8.2 Nonlinear MHD at large scales

Understanding the dynamical interactions of network and internetwork magnetic fields with large-scale flows is a difficult task, as the physical origin of these fields is itself rather uncertain. They may result from the emergence of magnetic fields generated far into the SCZ, or from small-scale turbulent dynamo action in the surface layers (Durney et al., 1993Petrovay and Szakaly, 1993Cattaneo, 1999Jump To The Next Citation PointVögler and Schüssler, 2007Jump To The Next Citation PointDanilovic et al., 2010Pietarila Graham et al., 2010), or from some reprocessing of the field of decaying active regions (Spruit et al., 1987), or from a mixture of all these processes.

8.2.1 Comparing energy spectra

Clues on the possible relationship and dynamical interactions between the large-scale dynamics of solar surface flows and magnetic fields may be obtained by analysing the combined shapes of the kinetic E (k) and magnetic EM (k) power spectra

⟨ ⟩ ∫ ∞ ⟨ 2 ⟩ ∫ ∞ 1v2 = E (k)dk, -B--- = EM (k)dk, (6 ) 2 0 2 ρμ0 0
and that of other spectral quantities, such as the cross-spectrum between velocity and magnetic field or the magnetic tension spectrum. As mentioned in Section 4.6.3, our knowledge of EM (k) is rather crude. We know that it is a growing function of scale at least in the 1 – 10 Mm range where E (k) is decreasing. A tentative comparison between available spectra obtained by various authors indicates that the spectral energy density of the vertical magnetic field13 is an order of magnitude smaller than the spectral energy density of horizontal motions at 10 Mm and is comparable to that of vertical motions at the same scale. This suggests that E(k ) and E (k) M cross at some large scale (as defined in k-space) around which magnetic effects become comparable to hydrodynamic effects (buoyancy, pressure, etc.), implying that the large-scale distribution of magnetic energy is not simply slaved to that of kinetic energy but affects it in some way.

8.2.2 Breaking the large-scale similarity of solar surface flows

We may subsequently wonder what should be the visible physical consequences of the large-scale dynamical interactions between the flow and the magnetic field and what the spectral crossover scale is. In the absence of any available major observational or theoretical constraint, we shall discuss the simple and possibly naive idea that the spectral crossover scale in the photospheric layers of the quiet Sun lies somewhere in the network-supergranulation range, implying that supergranulation would be a dynamical MHD scale. This suggestion is illustrated in Figure 15View Image, which depicts a possible spectral-space distribution of magnetic and kinetic energy in the quiet Sun, from the largest scales of interest in the context of this paper to the smallest dissipative scale (see legend for detailed warnings regarding the interpretation of the figure). Such a configuration breaks the self-similarity of the large-scale tail of the velocity power spectrum, as the crossover between the magnetic and kinetic power spectra now represents a special scale in terms of energetics. This represents an important change compared to the purely hydrodynamic view presented in Section 8.1.1.

View Image

Figure 15: A tentative log-log spectral-space description of nonlinear MHD turbulence in the quiet photosphere. The kinetic energy spectrum is represented by a full red line and the magnetic energy spectrum by a dashed blue line (k = 2π∕L). The thin dotted red line is representative of the results of state-of-the-art hydrodynamic simulations of supergranulation-scale convection (Stein et al., 2009a). The ordering of the small-scale cutoffs results from Pm ≪ 1 (see Section 2.2). A rather flat spectral slope for the magnetic spectrum has been represented in the range of scales smaller than 10 Mm in accordance with the data of Lee et al. (1997); Abramenko et al. (2001), and Harvey et al. (2007). The shape of the small-scale part of the spectrum below 0.5 – 1 Mm and of the large-scale part of the spectrum beyond 10 Mm are very speculative (Section 4.6.3), as we do not know what kind of MHD processes are at work in these ranges of scales. As indicated in the text, the field geometry and most energetic scales at subgranulation scales are still controversial issues (e.g., López Ariste et al., 2010) and the field production mechanism itself is a matter of debate (Vögler and Schüssler, 2007Jump To The Next Citation Point). Finally, the relative amplitude of the magnetic and kinetic energy spectra is somewhat arbitrary but has been calibrated so as to comply with the argument developed in the text.

Without yet going into the details of dynamical magnetic feedback, note that the previous suggestion is simply that MHD turbulence in extended domains (in comparison to the typical injection scale of the turbulence) exhibits dynamical magnetic effects specifically enabled by the large extent of the domain. These nonlinearities very likely add up to more familiar MHD nonlinearities (quenching of specific flow scales, density evacuation) affecting scales in between the injection and dissipation scales of the turbulence (see, for instance, Schekochihin et al., 2004Jump To The Next Citation Point for a detailed account of saturation in turbulent dynamo simulations at large to moderate P m). In this context, supergranulation would probably be better interpreted as a by-product of the nonlinear saturation of MHD turbulence in the quiet Sun, not as its main cause.

8.2.3 Dynamical magnetic feedback: a tricky question

The main theoretical challenges to enforce the credibility of this scenario are to understand physically how the magnetic field feeds back on the large-scale flow and how the magnetic and kinetic power spectra form consistently. These two questions are currently almost completely open.

The most intuitive feedback mechanism that can be thought of in the light of our current knowledge of the dynamics of the quiet photosphere is that a large-scale distribution of strong magnetic flux tubes emerging from smaller-scale dynamics (like for instance in the n-body model of Crouch et al., 2007Jump To The Next Citation Point) collectively reinforces the flow at supergranulation scales, very much like strong sunspot fields support circulations towards the umbra in the far field (see Section 4.6.4, Figure 8View Image and Wang, 1988 for a similar suggestion). This picture qualitatively complies with the remark of Ustyugov (2009Jump To The Next Citation Point) that strong magnetic flux concentrations seem to play an important role in the scale-selection process in simulations of network formation. A very interesting physical and mathematical argument along this line was made by Longcope et al. (2003), whose calculations suggest that the dynamical feedback of a distribution of magnetic fibrils embedded into the solar plasma physically translates into a large-scale viscoelasticity of the plasma. We note that a central question in this problem is to determine whether one should expect a depletion or an increase in the kinetic energy at the supergranulation scale, as a result of the magnetic feedback.

Yet another possible magnetic feedback mechanism is through the interactions between magnetic fields and radiation14. Observations, theory, and simulations all suggest that magnetic concentrations tend to depress the opacity surfaces of the photosphere, which in turn is thought to channel radiation outwards (Spruit, 1976Vögler, 2005). Strong magnetic concentrations at network scales may thereby alter the convection process at supergranulation scale and consequently single this scale out in the energy spectrum.

Overall, we note that the difficulty to understand the physical nature of dynamical magnetic feedback in this problem is in no way an exception. Nonlinear MHD phenomena are notorious for defying simple handwaving arguments. For instance, current observations seem to rule out the possibility of a simple scale-by-scale equipartition of magnetic and kinetic energy down to the smallest observable scales. The most advanced simulations of the small-scale solar surface dynamo (Vögler and Schüssler, 2007Jump To The Next Citation Point) cannot answer the question of nonlinear dynamo saturation in a definite, asymptotic way as yet15. Finally, the scale-locality of dynamical interactions is not guaranteed in nonlinear MHD flows, including the small-scale dynamo (Schekochihin et al., 2004Jump To The Next Citation PointYousef et al., 2007), so the simple observation of an equipartition of energy at some scale is probably not sufficient to understand the physics of magnetic feedback fully consistently.

8.2.4 Comments on the equipartition argument

Equipartition has been discussed at length in the context of supergranulation (e.g., Parker, 1963Jump To The Next Citation PointSimon and Leighton, 1964Clark and Johnson, 1967Simon and Weiss, 1968Frazier, 1970Parker, 1974Frazier, 1976). The main concern with the argument has been that many flux concentrations in the network are known to exceed kG strengths and are therefore well above equipartition with the supergranulation flow field. Indeed, using the typical value for the velocity field at supergranulation scales given in Section 4.2.3 and an order of magnitude estimate for the plasma density in the first 1 Mm below τ = 1, we see that for the kinetic and magnetic energy densities to be comparable in the supergranulation peak range, an rms magnetic field strength of 100 G is required:

( ) ( )2 Ekin = 45 -----ρ------- ---V------ J m −3, (7 ) 10−3 kg m −3 300 m s−1
( B )2 Emag = 40 ------ J m − 3. (8 ) 100 G
Hence, the magnetic energy density of strong network elements appears to be roughly 100 times larger than that of the supergranulation flow.

This result mostly suggests that supergranulation-scale motions cannot themselves generate these flux tubes. Partial evacuation of density and vigorous localized motions such as granulation-scale motions seem to be required to obtain superequipartition fields (Webb and Roberts, 1978Spruit, 1979Spruit and Zweibel, 1979Unno and Ando, 1979Proctor, 1983Hughes and Proctor, 1988Bushby et al., 2008). This does not imply, however, that the supergranulation and network scales are not selected by nonlinear magnetic feedback processes. Actually, the existence of localized magnetic concentrations exceeding equipartition with the supergranulation flow certainly hints that magnetic effects cannot be bypassed to explain the dynamics of supergranulation.

Note finally that the energetics of the field is not the only important parameter of the problem. The curvature of magnetic field lines (the variation of the field along itself) is equally important to understand their dynamical role. In this respect, strong but straight localized flux tubes may not be particularly effective at interacting with the flow in comparison to weaker but significantly more tangled fields. Understanding the effective large-scale magnetic response at the surface of the quiet Sun therefore very likely requires considering the integrated dynamical contribution of the whole multiscale distribution of surface fields instead of the simple magnetic pressure estimate of individual magnetic elements populating the magnetic network.

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