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The effect of the large ion temperature differences on the solutions of diffusion and continuity equations
Journal
of Atmospheric
and Terrestrial
Printedin Great Britain.
Physics,
Vol. 50, No. I, pp. 3340,
CCJZI-9169/88 $3.00+ .oO 0 1988 PergamonJournalsLtd.
1988.
The effect of the large ion temperature differences on the solutions of diffusion and continuity equations JOANNA
GR~CHULSKA
Space Research Centre, Polish Academy of Sciences, 01-237 Warszawa, ul. Ordona 21, Poland (Received infinalform 30 June 1987)
Abstract-Diffusion and continuity equations for O+ and H+ ions were integrated along a magnetic field line between 500 and 3000 km. Three sets of equations were used: (1) the standard ones; (2) modified according to the formulae given by St-Maurice and Schunk; (3) those derived by Conrad and Schunk. The differences between solutions obtained for each set of equations and the importance of the modifications introduced in the diffusion equations are discussed.
introduced into his formulae were taken from BURGERS(1969), who calculated them assuming that the species drift velocity differences are small in comDuring the last few years modified equations suitable parison with the species thermal speeds and that the for modelling of the topside ionosphere have been species temperature differences are small in comproposed in a number of papers. One of the propparison with the average temperature of the gas ositions was given in 1977 by ST-MAURICE and mixture. The collision terms obtained making this SCHUNK (1977). They derived diffusion equations for assumption are referred to as Burgers’ ‘linear’ colthe mid-latitude topside ionospheric plasma composed of two major ions, electrons and a number of lision terms. There is one special situation when the collision terms can be calculated rigorously from minor ions, assuming that the temperature differences Boltzmann’s collision integral, namely the Maxwell between the components are small in comparison with the average temperature of the gas mixture. CONRAD molecule interactions for which the collision frequency is independent of velocity. For this case the and SCHUNK(1979) obtained more general diffusion collision terms derived from the 13-moment approxiequations allowing for large temperature differences mation are valid for both large temperature differbetween the components. Both these approaches are based on the system of transport equations for gas ences and large drift velocity differences between the interacting species. For all other interactions the colmixtures given by SCHUNK (1975). SCHUNK (1975) presented a system consisting of lision terms given in SCHUNK(1975) are appropriate only for low speed flows with small temperature and continuity, momentum, internal energy, stress tensor velocity differences. and heat flow equations--separately for each comST-MAURICEand SCHUNK(1977) obtained diffusion ponent of the gas mixture. Such an approach makes it possible to take into account the separate velocities equations for mid-latitude ionospheric plasma from and temperatures of the components. The above sys13-moment momentum and heat flow equations with ‘linear’ Burgers’ collision terms as given in SCHUNK tem of transport equations was obtained by multiplying Boltzmann’s equation-for each com(1975) by making several assumptions. They assumed ponent-by an appropriate function of velocity and steady state flow and neutrality of the plasma and then integrating over the velocity space. Equations neglected stress and non-linear acceleration terms, as obtained in this way do not constitute, however, a well as density and temperature gradients perclosed set. In order to close it we must choose a suitpendicular to geomagnetic field lines and currents parable expression for the velocity distribution function. allel to them. Omitting the small terms proportional to the ratios (m,/m,)“’ and (T,- T,)/T,, where m, and This function is also necessary to evaluate the collision terms, which appear on the right-hand-sides of the T, are species mass and temperature, they obtained diffusion equations modified with respect to the stantransport equations. Schunk used in his derivation Grad’s 13-moment approximation of the distribution dard ones. The modifications arise from the presence function (GRAD, 1949, 1958). The collision terms he of a correction term accounting for the heat flow INTRODUCTION
33
effect on ordinary
diffusion
and from the Lvzt that
process is represented by terms proportional to the gradient of the temperature ol each ion separately. In the paper of CONKAD and %HUNK (1979) the diffusion equations for fully and partially ionised plasma are derived assuming that the gas component drift velocity differences are again much smaller than the species thermal speeds, but allowing for large temperature differences between the interacting species. The authors derived the equations in a similar way and based on similar assumptions as in the work of ST-MAURICE and &HUNK (1977), except that they took Burgers’ ‘semilinear’ collision terms--valid also for the large temperature differences between the species-instead of ‘linear’ ones appropriate only for small differences. The obtained expressions are modified with respect to those of St-Maurice and Schunk, although the thermal diffusion process is again governed by the temperature gradients of each plasma component, and a correction factor accounting for heat flow is again present in the formulae for the ordinary diffusion coefficients. It is the aim of this work to investigate how important--for the modelhng of topside ionosphere dynamics--are modifications allowing for a more rigorous description of flows with large temperature differences between the interacting species. The integration of the diffusion and continuity equations for O+ and H+ ions was carried out taking alternately the diffusion equations corresponding to the ‘linear’ and ‘semilinear’ approximations of Burgers for collision terms. for the same parameters and boundary conditions in each case, so that the differences between the solutions found when integrating the formulae of St-Maurice and Schunk and those of Conrad and Schunk were due to the modifications introduced in the latter case. One has to confront the above approaches with that of SCHUNK and WALKER (1969), where it is assumed that the plasma components have a common, equal temperature. To this end the standard diffusion equations, including thermal diffusion with thermal diffusion coefficients derived by SCHUNKand WALKER (1969) were integrated, coupled with continuity equations for 0 + and H+ ions. The obtained solutions were compared with the ones found for the St-Maurice and Schunk equations. Such a comparison enables one to conclude how important the influence of heat flow is on ordinary diffusion and whether the different ways of representing thermal diffusion-by the approach of ST-MAURICEand SCHUNK(1977) or that of SCHUNK and WALKER (1969Fgive rise to large discrepancies between the solutions found in modelling of the topside ionosphere. the thermal
diffusion
The diffusion equations derived by ST-MAIIKKF and SCHI,NK (1977) and by CONRAD and SCHUNK (lY79), dcscrihing the flow of Hi and 0’ ion>. have the following
form
u, = u, - D /
1
1vn,n,
mic
I
Zf,
+
T,
VT +
7: Vn,
n,
T,
and ui = q-D,
Vn, - ;$
T Vn,
/
+ ; VT, + +
/
n
/
where n,, u,?,m,v and T, are number density, velocity, mass and temperature, respectively, for species s and s = i or j, where i refers to 0+ and j to H +. k is Boltzmann’s constant, G is the acceleration due to gravity and a$ and aI, are the thermal diffusion coefficients. Di and Di, the ordinary diffusion coefficients, are given below. D, =
D, =
kT,
1
wz,v,, (I 1 A<,) kT,
’
I
nz,t~,;(1 -A,,)
’
where v,~~ is the collision frequency between species s and t and A,, is a correction factor which accounts for the effect of heat flow on ordinary diffusion. The differences between the approach of St-Maurice and Schunk (MS) and that of Conrad and Schunk (CS) emerge in the expressions for A,, CL,, and c$. The formulae for Ar”, avS r, and a$ MSderived by St-Maurice and Schunk, assuming that the differences between the temperatures of interacting species are small and Ass a” and u *” found by Conrad and Schuik, for :a;ge”temperlture differences, are given in the papers cited. The standard diffusion equations commonly used to model 0 + and H + flow can be obtained from (1) and (2) if we remove the modifications introduced in both papers discussed above. To do so we neglect the influence of heat flow on ordinary diffusion, putting Aij equal to zero, and assume that O+ and H+ temperatures are equal. Ordinary diffusion coefficients then reduce simply to
Diffusion and continuity equations D,
=kr,
t?liVij
D.=?!L / nljv,
where Tp is plasma temperature. Thermal diffusion is then represented by one term only, proportional to the gradient of Tp. The thermal diffusion coefficients are usually those of SCHUNK and WALKER (1969), derived with one common temperature assumed for all ionic components. In many modelling attempts where the standard equations have been applied thermal diffusion was neglected, but whenever they are referred to in this paper it is tacitly assumed that they include this process. To find out how important are both the MS and CS modifications the integration of three different sets of continuity and diffusion equations for O+ and H+ was carried out downwards from a height of 3000 km, along the magnetic field line corresponding to L = 3.2, to a height of 500 km. The first set under consideration includes the diffusion equations in their standard form. The second and third sets correspond, respectively, to the MS and CS modifications of the diffusion equations. In all three sets the continuity equations for O+ and H+ had the following form
f&w =a--L,. This is the stationary continuity equation for ions flowing along a geomagnetic field line, where s is the arc length along it, A is the cross-sectional area of the magnetic flux tube and Qs and L, are the production and loss rates of ion s. The production of ion O+ is due to photoionisation of neutral atomic oxygen and to the charge exchange reaction H++O-+O++H.
(4)
The loss of O+ is due to reactions with molecular nitrogen and molecular oxygen and to the charge exchange reaction O++H+H++O.
(5)
The production of H+ is due to reaction (5) and loss to reaction (4). The rate coefficients needed in the continuity equations were taken from RAIrr et al. (1974). The neutral atmospheric model was taken from HEDIN et al. (1977a, b) and ion and electron temperatures from EVANSand HOLT (1978) (for low altitudes). At higher levels, above 1000 km, it was assumed that the plasma temperature gradient was equal to either 1K km-’ or 2K km-‘.
35
Integration of these sets of four ordinary differential equations was performed using the RungeKutta method of fourth order. It was assumed that at the upper boundary O+ velocity was equal to zero. Such an assumption does not infhtence the final conclusions, as the solutions are not sensitive to the value of O+ flux unless it is unusually large. At the lower boundary chemical equilibrium for H+ ions was assumed. The remaining boundary conditions were changed over a wide range of values to produce solutions corresponding to various possible situations in the topside ionosphere and plasmasphere. To obtain one set of solutions (2 concentration and 2 velocity altitude profiles) a ‘shooting’ method was used ; three conditions at the upper boundary were kept constant while the fourth one was varied in search of the solution satisfying appropriate conditions imposed at the lower boundary. The solution obtained while integrating the standard equations will be denoted from now on by the letter S, those found for the equations of St-Maurice and Schunk by the letters MS and the solutions of Conrad and Schunk’s equations by the letters CS.
RESULTSAND DISCUSSION A direct comparison of the MS and S solutions will allow us to estimate how important the modifications introduced by St-Maurice and Schunk (1977) are. The relative differences (%) found between the S and MS solutions will be referred to as S differences. From several calculations carried out for various ionospheric conditions two examples were chosen. The MS solutions describing the ionosphere and plasmasphere in two different states are shown in Figs. 1 and 2. These results were found in such a way that the values of O+ and H+ velocities and H + concentration were kept constant at the upper boundary, while the O+ concentration was varied in search of the solutions which fulfilled the required conditions at the lower boundary. In these calculations the electron and ion temperatures below 1000 km were taken from EVANS and HOLT (1978) and above this level it was assumed that the gradients of both temperatures were equal to 1K km - ’ . The 0 + and H + temperatures were assumed to be identical. Figure 1 shows concentration rr- and Fig. 2 flux F,(F, = n,,u,)- altitude profiles. The continuous curves correspond to calculations in which a neutral atmospheric model with an exospheric temperature of 1500K was used. In this case Fig. 1 reveals the transition height occurring as high as 3000 km, as a consequence of a very low H+ concentration, while Fig. 2 shows the corresponding downward flow of
low ihr the IiOOK atmosphere and nculral oxygct~ are high. As the lower boundar)~ COP dition requires chemical equilibrium for protons, the small valueof lhc ratio of the neutral hydrogen conx-c
H
concentrations
(Km
250c
2000
1500
1000
la’
Id
tt?
no, nH km-’ ) Fig. I. 0 + and H f concentration profiles obtained by integration of diffusion (ST-MAURICE and SCHUNK, 1977) and continuity equations. The continuous and dashed lines correspond to calculations using neutral atmospheric models with exospheric temperatures. resDectivelv,of 15OOKand 75k .
H IKmf 2500
Fig. 2. 0 + and Hi flux profiles corresponding to the concentrations shown in Fig. 1. Arrows indicate directions of flow. There is an H + flow direction reversal in the 750K case and an O+ one in the I SOOKcase.
protons for all altitudes, with the reversal of the O+ flow direction just below 1000 km. In both figures the dashed lines refer to the 750K neutral model. Here, in contrast, the ionosphere is rich in H+, the transition height occurs well below 1000 km and O+ flux is always directed upwards, while proton flux is upward above about 1300 km and downward below this level. Such a low H+ concentration in the 15OOK calculations, in spite of the proton influx from above, is due to the fact that neutral hydrogen concentrations
centration to that ofncutral oxygen leads to H ’ concenlratmns about 3 orders of magnitude smaller than those of 0 ‘. At higher altitudes the proton Ilow from the protonosphere results in an increase in i-l * concentration. However, this H downward flux (of the order of 1Ohcm ~’ s ! ) is too small to increase the H+ concentration su~~iently to move the transitjon height to lower altitudes. In the case of the 750K calculations the relatively large ratio of neutral hydrogen to neutral oxygen concentration ensures effective proton production in the entire region considered and in the case represented by the dashed lines there arc a lot of H + ions, in spite of proton outflow. Some of the S differences are shown in Fig.3 (750K case) and in Fig. 4 (1500K case). There is a pronounced dependence of the magnitude of the differences on the rate of change of the variable in question. An inspection oE Figs. 3 and 4 with reference to Figs. 1 and 2 reveals that whenever the absolute gradient value for any variable increases, this is foIlowed by a ~orrespollding growth in the respective differences. This is seen in Fig, 3 in the vicinity of H + flux direction reversal, where a steep increase of ,4FT, occurs. Similarly. near the lower boundary a fast decrease of H” and 0’ fluxes at 500 km results in a sharp increase of AFL and AF:. The bchaviour of the flux differences generally follows that of the concentrations, except in the neighbourhood of the flux direction reversal, where even a slight shift of the reversal point produces misleadingly large relative flux differences. Therefore, in Fig. 4 and further on only the concentrations are discussed. The obtained differences are much more important for 0 ‘. ions than for H+ ions. This effect is partly due to the computational procedure itself, where 0’ concentration was varied in search of the particular boundary condition imposed, while H + concentration and velocity, as well as 0’ velocity, were kept constant. Extensive calculations of this kind were performed for a broad range of pertinent parameters and boundary conditions. In particular, the consequences of assuming equal ion temperatures were investigated by varying the Ii’ to Of temperature ratio and it was found that even in the extreme case of this ratio reaching a value of 2 the final effect on the differences produced was small for ail plausible ionospheric conditions. Apart from this, some clear trends were discovered to govern the behaviour of the observed differences : for an upward proton flux an upper limit
31
Diffusion and continuity equations H
1
1
(Kill
,KHm,,
2500 .
25w
15%
25%
35%
I
I
I
In I I
H
45%
AF,S,AF,:A$ Fig. 3. S differences obtained by comparing the solutions of standard equations with those shown as dashed lines (750K case) in Figs. 1 and 2. AFSu,AFS,andAnt are H+ flux, Of flux and O+ concentration differences, respectively. A& the H+ concentration difference, was omitted as it was smaller than 6%.
Fig. 5. Of and H+ concentration profiles found from the MS equations for low (dashed lines) and high (continuous lines) ion density.
An: 2000 -
\ I I I I
\ 1 I ,
1500-
\ \
50%
15%
25%
Ant,
25%
45%
An:
Fig. 4. S differences for Of and H+ concentrations, corresponding to the MS solutions shown as continuous lines (1500Kcase) in Figs. 1 and 2.
of 60% was never exceeded, while in the case of proton flow directed downwards in the entire region considered the differences could grow well above this value ; in special circumstances values as high as 200% can be produced. An example of the latter situation is illustrated in Figs. 5 and 6. Two pairs of curves are shown on each figure. The MS concentrations drawn as dashed and continuous lines in Fig. 5 correspond to the respective drawings of differences in Fig. 6. The directions of the ion fluxes associated with the concentrations shown in the figures were in both cases similar: the Of flux was upward and the H+ one
100%
150%
An: , An: Fig. 6. S differences for H+ and O+ concentrations found for the two situations shown in Fig. 5 and marked correspondingly.
downward in the entire region. However, in the ‘continuous’ case the H+ flux at 3000 km reached a relatively large value of 5 x 10” cm-’ s-l, while the flux corresponding to the situation drawn as dashed lines was equal there to 1 x 10’ crnW2s-l. Both pairs of MS solutions shown in Fig. 5 were found using in the calculations the 750K neutral atmospheric model ; the electron and 0 + temperatures below 1000 km were taken from EVANS and HOLT (1978) and H+ temperature was assumed to be 2 times larger than that of Of. Above 1000 km the assumption was made that for O+ temperature the latitude gradient is equal to 1K km-’ and for H+ 2K km-‘. These solutions were found using the searching procedure with the H + concentration varied at 3000 km and therefore at higher altitudes An”, is larger than An:.
15
.I GKou~C
In l’ig. 6 we see that the ‘dashed’ A& reaches almost 200%, while the differcncea in the ‘continuous’ ease never exceed 30%. This illustrates the fact that the S differences usually mcreasc in cases when the 0’ concentration is comparable to that of H + over a wide range of altitudes. WC set that 111the ‘dashed’ case both ion concentrations are relatively close to one another, because 0 ’ concentration slowly decreases with altitude in the vicinity of the transition height and both ion species have an ample opportunity to interact, as opposed to the conditions in the ‘continuous’ case. It must be stressed that this effect was found to be independent of the absolute values of the concentrations themselves, as the extremely high H + concentration (continuous line) might perhaps suggest. The following conclusion can be drawn.from the analysis of all the computations performed : the MS modifications are mainly due to the presence of the A,, correction factor in the ordinary diffusion coefficients. The effect of A,, is from 5 to 30 times larger, depending on the parameters and boundary conditions, than that due to the differences in description of the thermal diffusion process itself. In particular, it was found that, in agreement with suggestion of ST-MAURICE and SCHUNK (1977), the thermal diffusion terms proportional to 0’ temperature are unimportant and lead to differences below the level of I %. Now we will analyse in turn the modifications of CONRAD and SCHUNK (1979) allowing for large differences between the temperatures of the interacting species. The solutions of the CS equations will be compared with those of St-Maurice and Schunk and the differences obtained will be referred to as C differences. For the mid-latitude ionosphere it seems reasonable to assume that the ratio of H+ to 0’ temperatures varies in the range 0.8 -2. In Fig. 7 the CS solutions are shown. They were obtained for the 750K neutral atmosphere, the O+ and electron temperatures were taken from EVANS and HOLT (1978) and the H+ temperature was assumed to be equal to 0.8 of that of 0.‘. The corresponding C differences are presented in Fig. 8. As 0’ concentration at 3000 km was the varied parameter in the searching procedure, the differences concerning this ion are the most pronounced at higher altitudes, but even in this case they do not exceed 5%. The sudden increase of AFE near 1000 km is due to Hi flux direction reversal there. A& is everywhere smaller than 1% and was not drawn in the figure. Similar calculations were repeated taking the ratio of H+ to 0’ temperature equal to 1.2, 1.5 and 2, for various parameters and boundary conditions, and it was found that An: and An:; never exceeded a value
l.SK \
ld
lo4
?d Icni’l
nOJnH Fig. 7. O+ and H+ concentration and flux profiles obtained by integration of diffusion (CONRADand SCHUNK,1979) and continuity equations. Arrows show the directions of ion flow. There is an H+ flux direction reversal above 1000 km. The H+ to O+ temperature ratio was equal to 0.8.
-
L-k, 25GfJ
An,C
2000
:
AF, , An;, AF; Fig. 8. C differences obtained comparing the MS solutions with those of Conrad and Schunk shown in the previous figure. A&, the H+ concentration difference. was smaller than 1% and is not shown.
of 10%. AF$ and AFS increased only in the vicinity of the flow direction reversals. At high latitudes the ion temperature ratio may become larger. A number of calculations were therefore performed for an H+ to Of temperature ratio increased up to a value of 10. It was found that even in this extreme case, over a broad range of ionospheric conditions, the C differences stay below the 30% level. In Fig. 9 an example of the altitude behaviour of A.ng is displayed for ion temperature ratios of 2,5 and 10. Corresponding Ani were much smaller. Figure 10
39
Diffusion and continuity equations .
H [Km)
H IKm)
1.
2500
5%
10%
1%
20%
25%
An:
An&AF,C, AFb
Fig. 9. C differences of O+ concentration calculated for H+ to 0+ temperature ratios equal to 2,5 and 10.
Fig. 10. C differences corresponding to calculations with an H+ to O+ temperature ratio equal to 5. An’, was small and is not shown in the figure.
shows the C differences calculated for this ratio equal to 5. In addition, a comparison of solutions corresponding to both approaches for various ion to electron temperature ratios was made, but it was found that if ion temperatures are equal then both sets of equations are equivalent. The final formulae of Conrad and Schunk are not affected by large differences between ion and electron temperatures and even if electron temperature deviates significantly from ion temperature the solutions of both sets of equations are identical, provided that O+ and H + temperatures are equal.
typically differ by 40-50% from those produced by the standard approach, and in some circumstances the difference could be as large as 200%. This effect is mostly due to the presence of the correction factor Au accounting for the influence of heat flow on ordinary diffusion. It was also found that the improvement of the diffusion equations proposed by CONRAD and SCHUNK (1979) allowing treatment of the flow of ions with large temperature differences between the interacting species in a more consistent way leads to a smaller effect. The concentrations found while integrating the MS equations differ from those calculated from the CS formulae by less than lo%, for conditions typical in the mid-latitude topside ionosphere (the Hf to O+ temperature ratio does not exceed 2). For situations when the ratio of ion temperatures grows to larger values the differences increase slightly and if the Ht temperature is 10 times larger than that of O+ they increase to 30%. These results have been confirmed in extensive calculations performed for various ionospheric conditions.
SUMMARY
The analysis of the solutions obtained by integration of three sets of equations defined in the previous paragraphs proves that the modifications introduced in the equations of ST-MAURICEand SCHUNK (1977) may give rise to a considerable effect. The ion concentrations calculated from their formulae may
REFERENCES BURGERSJ. M.
1969
CONRADJ. R. and SCHUNKR. W. EVANSJ. V. and HOLT J. M.
Flow Equations for Composite Gases. Academic Press, New York.
1979 1978 1949 1958 1977a
J. geophys. Rex 84,811. Planet. Space Sci. 26, 729. Commun. pure appl. Math. 2,33 1. Han&k Phys. XII, 205. J. geophys. Res. 82,2139.
GRAD H.
GRADH. HEDIN A. E., SALAHJ. E., EVANSJ. V., REBBRC. A., NEWTONG. P., SPENCER N. M.. KAYSERD. C., ALCAYDED., BA~JERP., C~GGERL. and MCCLUREJ. P.
40 HEDIN A. E., REBEK (‘. A., SPENCEKiv. W.. BRINTON H. C. and MAYR H. G. RAITT W. J.. SCHKJNKR. W. and BANKS P. M. SCHuNK R. W. SC-HUNKR. W. and WALKER J. C. G. ST-MAURICE J. P. and SCHI:NK R. W.
J. GKO(‘HI.l.SKA IY77h 1975 1975 I%9
IO77
Plunet. Spucr Sci. 23, Planer. Space Sci. 23, PlaneI. Space Sci. 17, Planet. Space Sri. 25,
1103 437.
853. 907.
of Atmospheric
and Terrestrial
Printedin Great Britain.
Physics,
Vol. 50, No. I, pp. 3340,
CCJZI-9169/88 $3.00+ .oO 0 1988 PergamonJournalsLtd.
1988.
The effect of the large ion temperature differences on the solutions of diffusion and continuity equations JOANNA
GR~CHULSKA
Space Research Centre, Polish Academy of Sciences, 01-237 Warszawa, ul. Ordona 21, Poland (Received infinalform 30 June 1987)
Abstract-Diffusion and continuity equations for O+ and H+ ions were integrated along a magnetic field line between 500 and 3000 km. Three sets of equations were used: (1) the standard ones; (2) modified according to the formulae given by St-Maurice and Schunk; (3) those derived by Conrad and Schunk. The differences between solutions obtained for each set of equations and the importance of the modifications introduced in the diffusion equations are discussed.
introduced into his formulae were taken from BURGERS(1969), who calculated them assuming that the species drift velocity differences are small in comDuring the last few years modified equations suitable parison with the species thermal speeds and that the for modelling of the topside ionosphere have been species temperature differences are small in comproposed in a number of papers. One of the propparison with the average temperature of the gas ositions was given in 1977 by ST-MAURICE and mixture. The collision terms obtained making this SCHUNK (1977). They derived diffusion equations for assumption are referred to as Burgers’ ‘linear’ colthe mid-latitude topside ionospheric plasma composed of two major ions, electrons and a number of lision terms. There is one special situation when the collision terms can be calculated rigorously from minor ions, assuming that the temperature differences Boltzmann’s collision integral, namely the Maxwell between the components are small in comparison with the average temperature of the gas mixture. CONRAD molecule interactions for which the collision frequency is independent of velocity. For this case the and SCHUNK(1979) obtained more general diffusion collision terms derived from the 13-moment approxiequations allowing for large temperature differences mation are valid for both large temperature differbetween the components. Both these approaches are based on the system of transport equations for gas ences and large drift velocity differences between the interacting species. For all other interactions the colmixtures given by SCHUNK (1975). SCHUNK (1975) presented a system consisting of lision terms given in SCHUNK(1975) are appropriate only for low speed flows with small temperature and continuity, momentum, internal energy, stress tensor velocity differences. and heat flow equations--separately for each comST-MAURICEand SCHUNK(1977) obtained diffusion ponent of the gas mixture. Such an approach makes it possible to take into account the separate velocities equations for mid-latitude ionospheric plasma from and temperatures of the components. The above sys13-moment momentum and heat flow equations with ‘linear’ Burgers’ collision terms as given in SCHUNK tem of transport equations was obtained by multiplying Boltzmann’s equation-for each com(1975) by making several assumptions. They assumed ponent-by an appropriate function of velocity and steady state flow and neutrality of the plasma and then integrating over the velocity space. Equations neglected stress and non-linear acceleration terms, as obtained in this way do not constitute, however, a well as density and temperature gradients perclosed set. In order to close it we must choose a suitpendicular to geomagnetic field lines and currents parable expression for the velocity distribution function. allel to them. Omitting the small terms proportional to the ratios (m,/m,)“’ and (T,- T,)/T,, where m, and This function is also necessary to evaluate the collision terms, which appear on the right-hand-sides of the T, are species mass and temperature, they obtained diffusion equations modified with respect to the stantransport equations. Schunk used in his derivation Grad’s 13-moment approximation of the distribution dard ones. The modifications arise from the presence function (GRAD, 1949, 1958). The collision terms he of a correction term accounting for the heat flow INTRODUCTION
33
effect on ordinary
diffusion
and from the Lvzt that
process is represented by terms proportional to the gradient of the temperature ol each ion separately. In the paper of CONKAD and %HUNK (1979) the diffusion equations for fully and partially ionised plasma are derived assuming that the gas component drift velocity differences are again much smaller than the species thermal speeds, but allowing for large temperature differences between the interacting species. The authors derived the equations in a similar way and based on similar assumptions as in the work of ST-MAURICE and &HUNK (1977), except that they took Burgers’ ‘semilinear’ collision terms--valid also for the large temperature differences between the species-instead of ‘linear’ ones appropriate only for small differences. The obtained expressions are modified with respect to those of St-Maurice and Schunk, although the thermal diffusion process is again governed by the temperature gradients of each plasma component, and a correction factor accounting for heat flow is again present in the formulae for the ordinary diffusion coefficients. It is the aim of this work to investigate how important--for the modelhng of topside ionosphere dynamics--are modifications allowing for a more rigorous description of flows with large temperature differences between the interacting species. The integration of the diffusion and continuity equations for O+ and H+ ions was carried out taking alternately the diffusion equations corresponding to the ‘linear’ and ‘semilinear’ approximations of Burgers for collision terms. for the same parameters and boundary conditions in each case, so that the differences between the solutions found when integrating the formulae of St-Maurice and Schunk and those of Conrad and Schunk were due to the modifications introduced in the latter case. One has to confront the above approaches with that of SCHUNK and WALKER (1969), where it is assumed that the plasma components have a common, equal temperature. To this end the standard diffusion equations, including thermal diffusion with thermal diffusion coefficients derived by SCHUNKand WALKER (1969) were integrated, coupled with continuity equations for 0 + and H+ ions. The obtained solutions were compared with the ones found for the St-Maurice and Schunk equations. Such a comparison enables one to conclude how important the influence of heat flow is on ordinary diffusion and whether the different ways of representing thermal diffusion-by the approach of ST-MAURICEand SCHUNK(1977) or that of SCHUNK and WALKER (1969Fgive rise to large discrepancies between the solutions found in modelling of the topside ionosphere. the thermal
diffusion
The diffusion equations derived by ST-MAIIKKF and SCHI,NK (1977) and by CONRAD and SCHUNK (lY79), dcscrihing the flow of Hi and 0’ ion>. have the following
form
u, = u, - D /
1
1vn,n,
mic
I
Zf,
+
T,
VT +
7: Vn,
n,
T,
and ui = q-D,
Vn, - ;$
T Vn,
/
+ ; VT, + +
/
n
/
where n,, u,?,m,v and T, are number density, velocity, mass and temperature, respectively, for species s and s = i or j, where i refers to 0+ and j to H +. k is Boltzmann’s constant, G is the acceleration due to gravity and a$ and aI, are the thermal diffusion coefficients. Di and Di, the ordinary diffusion coefficients, are given below. D, =
D, =
kT,
1
wz,v,, (I 1 A<,) kT,
’
I
nz,t~,;(1 -A,,)
’
where v,~~ is the collision frequency between species s and t and A,, is a correction factor which accounts for the effect of heat flow on ordinary diffusion. The differences between the approach of St-Maurice and Schunk (MS) and that of Conrad and Schunk (CS) emerge in the expressions for A,, CL,, and c$. The formulae for Ar”, avS r, and a$ MSderived by St-Maurice and Schunk, assuming that the differences between the temperatures of interacting species are small and Ass a” and u *” found by Conrad and Schuik, for :a;ge”temperlture differences, are given in the papers cited. The standard diffusion equations commonly used to model 0 + and H + flow can be obtained from (1) and (2) if we remove the modifications introduced in both papers discussed above. To do so we neglect the influence of heat flow on ordinary diffusion, putting Aij equal to zero, and assume that O+ and H+ temperatures are equal. Ordinary diffusion coefficients then reduce simply to
Diffusion and continuity equations D,
=kr,
t?liVij
D.=?!L / nljv,
where Tp is plasma temperature. Thermal diffusion is then represented by one term only, proportional to the gradient of Tp. The thermal diffusion coefficients are usually those of SCHUNK and WALKER (1969), derived with one common temperature assumed for all ionic components. In many modelling attempts where the standard equations have been applied thermal diffusion was neglected, but whenever they are referred to in this paper it is tacitly assumed that they include this process. To find out how important are both the MS and CS modifications the integration of three different sets of continuity and diffusion equations for O+ and H+ was carried out downwards from a height of 3000 km, along the magnetic field line corresponding to L = 3.2, to a height of 500 km. The first set under consideration includes the diffusion equations in their standard form. The second and third sets correspond, respectively, to the MS and CS modifications of the diffusion equations. In all three sets the continuity equations for O+ and H+ had the following form
f&w =a--L,. This is the stationary continuity equation for ions flowing along a geomagnetic field line, where s is the arc length along it, A is the cross-sectional area of the magnetic flux tube and Qs and L, are the production and loss rates of ion s. The production of ion O+ is due to photoionisation of neutral atomic oxygen and to the charge exchange reaction H++O-+O++H.
(4)
The loss of O+ is due to reactions with molecular nitrogen and molecular oxygen and to the charge exchange reaction O++H+H++O.
(5)
The production of H+ is due to reaction (5) and loss to reaction (4). The rate coefficients needed in the continuity equations were taken from RAIrr et al. (1974). The neutral atmospheric model was taken from HEDIN et al. (1977a, b) and ion and electron temperatures from EVANSand HOLT (1978) (for low altitudes). At higher levels, above 1000 km, it was assumed that the plasma temperature gradient was equal to either 1K km-’ or 2K km-‘.
35
Integration of these sets of four ordinary differential equations was performed using the RungeKutta method of fourth order. It was assumed that at the upper boundary O+ velocity was equal to zero. Such an assumption does not infhtence the final conclusions, as the solutions are not sensitive to the value of O+ flux unless it is unusually large. At the lower boundary chemical equilibrium for H+ ions was assumed. The remaining boundary conditions were changed over a wide range of values to produce solutions corresponding to various possible situations in the topside ionosphere and plasmasphere. To obtain one set of solutions (2 concentration and 2 velocity altitude profiles) a ‘shooting’ method was used ; three conditions at the upper boundary were kept constant while the fourth one was varied in search of the solution satisfying appropriate conditions imposed at the lower boundary. The solution obtained while integrating the standard equations will be denoted from now on by the letter S, those found for the equations of St-Maurice and Schunk by the letters MS and the solutions of Conrad and Schunk’s equations by the letters CS.
RESULTSAND DISCUSSION A direct comparison of the MS and S solutions will allow us to estimate how important the modifications introduced by St-Maurice and Schunk (1977) are. The relative differences (%) found between the S and MS solutions will be referred to as S differences. From several calculations carried out for various ionospheric conditions two examples were chosen. The MS solutions describing the ionosphere and plasmasphere in two different states are shown in Figs. 1 and 2. These results were found in such a way that the values of O+ and H+ velocities and H + concentration were kept constant at the upper boundary, while the O+ concentration was varied in search of the solutions which fulfilled the required conditions at the lower boundary. In these calculations the electron and ion temperatures below 1000 km were taken from EVANS and HOLT (1978) and above this level it was assumed that the gradients of both temperatures were equal to 1K km - ’ . The 0 + and H + temperatures were assumed to be identical. Figure 1 shows concentration rr- and Fig. 2 flux F,(F, = n,,u,)- altitude profiles. The continuous curves correspond to calculations in which a neutral atmospheric model with an exospheric temperature of 1500K was used. In this case Fig. 1 reveals the transition height occurring as high as 3000 km, as a consequence of a very low H+ concentration, while Fig. 2 shows the corresponding downward flow of
low ihr the IiOOK atmosphere and nculral oxygct~ are high. As the lower boundar)~ COP dition requires chemical equilibrium for protons, the small valueof lhc ratio of the neutral hydrogen conx-c
H
concentrations
(Km
250c
2000
1500
1000
la’
Id
tt?
no, nH km-’ ) Fig. I. 0 + and H f concentration profiles obtained by integration of diffusion (ST-MAURICE and SCHUNK, 1977) and continuity equations. The continuous and dashed lines correspond to calculations using neutral atmospheric models with exospheric temperatures. resDectivelv,of 15OOKand 75k .
H IKmf 2500
Fig. 2. 0 + and Hi flux profiles corresponding to the concentrations shown in Fig. 1. Arrows indicate directions of flow. There is an H + flow direction reversal in the 750K case and an O+ one in the I SOOKcase.
protons for all altitudes, with the reversal of the O+ flow direction just below 1000 km. In both figures the dashed lines refer to the 750K neutral model. Here, in contrast, the ionosphere is rich in H+, the transition height occurs well below 1000 km and O+ flux is always directed upwards, while proton flux is upward above about 1300 km and downward below this level. Such a low H+ concentration in the 15OOK calculations, in spite of the proton influx from above, is due to the fact that neutral hydrogen concentrations
centration to that ofncutral oxygen leads to H ’ concenlratmns about 3 orders of magnitude smaller than those of 0 ‘. At higher altitudes the proton Ilow from the protonosphere results in an increase in i-l * concentration. However, this H downward flux (of the order of 1Ohcm ~’ s ! ) is too small to increase the H+ concentration su~~iently to move the transitjon height to lower altitudes. In the case of the 750K calculations the relatively large ratio of neutral hydrogen to neutral oxygen concentration ensures effective proton production in the entire region considered and in the case represented by the dashed lines there arc a lot of H + ions, in spite of proton outflow. Some of the S differences are shown in Fig.3 (750K case) and in Fig. 4 (1500K case). There is a pronounced dependence of the magnitude of the differences on the rate of change of the variable in question. An inspection oE Figs. 3 and 4 with reference to Figs. 1 and 2 reveals that whenever the absolute gradient value for any variable increases, this is foIlowed by a ~orrespollding growth in the respective differences. This is seen in Fig, 3 in the vicinity of H + flux direction reversal, where a steep increase of ,4FT, occurs. Similarly. near the lower boundary a fast decrease of H” and 0’ fluxes at 500 km results in a sharp increase of AFL and AF:. The bchaviour of the flux differences generally follows that of the concentrations, except in the neighbourhood of the flux direction reversal, where even a slight shift of the reversal point produces misleadingly large relative flux differences. Therefore, in Fig. 4 and further on only the concentrations are discussed. The obtained differences are much more important for 0 ‘. ions than for H+ ions. This effect is partly due to the computational procedure itself, where 0’ concentration was varied in search of the particular boundary condition imposed, while H + concentration and velocity, as well as 0’ velocity, were kept constant. Extensive calculations of this kind were performed for a broad range of pertinent parameters and boundary conditions. In particular, the consequences of assuming equal ion temperatures were investigated by varying the Ii’ to Of temperature ratio and it was found that even in the extreme case of this ratio reaching a value of 2 the final effect on the differences produced was small for ail plausible ionospheric conditions. Apart from this, some clear trends were discovered to govern the behaviour of the observed differences : for an upward proton flux an upper limit
31
Diffusion and continuity equations H
1
1
(Kill
,KHm,,
2500 .
25w
15%
25%
35%
I
I
I
In I I
H
45%
AF,S,AF,:A$ Fig. 3. S differences obtained by comparing the solutions of standard equations with those shown as dashed lines (750K case) in Figs. 1 and 2. AFSu,AFS,andAnt are H+ flux, Of flux and O+ concentration differences, respectively. A& the H+ concentration difference, was omitted as it was smaller than 6%.
Fig. 5. Of and H+ concentration profiles found from the MS equations for low (dashed lines) and high (continuous lines) ion density.
An: 2000 -
\ I I I I
\ 1 I ,
1500-
\ \
50%
15%
25%
Ant,
25%
45%
An:
Fig. 4. S differences for Of and H+ concentrations, corresponding to the MS solutions shown as continuous lines (1500Kcase) in Figs. 1 and 2.
of 60% was never exceeded, while in the case of proton flow directed downwards in the entire region considered the differences could grow well above this value ; in special circumstances values as high as 200% can be produced. An example of the latter situation is illustrated in Figs. 5 and 6. Two pairs of curves are shown on each figure. The MS concentrations drawn as dashed and continuous lines in Fig. 5 correspond to the respective drawings of differences in Fig. 6. The directions of the ion fluxes associated with the concentrations shown in the figures were in both cases similar: the Of flux was upward and the H+ one
100%
150%
An: , An: Fig. 6. S differences for H+ and O+ concentrations found for the two situations shown in Fig. 5 and marked correspondingly.
downward in the entire region. However, in the ‘continuous’ case the H+ flux at 3000 km reached a relatively large value of 5 x 10” cm-’ s-l, while the flux corresponding to the situation drawn as dashed lines was equal there to 1 x 10’ crnW2s-l. Both pairs of MS solutions shown in Fig. 5 were found using in the calculations the 750K neutral atmospheric model ; the electron and 0 + temperatures below 1000 km were taken from EVANS and HOLT (1978) and H+ temperature was assumed to be 2 times larger than that of Of. Above 1000 km the assumption was made that for O+ temperature the latitude gradient is equal to 1K km-’ and for H+ 2K km-‘. These solutions were found using the searching procedure with the H + concentration varied at 3000 km and therefore at higher altitudes An”, is larger than An:.
15
.I GKou~C
In l’ig. 6 we see that the ‘dashed’ A& reaches almost 200%, while the differcncea in the ‘continuous’ ease never exceed 30%. This illustrates the fact that the S differences usually mcreasc in cases when the 0’ concentration is comparable to that of H + over a wide range of altitudes. WC set that 111the ‘dashed’ case both ion concentrations are relatively close to one another, because 0 ’ concentration slowly decreases with altitude in the vicinity of the transition height and both ion species have an ample opportunity to interact, as opposed to the conditions in the ‘continuous’ case. It must be stressed that this effect was found to be independent of the absolute values of the concentrations themselves, as the extremely high H + concentration (continuous line) might perhaps suggest. The following conclusion can be drawn.from the analysis of all the computations performed : the MS modifications are mainly due to the presence of the A,, correction factor in the ordinary diffusion coefficients. The effect of A,, is from 5 to 30 times larger, depending on the parameters and boundary conditions, than that due to the differences in description of the thermal diffusion process itself. In particular, it was found that, in agreement with suggestion of ST-MAURICE and SCHUNK (1977), the thermal diffusion terms proportional to 0’ temperature are unimportant and lead to differences below the level of I %. Now we will analyse in turn the modifications of CONRAD and SCHUNK (1979) allowing for large differences between the temperatures of the interacting species. The solutions of the CS equations will be compared with those of St-Maurice and Schunk and the differences obtained will be referred to as C differences. For the mid-latitude ionosphere it seems reasonable to assume that the ratio of H+ to 0’ temperatures varies in the range 0.8 -2. In Fig. 7 the CS solutions are shown. They were obtained for the 750K neutral atmosphere, the O+ and electron temperatures were taken from EVANS and HOLT (1978) and the H+ temperature was assumed to be equal to 0.8 of that of 0.‘. The corresponding C differences are presented in Fig. 8. As 0’ concentration at 3000 km was the varied parameter in the searching procedure, the differences concerning this ion are the most pronounced at higher altitudes, but even in this case they do not exceed 5%. The sudden increase of AFE near 1000 km is due to Hi flux direction reversal there. A& is everywhere smaller than 1% and was not drawn in the figure. Similar calculations were repeated taking the ratio of H+ to 0’ temperature equal to 1.2, 1.5 and 2, for various parameters and boundary conditions, and it was found that An: and An:; never exceeded a value
l.SK \
ld
lo4
?d Icni’l
nOJnH Fig. 7. O+ and H+ concentration and flux profiles obtained by integration of diffusion (CONRADand SCHUNK,1979) and continuity equations. Arrows show the directions of ion flow. There is an H+ flux direction reversal above 1000 km. The H+ to O+ temperature ratio was equal to 0.8.
-
L-k, 25GfJ
An,C
2000
:
AF, , An;, AF; Fig. 8. C differences obtained comparing the MS solutions with those of Conrad and Schunk shown in the previous figure. A&, the H+ concentration difference. was smaller than 1% and is not shown.
of 10%. AF$ and AFS increased only in the vicinity of the flow direction reversals. At high latitudes the ion temperature ratio may become larger. A number of calculations were therefore performed for an H+ to Of temperature ratio increased up to a value of 10. It was found that even in this extreme case, over a broad range of ionospheric conditions, the C differences stay below the 30% level. In Fig. 9 an example of the altitude behaviour of A.ng is displayed for ion temperature ratios of 2,5 and 10. Corresponding Ani were much smaller. Figure 10
39
Diffusion and continuity equations .
H [Km)
H IKm)
1.
2500
5%
10%
1%
20%
25%
An:
An&AF,C, AFb
Fig. 9. C differences of O+ concentration calculated for H+ to 0+ temperature ratios equal to 2,5 and 10.
Fig. 10. C differences corresponding to calculations with an H+ to O+ temperature ratio equal to 5. An’, was small and is not shown in the figure.
shows the C differences calculated for this ratio equal to 5. In addition, a comparison of solutions corresponding to both approaches for various ion to electron temperature ratios was made, but it was found that if ion temperatures are equal then both sets of equations are equivalent. The final formulae of Conrad and Schunk are not affected by large differences between ion and electron temperatures and even if electron temperature deviates significantly from ion temperature the solutions of both sets of equations are identical, provided that O+ and H + temperatures are equal.
typically differ by 40-50% from those produced by the standard approach, and in some circumstances the difference could be as large as 200%. This effect is mostly due to the presence of the correction factor Au accounting for the influence of heat flow on ordinary diffusion. It was also found that the improvement of the diffusion equations proposed by CONRAD and SCHUNK (1979) allowing treatment of the flow of ions with large temperature differences between the interacting species in a more consistent way leads to a smaller effect. The concentrations found while integrating the MS equations differ from those calculated from the CS formulae by less than lo%, for conditions typical in the mid-latitude topside ionosphere (the Hf to O+ temperature ratio does not exceed 2). For situations when the ratio of ion temperatures grows to larger values the differences increase slightly and if the Ht temperature is 10 times larger than that of O+ they increase to 30%. These results have been confirmed in extensive calculations performed for various ionospheric conditions.
SUMMARY
The analysis of the solutions obtained by integration of three sets of equations defined in the previous paragraphs proves that the modifications introduced in the equations of ST-MAURICEand SCHUNK (1977) may give rise to a considerable effect. The ion concentrations calculated from their formulae may
REFERENCES BURGERSJ. M.
1969
CONRADJ. R. and SCHUNKR. W. EVANSJ. V. and HOLT J. M.
Flow Equations for Composite Gases. Academic Press, New York.
1979 1978 1949 1958 1977a
J. geophys. Rex 84,811. Planet. Space Sci. 26, 729. Commun. pure appl. Math. 2,33 1. Han&k Phys. XII, 205. J. geophys. Res. 82,2139.
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GRADH. HEDIN A. E., SALAHJ. E., EVANSJ. V., REBBRC. A., NEWTONG. P., SPENCER N. M.. KAYSERD. C., ALCAYDED., BA~JERP., C~GGERL. and MCCLUREJ. P.
40 HEDIN A. E., REBEK (‘. A., SPENCEKiv. W.. BRINTON H. C. and MAYR H. G. RAITT W. J.. SCHKJNKR. W. and BANKS P. M. SCHuNK R. W. SC-HUNKR. W. and WALKER J. C. G. ST-MAURICE J. P. and SCHI:NK R. W.
J. GKO(‘HI.l.SKA IY77h 1975 1975 I%9
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