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:

Lorentz force associated with the mean magnetic field directly affecting large-scale flow (sometimes called the “Malkus-Proctor effect”, after the groudbreaking numerical investigations of Malkus and Proctor (1975 )).
Large-scale magnetic field indirectly affecting large-scale flow via effects on small-scale turbulence and associated Reynolds stresses (sometimes called “ $/\$ -quenching”, see, e.g., Kitchatinov and Rüdiger, 1993 ).
Maxwell stresses of small-scale magnetic field directly or indirectly affecting large-scale flow.

The $a$ -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 $a$ -effect¹¹ . 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 saturates¹² . 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 $a$ -effect can be profound (see Pouquet et al., 1976 ; Durney et al., 1993 ). 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 “ $/\$ -quenching” formulae akin to $a$ -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 (2 ), 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, 1997 ; Moss and Brooke, 2000 ; Thelen, 2000b ):

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

$' @U-- = -/\--( \~/ × <B >) × <B > + P \~/ 2U (37) @t 4pr m$

where time has been scaled according to the magnetic diffusion time $2 t = R o. /jT$ as before. Two dimensionless parameters appear in Equation (37

). The first ( $/\$ ) is a numerical parameter measuring the influence of the Lorentz force, and which can be set to unity without loss of generality (cf. Tobias, 1997

; Phillips et al., 2002

). The second, $Pm = n/j$ , 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., 1998

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). Knobloch et al. (1998 ) have argued that amplitude modulation in such models can be divided into two main classes. Type-I modulation corresponds to a nonlinear interaction between modes of different parity, with the Lorenz force-mediated flow variations controlling the transition from one mode to another. Type-II modulation refers to an exchange of energy between a single dynamo mode (of some fixed parity) with the flow field. This leads to quasiperiodic modulation of the basic cycle, with the modulation period controlled by the magnetic Prandtl number. Both types of modulation can co-exists in a given dynamo model, leading to a rich overall dynamical behavior.

Figure 20 shows two butterfly diagrams produced by the nonlinear mean-field interface model of Tobias (1997 ) (see also Beer et al., 1998 ). 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. Note how the dynamo can emerge from such epochs with strong hemispheric asymmetries (top panel), or with a different parity (bottom panel).

Figure 20:

Amplitude and parity modulation in a 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 “ $/\$ -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., 1999 ; Pipin, 1999 ). This type of models stand or fall with the turbulence model they use to compute the various mean-field coefficients, and it is not yet clear which aspects of the results are truly generic to $/\$ -quenching.

To date, dynamical backreaction on large-scale flows has only been studied in 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 estimate 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. Even though meridional circulation in the convective envelope is a “weak” flow, in the sense of being much slower than differential rotation, it arises from a small imbalance between forces that are in themselves quite large; the importance of magnetically-mediated backreaction on meridional circulation is likely to be a complex affair.

5.3.2 Dynamical $a$ -quenching

A number of authors have attempted to bypass the shortcomings of $a$ -quenching by introducing into dynamo models an additional, physically-inspired partial differential equation for the $a$ -coefficient itself (e.g., Kleeorin et al., 1995 ; Blackman and Brandenburg, 2002 , 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 $a$ -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 ; Jenning 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.

Covas et al. (1998 ) have incorporated dynamical $a$ -quenching into a two-dimensional spherical axisymmetric mean-field dynamo model, and compared the resulting behavior to that produced by classical algebraic $a$ -quenching. These authors found, perhaps not surprisingly, both quantitative and qualitative differences between solutions computed using dynamical rather than algebraic $a$ -quenching, with the nonlinear behavior in the former case depending sensitively on model parameters, so it is not clear which class of models can be considered “more realistic”.