Astron. Astrophys. 327, 813-824 (1997)

3. The comparison of analytical and numerical solutions

With the aim to test the correctness of the numerical solutions we shall compare them with the analytical linear solutions of the Paper I. First, we shall apply the initial state of a small amplitude, which is given by the linear solution with a wavenumber k and an amplitude . Then we shall observe the temporal evolution of the numerical and analytical solutions which both start from the same initial state. We expect that before the nonlinear effects become important, these two solutions should evolve accordingly. We are going to compare the linear and the nonlinear amplitudes of displacements of the flux tube gas in the three spatial directions at a given moment . The linear solution will be represented by , and . The numerical nonlinear solution will be represented by the appropriate quantities , and , where , and are the Lagrangian displacements of an element of the flux tube with respect to its equilibrium position at the time . The maxima are taken over a full range of variability of the length parameter s. An exemplary result obtained with the angular velocity typical for our Galaxy , the Oort constant , the ratios of cosmic ray and magnetic pressures to the gas pressure , the aerodynamic drag coefficient and the wavelengths of the perturbation (representing the most unstable mode) is shown in Fig. 1.

Fig. 1. The comparison of the nonlinear numerical solution (full line) to the linear analytical solution (dotted line). The amplitudes of displacements in three directions vs. time are plotted. The basic parameters are: , , , and

Throughout the paper the initial height of the azimuthal flux tube is , which is comparable to a half of the typical vertical scaleheight (). The other parameters, which are not mentioned explicitly are the same as those assumed in Paper I for our Galaxy.

We note that the solutions are in a very good agreement especially concerning the coordinates x and y. However, a small discrepancy in the vertical direction is remarkable. The reason for this discrepancy is in the linear approximation of expression for the magnetic tension. Since in the linear approximation the curvature vector has only the linear component, then the coefficient contributes only in the zeroth order. This means that we take in linear approximation instead of , which is overestimated at the upward displaced parts of the flux tube, inflated due to the cosmic rays. Then the vertical component of the magnetic tension is diminished in the nonlinear regime with respect to the linear approximation. A similar argumentation applies also to the buoyancy term. This term is underestimated in the linear approximation. This explains why the top of the flux tube rises faster in numerical simulations than in the linear approximation. Thus, we can say that the numerical and analytical linear solutions are mutually consistent for relatively small vertical amplitudes . This allows us to expect that the basic conclusions of Paper I, essential for the dynamo theory remain invariable in the nonlinear regime for . We shall demonstrate in the next sections of this paper that this expectation is correct. Moreover, we shall be able to take into account some effects, which have not been tractable within the frame of the linear approximation.

As an example we shall examine the effect of aerodynamic drag force. HL'93 proposed that the aerodynamic drag force plays an important role for the dynamo transport coefficients, especially for the turbulent diffusivity. It was argued in that paper that the aerodynamic drag force slows down the vertical motions of fluxtubes and thereby diminishes the vertical diffusion of magnetic field. The discussion of the effect of the drag force has been omitted in Paper I since we focused our attention on the linear approximation and the resulting properties. Solving the flux tube equations by means of the nonlinear numerical simulations we are able to take the drag force into account. A corresponding solution with the parameter set as in Fig. 1 but with the aerodynamic drag force determined by the aerodynamic drag coefficient and the flux tube radius , is presented in Fig. 2.

Fig. 2. The comparison of the nonlinear numerical solution (full line) to the linear analytical solution (dotted line). The amplitudes of displacements in three dimensions vs. time are plotted. Parameters like in Fig. 1 but and .

It is apparent in Fig. 2 that with the given values of parameters, the aerodynamic drag force influences motion of the flux tube only moderately. One should be aware however that the assumed values of the drag coefficient and the flux tube radius are essentially model dependent. Moreover, the assumed physics underlying the aerodynamic drag effect may appear to be too simple for the case of galactic flux tubes. Nevertheless, one can notice that according to our expectations the drag force slows down the components of velocity perpendicular to the flux tube axis retaining the longitudinal velocity almost unchanged.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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