The Astrophysical Journal, 853:65 (18pp), 2018 January 20 https://doi.org/10.3847/1538-4357/aaa4f4 © 2018. The American Astronomical Society. All rights reserved. Magnetic Buoyancy and Rotational Instabilities in the Tachocline Peter A. Gilman High Altitude Observatory, National Center for Atmospheric Research, 3080 Center Green, Boulder, CO 80307-3000, USA; [email protected] Received 2017 September 7; revised 2017 November 27; accepted 2018 January 1; published 2018 January 24 Abstract We present results from an analytical model for magnetic buoyancy and rotational instabilities in a full spherical shell tachocline that includes rotation, differential rotation close to that observed helioseismically, and toroidal field. Perturbation solutions are found for the limit of large latitudinal wave number, a limit commonly used to maximize instability due to magnetic buoyancy. We find that at all middle and high latitudes vigorous rotational instability is induced by weak toroidal fields, particularly for high longitudinal wave number, even when the vertical rotation gradient is marginally stable without toroidal field. We infer that this instability will prevent much storage of toroidal fields in the tachocline at these latitudes, but could be responsible for the appearance of ephemeral active regions there. By contrast, the low-latitude vertical rotation gradient, opposite in sign to that at high latitudes, is not only stable itself but also prevents magnetic buoyancy instability until the peak toroidal field is raised above a threshold of about 9 kG at the equator, declining to zero where the vertical rotation gradient changes sign, at 32.3 in our model. Thus this rotation gradient provides a previously unnoticed mechanism for storage of toroidal fields until they amplify by dynamo action to order 10 kG, whereupon they can overcome the rotation gradient to emerge as sunspots. These results provide a new explanation for why sunspots are seen only at low latitudes. The purely rotational instability at latitudes above 50°, even without toroidal fields, also suggests that the high-latitude tachocline should be much thicker, due to HD turbulence, than has been inferred for lower latitudes from helioseismic measurements. Key words: stars: activity – stars: rotation – Sun: interior – Sun: magnetic fields – Sun: rotation – sunspots 1. Introduction dynamo models and simulations (Parker 1975, 1977; Cline et al. 2003). Limits to magnetic buoyancy instability in dynamos Parker (1955) introduced the concept of magnetic buoyancy because of j×B feedbacks on shear flow, representing to explain how magnetic flux generated by the solar dynamo differential rotation, have also been simulated (Vasil & can come to the photosphere and be seen as sunspots. Magnetic Brummell 2008, 2009). Hughes (2007) has recently reviewed buoyancy occurs in a flux tube when the tube is in total (gas ) the status of buoyancy instabilities in the tachocline. plus magnetic pressure equilibrium with its surroundings, One feature common to almost all the work cited above is coupled with the tendency for the temperature inside and that it was done in Cartesian geometry, in order to focus outside the tube to equilibrate due to radiative diffusion. This more on the physical processes involved. Furthermore, all condition is always met if the cross-section of the tube is small the early studies were done before the existence of the solar enough in at least one dimension perpendicular to the local axis tachocline was discovered, so differential rotation in radius of the tube. ’ similar to the tachocline was not considered. Our results for Since Parker s original work, magnetic buoyancy has been instability of combinations of magnetic fields and differential invoked in many contexts. It has been applied both to tubes rotation are done in spherical geometry, and for tachocline fi embedded in a plasma with much weaker magnetic elds differential rotation that is set by helioseismic observations. exterior to it, and to magnetic layers from which tubes might But to get solutions we return to the traditional eigenvalue arise. The study of magnetic buoyancy as a mechanism for instability problem, approximations for which are discussed fi producing instability of a magnetic layer was rst done in Gilman (1970) and more recently in Mizerski et al. (2013). ( ) systematically by Gilman 1970 . The study of magnetic One study that was done including spherical geometry buoyancy instability of magnetic layers has continued ever and differential rotation was that of Acheson (1978), since (Acheson 1978; Acheson & Gibbons 1978; Hughes which did use local analysis in latitude and radius as we 1985a, 1985b;Cattaneoetal.1990;Thelen2000; Tobias & do, but it contains very few numerical results and did not Hughes 2004; Kersale et al. 2007; Davies & Hughes 2011; have the benefit of knowledge of the existence of the Barker et al. 2012;Mizerskietal.2013), as has the study of tachocline. It also placed heavy emphasis on results with magnetic buoyancy instability of isolated toroidal flux tubes finite diffusion. Therefore, unfortunately there are few or rings (Schüssler 1979, 1980; Spruit & van Ballegooijen 1982; detailed points of comparison that can be made with the van Ballegooijen 1982; Moreno Insertis et al. 1992; Ferriz-Mas results presented here. & Schüssler 1993). Magnetic buoyancy has been modeled as a There is also a relevant literature on the destabilization of component of 3D magnetohydrodynamic (MHD) simulations of solar and stellar tachocline rotation gradients, particularly in magneto-convection applied to the solar convection zone the vertical, by weak toroidal magnetic fields (Balbus 1995; (Fan 2001, 2008; Nelson et al. 2014;Weber&Fan2015) and Menou et al. 2004; Parfrey & Menou 2007; Kagan & studied as a form of double-diffusive instability (Schmitt & Wheeler 2014), sometimes referred to as magneto-rotational Rosner 1983; Hughes & Weiss 1995; Skinner & Silvers 2013). instability. It appears that magnetic buoyancy effects are not Magnetic buoyancy effects have been modeled explicitly in solar included in any of these studies, so it is not possible to 1 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman examine the interplay between magnetic buoyancy and the instability, so the difference might not be easily detected rotational instabilities in these results. We can see the in observations, but it can have profound implications for the absence of magnetic buoyancy effects in the eigenvalue solar dynamo and for the location of emergence of magnetic equations for unstable modes in these studies; they contain flux to the surface (Dikpati & Gilman 2005). no terms involving the vertical gradient of toroidal fields, The same rationale can be used to justify local instability which is essential for magnetic buoyancy instability to exist studies of the radial gradient of rotation, as we do below. In this (Gilman 1970). In addition, only Kagan & Wheeler (2014) case, the starting point for instability analysis is the observation allow for non-axisymmetric instability modes, which are of the radial rotation gradient via helioseismic methods. But known to be important for magnetic buoyancy instability this is difficult to do at high latitudes; in fact we do not (Gilman 1970), particularly when rotation is present. currently know very well what the tachcocline thickness is The approach taken in Kagan & Wheeler (2014) was to there. We only have an estimate of the total change in rotation include full diffusive effects in the stability analysis and across the tachocline, not the radial rotation gradient itself. We differentiate domains in latitude and radius according to the will see that whether the high latitude local radial rotation type of instability found there; we will take the opposite gradient inferred from helioseismic measurements is unstable approach, keeping our model as simple as possible and to local perturbations is crucially dependent on this thickness. building up results first from physically meaningful special Therefore, in this case the instability calculation acquires a cases and limits, which form a basis for interpreting our full different role, namely leading to questioning assumptions made results for magnetic buoyancy and rotational instabilities in a about tachocline thickness at high latitudes. In addition, the differentially rotating spherical shell that represents the timescale for growth of rotational instability may not be very tachocline. different from the timescale to re-establish the radial rotation — Before presenting the details of our analysis of magnetic gradient by angular momentum transport from above unlike buoyancy and rotational instability, some more general the case of coronal mass ejections in the solar atmosphere. discussion is appropriate. Instability calculations can have Only turbulence calculations beyond the scope of this paper different purposes and goals, depending on the physics thought may be able to address the nonlinear competition between to be relevant and the observations that are available. For the rotational instability and restoration of the radial rotation Sun, there are examples of instability in action and observed, gradient. such as the sudden change in magnetic field in the photosphere, By contrast, no reliable direct observations of toroidal magnetic fields inside the Sun have ever been possible, but chromosphere, and corona that quickly leads to eruption of the existence of erupting solar activity and Hale’s magnetic flares and coronal mass ejections.