The rotational influence on solar convection

Several decades of observations have revealed much about solar surface convection [28, 29]. Driven by fast radiative cooling, granulation dominates the radial motion at the solar surface. Roughly 1 Mm in horizontal size, granulation produces a strong power-spectrum peak at spherical-harmonic degree $\approx\!\text{1000}$ in radial Dopplergrams [30, 31]. Significant power also exists at the $\approx\!\text{30\,Mm}$ supergranular scale, whose associated motions are mainly horizontal and best observed in limb Dopplergram spectra [32].

Photospheric power decreases monotonically for scales larger than supergranulation. From non-rotating intuition [33], we might expect power to peak at $\approx\!\text{100-200\,Mm}$, rather than the $\approx\!\text{30\,Mm}$ supergranular scale. The results of [34, 35] indicate that deep-rooted fluid motions do persist on larger scales. Against expectations, however, these motions appear weak compared to the smaller-scale supergranular and granular flows. A great deal of theory and simulation work has attempted to solve the supergranulation problem. To date, no model self-consistently demonstrates how the supergranular scale might arise. We direct the reader to the recent review by Rincon et al. 2017 [36] for a thorough discussion of this topic. Our focus here is on the apparent lack of large-scale power as expected from non-rotating convection. We suggest, as have others [37, 5], that the supergranular scale results from suppression of power on large scales, rather than through preferential driving at that spatial scale.

Local helioseismic techniques can probe subsurface convection directly (e.g., time-distance [38], ring-diagram analysis [39], holography [40]). Historically, these methods have largely been limited to $\approx\!\text{30\,Mm}$ depth and do not sample flow below the near-surface shear layer. As a result, numerical simulations play a substantial role in describing the dynamical balances in the deep convection zone.

Initially, nonlinear simulations of the full, rotating solar convection zone seemed to reproduce the Sun’s differential rotation profile. Those results suggested (as expected from non-rotating intuition) that convective power peaks at $\approx\!\text{100\,Mm}$ scales with $\approx\!\text{100\,m/s}$ flow-speed amplitudes [41, 42]. Limitations of those results began to appear, however, with systematic magnetohydrodynamic studies in the ensuing decade. These studies found that only systems with $\approx\!\text{10}\times$ weaker flows or equivalently, those which rotated $\approx\!\text{10}\times$ faster, were able to produce coherent magnetic fields and periodic magnetic cycles in analogue to the Sun [43, 44, 45, 46, 47, 48]. Moreover, simulations with more extreme parameters and large-scale power can generate anti-solar differential rotation, with slow equator and fast poles, [11, 49, 50, 51, 14, 52]. We also note that some recent observations suggest the Sun may lie near a boundary between these two basins in parameter space [53].

While most local helioseismic analyses focus on the near-surface shear layers, techniques have been developed to probe more deeply-rooted flow structures, such as solar meridional circulation [54, 55]. A notable puzzle arose following the deep-focusing time-distance analysis of Hanasoge et al. 2012 [1]. This work placed a roughly $1\,\text{m/s}$ upper limit on the $\approx\!\text{100\,Mm}$ scale flow amplitudes at a depth $\approx\!\text{60\,Mm}$. Subsequently, Greer et al. 2016 [2] sampled deeply enough to compare directly against the time-distance results at a depth of 30 Mm. Rather than a nondetection, this effort yielded measured flows that were 10–100$\times$ larger on those spatial scales. The disagreement between these results remains unresolved.

As an alternative to helioseismic measurement, the gyroscopic pumping effect [4] could, in principle, map the structure of deep convection. However, the technique requires accurate measurement of differential rotation and deep meridional circulation. Unfortunately, the current ambiguity in meridional circulation measurements makes this strategy presently impossible [15, 16, 17, 18, 19, 20].

As a summary, we believe the following are all closely related questions

• •

  Where does supergranulation come from?

• •

  Why are classic “giant cells” not observed?

• •

  Why and exactly how do observations seem to contradict numerical models?

These questions all essentially ask the same thing: “Where is the large-scale convective power?” Featherstone and Hindman 2016 [5] pointed out that rotational effects provide a natural explanation for all three of these questions. Based on rotational effects, they suggested that the horizontal scale of deep convection must be no larger than the supergranular scale. They also estimated interior convective speeds significantly weaker than previously predicted.

That work was numerical in nature, however. It did not capture, nor describe theoretically, the transition between near-surface and deep-seated convection. Here, we carry out a careful theoretical analysis of rotational effects and demonstrate how, indeed, the influence of rotation provides a striking response to these questions of solar convection.

In a coordinate frame with rotation rate, $\Omega$, around the $\hat{z}$ axis, the following is an exact reformulation of the fully compressible inviscid momentum equations

$\displaystyle\partial_{t}u+(\nabla\times u+2\Omega\,\hat{z})\times u+\nabla\varpi\ =\ T\,\nabla s.$

(1)

The variable $u$ is the fluid velocity, $T$ is temperature and $s$ is entropy per unit mass. Eq. (1) replaces the pressure, $p$, with the Bernoulli function, $\varpi$,

$\displaystyle\frac{\,\text{d}{p}}{\rho}\ =\ \,\text{d}{h}-T\,\text{d}{s},\qquad\varpi\ \equiv\ \frac{|u|^{2}}{2}+h+\phi,$

(2)

where $h$ is the enthalpy per unit mass. Because the convection zone only contains 2% of the solar mass, $M_{\odot}$ [56], the gravitational potential $\phi\approx-GM_{\odot}/r$, where $G$ is Newton’s gravitational constant.

A well-mixed convection zone implies a well-defined adiabatic stratification [57], which varies only with the gravitational potential,

$\displaystyle\,\text{d}{p_{0}}\ =\ -\rho_{0}\,\text{d}{\phi},\quad\,\text{d}{h_{0}}\ =\ -\,\text{d}{\phi},\quad\,\text{d}{s_{0}}=0.$

(3)

The exception happens in the upper $\approx\!\text{2\%}$, where strong super-adiabatic stratification drives flow close to the speed of sound.

Assuming an ideal gas law, $h_{0}=c_{p}T_{0}=\gamma p_{0}/\rho_{0}/(\gamma-1)$. The detailed specific heat depends on the ionisation fraction and elemental abundances, which only become significant effects very near the solar surface. We assume constant values for $c_{p}$ and $\gamma=5/3$; see Table 1. Near the bottom of the convection zone $T_{0}(R_{\text{c.z.}\!})\approx 2.3\times 10^{6}\,$K. Near the surface $T_{0}(R_{\odot})/T_{0}(R_{\text{c.z.}\!})\approx 10^{-3}$, which vanishes to a good approximation. Therefore

$\displaystyle T_{0}(r)\ \approx\ \frac{\phi(R_{\odot})-\phi(r)}{c_{p}}\ =\ T_{0}(R_{\text{c.z.}\!})\frac{R_{\text{c.z.}\!}}{H_{\text{c.z.}\!}}\frac{R_{\odot}-r}{r}.$

(4)

For an adiabatic reference state⁴⁴4All quantities with a ‘0’ subscript denote adiabatic background reference states., $\rho_{0}(r)\propto T_{0}(r)^{3/2}$, $p_{0}(r)\propto T_{0}(r)^{5/2}$. The values at the base of the convection zone fix the constants of proportionality.

In the bulk of the convection zone, all thermodynamic variables fluctuate from their reference values by $\approx\!\text{}10^{-6}$, which permits the anelastic approximation [58, 59, 60]. Therefore,

$\displaystyle\partial_{t}u+(\nabla\times u+2\Omega\,\hat{z})\times u+\nabla\varpi\ =\ T_{0}\nabla s.$

(5)

The only difference between the fully compressible and anelastic momentum equation is the replacement $T\to T_{0}$. We omit an explicit treatment of magnetism in our analysis. At most we expect $|B|\lesssim\sqrt{\mu_{0}\rho_{0}}|u|$, which does not alter any of our conclusions. The anelastic approximation also implies

$\displaystyle\nabla\cdot\left(\rho_{0}u\right)\ =\ 0.$

(6)

Energy transport closes the system,

$\displaystyle\rho_{0}\,\partial_{t}\!\left(T_{0}s+|u|^{2}/2\right)+\nabla\cdot\left(\rho_{0}\varpi u\right)\ =\ \nabla\cdot\left(K_{0}\nabla T_{0}\right).$

(7)

Eq. (7) omits the diffusion of thermal fluctuations the same way Eq. (5) omits viscous dissipation. These effects are essential in a turbulent fluid, but only become important for microscopic scales. We cannot, however, ignore the radiative diffusion of the background, which provides the heat that drives buoyancy [61].

On average, energy transport requires

$\displaystyle\langle\varpi u_{r}\rangle\ =\ \frac{1}{\rho_{0}}\left[\frac{L_{\odot}}{4\pi r^{2}}+K_{0}\frac{dT_{0}}{dr}\right]\ \equiv\ F_{0}(r).$

(8)

The angled bracket on the left-hand side of Eq. (8) represents an average over time and horizontal spherical surfaces. We model the conductivity with Kramers’ opacity [62] law,

$\displaystyle K_{0}(r)\ =\ \frac{c_{p}L_{\odot}}{4\pi GM_{\odot}}\!\left[\frac{T_{0}(r)}{T_{0}(R_{\text{c.z.}\!})}\right]^{7/2}\!,$

(9)

which assumes $\rho_{0}^{2}\propto T_{0}^{3}$. The two terms within the square brackets in Eq. (8) partially cancel because $K_{0}\,dT_{0}/dr<0$. By definition $F_{0}(R_{\text{c.z.}\!})=0$. Eq. (8) is the foundation for all estimates; the right-hand side contains only known parameters.

The quantity $F_{0}(r)$ represents a radiative “flux debt”. Below the convection zone, radiative diffusion carries the entire luminosity. At some point near $R_{\text{c.z.}\!}$, the temperature gradient needed to carry the total flux becomes larger than the adiabatic gradient and the system becomes unstable. Convection transports the remainder of the heat after the background relaxes to an almost adiabatic state. Notably, $F_{0}(r)$ is not constant in radius, which implies the following two comments:

• •

  No matter what, convective time scales are much slower than rotational timescales within some distance of the radiative zone.

• •

  The important question is how far into the convection zone can rotation dominate buoyancy?

Eq. (8) carries dimensional units of velocity cubed. Standard mixing-length theory assumes $|u|\sim F_{0}^{1/3}$ [63]. While sometimes $F_{0}^{1/3}$ gives the actual flow speed, more freedom exist in the rapidly rotating regime. We require a better understanding of $\varpi$.

The Rossby number measures the non-dimensional ratio of convective-to-rotational acceleration. A common definition is the Bulk Rossby number

$\displaystyle\text{Ro}_{\text{b.}\!}\ \equiv\ \frac{|u|}{2\Omega\,\mathrm{\textsl{H}}_{\rho}},$

(10)

where $|u|$ represents a typical velocity amplitude and $\mathrm{\textsl{H}}_{\rho}$ a density scale height,

$\displaystyle\mathrm{\textsl{H}}_{\rho}\equiv\frac{c^{2}}{g}\ \approx\ (\gamma-1)\frac{(R_{\odot}-r)\,r}{R_{\odot}},$

(11)

where $c^{2}=\gamma p_{0}/\rho_{0}$ and $g=GM_{\odot}/r^{2}$. Many references alternatively use the convection zone depth $\mathrm{\textsl{H}}_{\text{c.z.}\!}$ in place of the scale height [5], e.g, $\mathrm{\textsl{H}}_{\rho}(R_{\text{c.z.}\!})/\mathrm{\textsl{H}}_{\text{c.z.}\!}\approx 0.46$. Eq. (10) represents the ratio of relative-to-background vorticity assuming convective flows vary on a length scale comparable to $\mathrm{\textsl{H}}_{\rho}$ (or $\mathrm{\textsl{H}}_{\text{c.z.}\!}$). Flows undoubtedly fluctuate over entire domain, but $\text{Ro}_{\text{b.}\!}$ overestimates rotational influence if the actual energy-containing scale is smaller.

Motivated by buoyancy considerations, the Convective Rossby number is

$\displaystyle\text{Ro}_{\text{c.}\!}\ \equiv\ \frac{\sqrt{T_{0}|\!\left<s-s_{0}\right>\!|}}{2\Omega\,\mathrm{\textsl{H}}_{\rho}},$

(12)

where $\left<s-s_{0}\right>$ represents characteristic mean entropy differences. This definition is equivalent to that used in the Boussinesq setting and defined in terms of the Rayleigh, Taylor and Prandtl numbers, $\text{Ro}_{\text{c.}\!}\sim\sqrt{\mathrm{Ra}/\mathrm{Ta}/\mathrm{Pr}}$ [64].

Assuming $\ell<\mathrm{\textsl{H}}_{\rho}$ represents the actual (yet unknown) dynamical scale, we define an alternative Dynamical Rossby number

$\displaystyle\text{Ro}_{\text{d.}\!}\ \equiv\ \frac{|u|}{2\Omega\,\ell}\ \approx\ \frac{|\nabla\times u|}{2\Omega}.$

(13)

The dynamical Rossby number is the closest estimate of the actual ratio between local and background vorticity. All definitions relate to each other in some way and will coincide in the non-rotating regime.

The Taylor-Proudman theorem [65, 66] is an important constraint for rapidly rotating systems. The formal result comes from taking the curl of Eq. (5) and neglecting all effects other than rotation,

$\displaystyle 2\Omega\,\hat{z}\,\cdot\nabla u\ \approx\ 0,$

(14)

Strictly speaking, Eq. (14) is simply not true for convection. A generic system with boundaries cuts off the dynamics to fit within the domain; hence $|\hat{z}\,\cdot\nabla u|\sim|u|/\mathrm{\textsl{H}}_{\text{c.z.}\!}$. In a deep shell, $|\hat{z}\,\cdot\nabla u|\sim|u|/\mathrm{\textsl{H}}_{\rho}$.

The more subtle view of the Taylor-Proudman constraint comes from considering the magnitude of the neglected terms. For $\text{Ro}_{\text{d.}\!}<1$, there is substantial degree of anisotropy between the $\hat{z}$ and perpendicular directions,

$\displaystyle\frac{\ell}{\mathrm{\textsl{H}}_{\rho}}\approx\text{Ro}_{\text{d.}\!}\quad\iff\quad\ell\approx\sqrt{\frac{|u|\mathrm{\textsl{H}}_{\rho}}{2\Omega}}.$

(15)

Eq. (15) results from taking the curl of the momentum equations and balancing the resulting Coriolis terms against the nonlinear terms, i.e.,

$\displaystyle 2\Omega\left(\hat{z}\,\nabla\cdot u-\hat{z}\cdot\nabla u\right)\ \sim\ \nabla\times[(\nabla\times u)\times u],$

(16)

which is the CI part of the CIA balance. The curl eliminates leading-order geostrophic terms. Eq. (15) follows from estimating the left-hand side of Eq. (16) with $2\Omega|u|/\mathrm{\textsl{H}}_{\rho}$ and the right-hand side with $|u|^{2}/\ell^{2}$.

Eq. (15) gives a direct link between flow speed and scale in a rapidly rotating system. Determining if this relation holds requires considering convective energy and momentum balances in detail.

This anisotropic picture of rotating convection has existed for many years in the Boussinesq convection community. The anisotropy principle is present in the linear-stability results in Chandrasekhar’s famous book on hydrodynamic stability [67]. Stevenson 1979 [68] expanded the understanding using linear theory. Ingersoll and Pollard 1982 [69] made a similar analysis applied to the interiors of planets. Sakai 1997 observed the phenomena clearly in laboratory experiments [70]. In a fully nonlinear and turbulent setting, Julien and coworkers have refined and promoted the principles of anisotropy and local Taylor-Proudman balance for strongly nonlinear flows since the mid-1990s [64, 71, 72, 73, 74, 75]; see Aurnou et al. 2020 [8] for a current summary including experimental evidence. Detailed notions of rotational anisotropy have been slow to gain traction in the stratified stellar modelling community and astrophysics in general; although that has been changing in recent years [76, 77, 78, 79, 80, 81]. Overall, it seems that a consensus is forming surrounding the proper mechanical and thermal balances in rapidly rotating convection.

If we know the size of $\varpi$ in terms $\ell$ and $|u|$, then Eqs. (8) & (15) give the characteristic magnitude of everything else. In fully compressible dynamics, the pressure (equivalently $\varpi$) is a genuinely dynamical variable; it requires its own initial condition. In low-Mach-number flows, the pressure becomes a Lagrange multiplier [60] that enforces the divergence-free condition by cancelling locally compressive terms. Finding an equation for $\varpi$ requires multiplying Eq. (5) by $\rho_{0}$ and taking the divergence, which eliminates the time-evolution term and leaves an elliptic equation for $\varpi$. Because $\varpi$ appears linearly, we can decompose the total Bernoulli function as a sum of three “partial pressures”,

$\displaystyle\varpi\ \equiv\ \varpi_{\mathrm{Kin.}}+\varpi_{\mathrm{Ther.}}+\varpi_{\mathrm{Rot.}}.$

(17)

Each partial pressure responds to three separate sources

	$\displaystyle\nabla\cdot\left(\rho_{0}\nabla\varpi_{\mathrm{Kin.}}\right)$	$\displaystyle=$	$\displaystyle\nabla\cdot(\rho_{0}\,u\times(\nabla\times u)),$		(18)
	$\displaystyle\nabla\cdot\left(\rho_{0}\nabla\varpi_{\mathrm{Ther.}}\right)$	$\displaystyle=$	$\displaystyle\nabla\cdot(\rho_{0}\,T_{0}\nabla s),$		(19)
	$\displaystyle\nabla\cdot\left(\rho_{0}\nabla\varpi_{\mathrm{Rot.}}\right)$	$\displaystyle=$	$\displaystyle\nabla\cdot(2\Omega\,\rho_{0}\,u\times\hat{z}).$		(20)

Solving Eqs. (18)–(20) in practice also requires appropriate boundary conditions; each of the same magnitude as its corresponding bulk source.

We can estimate the size of each partial pressure from its source. First,

$\displaystyle|\varpi_{\mathrm{Kin.}}|\ \sim\ |u|^{2},\qquad|\varpi_{\mathrm{Ther.}}|\ \sim\ T_{0}|s|.$

(21)

At this point, the magnitude of $|s|$ is unknown, but it cannot dominate all other terms. If all three partial pressure are assumed roughly equal then $|u|\sim F_{0}^{1/3}$. This has been the traditional assumption in mixing-length theory. But it could also happen that the rotational (geostrophic) pressure dominates the kinetic pressure.

Eq. (20) differs from Eqs. (18) & (19) in that the left- and right-hand sides contain different numbers of derivatives. Geostrophy selects a length scale,

$\displaystyle|\varpi_{\mathrm{Rot.}}|\ \sim\ 2\Omega\,|u|\ell\ \sim\ \sqrt{2\Omega\,\mathrm{\textsl{H}}_{\rho}\,|u|^{3}}.$

(22)

Altogether, we find two separate cases: CASE I in the slowly rotating scenario, and CASE II in the rapidly rotating scenario. See Table 2 for a summary.

The magnitude of buoyancy variation poses a subtle question. The direct entropy evolution satisfies

$\displaystyle\rho_{0}T_{0}\left(\partial_{t}s+u\cdot\nabla s\right)\ =\ \nabla\cdot(K_{0}\nabla T_{0}).$

(23)

The right-hand side radiative heating varies only in radius. The deviations from an adiabatic background therefore contain both mean and fluctuating parts; $s=\left<s\right>\!(r)+\tilde{s}(r,\theta,\varphi)$.

Convection happens because large-scale entropy gradients become unstable to growing fluctuations. The instability saturates from turbulent transport counteracting the advection of the background. Specifically, $u\cdot\nabla\left<s\right>\sim u\cdot\nabla\tilde{s}$, or

$\displaystyle\ell\,\left<s-s_{0}\right>\ \sim\ \mathrm{\textsl{H}}_{\rho}\ \tilde{s}.$

(24)

An estimate for $\tilde{s}$ follows from the curl of the momentum equations,

$\displaystyle\nabla\times[(\nabla\times u)\times u]\ \sim\ \nabla T_{0}\times\nabla\tilde{s},$

(25)

which is the IA part of CIA balance. Therefore

$\displaystyle T_{0}\,\tilde{s}\ \sim\ \frac{|u|^{2}}{\text{Ro}_{\text{d.}\!}},\qquad T_{0}\left<s-s_{0}\right>\ \sim\ \frac{|u|^{2}}{\text{Ro}_{\text{d.}\!}^{2}}.$

(26)

Putting everything together, in the rapidly rotating regime [8],

$\displaystyle\text{Ro}_{\text{c.}\!}\ \sim\ \frac{\text{Ro}_{\text{b.}\!}}{\text{Ro}_{\text{d.}\!}}\ \sim\ \text{Ro}_{\text{d.}\!}.$

(27)

With all the theoretical elements in place, we apply our analysis to the solar convection zone.

A question remains how the misalignment of gravity and the rotation axis affects the estimates. Several studies investigated the dependence on the local latitude in Cartesian domains [82, 83, 84, 85, 80]. No results seem definitive and much of the effort is still ongoing. Achieving rotational constraint at solar parameters requires a domain large enough to experience the variation of background forces. A sizable energy-containing eddy near the equator will feel off-equator effects.

Table 2 depends on $H_{\rho}$, which entered the analysis by assuming $\hat{z}\cdot\nabla\sim 1/\mathrm{\textsl{H}}_{\rho}\lesssim 1/\mathrm{\textsl{H}}_{\text{c.z.}\!}$. Variation along the rotation axis is the important overall reference length scale. The poles pose no challenge, where $\hat{r}\approx\hat{z}$. But starting from the equator, the distance from $r=R_{\text{c.z.}\!}$ to $r=R_{\odot}$ in the $\hat{z}$ direction is

$\displaystyle H_{z}\ =\ \mathrm{\textsl{H}}_{\text{c.z.}\!}\sqrt{2R_{\text{c.z.}\!}/\mathrm{\textsl{H}}_{\text{c.z.}\!}+1}\ \approx\ 5\mathrm{\textsl{H}}_{\rho}.$

(29)

We therefore expect at low latitudes modest increases to the true length scale appearing in Table 2, $\ell\propto H_{z}^{2/5}\approx 1.9\mathrm{\textsl{H}}_{\rho}^{2/5}$. However $\text{Ro}_{\text{d.}\!}\propto H_{z}^{-3/5}\approx 0.4\mathrm{\textsl{H}}_{\rho}^{-3/5}$ and $|u|\propto H_{z}^{-1/5}\approx 0.7\mathrm{\textsl{H}}_{\rho}^{-1/5}$. Moreover, these estimates are the extreme case. Therefore, near the equator, we might expect Fig. 2 to show a slight increase in scale (perhaps up to $\approx\!\text{50-60\,Mm}$ for $r\approx R_{\text{c.z.}\!}$, but this would be accompanied by slower flows and additional rotational constraint.

Helioseismology provides a few additional checks on the overall rectitude of our estimates. Even with some differences, convection and shear provide proxies for one another [81]. Helioseismology, therefore, presents a consistency check for some of our estimates.

As a function of radius, $r$, and co-latitude, $\theta$, the local rotation rate $\Omega(r,\theta)$ implies an angular inertial-frame bulk flow $u=r\sin\theta\,\Omega\,\hat{\varphi}$, which implies a total vorticity,

$\displaystyle\nabla\times(r\sin\theta\,\Omega\,\hat{\varphi})\ =\ 2\Omega\,\hat{z}+r\!\sin\theta\,\nabla\Omega\times\hat{\varphi}.$

(30)

Assuming the differential rotation is strongly coupled to the dynamics,

$\displaystyle\text{Ro}_{\text{d.}\!}\ \approx\ \frac{r\!\sin\theta|\nabla\Omega|}{2\Omega}.$

(31)

Fig. 3 shows the local Rossby number of the differential rotation as a function of latitude and depth in the convection zone. The data comes from⁵⁵5Electronic Supplementary Material for the article “Global-Mode Analysis of Full-Disk Data from the Michelson Doppler Imager and Helioseismic and Magnetic Imager.” the local rotation rate in Larson and Schou 2018 [86]. We compute the gradient with a 4th-order finite difference derivative. A few pertinent observation are in order. The Rossby number is never more than 0.4 for $r/R_{\odot}<0.95$. The tachocline ($0.65<r/R_{\odot}<0.75$) dominates the picture in the deep interior. Above $r/R_{\odot}\approx 0.95$ the rapid increase of the Rossby number indicates the start of the near-surface shear layer, which is poorly understood. But for much of the bulk convection zone, the differential rotation Rossby number hovers consistently around $\approx\!\text{0.1}$.

Also intriguing, the half-width of the tachocline bump is $\approx\!\text{35\,Mm}$, which is consistent with the $\approx\!\text{30\,Mm}$ estimate for the convection. If convection maintains the tachocline, it seems reasonable that their sizes should match. We are aware that the resolution of helioseismology degrades with depth and tachocline widths are upper bounds on the actual thickness. Even so, the data from helioseismology accords with simple dynamical estimates.

The above picture of differential rotation is fully consistent with the large-scale thermal wind model of Balbus and coworkers (e.g., [87]). The above scalings allow a significant balance between,

$\displaystyle r^{2}\sin(\theta)\,\partial_{z}\Omega^{2}\ \approx\ -T_{0}^{\prime}(r)\,\partial_{\theta}\tilde{s}.$

(32)

Eq. (32) does not necessarily give the exact large-scale entropy profile (e.g., due to possible Reynolds- and Maxwell-stress corrections), but it gives and good indication and provides a consistency check. It is also becoming clear how sensitive differential rotation can be to large-scale thermal gradients [88].

Fig. 4 shows a solution to Eq. (32). We integrate the right-hand side over $\theta$ using the trapezoidal rule. As Balbus et al. 2012 [87] pointed out, we can freely add any radial function to the solution. We set the integration constant by $\int_{0}^{\pi}\tilde{s}(r,\theta)\sin(\theta)\,\text{d}{\theta}=0$. The entropy state needed to maintain differential rotation is roughly the same needed to drive rotationally constrained turbulent convection. Also intriguing, the entropy is better mixed near the equator than near the poles.