Figure 1 (from Gough and McIntyre 1998, 'Inevitability of a magnetic field in the Sun's radiative interior', Nature, 394, 755--757.) Full text downloadable from here as gzipped postscript (68Kbyte), or as uncompressed postscript (326Kbyte). A followup paper tying up the fluid-dynamical arguments, and drawing still stronger conclusions about tachocline and deep-interior rotation, is downloadable from here as acrobat (.pdf) (0.7Mbyte), as gzipped postscript (0.8Mbyte), or as uncompressed postscript (2.6Mbyte). This is chapter 8 in the Douglas Gough Festschrift Stellar Astrophysical Fluid Dynamics (Cambridge, 2003) edited by M. J. Thompson and J, Christensen-Dalsgaard. (It now looks as if the my old speculations about QBO-type phenomena in the Sun's interior are dead in the water.) Here's an animated version of Figure 8.1 of the Festschrift paper, showing the real stratosphere doing its anti-frictional thing.

More recent work has been radically changing the picture, through various twists and turns.... here's the pdf (270K) of a preprint reporting progress up to 2006, now published as Chapter 8 in The solar tachocline, ed. D. W. Hughes, R. Rosner and N. O. Weiss, Cambridge University Press, 2007, pages 183-212. CORRIGENDUM: After equation (8.6) in the published version, please replace "where C is a constant, provided also that..."   by   "where C is a constant. We have also assumed that..." (It's correct in the preprint.)

STILL MORE RECENTLY, Toby Wood and I discovered a new set of exact solutions helping to solve the magnetic confinement problem. They show how the weak downwelling expected in high latitudes can confine the interior field. The dynamics involves a nontrivial interplay between microscopic magnetic diffusion and the Lorentz and Coriolis forces. A remarkable feature of these exact confinement-layer solutions is that the extreme smallness of the flow velocities makes the flows stable to the known hydrodynamic and magnetohydrodynamic instabilities. So although they describe very simple, strictly laminar flows they may well provide us with a realistic high-latitude piece of the confinement jigsaw. A short paper presenting the confinement-layer solutions (pdf, 300K) has now been published, pp. 303-308 of the proceedings of the July 2007 conference on Unsolved Problems in Stellar Physics, edited by R. J. Stancliffe, J. Dewi, G. Houdek, R. G. Martin, and C. A. Tout, ISBN 978-0-7354-0462-5, ISSN 0094-243X, 465pp., ©2007 American Institute of Physics, AIP Conf. Proc. 948 (also arXiv:0709.1377 [astro-ph]).   STOP PRESS (10 April 2008): A coding error has been discovered which, however, leaves the main conclusions unchanged. Profile shapes are qualitatively the same as before, but numerical values need changing. The upshot is mostly to strengthen the conclusions. In particular, there is an increase in the range of permissible downwelling velocities U.  Also, we now know that the dynamics of the tachopause slip layer is not Ekman-like. Corrections and clarifications will be incorporated into a further paper in preparation for the Astrophysical Journal.

The figure above (detail in green layer now superseded), from the original Nature paper, shows a schematic representation of a meridional quadrant of the sun. The arrows represent the tachocline ventilation circulation, which follows surfaces S of constant specific angular momentum in the (green) body of the tachocline (whose thickness has been exaggerated by a factor 5), and is deflected by the magnetic field in the (blue) diffusive boundary layer (whose thickness has been exaggerated by a factor 50). The inclinations of the S-surfaces, which, owing to the exaggeration of the tachocline thickness, are not drawn accurately, follow from the observation that the interior angular velocity Omega_i lies between the angular velocities at the equator and at the poles in the (orange) convection zone. Moreover, the centre of upwelling should be at a latitude of about 30 degrees (where, incidentally, sunspots emerge at the start of a new cycle). We are unable to draw the return flow in the convection zone without knowledge of the Reynolds stresses; details in the midlatitude upwelling region are also uncertain, obeying severely nonlinear dynamics, and may well be unsteady. The red lines represent the magnetic field in the (purple and white) radiative interior, which is assumed to be the dipole relic of a primordial field, arguably the most likely possibility (for simplicity, aligned with the rotation axis); we are unsure of the geometry of the field near the centre of upwelling, where the field lines are either dashed or absent. North-south asymmetry, as seen in the sunspot distributions observed in the Maunder minimum²⁴, may be related to the non-reversing interior dipole field. At the base of the tachocline the interior field vanishes on the rotation axis, where the magnetic boundary-layer theory suggests a singularity in the tachocline depth. The corresponding physical reality, which would again require nonlinear theory to describe it, would be relatively deep penetration of the tachocline circulation into and out of a `polar pit', which might, conceivably, extend deep enough for lithium and beryllium to be destroyed by nuclear reactions. Recent inversions of SOI seismic data suggest such a pit in the angular-velocity variation. [Note added 2005: BUT it now seems that these pits are dynamically impossible. See above, `preprint reporting progress up to 2006'.]

An earlier paper, the one speculating about QBO-type phenomena, is downloadable from here.