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Astron. Astrophys. 359, 759-765 (2000)

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4. Numerical simulations

In this section we present results of numerical solutions of Eqs. (2) and (3). These simulations are performed using the CLAWPACK code (LeVeque, 1997b). This code is a collection of Fortran routines for solving a hyperbolic system of conservation laws. The general structure of the code is described in detail in the user notes written by LeVeque (1997a). The applied method is a finite volume method on a uniform rectangular grid. In this method, Eqs. (2) and (3) are discretized on a set of points: [FORMULA], where j denotes the numerical cell and n corresponds to the time step. Additionally, a Riemann solver which decomposes data at cell edges into a set of waves and wave speeds is adopted. Problems with the source terms like [FORMULA] are generally solved using the Godunov splitting in which the solution of the homogeneous conservation law [FORMULA] is alternated with the solution of the ordinary differential equation [FORMULA] in each time step.

In the CLAWPACK code two ghost cells are generated on each side of the computational domain which in the present approach is divided into uniform grid of [FORMULA] [FORMULA]. The boundaries of the simulation box are typically placed at [FORMULA] and [FORMULA] [FORMULA]. A periodic driver, which acts with its frequency [FORMULA], is set at the left boundary. In the present model the right boundaries are entirely open. So, any signal can easily cross them.

Initially, at [FORMULA], there are no signals in V and p while the flow [FORMULA] is set space-dependent. This flow has been chosen in such way that the [FORMULA] profiles are piecewise constants (Fig. 3). As a consequence of that the flow is piecewise linear. In the regions, where [FORMULA] ([FORMULA]), the sound waves are damped (amplified).

[FIGURE] Fig. 3. Spatial derivative of a random flow, [FORMULA], for a typical realization of a medium. The spatial coordinate x is in units of the correlation length [FORMULA] and the flow [FORMULA] is normalized by the sound speed [FORMULA].

4.1. Numerical results and discussion

Figs. 4 and 5 present the wave profiles in the case of a periodic driver with unit amplitude and frequency [FORMULA] (top panels) and [FORMULA] (bottom panels) for a random flow that is shown in Fig. 3. Fig. 4 corresponds to a time-signature of the signal which is collected at [FORMULA] [FORMULA]. As the amplitude of this signal grows in time in the case of [FORMULA] (top panel) the sound wave is unstable. In the case of [FORMULA] (bottom panel) the amplitude of the signal oscillates around [FORMULA] [FORMULA]. Fig. 5 displays the corresponding spatial profiles at the time [FORMULA] [FORMULA]. The effect of damping is seen well in the bottom panel; the wave amplitude decreases with the coordinate x. In the case of [FORMULA] (top panel) local amplification and damping are discernible.

[FIGURE] Fig. 4. Time signature of a signal which is collected at the point [FORMULA] [FORMULA] in the case of a random flow with its variance [FORMULA] [FORMULA]. A periodic driver, set at [FORMULA], acts with the frequency [FORMULA] (top panel) and [FORMULA] (bottom panel).

[FIGURE] Fig. 5. Spatial wave profile at [FORMULA] [FORMULA] in the case of a random flow with its variance [FORMULA] [FORMULA]. A periodic driver, set at [FORMULA], acts with the frequency [FORMULA] (top) and [FORMULA] (bottom).

The wave profiles of Figs. 4 and 5 can be analyzed spectrally to obtain the normalized frequency [FORMULA] and wavevector K, respectively. Figs. 6-8 display the results of this analysis. In particular, Fig. 6 illustrates the normalized frequency difference, [FORMULA] as a function of K for a number realizations of the random flow in the case of [FORMULA] (top panel) and [FORMULA] (bottom panel). As a result of random flows, the frequency difference for various realizations of the flow attain stochastic values; there are few realizations for which random flows speed up the sound wave but most realizations lead to a frequency decrease. There are waves of various frequencies and wavevectors being excited in different realizations. The (ensemble) averaged [FORMULA] is represented by the asterisk. This value is much higher than the value obtained analytically which is represented by the triangle. In the case of [FORMULA] the ensemble averaged data (asterisk) lies closer to the analytical data (triangle) than in the case of [FORMULA].

[FIGURE] Fig. 6. Normalized frequency difference [FORMULA] versus dimensionless wavevector K for a random flow with its variance [FORMULA] [FORMULA]. The data have been obtained by numerical integration of Eqs. (2) and (3) for [FORMULA] (top panel) and [FORMULA] (bottom panel) and a number of realizations of the medium (the crosses). The average of the complete set of data is denoted by the asterisk. The data obtained using Eq. (16) are represented by the triangles.

[FIGURE] Fig. 7. Normalized frequency difference [FORMULA] versus dimensionless variance, [FORMULA]. These data have been obtained by numerical integration of Eqs. (2) and (3) for the driving frequency [FORMULA] and one realization of the medium (the crosses). The data obtained using the Howe method (16) are represented by the triangles.

[FIGURE] Fig. 8. Normalized frequency difference [FORMULA] versus dimensionless wavevector K for a random flow with its variance [FORMULA] [FORMULA]. These data have been obtained by numerical integration of Eqs. (2) and (3) for several values of the driving frequency [FORMULA] and one realization of the medium (the crosses). The data obtained using Eq. (16) are represented by the triangles.

The difference between the analytical and numerical results is due to the approximation of a weak random flow applied in the method of Howe (1971). We expect that the results would be closer for a smaller value of [FORMULA] for which the assumption of a weak random flow is more appropriate. Indeed, Fig. 7 shows that for low [FORMULA] the analytical data (triangles) are closer to the numerical results (crosses) in the case of one realization of the flow.

Fig. 8 illustrates that the agreement between the numerical and analytical results depends also on the wavevector K. For low K this agreement is better. However, for [FORMULA] these two data depart in the way that numerical data fall down abruptly in comparison to the analytical data.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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