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Astron. Astrophys. 318, 963-969 (1997)

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3. The plume model

As discussed in the introduction, plumes appear to be associated with magnetic field concentrations at the coronal base. In this case the potential unperturbed field could be modelled by solving Eq. (6) with an appropriate boundary condition at [FORMULA]. However, observations show that plumes structure is mainly radial from [FORMULA] onwards (Fisher & Guhathakurta, 1995), hence the simple radial case will be assumed first as a starting approximation in order to investigate more easily the physical implications of the model. The analysis of the general case will be done in Sect. 3.2, where the resulting plume structure near the coronal base will be compared with observational data.

3.1. The radial case

Consider the zeroth order radial field


in a region around the plume axis [FORMULA] (where [FORMULA]). Through the definition of the Mach number


and using the continuity equation to eliminate [FORMULA], the radial derivative of the Bernoulli equation yields (the prime denotes a derivative in respect to r)


This is simply the famous Parker equation for radial, isothermal winds (Parker, 1958). It is well known that the corresponding phase plane [FORMULA] contains four different regions depending on the position relative to the sonic point [FORMULA]. The only physically relevant solution for the solar wind problem is the one crossing the sonic point with [FORMULA] (Parker or transonic solution) and eventually connecting via a shock to the interstellar medium. Recently the breeze solutions (i.e. those always subsonic in the phase diagram) have been shown to be unstable (Velli, 1994), thus confirming the necessity of the transonic solution for steady, isothermal outflows.

In the present model the temperature is a function of the field lines, hence the sonic radius [FORMULA] will be a function of the field lines too. This means that the flow becomes supersonic at different radii for different values of [FORMULA]. Imposing the transonic condition, the equation for M can be integrated again to give


where [FORMULA] is the magnitude of the magnetic field and [FORMULA]. Notice that, because of the transonic condition, the function [FORMULA] must be now derived from Eq. (10) and hence it is no longer free.

The density is related to the Mach number through the Bernoulli equation, which yields


where the relationship [FORMULA] has been assumed and where [FORMULA] gives the density profile at [FORMULA] in the static case (for solar values [FORMULA], hence the dynamic effects are actually negligible at the base of the corona). Note that for a constant temperature everywhere the Mach number does not depend upon [FORMULA] (from Eq. (10)) and therefore the density profile across the field lines remains the same at all heights.

In order to investigate the behaviour of the physical quantities in our model, the shapes of the two arbitrary functions [FORMULA] and [FORMULA] have to be chosen. Here the following functional forms will be assumed:



where the density (temperature) is considered to be non-dimensionalized against its value in the inter-plume region at the coronal hole base, so that [FORMULA] ([FORMULA]) gives the ratio between the densities (temperatures) on the plume axis and in the background coronal hole. Observed values of [FORMULA] are in the range 3-5. Expression (12) has been chosen following Ahmad & Withbroe (1977), where a gaussian-like density profile is shown to provide the best fit to the observed EUV intensities when the temperature is constant (the same analysis has been applied in X-rays by Ahmad & Webb, 1978). An example of the resulting 2-D density structure near the plume base is shown in Fig. 1, in which the radial decaying behaviour and the conservation of the [FORMULA] profile at all heights (for a constant temperature) are clearly visible.

[FIGURE] Fig. 1. The density [FORMULA], non-dimensionalized against its value at the base of the coronal hole, as a function of [FORMULA] and r. The parameters are [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Here [FORMULA] is defined as the characteristic angular half width at which the density drops by a factor [FORMULA] in respect to the corresponding axial value.

As pointed out earlier, the main effect of a variable temperature is that the sonic point becomes a function of [FORMULA], thus affecting also the radial density decay. Assuming a background coronal hole temperature of [FORMULA] the resulting sound speed is [FORMULA] and [FORMULA]. In Fig. 2 the sonic point position is shown as a function of [FORMULA] for different values of [FORMULA], whereas number density and velocity radial variations are given in Fig. 3 at the plume axis (PL) and for the coronal hole (CH).

[FIGURE] Fig. 2. The position of the sonic point as a function of [FORMULA] in units of [FORMULA] (again [FORMULA] and [FORMULA]). Three values of [FORMULA] are shown: for [FORMULA] ([FORMULA], solid line), for [FORMULA] ([FORMULA], dashed line) and for [FORMULA] ([FORMULA], dotted line). Note that the sonic point is closer to the Sun for a hot plume and further for a cold plume.
[FIGURE] Fig. 3. The number density [FORMULA] (in units of [FORMULA]) in logarithmic scale and the velocity V (in units of [FORMULA]). The solid lines refer to the plume axis (PL) whereas the dashed lines refer to the background coronal hole (CH). The parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

Note that the value of the temperature is a crucial parameter for the density and velocity behaviour at large distances. A plume to background temperature ratio as small as [FORMULA] implies a variation of [FORMULA] in the sonic point position and a density ratio which increases quite rapidly with r. Unfortunately, as Habbal et al. (1993) pointed out in an interesting review of previous observations, temperature measurements in coronal holes are affected by so many unknown parameters (temperature values can only be inferred using some models, where it is usually supposed to be constant across the plume) and uncertain quantities (like element abundances), that the accuracy in the measurements cannot be better than 20%. Therefore, it is obvious that there is no way to deduce our temperature profile in Eq. (13) from observations (there is not even an agreement whether a plume should be cooler or hotter than the surroundings), hence the comparison with observational data in the next sub-section will be done assuming [FORMULA]. On the contrary, the present model could be used to calculate the expected emission, for given values of the parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

The results shown so far for the radial case may be considered as simple applications of the hydrodynamic theory of isothermal winds, since the magnetic effects have not been taken into account yet. The last step left in our radial case analysis is to calculate the modifications to the zeroth order radial magnetic field, due to the unbalanced pressure gradient across the field lines. In fact, as gravity and inertial forces act radially, Eq. (7) becomes simply


where [FORMULA] is the operator defined in Eq. (6) and the pressure P has been defined in Eq. (5). Making use of the expressions for [FORMULA] and M, the equation for [FORMULA] can be written in the form


where the approximation [FORMULA] has been used. Note that the sonic singularity is removed from the right-hand side thanks to the choice of the arbitrary function [FORMULA] corresponding to the transonic solution.

Eq. (14) has been integrated numerically on a square grid [FORMULA], [FORMULA] with the condition [FORMULA] on all the boundaries. The solution automatically satisfies the symmetry condition [FORMULA] at [FORMULA]. The numerical technique implemented is a linear multigrid solver using a v-cycle (see, for example, Wesseling (1992) for general theory and Fiedler (1992), Longbottom et al. (1996) for specific solar applications of multigrid methods). The multigrid scheme used here results in the expected multigrid behaviour over classical iterative schemes, i.e., the number of iterations to achieve convergence to round off is independent of the number of grid points.

As expected, the modifications to the field lines are very small as long as the condition [FORMULA] holds, and this also defines the range within which our model retains its validity. In Fig. 4 the plasma beta, both on the plume axis and in the inter-plume region, is plotted together with the angular displacement of the corrected field lines, given by


[FIGURE] Fig. 4. The plasma beta is shown at the plume axis (PL) and in the coronal hole region (CH) for [FORMULA]. The field line displacement [FORMULA] is also shown as a function of r at the plume half width [FORMULA]. The values of the other parameters are the same as in Fig. 3.

It is interesting to notice that, apart from the line-tying effect at the coronal base (the field lines are supposed to be anchored in the sub-photospheric high-beta plasma), along each field line the behaviour of [FORMULA] follows exactly that of the plasma [FORMULA]. This may be seen from a simple dimensional analysis of the equation for [FORMULA], since [FORMULA] and [FORMULA], thus [FORMULA].

3.2. Flux concentration at the plume base

Although a background radial field is an excellent approximation at large distances, observations show evidence for a super-radial diverging field close to the plume base (see the introduction). As discussed briefly at the beginning of this section, the zeroth order potential field can be modelled by choosing a function [FORMULA] giving the non-radial contribution to [FORMULA] at the coronal base. A possible choice is


where b is a free parameter ([FORMULA] gives the purely radial case) and where the angular width [FORMULA] is chosen to be the same as in Eqs. (12) and (13) ([FORMULA]). Hence, the radial field component and the flux function at [FORMULA] are



giving a radial field outside the plume for [FORMULA].

Since [FORMULA] has a negative minimum at [FORMULA], where its value is [FORMULA], [FORMULA] can be negative if [FORMULA], thus giving a region of negative emerging flux around [FORMULA]. In Fig. 5 an example is given with a large value of b. Note that this situation resembles very closely the proposed scenario for plume formation, with close loops interacting with a stronger open flux concentration located at a supergranular junction. The required heating might be provided in the X-point region above the bipole, where a current sheet could form in response to photospheric motions of the bipole.

[FIGURE] Fig. 5. The field lines of the potential field calculated using Eq. (16) as lower boundary condition. The values of the parameters are [FORMULA] and [FORMULA]. Since [FORMULA], closed structures are present (a large value of b has been chosen in order to enhance the effect). The dashed line indicates the X-point region where a current sheet might form in response to photospheric motions of the bipole.

The main feature of our solution, characteristic of a potential analysis, is that all the modifications to the radial field occur only at low heights, on a scale corresponding to that defined by the plume width: at larger distances the contributions of the higher order multipoles of the photospheric field decay away and the field assumes a radial configuration. This is the same result found by Suess (1982) and the conclusion that can be drawn is that the observed super-radial expansion is indeed due to a magnetic effect, rather than a pressure or inertial one. However, Suess's model does not include any relationship between the density and the magnetic field, necessary to compare the model with the observations, while this comes out quite naturally and in a self-consistent way from our model. Notice that similar results are found in coronal hole models, where the super-radial expansion occurs out to much greater distances (2-3 [FORMULA], see, for example, Wang & Sheeley, 1990) than in plumes, but where the angular width of the structure is also larger by a corresponding factor.

The best values for the two parameters [FORMULA] and b, which determine the shape of the non-radial potential field through Eqs. (15) and (16), may be obtained by fitting the density structure derived from the theoretical model with some observational data. In order to achieve this, the Bernoulli equation has to be solved numerically for the transonic flow making use of the non-radial, potential background field. However, since the non-radial behaviour is confined to the coronal base, the position of the sonic points remains unaltered and the Mach number is still given by Eq. (10), where now [FORMULA] refers to the general potential solution. The modifications to the velocity are shown in Fig. 6 and these result in a slight enhancement of the flow due to the field concentration at the base.

[FIGURE] Fig. 6. The flow speed on the axis (solid line) and in the background coronal hole, where the field is radial (dashed line). The modifications due to the non-radial potential field appear only very close to the coronal base, whereas after [FORMULA] the velocity follows exactly the behaviour expected for a purely radial field. The parameters used here are [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

The density distribution may be still derived from Eq. (11) and the results are shown in Fig. 7, where a contour plot of the density is presented together with the unperturbed (dashed) and corrected (solid) field lines. The density contours are clearly distorted by the field line concentration through the function [FORMULA]. For a fully isothermal atmosphere ([FORMULA]) and neglecting the effects of the flow in the low corona, this function is proportional to the ratio of the density with its axial value at the same height r:


thus providing a means to compare density data with the magnetic field used in the model. In Fig. 7 the thicker solid line refers to a value [FORMULA] in the density ratio, defined to be the half angular width of the plume, whereas the diamonds are the observed values taken from the analysis by Ahmad & Withbroe (1977). A good fit appears to be obtained for the values [FORMULA] and [FORMULA].

[FIGURE] Fig. 7. The corrected field lines (solid), the unperturbed field lines (dashed) and a grey-scale density contour map (denser regions are darker). The thicker line corresponds to the theoretical plume width, defined as the angular distance at which the density drops by a factor [FORMULA] with respect to the axial value at the same height, whereas the diamonds are taken from the EUV observations by Ahmad & Withbroe (1977) (the data refers to the NP1 plume in their paper). The parameters used are [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

In spite of the impossibility of deriving with precision the shape of the field lines from the data (a straight line would appear to fit the data just as well!), it is important to remember that observations of plumes taken at larger distances yield a radial behaviour. For example, Fisher & Guhathakurta (1995) found that the density FWHM of polar plumes remains constant in angular width as a function of height extending from 1.16 to 5 [FORMULA]. This observational evidence clearly indicates that the super-radial expansion vanishes on a scale comparable with the width of the plume, thus supporting our potential model.

The modification to the zeroth order field has been worked out by solving directly Eq. (7) and deriving the function [FORMULA] from the knowledge of M and [FORMULA] ([FORMULA]). Notice that, even for not very small values of the plasma beta ([FORMULA] in Fig. 7), the corrections to the field lines remain extremely small, thus justifying our method of linearisation with respect to the magnetic field.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998