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Turbulent thermal convection dominates the energy transport throughout the Sun’s outer envelope. In recent years, puzzling disagreements have arisen between observations, models, and theory regarding the amplitude and structure of convection (1⇓–3). Some work has begun to address the situation (4, 5). However, the discrepancy is not settled and has come to be called the convective conundrum (6).
In this paper, we demonstrate how the dominant dynamical balances in the solar interior make firstprinciple predictions for the spatial scale and amplitude of deep solar convection. Our analysis shows that interior flows likely exist in a quasigeostrophic (QG) state, with a joint Coriolis–inertial–Archimedean (CIA) balance (e.g., refs. 7⇓⇓–10) remaining after accounting for leadingorder geostrophy.*Rotation strongly influences solar convection as a result. Our results put on a firm theoretical footing the earlier suggestions of Miesch et al. (4) and Featherstone and Hindman (5). We also corroborate and provide context to the observations of Hanasoge et al. (1, 3), as well as more recent observations (11). Our estimates leave the flow amplitude only somewhat smaller than previous mixinglength models, which ignore rotational effects. However, we demonstrate that rotational influence in the Sun most prominently affects the dominant flow length scale.
1. Solar Convective Processes
Apart from negligible friction, a turbulent fluid conserves angular momentum as it traverses the solar envelope. Angularmomentum transport occurs throughout the interior and generates differential rotation comprising a fast equator and slow poles (12, 13). Angularmomentum redistribution also drives a largescale north–south meridional circulation (14⇓⇓–17). Longterm observations document the nearsurface meridional flow. However, its depth dependence is much less clear, with different helioseismic techniques yielding different results (18⇓⇓⇓⇓–23).
Largescale plasma motions must play a pivotal role in the stellar dynamo process. The latitudinal and radial shear provides a poloidaltotoroidal conversion mechanism (i.e., the Ω effect) (24). Meridional circulation modulates the distribution of sunspots and may also establish the cycle timing (25⇓–27). Helical flow generates a mean electromotive force (i.e., the α effect), which provides a toroidaltopoloidal dynamo feedback (28⇓–30). Any new information concerning interior fluid motions will produce valuable insight into the operation of the Sun’s magnetic cycle.
Photospheric Convection.
Several decades of observations have revealed much about solar surface convection (31, 32). Driven by fast radiative cooling, granulation dominates the radial motion at the solar surface. Roughly 1Mm in horizontal size, granulation produces a strong powerspectrum peak at spherical harmonic degree ≈1,000 in radial Dopplergrams (33, 34). Significant power also exists at the ≈30Mm supergranular scale, whose associated motions are mainly horizontal and best observed in limb Dopplergram spectra (35).
Photospheric power decreases monotonically for scales larger than supergranulation. From nonrotating intuition (36, 37), we might expect convective power to peak at ≈100to200Mm (comparable with the convectivelayer depth), rather than the ≈30Mm supergranular scale. The results of refs. 38 and 39 indicate that deeprooted fluid motions do persist on larger scales. Against expectations, however, these motions appear weak compared with the smallerscale supergranular and granular flows. A great deal of theory and simulation work has attempted to solve the supergranulation problem, including direct formation mechanisms (6, 37, 40⇓–42). To date, no model selfconsistently demonstrates how the supergranular scale might arise. We direct the reader to the recent review by Rincon et al. (43) for a thorough discussion of this topic. Our focus here is on the apparent lack of largescale power as expected from nonrotating convection. We concur with results of past work (5, 44) that the supergranular scale results from suppression of power on large scales, rather than through preferential driving at that spatial scale.
Subphotospheric Convection.
Local helioseismic techniques can probe subsurface convection directly [e.g., time distance (45), ringdiagram analysis (46), holography (47)]. Historically, these methods have largely been limited to ≈30Mm depth and do not sample flow below the nearsurface 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 nonrotating intuition) that convective power peaks at ≈100Mm scales with ≈100m/s flowspeed amplitudes (48, 49). Limitations of those results began to appear, however, with systematic magnetohydrodynamic studies in the ensuing decade. These studies found that only systems with ≈10× weaker flows or equivalently, those that rotated ≈10× faster, were able to produce coherent magnetic fields and periodic magnetic cycles in analogue to the Sun (50⇓⇓⇓⇓–55). Moreover, simulations with more extreme parameters and largescale power can generate antisolar differential rotation, with slow equator and fast poles (14, 16, 17, 56⇓–58). We also note that some recent observations suggest the Sun may lie near a boundary between these two basins in parameter space (59).
While most local helioseismic analyses focus on the nearsurface shear layers, techniques have been developed to probe more deeply rooted flow structures, such as solar meridional circulation (60, 61). A notable puzzle arose following the deepfocusing time–distance analysis of Hanasoge et al. (1). This work placed a roughly 1m/s upper limit on the ≈100Mmscale flow amplitudes at a depth ≈60Mm. Subsequently, Greer et al. (2) sampled deeply enough to compare directly against the time–distance results at a depth of 30Mm. Rather than no detection, this effort yielded measured flows that were 10 to 100× larger on those spatial scales. The discrepancies are still surprising even if they are measured at different depths. The disagreement between these results remains unresolved (11).
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 (18⇓⇓⇓⇓–23).
The Convective Conundrum.
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 largescale convective power?” Some authors have recently studied possible direct formation mechanisms for supergranulation (6, 37, 40⇓–42). In particular, recent work by Schumacher and coworkers (6, 37, 42) finds a continual amalgamation of convection on the largest scales admitted by the computational domain. Featherstone and Hindman (5) pointed out that rotational effects provide a natural explanation for the last two 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 describe the nature of deepseated convection theoretically.
2. Analysis
Our goal is to estimate the dominant forces and their relative magnitudes. Our program is to manipulate the equations of motion into a form that exposes the principal dynamical balances as much as possible. We first define a general system of equations (under the anelastic approximation) that includes rotation (i.e., background profiles, momentum–mass–energy conservation, and radiation transport). We then isolate the place where rotational effects require a decision tree (i.e., the radiation flux balance). We define the tools…
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Read More:Rotation suppresses giantscale solar convection