6.3 Differential rotation

The helioseismic and surface observations of the solar differential rotation reviewed in Section 3 present several compelling challenges to theoretical and numerical modelers:

a monotonic decrease of angular velocity with latitude,
an angular velocity contrast of about $20%$ ( $~ 90 nHz$ ) between the equator and latitudes of $o ± 60$ ,
nearly radial angular velocity contours at mid-latitudes throughout the bulk of the convection zone,
narrow layers of strong vertical shear in the angular velocity near the top and bottom of the convection zone,
periodic and non-periodic temporal variations.

So how are we doing? Results from a recent simulation are shown in Figure 13 . On the positive side, the angular velocity exhibits a realistic latitudinal variation and contrast (Challenges 1 and 2), with little radial variation above mid-latitudes (Challenge 3). On the negative side, the low-latitude angular velocity contours are somewhat more cylindrical than suggested by helioseismology, with more radial shear. Furthermore, at present there is little tendency for simulations such as these to form rotational shear layers near the top and bottom of the convection zone (Challenge 4). Although these simulations do exhibit non-periodic angular velocity fluctuations of about the right amplitude relative to helioseismic inversions (a few percent; see Miesch, 2000 ; Brun and Toomre, 2002 ), there is currently little evidence for systematic behavior such as torsional oscillations (Challenge 5). We will now proceed to discuss the implications of these results in a little more detail.

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

illustrates how the differential rotation in Case M3 is maintained in terms of the angular momentum balance expressed by Equation (5

). The Reynolds stress (RS) moves angular momentum outward and equatorward, maintaining the differential rotation against viscous dissipation (VD). The advection of angular momentum by the meridional circulation (MC) also plays an important role, enhancing the outward transport by the Reynolds stress but opposing their latitudinal transport, moving angular momentum toward the poles. As might be expected, magnetic tension tends to suppress the rotational shear in both radius and latitude, but at least in this simulation, the Maxwell stress (MS) is much more effective at this than the mean poloidal field (MT) (see Section 6.5).

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 ( $r$ , $f$ ) 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 ( $g s- 1$ ), normalized by $1015r2 2$ , where $r 2$ is the outer radius of the shell.

Even in the most turbulent parameter regimes, a persistent feature of global-scale simulations of rotating convection has been the presence of extended downflow lanes at low latitudes aligned in a north-south orientation (see Section 6.2). Such flow structures naturally give rise to prograde equatorial differential rotation as demonstrated in panel a of Figure 15

. The Coriolis force tends to divert eastward (prograde) flows toward the equator and westward (retrograde) flows toward the poles, leading to positive $< v'v'> h f$ correlations which transport angular momentum toward the equator via the Reynolds stress [see Equation (70

)]. This is reflected by the Reynolds stress contribution in panel b of Figure 14

, which is efficient enough to maintain the differential rotation against meridional circulation, magnetic tension, and viscous diffusion. Similar Coriolis-induced correlations also produce radially outward transport by the Reynolds stress, but these are generally less efficient (Figure 14

, panel a).

Figure 15:

(a) Schematic diagram showing the influence of the Coriolis force on horizontal motions which converge into a north-south 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 anti-cyclonic vorticity (blue). Their remaining horizontal momentum is predominantly equatorward (see text).

Of the challenges listed at the beginning of this section, the first has been particularly difficult. Many simulations of rotating convection in spherical shells exhibit a polar vortex; prograde rotation in the polar regions which arises due to the tendency for flows to conserve angular momentum as they approach the rotation axis. Axisymmetric meridional circulations, in particular, tend to efficiently spin up the poles (see Section 4.3) as reflected by their poleward contribution in panel b of Figure 14

. The Reynolds stress must oppose this tendency in order to produce a monotonic decrease in angular velocity with latitude as is apparently the case in the Sun⁶ . This is more easily accomplished if the circulation does not extend all the way to the poles. Indeed, a common feature of those simulations which exhibit slow polar rotation, such as Case M3, is the absence of a single-celled meridional circulation which extends from low to high latitudes (Brun and Toomre, 2002

). This may have important implications for solar dynamo models (see Section 6.4). Thus, a polar vortex can be avoided if the meridional circulation is confined mainly to low and mid-latitudes. This will be discussed further in Section 6.4. Alternatively, if the north-south downflow lanes which are primarily responsible for equatorward angular momentum transport were to extend to higher latitudes, they may help spin down the poles. This occurs if the convection zone is made deeper, moving the tangent cylinder closer to the rotation axis (Gilman, 1979 ; Glatzmaier, 1987 ). Although this may not be very relevant for the Sun (the convection zone base is reasonably well established from helioseismic inversions, see Section 3.6), it may have implications for less massive stars which have deeper convective envelopes.

An additional complication to the problem of polar spin-up occurs when the convection is allowed to penetrate into an underlying stable region, as demonstrated in panel b of Figure 15 . In turbulent parameter regimes, the convection is dominated by downflow plumes and lanes which acquire cyclonic vorticity in the upper convection zone due to the tendency for converging horizontal flows to conserve their angular momentum (see Section 6.2). As these plumes move deeper into the convection zone, they may converge further due to the density stratification and thus spin up even more (although this convergence may be partially suppressed by entrainment, which has a spreading effect). When the plumes reach the overshoot region, they are decelerated by buoyancy and mass is spread out horizontally and redirected into upflows. The Coriolis force acting on these diverging downflows induce anticyclonic vorticity, leading to a sign reversal of the helicity (Miesch et al., 2000 ).

These downflow plumes are not purely radial. Rather, the influence of the Coriolis force tends to orient them toward the rotation axis in a process known as turbulent alignment (Brummell et al., 1996 ). Thus, when buoyancy removes the plumes’ vertical momentum in the overshoot region, they have a residual horizontal momentum which diverts them toward the equator. The combination of anticyclonic vorticity and equatorward circulation gives rise to a convergence of angular momentum flux from the Reynolds stress, $F RS$ , at high latitudes, which tends to spin up the poles. In other words, angular momentum transport in the overshoot region is generally poleward. The meridional circulation component, $F MC$ , enhances this poleward transport. As a result, sufficiently turbulent global-scale simulations of solar convection which include convective penetration tend to exhibit relatively fast polar rotation (Miesch et al., 2000 , 2004 ). Thus, the slow polar rotation in the Sun remains somewhat enigmatic, although some non-penetrative simulations like Case M3 do a reasonably good job. One possibility is that the transition from sub-adiabatic to super-adiabatic stratification in the penetrative convective simulations is not yet sharp enough (see Section 7.1).

The second challenge listed above has been less problematic; many simulations exhibit an angular velocity contrast between the equator and higher latitudes of about the right amplitude relative to the Sun ( $~ 20- 30%$ ). However, the third challenge, that of nearly radial angular velocity contours, has proven every bit as difficult as the first. As discussed in Section 4.3, there are two ways to break the tendency for cylindrical angular velocity contours: the Reynolds stress (i.e., the effective Rossby number is not small), and baroclinic driving (latitudinal entropy gradients), which can establish a thermal wind.

Figure 16 illustrates the relative importance of these two contributions in a simulation which exhibits a solar-like rotation profile (Challenges 1-3 are nearly met). This is Case AB of Brun and Toomre (2002 ), which is a close relative of Case M3 but is non-magnetic. Frames (a) and (b) illustrate the mean zonal velocity and its gradient along the rotation axis. If the differential rotation were in thermal wind balance, then this axial gradient (Figure 16 , panel b) would be equal to the baroclinic term on the left-hand-side of Equation (11 ), which is shown in panel c of Figure 16 ⁷ . The departure from thermal wind balance is demonstrated in panel d of Figure 16 .

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 $< vf >$ , (b) the zonal velocity gradient parallel to the rotation axis, $_O_ . \~/ <v > 0 f$ , (c) the baroclinic contribution to $_O_ . \~/ <v > 0 f$ 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).

The conclusion to be drawn from Figure 16

is that the non-cylindrical component of the angular velocity profile satisfies thermal wind balance in the lower convection zone, but not in the upper convection zone. There the Reynolds stress is responsible for the axial angular velocity gradients. Thus, simulations which come closest to meeting Challenge 3 above do so both by redistributing angular momentum via the Reynolds stress and by establishing latitudinal entropy gradients via anisotropic convective heat transport.

We emphasize that the ASH code was not tuned in any way to achieve the results shown in Figure 13 and elsewhere. The simulations typically begin from uniform rotation or from previous simulations with different parameter values. Boundary conditions are generally stress-free so angular momentum is conserved and uniform-flux or uniform-entropy so a thermal wind is not artificially driven. The subgrid-scale models are purely diffusive. Mean flows and thermal gradients are established solely via momentum and entropy transport by turbulent convection under the influence of rotation. Still, some parameter regimes and boundary conditions do marginally better than others. Low Prandtl numbers ( $~ 0.25$ ) tend to produce the most solar-like angular velocity contrasts (Challenge 2) and tend to avoid large-scale meridional circulations which can spin up the poles (Challenge 1). Fixing the heat flux at the boundaries rather than the entropy is more conducive to establishing latitudinal entropy gradients which can help to break the tendency for cylindrical rotational profiles, as discussed above (Challenge 3). Although recent results show a substantial improvement over the early, relatively low-resolution simulations by Gilman and Glatzmier (see Section 6.1), higher Reynolds and Rayleigh numbers do not necessarily yield more solar-like profiles. This may be because the north-south downflow lanes in more turbulent simulations are more confined to lower latitudes than in laminar simulations. Since these structures are primarily responsible for equatorward angular momentum transport as discussed above, the net result is often relatively fast polar rotation and a reduced angular velocity contrast. For further elaboration see Miesch (2000 ), Elliott et al. (2000 ) and Brun and Toomre (2002 ). See also Section 7 where possible resolutions to these issues are discussed.

Global convection simulations are only beginning to address the complicated issues surrounding challenge number 4 above, regarding rotational shear layers. Still, some progress has been made in understanding the speedup of angular velocity below the photosphere, inferred from helioseismic inversions and previously from tracer measurements (see Section 3.1). A plausible origin for this layer is in the tendency for the more vigorous convection in the solar surface layers to conserve angular momentum, spinning up as it approaches the rotation axis. This was first suggested by Foukal and Jokipii (1975 ) and has generally been borne out in convection simulations by Gilman and Foukal (1979 ) and more recently by DeRosa et al. (2002 ). However, these simulations were confined to the upper convection zone; deep shell simulations thus far show little tendency to form near-surface shear layers. Global convection simulations also have yet to form strong shear layers near the base of the convection zone which are comparable in structure to the solar tachocline. This may be because the viscous diffusion is too large and the spatial resolution is insufficient to capture small-scale dynamics occurring in the overshoot region (Section 7). Furthermore, the simulations may have insufficient temporal duration to capture the possibly long-term dynamics which drive the radiative interior toward uniform rotation (see Section 8.5).

Global simulations do not yet exhibit periodic temporal variations such as the solar torsional oscillations discussed in Section 3.3. However, similar torsional oscillations do arise naturally in mean-field dynamo models when the back reaction of the Lorentz force on the differential rotation is taken into account (Yoshimura, 1981 ; Schüssler, 1981 ; Kitchatinov et al., 1999 ; Durney, 2000b ; Covas et al., 2001 , 2004 ; Bushby and Mason, 2004 ). An alternative possibility was recently proposed by Spruit (2003 ) who argues that torsional oscillations may be a surface phenomenon which arise as a geostrophic flow response to thermally-induced latitudinal pressure gradients associated with belts of magnetic activity. Shorter-period tachocline oscillations may arise from the spatiotemporal fragmentation of torsional oscillations (Covas et al., 2001 , 2004 ) or from the interaction of gravity waves with differential rotation (Section 8.4). Oscillatory shear instabilies may also play a role (Section 8.2).

As a final comment to close this section, we note that the five challenges posed here are in all likelihood intimately connected. Since the radiative interior possesses much more mechanical and thermal inertia than the convective envelope, the differential rotation in the convection zone may be sensitive to the complex dynamics occuring in the tachocline. In other words, we may not fully understand the rotation profile in the convection zone until we get the tachocline right. A realistic tachocline is probably also a prerequisite to achieving the solar-like dynamo cycles and wave-mean flow interactions which appear to be responsible for torsional and tachocline oscillations. These issues will be discussed further in Section 7.3.