- Email: [email protected]
Acceleration of the solar wind by whistler waves from coronal holes
LUO
of a medium-sized flare. Admitedly, this estimate is very rough but it does show that it is by no means unthinkable for spotless regions to produce flares, even large flares. The important thing is that there must be a field of opposite polarities and this field must be disturbed sufficiently to have its intensity increased to several tens or hundreds G. The activation of quiescent dark filaments floating on the neutral line is a reflection of the disturbance in the local field and, as such, can be taken as a sign of impending flare eruptions in the spotless region. 2. Many researchers like Dodson /l/ and Svestka /3/ seem to regard spotless flares as a product in the late stage of evolution of a sunspot activity region (when the sunspots have declined to below 100 units or even vanished). In the 20 cases studied here, there are certainly such examples, but also we have 10 cases which are definitely independent of sunspots either in time (the rotation history) or in space. Also, as shown in the foregoing paragraph, the necessary condition for producing flares is not the existence of sunspots, rather, it is the existence of magnetic fields of opposite polarities of sufficient strength. The tortes actrng m me acrrve region can be seen from the variation in the angle 3. between a fibril and the main filament before the flare of 1980 January 8, 0359 UT. Fig. 3 shows that on Jan 6, this angle is about ho', and on Jan 8, it is only about 30". (Data lacking on Jan 7). Thus, a pressure gradually transformed into a shear. It was a sign that the active region passed from a state of energy storage to one of sudden energy release.
REFERENCES [ 1] [9]
Dodson, H. W., Hedeman E. R., Solar Phys., Dodson, II. W.,
dings and Workshop [
3]
13 (1970), 401417.
Hcdeman, E. R., Nlohler 0. C., Intenational Solar-Terrestrial Predictions ProceeProgram. Preprint No. 12.
Svesth, Z., Puliishingco. Solar Flares. 19%
Chin. Astron. Astrophys. 5 (1982) 192-198 102-209 -23 (1982)
Act.Astron.Sin.
ACCELERATION
Pergamon Press. Printed in Great Britain 0275-1062/82/030192-07$07.50/O
OF THE SOLAR WIND BY WHISTLER WAVES FROM CORONAL HOLES
ZHANG Zhen-da, H~ANG You-ran, Department of Astronomy, Observatory, LI Xiao-qing, Purple Mountain Observatory,
Nanjing Academia
Sinica Received
1981
February
23.
ABSTRACT We investigate the possibility of an additional acceleration of the high speed solar wind by whistler waves propagating outward from a coronal hole. We consider a stationary, spherically symmetric model and assume a radial wind flow as well as a radial magnetic field. The energy equation consists of (a) energy transfer of the electron beam which excites the whistler waves, and (b) energy transfer of the whistler waves described by conservation of wave action density. The momentum conservation equation includes the momentum transfer of two gases (a thermal gas and an electron beam). The variation of the temperature is described by a polytropic law. The variation of solar wind velocity with the radial distance is calculated for different values of energy density of the whistler waves. It is shown that the acceleration of high speed solar wind in
Solar Winds and Whistler
Waves
193
the coronal hole due to the whistler waves is very important. We have calculated that the solar wind velocity at the earth's orbit is about equal to 670 km/set (for wave energy density about lo-& erg cm-' at l.lR,). It is in approximate agreement with the observed values.
1.
INTRODUCTION
Coronal holes are regions in the corona with relatively low density and temperature and They are located within large units of the samemagnetic are large-scale open field regions. where the field diverges from the inner corona and extends towards the interpolarity, planetary space. Observations show that if a large coronal hole appears in the vicinity of the Sun's equator then solar wind will be observed at the Earth's orbit, with a flow speed reaching 600-800 km/s /l/, and the same magnetic polarity as the coronal hole, showing that the latter is the source for maintaining the solar wind. Much work has been done in the past on the mechanism of accelerating the solar wind. Parker's early modelofexpansionof the solar corona /z/was obviously unsuitable when explaining Alazarki et al. /3/ used Alfven waves to explain the solarwindand the high-speed solar wind. Jacques /4/ discussed the effect of waves (Alfven and obtained rather low velocities. acoustic) on the momentum and energy transfer of the wind from the quiet Sun and coronal We may say that the problem of the high-speed solar wind holes and on the critical point. is still one of the unsolved problems of solar physics. We believe that whistler waves in the coronal hole regions may play an important role in A hint in this direction is already in the observationa the acceleration of the solar wind. data of the wind at the distance of 1 AU, /5/. Since coronal holes occupy considerable portions of the Sun's surface (a single one may account for l-5% or even 10%) and the field atmosphere is in a state of continual there is basically open, and since theunderlying disturbance (e.g., tell-tale features like spicules occur in coronal hole regions), we can suppose that there is a continuous upward injection of electron beams across the bottom of a As an electron beam passes through the background gas, whistler waves are coronal hole. excited, which cause momentum transfer in the hot gas and heat the gas at the same time, thus providing an energy source for accelerating the high-velocity solar wind. Since the We shall consider a steady-state, spherically symmetric model of thesolarwind. coronal hole is taken to be the source of the solar wind, we shall regard, as in /4/, the wind to be radial, and the open field to be also radial (neglecting the effect of the Sun's In this case, because of the frozen-in property, field. rotation), that is, a one-dimensional the electron beam that produces whistler waves moves along the field with an average speed (v is the solar wind velocity) and the beam will not undergo diffusion, just as ve =v+v in Ref. 169 w h'ere electron beams (cosmic rays) pass through a spherical symmetric system and induce Alfven waves.
2. BASIC EQUATIONS Under the above assumption that the solar wind is a steady radial flow and the assumption of radial perturbations, we can derive a set of differential equations containing only one B be respectively the density, pressure, temperature spatial coordinate. Let P, P, T and and field intensity of the background hot gas, and let u, N and 71 be respectively the velocity, number density and pressure of the electron beam, and let w be the energy density of the whistler waves. The whistler waves excited by the electron beam produce 1. Energy Equation. transfer in the hot gas, which we think should obey the principle of conservation action density w/w,, 171,
%(E) + v . [(u,.+ u>E] = 0. In a steady-state,
we have
energy of the wave
(1)
ZHANG et al.
194
[(u,.+ u>21= 0.
v.
(2)
In spherical coordinates, this is d
x
[
P(V& + Y> ” -0
1-
0. (3)
where WO, vse and V@O are the frequency, group and phase velocities of whistlers in a static medium, la/,
Here, CB coHe=WC
G$_JG$
,mp*=
JG$F
J
are respectively, the gyro-frequencyof the electron, that of the hydrogen ion, and the electron plasma frequency. 'Iheother symbols have their usual meanings. The wave number k of the whistlers has the following bounds, /8/,
We take k = cwp,/c i.e. l/43 < a x 1, and
where vA= B/~/CP
is the velocity of Alfven waves, ~,-43av~; roe-d%,-
1.7X 10'Bd.
Substituting in (3) gives w r'(86avn + Y) 1.76X IO'&?aj const.,
and we obtain the energy density of the whistlers, W=
Baa r3(86avA + “> ’
(4)
where ?;(86av", + UO) 11,) = wrJ* BO In the one-dimensional radial case, the electron beam generates whistler waves which satisfy the Cerenkov condition, # - YP= B + v%=" + 43LxV".
(5)
The electron beam transfers part of its energy to the waves, and the change in its energy is V-[NLJ(ue+ u2/2 + n/N)], where Ue is internal energy. Because the mass density N is very small, we can neglect the uz/2 term and only two terms remain. NuU,+ z?zz=zzlN(Y - 1)-Q/N+ III- I?u[(u - 1)-'-t11.
Solar Winds and Whistler Waves
195
For a relativistic electron beam, y = 413, for a thermal beam, y = 513. Considering that our beam is between these two and closer to the latter, we shall take y = 1.5, hence(y-l)-'+1=3 and we have -v * [3I?Ul -v
(6)
- (VP” + U’P).
This is the equation of energy conservation, in which P is the wave pressure tensor. Similar to the Alfven wave pressure /4/, it was proved in Ref. /li that, for whistler waves propagating in the radial direction in the spherical symmetric case, we have the relation p* - w&, that is,
(7) where
Momentum Equation. For the two-component gas consisting of a thermal component and 2. an electron beam the latter having a very small number density, the momentum equation is, /6,9/,
where u = 0.619 constant. 3.
mH is
the mean mass per particle in the solar atmosphere and
KB
isBoltzmann's
Continuity Equation. This is V*(pV) = 0 and V*(W) = 0, that is
4. Polytropic Process. When there is no heating bv waves or any other external source. _ the energy equation of a thermal gas is simply the adiabatic equation: but when there are waves and electron beams present, it can generally be described by the polytropic process
00 OIZ T
(10) =
Top;-“p”-L
E
~p’l-‘,
in which the polytropic index u is used to describe approximately the thermal flux causing non-abiabatic expansion. Conservation of Magnetic Flux. This is div B = 0, that is,
5.
B
_
Bad
r= *
(11)
In (7), (9), (10) end (11) above, the integration constants bo, de, eo and Bore2 are to be determined by the conditions at the bottom of the coronal hole. Thus, altogether we have 7 unknowns, the solar wind velocity v, mass density p, temperature T, magnetic field intensity B, particle mass density N and pressure n in the electron beam and the whistler wave energy density W.
ZHANG et al.
196
Equation of the Solar Wind. 6. differential equation
v=
2gDp*-‘“+”
dr
-
L, -
dor-’ -
Hr--‘v-’
The above equations
(7)-(11) can be reduced
to one simple
L,
+
’
Dar-,oy-~o+l~
(12) where
H = GM
ada,
4&$-"-i-VLZ
L, = 3
i&/112
f
Zb,,r+
+ #
( 2
+ 3
)
I
_5 +--lym + ” ( 2
,
>
Once (12) is integrated, i.e., the variation of v with 1: is found, we can find from (4), (5), (9) and (10) the variations with r of w, u, p and T.
3.
NUMERICAL
1.
Determination
When both equations.
CALCULATION of the Critical
the numerator
Point and Numerical
and denominator
Solution
of Differential
Equations
of Eqn (12) are zero, we have a pair of non-linear
The ro and vo that satisfy (13) define the critical point. What we seek is the solution of Eqn(l.2) that passes the critical point/LO/. To find the critical point, we fix a set of boundary values PO, 30, TO, 710, WO, r~ and vO, and find first the critical point of the Parker This is then taken as a first approximation in a solution of (13) gas (wg = 0, La = Lr = 0). by method of descent to get the critical point (ro,vo) of the non-Parker gas. Then we solve (12) numerically starting from a neighboring point of the critical point and working towards the initial point rQ, to obtain the value vO' at rO. If vg' is sufficiently close to VO, then the above critical point is what we seek. 2.
Initial and Boundary
Values
We took r. = l.lR, as the starting distance of our integration, that is, the base of the coronal hole. At this distance, we took B. = 8G, in agreement with the range 6-12G given in We took p. = 1.36 x 10-'6g/cm3 and To = 1.7 x 10% (cf. 1.5 - 2.0 x 10'K given in Ref.lll. 1111 have shown that the flow velo/Ii). We took v0 = Skmfs, since ultraviolet observations city at the base of a coronal hole may be as high as Ibkm/s. Next, following f31, we took a fixed value o = 1.18, using a fixed value greatly facilitates the calculations, and this We found by calculparticular value agrees well with the observed temperature variations. ation that the solar wind velocity in fact did not change much for different values of LS. We pointed
out earlier
that 1/43
In our calculation,
we tooka=l/lO.
The
Solar Winds and Whistler Waves
197
maxinum value of w0 was taken to be 5 x lO-"dynitm', approximately corresponding to I& = m x 106cm-3, u* = 8 x 10scm/s. This Nc value is 2 orders of magnitude below the mean partic& mass density in a coronal hole. From the observations of spicules in coronal holes ill, it is entirely possible for the solar atmosphere below the base of the coronal hole to provide an electron beam kinetic energy of this order of magnitude. We took w. to be l/5 of R*, i.e., we assumed that l/5 of the kinetic energy of the high-energy electrons is converted to the energy of the whistler waves.
4.
RESULTS AND CALCULATIONS
We calculated the variation of v with I‘for w0 = 0, 10e6, lo-", lo-" erg/cm' (correspondingly li0= 0, 5 x 10v6, 5 x lCl-I,5 x lo-' erg/cm'). The results are shown in Fig. 1. m0 corresponds to the Parker gas. Crosses mark the critical values, which can be seen to move to smaller distances with increasing wave energy. This is in agreement with the results of /3/ and 141. Fig. 1 shows that the wave energy density exerts an important influence on the acceleration of the solar wind. In our single-wave model where the electron beam is the onlysource for whistler waves and there are no other sources exciting MHD waves, we need only require a whistler wave with an energy density of lo-"erg/cm3 capable of transferring its energy to the backgroung gas in order to accelerate the solar wind to a speed of 670 km/s at the distance of the Earth orbit and this last value is in agreement with space observations /II. According to the results in /12/, the mechanical energy flux passing from the chromosphere to the lower part of the coronal hole is about 8 x 105erg cm-as-f. Using F = w'(V i-V )=w V we estimate the mechanical energy dens1 -0y to 8: w' = 8 x 10sf109 = 8 x lo-' erg/cm'. Thus, our w0 value is far less than w', that is, the whistler wave energy density used in accelerating the hot gas need only be 1/g of the mechanical energy density. This shows that the Fig. 1 Variation of solar wind velocity above value of wg is reasonable. Also, from v with radial distance r. the linear theory of the interactionbetween Curves 1,2,3,4 correspond respectively the waves and the particles, we know that to w0 = 0, 10e6, 10w5, lo-* erg/cm'. the energy of the wave excited by the particles can reach l/3 the kinetic energy of the latter /13/. Our calculations show that the wind velocity increase tapers off with distance (note the distance scale of Fig. 1 is logarithmic) so that after 1 AU there will be no large increases. This does not contradict observations. TABLE 1
The Calculated Solar Wind Speed v(r) at various distances (w~=lO-loerg cmw3, vO= 8 km/s)
distancerth
10
20
40
50
100
150
160
170
180
190
200
352
390
523
606
619
632
643
655
665
30 _---__--
km/s
166
249
306
Figs. 2-5 show our calculated variations with f of the wave energy W, the electron beam velocity u, and the temperature T and density o of the solar wind. All calculations are for the case wg = 10-"erg/cm'. Fig.Z,calculated from Eqn(4), shows that the wave energy w changes drastically near the Sun, but the change becomes much slower when reaching the Earth's orbit. This shows that, near the Sun, the energy of the whistlers is rapidLy passed to the solar wind, and exerts an important effect on the latter's acceleration.
ZHANG et al.
198
Pie.
3
Variation of electron velocity u with r.
beam
Fig.4
Fig.2
f
Variation of wave energy with distance r.
Variation of solar wind density p with=.
4
W
z3 z2
Fig. 5 Variation temperature
of solar wind T with r.
Figs. 3 and 4 were calculated from Eqns (5) and (4). These show a particle numberdensity at Earth's orbit of about 25cm'-a and a particle flux density of about 16 x 10' particles cm-2s-1. These are somewhat higher than the observed values of 5-10 crne3 and 4.5 x lo8 particles cm-'s-l, but we can make the results match by adjusting PO in our calculations. The effect of adjustingp, on the calculated v(r) is slight, that is, any adjustment of p0 will not alter greatly the above results. Fig. 5 was calculated from Eqn(l0). the same order of magnitude as observed. very small.
It shows that at Earth's orbit, T 5 10'K. This is of The effect of initial value To on v(r) is also
REFERENCES III 121
Zirker, J. B.
(ed),
“Cmonal
[31
Alazraki, G., et al., Astron. dp.,
[41
Jacques, 8. A., Ap. J., 215 (19i7).
r51
Gary, S. P., et al., J:Grophys.
[61
Ipavieb. F. Ma, Ap. J., 196 (1975),
Parker, E. N., Ap. J.,
128
Holes and
(1958),
High Speed
13 (1971),
380.
94% 82 (19:7),
Bes.,
1087.
107.
[71
Bretherton. F. P.. Proc. Bog. Sm. A., 302 (1969).
[31
Kaplan,
[91
Li Xix-qing
[lOI
Parker, E. N., AP.
[III
Cushman, G. W., et al.. Ap. J., 207 (1976),
[I31
Romer, R. et &I., BP. J., 216 (1977), Tsytoviob. V. N., “Nonlinear Effect
[I31
wind strenms”,
664.
529.
S. A., nud Taytovieh, V. ZT., Plasma Astropkgaies, 1973. and Sap J.,
?du-tao. Solor PAys.. 75(1982), 132 (1960).
83.
821. L61-L62,
~57.
141. in Plasua”,
1970.
1977.
p. 13,
240.
of a medium-sized flare. Admitedly, this estimate is very rough but it does show that it is by no means unthinkable for spotless regions to produce flares, even large flares. The important thing is that there must be a field of opposite polarities and this field must be disturbed sufficiently to have its intensity increased to several tens or hundreds G. The activation of quiescent dark filaments floating on the neutral line is a reflection of the disturbance in the local field and, as such, can be taken as a sign of impending flare eruptions in the spotless region. 2. Many researchers like Dodson /l/ and Svestka /3/ seem to regard spotless flares as a product in the late stage of evolution of a sunspot activity region (when the sunspots have declined to below 100 units or even vanished). In the 20 cases studied here, there are certainly such examples, but also we have 10 cases which are definitely independent of sunspots either in time (the rotation history) or in space. Also, as shown in the foregoing paragraph, the necessary condition for producing flares is not the existence of sunspots, rather, it is the existence of magnetic fields of opposite polarities of sufficient strength. The tortes actrng m me acrrve region can be seen from the variation in the angle 3. between a fibril and the main filament before the flare of 1980 January 8, 0359 UT. Fig. 3 shows that on Jan 6, this angle is about ho', and on Jan 8, it is only about 30". (Data lacking on Jan 7). Thus, a pressure gradually transformed into a shear. It was a sign that the active region passed from a state of energy storage to one of sudden energy release.
REFERENCES [ 1] [9]
Dodson, H. W., Hedeman E. R., Solar Phys., Dodson, II. W.,
dings and Workshop [
3]
13 (1970), 401417.
Hcdeman, E. R., Nlohler 0. C., Intenational Solar-Terrestrial Predictions ProceeProgram. Preprint No. 12.
Svesth, Z., Puliishingco. Solar Flares. 19%
Chin. Astron. Astrophys. 5 (1982) 192-198 102-209 -23 (1982)
Act.Astron.Sin.
ACCELERATION
Pergamon Press. Printed in Great Britain 0275-1062/82/030192-07$07.50/O
OF THE SOLAR WIND BY WHISTLER WAVES FROM CORONAL HOLES
ZHANG Zhen-da, H~ANG You-ran, Department of Astronomy, Observatory, LI Xiao-qing, Purple Mountain Observatory,
Nanjing Academia
Sinica Received
1981
February
23.
ABSTRACT We investigate the possibility of an additional acceleration of the high speed solar wind by whistler waves propagating outward from a coronal hole. We consider a stationary, spherically symmetric model and assume a radial wind flow as well as a radial magnetic field. The energy equation consists of (a) energy transfer of the electron beam which excites the whistler waves, and (b) energy transfer of the whistler waves described by conservation of wave action density. The momentum conservation equation includes the momentum transfer of two gases (a thermal gas and an electron beam). The variation of the temperature is described by a polytropic law. The variation of solar wind velocity with the radial distance is calculated for different values of energy density of the whistler waves. It is shown that the acceleration of high speed solar wind in
Solar Winds and Whistler
Waves
193
the coronal hole due to the whistler waves is very important. We have calculated that the solar wind velocity at the earth's orbit is about equal to 670 km/set (for wave energy density about lo-& erg cm-' at l.lR,). It is in approximate agreement with the observed values.
1.
INTRODUCTION
Coronal holes are regions in the corona with relatively low density and temperature and They are located within large units of the samemagnetic are large-scale open field regions. where the field diverges from the inner corona and extends towards the interpolarity, planetary space. Observations show that if a large coronal hole appears in the vicinity of the Sun's equator then solar wind will be observed at the Earth's orbit, with a flow speed reaching 600-800 km/s /l/, and the same magnetic polarity as the coronal hole, showing that the latter is the source for maintaining the solar wind. Much work has been done in the past on the mechanism of accelerating the solar wind. Parker's early modelofexpansionof the solar corona /z/was obviously unsuitable when explaining Alazarki et al. /3/ used Alfven waves to explain the solarwindand the high-speed solar wind. Jacques /4/ discussed the effect of waves (Alfven and obtained rather low velocities. acoustic) on the momentum and energy transfer of the wind from the quiet Sun and coronal We may say that the problem of the high-speed solar wind holes and on the critical point. is still one of the unsolved problems of solar physics. We believe that whistler waves in the coronal hole regions may play an important role in A hint in this direction is already in the observationa the acceleration of the solar wind. data of the wind at the distance of 1 AU, /5/. Since coronal holes occupy considerable portions of the Sun's surface (a single one may account for l-5% or even 10%) and the field atmosphere is in a state of continual there is basically open, and since theunderlying disturbance (e.g., tell-tale features like spicules occur in coronal hole regions), we can suppose that there is a continuous upward injection of electron beams across the bottom of a As an electron beam passes through the background gas, whistler waves are coronal hole. excited, which cause momentum transfer in the hot gas and heat the gas at the same time, thus providing an energy source for accelerating the high-velocity solar wind. Since the We shall consider a steady-state, spherically symmetric model of thesolarwind. coronal hole is taken to be the source of the solar wind, we shall regard, as in /4/, the wind to be radial, and the open field to be also radial (neglecting the effect of the Sun's In this case, because of the frozen-in property, field. rotation), that is, a one-dimensional the electron beam that produces whistler waves moves along the field with an average speed (v is the solar wind velocity) and the beam will not undergo diffusion, just as ve =v+v in Ref. 169 w h'ere electron beams (cosmic rays) pass through a spherical symmetric system and induce Alfven waves.
2. BASIC EQUATIONS Under the above assumption that the solar wind is a steady radial flow and the assumption of radial perturbations, we can derive a set of differential equations containing only one B be respectively the density, pressure, temperature spatial coordinate. Let P, P, T and and field intensity of the background hot gas, and let u, N and 71 be respectively the velocity, number density and pressure of the electron beam, and let w be the energy density of the whistler waves. The whistler waves excited by the electron beam produce 1. Energy Equation. transfer in the hot gas, which we think should obey the principle of conservation action density w/w,, 171,
%(E) + v . [(u,.+ u>E] = 0. In a steady-state,
we have
energy of the wave
(1)
ZHANG et al.
194
[(u,.+ u>21= 0.
v.
(2)
In spherical coordinates, this is d
x
[
P(V& + Y> ” -0
1-
0. (3)
where WO, vse and V@O are the frequency, group and phase velocities of whistlers in a static medium, la/,
Here, CB coHe=WC
G$_JG$
,mp*=
JG$F
J
are respectively, the gyro-frequencyof the electron, that of the hydrogen ion, and the electron plasma frequency. 'Iheother symbols have their usual meanings. The wave number k of the whistlers has the following bounds, /8/,
We take k = cwp,/c i.e. l/43 < a x 1, and
where vA= B/~/CP
is the velocity of Alfven waves, ~,-43av~; roe-d%,-
1.7X 10'Bd.
Substituting in (3) gives w r'(86avn + Y) 1.76X IO'&?aj const.,
and we obtain the energy density of the whistlers, W=
Baa r3(86avA + “> ’
(4)
where ?;(86av", + UO) 11,) = wrJ* BO In the one-dimensional radial case, the electron beam generates whistler waves which satisfy the Cerenkov condition, # - YP= B + v%=" + 43LxV".
(5)
The electron beam transfers part of its energy to the waves, and the change in its energy is V-[NLJ(ue+ u2/2 + n/N)], where Ue is internal energy. Because the mass density N is very small, we can neglect the uz/2 term and only two terms remain. NuU,+ z?zz=zzlN(Y - 1)-Q/N+ III- I?u[(u - 1)-'-t11.
Solar Winds and Whistler Waves
195
For a relativistic electron beam, y = 413, for a thermal beam, y = 513. Considering that our beam is between these two and closer to the latter, we shall take y = 1.5, hence(y-l)-'+1=3 and we have -v * [3I?Ul -v
(6)
- (VP” + U’P).
This is the equation of energy conservation, in which P is the wave pressure tensor. Similar to the Alfven wave pressure /4/, it was proved in Ref. /li that, for whistler waves propagating in the radial direction in the spherical symmetric case, we have the relation p* - w&, that is,
(7) where
Momentum Equation. For the two-component gas consisting of a thermal component and 2. an electron beam the latter having a very small number density, the momentum equation is, /6,9/,
where u = 0.619 constant. 3.
mH is
the mean mass per particle in the solar atmosphere and
KB
isBoltzmann's
Continuity Equation. This is V*(pV) = 0 and V*(W) = 0, that is
4. Polytropic Process. When there is no heating bv waves or any other external source. _ the energy equation of a thermal gas is simply the adiabatic equation: but when there are waves and electron beams present, it can generally be described by the polytropic process
00 OIZ T
(10) =
Top;-“p”-L
E
~p’l-‘,
in which the polytropic index u is used to describe approximately the thermal flux causing non-abiabatic expansion. Conservation of Magnetic Flux. This is div B = 0, that is,
5.
B
_
Bad
r= *
(11)
In (7), (9), (10) end (11) above, the integration constants bo, de, eo and Bore2 are to be determined by the conditions at the bottom of the coronal hole. Thus, altogether we have 7 unknowns, the solar wind velocity v, mass density p, temperature T, magnetic field intensity B, particle mass density N and pressure n in the electron beam and the whistler wave energy density W.
ZHANG et al.
196
Equation of the Solar Wind. 6. differential equation
v=
2gDp*-‘“+”
dr
-
L, -
dor-’ -
Hr--‘v-’
The above equations
(7)-(11) can be reduced
to one simple
L,
+
’
Dar-,oy-~o+l~
(12) where
H = GM
ada,
4&$-"-i-VLZ
L, = 3
i&/112
f
Zb,,r+
+ #
( 2
+ 3
)
I
_5 +--lym + ” ( 2
,
>
Once (12) is integrated, i.e., the variation of v with 1: is found, we can find from (4), (5), (9) and (10) the variations with r of w, u, p and T.
3.
NUMERICAL
1.
Determination
When both equations.
CALCULATION of the Critical
the numerator
Point and Numerical
and denominator
Solution
of Differential
Equations
of Eqn (12) are zero, we have a pair of non-linear
The ro and vo that satisfy (13) define the critical point. What we seek is the solution of Eqn(l.2) that passes the critical point/LO/. To find the critical point, we fix a set of boundary values PO, 30, TO, 710, WO, r~ and vO, and find first the critical point of the Parker This is then taken as a first approximation in a solution of (13) gas (wg = 0, La = Lr = 0). by method of descent to get the critical point (ro,vo) of the non-Parker gas. Then we solve (12) numerically starting from a neighboring point of the critical point and working towards the initial point rQ, to obtain the value vO' at rO. If vg' is sufficiently close to VO, then the above critical point is what we seek. 2.
Initial and Boundary
Values
We took r. = l.lR, as the starting distance of our integration, that is, the base of the coronal hole. At this distance, we took B. = 8G, in agreement with the range 6-12G given in We took p. = 1.36 x 10-'6g/cm3 and To = 1.7 x 10% (cf. 1.5 - 2.0 x 10'K given in Ref.lll. 1111 have shown that the flow velo/Ii). We took v0 = Skmfs, since ultraviolet observations city at the base of a coronal hole may be as high as Ibkm/s. Next, following f31, we took a fixed value o = 1.18, using a fixed value greatly facilitates the calculations, and this We found by calculparticular value agrees well with the observed temperature variations. ation that the solar wind velocity in fact did not change much for different values of LS. We pointed
out earlier
that 1/43
In our calculation,
we tooka=l/lO.
The
Solar Winds and Whistler Waves
197
maxinum value of w0 was taken to be 5 x lO-"dynitm', approximately corresponding to I& = m x 106cm-3, u* = 8 x 10scm/s. This Nc value is 2 orders of magnitude below the mean partic& mass density in a coronal hole. From the observations of spicules in coronal holes ill, it is entirely possible for the solar atmosphere below the base of the coronal hole to provide an electron beam kinetic energy of this order of magnitude. We took w. to be l/5 of R*, i.e., we assumed that l/5 of the kinetic energy of the high-energy electrons is converted to the energy of the whistler waves.
4.
RESULTS AND CALCULATIONS
We calculated the variation of v with I‘for w0 = 0, 10e6, lo-", lo-" erg/cm' (correspondingly li0= 0, 5 x 10v6, 5 x lCl-I,5 x lo-' erg/cm'). The results are shown in Fig. 1. m0 corresponds to the Parker gas. Crosses mark the critical values, which can be seen to move to smaller distances with increasing wave energy. This is in agreement with the results of /3/ and 141. Fig. 1 shows that the wave energy density exerts an important influence on the acceleration of the solar wind. In our single-wave model where the electron beam is the onlysource for whistler waves and there are no other sources exciting MHD waves, we need only require a whistler wave with an energy density of lo-"erg/cm3 capable of transferring its energy to the backgroung gas in order to accelerate the solar wind to a speed of 670 km/s at the distance of the Earth orbit and this last value is in agreement with space observations /II. According to the results in /12/, the mechanical energy flux passing from the chromosphere to the lower part of the coronal hole is about 8 x 105erg cm-as-f. Using F = w'(V i-V )=w V we estimate the mechanical energy dens1 -0y to 8: w' = 8 x 10sf109 = 8 x lo-' erg/cm'. Thus, our w0 value is far less than w', that is, the whistler wave energy density used in accelerating the hot gas need only be 1/g of the mechanical energy density. This shows that the Fig. 1 Variation of solar wind velocity above value of wg is reasonable. Also, from v with radial distance r. the linear theory of the interactionbetween Curves 1,2,3,4 correspond respectively the waves and the particles, we know that to w0 = 0, 10e6, 10w5, lo-* erg/cm'. the energy of the wave excited by the particles can reach l/3 the kinetic energy of the latter /13/. Our calculations show that the wind velocity increase tapers off with distance (note the distance scale of Fig. 1 is logarithmic) so that after 1 AU there will be no large increases. This does not contradict observations. TABLE 1
The Calculated Solar Wind Speed v(r) at various distances (w~=lO-loerg cmw3, vO= 8 km/s)
distancerth
10
20
40
50
100
150
160
170
180
190
200
352
390
523
606
619
632
643
655
665
30 _---__--
km/s
166
249
306
Figs. 2-5 show our calculated variations with f of the wave energy W, the electron beam velocity u, and the temperature T and density o of the solar wind. All calculations are for the case wg = 10-"erg/cm'. Fig.Z,calculated from Eqn(4), shows that the wave energy w changes drastically near the Sun, but the change becomes much slower when reaching the Earth's orbit. This shows that, near the Sun, the energy of the whistlers is rapidLy passed to the solar wind, and exerts an important effect on the latter's acceleration.
ZHANG et al.
198
Pie.
3
Variation of electron velocity u with r.
beam
Fig.4
Fig.2
f
Variation of wave energy with distance r.
Variation of solar wind density p with=.
4
W
z3 z2
Fig. 5 Variation temperature
of solar wind T with r.
Figs. 3 and 4 were calculated from Eqns (5) and (4). These show a particle numberdensity at Earth's orbit of about 25cm'-a and a particle flux density of about 16 x 10' particles cm-2s-1. These are somewhat higher than the observed values of 5-10 crne3 and 4.5 x lo8 particles cm-'s-l, but we can make the results match by adjusting PO in our calculations. The effect of adjustingp, on the calculated v(r) is slight, that is, any adjustment of p0 will not alter greatly the above results. Fig. 5 was calculated from Eqn(l0). the same order of magnitude as observed. very small.
It shows that at Earth's orbit, T 5 10'K. This is of The effect of initial value To on v(r) is also
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