by
Thomas Smid
Copyright Thomas Smid
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Historically, much of the theory of radar backscatter from the ionosphere can be considered as a chain of original mispredictions which only subsequently were corrected through further theoretical arguments. This started with the observation that incoherent scatter lines are usually much narrower than predicted by Gordon (1958) on the basis of the Thomson scattering theory and includes for example misconceptions like the Jicamarca radar facility which was positioned near the magnetic equator in order to observe the theoretically expected gyro-resonances but failed to detect these (see Evans (1969) and references therein). These failures have not led to a re-consideration of the basic theoretical assumptions, but ionospheric theory seems to have disintegrated instead into a wealth of incoherent concepts each invented to explain individual measured features a posteriori (as an indication, one may take the fact that more than hundred different 'plasma instabilities' and corresponding 'waves' are claimed to be known (Giraud and Petit (1978)), many of which have entered into ionospheric theory at convenience. From a scientific point of view, this over-abundance of individual ad-hoc concepts is not only unsatisfying but indicates clearly that the original theoretical propositions are incorrect.
On this background, the work of Smid (1993) establishes the theory of scattering of electromagnetic waves by highly excited atoms formed by recombination from a plasma and proposes that this mechanism accounts for the scattering of radio waves in the ionosphere in general. This conclusion can be reached by theoretical arguments and is supported by the numerical results presented in Smid (1993) which show good quantitative agreement between the corresponding theoretical scattering cross section and experimentally derived values and give for instance also a straightforward explanation for the time scales related to the so called 'short wave fadeout' in connection with solar bursts. In contrast, the more established theories do in both cases not provide quantitative answers in an absolute sense.
It is apparent that with the scattering being due to atoms instead of electrons, the order of magnitude for the normally observed line width for incoherent scatter is readily explained. In the F- region, the exact line profile should however correspond to the ion velocity distribution function rather than the (ground state) neutrals, as the life time of the excited neutral states (being formed by ion-electron recombination) is short compared to the collisional relaxation time (see Fig.1 in Smid (1993)). Only for smaller heights are elastic atom-atom collisions so frequent that the excited states attain a Maxwellian distribution, in qualitative agreement with the observed height dependence of incoherent scatter line profiles.
It is therefore obviously vital to compute the ion velocity distribution function in the F- region ionosphere consistently through a solution of the Boltzmann equation.
As far as the applicant is aware, this problem has not been formulated and solved yet.
In fact, known theories of plasma diffusion in the undisturbed ionospheric F- layer generally assume Maxwellian velocity distributions both for the ions and electrons (i.e. LTE) despite the essentially collisionless nature of the problem. The inconsistency of this approach is for instance apparent from the treatment in the textbook of Rishbeth and Garriott (1969) (Chpt. 4.3) : the assumption of quasi neutrality (ni(h) =ne(h) for all heights h) implicates obviously an electric polarization field E(h)=0, in contradiction to the value derived from the suggested force equations for the electrons and ions. More important, the assumed Maxwellian distribution results in identical upward and downward fluxes of both constituents at any height, i.e. despite a spatial gradient no polarization field would be necessary as the diffusion terms would vanish identically, again in inconsistency with the derived value.
From these paradoxical results in case of an LTE- approach applied to an almost collisionless inhomogeneous plasma, it is apparent that only a consistent solution of the Boltzmann equation is capable of describing the plasma diffusion problem for the ionospheric F-region correctly.
Non- Maxwellian distribution functions have been derived for certain special cases of upper atmospheric physics: the velocity distribution for neutral hydrogen in the collisionless exosphere has been considered by Chamberlain (1963), Bertaux (1978), Fahr and Paul (1976), Fahr and Weidner (1977). For ions at high latitudes, Schunk and Walker (1972), St- Maurice and Schunk (1979) obtained deviations from LTE for electric fields E= 10mV/m, neglecting however spatial gradients in the Boltzmann equation. Electric field strengths of this order of magnitude can only be produced by high energy particles entering the atmosphere from interplanetary space in the region of open field lines and are therefore not present in the undisturbed low - or mid- latitude F- region. For the latter, diffusion over distances of the order of one atmospheric scale height should however produce significant effects even for the much smaller steady state electric field related to the intrinsically ionospheric electrons. Spatial gradients can then not be neglected any more but are, on the contrary, the essential aspect of the problem. A solution for the corresponding Boltzmann equation, which in this generality has not been considered yet for the ionospheric plasma diffusion problem, is suggested in Section 2. Despite severe problems of evaluating the formal solution numerically, preliminary results illustrating the diffusive ion dynamics have already been obtained. They indicate that the corresponding distribution function in the ionospheric F- region is in fact strongly non-Maxwellian and could account for the details of the observed 'incoherent scatter ion line' shape in terms of the the scattering theory proposed by Smid (1993) (note that for the latter the scattered line shape reflects directly the velocity distribution function, in contrast to the accepted theory where non- Maxwellian distribution functions are claimed to lead to the disappearance of the familiar 'double-humped' profile and the establishments of a 'single humped' line in connection with auroral radar backscatter (Swift, 1975; Venkat Raman et al., 1981; Hubert, 1984; Farmer et al., 1988) ).
Within the proposed research program, the numerical approach outlined in Sect.2 shall be improved in a way that a quantitative comparison with experimentally obtained line profiles is directly possible. For this purpose, other than plasma diffusion effects which might determine the backscatter profile shall also be considered. These points are discussed in more detail in the concluding Sect.3.
2. Description of Previous Work
As already indicated in the introductory section, the problem of plasma diffusion in an inhomogeneous atmosphere has not been considered yet in the literature (at least not in the context of ionospheric physics) and therefore no theoretical basis apart from the familiar Boltzmann equation in its general form is available.
In the following, a so far unpublished approach by the applicant to solve the latter is therefore described. Preliminary numerical results are presented which have been obtained for the ion velocity distribution function in a first order approximation and/or for simplified boundary conditions. The insight gained from these concerning the non-LTE plasma diffusion problem shall serve as a basis for a more consistent numerical scheme in order to obtain velocity distribution functions which are quantitatively directly comparable to measurements of ionospheric backscatter lines.
2.1 Formal Solution of the Boltzmann Equation for the Ionospheric Plasma
The Boltzmann equations for the electrons and ions in a stationary state can generally be written as
v.Ñrne(r,v) + Fe/me .Ñvne(r,v) = qeIon(r,v) -leRec(r,v) + Ce(r,v) (1)
v.ÑrnI(r,v) + FI/MI .ÑvnI(r,v) = qIIon(r,v) -lIRec(r,v) + CI(r,v) (2)
The left hand side of the equations describes the production and loss rates for the velocity specific density due to convection in geometrical and velocity space, whereas the right hand side contains the local production and loss rates due to ionization, recombination and collisions.
For the present case of plasma convection in the ionosphere, transport is effectively confined to a direction along the magnetic field lines and the transverse component can be considered to be dynamically decoupled from this and isotropic due to the small larmor radii of both constituents in comparison to the atmospheric scale height.
One can therefore restrict the problem to the convection along the magnetic field line by considering the one dimensional equations
v . ¶ne(s,v)/¶s + ae(s) .¶ne(s,v)/¶v = qeIon(s,v) -leRec(s,v) + Ce(s,v) (3)
v . ¶nI(s,v)/¶s + aI(s) .¶nI(s,v)/¶v = qIIon(s,v) -lIRec(s,v) + CI(s,v) , (4)
where
ae(s) = g(s) - e.E(s)/me , (5)
aI(s) = g(s) + e.E(s)/MI (6)
are the accelerations of the electrons and ions due to gravitation and the electric plasma polarization field. The latter has in principle to be determined consistently together with Eqs.(3) and (4) by adding Gauss's equation. Due to the additional problems introduced by this circumstance and because the main intention is to compute the ion velocity distribution function, it is reasonable to obtain first numerical solutions only for Eq.(4) with the polarization field treated as a free parameter. In order to do this, one has to specify the local production and loss terms on the right hand side of the equation, as only the production due to ionization qIIon is an independent quantity. The loss rate due to recombination can be written as
with the specific recombination coefficient given by
¥
aIRec(s,v) = òdv’ fe(s,v’) . sRec(vRec(s,v,v’)) .|vRec(s,v,v’)| , (8)
-¥
fe(s,v) = ne(s,v)/Ne(s) (9)
the electron velocity distribution function and
¥
Ne(s) = ò dv’ ne(s,v’) . (10)
-¥
For the present consideration, aIRec(s,v) is practically independent of v as vRec is determined by the much higher electron velocities v' in Eq.(8) . Assuming furthermore that the local deviations from charge neutrality connected to the plasma polarization field are so small that
¥
NI(s) = ò dv’ nI(s,v’) » Ne(s) . (11)
-¥
Eq.(7) can be written as
lIRec(s,v) = nI(s,v) .NI(s) . aIRec(s) . (12)
The collision term in Eq.(4) has to be separated into a production and loss term, i.e.
CI(s,v) = qIc(s,v) - lIc(s,v) . (13)
If only elastic collisions with neutrals are relevant, these are given by
¥
qIc(s,v) = 1/Öp/v0 .e-(v/v0)2.NN(s). sc . ò dv’ nI(s,v’) .vc(s,v’) (14)
-¥
with the collision cross section sc assumed here to have a fixed value of 2.10-16 cm2 independent of the relative velocity of the ions and neutrals which schematically has been defined here through
vc(s,v) = Ö(v0,N2(s) +v2) , (16)
where v0,N is the thermal velocity dispersion of the neutrals (the above expression shall approximately describe the situation for an isotropic distribution of velocity vectors of the neutrals).
The form of Eq.(14) assumes that each collision of an ion leads to a complete redistribution in velocity space according to the (height independent) Maxwellian velocity distribution function of the neutrals. This approximation appears to be reasonable for the present purpose but a more accurate quantitative theory should take the differential collision cross section consistently into account.
Through rearrangement of the terms, Eq.(4) can now formally be written as a first order linear differential equation in either the variable s or v, i.e.
¶nI(s,v)/¶s + nI(s,v) . Ps(s,v) - Qs(s,v) = 0 (17)
¶nI(s,v)/¶v + nI(s,v) . Pv(s,v) - Qv(s,v) = 0 , (18)
where
Ps(s,v) = 1/v. [ nl(s,v)+ aI(s)/nI(s,v).¶nI(s,v)/¶v ] (19)
Qs(s,v) = 1/v . [ qIIon(s,v) + qIc(s,v) ] (20)
Pv(s,v) = 1/aI(s). [ nl(s,v) + v/nI(s,v).¶nI(s,v)/¶s ] (21)
Qv(s,v) = 1/aI(s) . [ qIIon(s,v) + qIc(s,v) ] (22)
with the collision frequency nl given by
nl(s,v) = [ NI(s). aIRec(s) + NN(s). sc.vc(s,v) ] . (23)
The formal solutions of Eqs.(17) and (18) determine nI(s,v) by integration with regard to either s or v in the form (Smirnov, 1964)
s s s
nI(s,v) = nI(0,v) . exp[ - òds’Ps(s’,v)] + òds’ Qs(s’,v).exp[ -òds’’Ps(s’’,v)] (24)
0 0 s’
nI(s,v) = nI(s,v0). exp[ -òdv’Pv(s,v’)] + òdv’ Qv(s,v’).exp[ -òdv’’Pv(s,v’’)] (25)
v0 v0 v'
Through the dependence of P and Q on nI(s,v) (see Eqs.(19)-(23), 11,14), the above expressions represent non-linear integral equations which have to be solved numerically. Several problems arise from the specific forms of P and Q :
the factors 1/v and 1/aI(s) in Eqs.(19) -(22) cause singularities for zero velocities or accelerations. Furthermore, the appearance of the gradient expressions in P tends to lead to a numerical instability which is notorious for hyperbolic differential equations as given by the original Eqs.(3) and (4) (see for instance Smith, 1984 ) and may be considered as one reason why a consistent solution of the Boltzmann equation has not been achieved yet.
Since the production rate due to ionization qIIon(s,v) is the only invariant quantity ('kernel') in Eqs.(24) or (25) and the boundary values nI(0,v) or nI(s,v0) are in general not given but have to be determined consistently together with the other function values, either of the equations has to be solved by iterative rather than simple integration. This introduces the problem that the initial assumption for nI(s,v) differs usually so strongly from the consistent solution that the absolute value of the coefficient P gets easily so large that the exponentials in Eq.(24) and (25) become numerically intractable. Problems in this sense occur in particular due to the discontinuities of the medium at the physical boundaries and the numerical mesh points as the spatial gradient becomes ill defined here.
Several measures have been taken in order to improve the numerical stability with regard to the above mentioned aspects: special subroutines extend the numerical range and reduce thus the likelihood of computational over/- underflow for large arguments of the exponent in Eqs.(24) or (25). Although this procedure widened the range of applicability, convergence of the iteration process was still not obtained if the convection term became dominant in Ps or Pv as long as just one of the equations was iterated (i.e. as long as integration was only performed in one of the independent variables with the other rather treated as a parameter). Stability was only reached by (inconsistently) assuming the gradient with regard to the complementary variable as given, since then the propagation of errors in variable space typical for hyperbolic differential equations does not occur. Consistency could be re-established however by iterating Eqs.(24) and (25) under this assumption alternatingly, where Eq.(24) integrated for the total ion density with the velocity distribution function fixed and vice versa for Eq.(25), i.e.
s
NI(s) = 1/fI(s,v). [NI(0).fI(0,v) .Ts(0,s,v) + òds’ Qs(s’,v). Ts(s’,s,v) ] (26)
0
v
fI(s,v) = fI(s,v0) .Tv(v0,v,s) +1/NI(s). òdv’ Qv(s,v’). Tv(v’,v,s) (27)
v0
where in correspondence to Eqs.(9)-(11)
fI(s,v) = nI(s,v)/NI(s) (28)
and furthermore the abbreviations
s
Ts(s’,s,v) = exp[ -òds’’Ps(s’’,v)] (29)
s’
v
Tv(v’,v,s) = exp[ -òdv’’Pv(v’’,s)] (30)
v’
have been used.
Eq.(26) and (27) provide now two coupled non-linear integral equations with regard to the total ion density and the ion velocity distribution function.
By iterating these equations alternatingly, the inherent instability of hyperbolic differential equations was eliminated and convergence of the iteration was also achieved if the convection term dominated the local production and loss terms (at least for physical parameters corresponding to ionospheric conditions). However, the convergence problems due to the discontinuities of a finite medium and the numerical discretization of Eqs.(26) and (27) were not removed by this method. Therefore, so far only first order solutions for the convection problem (which neglect the spatial gradient of the velocity distribution function in comparison to the gradient of the total ion density) could be computed under the assumption of an infinite homogeneous continuation of the medium beyond the boundaries of the inhomogeneous region. With the correct physical boundary conditions, illustrative results could still be obtained for the free diffusion problem (i.e. without any force field) and, as already indicated above, for the case of a given (though not necessarily Maxwellian) velocity distribution function.
The results of these computations are discussed in the next section.
2.2 Numerical results
All results presented in this section are formulated in dependence of a height variable h rather than the line element variable s used in Sect.2.1 as they assume for simplicity a vertical magnetic field line in order to be directly comparable with the classical (LTE) plasma diffusion theory as for instance presented in Rishbeth and Garriott (1969) (Chpt.4.3).
2.2.1 Recombination Effects on the Ion Velocity Distribution Function
in an Inhomogeneous Medium.
Resultant Plasma Polarization Field
An essential feature of the Boltzmann equation for a plasma as formulated in Eqs.(1) and (2) in Sect.2.1 is the appearance of the ionization production and recombination loss terms which are commonly neglected in the classical formulation of the Boltzmann equation. Just these quantities however are responsible for the inhomogeneities of the ionospheric plasma and affect therefore the velocity distribution function through the convection terms in the equations.
In this sense, Fig.1 illustrates the effect which an inhomogeneous ionization rate and the recombination has on the ion velocity distribution function in the absence of an external force field and collisions (only Eq.(24) needed therefore to be iterated; see discussion above Eq.(26)). The spatial gradient of the initial ionization rate is assumed here to correspond to the vertical ionospheric density distribution of atomic oxygen between a height h=200 and 500 km ( òdv qIIon(200,v) =500 cm-3sec-1) with reflecting boundaries at these heights (a reflective upper boundary shall schematically represent the closed situation due to the earths magnetic field structure). In order to exaggerate the recombination effects for the ion velocity distribution function (which is shown here at height intervals Dh=30 km with the length of the dashes increasing from the lower (h*=0 km) to the upper (h*=300 km) boundary), the recombination coefficient has however been taken a factor 10 higher than the actual ionospheric value (a height independent value aIRec = 5.10-8 cm3/sec was used here).
¥ ¥
DFI(h)/FI(h) = òdv’ v’ .fI(h,v’) / òdv’ |v’| .fI(h,v’) (31)
-¥ -¥
Fig.2 shows also the related total ion (plasma) density (dashed curve, left ordinate). It is obvious that the diffusion of the plasma between the two boundaries leads to a considerable smoothing of the original spatial inhomogeneities (as given by the ion production rate) if no external forces are present (the solid curve shows the density height profile if a local ionization/recombination balance without any diffusion is assumed). The density is still not constant within the medium, however, due to the finite mean free path with regard to recombination.
Fig.3 shows the ion velocity distribution functions for the actual ionospheric recombination coefficient aIRec = 5.10-9 cm3/sec (again taken as height independent) and including now elastic collisions with neutrals (NN= 109 cm-3 at the lower boundary, decreasing upwards with a scale height according to atomic oxygen; sc = 2.10-16 cm2 (see Sect.2.1, Eq.(23) ) ).
Due to the smaller recombination probability and the collisional relaxation, the curves differ much less strongly from a Maxwellian than in Fig.1, with the flux asymmetry DFI/FI having a maximum value of about 8% (see Fig.4, long-short dashed curve). The fact that the height gradient of the total plasma density differs hardly from that in Fig.2 despite the much smaller recombination coefficient is explained by the effect of the elastic collisions which tend to trap the ions locally and prevent their free flight.
As already indicated in Sect.1, the asymmetry of the distribution function and the related quantity DF/F determines directly the magnitude of the plasma polarization field. A crude estimate of the latter is obtained from the relationship
Ep = We/e/Hp .DFe/Fe (32)
where We is the average energy of the electrons, e the elementary charge and Hp the average plasma scale height (all to be taken in cgs-units).
If one assumes that the electron flux asymmetry DFe/Fe is approximately equal to DFI/FI (which should be a reasonable assumption as the electrons should tend to adapt to the dynamics of the ions due to their much smaller mass) and that furthermore the gravitational force field does not alter the quantity DFI/FI substantially (as the latter is causally determined by recombination), one finds with We »7 eV (Smid, 1987), Hp = 126 km and the average flux asymmetry from Fig.4 (DFI/FI »5.5%) a polarization field
Ep »10-10 stV/cm =
= 3.10-6 V/m . (33)
(It would of course be desirable to have a self-consistent-field calculation for Ep. This would require a simultaneous solution for the electron convection problem (i.e. of Eq.(3) ) which could be attempted if the numerical problems associated with a consistent numerical computation of the ion convection in a given electric field have been overcome).
2.2.2 Effects of the Plasma Polarization Field on the
Ion Velocity Distribution Function
As discussed in Sect.2.1, both Eqs.(26) and (27) have to be iterated together in order to achieve numerical stability if the convective force term becomes significant. Even then, however, the problem arises that the procedure becomes unstable at those points where the spatial gradient is ill defined, that is in particular at the boundaries of the medium. The effects of the combined gravitational and plasma polarization field could therefore only be considered by making the first order approximation ¶fI(h,v)/¶h<< ¶NI(h)/¶h, taking fI(h,v) to be Maxwellian at the lower boundary and assuming furthermore an infinite homogeneous continuation of the medium beyond the boundaries of the actual finite column.
The latter assumption, however, leads obviously to an incorrect symmetry of the problem as the property of reflecting boundaries (taking schematically the closed magnetic field line situation into account) is removed. As a consequence, the resultant ion velocity distribution functions do not show the correct symmetry and there is a corresponding net mass outflow into the direction of acceleration as given by the assumed force field.
One should note that with the value for the polarization field as calculated at the end of the last section, the gravitational force acting on the ions is overcompensated and leads to an upward acceleration of a magnitude approximately equal to the gravitational (this circumstance alone is already in contradiction to the (inconsistent) classical LTE- plasma diffusion theory where the plasma polarization field compensates just for half the gravitational acceleration and the resulting force is still downward (see for instance Rishbeth and Garriott (1969), Chpt.4.3) ).
Fig.5 shows the resultant ion velocity distribution fI(h,v) (dashed curves) plotted for various heights between the lower (h*=0 km) and upper (h*=300 km) boundary (i.e. h»200-500 km) at intervals Dh*=30 km with the length of the dashes increasing with height (the thick solid curve gives a Maxwellian for comparison). As it is apparent, fI(h,v) becomes systematically broadened and displaced towards positive (i.e upwards directed) velocities with increasing height due to the acceleration of the ions by the plasma polarization field (which was taken as height independent here for simplicity). Fig.6 gives the related height dependence of the total ion (plasma) density (dashed curve). It is seen that the ëconsistentí (within the assumed model) velocity distribution function does qualitatively result in the required decrease of NI with height despite the upwards acceleration of the ions, in contrast to the isothermal Maxwellian case which yields an increasing density with height (solid curve; in order to enable a better comparison of both curves, the loss of ions due to the assumption of open boundaries has been artificially eliminated here by scaling the density height profiles such that the total recombination rate within the whole column equals the total ionization rate (as it should be the case for a closed system); this procedure does not affect the velocity distribution functions).
Both figures should however only be considered as qualitative results as they are obviously not consistent with the original first order approximation ¶fI(h,v)/¶h<< ¶NI(h)/¶h : for velocities smaller than the velocity corresponding to the maximum of fI(h,v), the latter varies with height in the same sense (and stronger) as NI(h) whereas for greater it changes in the opposite sense. A higher order approximation should therefore lead to even stronger deviations from a Maxwellian for small ion velocities, whereas the enhancement of fI(h,v) for higher velocities becomes reduced although not height independent as for LTE (it is obvious that in a collisionless plasma the quantity nI(h,v) = fI(h,v).NI(h) must become independent of height for sufficiently high velocities, since the ion velocity becomes relatively unaffected by the force field and, furthermore, the mean free path with regard to recombination increases, i.e. fI(h,v) varies with height inversely to NI(h) in this limit; the additional effect of elastic collisions should result in a situation between this case and LTE) .
If the correct spatial symmetry could be applied here numerically (as it was possible for the free diffusion problem, Figs.1-4) the ion velocity distribution function should display a double humped feature rather than the one sided displacement resulting from the assumption of open boundaries.
Figs.7 and 8 demonstrate that a double humped feature can also result in the necessary decrease of the total ion (plasma) density with height in the presence of an upward accelerating force if the central depression of the distribution function is deep enough. Fig.7 shows a profile of the form fI(h,v) ~ exp[ -(v/v0)2 .t . e-(v/v0)2] where t = 3.5 here (dashed curve; the thick solid curve gives a Maxwellian for comparison). Fig. 8 gives the corresponding total ion density NI(h) as obtained by iterating Eq.(26) for this fixed distribution function. Since this form for fI(h,v) was adopted here completely arbitrary and without any physical justification, it is obvious that a correspondingly modified double humped function can in fact be consistent with the actual height dependence for the total ion density in contrast to the Maxwellian case (fully drawn; the long dashed curve represents the 'classical' height dependence of the ionospheric plasma as given by twice the neutral scale height).
The results indicate therefore that the familiar double humped shape of 'incoherent scatter ion lines' may directly reflect the non-Maxwellian ion velocity distribution produced by convection in the plasma polarization field of the undisturbed ionosphere (the relevant velocity distribution is actually that of the highly excited neutral atoms which, however, can be assumed to be identical with the ion velocity distribution function in the F-region (see Sects. 1 and 3)) (for a quantitative comparison with observations, the component of the velocity distribution function perpendicular to the magnetic field, which is dynamically unaffected by the convection, has obviously to be added with a relative weight according to the magnetic aspect angle of the observation; this will in general lead to a reduction of the 'double humped' appearance).
3. Conclusions and Proposal for Further Theoretical Investigation
The results obtained with the present non-LTE model for the vertical ion convection in the average ionosphere demonstrate that deviations of the velocity distribution function from a Maxwellian due to the effect of spatial density gradients are substantial. They could account directly for the observed double humped feature of the so called ëincoherent scatter ion linesí in the F region if one assumes that the scattering is actually due to atoms in highly excited states formed by recombination from the plasma rather than free electrons (Smid, 1993). As the lifetime of the corresponding quantum states (n=200-300) is of the order of 0.1 sec, the distribution function of those atoms is therefore largely unaffected by elastic collisions with neutrals in the F-region, whereas at lower altitudes collisional relaxation of the distribution function towards a Maxwellian takes place, in agreement with the observed height dependence of 'incoherent scatter ion lines'.
These relationships could be established in a direct quantitative way if the numerical scheme for solving the Boltzmann equation is modified such as to overcome the computational problems mentioned in the previous sections which prevented a completely consistent solution so far.
This could be achieved by using higher order approximations with regard to the discretization of the integrals in Eq.(26) and (27) (simple step functions were assumed for the calculations presented here) which remove the discontinuities and make nI(h,v) differentiable throughout phase space.
A more sophisticated one dimensional model which simulates the ion convection on the actually curved magnetic field line (taking therefore the variation of the gravitational and plasma polarization field on this line into account) would remove the necessity of an upper boundary and the associated discontinuity problems at this point (the problems at the lower boundary are less severe as one can assume fI(hmin,v) to be Maxwellian if one chooses hmin such that this height is collision dominated).
Also, the electron convection should be included into the scheme in order that the plasma polarization field is determined self-consistently.
For a quantitative comparison with observed backscatter lines, the component of the velocity distribution function perpendicular to the magnetic field (which is dynamically unaffected by the plasma convection), has finally to be added with a relative weight according to the magnetic aspect angle of the observation.
The main emphasis of the proposed research shall be focused on these aspects. Other effects which may affect the shape of ionospheric radar backscatter lines shall be investigated as well, however. Of implicit relevance could be the actual differential cross section for elastic collisions with neutrals and/or charge exchange which might lead to significant deviations from the Maxwellian velocity redistribution model assumed for the present calculations. Of further, explicit, importance may be the directional dependence of the scattered signal for conditions of specular scattering as it manifests itself for instance in the well known 'magnetic aspect sensitivity' of ionospheric radar backscatter. As it is apparent from the results presented in Smid(1993), the transition region between specular (i.e unisotropic) and non-specular (isotropic) scattering occurs just within the frequency band commonly used for probing the ionosphere, and it is possible that this situation of 'near specular' scattering results in an effective backscatter coefficient varying within the line and causes therefore deviations of the observed line shape from the profile corresponding to the velocity distribution function of the Rydberg atoms.
It should be mentioned that the so called 'electronic component' of incoherent backscatter can of course hardly be explained by the convection effects on the ion dynamics and the resultant distribution function of the atomic Rydberg states. The radiowave-atom- and radiowave-plasma interaction as proposed in Smid (1993) and Smid (1992) respectively will therefore have to be investigated in more detail in order to give a complete description for the radar backscatter. This can probably only be done within a separate research program.
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