Figure 1:
Angular velocity profile in the solar interior inferred from helioseismology (after Thompson et al., 2003). In panel (a), a 2D (latituderadius) rotational inversion is shown based on the subtractive optimally localized averaging (SOLA) technique. In panel (b), the angular velocity is plotted as a function of radius for several selected latitudes, based on both SOLA (symbols, with 1 error bars) and regularized least squares (RLS; dashed lines) inversion techniques. Dashed lines indicate the base of the convection zone. All inversions are based on data from the Michelson Doppler Imager (MDI) instrument aboard the SOHO spacecraft, averaged over 144 days. Inversions become unreliable close to the rotation axis, represented by white areas in panel (a). Note also that global modes are only sensitive to the rotation component which is symmetric about the equator (courtesy M.J. Thompson & J. ChristensenDalsgaard). 

Figure 2:
Spatial and temporal variation of the meridional circulation in the surface layers of the Sun. (a) The colatitudinal velocity in the solar photosphere obtained from Doppler measurements, averaged over longitude and time. Positive values represent southward flow and different curves correspond to adjacent 6month averaging intervals between 1992 and 1995 (from Hathaway, 1996b). (b) as a function of latitude and depth inferred from ringdiagram analysis. Each inversion is averaged over a 3month interval and results are shown for 1997, 1999, and 2001. Grey and white regions represent southward and northward flow, respectively. A contour plot of the velocity amplitude underlies the arrow plots, with contours labeled in . Flow near the surface and in the southern hemisphere is generally poleward but beginning in 1998, equatorward circulation is found in the northern hemisphere at depths below (from Haber et al., 2002). 

Figure 3:
Shown is a synoptic horizontal flow map below the photosphere inferred from ring diagram analysis (Haber et al., 2002; Hindman et al., 2004). Vectors indicate flow speed and direction while the underlying image represents the radial magnetic field strength (red and green denote opposite polarity). Characteristic velocity amplitudes are . These inversions are based on MDI data averaged over 7 days and sampled over square horizontal patches, each spanning 15 in latitude and longitude. The data shown have not been corrected for inclination (pangle) effects which would shift velocities by about (courtesy D. Haber). 

Figure 4:
Shown is a potentialfield extrapolation of the radial magnetic field measured in the photosphere with the MDI instrument aboard the SOHO spacecraft (Schrijver and DeRosa, 2003, ; see also http://www.lmsal.com/forecast). White lines denote closed loops while green and magenta lines denote open fields of positive and negative polarity, respectively (courtesy M. DeRosa). 

Figure 5:
Schematic diagram illustrating the energy flow in an anelastic model. The thermal energy incorporates both the internal energy of the plasma and the gravitational potential energy as described in the text. The buoyancy force and compression can transfer energy among the thermal and kinetic energy reservoirs while the Lorentz force can transfer energy among the kinetic and magnetic energy reservoirs. Viscous and Ohmic heating can also convert kinetic and magnetic energy to thermal energy. 

Figure 6:
(a) Angular velocity profile based on helioseismic inversions. This is a 2D SOLA inversion based on MDI data similar to that shown in panel a of Figure 1. Solid and dotted lines denote prograde and retrograde rotation relative to . (b) The specific angular momentum profile given by the rotation profile in (a). (c) A hypothetical meridional circulation pattern, illustrated in terms of the massflux streamfunction defined in Equation (13). The circulation in the northern hemisphere is counterclockwise. (d) Divergence of the angular momentum flux carried by the hypothetical meridional circulation. Solid and dotted lines denote positive and negative values respectively. If Equation (8) were satisfied, this would be equal to the convergence of the angular momentum transport by the Reynolds stress, . 

Figure 7:
Shown are the thermal variations implied by Equation (11) based on an angular velocity profile, , obtained from helioseismic inversions (Figure 6, panel a), and on other parameters (, , ) obtained from a solar structure model (model S of ChristensenDalsgaard, 1996). Frame (a) illustrates the normalized latitudinal entropy gradient, , consistent with thermal wind balance. Frame (b) illustrates the corresponding temperature perturbation, assuming (cf. Equation (43) in Appendix A.2). 

Figure 8:
Schematic illustration of the solar dynamo. Numbers indicate particular processes as described in the text (courtesy N. Brummell). 

Figure 9:
The radial velocity near the top of the simulation domain is shown for Case M3 (Brun et al., 2004), Case F (Brun et al., 2005), and Case D2 (DeRosa et al., 2002). Bright and dark tones denote upflow and downflow as indicated by the color tables. Orthographic projections are shown with the north pole tilted toward the observer. The equator is indicated with a solid line. Magnified areas shown in the lower panels correspond to square patches which extend from latitudes of . 

Figure 10:
Movie showing the temporal evolution of the radial velocity near the top of the shell () in Case F is shown in an orthographic projection as in Figure 9. The movie covers a time span of 7 days. 

Figure 11:
The temperature (a), radial vorticity (b), and horizontal divergence (c) near the top of the convection zone in Case F. The time instance and projection are as in Figure 9. 

Figure 12:
The radial velocity (a), temperature (b), and enstrophy (c) are shown for Case F in the mid convection zone. The time instance and projection are as in Figure 9. 

Figure 13:
The angular velocity in Case M3 (Brun et al., 2004) is shown averaged over longitude and time, both as a 2D profile (a) and as a function of radius at selected latitudes (b). Compare with Figure 1 (from Brun et al., 2004). 

Figure 14:
The angular momentum fluxes defined in Appendix A.4, Equations (69)(73) are plotted for case M3 as a function of radius, integrated over horizontal surfaces (a), and as a function of latitude, integrated over conical (, ) surfaces (b). All data are averaged over time. Linestyles denote different components as indicated and solid lines denote the sum of all components. Fluxes are in cgs units (), normalized by , where is the outer radius of the shell. 

Figure 15:
(a) Schematic diagram showing the influence of the Coriolis force on horizontal motions which converge into a northsouth aligned downflow lane (vertical black line). Eastward and westward flows (red) are diverted toward the south and north, respectively (blue) (cf. Gilman, 1986). (b) Schematic diagram illustrating the dynamics of downflow plumes (after Miesch et al., 2000). In the upper convection zone, horizontal flows converge into the plume, acquiring cyclonic vorticity due to the influence of the Coriolis force (red). Near the base of the convection zone (black line), plumes are decelerated by negative buoyancy and diverge, acquiring anticyclonic vorticity (blue). Their remaining horizontal momentum is predominantly equatorward (see text). 

Figure 16:
The following results are shown for Case AB, averaged over longitude and time (from Brun and Toomre, 2002). (a) The mean zonal velocity , (b) the zonal velocity gradient parallel to the rotation axis, , (c) the baroclinic contribution to as defined by Equation (11), and (d) the remainder after subtracting profile (c) from profile (b). The color bar on the left refers to frame (a) and the color bar on the right to frames (b)(c). 

Figure 17:
Streamlines are shown for the mean meridional mass flux in Case M3 (a) and Case P (b), as defined by the streamfunction in Equation (13). Red/orange tones and black contours denote clockwise circulation whereas blue tones and green contours denote counterclockwise circulations. The right frames show the corresponding latitudinal velocity (positive southward) near the top (c) and bottom (d) of the convection zone for each simulation (represented by blue and red lines, respectively). All results are averaged over longitude and time (60 days for Case M3 and 72 days for case P). 

Figure 18:
Movie showing streamlines for the longitudinallyaveraged mass flux in Case M3 are shown evolving over the course of 60 days. Contours are indicated as in panel a of Figure 17, which represents a temporal average of this sequence of images. The inset illustrates the mean latitudinal velocity near the top of the domain () as in the temporal average of panel c in Figure 17. 

Figure 19:
(a) Power spectra are shown for the mass flux vorticity (red) and the streamfunction (blue) for case P, averaged over radius and time [see Equations (12) and (13)]. The former curve (red) is equivalent to in Equation (24). Spectra are normalized such that they sum to unity. Exponential fits to each curve are also shown for comparison. Frame (b) exhibits the same curves as in frame (a) but with a linear vertical axis and a logarithmic horizontal axis. Frame (c) shows the relative contributions of the maintenance terms in Equation (24), using the same normalization as for in frames (a) and (b). In frame (d), the Reynold stress contribution, represented by the blue curve in (c), is decomposed into contributions from radial advection, radial tipping, and latitudinal transport as described in the text. The plots in (b)(d) extend only to as contributions beyond this point are negligible. 

Figure 20:
The radial velocity (a), the radial magnetic field, (b), and the toroidal magnetic field, (c), are shown near the top of the computational domain () for Case M3 of Brun et al. (2004). White and yellow tones denote outward flow (a), outward field (b), and eastward field (c) as indicated by the color tables. 

Figure 21:
(a) Potentialfield extrapolation of the radial magnetic field at the outer boundary of Case M3. White lines represent closed loops while green and magenta lines indicate field which is outward and inward, respectively, at , the boundary of the extrapolation domain. (b) Volume rendering of the toroidal field of Case M3 in a narrow latitude band centered at the the equator. The equatorial plane is tilted slightly with respect to the line of sight. Typical field amplitudes are 1000 and 3000 G in frames (a) and (b), respectively (from Brun et al., 2004). 

Figure 22:
A schematic diagram illustrating the radial entropy gradient, , the convective enthalpy flux, , and the radiative heat flux near the base of the convection zone (see Equation (3) and Appendix A.3). Each quantity is plotted on a horizontal axis (increasing toward the right) as a function of radius (vertical axis). The radiative flux is normalized with respect to the total solar flux, . Four regimes are indicated as discussed in the text (after Zahn, 1991). 

Figure 23:
Movie. The vorticity field is shown in a simulation of penetrative convection in a circular annulus (from Rogers and Glatzmaier, 2005b) (courtesy T. Rogers). 

Figure 24:
Schematic illustration of (a) , (b) , and (c) instabilities for a toroidal band of flux on a 2D spherical surface in the presence of a latitudinal differential rotation (from Dikpati et al., 2004). 

Figure 25:
Growth rates for magnetic shear instabilities are plotted as a function of the initial latitude (vertical axes) and field strength (horizontal axes) of a toroidal band. Shaded areas indicate instability (growth rates for one or more modes ). The left and right columns correspond to parameter regimes characteristic of the overshoot region and lower tachocline, respectively. The lower plots represent cases in which a zonal jet contributes to the initial force balance as discussed in the text. Cases represented in the upper plots have no such jet. Contour lines represent and symmetric (S) and antisymmetric (A) modes as indicated. The nondimensional model is normalized such that a growth rate of 0.01 corresponds to an efolding growth time of 1 year. The parameter is the fractional angular velocity contrast between equator and pole and, in our notation, the reduced gravity (from Dikpati et al., 2003). 

Figure 26:
Results are shown from simulations of freelyevolving stablystratified turbulence with imposed shear. The effective turbulent viscosity (black lines) and turbulent thermal diffusivity (red lines) are plotted as a function of time for simulations with vertical shear (solid lines) and horizontal shear (dashed lines). The time is normalized with respect to the shear rate, , and and are normalized with respect to the molecular values. Frames (a) and (b) correspond respectively to moderately stratified () and strongly stratified () cases (from Jacobitz, 2002). 

Figure 27:
Resulting dynamics when an internal gravity wave encounters (a) a critical layer and (b) a trapping plane , indicated by dashed lines (see text). Curved lines represent ray paths while thin and bold arrows indicate the wavevector and the group velocity including advection by the background flow where . Ray paths are everywhere parallel to . In (a) the zonal velocity gradient is vertical, and the perspective shows a longitudedepth (,) plane. Two ray paths are shown. As each wave asymptotically approaches the vertical wavenumber increases and the group velocity becomes parallel to . In (b) the zonal velocity gradient is latitudinal and the perspective shows a horizontal (,) plane. The and vectors are shown at several points along a single ray path. As is approached, again becomes parallel to (from Staquet and Sommeria (2002); see also Staquet and Huerre (2002)). 

Figure 28:
An oscillating zonal flow driven by gravity waves is shown based on the twowave model described by Kim and MacGregor (2001) and MacGregor (2003). The left column illustrates the zonal velocity as a function of height at several instants in the evolution, with time increasing downward as indicated. The right column illustrates the corresponding rate of change of induced by prograde waves (solid lines), retrograde waves (dashed lines), and viscous dissipation (dotted lines). All quantities are normalized with respect to a characteristic velocity and vertical length scales and . As time proceeds, waves propagating with the same sense as accelerate the flow in such a way that velocity extrema shift upward (toward the right) while new extrema appear deeper down. Vertical dotted lines in the left column are included as a reference point to illustrate the phase of the oscillation (courtesy K. MacGregor). 

Figure 29:
Schematic diagram from Gough and McIntyre (1998) (http://www.nature.com), illustrating the proposed tachocline structure. A meridional circulation (black lines) is driven by gyroscopic pumping in the convective envelope (orange) and penetrates into the tachocline (green) along lines of constant specific angular momentum . A poloidal magnetic field (red lines) in the radiative interior (blue) halts the downward spread of this circulation in a thin boundary layer called the tachopause. In upwelling regions, the field structure is uncertain (dotted lines). The width of the tachocline is exaggerated in this perspective. 
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