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Behavior of ionized plasma in the high latitude topside ionosphere: The polar wind
Pianer. Printed
Space Sci., Vol. 35, No. 6, pp. 7Og7-713, 1987 in Great Britain.
0
~32~433/~7$3.~+0.~ 1987 Pergaxm Journals
Ltd.
BEHAVIOR OF IONIZED PLASMA IN THE HIGH LATITUDE TOPSIDE IONOSPHERE: THE POLAR WIND SUPRIYA B. GANGULI
Science Applications
International
and H. G. MITCHELL, Jr. Corporation, Mclean, VA 22102, U.S.A.
P. J. PALMADESSO
Naval
Research
Laboratory,
Washington,
(Received in.~~ai,f~~~
DC 20375, U.S.A.
22 October 1986)
Abstract-We have developed
a numerical model to study the steady-state behavior of a fully ionized plasma (H+, Of and the electrons) encompassing the geomagnetic field lines extending from 1500 km to IOR,. The theoretical formulation is based on the 16 moment system of transport equations. The 16 moment equations, which allow transverse and parallel thermal energy to be transported separately and are based on the assumption that the distribution function is nearly bi-Maxwellian, are expected to simulate anisotropy for a collisionless plasma better than the 13 moment equations, used in some previous studies of the polar wind, which allow only a single heat flow per species and are based on simple Maxwellian distributions. We find that in the cases studied the results of 13 and 16 moment simulations are surprisingly similar, even though the temperature anisotropy is quite large. Our studies of the polar wind show that the electron temperature anisotropy develops around 2500 km with Tl > r,, Below 2500 km the collisions keep the electron temperature isotropic. The H+ ion temperature exhibits adiabatic cooling with anisotropy rfl > r,. The results of our polar wind simulations are in good agreement with recent experimental observations. These, to our knowledge, are the first successful steady-state solutions of the 16 moment system of transport equations for the polar wind.
finite system of moment equations does not constitute a closed set since the equation governing the velocity moment of order r contains the velocity moment of order r + 1. It is therefore necessary to adopt a closure approximation, such as an analytic expression for the species velocity distribution function to close the set of equations. In the 13 moment treatment the distribution function is represented as the product of a Maxwellian and a truncated series of orthogonal polynomials, with a total of 13 free parameters in the general case (for space physics applications the gyrotropic approximation is used to reduce the number of free variables). The resulting system of equations includes continuity, momentum, energy, pressure tensor and heat flow equations for each species. The closed system of equations can be applied to both collision-dominated and collisionless regions thus providing a smooth transition between the two regions. In the collision-dominated limit, the pressure tensor and heat flow equations yield Navier-Stokes expressions for the stress tensor and the heat flow vector. In the collisionless limit, the equations can be reduced to the Chew-Goldberger-Low equations (Chew et al., 1956). Transport equations based on a bi-Maxwe~Iian species distribution function were first derived by Chew er (II. (1956) for a collisionless plasma. The heat
INTRODUCTION
The behavior of plasma on the aurora1 magnetic field lines has been the subject of a number of studies in recent years. The existence of ionospheric plasma outflow from polar regions along open magnetic field lines was originally suggested by Axford (1968) and Banks and Holzer (1968). The term “polar wind” was suggested by Axford (1968) who compared this flow with the solar wind flow. Theoretical models of the polar wind including hydrodynamic, hydromagnetic, kinetic and generalized transpo~ equation models have been developed by Banks and Holzer (1968, 1969), Holzer et al. (1971), Lemaire and Scherer (1973), Schunk and Watkins (1981, 1982), and Ganguli et al. (1985a). Time-dependent solutions of polar wind were obtained by Mitchell and Palmadesso (1983) and Gombosi et al. (1985). The effects of hot electron populations on the polar wind were studied by Lemaire and Scherer (1978, 1983) and Barakat and Schunk (1984). The effects of field-aligned return currents on the polar wind were investigated by Ganguli et al. (1985b) and Ganguli (1986). A set of transport equations for aeronomy and space physics. based on the 13 moment approximation of Grad (1949, 1958), was derived by Schunk (1974, 1977) and used to study the polar wind. In general. a 703
704
S. B. GANGULI
flows were not considered. This work was extended by several other authors, e.g. Oraevskii et al. (1968). Chodura and Pohl (1971) derived a system of transport equations for an anisotropic plasma based on a bi-Maxwellian species distribution and including the transport effects as heat flow and viscosity. The Coulomb collisions between the interacting species were taken into account. The method used to derive this system of equations was an extension of Grad’s method and corresponds to a 16 moment approximation for the species distribution function. Demars and Schunk (1979) extended this work by considering anisotropic plasma of arbitrary degree of ionization. The collision terms for the important charge exchange interaction between an ion and its parent neutral were also calculated. Barakat and Schunk (1982a) in their review paper presented a unified approach to the study of transport phenomena in multicomponent anisotropic space plasmas. A generalized system of equations was presented which can describe subsonic and supersonic flows, collision-dominated and collisionless flows, plasma flows in rapidly changing magnetic field configurations, multicomponent plasma flows with large temperature differences between the interacting species and plasma flows that contain anisotropic temperature distributions. Comparison studies of transport equations based on Maxwellian and bi-Maxwellian distributions for anisotropic plasmas were performed by Barakat and Schunk (1982b). They have considered the flow of a homogeneous, weakly-ionized plasma subjected to homogeneous electric and magnetic fields. The problem has an analytic solution and the velocity distribution functions obtained from different series expansions were compared with the exact solution. For the Maxwellian expansion, Grad’s 5, 13 and 20 moment approximations were considered and for the bi-Maxwellian expansion 6 and 16 moment approximations were considered. Emphasis was given to modelling the anisotropic character of the distribution function. Their results show that the 20 moment Maxwellian expansion can describe larger nonMaxwellian deviations than the 5 and 13 moments because of the greater number of terms retained at that level of approximation. For the bi-Maxwellian expansions the 16 moment approximation is better than the 6 moment. As expected, an expansion based on a bi-Maxwellian distribution is better suited to describe anisotropic plasmas than the expansion based on a Maxwellian, even if fewer terms are retained in the bi-Maxwellian expansion. Barakat and Schunk (1982~) have also performed Monte Carlo simulations of the above-mentioned problem (anisotropic plasmas) and compared the simulation results
et al.
with Maxwellian and bi-Maxwellian expansions. The conclusions remain the same as in their previous study using analytic solutions. Schunk and Watkins (1981, 1982) conducted a study of the steady state flow of a fully ionized H+Ofelectron plasma along geomagnetic field lines extending from 1500 to 12000 km, using the 13 moment system of transport equations of Schunk (1977). In their first study of the polar wind (1981) they used a 1-D gyrotropic version of the 13 moment equations for the electron gas and a simplified set of transport equations for the hydrogen and oxygen ions. Their results show that electron temperature anisotropy develops at altitudes above 2500 km, with TL > T,, Below 2500 km the electron gas is essentially collision dominated. In their next study of the polar wind (1982) they used the 13 moment system of equations for the hydrogen ions also. The results show that for supersonic flow at high altitudes the hydrogen ion temperature parallel to the field line (r,,,) is greater than the temperature perpendicular to the field line (T,,). The reverse is true for subsonic flow. The results of the 13 moment system of equations were compared to the standard solutions. The difference is small within the plasmasphere where the electron density is high but outside the plasmasphere and in the ionosphere above SAR-arcs where the electron density is lower and substantial electron and proton heat flow occurs, the difference between the two sets of solutions is significant. Mitchell and Palmadesso (1983) used the same 13 moment system of equations and developed a dynamic numerical model of the plasma along an aurora1 field line extending from 800 km to 10 R,. The plasma consists of the electrons, hydrogen and oxygen ions. The electrons and the hydrogen ions are the dynamic species in the model. They have performed simulations for the case of a current-free polar wind and the case in which an upward field-aligned current was applied along the field line. The results of the polar wind simulations have been compared to those of Schunk and Watkins (1981, 1982). It is seen that for low electron temperature and supersonic hydrogen ion outflow, both exhibit an anisotropic hydrogen ion cooling and both have a region in which the parallel temperature increases with altitude before adiabatic cooling dominates. The electron temperature profiles, however, cannot be directly compared. The lower boundary of Mitchell and Palmadesso (1983) was held at a constant temperature and the simulation was run until a steady state was achieved. The outflow condition treats the upper boundary as an electron heat sink and therefore the electron temperature decreases with altitude. Schunk and Watkins (1981)
Ionized plasma in high latitude topside ionosphere specified the electron temperature and a positive electron temperature gradient at the lower end, as the initial condition to study the electron temperature anisotropy. The upper boundary need not be specified in this case. In this paper we shall present solutions to a system of transport equations obtained from the bi-Maxwellian-based 16 moment equations by taking the gyrotropic limit. We have studied the steady-state behavior of the plasma encompassing geomagnetic field lines extending from 1500 km to 10 R, with three different sets of boundary conditions in order to exhibit the sensitivity of the solutions to these conditions. These, to our knowledge, are the first successful steady-state solutions to the 16 moment system of transport equations for the polar wind. After describing the model in more detail, we present the results of our simulations and compare these with previous studies of the polar wind and the experimental observations of Nagai et al. (1984).
105
magnetic field line, temperatures parallel and perpendicular to the field line and two heat flows (parallel and perpendicular thermal energy) along the field line. The resulting transport equations used for the hydrogen ions and the electrons are given as follows : continuity
: an ano X+lr’dna=g
momentum
au
au
z+vayfp
(1)
: k &IT,,
GM
eE
nln~+7-m
parallel energy : kz
+kz$
+ f $
+2k(h,,
-h,)
+?kT,,;= THE MODEL
The model (Ganguli et al., 1985a) simulates the steady-state behavior of a fully ionized plasma (H+, O+ and the electrons) along the geomagnetic field lines in the high latitude topside ionosphere (extending from 1500 km to 10 R,). The theoretical formulation is based on the 16 moment system of transport equations of Barakat and Schunk (1982a). The 13 moment system of transport equations allows for different species temperature parallel and perpendicular to the field line, but allows only a single heat flow per species. The 16 moment equations allow transverse and parallel thermal energy to be transported separately. The electrons and the hydrogen ions are the dynamic species in the model and the oxygen ions form a static background population at a constant temperature. The H+ ion and electron temperature anisotropies and heat flows are not sensitive to the oxygen temperature and heat flows, as noticed by Schunk and Watkins (198 1, 1982). Thus maintaining the O+ ions at a constant temperature is not expected to introduce appreciable errors as far as the H+ ions and electrons are concerned. For our present purposes, the problem of plasma flow along the geomagnetic field lines can be reduced to one dimension. We do not expect curvature effects to be important for the relatively cold ionospheric particles. The distribution function is considered to be gyrotropic about the field line direction, which reduces the 16 moments to six moments: number density, velocity of the species parallel to the geo-
perpendicular
kz
(3)
energy :
for parallel energy :
heat flow equation
dv
3 k2T,, aT,,
6h,, (5)
heat flow equation
for perpendicular
dh, ah, ,t+vdr+hlay+
au
charge neutrality
:
energy :
k2T,, aT, -__ m ar
II, = n,+n,
(7)
where, n = number density, v = species velocity, A = cross-sectional area of a flux tube (proportional to 1 /B), k = fag
= - kag
= ;, B = magnetic
field
of the Earth, E = electric field parallel to the field line, G = gravitational constant, M = mass of the Earth, m = mass of the particular species, T,, = temperature of the species parallel to the field line, T, = temperature of the species perpendicular to the field line, h,, = heat flow parallel to the field line, h, = heat flow perpendicular to the field line, k = Boltzmann constant, e = electrons, p = hydrogen ions, o = oxygen ions.
S. B.
706
Gnrioutt et at.
For a given moment F of the distribution function, at;/& represents the ci-iange in F due to the effects of collisions and may also include anomalous transport effects associated with plasma turbulence. The collision terms used in this model are Burgers’ (1969) collision terms for the case of Coulomb collisions with corrections for finite species velocity differences. The collision terms are given in Appendix A. Using equations (1) and (2), the oxygen ion density is calculated. We have assumed that the total flux tube current f remains constant along the tube I = ed(n, VP- nc V,)
tion of integration is very important for these equations since the stability of the computation depends on the directian of integration. For example, the velocity equations can be integrated starting from the lower boundary but the electron temperature equations are integrated starting from the upper boundary. First an “initial guess” for each moment profile was obtained, based on observations. The equations were then solved iteratively until a new solution that satisfies all the boundary conditions in steady state was obtained. Unphysical solutions do not satisfy the boundary conditions and are rejected.
@I
which implies
In this paper the current I is assumed to be zero. Using equations (2) (7) and (8), the electric field E parallel to the field line is calculated.
We will consider the steady-state solutions of the equations (l)-(6). This set of ordinary differential equations is solved by first algebraically eliminating al1 derivatives in a given equation except one and then integrating the resulting quantities numericalIy along the field line. The simulation is carried out on an unequally spaced grid with 100 eells, with smalier cell sizes at the lower end of the flux tube. This enables us to determine the plasma dynamics in the presence of large density gradients due to the small scale height of oxygen. The equations we solve here represent a system of stiff differential equations. A typical solution of a stiff system has a short initial interval in which it changes rapidly (called an initial transient) after which it settles down to a comparatively slowly varying state, Methods of solving stiff ordinary differcntiat equations have been discrsssed by Gear et c& (1969) and Shampine and Gear (1979). We have used the general procedure employed by Watkins (198fa,b). The direc-
&hunk and Watkins (1982) have performed 13 moment polar wind simulations for both supersonic and subsonic hydrogen ion outflow. Until recently it was not clear if the polar wind is supersonic or subsonic. Ciurgiolo and Burch (1982) have reported the supersonic nature of the polar wind. The data for this study were obtained from the DE 1 satellite. Magai eb al. (1984) also reported the supersonic nature of the polar wind along polar cap field lines. These observations dearly show that the hydrogen ion outflow in the poIar ionosphere is supersonic in nature. Subsonic polar wind has not yet been observed. The theoretical study of Raitt et af. (1975, 1977) showed that if the H” ion outflow changes from subsonic to supersonic the transition is always below 1500 km. We therefore use 1500 km as our lower boundary, as did Schunk and Watkins (1981). We shall thus consider the hydrogen ion outflow as supersonic. As noted above, solutions have been obtained for the current-free case. The boundary conditions selected are similar to those Schunk and Watkins (1981, 1982). At the lower boundary (1500 km) we set the II+ ion velocity at 16 kms-‘, II+ ion density at 80cn~-~ and oxygen ion density at 5OOOcm-“. The hydrogen ion temperature at the lower boundary is r,,, = T,, = 3500 K while the electron temperature is T,,, = 7’,, = 1000 K. The oxygen ion temperature was kept constant at 1200 K along the entire length of the flux tube. The lower boundary heat flows are qe,,=0.1396x 10e7 erg-cm s-‘, qei =0.1749x IO-’ erg-cm s- ‘, qp,,= 0.6487 x lo-’ erg-cm s- ‘, and qpl = 0.9794 x lo-’ erg-cm s- ‘. The results of this current-free polar wind simulation are shown in Figs Ia-Ig. Figure la shows oxygen ian density, tb shows II’ ion density, lc shows electron density, and Id shows hydrogen ion velocity. oxygen ions are the dominant species only up to an altitude of?500 km. The heavier ions O* are bound to Earth
Ionized plasma in high latitude topside ionosphere by gravity while the ambipolar electric field tries to hold the hydrogen ions and electrons together, with the result that H+ ions are accelerated upwards. Supersonic H + ions flow along divergent magnetic field lines. The hydrogen ion momentum balance equation shows that it is the ambipolar electric field which produces the sharp increase in the H+ ion velocity seen at the lower end of the tube in Fig. Id. The electron velocity also increases with altitude as is evident from equation (9) and is shown in Fig. le. The H+ ion and electron velocities in the figures are plotted on a logarithmic scale to facilitate comparison with Schunk and Watkins (1981). For the H+ ions, the total flux along the flux tube is conserved, i.e., nVu = constant. Therefore, as the velocity and the area increase the density should decrease. The hydrogen ion density decreases as it flows through the diverging flux tube. The electron density also decreases, as is evident from the charge neutrality equation (7). A study of equation (12) reveals that at the upper end of the flux tube the velocity variation is driven mainly by the terms containing the hydrogen ion temperature anisotropy, electron temperature anisotropy, hydrogen ion temperature and electron temperature parallel to the field line, i.e. mirror forces and parallel pressure gradients. Figure 1f shows the hydrogen ion temperature parallel and perpendicular to the geomagnetic field. The hydrogen ion temperature exhibits two basic characteristics : (1) adiabatic cooling-supersonic ion gas cools down as it expands in a diverging magnetic field, (2) temperature anisotropy. These effects were also displayed by the previous studies of the polar wind using the 13 moment system of equations by Schunk and Watkins (1982) and Mitchell and Palmadesso (1983). Like Schunk and Watkins (1982) we obtained solutions for only positive H+ heat flow at the lower boundary. This implies that an upward flow of heat from the lower ionosphere is required for a supersonic hydrogen ion outflow. This upward flow of heat is associated with a negative H+ temperature gradient. At the lower end of the flux tube, due to rapid expansion of the supersonic ion gas, both TL and T,, decrease with altitude. This was also noted by Schunk and Watkins (1982). The hydrogen ion temperature perpendicular to the field line decreases continuously. Appreciable temperature anisotropy develops around 2500 km. The temperature anisotropy is more prominent in the H* ions than in the electrons. This is because of the fact that the hydrogen ions have supersonic velocities and in the collisionless region heat
707
transport is not efficient for H+ ions, compared to that for electrons. As the particles move up in a decreasing magnetic field, V, decreases in order to keep the adiabatic invariant p = mv2/2B constant (on the collisionless high altitude portion of the flux tube). As the total energy of the particle is conserved, the decrease in (1/2)mv: results in an increase of (1/2)mvf. That is, perpendicular energy gets converted to parallel energy. A study of equations (13) and (14) shows that the temperature variation is produced mainly as an interaction between convective and conductive heat transport processes and the adiabatic effects. The collisions play an important role below 2500 km. In equations (15) and (16) the temperature gradient terms are very important. Electron temperatures parallel and perpendicular to the geomagnetic field lines are shown in Fig. lg. The most important effects exhibited by electron temperature are : (1) temperature anisotropy ; (2) dominance of collisional effects below 2500 km. “Collision dominated” implies that the collisional relaxation rate is greater than the rate associated with the driving term that produces the anisotropy. Appreciable temperature anisotropy develops above 2500 km. Both parallel and perpendicular temperatures increase with altitude. Heat flow is downward here (that is, heat flow from the magnetosphere to the ionosphere) as considered by Schunk and Watkins (1981) with positive temperature gradients. Also, as noticed by Schunk and Watkins (1981), the electron temperature ratio at the upper end is TL/ T,, - 2. The contributions from the collision terms are very important below 2500 km in equations (13) and (14). Since the collision terms are important in this region, the electron temperature anisotropy observed in this region is very small. The anisotropy increases with altitude. The electron temperature variation is produced mainly as a balance between heat flow processes and the mirror force process. In order to preserve the adiabatic invariant in an increasing magnetic field (downward heat flow), the electron perpendicular energy increases with the decrease in parallel energy. The parallel and perpendicular temperature gradients are positive for this case and these terms are important at all altitudes in equations (15) and (16). The electron species velocity is less than the electron thermal velocity. Our results here are in good agreement with Schunk and Watkins’ (1981) low electron temperature case. Schunk and Watkins (198 1) varied the boundary electron temperature gradients to see the extent to which
S. B.
708
lm,
am
Ym
7aYJ
GANGULI
et
al.
xc0
ALTITUDE km1
FIG. 1. (a-f). Caption on next page.
Ionized plasma in high latitude topside ionosphere
FIG. 1. POLAR WIND PROFILESvs ALTITUDE.
The boundary conditions at 1SOOkm are n, = SOOOcm-“, n, = 80cmw3, VP = 16kms-‘, TP,, = TPL = 3500K, T.,, = T,, = lOOOK, T, = 1200K. (a) Oxygen ion number density, (b) H+ ion number density, (c) electron number density, (d) H+ ion velocity, (e) electron velocity, (f) H+ ion temperatures parallel and perpendicular to the geomagnetic field and (g) electron temperatures parallel and perpendicular to the geomagnetic field.
their solutions are valid. Their results show that with the increase in boundary electron temperature gradient the electron temperature is increased but the direction of anisotropy remained the same. It is also clearly explained by Schunk and Watkins (1981) that this observed anisotropy is opposite to that predicted by the Chew-Goldberger-Low (double adiabatic) energy equations. The tendency of the adiabatic terms to produce an anisotropy with T,, > T, is also present in the transport equations, but the heat flow processes dominate over the adiabatic processes and we observe an anisotropy with T1 > T,,. For very strong heat flows the results of the 13 moment and 16 moment systems of equations are expected to be significantly different. The electron temperature profiles cannot be directly compared to Mitchell and Palmadesso (1983). They treated the upper boundary as an electron heat sink, which decreases the electron temperature with altitude. We have compared our solution of the equations based on the 16 moment system with that of Schunk and Watkins’ (1982) solution of the 13 moment equations. The general behavior of the H+ ion temperature parallel and pe~endicular to the field line remains the same. The H+ ion temperature perpendicuIar to the field line decreases continuously with altitude. The
709
parallel temperature first decreases and then increases and finally tends to remain constant with altitude. We observe an anisotropy with TL > T,, below 2.500km. Schunk and Watkins (1982) observed a similar anisotropy for the low electron temperature case when they lowered the H+ ion temperature and heat flow and when the oxygen ion density was increased at the lower end. They have pointed out that the results are sensitive to the boundary conditions used, in the sense that the qualitative picture remains the same but the quantitative behavior changes. The heat flows observed in these simulations are relatively small but the temperature anisotropies are quite large. This situation should be within the limits of validity of the bi-~axwellian-based 16 moment equations but should be very stressful for the isotropic-Maxwellian-based 13 moment equations ; in view of this the differences between the results of the 13 and 16 moment simulations are surprisingly small. We have also compared our results with those of Mitchell and Palmadesso (1983) and Holzer et al. (1971). Mitchell and Palmadesso (1983) started their simulation at 800 km at the lower boundary. At this level the H+ ion temperature increases with altitude due to Joule heating caused by H+ ion collisions with oxygen ions. This effect rapidly decreases with altitude. The general qualitative picture here is in good agreement with our results. Holzer et ai. (1971) have compared the solutions of hydrodynamic and kinetic equations for supersonic polar wind outflow in a collisionless regime. Both hydrodynamic and kinetic solutions show that T,_ decreases continuously. T,, for both cases decreases rapidly with altitude and then tends to become constant. For the hydrodynamic case at all altitudes the hydrogen ion temperature parallel to the field line was greater than that perpendicular to the field line. The kinetic solutions showed a region at the lower end where TL was greater than T,,. The cross-over region was noticed at the lower end. In these cases also we notice that our solutions are in good agreement with Holzer et al. (1971). Finally, we compared our results with Nagai et al. (1984). Nagai et al. (198 1) have used the data from Dynamics Explorer (DE) to demonstrate the supersonic nature of the polar wind. The observations were obtained from 65” to 81” invariant latitude and altitude near 2R,. The results show that for an estimated range of spacecraft potential of + 3 to + 5 V, a temperature range of 0.1.-0.2 eV was obtained corresponding to flow velocities of 25-16 km s- i. Calculated Mach numbers are 5.1 for a velocity of 25kms-‘, 3.9 for 21 kms-’ and 2.6 for 16kms-‘. There are different ways to define the H+ ion Mach
Ko3
lKK0 ALTITUDE I&m,
ALTITUDE km,
FIG. 2. POLAR WIND PROFILES vs ALTITUDE. boundary conditions at 1500km are n,= 5~Ocm~3, np = lOO~rn_~, VP= IOkms-‘, Tpii = Z’,, = 38OOK, TCi,= T,, = lOOOK, 7’, = 1200K. (a) H’ ion number density, (b) electron number density, (c) H+ ion velocity, (d) electron velocity, (e) H+ ion temperatures parallei and perpendicular to the geomagnetic field and (f) electron temperatures parallel and perpendicular to the geomagnetic field. The
Ionized plasma
in high latitude
2.0 ALTITUDE
(REI
0
I
2.0
IRE)
ALTITUDE
J 6.0 IRE)
2.0
7.5
6.0
7.5
6.0
7.5
(REI
4.0 ALTITUDE
I
4.0
4.0 ALTITUDE
1mm ALTITUDE
711
topside ionosphere
IREI
FIG. 3. POLAR WIND PROFILES VS ALTITUDE. The boundary conditions at 1500 km are n, = 5000cm-‘, VP = 16kms-‘, rzP= 100cmm3, Tr,, = Trl = 3500K, T,,, = T,, = 1000 K, T, = 1200K. The upper boundary is extended to 10 R,. (a) Oxygen ion number density, (b) H+ ion (dotted curve) and electron (solid curve) number densities, (c) H+ ion (dotted curve) and electron (solid curve) velocities, (d) H+ ion temperatures parallel (dotted curve) and perpendicular (solid curve) to the geomagnetic field and (e) electron temperatures parallel (dotted curve) and perpendicular (solid curve) to the geomagnetic field.
number. Here, the single ion Mach number definition is used, i.e. M = V~/~k~~~~~~‘j~.The Mach numbers calculated from our results corresponding to a flow velocity of I6 km s- ’ is 2.7 and 4.1 for a veIocity of 21 kms-‘. We have varied slightly the H+ ion velocity, density and temperature at the lower end to study the sensitivity of the solutions to these parameters. At the lower end the hydrogen ion density is increased to 1OOcm” ‘, velocity is decreased to lOkms_ ’ and the temperature is raised to 3800 K. The results of these changes are presented in Figs 2a-2F. The qualitative picture and relative importance of the contributing terms remain the same as before. The quantitative change is obvious as the initial conditions are changed. The Mach number calculated in this case of flow velocity of 16 km s- ’ is 2.68. This is in good agreement with the value 2.6 calculated by Nagai et ul. (1984). Next, we extend our upper boundary to 10 R,. The electron temperatures parallel and perpendicular to the magnetic field lines are set equal to Z’,,,= r,, = 6000 K at this boundary, and we abandon the conditions on electron heat flow at the lower boundary. Actual measurements of the temperature of outflowing ionospheric electrons at the upper end of the flux tube are not available at this time and 6000K is chosen arbitrarily to suit other measured quantities used as the boundary conditions in this steady-state model. The rest&s of these simulations are presented in Figs 3a-3e. The qualitative picture and the relative importance of the terms remain the same as before. The change in electron temperature profile is expected as now the upper end is held at a fixed temperature. The ratio TJ T,, can vary depending on the upper boundary conditions. The Mach numbers calculated From our results are 4.1, corresponding to a Aow velocity of 21 km s- ‘, and 2.7 for avelocityof16kmsP’.
function, as compared with the 13 moment system with a Maxwellian distribution function, have been discussed. The results of this simulation are compared with the above-mentioned previous studies of the polar wind and recent experimental results based on DE1 satellite observations. In this paper we have shown that in cases of low heat flow the results of 13 and 16 moment equations are surprisingly similar, indicating that the 13 moment equations provide an adequate description of the polar wind physics even though the temperature anisotropies calculated are large enough to stress the ~‘near-Ma~wellian” approximation used to derive them, within the parameter ranges considered here. The asymptotic behavior of the solutions of the transport equations at high altitudes is quite sensitive to the boundary conditions applied at the lower end of the flux tube, This, coupled with absence of any standard set of assumptions for specifying those dynamic variables for which upper boundary conditions must be supplied, leads to considerable variation of the results at high altitudes in the studies referenced above. In essence, one finds similar ionospheres and widely different magnetospheres in this group of calculations (i.e. different characteristics for the up~owing pIasma component of~onos~he~~ origin in the magnetosphere). While this makes quantitative comparison of results somewhat difficult, it is CkXr that qualitative agreement among the various calcuiations is good. Our results are in good agreement with the observations of Nagai ef al. (1984). Acknowledgements-.Wc would like to thank Dr R. W. Schunk of Utah State University, Dr S. J, Marsh of SachsFreeman Associates and Dr 1. Haber of the Naval Research Laboratory for useful discussions. This work is supported by the Office of Naval Research and the National Aeronautics and Space Adlni~jstr~tion.
REFERENCES
We have studied the steady-state behavior of the plasma encompassing the geomagnetic field lines using equations based on the 16 moment system of transport equations. These are the first successful steady-state solutions to the 16 moment. set of transport equations for the polar wind (Ganguli et al., 19SSa). Previous theoretical works on the polar wind including hydrodynamic, adiabatic and transport equations have been discussed and compared for advantages and disadvan~ges of each procedure and in order to validate our model. In a generalized system of transport equations, the advantages of using the 16 moment system with a bi-Maxwellian dis~~bution
Axford, W. I. (l968) The polar wind and the terrestrial helium budget. J. ger@J~ Res. 73, 6855. Banks, P. M. and Holtzer, T. E. (1968) The polar wind, J. geophys. Res. 73,6846. Banks, P. M. and Holtzer, T. E. (1969) High-latitude plasma transport : the polar wind. .I. geophys. Res. 74, 63 17. Barakat, A. R. and Schunk, R. W. (1982a) Transport equations for multicamponent anisotropic space plasmas: a review. Plasma PhJs. 24, 389. Barakat, A. R. and Schunk, R. W. (1982b) Comparison of transport equations based on Maxwellian and bi-Maxwellian distributions for anisotropic plasmas. J. Ph_ys. D: A&.
Phys. X5, I 195.
Barakat, A. R. and Chunk, R. W. (1982~) Comparison of M~well~an and bi-Maxwellian expansions with Monte Carlo simulations for anisotropic plasmas. X P&w. D: Appl. Phys. 15,2189.
Ionized plasma
in high latitude
Barakat, A. R. and Schunk, R. W. (1984) Effect of hot electrons on the polar wind. J. geophys. Res. @, 9771. Burgers, J. M. (1969) Flow Equations&r Compode Gases. Academic Press, New York. Chew, G. F., Goldberger, M. L. and Low, F. E. (1956) The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. Roy. SOL, Ser. A 236, 112. Chodura, R. and Pohl, F. (197 1) Hydrodynamic equations for anisotropic plasmas in magnetic fields--II. Transport equations including collisions. Plasma Phys. 13, 645. Demars, H. G. and Schunk, R. W. (1979) Transport equations for multispecies plasmas based on individual biMaxwellian distributions. f. upp/. Whys. 12, 1051. Ganguli, S. B. (1986) Plasma transport in the aurora1 return current region. Ph.D. Thesis, Boston College. Ganguli, S. B., Mitchell, H. G., Jr. and Palmadesso, P. J. (1985a) Behavior of ionized plasma in the high latitude topside ionosphere. NRL Memorandum Report 5623, 1985a. Ganguli, S. B., Mitchell, H. G., Jr. and Palmadesso, P. J. (1985b) Dynamics of the large scale return currents on aurora1 field lines. NRL Memorandum Report 5673, 1985b. Gear, C. W. (1969) Numerical Initial Value Problems in Ordinary DifferentialEquation.F. Prentice-Hall, EngIewood Cliffs, New Jersey. Gombosi, T. I., Cravens, T. E. and Nagi, A. F. (1985) A time dependent theoretical model of the polar wind : preliminary results. Geophys. Res. Mt. 12, 167. Grad, H. (1949) On the kinetic theory of rarefied gases. Co~n~un~ pure appl. Math. 2, 331. Grad, H. (1958) Principles of the kinetic theory of gases, in ~un~b~ch der P&d. XII, p. 205. Springer, Berlin. Gurgiolo, C. and Burch, J. L. (1982) DE1 observations of the polar wind-a heated and an unheated component. Geophys. Res. Lett. 9, 945. Holtzer, T. E., Fedder, J. A. and Banks, P. M. (1971) A comparison of kinetic and hydrodynamic models of an expanding ion-exosphere. J. geophys. Res. 76, 2453. Lemaire, J. and Scherer, M. (1973) Kinetic models of the solar and polar winds. Rev. Geophys. Space Phys. 1 I, 427. Lemaire, J. and Scherer, M. (1978) Field-aligned distribution of plasma mantle and ionospheric plasmas. J. atmos. terr. Phvs. 40, 337. Lemaire, J. and Scherer, M. (1983) Field-aligned current density vs electric potential characteristics for magnetospheric flux tubes. Ann. Geophys, I, 91. Mitchell, H. G. and Palmadesso, P. J. (1983) A dynamic model for the amoral field line plasma in the presence of tield-aligned current. .K geophys. Res. 88, 2 13 1. Nagai, T., Waite, J. H., Jr., Green, J. L., Chappell, C. R., Olsen, R. C. and Comfort, R. II. (1984). First measurements of supersonic polar wind in the polar magnetosphere, 11, 669. Oraevskii, V., Chodura, R. and Feneberg, W. (1968) Hydrodynamic equations for plasmas in strong magnetic fieldI. Collisionless approximation. Plasma Ph,vs. IO, 8 19. Raitt, W. J., Schunk, R. W. and Banks, P. M. (1975) A comparison of the temperature and density structure in the high and low speed thermal proton ffows. Planet. Space Sci. 23, 1103. Raitt, W. J., Schunk, R. W. and Banks, P. M. (1971) The influence of convection electric fields on thermal proton outflow from the ionosphere. Planer. Space Sci. 25, 291.
topside ionosphere
713
&hunk, R. W. (1974) Transport equations for aeronomy. Planet. Space Sci. 23, 337. Schunk, R. W. (1977) Mathematical structure of transport equations for multispecies flows. Ret;. Geophys. Space Phys. 15, 429. Schunk, R. W. and Watkins, D. S. (1981) Electron temperature anisotropy in the polar wind. J. geophys. Res. 86, 91. Schunk, R. W. and Watkins, D. S. (1982) Proton temperature anisotropy in the polar wind. J. geophJx Res. 87, 171. Shampine, L. F. and Gear, C. W. (1979) A user’s view of solving stiff ordinary differential equations. Siam Rev. 21, 1. Watkins, D. S. (19SIa) Efficient initialization of stiff systems with one unknown initial condition. Siam J. Ntlmer. Anal. 18, 794. Watkins, D. S. (198lb) Determining initial values for stiff systems of ordinary differential equations. Siam J. Numer. Anal. 18, 13.
APPENDIX
The collision terms used in this paper for the 16 moment system of transport equations are shown here. As explained before, these are Burgers’ (1969) coIlision terms for the case of Coulomb collisions with corrections for finite species velocity differences. The subscripts band a represent the species, in this case either the electrons or H+ or O+.
(AI)
Each sum includes all charged particles lation. The velocity-corrected Coulomb vi,*is given by :
species in the simucollision frequency
n,(32H)‘~2e~e~(mm,+m~)lnAexpf-x~~) vha = 3m,2m,&
.--.
(A7)
S. B.
714 In A is the Coulomb
logarithm
GANGULI et al.
and
T,=fT”,,+fGp
648) 649)
Space Sci., Vol. 35, No. 6, pp. 7Og7-713, 1987 in Great Britain.
0
~32~433/~7$3.~+0.~ 1987 Pergaxm Journals
Ltd.
BEHAVIOR OF IONIZED PLASMA IN THE HIGH LATITUDE TOPSIDE IONOSPHERE: THE POLAR WIND SUPRIYA B. GANGULI
Science Applications
International
and H. G. MITCHELL, Jr. Corporation, Mclean, VA 22102, U.S.A.
P. J. PALMADESSO
Naval
Research
Laboratory,
Washington,
(Received in.~~ai,f~~~
DC 20375, U.S.A.
22 October 1986)
Abstract-We have developed
a numerical model to study the steady-state behavior of a fully ionized plasma (H+, Of and the electrons) encompassing the geomagnetic field lines extending from 1500 km to IOR,. The theoretical formulation is based on the 16 moment system of transport equations. The 16 moment equations, which allow transverse and parallel thermal energy to be transported separately and are based on the assumption that the distribution function is nearly bi-Maxwellian, are expected to simulate anisotropy for a collisionless plasma better than the 13 moment equations, used in some previous studies of the polar wind, which allow only a single heat flow per species and are based on simple Maxwellian distributions. We find that in the cases studied the results of 13 and 16 moment simulations are surprisingly similar, even though the temperature anisotropy is quite large. Our studies of the polar wind show that the electron temperature anisotropy develops around 2500 km with Tl > r,, Below 2500 km the collisions keep the electron temperature isotropic. The H+ ion temperature exhibits adiabatic cooling with anisotropy rfl > r,. The results of our polar wind simulations are in good agreement with recent experimental observations. These, to our knowledge, are the first successful steady-state solutions of the 16 moment system of transport equations for the polar wind.
finite system of moment equations does not constitute a closed set since the equation governing the velocity moment of order r contains the velocity moment of order r + 1. It is therefore necessary to adopt a closure approximation, such as an analytic expression for the species velocity distribution function to close the set of equations. In the 13 moment treatment the distribution function is represented as the product of a Maxwellian and a truncated series of orthogonal polynomials, with a total of 13 free parameters in the general case (for space physics applications the gyrotropic approximation is used to reduce the number of free variables). The resulting system of equations includes continuity, momentum, energy, pressure tensor and heat flow equations for each species. The closed system of equations can be applied to both collision-dominated and collisionless regions thus providing a smooth transition between the two regions. In the collision-dominated limit, the pressure tensor and heat flow equations yield Navier-Stokes expressions for the stress tensor and the heat flow vector. In the collisionless limit, the equations can be reduced to the Chew-Goldberger-Low equations (Chew et al., 1956). Transport equations based on a bi-Maxwe~Iian species distribution function were first derived by Chew er (II. (1956) for a collisionless plasma. The heat
INTRODUCTION
The behavior of plasma on the aurora1 magnetic field lines has been the subject of a number of studies in recent years. The existence of ionospheric plasma outflow from polar regions along open magnetic field lines was originally suggested by Axford (1968) and Banks and Holzer (1968). The term “polar wind” was suggested by Axford (1968) who compared this flow with the solar wind flow. Theoretical models of the polar wind including hydrodynamic, hydromagnetic, kinetic and generalized transpo~ equation models have been developed by Banks and Holzer (1968, 1969), Holzer et al. (1971), Lemaire and Scherer (1973), Schunk and Watkins (1981, 1982), and Ganguli et al. (1985a). Time-dependent solutions of polar wind were obtained by Mitchell and Palmadesso (1983) and Gombosi et al. (1985). The effects of hot electron populations on the polar wind were studied by Lemaire and Scherer (1978, 1983) and Barakat and Schunk (1984). The effects of field-aligned return currents on the polar wind were investigated by Ganguli et al. (1985b) and Ganguli (1986). A set of transport equations for aeronomy and space physics. based on the 13 moment approximation of Grad (1949, 1958), was derived by Schunk (1974, 1977) and used to study the polar wind. In general. a 703
704
S. B. GANGULI
flows were not considered. This work was extended by several other authors, e.g. Oraevskii et al. (1968). Chodura and Pohl (1971) derived a system of transport equations for an anisotropic plasma based on a bi-Maxwellian species distribution and including the transport effects as heat flow and viscosity. The Coulomb collisions between the interacting species were taken into account. The method used to derive this system of equations was an extension of Grad’s method and corresponds to a 16 moment approximation for the species distribution function. Demars and Schunk (1979) extended this work by considering anisotropic plasma of arbitrary degree of ionization. The collision terms for the important charge exchange interaction between an ion and its parent neutral were also calculated. Barakat and Schunk (1982a) in their review paper presented a unified approach to the study of transport phenomena in multicomponent anisotropic space plasmas. A generalized system of equations was presented which can describe subsonic and supersonic flows, collision-dominated and collisionless flows, plasma flows in rapidly changing magnetic field configurations, multicomponent plasma flows with large temperature differences between the interacting species and plasma flows that contain anisotropic temperature distributions. Comparison studies of transport equations based on Maxwellian and bi-Maxwellian distributions for anisotropic plasmas were performed by Barakat and Schunk (1982b). They have considered the flow of a homogeneous, weakly-ionized plasma subjected to homogeneous electric and magnetic fields. The problem has an analytic solution and the velocity distribution functions obtained from different series expansions were compared with the exact solution. For the Maxwellian expansion, Grad’s 5, 13 and 20 moment approximations were considered and for the bi-Maxwellian expansion 6 and 16 moment approximations were considered. Emphasis was given to modelling the anisotropic character of the distribution function. Their results show that the 20 moment Maxwellian expansion can describe larger nonMaxwellian deviations than the 5 and 13 moments because of the greater number of terms retained at that level of approximation. For the bi-Maxwellian expansions the 16 moment approximation is better than the 6 moment. As expected, an expansion based on a bi-Maxwellian distribution is better suited to describe anisotropic plasmas than the expansion based on a Maxwellian, even if fewer terms are retained in the bi-Maxwellian expansion. Barakat and Schunk (1982~) have also performed Monte Carlo simulations of the above-mentioned problem (anisotropic plasmas) and compared the simulation results
et al.
with Maxwellian and bi-Maxwellian expansions. The conclusions remain the same as in their previous study using analytic solutions. Schunk and Watkins (1981, 1982) conducted a study of the steady state flow of a fully ionized H+Ofelectron plasma along geomagnetic field lines extending from 1500 to 12000 km, using the 13 moment system of transport equations of Schunk (1977). In their first study of the polar wind (1981) they used a 1-D gyrotropic version of the 13 moment equations for the electron gas and a simplified set of transport equations for the hydrogen and oxygen ions. Their results show that electron temperature anisotropy develops at altitudes above 2500 km, with TL > T,, Below 2500 km the electron gas is essentially collision dominated. In their next study of the polar wind (1982) they used the 13 moment system of equations for the hydrogen ions also. The results show that for supersonic flow at high altitudes the hydrogen ion temperature parallel to the field line (r,,,) is greater than the temperature perpendicular to the field line (T,,). The reverse is true for subsonic flow. The results of the 13 moment system of equations were compared to the standard solutions. The difference is small within the plasmasphere where the electron density is high but outside the plasmasphere and in the ionosphere above SAR-arcs where the electron density is lower and substantial electron and proton heat flow occurs, the difference between the two sets of solutions is significant. Mitchell and Palmadesso (1983) used the same 13 moment system of equations and developed a dynamic numerical model of the plasma along an aurora1 field line extending from 800 km to 10 R,. The plasma consists of the electrons, hydrogen and oxygen ions. The electrons and the hydrogen ions are the dynamic species in the model. They have performed simulations for the case of a current-free polar wind and the case in which an upward field-aligned current was applied along the field line. The results of the polar wind simulations have been compared to those of Schunk and Watkins (1981, 1982). It is seen that for low electron temperature and supersonic hydrogen ion outflow, both exhibit an anisotropic hydrogen ion cooling and both have a region in which the parallel temperature increases with altitude before adiabatic cooling dominates. The electron temperature profiles, however, cannot be directly compared. The lower boundary of Mitchell and Palmadesso (1983) was held at a constant temperature and the simulation was run until a steady state was achieved. The outflow condition treats the upper boundary as an electron heat sink and therefore the electron temperature decreases with altitude. Schunk and Watkins (1981)
Ionized plasma in high latitude topside ionosphere specified the electron temperature and a positive electron temperature gradient at the lower end, as the initial condition to study the electron temperature anisotropy. The upper boundary need not be specified in this case. In this paper we shall present solutions to a system of transport equations obtained from the bi-Maxwellian-based 16 moment equations by taking the gyrotropic limit. We have studied the steady-state behavior of the plasma encompassing geomagnetic field lines extending from 1500 km to 10 R, with three different sets of boundary conditions in order to exhibit the sensitivity of the solutions to these conditions. These, to our knowledge, are the first successful steady-state solutions to the 16 moment system of transport equations for the polar wind. After describing the model in more detail, we present the results of our simulations and compare these with previous studies of the polar wind and the experimental observations of Nagai et al. (1984).
105
magnetic field line, temperatures parallel and perpendicular to the field line and two heat flows (parallel and perpendicular thermal energy) along the field line. The resulting transport equations used for the hydrogen ions and the electrons are given as follows : continuity
: an ano X+lr’dna=g
momentum
au
au
z+vayfp
(1)
: k &IT,,
GM
eE
nln~+7-m
parallel energy : kz
+kz$
+ f $
+2k(h,,
-h,)
+?kT,,;= THE MODEL
The model (Ganguli et al., 1985a) simulates the steady-state behavior of a fully ionized plasma (H+, O+ and the electrons) along the geomagnetic field lines in the high latitude topside ionosphere (extending from 1500 km to 10 R,). The theoretical formulation is based on the 16 moment system of transport equations of Barakat and Schunk (1982a). The 13 moment system of transport equations allows for different species temperature parallel and perpendicular to the field line, but allows only a single heat flow per species. The 16 moment equations allow transverse and parallel thermal energy to be transported separately. The electrons and the hydrogen ions are the dynamic species in the model and the oxygen ions form a static background population at a constant temperature. The H+ ion and electron temperature anisotropies and heat flows are not sensitive to the oxygen temperature and heat flows, as noticed by Schunk and Watkins (198 1, 1982). Thus maintaining the O+ ions at a constant temperature is not expected to introduce appreciable errors as far as the H+ ions and electrons are concerned. For our present purposes, the problem of plasma flow along the geomagnetic field lines can be reduced to one dimension. We do not expect curvature effects to be important for the relatively cold ionospheric particles. The distribution function is considered to be gyrotropic about the field line direction, which reduces the 16 moments to six moments: number density, velocity of the species parallel to the geo-
perpendicular
kz
(3)
energy :
for parallel energy :
heat flow equation
dv
3 k2T,, aT,,
6h,, (5)
heat flow equation
for perpendicular
dh, ah, ,t+vdr+hlay+
au
charge neutrality
:
energy :
k2T,, aT, -__ m ar
II, = n,+n,
(7)
where, n = number density, v = species velocity, A = cross-sectional area of a flux tube (proportional to 1 /B), k = fag
= - kag
= ;, B = magnetic
field
of the Earth, E = electric field parallel to the field line, G = gravitational constant, M = mass of the Earth, m = mass of the particular species, T,, = temperature of the species parallel to the field line, T, = temperature of the species perpendicular to the field line, h,, = heat flow parallel to the field line, h, = heat flow perpendicular to the field line, k = Boltzmann constant, e = electrons, p = hydrogen ions, o = oxygen ions.
S. B.
706
Gnrioutt et at.
For a given moment F of the distribution function, at;/& represents the ci-iange in F due to the effects of collisions and may also include anomalous transport effects associated with plasma turbulence. The collision terms used in this model are Burgers’ (1969) collision terms for the case of Coulomb collisions with corrections for finite species velocity differences. The collision terms are given in Appendix A. Using equations (1) and (2), the oxygen ion density is calculated. We have assumed that the total flux tube current f remains constant along the tube I = ed(n, VP- nc V,)
tion of integration is very important for these equations since the stability of the computation depends on the directian of integration. For example, the velocity equations can be integrated starting from the lower boundary but the electron temperature equations are integrated starting from the upper boundary. First an “initial guess” for each moment profile was obtained, based on observations. The equations were then solved iteratively until a new solution that satisfies all the boundary conditions in steady state was obtained. Unphysical solutions do not satisfy the boundary conditions and are rejected.
@I
which implies
In this paper the current I is assumed to be zero. Using equations (2) (7) and (8), the electric field E parallel to the field line is calculated.
We will consider the steady-state solutions of the equations (l)-(6). This set of ordinary differential equations is solved by first algebraically eliminating al1 derivatives in a given equation except one and then integrating the resulting quantities numericalIy along the field line. The simulation is carried out on an unequally spaced grid with 100 eells, with smalier cell sizes at the lower end of the flux tube. This enables us to determine the plasma dynamics in the presence of large density gradients due to the small scale height of oxygen. The equations we solve here represent a system of stiff differential equations. A typical solution of a stiff system has a short initial interval in which it changes rapidly (called an initial transient) after which it settles down to a comparatively slowly varying state, Methods of solving stiff ordinary differcntiat equations have been discrsssed by Gear et c& (1969) and Shampine and Gear (1979). We have used the general procedure employed by Watkins (198fa,b). The direc-
&hunk and Watkins (1982) have performed 13 moment polar wind simulations for both supersonic and subsonic hydrogen ion outflow. Until recently it was not clear if the polar wind is supersonic or subsonic. Ciurgiolo and Burch (1982) have reported the supersonic nature of the polar wind. The data for this study were obtained from the DE 1 satellite. Magai eb al. (1984) also reported the supersonic nature of the polar wind along polar cap field lines. These observations dearly show that the hydrogen ion outflow in the poIar ionosphere is supersonic in nature. Subsonic polar wind has not yet been observed. The theoretical study of Raitt et af. (1975, 1977) showed that if the H” ion outflow changes from subsonic to supersonic the transition is always below 1500 km. We therefore use 1500 km as our lower boundary, as did Schunk and Watkins (1981). We shall thus consider the hydrogen ion outflow as supersonic. As noted above, solutions have been obtained for the current-free case. The boundary conditions selected are similar to those Schunk and Watkins (1981, 1982). At the lower boundary (1500 km) we set the II+ ion velocity at 16 kms-‘, II+ ion density at 80cn~-~ and oxygen ion density at 5OOOcm-“. The hydrogen ion temperature at the lower boundary is r,,, = T,, = 3500 K while the electron temperature is T,,, = 7’,, = 1000 K. The oxygen ion temperature was kept constant at 1200 K along the entire length of the flux tube. The lower boundary heat flows are qe,,=0.1396x 10e7 erg-cm s-‘, qei =0.1749x IO-’ erg-cm s- ‘, qp,,= 0.6487 x lo-’ erg-cm s- ‘, and qpl = 0.9794 x lo-’ erg-cm s- ‘. The results of this current-free polar wind simulation are shown in Figs Ia-Ig. Figure la shows oxygen ian density, tb shows II’ ion density, lc shows electron density, and Id shows hydrogen ion velocity. oxygen ions are the dominant species only up to an altitude of?500 km. The heavier ions O* are bound to Earth
Ionized plasma in high latitude topside ionosphere by gravity while the ambipolar electric field tries to hold the hydrogen ions and electrons together, with the result that H+ ions are accelerated upwards. Supersonic H + ions flow along divergent magnetic field lines. The hydrogen ion momentum balance equation shows that it is the ambipolar electric field which produces the sharp increase in the H+ ion velocity seen at the lower end of the tube in Fig. Id. The electron velocity also increases with altitude as is evident from equation (9) and is shown in Fig. le. The H+ ion and electron velocities in the figures are plotted on a logarithmic scale to facilitate comparison with Schunk and Watkins (1981). For the H+ ions, the total flux along the flux tube is conserved, i.e., nVu = constant. Therefore, as the velocity and the area increase the density should decrease. The hydrogen ion density decreases as it flows through the diverging flux tube. The electron density also decreases, as is evident from the charge neutrality equation (7). A study of equation (12) reveals that at the upper end of the flux tube the velocity variation is driven mainly by the terms containing the hydrogen ion temperature anisotropy, electron temperature anisotropy, hydrogen ion temperature and electron temperature parallel to the field line, i.e. mirror forces and parallel pressure gradients. Figure 1f shows the hydrogen ion temperature parallel and perpendicular to the geomagnetic field. The hydrogen ion temperature exhibits two basic characteristics : (1) adiabatic cooling-supersonic ion gas cools down as it expands in a diverging magnetic field, (2) temperature anisotropy. These effects were also displayed by the previous studies of the polar wind using the 13 moment system of equations by Schunk and Watkins (1982) and Mitchell and Palmadesso (1983). Like Schunk and Watkins (1982) we obtained solutions for only positive H+ heat flow at the lower boundary. This implies that an upward flow of heat from the lower ionosphere is required for a supersonic hydrogen ion outflow. This upward flow of heat is associated with a negative H+ temperature gradient. At the lower end of the flux tube, due to rapid expansion of the supersonic ion gas, both TL and T,, decrease with altitude. This was also noted by Schunk and Watkins (1982). The hydrogen ion temperature perpendicular to the field line decreases continuously. Appreciable temperature anisotropy develops around 2500 km. The temperature anisotropy is more prominent in the H* ions than in the electrons. This is because of the fact that the hydrogen ions have supersonic velocities and in the collisionless region heat
707
transport is not efficient for H+ ions, compared to that for electrons. As the particles move up in a decreasing magnetic field, V, decreases in order to keep the adiabatic invariant p = mv2/2B constant (on the collisionless high altitude portion of the flux tube). As the total energy of the particle is conserved, the decrease in (1/2)mv: results in an increase of (1/2)mvf. That is, perpendicular energy gets converted to parallel energy. A study of equations (13) and (14) shows that the temperature variation is produced mainly as an interaction between convective and conductive heat transport processes and the adiabatic effects. The collisions play an important role below 2500 km. In equations (15) and (16) the temperature gradient terms are very important. Electron temperatures parallel and perpendicular to the geomagnetic field lines are shown in Fig. lg. The most important effects exhibited by electron temperature are : (1) temperature anisotropy ; (2) dominance of collisional effects below 2500 km. “Collision dominated” implies that the collisional relaxation rate is greater than the rate associated with the driving term that produces the anisotropy. Appreciable temperature anisotropy develops above 2500 km. Both parallel and perpendicular temperatures increase with altitude. Heat flow is downward here (that is, heat flow from the magnetosphere to the ionosphere) as considered by Schunk and Watkins (1981) with positive temperature gradients. Also, as noticed by Schunk and Watkins (1981), the electron temperature ratio at the upper end is TL/ T,, - 2. The contributions from the collision terms are very important below 2500 km in equations (13) and (14). Since the collision terms are important in this region, the electron temperature anisotropy observed in this region is very small. The anisotropy increases with altitude. The electron temperature variation is produced mainly as a balance between heat flow processes and the mirror force process. In order to preserve the adiabatic invariant in an increasing magnetic field (downward heat flow), the electron perpendicular energy increases with the decrease in parallel energy. The parallel and perpendicular temperature gradients are positive for this case and these terms are important at all altitudes in equations (15) and (16). The electron species velocity is less than the electron thermal velocity. Our results here are in good agreement with Schunk and Watkins’ (1981) low electron temperature case. Schunk and Watkins (198 1) varied the boundary electron temperature gradients to see the extent to which
S. B.
708
lm,
am
Ym
7aYJ
GANGULI
et
al.
xc0
ALTITUDE km1
FIG. 1. (a-f). Caption on next page.
Ionized plasma in high latitude topside ionosphere
FIG. 1. POLAR WIND PROFILESvs ALTITUDE.
The boundary conditions at 1SOOkm are n, = SOOOcm-“, n, = 80cmw3, VP = 16kms-‘, TP,, = TPL = 3500K, T.,, = T,, = lOOOK, T, = 1200K. (a) Oxygen ion number density, (b) H+ ion number density, (c) electron number density, (d) H+ ion velocity, (e) electron velocity, (f) H+ ion temperatures parallel and perpendicular to the geomagnetic field and (g) electron temperatures parallel and perpendicular to the geomagnetic field.
their solutions are valid. Their results show that with the increase in boundary electron temperature gradient the electron temperature is increased but the direction of anisotropy remained the same. It is also clearly explained by Schunk and Watkins (1981) that this observed anisotropy is opposite to that predicted by the Chew-Goldberger-Low (double adiabatic) energy equations. The tendency of the adiabatic terms to produce an anisotropy with T,, > T, is also present in the transport equations, but the heat flow processes dominate over the adiabatic processes and we observe an anisotropy with T1 > T,,. For very strong heat flows the results of the 13 moment and 16 moment systems of equations are expected to be significantly different. The electron temperature profiles cannot be directly compared to Mitchell and Palmadesso (1983). They treated the upper boundary as an electron heat sink, which decreases the electron temperature with altitude. We have compared our solution of the equations based on the 16 moment system with that of Schunk and Watkins’ (1982) solution of the 13 moment equations. The general behavior of the H+ ion temperature parallel and pe~endicular to the field line remains the same. The H+ ion temperature perpendicuIar to the field line decreases continuously with altitude. The
709
parallel temperature first decreases and then increases and finally tends to remain constant with altitude. We observe an anisotropy with TL > T,, below 2.500km. Schunk and Watkins (1982) observed a similar anisotropy for the low electron temperature case when they lowered the H+ ion temperature and heat flow and when the oxygen ion density was increased at the lower end. They have pointed out that the results are sensitive to the boundary conditions used, in the sense that the qualitative picture remains the same but the quantitative behavior changes. The heat flows observed in these simulations are relatively small but the temperature anisotropies are quite large. This situation should be within the limits of validity of the bi-~axwellian-based 16 moment equations but should be very stressful for the isotropic-Maxwellian-based 13 moment equations ; in view of this the differences between the results of the 13 and 16 moment simulations are surprisingly small. We have also compared our results with those of Mitchell and Palmadesso (1983) and Holzer et al. (1971). Mitchell and Palmadesso (1983) started their simulation at 800 km at the lower boundary. At this level the H+ ion temperature increases with altitude due to Joule heating caused by H+ ion collisions with oxygen ions. This effect rapidly decreases with altitude. The general qualitative picture here is in good agreement with our results. Holzer et ai. (1971) have compared the solutions of hydrodynamic and kinetic equations for supersonic polar wind outflow in a collisionless regime. Both hydrodynamic and kinetic solutions show that T,_ decreases continuously. T,, for both cases decreases rapidly with altitude and then tends to become constant. For the hydrodynamic case at all altitudes the hydrogen ion temperature parallel to the field line was greater than that perpendicular to the field line. The kinetic solutions showed a region at the lower end where TL was greater than T,,. The cross-over region was noticed at the lower end. In these cases also we notice that our solutions are in good agreement with Holzer et al. (1971). Finally, we compared our results with Nagai et al. (1984). Nagai et al. (198 1) have used the data from Dynamics Explorer (DE) to demonstrate the supersonic nature of the polar wind. The observations were obtained from 65” to 81” invariant latitude and altitude near 2R,. The results show that for an estimated range of spacecraft potential of + 3 to + 5 V, a temperature range of 0.1.-0.2 eV was obtained corresponding to flow velocities of 25-16 km s- i. Calculated Mach numbers are 5.1 for a velocity of 25kms-‘, 3.9 for 21 kms-’ and 2.6 for 16kms-‘. There are different ways to define the H+ ion Mach
Ko3
lKK0 ALTITUDE I&m,
ALTITUDE km,
FIG. 2. POLAR WIND PROFILES vs ALTITUDE. boundary conditions at 1500km are n,= 5~Ocm~3, np = lOO~rn_~, VP= IOkms-‘, Tpii = Z’,, = 38OOK, TCi,= T,, = lOOOK, 7’, = 1200K. (a) H’ ion number density, (b) electron number density, (c) H+ ion velocity, (d) electron velocity, (e) H+ ion temperatures parallei and perpendicular to the geomagnetic field and (f) electron temperatures parallel and perpendicular to the geomagnetic field. The
Ionized plasma
in high latitude
2.0 ALTITUDE
(REI
0
I
2.0
IRE)
ALTITUDE
J 6.0 IRE)
2.0
7.5
6.0
7.5
6.0
7.5
(REI
4.0 ALTITUDE
I
4.0
4.0 ALTITUDE
1mm ALTITUDE
711
topside ionosphere
IREI
FIG. 3. POLAR WIND PROFILES VS ALTITUDE. The boundary conditions at 1500 km are n, = 5000cm-‘, VP = 16kms-‘, rzP= 100cmm3, Tr,, = Trl = 3500K, T,,, = T,, = 1000 K, T, = 1200K. The upper boundary is extended to 10 R,. (a) Oxygen ion number density, (b) H+ ion (dotted curve) and electron (solid curve) number densities, (c) H+ ion (dotted curve) and electron (solid curve) velocities, (d) H+ ion temperatures parallel (dotted curve) and perpendicular (solid curve) to the geomagnetic field and (e) electron temperatures parallel (dotted curve) and perpendicular (solid curve) to the geomagnetic field.
number. Here, the single ion Mach number definition is used, i.e. M = V~/~k~~~~~~‘j~.The Mach numbers calculated from our results corresponding to a flow velocity of I6 km s- ’ is 2.7 and 4.1 for a veIocity of 21 kms-‘. We have varied slightly the H+ ion velocity, density and temperature at the lower end to study the sensitivity of the solutions to these parameters. At the lower end the hydrogen ion density is increased to 1OOcm” ‘, velocity is decreased to lOkms_ ’ and the temperature is raised to 3800 K. The results of these changes are presented in Figs 2a-2F. The qualitative picture and relative importance of the contributing terms remain the same as before. The quantitative change is obvious as the initial conditions are changed. The Mach number calculated in this case of flow velocity of 16 km s- ’ is 2.68. This is in good agreement with the value 2.6 calculated by Nagai et ul. (1984). Next, we extend our upper boundary to 10 R,. The electron temperatures parallel and perpendicular to the magnetic field lines are set equal to Z’,,,= r,, = 6000 K at this boundary, and we abandon the conditions on electron heat flow at the lower boundary. Actual measurements of the temperature of outflowing ionospheric electrons at the upper end of the flux tube are not available at this time and 6000K is chosen arbitrarily to suit other measured quantities used as the boundary conditions in this steady-state model. The rest&s of these simulations are presented in Figs 3a-3e. The qualitative picture and the relative importance of the terms remain the same as before. The change in electron temperature profile is expected as now the upper end is held at a fixed temperature. The ratio TJ T,, can vary depending on the upper boundary conditions. The Mach numbers calculated From our results are 4.1, corresponding to a Aow velocity of 21 km s- ‘, and 2.7 for avelocityof16kmsP’.
function, as compared with the 13 moment system with a Maxwellian distribution function, have been discussed. The results of this simulation are compared with the above-mentioned previous studies of the polar wind and recent experimental results based on DE1 satellite observations. In this paper we have shown that in cases of low heat flow the results of 13 and 16 moment equations are surprisingly similar, indicating that the 13 moment equations provide an adequate description of the polar wind physics even though the temperature anisotropies calculated are large enough to stress the ~‘near-Ma~wellian” approximation used to derive them, within the parameter ranges considered here. The asymptotic behavior of the solutions of the transport equations at high altitudes is quite sensitive to the boundary conditions applied at the lower end of the flux tube, This, coupled with absence of any standard set of assumptions for specifying those dynamic variables for which upper boundary conditions must be supplied, leads to considerable variation of the results at high altitudes in the studies referenced above. In essence, one finds similar ionospheres and widely different magnetospheres in this group of calculations (i.e. different characteristics for the up~owing pIasma component of~onos~he~~ origin in the magnetosphere). While this makes quantitative comparison of results somewhat difficult, it is CkXr that qualitative agreement among the various calcuiations is good. Our results are in good agreement with the observations of Nagai ef al. (1984). Acknowledgements-.Wc would like to thank Dr R. W. Schunk of Utah State University, Dr S. J, Marsh of SachsFreeman Associates and Dr 1. Haber of the Naval Research Laboratory for useful discussions. This work is supported by the Office of Naval Research and the National Aeronautics and Space Adlni~jstr~tion.
REFERENCES
We have studied the steady-state behavior of the plasma encompassing the geomagnetic field lines using equations based on the 16 moment system of transport equations. These are the first successful steady-state solutions to the 16 moment. set of transport equations for the polar wind (Ganguli et al., 19SSa). Previous theoretical works on the polar wind including hydrodynamic, adiabatic and transport equations have been discussed and compared for advantages and disadvan~ges of each procedure and in order to validate our model. In a generalized system of transport equations, the advantages of using the 16 moment system with a bi-Maxwellian dis~~bution
Axford, W. I. (l968) The polar wind and the terrestrial helium budget. J. ger@J~ Res. 73, 6855. Banks, P. M. and Holtzer, T. E. (1968) The polar wind, J. geophys. Res. 73,6846. Banks, P. M. and Holtzer, T. E. (1969) High-latitude plasma transport : the polar wind. .I. geophys. Res. 74, 63 17. Barakat, A. R. and Schunk, R. W. (1982a) Transport equations for multicamponent anisotropic space plasmas: a review. Plasma PhJs. 24, 389. Barakat, A. R. and Schunk, R. W. (1982b) Comparison of transport equations based on Maxwellian and bi-Maxwellian distributions for anisotropic plasmas. J. Ph_ys. D: A&.
Phys. X5, I 195.
Barakat, A. R. and Chunk, R. W. (1982~) Comparison of M~well~an and bi-Maxwellian expansions with Monte Carlo simulations for anisotropic plasmas. X P&w. D: Appl. Phys. 15,2189.
Ionized plasma
in high latitude
Barakat, A. R. and Schunk, R. W. (1984) Effect of hot electrons on the polar wind. J. geophys. Res. @, 9771. Burgers, J. M. (1969) Flow Equations&r Compode Gases. Academic Press, New York. Chew, G. F., Goldberger, M. L. and Low, F. E. (1956) The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. Roy. SOL, Ser. A 236, 112. Chodura, R. and Pohl, F. (197 1) Hydrodynamic equations for anisotropic plasmas in magnetic fields--II. Transport equations including collisions. Plasma Phys. 13, 645. Demars, H. G. and Schunk, R. W. (1979) Transport equations for multispecies plasmas based on individual biMaxwellian distributions. f. upp/. Whys. 12, 1051. Ganguli, S. B. (1986) Plasma transport in the aurora1 return current region. Ph.D. Thesis, Boston College. Ganguli, S. B., Mitchell, H. G., Jr. and Palmadesso, P. J. (1985a) Behavior of ionized plasma in the high latitude topside ionosphere. NRL Memorandum Report 5623, 1985a. Ganguli, S. B., Mitchell, H. G., Jr. and Palmadesso, P. J. (1985b) Dynamics of the large scale return currents on aurora1 field lines. NRL Memorandum Report 5673, 1985b. Gear, C. W. (1969) Numerical Initial Value Problems in Ordinary DifferentialEquation.F. Prentice-Hall, EngIewood Cliffs, New Jersey. Gombosi, T. I., Cravens, T. E. and Nagi, A. F. (1985) A time dependent theoretical model of the polar wind : preliminary results. Geophys. Res. Mt. 12, 167. Grad, H. (1949) On the kinetic theory of rarefied gases. Co~n~un~ pure appl. Math. 2, 331. Grad, H. (1958) Principles of the kinetic theory of gases, in ~un~b~ch der P&d. XII, p. 205. Springer, Berlin. Gurgiolo, C. and Burch, J. L. (1982) DE1 observations of the polar wind-a heated and an unheated component. Geophys. Res. Lett. 9, 945. Holtzer, T. E., Fedder, J. A. and Banks, P. M. (1971) A comparison of kinetic and hydrodynamic models of an expanding ion-exosphere. J. geophys. Res. 76, 2453. Lemaire, J. and Scherer, M. (1973) Kinetic models of the solar and polar winds. Rev. Geophys. Space Phys. 1 I, 427. Lemaire, J. and Scherer, M. (1978) Field-aligned distribution of plasma mantle and ionospheric plasmas. J. atmos. terr. Phvs. 40, 337. Lemaire, J. and Scherer, M. (1983) Field-aligned current density vs electric potential characteristics for magnetospheric flux tubes. Ann. Geophys, I, 91. Mitchell, H. G. and Palmadesso, P. J. (1983) A dynamic model for the amoral field line plasma in the presence of tield-aligned current. .K geophys. Res. 88, 2 13 1. Nagai, T., Waite, J. H., Jr., Green, J. L., Chappell, C. R., Olsen, R. C. and Comfort, R. II. (1984). First measurements of supersonic polar wind in the polar magnetosphere, 11, 669. Oraevskii, V., Chodura, R. and Feneberg, W. (1968) Hydrodynamic equations for plasmas in strong magnetic fieldI. Collisionless approximation. Plasma Ph,vs. IO, 8 19. Raitt, W. J., Schunk, R. W. and Banks, P. M. (1975) A comparison of the temperature and density structure in the high and low speed thermal proton ffows. Planet. Space Sci. 23, 1103. Raitt, W. J., Schunk, R. W. and Banks, P. M. (1971) The influence of convection electric fields on thermal proton outflow from the ionosphere. Planer. Space Sci. 25, 291.
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713
&hunk, R. W. (1974) Transport equations for aeronomy. Planet. Space Sci. 23, 337. Schunk, R. W. (1977) Mathematical structure of transport equations for multispecies flows. Ret;. Geophys. Space Phys. 15, 429. Schunk, R. W. and Watkins, D. S. (1981) Electron temperature anisotropy in the polar wind. J. geophys. Res. 86, 91. Schunk, R. W. and Watkins, D. S. (1982) Proton temperature anisotropy in the polar wind. J. geophJx Res. 87, 171. Shampine, L. F. and Gear, C. W. (1979) A user’s view of solving stiff ordinary differential equations. Siam Rev. 21, 1. Watkins, D. S. (19SIa) Efficient initialization of stiff systems with one unknown initial condition. Siam J. Ntlmer. Anal. 18, 794. Watkins, D. S. (198lb) Determining initial values for stiff systems of ordinary differential equations. Siam J. Numer. Anal. 18, 13.
APPENDIX
The collision terms used in this paper for the 16 moment system of transport equations are shown here. As explained before, these are Burgers’ (1969) coIlision terms for the case of Coulomb collisions with corrections for finite species velocity differences. The subscripts band a represent the species, in this case either the electrons or H+ or O+.
(AI)
Each sum includes all charged particles lation. The velocity-corrected Coulomb vi,*is given by :
species in the simucollision frequency
n,(32H)‘~2e~e~(mm,+m~)lnAexpf-x~~) vha = 3m,2m,&
.--.
(A7)
S. B.
714 In A is the Coulomb
logarithm
GANGULI et al.
and
T,=fT”,,+fGp
648) 649)