The simplest model for the emergence of a set of special scales is that of multiple steady linear or weakly
nonlinearly interacting modes of thermal convection forced at different depths. The first theoretical
argument of this kind is due to Simon and Leighton (1964), who suggested that supergranulation-scale
motions corresponded to simple convection cells driven at the depth of He^{++} recombination and just
advecting granulation-scale convection. Schwarzschild (1975) invoked an opacity break, He^{+} and H^{+}
recombinations as the drivers of supergranulation-scale convection. Simon and Weiss (1968) and
Vickers (1971), on the other hand, suggested that deep convection in the Sun had a multilayered structure
composed of deep, giant cell circulations extending from the bottom of the convection zone to
40 Mm deep, topped by a shallower circulation pattern corresponding to supergranulation. In this
second theory, recombination is not a necessary ingredient. Bogart et al. (1980) attempted
to match a linear combination of convective eigenmodes to the solar convective flux but did
not find that supergranulation came out as a preferred scale of convection in this quasilinear
framework.

Antia et al. (1981) argued that turbulent viscosity and diffusivity should be taken into account in linear calculations, as they alter the growth and scales of the most unstable modes of convection. In their linear calculation with microscopic viscosity and thermal diffusivity coefficients replaced by their turbulent counterparts, granulation, and supergranulation show up as the two most unstable harmonics of convection. Calibrating the amplitudes of a linear superposition of convective modes to match mixing-length estimates of the solar convective flux in the spirit of Bogart et al. (1980), Antia and Chitre (1993) further argued that they could reproduce the main characteristics of the power spectrum of solar surface convection.

Gierasch (1985) devised a one-dimensional energy model for the upper solar convection zone from which he argued that turbulent dissipation takes place and deposits thermal energy at preferred depths, thereby intensifying convection at granulation and supergranulation scales. On this subject, we also mention the work of Wolff (1995), who calculated that the damping of r-modes in the Sun should preferentially deposit heat 50 Mm below the surface as a result of the ionisation profile in the upper solar convection zone. This process might in turn result in convective intensification at similar horizontal scales.

An interesting theoretical suggestion on the problem of supergranulation was made by Van der Borght (1974), who considered the case of steady finite-amplitude thermal convection cells in the presence of fixed heat flux boundary conditions imposed at the top and bottom of the layer. He showed that the convection pattern in this framework has much smaller temperature fluctuations than in the standard Rayleigh–Bénard model with fixed temperature boundary conditions. This makes this case quite interesting for the supergranulation problem, considering that intensity fluctuations at supergranulation scales are rather elusive (see Section 4.3).

Even more interestingly, fixed heat flux boundary conditions naturally favour marginally stable convection cells with infinite horizontal extent compared to the layer depth, or convection cells with a very large but finite horizontal extent when a weak modulation of the heat flux is allowed for (Sparrow et al., 1964; Hurle et al., 1967; Van der Borght, 1974; Busse and Riahi, 1978; Chapman and Proctor, 1980; Depassier and Spiegel, 1981). This case is therefore very different from the standard Rayleigh–Bénard case with fixed temperature boundary conditions, which gives rise to cells with aspect ratio of order unity. In this framework, there is no need to invoke deep convection to produce supergranulation-scale convection.

This idea was carried on with the addition of a uniform vertical magnetic field threading the convective layer. Contrary to the hydrodynamic case described above, where zero-wavenumber solutions are preferred linearly (albeit with a zero growth-rate), convection cells with a long but finite horizontal extent dominate in the magnetised case, provided that the magnetic field exceeds some threshold amplitude. The horizontal scale of the convection pattern in the model is subsequently directly dependent on the magnetic field strength. Murphy (1977) was the first to suggest that this model might be relevant to supergranulation. The linear problem in the Boussinesq approximation was solved by Edwards (1990). Rincon and Rieutord (2003) further solved the fully compressible linear problem numerically and revisited it in the context of supergranulation. Using typical solar values as an input for their model parameters (density scale height, turbulent viscosity etc.) they showed that the magnetic field strength (measured in the nondimensional equations by the Chandrasekhar number Q) required for compressible magnetoconvection with fixed heat flux to produce supergranulation-scale convection was of the order 100 G.

The discovery by Gizon et al. (2003) that supergranulation has wave-like properties (Section 4.5) opened some new perspectives for theoretical speculation. In particular, it offered an opportunity to revive the interest for several important theoretical findings pertaining to the issue of oscillatory convection, which we now attempt to describe.

The existence of time-dependent oscillatory modes of thermal convection has been known for a long time
(Chandrasekhar, 1961 provides an exhaustive presentation of linear theory on this topic). In many cases,
such a behaviour requires the presence of a restoring force acting on the convective motions
driven by buoyancy. It can be provided by Coriolis effects (rotation) or magnetic field tension for
instance. The existence of oscillatory solutions is also known to depend very strongly on how
various dissipative processes (viscous friction, thermal diffusion, and ohmic diffusion) compete in
the flow. This is usually measured or parametrised in terms of the thermal Prandtl number
, where is the kinematic viscosity and is the thermal diffusivity, the magnetic
Prandtl number , where is the magnetic diffusivity, and the “third” Prandtl
number^{7}
. In the Sun, , (see Section 2.2), and
at the photosphere.

As mentioned in Section 4.5, the supergranulation flow seems to be weakly influenced by the global solar rotation. In the presence of a vertical rotation vector, overstable oscillatory convection is preferred to steady convection provided that is small (Chandrasekhar, 1961). In more physical terms, an oscillation is only possible if inertial motions are not significantly damped viscously on the thermalization timescale of rising and sinking convective blobs.

Busse (2004, 2007) suggested on the basis of a local Cartesian analysis that the drift of supergranulation could be a signature of weakly nonlinear thermal convection rotating about an inclined axis and found a phase velocity consistent with the data of Gizon et al. (2003), assuming an eddy viscosity prescription consistent with solar estimates (based on the typical sizes and velocity of granulation). Earlier work on the linear stability of a rotating spherical Boussinesq fluid layer heated by internal heat sources showed that the most rapidly growing perturbations are oscillatory and form a prograde drifting pattern of convection cells at low Prandtl number in high Taylor number regimes corresponding physically to large rotation (Zhang and Busse, 1987).

A directly related issue is that of the influence of differential rotation on supergranulation. Green and Kosovichev (2006) considered the possible role of the solar subsurface shear layer (Schou et al., 1998) by looking at the effect of a vertical shear flow on the onset of convection in a strongly stratified Cartesian layer using linear theory. They found that convective modes in the non-sheared problem become travelling when a weak shear is added. Some previous work found that this behaviour is possible either at low (Kropp and Busse, 1991) or if some form of symmetry breaking is present in the equations (Matthews and Cox, 1997). Since linear shear alone cannot do the job, it is likely that density stratification plays an important role in obtaining the result. Green and Kosovichev (2006) also report that the derived phase speeds for their travelling pattern are significantly smaller than those inferred from observations by Gizon et al. (2003).

Note that the relative orientations and amplitudes of rotation, shear, and gravity are fundamental parameters in the sheared rotating convection problem. It should therefore be kept in mind that the results (e.g., the pattern phase velocity and wavelength) of local Cartesian theoretical models of supergranulation incorporating solar-like rotation effects are expected to depend on latitude, as the orientation of the rotation vector changes from horizontal at the equator to vertical at the pole, and the subsurface rotational velocity gradient varies with latitude in the Sun. We recall that there is as yet no conclusive observational evidence for a latitudinal dependence of the scales of supergranulation (see Section 4.5), so it is unclear if local models of sheared rotating convection can help solve the problem quantitatively. Global spherical models do not necessarily suffer from this problem, as they predict global modes with a well-defined phase velocity.

Magnetoconvection in a uniform vertical magnetic field is also known to preferentially take on the form of time-oscillations at onset provided that (e.g., Chandrasekhar, 1961; Proctor and Weiss, 1982), a process referred to as magnetic overstability. Oscillatory magnetoconvection is also known to occur for non-vertical magnetic fields (e.g., Matthews et al., 1992; Hurlburt et al., 1996; Thompson, 2005, and references therein). Physically, field lines can only be bent significantly by convective motions and act as a spring if they do not slip too much through the moving fluid, which requires, in this context, that the magnetic diffusivity of the fluid be small enough in comparison to its thermal diffusivity. Since in the quiet photosphere, oscillatory magnetoconvection represents a possible option to explain the wavy behaviour of supergranulation. On this topic, Green and Kosovichev (2007) recently built on the work of Green and Kosovichev (2006) and considered the linear theory of sheared magnetoconvection in a uniform horizontal (toroidal) field shaped by the subsurface shear layer. They report that the phase speed of the travelling waves increases in comparison to the hydrodynamic case studied by Green and Kosovichev (2006) and argue that the actual phase speed measured by Gizon et al. (2003) can be obtained for a uniform horizontal field of 300 G.

Finally, it is known theoretically and experimentally that even in the absence of any effect such as magnetic couplings, rotation or shear, the value of the thermal Prandtl number can significantly affect the scales and time evolution of convection, both in the linear and nonlinear regimes. Its value notably controls the threshold of secondary oscillatory instabilities of convection rolls (Busse, 1972). At very low Prandtl numbers, Thual (1992) showed that a very rich dynamical behaviour resulting from the interactions between the primary convection mode and the secondary oscillatory instability takes place close to the convection threshold. This includes travelling and standing wave convection.

Most theoretical studies of supergranulation to date have been either ideal (no dissipation) or for . For this reason, some important physical effects relevant to supergranulation-scale convection may well have been overlooked until now.

The previous models are interesting in many respects but it should be kept in mind that they all
have very important shortcomings. First, they rely on linear or weakly nonlinear calculations,
which is hard to justify considering that the actual Reynolds number in the solar photosphere is
over 10^{10} and that power spectra of solar surface flows show that the dynamics is spread over
many scales. A classical mean-field argument is that small-scale turbulence gives rise to effective
turbulent transport coefficients, justifying that the large-scale dynamics be computed from linear
or weakly nonlinear theory. Even if it is physically appealing, this argument still lacks firm
theoretical foundations. Assuming that turbulent diffusion can be parametrised by using the same
formal expression as microscopic diffusion is a strong assumption, and so is the neglect of direct
nonlocal, nonlinear energy transfers between disparate scales. Dedicated numerical simulations of
this problem are therefore more than ever required to justify or to discard using this kind of
assumptions.

Models with poor thermally conducting boundaries have the interesting feature of producing fairly shallow convection cells, with a large horizontal extent in comparison to their vertical extent. If it is confirmed that supergranulation is indeed a shallow flow (Section 4.4), investigating the theoretical basis of this assumption may prove useful to understand the origins of supergranulation. Of course, the main problem is that it remains to be demonstrated that fixed heat flux boundaries represent a good approximation of the effect of granulation-scale convection on larger-scale motions beneath the solar surface.

Finally, magnetoconvection models all assume the presence of a uniform field (either horizontal or vertical) threading the convective layer, which is certainly an oversimplified zeroth-order prescription for the magnetic field geometry in the quiet Sun. Numerical modelling probably provides the only way to incorporate more complex magnetic field geometries, time-evolution and dynamical feedback in supergranulation models.

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