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Astron. Astrophys. 354, 321-327 (2000)

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4. Density calculations

The time averaged density of contaminant particles in the flow can be calculated analytically for the case in which the source of particles is constant and infinite in extent in the manner described by PD96. Density calculations with time varying or finite sources require the use of numerical methods. We describe each case separately below.

4.1. Infinite source

Since we are considering a different case of the magnetic field direction and flow velocity configuration with different expressions for the particles trajectories, we reproduce the derivation of PD96 with the corresponding modifications.

The continuity equation of the mass flow for this case is given by

[EQUATION]

where n is the density of pick-up ions and v is the average velocity at some point, Q is the source rate of creation of particles and L their loss rate. We will assume steady state conditions and that the particles enter our region of study from abroad; that is they are born at [FORMULA], and [FORMULA] at [FORMULA]. The continuity equation thus becomes:

[EQUATION]

[EQUATION]

where [FORMULA],[FORMULA] and [FORMULA] are the average of the x, y and z components of the velocity for all the particles that move through the point [FORMULA]. The average value of the velocity at point [FORMULA] is obtained by integrating the velocity of the particles that in oscillatory motion move through [FORMULA];

[EQUATION]

where [FORMULA] is the most distant position that produces particles that pass through [FORMULA]. This value can be calculated considering Eq. (22). The most distant position will occur at positions [FORMULA] located at the top of the cycloid of the particles trajectory, that is to say when [FORMULA]; this is:

[EQUATION]

On the other hand taking the time derivative of Eq. (21) leads to:

[EQUATION]

Combining Eqs. (22) and (31) we have:

[EQUATION]

With this expression Eq. (29) turns out to be:

[EQUATION]

Substituting Eq. (30) in (33)

[EQUATION]

Since, by assumption, the system properties do not change along the x-axis, then:

[EQUATION]

On the other hand due to the fact that the z component consists only of periodic motions we have

[EQUATION]

and therefore the continuity Eq. (28) can be integrated as

[EQUATION]

whereFistheaveragemassflowofpick-upionsdueto the source assumed constant. Since the source is also assumed infinite and uniform in the z direction we can apply this equation to the region inside and outside the velocity shear to obtain,

[EQUATION]

where [FORMULA] and [FORMULA] are the velocity and density of the flow in the region outside the velocity shear. The density profile can be written as a function of [FORMULA] by using [FORMULA] and Eq. (30),

[EQUATION]

which can be substituted in the Eq. (34) to give:

[EQUATION]

This equation implies that the assimilated particles flow with a velocity different from the containing flow as noted by PD96. On the other hand, substituting Eq. (39) in Eq. (38) yields,

[EQUATION]

Substituting [FORMULA] and [FORMULA] in Eq. (41), becomes:

[EQUATION]

where [FORMULA] is the width of the zone of the velocity shear. This equation maybe written in term of the system parameters,

[EQUATION]

Figs. 3 and 4 show the density profile obtained as a function of [FORMULA] and the angle [FORMULA] respectively.

[FIGURE] Fig. 3. Density profiles calculated from Eq. (42) with [FORMULA], [FORMULA], [FORMULA], for different values of [FORMULA].

[FIGURE] Fig. 4. Density profile calculated of Eq. (42). For this case we consider [FORMULA], [FORMULA],[FORMULA] for different values of [FORMULA] and [FORMULA]

Fig. 3 depicts the normalized density expected when [FORMULA], [FORMULA], and [FORMULA], [FORMULA] or [FORMULA] km. Larger particle densities are obtained when the height the of velocity shear is larger. This is due to the fact that at a given height ([FORMULA]) the average bulk velocity reduces as the width of the velocity shear zone decreases and therefore the particles remain closer to the region in which they are born. Fig. 4 indicates a similar behavior for cases in which [FORMULA] is constant and [FORMULA], [FORMULA] and [FORMULA].

For comparation we can consider the limiting case when the angle between the magnetic field and the direction of the flow is [FORMULA]; that is to say when [FORMULA]. For this particular case Eq. (43) is reduced to:

[EQUATION]

where [FORMULA] and [FORMULA]. This expression was presented by PD96.

4.2. Finite source

We will analyze here the trajectories of pick-up ions in a system in which the magnetic field is in the direction perpendicular to the plane of the flow. As it was described in Sect. 2.3, we find that the trajectory of the particles is cycloidal and therefore their velocity will be smaller near the apex, i.e. the particles will spend more time in that region. If we consider that the ion source is continuous in time then the longer transit time of the ions in certain regions will translate into a larger number of particles in those regions, as is illustrated in Fig. 5. We will refer, in particular, to cases in which the source extent is not limited in the directions x and z (the direction of the field and the velocity shear), but only in the direction of the flow where and it is confined to a small region of width [FORMULA] such that,:

[EQUATION]

where [FORMULA] is given by Eq. (23) and [FORMULA] it is the width of the of the velocity shear,

[EQUATION]

Due to the fact that the period of the trajectories does not depend on the velocity of the flow, the period is the same throughout the velocity shear. If we trace a line through the apexes for each period, it will indicate the zones where the particle density is larger. In cases in which there is no velocity shear, these lines will be in the z direction (perpendicular to the direction of the flow) as is indicated in Fig. 6.

[FIGURE] Fig. 5. Cycloidal trajectory of a particle calculated from Eqs. (24)-(25). The trajectory shows regions near the apexes where the particles become accumulated.

[FIGURE] Fig. 6. Trajectories for particles assimilated that move at different heights. The thick solid lines show the larger density zones across the cycloids. In this case there is no velocity shear.

In the presence of a velocity shear, the lines tracing the zones of maximum density will be tilted with respect to the direction of the flow. The inclination angle, measured from the y-axis will be:

[EQUATION]

and the length of the n-th line of maximum accumulation is given by:

[EQUATION]

as is shown in Fig. 7.

[FIGURE] Fig. 7. Trajectory of the particles assimilated by a flow with a velocity shear. The different paths are related to particles born at different heights. The straight thick, solid lines join the larger density zones through the various cycloids.

In order to exemplify some of the consequences of this result, we consider a system with the following characteristics [FORMULA], [FORMULA], [FORMULA]. In Fig. 8 we present the position of a collection of particles calculated from Eqs. (25) and (26), after a time sufficiently large for a particle that was born at half the height of the shear zone to cross the entire length of the plot. [FORMULA] particles are considered only for illustration pourpose, we are unable to compute the case of realistic densities requiring [FORMULA] particles. In this graph the formation of filamentary structures is easily observed.

[FIGURE] Fig. 8. Particles distribution for a source confined in the direction y; the width of the source is [FORMULA].

We consider a grid whose elements have the same dimensions in both directions, we choose a given value of y and we count how much particles there is in each element of the grid that corresponds to that value of y. This gives us an idea of as varies the density as function of z. In Fig. 9 is presented the density profile along the z direction, calculated in the way described previously, for the case presented in Fig. 8 with [FORMULA] and [FORMULA]. In this case one can observe that a finite size of the source, leads to an increase in the particle density in certain regions with respect to the corresponding values when the source is infinite.

[FIGURE] Fig. 9. Density profile as a function of z a [FORMULA]. The solid line correspond to the finite source case shown in Fig. 8, the dotted line is the analytic result for an infinite source.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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