Outside the Sun, we can observe only those photons that are emitted into the solar photosphere, where the density of atoms is sufficiently low and the photons can finally escape from the Sun without colliding with another atom or ion. As an analogy, imagine that you are attending a big campus rally and have found a prime spot near the center of the action.
Your friend arrives late and calls you on your cell phone to ask you to join her at the edge of the crowd. You decide that friendship is worth more than a prime spot, and so you work your way out through the dense crowd to meet her. You can move only a short distance before bumping into someone, changing direction, and trying again, making your way slowly to the outside edge of the crowd. All this while, your efforts are not visible to your waiting friend at the edge.
So too photons making their way through the Sun are constantly bumping into atoms, changing direction, working their way slowly outward, and becoming visible only when they reach the atmosphere of the Sun where the density of atoms is too low to block their outward progress.
The photosphere looks sharp only from a distance. If you were falling into the Sun, you would not feel any surface but would just sense a gradual increase in the density of the gas surrounding you. It is much the same as falling through a cloud while skydiving. From far away, the cloud looks as if it has a sharp surface, but you do not feel a surface as you fall into it.
One big difference between these two scenarios, however, is temperature. The Sun is so hot that you would be vaporized long before you reached the photosphere.
Figure 5. Granulation Pattern: The surface markings of the convection cells create a granulation pattern on this dramatic image left taken from the Japanese Hinode spacecraft. You can see the same pattern when you heat up miso soup. We might note that the atmosphere of the Sun is not a very dense layer compared to the air in the room where you are reading this text. Observations with telescopes show that the photosphere has a mottled appearance, resembling grains of rice spilled on a dark tablecloth or a pot of boiling oatmeal.
This structure of the photosphere is called granulation see Figure 5 Granules, which are typically to kilometers in diameter about the width of Texas , appear as bright areas surrounded by narrow, darker cooler regions. The lifetime of an individual granule is only 5 to 10 minutes. Even larger are supergranules, which are about 35, kilometers across about the size of two Earths and last about 24 hours. The motions of the granules can be studied by examining the Doppler shifts in the spectra of gases just above them see The Doppler Effect.
The bright granules are columns of hotter gases rising at speeds of 2 to 3 kilometers per second from below the photosphere. As this rising gas reaches the photosphere, it spreads out, cools, and sinks down again into the darker regions between the granules. Measurements show that the centers of the granules are hotter than the intergranular regions by 50 to K. Figure 6. Because they are transparent to most visible radiation and emit only a small amount of light, these outer layers are difficult to observe.
Until this century, the chromosphere was visible only when the photosphere was concealed by the Moon during a total solar eclipse see the chapter on Earth, Moon, and Sky. Observations made during eclipses show that the chromosphere is about to kilometers thick, and its spectrum consists of bright emission lines, indicating that this layer is composed of hot gases emitting light at discrete wavelengths.
The reddish color of the chromosphere arises from one of the strongest emission lines in the visible part of its spectrum—the bright red line caused by hydrogen, the element that, as we have already seen, dominates the composition of the Sun.
In , observations of the chromospheric spectrum revealed a yellow emission line that did not correspond to any previously known element on Earth. It took until for helium to be discovered on our planet. Today, students are probably most familiar with it as the light gas used to inflate balloons, although it turns out to be the second-most abundant element in the universe.
The temperature of the chromosphere is about 10, K. This means that the chromosphere is hotter than the photosphere, which should seem surprising. In all the situations we are familiar with, temperatures fall as one moves away from the source of heat, and the chromosphere is farther from the center of the Sun than the photosphere is. Figure 7. Temperatures in the Solar Atmosphere: On this graph, temperature is shown increasing upward, and height above the photosphere is shown increasing to the right.
Note the very rapid increase in temperature over a very short distance in the transition region between the chromosphere and the corona. The increase in temperature does not stop with the chromosphere. Above it is a region in the solar atmosphere where the temperature changes from 10, K typical of the chromosphere to nearly a million degrees. The hottest part of the solar atmosphere, which has a temperature of a million degrees or more, is called the corona.
Appropriately, the part of the Sun where the rapid temperature rise occurs is called the transition region. It is probably only a few tens of kilometers thick. Figure 7 summarizes how the temperature of the solar atmosphere changes from the photosphere outward. IRIS is the first space mission that is able to obtain high spatial resolution images of the different features produced over this wide temperature range and to see how they change with time and location Figure 8.
Figure 3 and the red graph in Figure 7 make the Sun seem rather like an onion, with smooth spherical shells, each one with a different temperature. This pattern is called granulation, and is associated with large scale fluid motions at and just below the photosphere; the brighter, central regions correspond to rising hotter fluid, and the darker, narrow lanes to sinking, colder fluid.
Typical speeds in granular flows are of the order of a few kilometers per second. High Altitude Observatory. Search form Search. Another way to proceed is to use the equation of mass conservation of mean flows. When high-frequency acoustic waves are filtered out, one may use the anelastic approximation and write. From this equation, we see that a measure of the horizontal divergence and the vertical velocity together with a value of the density scale height given by a model , allow for an estimation of the vertical velocity scale height.
Combining Dopplergrams and correlation tracking inferences with the above considerations on the continuity equation, November made the noteworthy prediction that the supergranulation flow should disappear at depths larger than 2.
Note that his suggestion that the mesogranulation signal detected in power spectra at a horizontal scale of 7 Mm corresponded to the vertical flow component of convective supergranulation cells was part of the same argument.
More recently, Rieutord et al. The advent of local helioseismology in the late s made it possible to probe the supergranulation flow at optically-thick levels. Duvall Jr et al. Duvall Jr further estimated that the depth of supergranulation was 8 Mm. Zhao and Kosovichev reported evidence for converging flows at 10 Mm and estimated the supergranulation depth to be 15 Mm.
Woodard reported a detection of the flow pattern down to 5 Mm corresponding to the deepest layers accessible with their data set. Using new Hinode data, Sekii et al. The existence of a return flow at depths larger than 5 Mm has also been suggested but remains unclear Duvall Jr, ; Zhao and Kosovichev, Note that imaging deep convection using helioseismic techniques is not an easy task.
Braun and Lindsey and Lindsey and Braun provide a detailed description of the shortcomings and artefacts of helioseismic inversions in this context see also Gizon and Birch, To summarise, the determination of the vertical extent of the supergranulation below the surface is still in a preliminary phase. The few results mentioned above point to a shallow structure but they are affected by large uncertainties associated with both the intrinsic difficulty to perform such measurements and with their weak statistical significance.
It is clear that a decisive step forward regarding this problem requires a careful study of the systematics and the processing of a very large amount of data to reduce the impact of the fluctuating nature of the flows.
This is not a large value, indicating that the Coriolis acceleration should have an effect on the dynamics of supergranules. This effect has been observed by Gizon and Duvall Jr , who showed Figure 5a that the correlation between vertical vorticity and horizontal divergence of supergranules changes sign at the equator: it is negative in the northern hemisphere and positive in the southern one. These anticyclones are surrounded by cyclonic vorticity associated with downward flows; because these downdrafts have a somewhat smaller scale, this cyclonic vorticity is less conspicuous in measurements than the anticyclonic contribution of supergranules, but it has actually been singled out in the work of Komm et al.
The first reports on the rotational properties of supergranulation focused on the rotation rate of the supergranulation pattern Duvall Jr, ; Snodgrass and Ulrich, This is now referred to as the superrotation of supergranules. In recent years, local helioseismology has proven extremely useful to study the rotational properties of supergranules.
Their superrotation was confirmed by Duvall Jr and Gizon using the time-distance technique applied to f -modes. Beck and Schou estimated that the supergranulation rotation rate is larger than the solar rotation rate at any depth probed by helioseismology. Analysing time series of divergence maps inferred from time-distance helioseismology applied to MDI data, Gizon et al.
They showed that the power spectrum of the supergranulation signal close to the equator presented a power excess in the prograde direction with a slight equatorwards deviation in both hemispheres , thus explaining the anomalous superrotation rate of the pattern. The dispersion relation for the wave appears to be only weakly dependent on the latitude Gizon and Duvall Jr, Schou confirmed these findings with direct Doppler shift measurements and found that wave motions were mostly aligned with the direction of propagation of the pattern.
However, Rast et al. According to Gizon and Birch , this interpretation is not supported by observations. They argue that the finding of Lisle et al. Even more recently, Hathaway et al.
Using correlation tracking of divergence maps derived from intensity maps Meunier et al. For a detailed discussion on the identification of supergranulation rotational properties with helioseismology, we refer the reader to the review article by Gizon and Birch on local helioseismology. For the sake of completeness on the topic of rotation, we finally mention the observations by Kuhn et al. Recently, Williams et al. As shown in Section 2. The particular role played by magnetic fields in the supergranulation problem and the large amount of observational information available on this topic justify a dedicated subsection.
In the following, we first look at the correlations between supergranulation and the magnetic network and then describe the properties of internetwork fields, whose dynamics can hardly be dissociated from the formation of the magnetic network. After a short detour to the observations of the interactions between supergranulation and active regions, we finally review several studies of the dependence of supergranulation on the global solar magnetic activity.
Such a spectroheliogram is shown in Figure 6. Leighton et al. For this reason, both magnetograms and spectroheliograms are used to trace supergranulation e. It should be kept in mind, however, that the dynamical interactions between magnetic fields and supergranulation are actually not well understood theoretically. This problem will be discussed at length in Section 8.
The magnetic network refers to a distribution of magnetic field concentrations associated with bright points in spectroheliograms with typical field strengths of the order of 1 kG see reviews by Solanki, ; de Wijn et al.
The magnetic network is not regularly distributed on the boundaries of supergranulation cells but rather concentrates into localised structures see Figure 7. Estimates for the lifetime and size of supergranules inferred from magnetograms or spectroheliograms are significantly smaller than those based on direct velocimetric measurements Wang and Zirin, ; Schrijver et al.
For instance, Hagenaar et al. Krijger and Roudier found that the chromospheric network is well reproduced by letting magnetic elements that are emerging be passively advected by the surface supergranulation flow field. Magnetic field distribution grey scale levels on the supergranulation boundaries. The black dots show the final positions of floating corks that have been advected by the velocity field computed from the average motion of granules.
The distribution of corks very neatly matches that of the magnetic field. These results suggest that the formation of the magnetic network is in some way related to the supergranulation flow. It is however probably too simplistic and misleading to make a one-to-one correspondence between the single scale of supergranulation and the network distribution of magnetic bright points. Several studies with the Swedish Solar Telescope at La Palma observatory indicate that strong correlations between flows at scales comparable to or smaller than mesoscales i.
A recent study by Roudier et al. One of the major advances on solar magnetism in the last ten years has been the detection of quiet Sun magnetic fields at scales much smaller than that of granulation e. The ubiquity of these fields and their energetics suggest that the dynamics of internetwork fields could also be an important piece of the supergranulation puzzle see also Section 4. It is therefore useful to recall their main properties before discussing the physics of supergranulation in the next sections.
Note that the following summary is not meant to be exhaustive. For a dedicated review, we refer the reader to the recent work of de Wijn et al. Internetwork fields refer to mixed-polarity fields that populate the interior of supergranules. Their strength is on average thought to be much weaker than that of network fields, but magnetic bright points are also observed in the internetwork, e.
Besides, network and internetwork fields are known to be in permanent interaction e. In the light of nowadays high-resolution observations, the historical dichotomy between network and internetwork fields appears to be rather blurred this point will be further discussed in Section 8. Internetwork magnetism was originally discovered by Livingston and Harvey , and subsequently studied by many authors e. The strength of internetwork fields, their distribution at granulation and subgranulation scales and their preferred orientation are still a matter of debate.
Almost every possible value in the 5 — G range can be found in literature for the typical field strengths within the internetwork Martin, ; Keller et al. This wide dispersion is explained by several factors. The most important one is certainly that Zeeman spectropolarimetry, one of the most frequently used tools to study solar magnetism, is affected by cancellation effects when the magnetic field reverses sign at scales smaller than the instrument resolution Trujillo Bueno et al.
Hence, very small-scale fields still partially escape detection via this method. Recent Zeeman spectropolarimetry estimates of the average field strength based on Hinode observations Lites et al. On the side of Hanle spectropolarimetry, Trujillo Bueno et al. The scale-by-scale distribution of magnetic energy and the power spectrum of magnetic fields in the quiet photosphere are other important quantities to look at, as they may give us some clues on the type of MHD physics at work in the subgranulation to supergranulation range.
Based on various types of analysis structure statistics, wavelets, etc. Explicit studies of the power spectrum of the quiet Sun are currently limited to the range 1 — Mm and to the line-of-sight component of the magnetic field. We have been unable to find any study of the magnetic power spectrum of the quiet Sun covering scales well below 1 Mm, at which internetwork fields can now be detected with Hinode. At scales below 10 Mm, the magnetic power spectrum of the quiet photosphere has been found to be rather flat and decreasing with decreasing scales.
Proceeding along the description of the interactions between supergranulation and magnetic fields, one may also consider the properties of surface flows at scales comparable to supergranulation within active regions and in the vicinity of sunspots. The reason for this is twofold. First, we may wonder how the supergranulation pattern evolves locally during the formation or decay of an active region.
Second, the properties of flows around sunspots may give us some hints of the effect of strong magnetic flux concentrations on the flow dynamics in the quiet photosphere. As far as the first point is concerned, the information is fairly scarce at the moment. Rieutord et al. While the pores of a size comparable to that of a granule are emerging, the supergranulation flow becomes very weak just like if the surrounding magnetic flux associated with the pores had a significant impact on the flow.
A related observation by Hindman et al. On the second point, many studies in the past have focused on the detection and characterisation of intrinsic flows associated with sunspot regions see Solanki, and Thomas and Weiss, for exhaustive descriptions of sunspot structure and dynamics and significant observational progress has been made on this problem in recent years thanks to local helioseismology Lindsey et al.
The general picture that has progressively emerged is the following see Hindman et al. There is a corresponding return flow at depths smaller than 2 Mm, so the moat circulation is fairly shallow. Structure of flows surrounding a sunspot, as inferred from helioseismology from Hindman et al. Several authors studied the structure of the moat flow using Doppler signal Sheeley Jr and Bhatnagar, ; Sheeley Jr, , by tracking surface features, such as granules Muller and Mena, ; Shine et al.
It is however unclear whether or not this flow has anything to do with the regular supergranulation, as the outflow is centred on a strong field region in that case whereas it is the supergranulation inflow vertices that coincide with magnetic flux concentrations in the quiet Sun. As far as supergranulation is concerned, nevertheless, the lesson to be learned from helioseismology of sunspot regions is that magnetoconvection in strong fields has the naturally ability to produce a variety of coherent outflows and inflows at various horizontal and vertical scales in the vicinity of regions of strong magnetic flux.
This phenomenology may be worth exploring further in the somewhat scaled-down system consisting of the supergranulation flow and local flux concentrations associated with the magnetic network in the quiet Sun see Section 8. In view of the association between supergranulation and the magnetic network, it is finally natural to wonder if and how the size of supergranules varies with solar activity.
Singh and Bappu , studying spectroheliograms spanning a period of seven solar maxima, found a decrease of the typical size of the chromospheric network between the maxima and the minima of the cycle.
Their results are in line with those of Kariyappa and Sivaraman , Berrilli et al. These somehow contradicting results show that magnetic tracers must be used with care for this kind of measurements. The results are indeed sensitive to the thresholds used to identify the various field components e. Disentangling all these effects is not an easy task.
Recent studies have thus attempted to use proxies independent of magnetic tracers of supergranulation to measure its size, notably velocity features like positive divergences. DeRosa and Toomre , using two data sets obtained at periods of different levels of magnetic activity, found smaller supergranulation cell sizes in the period of high activity.
A similar conclusion was reached by Meunier et al. Meunier et al. Hence, it seems that a negative or a positive correlation can be obtained, depending on whether the level of magnetic activity is defined with respect to internetwork or network fields. We refer the reader to Meunier et al. Finally, on the helioseismic side, the dispersion relation for the supergranulation oscillations found by Gizon et al. However, the same authors reported a decrease in the lifetime and power anisotropy of the pattern from solar minimum to solar maximum.
Most of its main observational properties, like its size, lifetime and the strength of the associated flows are now well determined. However, other aspects of supergranulation dynamics, like the vertical dependence of the flow, the vertical component of the velocity at the edge of supergranules and the connections between supergranulation and magnetic fields are still only very partially constrained by observations.
They all require further investigations. As far as velocity measurements are concerned, we may anticipate progress in the near future on the question of the depth of supergranulation thanks to local helioseismology applied to higher-resolution observations. However, characterising vertical flows at the supergranulation scale is a more complex task since such flows are faint and very localised in space.
Improving the diagnostics of the latitudinal dependence of the supergranulation pattern may also prove useful, in particular to help understand if subsurface shear plays a significant role in shaping the supergranulation flow. Even more accurate studies of this kind could become feasible soon by using tracking techniques applied to images obtained with wide-field cameras imaging the full solar disc with sub-granulation resolution.
The case of the interactions with magnetic fields deserves a lot of further attention on the observational side in our view. It is now well established that flows at scales larger than granulation advect internetwork fields and tend to concentrate magnetic elements into the network, along the boundaries of supergranules. This process is essentially kinematic, in the sense that the magnetic field only has a very weak feedback on the flow.
But what we observe ultimately is probably a nonlinear statistically steady magnetised state, in which the magnetic field provides significant feedback on the flow. Hence, it would be useful to have more quantitative observational results on the relations between the properties of supergranules and the surrounding magnetic fields internal and boundary flux, filling factors, strength, size to characterise this feedback more precisely we refer the reader to Section 8 for an exhaustive discussion on supergranulation and MHD.
Most importantly, a precise determination of the magnetic energy spectrum of the quiet Sun over a very wide range of scales would be extremely precious to understand the nature of MHD interactions between supergranulation, network and internetwork fields. Finally, it would also be interesting to have more documented observational examples of the interactions of supergranulation with magnetic regions of various strengths active regions, polar regions to gain some insight into the dynamical processes at work in the problem.
This latter point is important from the perspective of the global solar dynamo problem, as it would help better constrain the transport of magnetic field by turbulent diffusion at the surface of the Sun.
We now turn to the description of existing theoretical models of supergranulation. These models are basically of two types: those that postulate that supergranulation has a convective origin i.
We then review various thermal convection models of supergranulation Section 5. A few concluding remarks follow. Before we start, it is perhaps useful to mention that most of these models are unfortunately only very qualitative, in the sense that they either rely on extremely simplified theoretical frameworks like linear or weakly nonlinear theory in two dimensions, or simple energetic arguments or on simple dynamical toy models designed after phenomenological considerations.
The looseness of theoretical models, combined with the incompleteness of observational constraints and shortcomings of numerical simulations, has made it difficult to either validate or invalidate any theoretical argument so far.
What numerical simulations tell us and how the theoretical models described below fit with numerical results and observations will be discussed in detail in Section 6. The simplest formulation of the problem of thermal convection of a fluid is called the Rayleigh. It describes convection of a liquid enclosed between two differentially heated horizontal plates, each held at a fixed temperature.
The equilibrium background state is a linear temperature profile with temperature decreasing from the bottom to the top of the layer. This case is in many respects different and simpler than the strongly stratified SCZ case, which treatment requires using more general compressible fluid and energy equations than those given below Nordlund, , but is sufficient to discuss many of the important physical Section 5. This set-up is shown on Figure 9. In nondimensional form, the equations for momentum and energy conservation, the induction equation, the equations for mass conservation and magnetic field solenoidality read.
This set of equations must be complemented by appropriate boundary conditions, most commonly fixed temperature or fixed thermal flux conditions on the temperature, no-slip or stress-free conditions on velocity perturbations, and perfectly conducting or insulating boundaries for the magnetic field. Several important numbers appear in the equations above, starting with the Rayleigh number.
The second important parameter above is the Chandrasekhar number. The relative importance of the Coriolis force in comparison to viscous friction is measured by the Taylor number ,.
The effects of magnetic fields and rotation on the linear stability analysis are discussed in the next paragraphs. It should be noted that Ra , Q , and Ta are all extremely large numbers in the Sun, if they are computed from microscopic transport coefficients Section 2.
So, in principle, there is no reason why solar convection should be close to the instability threshold. However, theoretical studies of large-scale convection such as supergranulation commonly assume that viscous, thermal, and magnetic diffusion at such scales are determined by turbulent transport, not microscopic transport.
Making this strong mean-field assumption serves to legitimate using the standard toolkits of linear and weakly nonlinear analysis to understand the large-scale behaviour of solar convection.
Following its discovery in the s and further studies in the s, supergranulation was quickly considered to have a convective origin, very much like the solar granulation. Since then, many theoretical models relying on the basic phenomenology of thermal convection sketched in Section 5. 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. Simon and Weiss and Vickers , 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. Antia et al. 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.
Gierasch 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 , 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 , 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.
This makes this case quite interesting for the supergranulation problem, considering that intensity fluctuations at supergranulation scales are rather elusive see Section 4. 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.
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 was the first to suggest that this model might be relevant to supergranulation. The linear problem in the Boussinesq approximation was solved by Edwards Rincon and Rieutord 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. The discovery by Gizon et al.
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, 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.
As mentioned in Section 4. In the presence of a vertical rotation vector, overstable oscillatory convection is preferred to steady convection provided that Pr is small Chandrasekhar, 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 , suggested on the basis of a local Cartesian analysis that the drift of super-granulation 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. 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, A directly related issue is that of the influence of differential rotation on supergranulation.
Green and Kosovichev considered the possible role of the solar subsurface shear layer Schou et al. They found that convective modes in the nonsheared problem become travelling when a weak shear is added. Some previous work found that this behaviour is possible either at low Pr Kropp and Busse, or if some form of symmetry breaking is present in the equations Matthews and Cox, 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 also report that the derived phase speeds for their travelling pattern are significantly smaller than those inferred from observations by Gizon et al. 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. We recall that there is as yet no conclusive observational evidence for a latitudinal dependence of the scales of supergranulation see Section 4.
Global spherical models do not necessarily suffer from this problem, as they predict global modes with a well-defined phase velocity. Oscillatory magnetoconvection is also known to occur for non-vertical magnetic fields e. 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.
On this topic, Green and Kosovichev recently built on the work of Green and Kosovichev 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 and argue that the actual phase speed measured by Gizon et al.
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, At very low Prandtl numbers, Thual 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. 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. 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.
Besides thermal convection scenarios, a few other theoretical arguments have been put forward to explain the origin of the solar supergranulation. These ideas are all based on the possible collective effects of small-scale structures such as granules, which might lead to a large-scale instability injecting energy into the supergranulation range of scales.
The first work along this line of thought was published by Cloutman He proposed to explain the origin of supergranulation using the physical picture of rip currents on the beaches of oceans: the repeated breaking of waves on beaches induces currents rip currents flowing parallel to the coast line. On the Sun, he identified breakers with the rising flows of granules breaking into the stably stratified upper photosphere.
The rip current model provides an illustration of the suggestion of Rieutord et al. Asymptotic theory on simple prescribed vortical flows can be performed under the assumption of scale separation Dubrulle and Frisch, between the basic periodic flow and the large-scale instability mode. In such theories, the sign and amplitudes of the turbulent viscosities is found to be a function of the Reynolds number.
For instance, an asymptotic theory based on a large aspect ratio expansion was developed by Newell et al. In this problem, large-scale instabilities take on the form of a slow, long-wavelength modulation of convection roll patterns. Their evolution is governed by a phase diffusion equation with tensorial viscosity. In the case of negative effective parallel diffusion with respect to the rolls orientation , the Eckhaus instability sets in, while the zigzag instability is preferred in the case of negative effective perpendicular diffusion.
Another way of explaining the origin of supergranulation assumes that the pattern results from the collective interaction of plumes. The word plume usually refers to buoyantly driven rising or sinking flows. Plumes can be either laminar or turbulent, however the turbulent ones have by far received most of the attention because of their numerous applications see Turner, The first numerical simulations of compressible convection at high enough Reynolds numbers e.
These results prompted Rieutord and Zahn to study in some details the fate of these downdrafts. Unlike the downflows computed in early simulations, solar plumes are turbulent structures, which entrain the surrounding fluid see Figure 1. As Rieutord and Zahn pointed it out, the mutual entrainment and merging of these plumes naturally leads to an increase of the horizontal scale as one proceeds deeper.
For some parameters typical of the solar granulation individual velocities and radius of the fountains notably , he argued that the clustering scales of the flow after a long evolution of the system resembled that of mesogranulation and supergranulation.
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