Astron. Astrophys. 329, 735-746 (1998)

3. Results

3.1. Spherically symmetric flow

Let us first consider spherically symmetric flow, i.e. . The electron density at the inner boundary, , where , satisfies Eq. (?? ). This means that the radiative losses at and above the top of the chromosphere, as well as the increase of the solar wind enthalpy flux through the transition region, are balanced by the inward heat flux. This procedure allows us to treat the formation of the corona and acceleration of the solar wind in a self consistent manner.

We want to study models where the corona is heated relatively close to the sun. We specify the mechanical energy flux density from the sun, , at the inner boundary. The mechanical energy flux, , in the flow tube is transferred to the corona as heat over a damping length :

This simple model of the coronal heating may not be realistic, but model studies with different damping lengths, , and different energy flux densities, , should help us understand how the location and amplitude of the coronal heating determine the structure of the corona and the solar wind proton flux and asymptotic flow speed. We choose a set of parameter values, , , , and , and vary them, one at a time, to see the effect of each parameter.

3.1.1. Variation of the damping length,

We choose the base values for all the parameters, but vary the damping length: The mechanical energy flux density from the sun is , the protons get of this energy flux while the electrons get the remaining 10%, and the proton heat conduction coefficient is given by Eq. (?? ) with . The damping length, , is varied from to . The results are plotted in Fig. 1.

Fig. 1. Results from the model with geometry. The proton heat conduction parameter is . The input mechanical energy flux density is , the protons get of this energy flux, and the damping length, , is varied. In the left column the flow speed, u, electron and proton temperatures, and , and electron density, n, are plotted versus heliocentric distance, r, for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the flow speed, , (solid line) and proton flux density, , (dashed line) at , the middle panel shows the electron density, , at the inner boundary, , and the bottom panel shows the fraction of the energy flux from the sun that is lost as heat conductive flux into the inner boundary (solid line) and as heat conductive flux at (dashed line), versus .

The left column of Fig. 1 shows the flow speed, u, the electron and proton temperature, and , and the electron density, n, as a function of heliocentric distance, r, for , , and . For the base values of the model parameters, i.e. for , we find that the proton temperature reaches a maximum of at . The electron temperature maximum is only . The solar wind has an asymptotic flow speed of about , and the solar wind proton flux density is at . The proton gas is assumed to be adiabatic in the outer solar wind, so the proton temperature is low, , at the orbit of Earth. The electron temperature at is .

We see that the proton temperature maximum increases with increasing damping length, from for to for , whereas the coronal electron temperature does not vary significantly when the damping length increases. A high coronal proton temperature is consistent with a high asymptotic flow speed, and we find that it is about for .

The density decrease, with heliocentric distance, in the inner corona is like the density fall-off in a static corona, and the density profile gradually approaches an -profile in the outer solar wind. The differences in the electron density profiles, displayed in the lower left panel, are caused by an increase of the asymptotic flow speed and a decrease of the solar wind proton flux with increasing damping length, .

The right column of Fig. 1 shows how some quantities vary with the damping length, . The upper right panel shows that the proton flux density at decreases from for to for . The flow speed at the orbit of Earth increases from for to for . The decreasing proton flux is consistent with an increase of the asymptotic flow speed because most of the energy flux deposited in the extended corona is lost as solar wind energy flux.

When the inner corona is heated, as it is when , the protons loose energy through inward heat conduction and to the electrons because of strong collisional coupling in the inner region. The lower right panel in Fig. 1 shows that in this case almost of the energy flux deposited in the corona is transported back to the sun as heat conductive flux. The outward heat conduction at is negligible. One-third of the inward heat flux goes into heating of the solar wind in the transition region, while two-third is radiated away. This determines the density at the inner boundary, , shown in the middle panel in the right column, and for it is . For an increasing damping length the inward heat conduction decreases rapidly. Since most of the inward heat conductive flux is lost as radiation, is approximately proportional to the inward heat flux. When the inward heat conductive flux is only of the input mechanical energy flux, and .

3.1.2. Variation of the energy input,

We have now studied how variations of the damping length, , affect the structure of the corona and the solar wind proton flux and flow speed. Let us see what influence the amplitude of the mechanical energy flux density, , has on the results. The other model parameters are set to their base values: , and . The results are presented in Fig. 2.

Fig. 2. Results from the model with geometry. The proton heat conduction parameter is . The mechanical energy flux' damping length is , of the energy flux goes into the protons, and the input mechanical energy flux density, , is varied. In the left column the flow speed, u, and the electron and proton temperatures, and , are plotted versus heliocentric distance, r, for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the flow speed, , (solid line) and proton flux density, , (dashed line) at , and the bottom panel shows the electron density, , at the inner boundary, , versus .

The left column in Fig. 2 shows the flow speed, u, (upper panel), and the electron and proton temperature, and , (lower panel) versus heliocentric distance, r, for , , and . We see only small variations in these quantities when the energy flux from the sun changes.

The upper right panel in Fig. 2 shows that the solar wind proton flux density at , , increases linearly from to when changes from to , and that the flow speed at , , increase from to in the same range of -values. The lower right panel shows that the electron density at the inner boundary, , also increases with , but the inward heat conductive flux increases somewhat slower than linearly with , and this trend is reflected in the increase of . For the increase of from to increases from to . Fig. 2 shows that the solar wind proton flux is propotional to the energy flux density, , whereas the variations in the temperature profiles and in the solar wind flow speed are relatively small.

3.1.3. Variation of the proton heat conduction, h

We know that the classical proton heat conductive flux is an overestimate of the proton heat flux in the outer corona and in the solar wind. In our model we allow for classical heat conduction in the inner corona, and make use of the parameter h to reduce the heat conduction coefficient with increasing heliocentric distance. Let us now study the effect of "cutting off" the heat conduction in the proton gas, at different heliocentric distances, by varying this parameter from to , i.e. we go from a model with classical proton heat conduction only in the very inner corona to a model with classical heat conduction out to several solar radii. The other parameters are set to their base values: , , and . The results are shown in Fig. 3.

Fig. 3. Results from the model with geometry. The input mechanical energy flux density is , the protons get of this energy flux, and the damping length is . The proton heat conduction parameter, h, is varied. In the left column the flow speed, u, and the electron and proton temperatures, and , are plotted versus heliocentric distance, r, for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the flow speed, , (solid line) and proton flux density, , (dashed line) at , and the bottom panel shows the electron density, , at the inner boundary, , versus h.

The left column shows the flow speed, u, (upper panel) and the electron and proton temperature, and , (lower panel) along the radial flow tube for , and . We see that the temperature profiles are similar in the very inner corona, but for the case with the proton temperature reaches a maximum of , whereas in the case the proton temperature reaches . In this case a significant fraction of the energy flux is deposited in the region where there is almost no proton heat conductive flux, so the heat flux from the corona and into the inner boundary, is reduced. Hence, the electron density at the inner boundary is reduced, and the solar wind proton flux is reduced. As the energy flux from the sun is fixed, , the asymptotic flow speed is increased, from for to for (see upper left panel). Notice that the increase of h from to is enough to reduce the asymptotic flow speed from to . A further increase of h has a relatively small effect on the asymptotic flow speed. The electron temperature profile is almost the same in all cases.

The right column shows the flow speed, , and the proton flux density, , at (upper panel) and the electron density, , at the inner boundary (lower panel). Again we see that there are significant changes for small values of h. With classical proton heat conduction only in the inner corona, and no proton heat flux in the outer corona, i.e. h small, the heat conductive flux into the transition region is small, and the electron density, , and the proton flux, , are low, while the asymptotic flow speed is high. For h increasing from 0.1 to , decreases from to while increases from to and increases from to .

3.1.4. Variation of the proton heating, y

Now we will vary the energy distribution between protons and electrons, which is determined by the parameter y. The other parameters are set to their base values: , and . The results are shown in Fig. 4.

Fig. 4. Results from the model with geometry. The mechanical energy flux density at the inner boundary is . The mechanical energy flux' damping length is , the proton heat conduction parameter , and the part of the energy flux absorbed by the protons, y, is varied. In the left column the flow speed, u, and the electron and proton temperatures, and , are plotted versus heliocentric distance, r, for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the flow speed, , (solid line) and proton flux density, , (dashed line) at , and the bottom panel shows the electron density, , at the inner boundary, , versus y.

The left column shows the flow speed, u, (upper panel) and the electron and proton temperatures, and , versus heliocentric distance, r, for , and . The figure shows that an increase of y leads to an increase of the maximum proton temperature and the asymptotic flow speed, and a decrease of the maximum electron temperature, but the flow speed profiles and temperature profiles in the corona, obtained for and , are not very different. With the only heating of the electrons is by collisions with protons. In this case the electrons become almost adiabatic at .

In the upper panel of the right column we show the solar wind flow speed, , and the proton flux density, , at the orbit of Earth, versus y, and in the lower panel the electron density, , at the inner boundary is plotted, also as a function of y. We see that both the flow speed and the proton flux density at increase when y increase from to . This is possible because the inward heat conductive flux from the corona decreases when y increases, and more of the energy flux is available for driving the solar wind. For the parameters used here the flow speed and proton flux do not change dramatically, even for the relatively large variations in y. If we had varied y in a model with a smaller h -value and a larger value of , we would have found a larger variation in the flow speed and the proton flux than in the model considered.

The lower panel of the right column shows the electron density, , at the inner boundary as a function of y. As the electrons easily conduct heat into the transition region, is largest when a large fraction of the mechanical energy flux from the sun is deposited in the electron fluid. For we have that , whereas for we find that is reduced to . But the high density for does not correspond to a large solar wind proton flux. The lower mean temperature in the inner corona makes the density scale height short, and the higher temperature in the case leads to a higher proton flux, in spite of the lower value of .

3.2. Rapidly expanding flow geometry

The results obtained for an geometry show that a small damping length, say , does not produce high speed solar wind. This result does not seem to be in agreement with the results found by McKenzie et al. 1995. They considered a two-fluid solar wind model, with proton heating and with no heat conduction in the proton gas, and they used a rapidly expanding flow geometry. Let us now study a model with their geometry and their coronal heating function.

The expansion of their radial flow tube is given by where

and , and their heating function is

In this model the expansion of the flow tube is approximately a factor 8 larger than the expansion of an flow tube. It should be pointed out that the characteristic length of variation, L, for the heating function is considerably shorter than a charateristic length of variation for the corresponding energy flux,

With the coronal heating function in Eq. (?? ), the mechanical energy flux, , in the flow tube is damped more slowly in the inner corona, and more rapidly in the outer corona than an energy flux with a constant damping length, . For , 0.30 and the energy flux from the sun, , is reduced by a factor e over a distance , 0.62 and respectively.

Notice that the energy flux density at the inner boundary in this model is

3.2.1. Variation of the damping length, L

Also in this model the inner boundary is taken at a level where the temperature is , and the electron density, , at the inner boundary is given by Eq. (?? ). The collisional coupling is strong at the inner boundary. In the rapidly expanding flow, considered here, the cross-section of the flow tube increases with a factor 8 compared to the cross-section of a flow tube in an flow geometry, so we increase the base value of the energy flux density at the inner boundary to . The base values for the other model parameters are , , and . Let us first vary the damping length L. The results are presented in Fig. 5.

Fig. 5. Results from the model with rapidly expanding geometry when the characteristic length, L, of the heating function, Q, is varied. The input mechanical energy flux density is , and of this energy flux is absorbed by the protons. The proton heat conduction parameter is . In the left column the radial profiles of the flow speed, u, and the electron and proton temperature, and , are shown for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the speed at the orbit of Earth, , (solid line) and proton flux density, , (dashed line), and the bottom panel shows the electron density, , at the inner boundary, at , versus L.

The left coulmn of Fig. 5 shows flow speed, u, (upper panel) and electron and proton temperature, and , (lower panel) versus heliocentric distance, r, for , , and . For the base value, , the coronal proton temperature reaches a maximum of . This is sufficient to accelerate the flow to an asymptotic flow speed of . For the smaller L -value, , the proton temperature is reduced and the solar wind reaches a lower speed, . For we find a maximum proton temperature of and an asymptotic flow speed of . The electron temperature in the outer solar wind changes when L is varied, but the changes of the coronal electron temperature are small. We see that even for the smallest L value the proton temperature reaches almost , and the solar wind is accelerated to more than at the orbit of Earth. So heating close to the sun produces a higher flow speed in a rapidly expanding flow geometry than in spherically symmetric outflow.

In the upper right panel we show the flow speed, , and proton flux, , at , versus the characteristic length, L, and in the lower right panel we show the variation of the coronal base electron density, . The tendencies are the same as when we varied in the spherically symmetric flow (cf. Fig. 1): Extended coronal heating leads to higher asymptotic flow speed, lower proton flux, and lower electron density at the inner boundary. More specifically gives , and , whereas gives , and .

For we obtain asymptotic flow speeds that are in fairly good agreement with observations of quasi-steady high speed solar wind (McComas et al. 1995; Phillips et al. 1995), and the high coronal proton temperatures, consistent with these high flow speeds, are in good agreement with the proton temperatures derived from the observations in large coronal holes (Kohl et al. 1996). Notice that the large flow speeds are obtained when a significant fraction of the energy flux from the sun is deposited in the region where the proton heat conductive flux is small. Let us also illustrate the effect of shifting the proton heating from the collision dominated region and into the collisionless region by holding L constant and varying the parameter h.

3.2.2. Variation of the proton heat conduction, h

The parameter h determines how far into the corona the classical expression can be used to describe the proton heat flux. For the proton heat flux is negligible. Let us now vary h. The other parameters have their base values: , , and . The results are shown in Fig. 6.

Fig. 6. Results from the model with rapidly expanding geometry. The mechanical energy flux density at the inner boundary is . The characteristic length, L, of the heating function, Q, is , of the energy flux goes into the protons, and the proton heat conduction parameter h is varied. In the left column the flow speed, u, and the electron and proton temperatures, and , are plotted versus heliocentric distance, r, for (dashed line), (solid line) and (dotted line). In the right column the top panel shows the flow speed, , (solid line) and proton flux density, , at , and the bottom panel shows the electron density, , at the inner boundary, , versus h.

The left column shows the flow speed, u, (upper panel) and proton and electron temperature, and , versus heliocentric distance, r, from the inner boundary and out to the orbit of Earth. We show results for , 0.3, and . For the base value, , we find an asymptotic flow speed of and a proton temperature maximum, at , of . A reduction of the region where proton heat conduction is important, by reducing h from to , leads to an enhanced coronal proton temperature, with a maximum of , and an enhanced asymptotic flow speed of . An extension of the region where heat conduction in the proton gas is important, by increasing h to , leads to a somewhat lower and broader proton temperature maximum in the corona, but the asymptotic flow speed is virtually unchanged.

In the right panel we show variations of some solar wind and corona parameters, as a function of h. The flow speed, , and the proton flux density, , at are shown in the upper right panel. In the lower right panel we show the electron density, , at the inner boundary. The asymptotic flow speed decreases and the solar wind proton flux increases with increasing values of h, but for the variations are relatively small. The heat conductive flux into the transition region is small in the case of small h -values, and the solar wind proton flux is small. However, in this rapidly expanding flow the electron density at the inner boundary is almost constant when h is varied from to (lower right panel). Most of the heat flux into the transition region goes into heating the flow. Thus, the solar wind proton flux increases with increasing h -values, but is almost unchanged.

These results show that we obtain more or less the same solar wind proton flux and asymptotic flow speed when the coronal protons are heated in a region where classical proton heat conduction can distribute the heat within the proton gas. When most of the heating takes place in a region where there is almost no proton heat conduction, and the coupling to the electrons is weak, the solar wind proton flux is reduced, the maximum coronal proton temperature is increased, and the asymptotic flow speed of the solar wind is also increased.

3.3. Constant density at the inner boundary

In most model studies of the solar wind the electron density is specified at an inner boundary, the coronal base, and it is not adjusted to the inward heat flux. Let us now consider such models to see how sensitive the solar wind flow speed and proton flux are to variations in the electron density at the inner boundary.

To make this as simple as possible, we set the proton heat flux equal to zero, , everywhere in the model. This assumption leads to higher proton temperatures and higher flow speeds than what we would get if a reasonable proton heat flux were included. The inner boundary is, as in our previous models, placed at , where , and we let the protons get of the energy flux from the sun. We consider both spherically symmetric flow, , where the energy flux density at the inner boundary is , and the rapidly expanding flow geometry with . The electron densities at the inner boundary is in the range to .

The left column in Fig. 7 shows results for the -model, the right column is for the rapidly expanding flow. The variation of flow speed, , (upper panels) and proton flux density, , (lower panels) at with increasing damping length, or L, are plotted for three values of the coronal base electron density, , , and . The results are qualitatively similar in these two models: For a given electron density in the inner corona, the asymptotic flow speed increases and the proton flux decreases with increasing values of or L. This is the same result as we found in the self-consistent models, where the electron density at the inner boundary was adjusted to the heat flux into the transition region (cf. Fig. 1 and Fig. 5); For a lower coronal base electron density the asymptotic flow speed is higher, and the solar wind proton flux is lower. The main difference between the two geometries is that in rapidly expanding flow one can obtain very large asymptotic flow speeds even for small damping lengths. For a coronal base density of we find for . ( corresponds to a damping length for the energy flux of .) For the same coronal base electron density and we find in the -model.

Fig. 7. Results from models where the electron density, , at the inner boundary, , is specified; it is not determined by the inward heat conductive flux. There is no proton heat conduction, and the protons get of the energy flux. The left column shows results from the model with geometry and a mechanical energy flux from the sun of . The right column shows results from the model with a more rapidly expanding geometry and . The flow speed, , (top panels) and proton flux density, , (bottom panels) at are plotted as functions of and L for (solid line), (dashed line), and (dotted line).

On the other hand, when increasing the electron density by a factor 10 to , only the large damping lengths give high speed solar wind. The small damping lengths and give flow speeds at of only and , respectively, but still the large damping lengths, and give speeds as high as and . The flow speeds above are well above what is observed in the quasi-steady wind, and it can be argued that these results are not relevant for the solar wind. However, the model study serves to illustrate how sensitive the speed of the solar wind may be to the choices of the electron density in the inner corona.

In the rapidly expanding flow the expansion rate is larger than the electron-proton collision rate in the inner corona when the density is low. Then the thermal coupling between the electrons and protons is very weak, and almost all the energy flux deposited in the proton gas is lost in the solar wind. The solar wind proton flux is small, due to the low density at the inner boundary, and the asymptotic flow speed is large. The large flow speed is consistent with a large proton temperature in the corona. When the length scale, L, is decreased, a larger fraction of the energy flux is deposited in the subsonic region of the flow. Thus, the solar wind proton flux increases and the asymptotic flow speed decreases. When the coronal base electron density is decreased, the solar wind proton flux is also decreased, and the asymptotic flow speed of the solar wind is increased. The results in Fig. 7 show that the coronal base electron density is very important for determining the fraction of the energy flux from the sun that is lost as kinetic and gravitational solar wind energy flux, in particular in rapidly expanding flow.

In spherically symmetric flow the expansion rate is reduced, and the electrons and protons are thermally coupled in the inner corona, also in the low density case, where . This coupling leads to a larger solar wind proton flux, a smaller coronal proton temperature, and a lower asymptotic flow speed of the solar wind.

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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