5.3 Dynamical nonlinearity

5.3.1 Backreaction on large-scale flows

The dynamo-generated magnetic field will, in general, produce a Lorentz force that will tend to oppose the driving fluid motions. This is a basic physical effect that should be included in any dynamo model. It is not at all trivial to do so, however, since in a turbulent environment both the fluctuating and mean components of the magnetic field can affect both the large-scale flow components, as well as the small-scale turbulent flow providing the Reynolds stresses powering the large-scale flows. One can thus distinguish a number of (related) amplitude-limiting mechanisms:

The α-quenching formulae introduced in Section 4.2.1 is a particularly simple – some would say simplistic – way to model the backreaction of the magnetic field on the turbulent fluid motions producing the α-effect10. In the context of solar cycle models, one could also expect the Lorentz force to reduce the amplitude of differential rotation until the effective dynamo number falls back to its critical value, at which point the dynamo again saturates11. The third class of quenching mechanism listed above has not yet been investigated in detail, but numerical simulations of MHD turbulence indicate that the effects of the small-scale turbulent magnetic field on the α-effect can be profound (see Pouquet et al., 1976; Durney et al., 1993; Brandenburg, 2009Jump To The Next Citation Point; Cattaneo and Hughes, 2009).

Introducing magnetic backreaction on differential rotation is a tricky business, because one must then also, in principle, provide a model for the Reynolds stresses powering the large-scale flows in the solar convective envelope (see, e.g., Kitchatinov and Rüdiger, 1993), as well as a procedure for computing magnetic backreaction on these. This rapidly leads into the unyielding realm of MHD turbulence, although algebraic “Λ-quenching” formulae akin to α-quenching have been proposed based on specific turbulence models (see, e.g., Kitchatinov et al., 1994). Alternately, one can add an ad hoc source term to the right hand side of Equation (2View Equation), designed in such a way that in the absence of the magnetic field, the desired solar-like large-scale flow is obtained. As a variation on this theme, one can simply divide the large-scale flow into two components, the first (U) corresponding to some prescribed, steady profile, and the second (′ U) to a time-dependent flow field driven by the Lorentz force (see, e.g., Tobias, 1997Jump To The Next Citation Point; Moss and Brooke, 2000Jump To The Next Citation Point; Thelen, 2000b):

′ u = U (x) + U (x,t,⟨B ⟩), (36 )
with the (non-dimensional) governing equation for U ′ including only the Lorentz force and a viscous dissipation term on its right hand side. If u amounts only to differential rotation, then U ′ must obey a (nondimensional) differential equation of the form
′ ∂U--= -Λ--(∇ × ⟨B ⟩) × ⟨B ⟩ + P ∇2U, (37 ) ∂t 4πρ m
where time has been scaled according to the magnetic diffusion time 2 τ = R ⊙∕ ηT as before. Two dimensionless parameters appear in Equation (37View Equation). The first (Λ) is a numerical parameter measuring the influence of the Lorentz force, and which can be set to unity without loss of generality (cf. Tobias, 1997Jump To The Next Citation Point; Phillips et al., 2002Jump To The Next Citation Point). The second, Pm = ν∕η, is the magnetic Prandtl number. It measures the relative importance of viscous and Ohmic dissipation. When Pm ≪ 1, large velocity amplitudes in U ′ can be produced by the dynamo-generated mean magnetic field. This effectively introduces an additional, long timescale in the model, associated with the evolution of the magnetically-driven flow; the smaller Pm, the longer that timescale (cf. Figures 4 and 10 in Brooke et al., 1998Jump To The Next Citation Point).

The majority of studies published thus far and using this approach have only considered the nonlinear magnetic backreaction on differential rotation. This has been shown to lead to a variety of behaviors, including amplitude and parity modulation, periodic or aperiodic, as well as intermittency (more on the latter in Section 5.6).

Figure 23View Image shows two butterfly diagrams produced by the nonlinear mean-field interface model of Tobias 1997Jump To The Next Citation Point (see also Beer et al., 1998Jump To The Next Citation Point; Bushby, 2006Jump To The Next Citation Point). The model is defined on a Cartesian slab with a reference differential rotation varying only with depth, and includes backreaction on the differential rotation according to the procedure described above. The model exhibits strong, quasi-periodic modulation of the basic cycle, leading to epochs of strongly reduced amplitude, with the modulation period controlled by the magnetic Prandtl number. Note how the dynamo can emerge from such epochs with strong hemispheric asymmetries (top panel), or with a different parity (bottom panel).

View Image

Figure 23: Amplitude and parity modulation in a 1D slab dynamo model including magnetic backreaction on the differential rotation. These are the usual time-latitude diagrams for the toroidal magnetic field, now covering both solar hemispheres, and exemplify the two basic types of modulation arising in nonlinear dynamo models with backreaction on the differential rotation (see text; figure kindly provided by S.M. Tobias).

It is not clear, at this writing, to what degree these behaviors are truly generic, as opposed to model-dependent. The analysis of Knobloch et al. (1998) suggests that generic behaviors do exist. On the other hand, a number of counterexamples have been published, showing that even in a qualitative sense, the nonlinear behavior can be strongly dependent on what one would have hoped to be minor modelling details (see, e.g., Moss and Brooke, 2000; Phillips et al., 2002).

The differential rotation can also be suppressed indirectly by magnetic backreaction on the small-scale turbulent flows that produce the Reynolds stresses driving the large-scale mean flow. Inclusion of this so-called “Λ-quenching” in mean-field dynamo models, alone or in conjunction with other amplitude-limiting nonlinearities, has also been shown to lead to a variety of periodic and aperiodic amplitude modulations, provided the magnetic Prandtl number is small (see Küker et al., 1999Jump To The Next Citation Point; Pipin, 1999; Rempel, 2006bJump To The Next Citation Point). This type of models stand or fall with the turbulence model used to compute the various mean-field coefficients, and it is not yet clear which aspects of the results are truly generic to Λ-quenching. Gizon and Rempel (2008) do show that information is present in subsurface measurements of the time-varying component of large-scale flows, which can be used to constrain the Λ-effect and its cycle-related variations.

To date, dynamical backreaction on large-scale flows has only been studied in detail in the context of dynamo models based on mean-field electrodynamics. Equivalent studies must be carried out in the other classes of solar cycle models discussed in Section 4. In particular, it is essential to model the effect of the Lorentz force on meridional circulation in models based on the Babcock–Leighton mechanism and/or hydrodynamical instabilities in the tachocline, since in these models the circulation is the primary determinant of the cycle period and enforces equatorward propagation in the butterfly diagram.

5.3.2 Dynamical α-quenching

A number of authors have attempted to bypass the shortcomings of α-quenching by introducing into dynamo models an additional, physically-inspired partial differential equation for the α-coefficient itself (e.g., Kleeorin et al., 1995; Blackman and Brandenburg, 2002Jump To The Next Citation Point, and references therein). The basic physical idea is that magnetic helicity must be conserved in the high-Rm regime, so that production of helicity in the mean field implies a corresponding production of helicity of opposite sign at the scales of the fluctuating components of the flow and field, which ends up acting in such a way as to reduce the α-effect. Most investigations published to date have made used of severely truncated models, and/or models in one spatial dimensions (see, e.g., Weiss et al., 1984; Schmalz and Stix, 1991; Jennings and Weiss, 1991; Roald and Thomas, 1997; Covas et al., 1997; Blackman and Brandenburg, 2002), so that the model results can only be compared to solar data in some general qualitative sense. Rich dynamical behavior definitely arises in such models, including multiperiodicity, amplitude modulation, and chaos, and some of these behaviors do carry over to into a two-dimensional spherical axisymmetric mean-field dynamo model (see Covas et al., 1998).

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