4.3 Interface dynamos

4.3.1 Strong α-quenching and the saturation problem

The α-quenching expression (23View Equation) used in the preceding section amounts to saying that dynamo action saturates once the mean, dynamo-generated field reaches an energy density comparable to that of the driving turbulent fluid motions, i.e., √ ---- Beq ∼ 4πρv, where v is the turbulent velocity amplitude. This appears eminently sensible, since from that point on a toroidal fieldline would have sufficient tension to resist deformation by cyclonic turbulence, and so could no longer feed the α-effect. At the base of the solar convective envelope, one finds Beq ∼ 1 kG, for 3 −1 v ∼ 10 cm s, according to standard mixing length theory of convection. However, various calculations and numerical simulations have indicated that long before the mean field ⟨B ⟩ reaches this strength, the helical turbulence reaches equipartition with the small-scale, turbulent component of the magnetic field (e.g., Cattaneo and Hughes, 1996Jump To The Next Citation Point, and references therein). Such calculations also indicate that the ratio between the small-scale and mean magnetic components should itself scale as 1∕2 Rm, where Rm = vℓ∕η is a magnetic Reynolds number based on the microscopic magnetic diffusivity. This then leads to the alternate quenching expression

--------α0--------- α → α(⟨B ⟩) = 1 + Rm (⟨B ⟩∕Beq)2, (31 )
known in the literature as strong α-quenching or catastrophic quenching. Since Rm ∼ 108 in the solar convection zone, this leads to quenching of the α-effect for very low amplitudes for the mean magnetic field, of order 10–1 G. Even though significant field amplification is likely in the formation of a toroidal flux rope from the dynamo-generated magnetic field, we are now a very long way from the 10 – 100 kG demanded by simulations of buoyantly rising flux ropes (see Fan, 2009).

A way out of this difficulty was proposed by Parker (1993), in the form of interface dynamos. The idea is beautifully simple: If the toroidal field quenches the α-effect, amplify and store the toroidal field away from where the α-effect is operating! Parker showed that in a situation where a radial shear and α-effect are segregated on either side of a discontinuity in magnetic diffusivity (taken to coincide with the core-envelope interface), the α Ω dynamo equations support solutions in the form of travelling surface waves localized on the discontinuity in diffusivity. The key aspect of Parker’s solution is that for supercritical dynamo waves, the ratio of peak toroidal field strength on either side of the discontinuity surface is found to scale with the diffusivity ratio as

( ) max-(B2)- η2- − 1∕2 max (B ) ∼ η , (32 ) 1 1
where the subscript “1” refers to the low-η region below the core-envelope interface, and “2” to the high-η region above. If one assumes that the envelope diffusivity η2 is of turbulent origin then η ∼ ℓv 2, so that the toroidal field strength ratio then scales as ∼ (vℓ∕η )1∕2 ≡ Rm1 ∕2 1. This is precisely the factor needed to bypass strong α-quenching (Charbonneau and MacGregor, 1996). Somewhat more realistic variations on Parker’s basic model were later elaborated (MacGregor and Charbonneau, 1997Jump To The Next Citation Point and Zhang et al., 2004), and, while differing in important details, nonetheless confirmed Parker’s overall picture.

Tobias (1996aJump To The Next Citation Point) discusses in detail a related Cartesian model bounded in both horizontal and vertical direction, but with constant magnetic diffusivity η throughout the domain. Like Parker’s original interface configuration, his model includes an α-effect residing in the upper half of the domain, with a purely radial shear in the bottom half. The introduction of diffusivity quenching then reduces the diffusivity in the shear region, “naturally” turning the model into a bona fide interface dynamo, supporting once again oscillatory solutions in the form of dynamo waves travelling in the “latitudinal” x-direction. This basic model was later generalized by various authors (Tobias, 1997Jump To The Next Citation Point; Phillips et al., 2002Jump To The Next Citation Point) to include the nonlinear backreaction of the dynamo-generated magnetic field on the differential rotation; further discussion of such nonlinear models is deferred to Section 5.3.1.

4.3.2 Representative results

The next obvious step is to construct an interface dynamo in spherical geometry, using a solar-like differential rotation profile. This was undertaken by Charbonneau and MacGregor (1997Jump To The Next Citation Point). Unfortunately, the numerical technique used to handle the discontinuous variation in η at the core-envelope interface turned out to be physically erroneous for the vector potential A describing the poloidal field7 (see Markiel and Thomas, 1999Jump To The Next Citation Point, for a discussion of this point), which led to spurious dynamo action in some parameter regimes. The matching problem is best avoided by using a continuous but rapidly varying diffusivity profile at the core-envelope interface, with the α-effect concentrated at the base of the envelope, and the radial shear immediately below, but without significant overlap between these two source regions (see Panel B of Figure 9View Image). Such numerical models can be constructed as a variation on the αΩ models considered earlier.

In spherical geometry, and especially in conjunction with a solar-like differential rotation profile, making a working interface dynamo model is markedly trickier than if only a radial shear is operating, as in the Cartesian models discussed earlier (see Charbonneau and MacGregor, 1997; Markiel and Thomas, 1999Jump To The Next Citation Point; Zhang et al., 2003aJump To The Next Citation Point). Panel A of Figure 9View Image shows a butterfly diagram for a numerical interface solution with C Ω = 2.5 × 105, C α = +10, and a core-to-envelope diffusivity contrast Δ η = 10 −2. The poleward propagating equatorial branch is precisely what one would expect from the combination of positive radial shear and positive α-effect according to the Parker–Yoshimura sign rule8. Here the α-effect is (artificially) concentrated towards the equator, by imposing a latitudinal dependency α ∼ sin(4𝜃 ) for π∕4 ≤ 𝜃 ≤ 3π∕4, and zero otherwise.

View Image

Figure 9: A representative interface dynamo model in spherical geometry. This solution has C Ω = 2.5 × 105, Cα = +10, and a core-to-envelope diffusivity contrast of 10–2. Panel A shows a sunspot butterfly diagram, and Panel B a series of radial cuts of the toroidal field at latitude 15°. The (normalized) radial profiles of magnetic diffusivity, α-effect, and radial shear are also shown, again at latitude 15°. The core-envelope interface is again at r∕R ⊙ = 0.7 (dotted line), where the magnetic diffusivity varies near-discontinuously. Panels C and D show the variations of the core-to-envelope peak toroidal field strength and dynamo period with the diffusivity contrast, for a sequence of otherwise identical dynamo solutions.

The model does achieve the kind of toroidal field amplification one would like to see in interface dynamos. This can be seen in Panel B of Figure 9View Image, which shows radial cuts of the toroidal field taken at latitude π∕8, and spanning half a cycle. Notice how the toroidal field peaks below the core-envelope interface (vertical dotted line), well below the α-effect region and near the peak in radial shear. Panel C of Figure 9View Image shows how the ratio of peak toroidal field below and above rc varies with the imposed diffusivity contrast Δ η. The dashed line is the dependency expected from Equation (32View Equation). For relatively low diffusivity contrast, − 1.5 ≤ log(Δ η) ≲ 0, both the toroidal field ratio and dynamo period increase as ∼ (Δη )− 1∕2. Below log(Δ η) ∼ − 1.5, the max (B )-ratio increases more slowly, and the cycle period falls, contrary to expectations for interface dynamos (see, e.g., MacGregor and Charbonneau, 1997). This is basically an electromagnetic skin-depth effect; the cycle period is such that the poloidal field cannot diffuse as deep as the peak in radial shear in the course of a half cycle. The dynamo then runs on a weaker shear, thus yielding a smaller field strength ratio and weaker overall cycle; on the energetics of interface dynamos (see Ossendrijver and Hoyng, 1997, also Steiner and Ferriz-Mas, 2005).

4.3.3 Critical assessment

So far the great success of interface dynamos remains their ability to evade α-quenching even in its “strong” formulation, and so produce equipartition or perhaps even super-equipartition mean toroidal magnetic fields immediately beneath the core-envelope interface. They represent the only variety of dynamo models formally based on mean-field electrodynamics that can achieve this without additional physical effects introduced into the model. All of the uncertainties regarding the calculations of the α-effect and magnetic diffusivity carry over from α Ω to interface models, with diffusivity quenching becoming a particularly sensitive issue in the latter class of models (see, e.g., Tobias, 1996a).

Interface dynamos suffer acutely from something that is sometimes termed “structural fragility”. Many gross aspects of the model’s dynamo behavior often end up depending sensitively on what one would normally hope to be minor details of the model’s formulation. For example, the interface solutions of Figure 9View Image are found to behave very differently if the α-effect region is displaced slightly upwards, or assumes other latitudinal dependencies. Moreover, as exemplified by the calculations of Mason et al. (2008), this sensitivity carries over to models in which the coupling between the two source regions is achieved by transport mechanisms other than diffusion. This sensitivity is exacerbated when a latitudinal shear is present in the differential rotation profile; compare, e.g., the behavior of the C > 0 α solutions discussed here to those discussed in Markiel and Thomas (1999). Often in such cases, a mid-latitude αΩ dynamo mode, powered by the latitudinal shear within the tachocline and envelope, interferes with and/or overpowers the interface mode (see also Dikpati et al., 2005Jump To The Next Citation Point).

Because of this structural sensitivity, interface dynamo solutions also end up being annoyingly sensitive to choice of time-step size, spatial resolution, and other purely numerical details. From a modelling point of view, interface dynamos lack robustness.

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