4.3 Interface dynamos

4.3.1 Strong $a$ -quenching and the saturation problem

The $a$ -quenching expression (24 ) 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., $V~ ---- Beq ~ 4pr v$ , 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 $a$ -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, 1996 , 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 $Rm = vl/j$ is a magnetic Reynolds number based on the microscopic magnetic diffusivity. This then leads to the alternate quenching expression

known in the literature as strong $a$ -quenching or catastrophic quenching. Since $Rm ~ 108$ in the solar convection zone, this leads to quenching of the $a$ -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, 2004 ).

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 $a$ -effect, amplify and store the toroidal field away from where the $a$ -effect is operating! Parker showed that in a situation where a radial shear and $a$ -effect are segregated on either side of a discontinuity in magnetic diffusivity (taken to coincide with the core-envelope interface, see Panel A of Figure 9 ), the $a_O_$ 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

where the subscript “1” refers to the low- $j$ region below the core-envelope interface, and “2” to the high- $j$ region above. If one assumes that the envelope diffusivity $j 2$ is of turbulent origin then $j2 ~ lv$ , so that the toroidal field strength ratio then scales as $1/2 1/2 ~ (vl/j1) =_ Rm$ . This is precisely the factor needed to bypass strong $a$ -quenching (Charbonneau and MacGregor, 1996 ). Somewhat more realistic variations on Parker’s basic models were later elaborated (MacGregor and Charbonneau (1997

); Zhang et al. (2004 ), see also Panels B and C of Figure 9

), and, while differing in important details, nonetheless confirmed Parker’s overall picture.

Figure 9:

Three interface dynamo models in semi-infinite Cartesian geometry. In Parker’s original model, the $a$ -effect occupies the space $z > 0$ and the radial shear $z < 0$ , with the magnetic diffusivity varying discontinuously from (low) $j 1$ to (high) $j 2$ across the $z = 0$ surface. The MacGregor and Charbonneau model is similar, except that the shear and $a$ -effect are spatially localized as $d$ -functions, located a finite distance away from $z = 0$ in their respective halves of the domain. The Zhang et al. model moves one step closer to the Sun by truncating vertically the Parker model, and sandwiching the dynamo layers between a vacuum layer (top) and very low diffusivity layer (bottom); this bottom layer represents the radiative core, while the shear region is then associated with the tachocline. Each one of these models supports travelling waves solutions of the form $exp(ikx - wt)$ , vertically localized about the core-envelope interface ( $z = 0$ ).

Tobias (1996a

) discusses in detail a related Cartesian model bounded in both horizontal and vertical direction, but with constant magnetic diffusivity $j$ throughout the domain. Like Parker’s original interface configuration, his model includes an $a$ -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, 1997

; Phillips et al., 2002

) 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 (1997 ). Unfortunately, the numerical technique used to handle the discontinuous variation in $j$ at the core-envelope interface turned out to be physically erroneous for the vector potential $A$ describing the poloidal field⁸ (see Markiel and Thomas, 1999 , 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 $a$ -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 10 ). Such numerical models can be constructed as a variation on the $a_O_$ models considered earlier, and stand somewhere between Parker’s original picture (see Panel A of Figure 9 ) and the models with spatially localized $a$ -effect and shear (see Panels B and C of Figure 9 ). 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, 1999 ). Panel A of Figure 10 shows a butterfly diagram for a numerical interface solution with $5 C_O_ = 2.5 × 10$ , $Ca = +10$ , and a core-to-envelope diffusivity contrast $Dj = 10-2$ . A magnetic energy time series for this solution is plotted in Panel D of Figure 8 , together with a solution with a smaller diffusivity contrast $Dj = 0.1$ (see Panel C of Figure 8 ). The poleward propagating equatorial branch is precisely what one would expect from the combination of positive radial shear and positive $a$ -effect according to the Parker-Yoshimura sign rule⁹ . Here the $a$ -effect is (artificially) concentrated towards the equator, by imposing a latitudinal dependency $a ~ sin(4h)$ for $p/4 < h < 3p/4$ , and zero otherwise.

Figure 10:

A representative interface dynamo model in spherical geometry. This solution has $C_O_ = 2.5× 105$ , $Ca = +10$ , 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 $o 15$ . The (normalized) radial profiles of magnetic diffusivity, $a$ -effect, and radial shear are also shown, again at latitude $o 15$ . The core-envelope interface is again at $r/Ro . = 0.7$ (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. Corresponding animations are available in Resource 2.

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 10

, which shows radial cuts of the toroidal field taken at latitude $p/8$ , and spanning half a cycle. Notice how the toroidal field peaks below the core-envelope interface (vertical dotted line), well below the $a$ -effect region and near the peak in radial shear. Panel C of Figure 10

shows how the ratio of peak toroidal field below and above $rc$ varies with the imposed diffusivity contrast $Dj$ . The dashed line is the dependency expected from Equation (33

). For relatively low diffusivity contrast, $- 1.5 < log(Dj) <~ 0$ , both the toroidal field ratio and dynamo period increase as $~ (Dj)- 1/2$ . Below $log(Dj) ~ - 1.5$ , 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 (cf. Panels C and D of Figure 8

; on the energetics of interface dynamos; see also Ossendrijver and Hoyng, 1997 )

Zhang et al. (2003a ) have presented results for a fully three-dimensional $a2_O_$ interface dynamo model where, however, dynamo solutions remain largely axisymmetric when a strong shear is present in the tachocline. They use an $a$ -effect spanning the whole convective envelope radially, but concentrated latitudinally near the equator, a core-to-envelope magnetic diffusivity contrast $Dj = 10 -1$ , and the usual algebraic $a$ -quenching formula. Unfortunately, their differential rotation profile is non-solar. However, they do find that the dynamo solutions they obtain are robust with respect to small changes in the model parameters. The next obvious step here is to repeat the calculations with a solar-like differential rotation profile.

4.3.3 Critical assessment

So far the great success of interface dynamos remains their ability to evade $a$ -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 $a$ -effect and magnetic diffusivity carry over from $a_O_$ 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 10 are found to behave very differently if either

the $a$ -effect region is displaced upwards by a mere $0.05Ro .$ , or
the $a$ -effect is less concentrated towards the equator, for example via the $~ sin2 hcosh$ form used in Section 4.2, or
the tachocline thickness is increased by 50%, leading to somewhat greater overlap between the $a$ -effect and shear source regions.

Compare also the behavior of the $Ca > 0$ solutions discussed here to those discussed in Markiel and Thomas (1999 ). Once again the culprit is the latitudinal shear. Each of these minor variations on the same basic model has the effect that a parallel mid-latitude dynamo mode, powered by the latitudinal shear within the tachocline and envelope, interferes with and/or overpowers the interface mode. This interpretation is not inconsistent with the robustness claimed by Zhang et al. (2003a ), since these authors have chosen to omit the latitudinal shear throughout the convective envelope in their model. 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.