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Astron. Astrophys. 358, 125-143 (2000)

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5. Nonlinear models

Our adopted expressions for [FORMULA] and [FORMULA] are specific to the present-day Galaxy. The true dynamo parameters have undoubtedly evolved in the course of time, being, in all likelihood, greater at the beginning, when the large-scale magnetic field was weaker and the SN rate was presumably higher. There is also the possibility raised by Ko & Parker (1989) that our Galaxy passes through repeated episodes of active star formation during which the dynamo parameters are enhanced. Thus, what the numerical results of Sect. 4 teach us pertains to the Galactic magnetic field at the present epoch (in particular, what does its large-scale structure look like, does it oscillate or remain stationary, can it be sustained by SN explosions[FORMULA]), not to its past history. 1

Calculating the exact temporal evolution of the large-scale magnetic field, [FORMULA], requires to know how the dynamo parameters themselves vary with time. Clearly, the latter depend on basic ISM parameters like the SN rate, the Galactic rotation velocity, the ambient interstellar pressure (including a magnetic component), the interstellar mass density, the gravitational field, the magnetic field[FORMULA], all of which vary with time in a complicated interrelated manner. In other words, the evolution of [FORMULA] is coupled to that of the other ISM parameters.

Although the full problem has never been solved self-consistently, there have been attempts to go beyond the purely kinematic (linear) approach in which the dynamo parameters are prescribed. These attempts consist of allowing [FORMULA] (and sometimes also [FORMULA]) to depend on [FORMULA] via a multiplicative quenching function, [FORMULA], meant to represent the adverse reaction of the Lorentz force on the turbulent motions responsible for the dynamo process. The quenching function is generally assigned a convenient form which has the expected property of decreasing to zero as the magnetic field strength increases to infinity. A common choice is


(e.g., Jepps 1975), where [FORMULA] fixes the level at which the growth of B begins to saturate and n is typically set to 2 (Brandenburg et al. 1992; 1993). More realistic expressions were derived by Rüdiger & Kichatinov (1993), based on the second-order correlation approximation and on Kichatinov's (1987) quasi-isotropic, inhomogeneous, and anelastic turbulence model; the formal expressions obtained in a first step were then rendered usable in practice with the help of the mixing-length approximation. It is, incidentally, the resulting quenching functions that Elstner et al. (1995) used in their "wind dynamo" simulations (described in Sect. 4.1.2).

For the purposes of illustration and comparison with previous work, we repeated the most representative axisymmetric runs in the nonlinear regime, utilizing a modified version of Eq. (14) rather than the more complicated expressions of Rüdiger & Kichatinov (1993), which, although better-founded, do not apply to SN-driven turbulence. Our precise choice for the quenching function relies on the following argument: in the spirit of existing nonlinear calculations, we start from the premise that all the ISM parameters except [FORMULA] are fixed to their present-day value, presumed to be given by the ISM model of Ferrière (1998a). The primary way in which the magnetic field growth affects the dynamo parameters is through an enhancement of the ambient interstellar pressure, [FORMULA], which directly opposes the expansion of SNRs and SBs and raises the critical value of the driving interior pressure at which they merge with the background ISM (Ferrière 1998b). In the idealized case when the bubbles' radius evolves according to a power law of time, the various components of [FORMULA] and [FORMULA] can be shown to decrease as an inverse power of [FORMULA]. Suppose, for instance, that they decrease as [FORMULA], and decompose [FORMULA] into a fixed (thermal + turbulent) gas pressure, [FORMULA], and a time-dependent magnetic pressure, [FORMULA]. The dynamo parameters are then equal to their reference value (given by Fig. 1) times the quenching factor


where [FORMULA] is the reference, i.e., present-day, magnetic field strength, and both [FORMULA] and [FORMULA] are provided by Ferrière's (1998a) ISM model. Note that Eq. (15) is equivalent to Eq. (14) with [FORMULA] and [FORMULA], renormalized at [FORMULA]. This equation, therefore, implies that saturation starts when magnetic pressure comes into equipartition with the interstellar gas pressure.

Since even and odd solutions may no longer be treated separately in the presence of nonlinear interactions, we extended the computation domain to a full-thickness disk and replaced the symmetry condition at [FORMULA] by a perfect conductor or vacuum boundary condition at [FORMULA] kpc.

We executed Run 1 (reference), Run 2 ([FORMULA]), Run 7 ([FORMULA]), and Run 8 (vacuum boundary conditions), for three sets of initial conditions, respectively appropriate for a strictly even magnetic field (S0), a strictly odd magnetic field (A0), and a mixed-parity configuration with equal amplitudes of the even and odd components (mixed). Each run was successively carried out with [FORMULA]-quenching only and with both [FORMULA]- and [FORMULA]-quenching. The results with [FORMULA]-quenching only are summarized in Table 3 and those with [FORMULA]- and [FORMULA]-quenching are summarized in Table 4. Both tables include the initial and final parities, the type of temporal behavior, the growth rate of the magnetic field strength at the initial time, and, for the growing solutions, a rough estimate of the time at which the field growth saturates (starting from a seed field of 0.01 µG at maximum) as well as the maximum value of the three field components after complete saturation.


Table 3. Descriptive parameters of a few axisymmetric runs in the nonlinear regime with [FORMULA]-quenching only


Table 4. Descriptive parameters of a few axisymmetric runs in the nonlinear regime with [FORMULA]- and [FORMULA]-quenching

Let us first discuss the implications of [FORMULA]-quenching. Initially, the large-scale magnetic field is everywhere very weak, so that the components of [FORMULA] are enhanced with respect to their reference value by a factor [FORMULA] (see Eq. (15)), which, in the region of interest, varies approximately between 1.1 and 1.3. In consequence, the initial field generally grows faster than in the corresponding linear calculations and the early magnetic structure looks slightly different. In the truly growing cases, the components of [FORMULA] become gradually quenched according to Eq. (15), and the field amplification slows down especially when [FORMULA] becomes comparable to [FORMULA]. Because of this differential saturation effect, the magnetized region tends to be less peaked and to extend farther out than in linear models.

In the reference run (Run 1), the magnetic field is monotonically amplified up to complete saturation. The initial growth is exponential and proceeds at a rate [FORMULA] Gyr-1 (see Table 3), which, as expected, is somewhat higher than the linear rate ([FORMULA] Gyr-1; see Table 1 for the S0 mode and Table 2 for the A0 mode). The magnetic field saturates after [FORMULA] Gyr, when its maximum amplitude is [FORMULA]G, roughly twice the maximum field strength corresponding to the gas pressure [FORMULA]. The odd solution saturates later and attains higher amplitudes than the even solution, consistent with the results of Sect. 4 which suggest that, when [FORMULA] and [FORMULA] are simultaneously decreased below their reference value, the even mode reaches marginal stability before the odd mode does. Regarding the overall magnetic morphology, [FORMULA] remains the dominant field component. In the pure-parity cases, [FORMULA] possesses again two distinct polarities on each side of the midplane, which now extend radially out to [FORMULA] kpc and vertically up to the boundary. Although the odd field alone grows stronger than the even field alone, the mixed-parity field evolves toward an even configuration. This surprising result can be attributed to nonlinear interactions between the even and odd components and to the nonmonotonous variation of the even mode's linear growth rate with the global scaling factor of [FORMULA], [FORMULA], when [FORMULA] is small (see Sect. 4.1.3).

The adopted boundary conditions have little impact on the early evolution of the large-scale magnetic field, but they turn out to influence its late-time geometry. In particular, if the perfectly conducting external medium is replaced by a vacuum (Run 8), the magnetic field near the upper and lower boundaries diffuses freely out of the Galaxy, thereby weakening the high-[FORMULA] polarities of [FORMULA] to the point that they may entirely vanish. The low-[FORMULA] polarities, in contrast, are little affected, except for a slight vertical expansion and a slight contraction in the radial direction. For illustration, Fig. 14 displays the magnetic topology of the mixed-parity field at 40 Gyr.

[FIGURE] Fig. 14. Contour lines of the three cylindrical components and poloidal lines of force of the large-scale Galactic magnetic field in a given meridional plane, for Run 8 (vacuum boundary conditions) in the nonlinear regime with [FORMULA]-quenching only, at 40 Gyr. The initial field has even and odd components of equal strength.

If the escape velocity is set to zero (Run 2), the even and odd fields become oscillatory, and their initial growth rate lies in the range predicted by linear computations with [FORMULA] (initial value of the quenching function near the peak of [FORMULA]). The field strength at all times stays [FORMULA], so the effect of quenching remains relatively unimportant and the magnetic structure throughout a dynamo cycle resembles that of the linear solutions. The mixed-parity field at first grows oscillatorily, with its odd component becoming rapidly preponderant (see Fig. 15). As [FORMULA]-quenching gradually reduces the amplitude of [FORMULA], the growth rate of the odd component steadily decreases, while the growth rate of the even component increases, turns positive, and eventually overtakes the odd component's rate (in accordance with the dependence of the linear rates on [FORMULA] when [FORMULA]; see third paragraph of Sect. 4.1.3 for the S0 mode and fifth paragraph of Sect. 4.2 for the A0 mode). From then on, the field evolves toward an even state, and its subsequent amplification soon becomes monotonous (in accordance with the change in temporal behavior described in the third paragraph of Sect. 4.1.3).

[FIGURE] Fig. 15. Time dependence of the total energy contained in the large-scale Galactic magnetic field, E, time dependence of its global parity, [FORMULA], and time sequence of its azimuthal component, for Run 2 ([FORMULA]) in the nonlinear regime with [FORMULA]-quenching only. The initial field has even and odd components of equal strength. Contrary to the time sequences displayed in Figs. 6, 9, and 11, each snapshot has independent contour levels at [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA] its maximum absolute value.

The effects of a vertical dependence in [FORMULA] (Run 7) are similar to those found in linear models: the magnetic field becomes oscillatory and decays at a rate close to the linear rate, while the magnetic pattern switches to a juxtaposition of alternating polarities migrating toward large R, small [FORMULA], along lines of constant [FORMULA]. For the mixed-parity field, the odd component survives longer than the even component, consistent with the relative value of their individual decay rate.

We now turn to simulations with [FORMULA]- and [FORMULA]-quenching (see Table 4). The solutions of Run 1 and Run 8 remain monotonically amplified, at a rate which is initially less than with [FORMULA]-quenching only (see Table 3). Because the growth of B diminishes both [FORMULA] and [FORMULA], the dynamo number, [FORMULA] (Eq. (13)), actually increases with time, so that field growth proceeds at an accelerated rate until either D gets large enough to induce oscillations or the components of [FORMULA] drop to the arbitrarily fixed value of the background diffusivity (which stays unquenched). The subsequent field growth slows down and eventually saturates at [FORMULA]G for the oscillatory odd solution and at [FORMULA]G for the steady even and mixed-parity fields. The latter becomes again roughly even (see Fig. 16), and, somewhat paradoxically, its distribution appears smoother than in the absence of [FORMULA]-quenching (compare with Fig. 14). The reason is that [FORMULA]-quenching erases the peaks existing in the reference diffusivities (Fig. 1), with the consequence that magnetic diffusion near the midplane becomes comparatively more efficient and manages to smooth out the smaller-scale features apparent in Fig. 14. The result would look totally different if [FORMULA]-quenching were not limited by the background diffusivity; in that case, highly-structured magnetic pockets would develop with extremely strong fields ([FORMULA]G at 100 Gyr).

[FIGURE] Fig. 16. Same as Fig. 14 with both [FORMULA]- and [FORMULA]-quenching.

The inclusion of [FORMULA]-quenching in Run 2 preserves the oscillatory character of the pure-parity fields, but modifies their initial growth rate: the even field's rate just passes above zero while the odd field's rate is reduced by a factor [FORMULA]. Oscillations prevent the accelerated amplification observed in the steady solutions of Run 1 and Run 8. As a result, both fields maintain [FORMULA] and their oscillatory pattern is similar to that obtained without quenching or with [FORMULA]-quenching only. The mixed-parity field also remains oscillatory; moreover, it rapidly loses the memory of its even component and basically suffers the same fate as the purely odd field. It should be noted that Schultz et al. (1994) and Elstner et al. (1996) noticed the same fundamental difference in behavior between steady and oscillatory fields subject to both [FORMULA]- and [FORMULA]-quenching, namely, the unrealistically high values reached by the former versus the limited intensities at which the latter saturate.

As for the solutions of Run 7, they are little affected by the inclusion of [FORMULA]-quenching. Again, this is because their negative growth rate ensures that [FORMULA] at all times.

We emphasize once more that too much reality should not be ascribed to the numerical results presented in this section. The two main reasons are that (1) the dependence of the dynamo parameters on ISM parameters other than the large-scale magnetic field, [FORMULA], is ignored, and (2) their dependence on [FORMULA] is approximated by a simplistic function (Eq. (15)) which is unable to take the full complexity of the magnetic feedback into account. This complexity arises from the multitude of factors at play (the effect of which does not reduce to a simple power law), the anisotropic nature of the dynamo process (each component of [FORMULA] and [FORMULA] should have its own quenching function; see Ziegler 1996), the anisotropic nature of the Lorentz force (the various quenching functions should be allowed to depend on the direction of [FORMULA]), and the nonlocal character of the SN-driven dynamo (any given point is affected by the properties of [FORMULA] throughout the domain of influence of neighboring SNRs and SBs).

The nonlinear approach must be regarded as a useful complement to, rather than an improvement over, linear calculations. Both have their advantages: linear solutions are inherently more transparent to the underlying physical mechanisms, while their nonlinear counterparts show how the back-reaction of the Lorentz force limits field growth in otherwise favorable regions and how the magnetic configuration evolves, through mode interactions, toward a final self-consistent state.

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Online publication: June 26, 2000