List Of Footnotes

1		Solar neutrinos are an exception to this maxim. Neutrinos which are generated in the core of the Sun can propagate unhindered through the solar interior and ultimately be detected on Earth. Such measurements provide important constraints on models of solar structure and evolution and they have some potential for probing magnetic fields near the base of the convection zone (Sturrock and Weber, 2002 ).
2		Gravity waves also exist in the Sun and are of potential importance in helioseismology (Christensen-Dalsgaard, 2002 ). However, they are confined to the deep radiative interior so they are much more difficult to observe and have not yet been unambiguously detected.
3		The Sun is a slow rotator in the sense that the centrifugal force is many orders of magnitude smaller than the gravitational force. Still, large-scale motions in the deep convection zone may be slow enough that the Coriolis force dominates over the inertial force in the rotating frame, which implies small Rossby numbers.
4		The cylinder which is aligned with the rotation axis and tangent to the base of the convection zone.
5		Counter-clockwise in the northern hemisphere and clockwise in the southern hemisphere.
6		Helioseismic measurements do not indicate a polar spin-up - on the contrary, they suggest the pole rotates even slower than expected based on a smooth extrapolation from lower latitudes (Section 3.1). However, inversions become unreliable near the pole so the angular velocity profile at the highest latitudes remains somewhat uncertain.
7		Note that $_O_0. \~/ <vf> = c_O_0. \~/ _O_$ , so the baroclinic contribution to the zonal velocity gradient, plotted in panel c of Figure 16 , is obtained by multiplying Equation (11 ) by $c/_O_0$ .
8		At the top of the shell in case TUR, $12 -1 nt = 3× 10 cm s$ and $13 -1 kt = 3× 10 cm s$ whereas for case P, $12 - 1 nt = 2.5× 10 cm s$ and $13 -1 kt = 1× 10 cm s$ . In both cases, $n$ and $k$ vary with depth in proportion to $--1/2 r$ . The resolution ( $Nh$ , $Nf$ , $NR$ ) is (256,512,98) and (512,1024,98) in Cases TUR and P, respectively.
9		Case P does not exhibit this tendency over the time interval shown in Figure 17 but it is present over other averaging intervals and in its progenitor, case TUR; see Figures 16 and 17 of Miesch et al. (2000 ).
10		Back-reaction of the dynamo-generated field on the flow via the Lorentz force does not significantly change the convective patterns in case M3 but it does tend to suppress the differential rotation; see Section 6.3.
11		More laminar simulations do exhibit global patterns, with positive and negative current helicity in the northern and southern hemispheres, respectively (Gilman, 1983 ; Glatzmaier, 1985a ).
12		We use the term $a$ -effect here in the general sense of a process which converts toroidal field energy to poloidal field energy, without necessarily implying quasi-linearity; see Section 4.5.
13		In mean-field parlance, the lack of scale separation in time implies that the Strouhal number is not necessarily small. The Strouhal number is the correlation time scale of the velocity fluctuations divided by the advective time scale of the mean flow.
14		The turnover frequency of a convective eddy is just its vorticity, $w$ . Since the vorticity spectrum generally increases toward smaller scales (positive slope versus wavenumber), there will come a point where the effective Rossby and Froude numbers, $w/2_O_ 0$ and $w/N$ become greater than unity ( $N$ is the Brunt-Väisälä frequency).
15		Another motivation for many of these models is to improve the accuracy and conservation properties of the nonlinear advection terms. Furthermore, the scaling of the computational workload in finite element and finite volume models with increasing resolution, $N$ , is much better than spectral models where Legendre transforms, which scale as $N 2$ , eventually dominate.
16		Of course, a tachocline can be imposed in a simulation through boundary conditions or body forces.
17		This may be derived on energetic grounds ( see, e.g., Drazin and Reid, 1981 ; Tritton, 1988 ).
18		...as anyone who flies in airplanes regularly can attest to.
19		In some parameter regimes, the largest growth rates occur for $m > 1$ modes but even here nonlinear calculations by Cally et al. (2003 ) indicate that the $m = 1$ tipping instability eventually dominates, at least for field strengths $>$ 50 KG.
20		On spherical surfaces, nonlinear interactions are no longer restricted to wavevector triads but inverse cascades still occur.
21		This is a simplification. The formation and maintenance of banded zonal flows in forced-dissipative flows may involve non-local spectral transfer or wave interactions which cannot be classified as cascade processes. However, the point is the same; nonlinear spectral transfer, be it local or non-local, can occur freely at low longitudinal wavenumbers but is suppressed at low latitudinal wavenumbers. For a thorough discussion see Rhines (1975 , 1994 ); Vallis and Maltrud (1993 ); Huang and Robinson (1998 ) and Vallis (2005 ). Alternatively, the formation of banded zonal flows can be viewed from the perspective of local potential vorticity mixing coupled with wave-induced momentum transport (McIntyre, 2003 ).
22		Acoustic waves are essential diagnostic tools of the solar interior, but they are neither generated near the base of the convection zone nor do they play a significant dynamical role because of the low Mach numbers thought to characterize the rotational shear and the convective motions.
23		Earlier attempts to account for the uniform rotation of the radiative interior by gravity waves (e.g., Kumar and Quataert, 1997 ; Zahn et al., 1997 ; Talon and Zahn, 1998 ) did not properly account for wave-induced angular momentum transport.