟩ ≈ L ∕h and low lying loops connecting opposite sides of granules must have larger average horizontal than vertical field.

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Figure 8: Photospheric magnetic field lines showing many low-lying, horizontally directed magnetic structure from a simulation from the upper convection zone to the corona. Image reproduced by permission from Abbett (2007Jump To The Next Citation Point), copyright by AAS.
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Figure 9: Image of Log horizontal B in vertical slice through saturated phase of dynamo simulation of Vögler and Schüssler (2007). Mean optical depth unity is at approximately 0.9 Mm. Note many loops of different sizes closing in the photosphere. Image reproduced by permission from Schüssler and Vögler (2008), copyright by ESO.

Global dynamos have been simulated by Brun et al. (2004), Miesch (2005), Dobler et al. (2006), Browning et al. (2006), and Brown et al. (2007, 2010). See reviews by Brandenburg and Dobler (2002), Ossendrijver (2003), Miesch and Toomre (2009), and Charbonneau (2010). Note that the fact that both the rate of magnetic flux emergence and the probability distribution of magnetic flux magnitudes are featureless power laws from 1016 – 1023 Mx suggests that the solar dynamo has no preferred scale, that it acts throughout the convection zone with each scale of convective motions generating new flux on that scale (Parnell et al., 2009; Thornton and Parnell, 2011). That is, all the surface magnetic features are produced by a common process (which can not be all dominated by surface effects since the sunspots and active regions clearly are not).


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