## 5. Nonlinear modelsOur adopted expressions for and
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), not to its past
history. Calculating the exact temporal evolution of the large-scale magnetic field, , 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, all of which vary with time in a complicated interrelated manner. In other words, the evolution of 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 (and sometimes also ) to depend on via a multiplicative quenching function, , 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 fixes
the level at which the growth of 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 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, , 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 and can be shown to decrease as an inverse power of . Suppose, for instance, that they decrease as , and decompose into a fixed (thermal + turbulent) gas pressure, , and a time-dependent magnetic pressure, . The dynamo parameters are then equal to their reference value (given by Fig. 1) times the quenching factor where is the reference, i.e., present-day, magnetic field strength, and both and are provided by Ferrière's (1998a) ISM model. Note that Eq. (15) is equivalent to Eq. (14) with and , renormalized at . 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 by a perfect conductor or vacuum boundary condition at kpc. We executed Run 1 (reference), Run 2
(), Run 7
(), 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
-quenching only and with both
- and
-quenching. The results with
-quenching only are summarized in
Table 3 and those with - and
-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
Let us first discuss the implications of -quenching. Initially, the large-scale magnetic field is everywhere very weak, so that the components of are enhanced with respect to their reference value by a factor (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 become gradually quenched according to Eq. (15), and the field amplification slows down especially when becomes comparable to . 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
Gyr 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- polarities of to the point that they may entirely vanish. The low- 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.
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 (initial value of the quenching function near the peak of ). The field strength at all times stays , 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 -quenching gradually reduces the amplitude of , 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 when ; 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).
The effects of a vertical dependence in
(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 We now turn to simulations with -
and -quenching (see Table 4).
The solutions of Run 1 and Run 8 remain monotonically
amplified, at a rate which is initially less than with
-quenching only (see Table 3).
Because the growth of
The inclusion of -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 . Oscillations prevent the
accelerated amplification observed in the steady solutions of
Run 1 and Run 8. As a result, both fields maintain
and their oscillatory pattern is
similar to that obtained without quenching or with
-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
- and
-quenching, namely, the
unrealistically high values reached by the former As for the solutions of Run 7, they are little affected by the inclusion of -quenching. Again, this is because their negative growth rate ensures that 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, , is ignored, and (2) their dependence on 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 and 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 ), and the nonlocal character of the SN-driven dynamo (any given point is affected by the properties of 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. © European Southern Observatory (ESO) 2000 Online publication: June 26, 2000 |