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Three-dimensional computer simulation of plasma cloud evolution in the ionosphere
~32~33~ $3.00+0.00 Pergamon Press plc
f’her. Sptrfe Sci., Vol. 38, No. 11, pp. 1375-1386, 1990 Printed in Great Britain.
THREE-DIMENSIONAL COMPUTER SIMULATION OF PLASMA CLOUD EVOLUTION IN THE IONOSPHERE V. A. ROZHANSKY,
I. YU. VFSELOVA
and S. P. VOSKOBOYNIKOV
Kalinin Polyteehnical Institute, Leningrad K-251, U.S.S.R. (Received infinalform
24 April 1990)
A~act-~~-Dimensions computer simulation is performed for the plasma inhomogeneities of the arbitrary non-linearity and form in the ionosphere. It is shown that small-scale (a,, 5 100 m) cloud evolution is governed by the diffusion. For the very strong inhomogeneities cloud evolution is determined by the non-linear deformation in the self-consistent electric field. Large-scale inhomogeneities’ evolution with a, - 3-5 km could be described by equipotential magnetic field line models. The intermediate case is also analysed. The simulation results are consistent with “‘Spolokb” experimental data.
1. INTRODUCITON
Many papers are devoted to the complicated of plasma
cloud
evolution
problem in the ionosphere. For
example, barium clouds are used to trace plasma convection and to determine the ambient ionospheric electric field. The main question of the interpretation is the analysis of the ambient field perturbation extent and so the possibility of the ambient plasma movement determination. The simple qualitative low-density perturbation dynamics model was suggested by Haerendel et al. (1967). It was shown that the disturbed electric field became strong when plasma density perturbation exceeded the ambient one by a large factor. The 2-D simulation of plasma cloud evolution was performed by Lloyd and Haerendel(1973) for the two-layer ionosphere model. Similar calculations for two- or multilayer models were made by several authors-Perkins et al. (1973), Scannapieco et al. (1974), Zabusky et al. (1973, 197.5, 1976), Doles III et al. (1976) and Pudovkin et al. (1987). In these models the magnetic field lines were considered to be equipotential and the diffusion terms in the transport equation were neglected, so that 3-D equations were reduced to the 2-D ones for the densities integrated along magnetic field B. Such reduction is only possible for inhomogeneities with the transverse scales a, 2 3-5 km (see below). For inhomogeneities with smaller scales a complete 3-D approach is necessary. The analytical solution for linear perturbation on the homogeneous background was obtained by Gurevich and Tsedilina (1967), and Rozhansky and Tsendin (1988). For strong ~rturbation only selfsimilar analytical solutions and results for special
limiting cases were obtained ; for further details see Rozhansky and Tsendin (1988). The direct computer simulation of the complete transport equation system is a rather difficult problem due to the drastic difference between longitudinal and transverse transport coefficients. In the ionosphere their ratio is ofthe order lo’--106. For model transport coefficients a 3-D simulation was performed by Voskoboynikov et al. (1987) for the case of a homogeneous background plasma. In the recent work of Drake et al. (1988) 3-D calculation of perturbed potential was performed for an initial density profile with a large spatial scale ratio along and across B. Computer simulation presented in this work is based on the reduced 3-D transport equation system suggested by Rozhansky (1985) where the main problem of drastic transport coefficient difference is overcome. For the first time simulation is performed for the inhomogeneities of arbitrary form and nonlinearity in the real ionosphere. It is shown that only for rather large-scale clouds of aI 2 3-5 km, magnetic field lines are close to equipotential ones. For al 5 1 km cloud evolution is essentially 3-D. Small-scale cloud deformation with a, 5 100 m is determined by diffusion and occurs in the same way as in homogeneous ambient plasma. For scales in the interval 100 m < a, < 1 km the strong inhomogeneity lifetime is governed by non-linear deformation in elf-insistent electric fields and ambipolar diffusion along magnetic field lines. The analytical formulae for maximal density time dependence and cloud velocity for arbitrary non-linearity are suggested and tested by comparison with the numerical calculations. The simulation results are in good agreement with the barium cloud experiments “Spolokh” data ; see Andreeva et al.
1375
1376
V. A.
ROZIUNSKY
(1984), and Dzubenko ef al. (1983). For a, > 1 km the calculations obtained are consistent with the 2-D model results. We consider the slow diffusion-type evolution of plasma cloud. For the barium clouds created by the explosion this means that the initial rapid expansion phase is over and the pressure gradient of the injected plasma is balanced by the neutral gas and magnetic pressure. For the experiments of the “Spolokh” type, for example, the duration of the initial phase is l-2 s at a height of 170 km. For the barium plasma clouds created by photoionization of neutral clouds the diffusion-type evolution begins after 20 s when the photoionization process is over. There are also many other different methods of producing strong plasma inhomogeneities-electron and ion beams, meteors, rocket traces, etc. In each case one should analyse the initial phase and obtain the criteria of validity of the diffusion-type approach.
et al.
cp(X,Y--f a) = 0; n,(x,y-,
co) = 0;
h is the height in the ionosphere. The injected density maximum is situated at z = 0 (h = h,). At h < 100 km (h = h J and at h = h2 > ho the transverse ionospheric conductivity is negligible compared with the ambient conductivity at z = 0 (h = ho) inside the cloud. So the boundary condition at h = h,, h2 corresponds to the condition on the dielectric surface where ion fluxes along B are equal to the electron ones : ren = ri,, + rll, . Neglecting longitudinal ion mobility with respect to the electron one (we consider at h 2 100 km b,,, >>bi,,ll, be1 <
l-e,,Ih-h,,h,= -D.,,z+b,“n$
Three-dimensional evolution of plasma cloud created by injection into the ionosphere of ions with different natures is described by the continuity equation system
V@,Vn, T &n,Vrp f &n,Eo)+L -R.,
(1)
where n, = particle densities, cpis the perturbed potential, o!= e, i, I, where subscript e corresponds to electrons, and subscripts i, I to the background and injected ions, respectively, E. is the ambient electric field in the neutral wind frame, 6. and d, are. the height-dependent mobility and diffusion tensors, and I,, &are ionization and recombination terms. Particle temperatures are considered to be constant and the magnetic field B vertical. The z axis is parallel to the magnetic field B, the y axis to the ambient electric field, and the x axis coincides with the E, x B (Hall) direction. In these coordinates mobility and diffusion tensors have the form
6, =
i
b
7b.H
0
kzH
b,,
0
0
bull I
0
,
TD.H
0
D,,
0
0
=O.
(3)
so
n, = ni+n,.
2. MODEL
at
_ h-h,.hl
We assume quasineutrality,
an e=
(2)
RCx,Y + 03) = nO(h)= nO(z)~
.
DC41
The boundary conditions at x, y + cc are
(4)
The injected ions’ initial profile was taken in the form M,Y,
z*
4 = h40> exp - q [
The longitudinal small compared ionosphere. According to by Rozhansky potential can be
(x"+y'> 2
aA
1
.
cloud scale ui is considered to be with the global vertical size of the the analytical and numerical results and Tsendin (1988) the perturbed taken in the form
rp = :
ln (+J
+ WX,Y, 4,
where Y is a function with large longitudinal I,, >>a,, . We assume also alI <
(5) scale
(6)
For non-boundary homogeneous plasma the characteristic scale I,, at t = 0 is simply a,,/b,,lb, and then increases with time. In the real ionosphere the I, value can also be determined by the vertical scale of no(z) and b,(z). If (6) is fulfilled the potential form (5) is justified for the homogeneous plasma but the algorithm presented revealed the Y longitudinal scale to be large enough in the real ionosphere also. According to equation (10) with the boundary conditions (2), (1 I), (12) only two scales for Y exist-u,dmand the vertical scale of the ionosphere. The ambient density perturbation can be obtained from (15) and has the same scale. So form (51 . I is correct.
1377
Three-dimensional computer simulation of plasma cloud evolution
We consider the strong density perturbation A >>1 so that near the injected cloud n, = n,. From (1) and (4) we have the equation for particle fluxes div I?, = div Fi + div I?, .
(7)
For the ambient ions at ]z] > a,, assuming n, = ni we obtain
(8) Let us introduce dimensionless variables
where hi,(O), b,,,(O) are the mobilities at z = 0 at the injected cloud height. Neglecting the longitudinal ambient ion flux in the ambient plasma with respect to the electron one we obtain the perturbed potential equation for the given density profile (Rozhansky, 1985) :
Equations (10) and (11) are obtained for the arbitrary profile of ni(z) if @‘““I >>ni. Relation (11) must be considered as a mixed boundat-y condition for equation (10) at 4 = 0. The potential Y at [ = 0 is a continuous function but the longitudinal electric field at [ -+ + 0 differs from the one at [ + -0. The boundary condition (3) has the form
Plasma density can be obtained from ion equation (1). After integrating the injected ions’ continuity equation over z up to a,, < ]zD]CCI, with 1, = RI = 0 we obtain (in the variables <, q, [)
+&fi,(V~‘I’--EoaJ
1 .
(13)
The equation is a 2-D one. The ni profile can be reconstructed in a practically important case when T, 3 Inn1 -auf az I-0 7-x----~
T, call
So the ~zi temporal evolution is determined by the Boltzmann part of potential (5) and corresponds to the 1-D ambipolar diffusion (c = Ti) (Rozhansky and Tsendin, 1988) :
f W) where
Ni exp {--2*/[4(1+
Kll = bell@)lbell Co); bL,El= &,H = Dd..li/biJ.(o>;
ii,
baLJitz)lbil
Co);
&~/~iJ(o)*
=
We assume mcV,i<<&vii+(m,, h are the electron and ion masses, and v,i,N the electron-ion and electronneutral collision frequencies) so that D,,, = (T,/e)b,,, . Inserting (5) into (7), integrating (7) over z up to a,, < Izoj <>ni and \y is practically constant here for z0 -+ 0, we have
nr =
T,l~:,)~,,,~+~~ll
J;;[a:+4(1+T,/~)D,,,t]‘/2
’
04)
This treatment is not valid in the transition region (n, < n, and ni N ni) so the far tails of the profile (14) are missing. The ambient density longitudinal perturbation scale is I, >>aW. Neglecting ambient ion shift along B 4Q,,r <
lie1 1(=+0-f&-0
= -V,
-V,[(6;+6:)~*,@,ul-Eoa1)]
(&--&)-I-
++&I)
1
v,&
+&&(3)@,Y_E,a,)
>I r=o’
(11)
where re,, =
ayl f&x
cu i%$=
s -03
dz
n,bJ2(0) iv,bi’!z(O) ~~(O)~~~‘(O)~~= a~~~(O)~~~‘(O).
1 +(&-Ri)al.
(15)
Equation (15) is also 2-D, and [-dependence is a parametric one. The equation system (lo), (13), (15) with the boundary conditions (2), (11) and (12) was simulated numerically. In contrast to the initial equation (1) our basic system (IO), (13), (15) reveals only transport coefficients’ height dependence and the density equations are 2-D. This system describes both plasma evolution in the strong ambient electric field and transverse diffusion processes.
V. A. ROZHANSKY ef al.
1378 3.PERTURBED
POTENTIALSIMULATION
OF A STRONG
AMBIENT
INTHE
ELECTRIC
CASE
FIELD
For large-scale inhomogeneities the diffusion terms in (IO), (1 I), (13) and (15)can be neglected for smooth density profiles. At t = 0 when the background plasma is undisturbed the potential equation is reduced to the Laplace-type one
with the coefficients depending on 1 only c,(C) = &n*,;
c&)
= blln”Q.
While deriving (16) we neglect the transverse electron diffusion coefficients. We also consider (in this section only) the Hall current at c = 0 to be zero. The perturbed potential distribution for the given density profile is described by 116) with the boundary condition (11) where the diffusion terms are neglected. According to (10) and (11) the potential Y is determined by the injected particles’ integrated density only
and depends on parameter A”= J&4. In the calculations presented the chosen values were : 1 km < al < 5 km, the cloud height h,, 160 km < ho < 250 km, the density non-linearity parameter A - 1 C A 6 100 for a, = a, = a. For at # a, the corresponding value of A”is A”= Aa, /aI. The lower boundary of the ionosphere was taken at 90 km, the upper one at h = 2SO300 km. The simulation parameters correspond to night and day aurora1 ionosphere ; see Gurevich et al. (1973). To solve equation (16) the finite-difference method was used. The arising linear algebraic equations system for Y was solved by the technique described in Voskoboynikov ef al. (1987). This technique includes the implicit method of minimal residuals in combination with a cyclic reduction algorithm. The typical equipotential lines Y = const. in the plane pe~ndicular to the magnetic field at I = 0 are presented in Fig. 1. The perturbed electric field EI height dependence is shown in Fig. 2. According to Fig. 2 E; is practically constant along magnetic field lines only for rather large-scale inhomogeneities a, k 5 km. In this case the perturbed potential value inside the cloud E,t is given by (17) obtained for the two-layer model at t = 0 for 6, = biH (Lloyd and Haerendel, 1973 ; see also Rozhansky and Tsendin, 1988)
-h---E; _ -&
1
1 +e NIbII12&’
FIG. 1. EQLJIPOTENTUL LINES'P = CoNSI'.IN THE 5Oq PLANE
WHERE THEINITIAL CLOUD IS SITUATED. Cloud height ho = 160 km, al = 1 km, A = 10. The parameters chosen correspond to the day ionosphere.
where xi, is the height-integrated total Pedersen conductivity of the ionosphere. According to (17) for A = 1, ho = 160 km, aA = 5 km and in the day-time E$E, = 2.75. 10m2. The value obtained in the 3-D numerical simulation is 2.52 * 10A2. For a, = 1 km the difference increases. For example in the day ionosphere for A = 10, ho = 160 km, from (17) we have E;/Eo = 5.4. 10e2, while the numerical simulation gives EJEo = 7.5 - lo- *. So for a, < 5 km the equipotential field lines approximation (17) underestimates the perturbed electric field value. This fact is
z? ”
-ii
5
i ‘W
loo
140 i60 180 200 220 240 h (km)
FIG.2. PERTURBED HEIGHT
(17)
120
ELECTMC
DEPENDENCE
FIELD IN THE VICINITY FOR NIGHT
[ = q =
0
IONOSPHERE.
Cloud height ho = 160 km, A = IO. (1) ai = 0.1 km; (2) a, =0.3km;(3)a,=1km;(4)a,=2.5km;(5)a,=5km.
1379
Three-dimensional computer simulation of plasma cloud evolution
connected with the Ei decrease in the low ionosphere where Pedersen conductivity is maximal. Accordingly the maximal Pedersen conductivity zone is “switched off” from the “influence region” where the ambient ion short-circuiting current is formed. The E; height asymmetry is clearly seen in Fig. 2. At h > 200 km the perturbed electric field remains practically constant due to the sharp Pedersen conductivity decrease ; so the simulation results are unsensitive to the upper boundary position. The perturbed electric field noticeably decreases under the cloud. For a, < 100 m the Ei distribution becomes symmetrical in the z direction, so we can use the formula obtained by Rozhansky (1985); see also Rozhansky and Tsendin (1988) for unboundary homogeneous plasma (E; = 0)
/I Co
h (km)
FIG. 3. PERTURBED ELECTRIC FIELD INSIDE THE CLOUD DIFFERENT INITIAL CJBUD HEIGHTS, A = 1, a,=2.5 NIGHT
(1)ho =
EI_ Eo -
FOR
km,
IONOSPHERE.
160 km; (2) h, = 200 km; (3) h, = 250 km.
-1
>.
(18)
For example, in the day ionosphere for A = 10, a, = 100 m, h,, = 160 km the simulation value is EJEo = 5.1. lo-‘, while from (18) we have EJEo = 6.8. 10m2. The equipotential magnetic field lines model (17) gives us E//E0 = 5.7. 10m3 which is 10 times less than (18). For a given non-linearity parameter A the perturbed electric field decreases with the cloud height ho. For the small-scale inhomogeneities with a, < 100 m this fact is connected with mobility height dependence, so that the “influence region” of the ambient ion shortcircuiting current with the scale al,/m increases. For the large-scale inhomogeneities a, > 5 km the transverse conductivity inside the cloud decreases with height while the region of the ion short-circuiting current remains practically constant. The perturbed electric field dependence on the cloud height ho for the intermediate case is presented in Fig. 3. In our calculations the magnetospheric inertial currents were neglected. It is justified for the inhomogeneities with a, 5 300 m because the perturbed electric field is localized in the ionosphere and practically vanishes at magnetospheric heights. For the large-scale inhomogeneities the presented model is valid if the wave magnetospheric conductivity c, = c2/47rvA (vA is the Alfven velocity) is less than the Pedersen integrated conductivity of the ionosphere. In this case the conductivity of the conjuction ionosphere must be taken into account also. The effects of the finite c,, were discussed by Rozhansky (1986) and Pudovkin et al. (1986). 4. THREEDIMENSIONAL
250
150
DENSITY EVOLUTION
The equation system (lo), (13), (15) with the boundary conditions (2), (1 1), (12) and initial condition
n,(x,~,z)l,=o = AdO) x exp
was solved numerically
by it splitting according to physical processes. The equations (lo), (13), (15) at every time step are considered as linear ones. The above-mentioned method was used for solving the elliptical potential equation. The parabolic density equations were solved by splitting according to the physical processes and directions. The method reduced the problem to the 1-D equations. For their solution the Godunov method (Godunov et al., 1973, 1976) was used and also the method of CrankNicolson. The diffusion equations were integrated by the explicit Euler method. The calculations were performed for a, = 0.1, 0.3, parameter and 1 km, a,, = (I~, the non-linearity A = 10,100,500, and the ambient electric field E, = 1 and 2 mV m- I. Ionization and recombination terms were taken into account for uI = 0.3 km. The initial cloud was situated at ho = 160 km, the low boundary height h, = 100 km, and the upper one h, = 190 km. It can be seen from numerical results that the ambient density perturbation is small enough at h > 190-200 km. So the h, variation causes no significant effects. The part of integrated Pedersen conductivity could be easily taken into account by renormalization of A (A becomes smaller). Such a procedure is necessary for a, 2 1 km only. The spatial step in the z direction was 5 km. It is smaller than both the characteristic scale of the perturbed electric field in the vertical direction and
1380
V.
PUENTIALPRDFlI&5 FIG. 4. PERTURBED
A.
ROZHANSKY
(MINUS
ALTO
et
al.
TRRM)
IN THE PiXXBSN
($I
AND
HALL (t)
DIRECTIONS.
Cloud height ho = 160 km, r = rb,&/a,. (a) A = 10, al = 0.1 km, T = 0, t = 0. (1) h = 160 km ; (2) h = 170 km; (3) h = 180 km; (4) h = 150 km; (5) h = 140 km. (b) A = 10, aI = 0.1 km, h = 160 km (1) ‘c= 0.0, q = 0.0;(2)x = 0.6, q = 0.5. (c)A = 500, a, = 0.3 km, h = 160 km, T = 0, 5 = 0.
the ambient density scale (the smallest vertical scale I,, for a, = 0.1 km was 13.3 km). In the XOY plane the grid with the spatial step 0.2%~~ was used covering the region (8 x @a,. The grid was moving in the Pedersen direction with the injected ions’ maximum density velocity. The calculations were performed for the day ionosphere and T, = q. (a) a, = 0.1 km The Y profiles (the perturbed potential minus Boltzmann part) are presented in Fig. 4. For al = 0.1 km ‘P decreases along z with the spatial scale I,, >>al (Rozhanslcy and Tsendin, 1988) lr = aLtber (0)lk!_("ll"'*
e EoaL
cp([ = 0) = T,/e(lnA-~2-tz)+Y
> 0.
When 5 = 0, q= 5 = 0 (more accurate when }zl >>a,,) the Boltzmann part turns zero and 50= Y c 0. So the potential is of quadrupole type; see Rozhansky and Tsendin (1988). The estimate for Y could be easily obtained from (7). The transverse ion flux divergence and the electron longitudinal one are of the same order, so we have
(19)
For the parameters taken I, = 133~~. The diffusion Diln,Jai and convective bilEgo ffuxes are of the same order when a, = 0.1 km. The corresponding parameter is
r, ff=------.
this case is of diffusion type. From (5) we see that at < = 0, t = 0 the total perturbed potential
(20)
For a,=O.l km and Eo=l mV m-’ a=0.77. According to Fig. 4a-4c the perturbed potential in
For A = 10 and #“@ = 0.13,9 = Y/(E,a,) is of the order of 0.1 according to estimate (21) in agreement with results presented in Fig. 4a4c. The dipole electric field which partially screens the ambient one is added to the diffusion-type quadrupole field; see Fig. 4a. The perturbed electric field value in the cloud centre for o! < 1 and bIH = beH is given by (18). To estimate
Three-dimensional computer simulation of plasma cloud evolution
1381
O.& T 0
I
1
0. I
0.2
I 0.3
I
I
0.4
0.5
I 06
I 0.5
t 1.0
I 1.5
tE,
FIG. 5. MAXI~LDENSITY
~~DE~ENDEN~~
= tbeHE&,.
Cloud height h, = 160 km. (a) ai = 0.1 km. (1) Calculation according to equation (22), 2,3-sim~la~on results for A = 10, and 100, correspondingly. (b) a, = 0.3 km, A = 500. Dotted line-calculation by (22), straight lines-simulation results. (1) IT,,= 1 mV m- ’ ; (2) E. = 2 mV m- ‘. (c) al = 1 km, A = 100. Dotted line-calculation according to (22).
for a, = 1 km we subtract the diffusion part of electric field (Y profile along 5) from the total perturbed electric field (Y profile along q). The EJE, calculated value even in the case G(w 1 agreed with (18) for A = 10 and 100 with an accuracy less than 30% (the same agreement was observed for the dipole E’ component
the calculations in Section 3). The potential distribution is similar to that obtained by Drake et al. (1988). It is shown in Rozhansky and Tsendin (1988) that the plasma inhomogeneity evolution is governed by shop-cir~iting currents in the background plasma.
1382
V.
FIG. 6. R,(q)
A.
HALF-WIDTH TIME DEPENDENCE, a, = 0.1
ROZHANSKY
km,
A = 10.
Crosseenumerical simulation ; straight line-calculation according to (22) : A: = (1 + S&l/a:).
Such a mechanism is effective if the relative density perturbation in the depletion regions h < 1. When @-’ << 1, h N @“-j cc 1. In this case the perturbed electric field is small, so the initial cloud drifts with the injected ions velocity in the ambient field v,, = &E,. Such a shift was observed for A = 10. Cloud decay is determined by injected ions’ unipolar diffusion
N,(5,v,t) =
NJ(0,0,0) 1+ 80,, t/a:
1
-(x-b*“Eet)2-(Y-bllE,t)2 .
x exp [
a:+8D*,t
(22)
The simulation results are compared with (22) in Figs 5 and 6. For A = 10 the agreement is good enough and the deviation from the straight line in Fig. 6 is caused by mesh finiteness. When A = 100 and RimaX)= 1.33, density perturbation in the depletion region becomes comparable with the ambient one (Fig. 7a), so cloud decay decelerates according to Rozhansky and Tsendin (1988) ; see Figs 5a and 9a. The perturbed electric field E; = 0.37E,, in agreement with (18) becomes of order E,, and the cloud velocity correspondingly decreases. The @i profile remains almost symmetrical while the background density profile presented in Fig. 7a is noticeably asymmetric. The negative ambient density perturbation at q = 2 exceeds the one at q = -2 due to the ambient ions’ shift in the (-E,) direction under the perturbed electric field influence. The difference in z direction is caused by transport coefficients’ height dependence.
et
al.
(b) a, = 0.3 km In this case the parameter a = 0.26 so the transverse diffusion term’s role is smaller than for a, = 0.1 km. Accordingly the Y character in Fig. 4c is closer to dipole-type than in Fig. 4a. As for al = 0.1 km, the perturbed potential longitudinal scale is given by (19) but the potential profile becomes asymmetric in the z direction-it is more sharp below the cloud. The perturbed electric field inside the cloud is given by (18) with accuracy less than 10 % . For A = 10 the electric field is slightly disturbed so the injected cloud moves practically with the velocity v0 = 61E,, and its decay is controlled by diffusion in accordance with (22). For A = 100 when h _ film@ N 1 cloud velocity and decay rate decrease. The most interesting case is fi$max)> 1 when depletion regions are deep enough. The parameters chosen (A = 500, a, = 0.3 km, h,, = 160 km) correspond to the barium cloud experiment “Spolokh” data ; seeAndreeva et al. (1984) and Dzubenko et al. (1983). In agreement with (18) the ambient electric field is almost completely screened inside the cloud (Fig. lc) and the cloud velocity strongly decreases (Figs 9b and 10a). Indeed at t = 3 the injected ion maximum shift in the [ direction is < = 1 instead of r = 3 for the unperturbed electric field. It is interesting to note that the cloud does not move in the E, direction while for the Hall to Pedersen mobility ratio bIH/bII = 3.5 the corresponding shift should be noticeable (-0.28 at 7 = 3). This fact is connected with perturbed electric field x-component E: > 0. For 6 = (bcH-blH)/bll << 1 (for parameters chosen at h,, = 160 km, 6 = 0.29) and #{maX)>> 1 according to Rozhansky and Tsendin (1988), E:/(E,- EJ = 6. Such perturbed field was obtained in numerical simulation. Taking into account E: we obtain the cloud velocity in they direction uv = - bIHE: + b,, (E, -E;) = (E, - E; )bn, (-6+
n>.
(23)
Since 6 - bI,/blH N 5 * IO- 3 the u,, is practically zero. The injected ion outflow occurs at the cloud side where & 5 1 in the 5 direction as can be seen in Fig. 10a. The front side of the cloud in the [E, x B] direction becomes more sloping while at the back steepening takes place. Such evolution is presented in Figs 9a and 10a. Equidensities with & < 1 elongate in the direction forming an angle with the r axis. Similar elongation was observed in the two-layer model numerical simulation of Scannapieco et al. (1974), Pudovkin et al. f 1987). etc. It is caused bv the electron
Three-dimensional computer simulation of plasma cloud evolution
‘30
_:
1,
-:
0
1383
II
I2
’ 3
130
& ,90
-3
-2
-I
0
-
II30
150-
140 -
130-
? Cc)
FIG. 7. AMBIENT ION EQUIDENSITTES IN THE 2 Y I’LANJZ WHERE THE fl, MAXIMUM IS SITUA~. Figures correspond to 64 = (q-~(z))/+,(z). (a ) nl = 0.1, A = 100, z = 2.0, C = 1.5; (b) a, = 0.3, A=500,s=1.0,~=0.25;(c)al=1.0;A=100,r=2.0,~=1.
and ion Hall mobilities difference ; the corresponding mechanism was discussed by Rozhansky and Tsendin (1988). Ambient ion density profiles presented in Figs 7b and 8 essentially differ from the case a, = 0.1 km, Fig.
7a. The depletion region is situated mainly on the right side where q > 0 due to ambient ions’ redistribution caused by the perturbed field E;. Positive % perturbation is displaced in the negative q direction. Cloud lifetime is one of its most important charac-
1384
V.
A. ROZHAN~KY ef al.
:t~3~ -3
-2
-I
0
FIG.8. AMBIENT IONEQUIDENSITJJB INTHE
teristics. If we assume the injected ions flow in the 5 direction with density #, = 1 and velocity v0 = CEO/B
then we obtain the linear density decrease law (see also Rozhanksy and Tsendin, 1988) ~rnEX)(~)= urns)
- KZ,
(24)
where 1cis a numerical coefficient. Such linear decay was obtained in our simulation, Fig. 5b. For E. = 2 mV m- ‘, K = 0.3. For E0 = 1 mV m-’ the decay time is smaller due to the relative diffusion influence increase. Naturally for I,, of order (19) the injected ion numbers flowing out due to diffusion is given by the estimate
E (bl FIG.
9. INJECTED IONS’ DENSITYEVOLUTION IN THE HALL DECECTION.
The parameter Sa~i~~~=~ is equal to 0.54 for & = 1 mV m- ’ and 0.27 for .!&,= 2 mV m- ‘. So in the second case & decreases more slowly-Fig. 5b. It should be noted that fir time decay is larger than that due to ion unipolar diffusion (22) corresponding to the case fl,(m”) -C l-the dotted line in Fig. 5b. So the cloud lifetime for A = 500, uI = 300 m is essentially determined by the non-linear deformation in the Hall direction and the density decrease law is given by (24). The simulation results are consistent with “Spolokh” experimental data-Andreeva et al. (19841, Dzubenko et al. (1983 )---where the bulk of the cloud was practically at rest and the injected particle numbers flowing from the main cloud into the tail was increasing linearly with time in accordance with (24). The equipotential magnetic field lines model (17) predicts for t = 0, beH = bin and A = 500 the cloud
Profites presented
include maximum & data (the corresponding tf was chosen). {a) al = 0.1 km, A = 100. (1) ~=0.0;~=0.0;(2)~=0.6;~=0.5;(3)~=2.0;~=1.5; (b)ua,=0.3km,A==500.(1)~=0.0;(2)~=2.0;~=3.0.
velocity in the c direction to be 1.5 times greater than obtained in 3-D simulation. (c) a, = 1 km
This case is an intermediate one so it reveals both features of homogeneous ambient plasma and equipotential magnetic field line models. Simulation was performed for E,, = 1 mV m- ‘. The non-linear parameter chosen, A = 100, lies between the strong nonlinear case when the depletion regions are deep enough
Three-dimensional
computer
simulation
of plasma -3
1385
cloud evolution -2
-I
0
I
2
3
-I
-I
0
0 I
I
6
I -3
I -2
-I
I<
0
I I
I 2
I 3
2
2
3
3
4
4
5
5
6 -3
-2
(al
-I
0
I
2
3
tb) FIG. 10. INJECTED
ION EQUIDENSmIE.5
IN THE @fj
PLANE.
(a)a,=0.3km,A=5OO,r=3.0;(b)a,=1km,A=lOO,r=2.O.
observed, similar to the two-layer model resultsand the perturbed electric field is of order E. and the situation when the perturbed electric field is small Fig. lob. compared with Eo. The cloud velocity in the Hall direction, as can be 5. CONCLUSIONS seen from Fig. lob, is smaller than for the unperturbed (1) For the first time 3-D computer simulation has electric field. It corresponds to the electric field perbeen performed for the inhomogeneities of arbitrary turbation calculated by both (17) and (18). The maxinon-linearity and form in the ionosphere. mal density linear decrease presented in Fig. 5c is in (2) Small-scale (al 6 100 m) cloud evolution is accordance with (24) with the coefficient JC= 0.13. It governed by diffusion ; it occurs in the same way as in is smaller than the one for A = 500 because the density homogeneous ambient plasma. maximum moves with noticeable velocity - 0.6cE,,/B, (3) Inhomogeneities with intermediate transverse sothe relative outflow velocity in the Hall direction is scales a, - 0.3-l km (for the ambient electric field of of order O.rlcE,,/B and K is two times less than in order l-2 mV m- ‘) are in the medium position the previous case. The injected cloud transverse scale between the homogeneous background plasma case increases faster than due to the unipolar diffusion and the equipotential magnetic field lines model. For because parameter CIis three times smaller than for the very strong inhomogeneities (i?, > 1), when the a, = 0.3 km. Equidensities with IjI(“‘@ 5 1 elongate depletion regions in the background plasma are deep in the direction forming an angle with the < axis. enough, maximal cloud density decreases with time This angle is smaller than in the two-layer model of according to the linear law. Cloud evolution in this Scannapieco et al. (1974) or in calculations performed case is determined by the non-linear deformation in by Pudovkin et al. (1987). Such an effect is produced the self-consistent electric field. by the 3-D character of our simulation so that in (4) Perturbed electric field and injected cloud velthe low ionosphere at heights of order 120 km the ocity for inhomogeneities with a, < 1 km are conmagnetic field lines are essentially non-equipotential sistent with the analytical formula (18) obtained for -SO the Hall current perturbation responsible for homogeneous non-boundary ambient plasma. the elongation character is smaller than in equipoten(5) Large-scale inhomogeneities’ evolution with tial magnetic field line models. (21 N 3-5 km could be described by the equipotential Depletion regions are more powerful than for magnetic field lines model. small-scale inhomogeneities a, = 100 m (Fig. 7a,c). Most of it is situated at q > 0 on the right side of Fig. 7c. Such a picture corresponds to perturbed potential dipole character while the quadrupole part is practically absent. Back-side cloud steepening at t; < 0 is
REFERENCES
Andreeva, L. A., Ivchenko, I. S., Milinevsky, G. P., Rozhansky, V. A., Rnzhin, Yu. Ya., Skomarovsky, V. S.
1386
V. A. ROVIANSKYel al.
and Tsendin, L. D. (1984) Dynamics of artificial plasma clouds in “Spolokh” experiments. Cloud deformation. Planet. Space Sci. 32, 1045. Doles, J. H., III, Zabusky, N. J. and Perkins, F. W. (1976) Deformations and striations of plasma clouds in the ionosphere. 3. Numerical simulation of a multilevel model with recombination chemistry. J. geophys. Res. 81,5987. Drake, J. F., Mulbrandon, M. and Huba, J. D. (1988) Threedimensional equilibrium and stability of ionospheric plasma clouds. Phys. Fluids 31, 3412.; Dzubenko, N. I., Zhilinsky, A. P., Zhuiin, I. A., Ivchenko, I. S., Molotai, A. A., Rozhansky, V. A., R&in, Yu. Ya., Skomorovsky, V. S. and Tsendin, L. D. (1983) Dynamics of artificial plasma clouds in “Spolokh” experiments : movement pattern. Planet. Space Sci. 31,849. Godunov, S. K. and Ryabenky, V. S. (1973) Dl@rence Scheme. Nauka, Moskow. Godunov, S. K., Zabrodin, A. V., Ivanov, M. Ya., Krajko, A. N. and Prokopov, G. P. (1976) Gas Dynamics Mullidimensional Problems Numerical ;Simulation. Nauka, Moskow. _$ Gurevich, A. V., Fishuk, D. I. and Tsedilina, E. E. (1973) Three-dimensional model of electron density distribution in calm ionosphere. Geomagn. Aeronomia 13, 31. Gurevich, A. V. and Tsedilina, E. E. (1967) Motion and diffusion of plasma inhomogeneities. Sou. Phys. Usp. 91, 609.
Haerendel, G., Lust, R. and Rieger, E. (1967) Motion of artificial ion clouds in the upper atmosphere. Planet. Space sci. 15, 1. Lloyd, K. H. and Haerendel, G. (1973) Numerical modelling of plasma clouds in the ionosphere. J. geophys. Res. 78, 7389. Perkins, F. W., Zabusky, N. J. and Doles, J. H., 111(1973) Deformation and striation of plasma clouds in the ionosphere. J. geophys. Res. 78, 697. Pudovkin, M. I., Golovchanskaya, I. V., Nikolaev, N. V. and Klimov, M. P. (1987) Two-layer model validity for
strong inhomogeneity Aeronomia
evolution description.
Geomagn.
27, 560.
Pudovkin, M. I., Lyatskaya, A. M. and Golovchanskaya, I. V. (1986) Longitudinal current structure over the bank of increased conductivity in the ionosphere. Geomagn. Aeronomia 26,767.
Rozhansky, V. A. (1985) Perturbed electric fields and evolution of ionosuheric inhomogeneities. Geomaqn. Aeronomia 25,906.
_
Rozhansky, V. A. (1986) Magnetospheric effects caused by the ionospheric inhomogeneities. Geomagn. Aeronomia 26, 780.
-
Rozhansky, V. A. and Tsendin, L. D. (1988) Collisional Transport in Partially Ionized Plasma. Energoatomizdat, Moskow. Scannapieco, A. J., Ossakov, S. L., Book, D. L., McDonald, B. E. and Goldman, S. R. (1974) Conductivity ratio effect on the drift and deformation of F-region barium clouds coupled to E-region ionosphere. J. geophys. Res. 79,2913. Voskobovnikov. S. P. and Rozhanskv. V. A. (1987) Threedimensional simulation of perturdkh elect& fields produced by the ionospheric inhomogeneities. Geomagn. Aeronomia 26,161.
Voskoboynikov, S. P., Rozhansky, V. A. and Tsendin, L. D. (1987) Numerical modelling of three-dimensional plasma cloud evolution in crossed E x B fields. Planet. Space Sci. 35, 835.
Zabusky, N. J. and Doles, J. H., III (1975) The effect of artificially close boundaries on numerical simulations of the instabilitv and evolution of barium clouds. J. _ aeophys. _. Res. 80, 1846.
Zabusky, N. J., Doles, J. H., III and Perkins, F. W. (1973) Deformation and striation of plasma clouds in the ionosphere. 2. Numerical simulation of a nonlinear two-dimensional model. J. ueophvs. Rex 78, 711. Zabusky, N. J., Doles; i H., III and Perkins, F. W. (1976) Deformation and striation of plasma clouds in the ionosphere. J. geophys. Res. 81,5987.
f’her. Sptrfe Sci., Vol. 38, No. 11, pp. 1375-1386, 1990 Printed in Great Britain.
THREE-DIMENSIONAL COMPUTER SIMULATION OF PLASMA CLOUD EVOLUTION IN THE IONOSPHERE V. A. ROZHANSKY,
I. YU. VFSELOVA
and S. P. VOSKOBOYNIKOV
Kalinin Polyteehnical Institute, Leningrad K-251, U.S.S.R. (Received infinalform
24 April 1990)
A~act-~~-Dimensions computer simulation is performed for the plasma inhomogeneities of the arbitrary non-linearity and form in the ionosphere. It is shown that small-scale (a,, 5 100 m) cloud evolution is governed by the diffusion. For the very strong inhomogeneities cloud evolution is determined by the non-linear deformation in the self-consistent electric field. Large-scale inhomogeneities’ evolution with a, - 3-5 km could be described by equipotential magnetic field line models. The intermediate case is also analysed. The simulation results are consistent with “‘Spolokb” experimental data.
1. INTRODUCITON
Many papers are devoted to the complicated of plasma
cloud
evolution
problem in the ionosphere. For
example, barium clouds are used to trace plasma convection and to determine the ambient ionospheric electric field. The main question of the interpretation is the analysis of the ambient field perturbation extent and so the possibility of the ambient plasma movement determination. The simple qualitative low-density perturbation dynamics model was suggested by Haerendel et al. (1967). It was shown that the disturbed electric field became strong when plasma density perturbation exceeded the ambient one by a large factor. The 2-D simulation of plasma cloud evolution was performed by Lloyd and Haerendel(1973) for the two-layer ionosphere model. Similar calculations for two- or multilayer models were made by several authors-Perkins et al. (1973), Scannapieco et al. (1974), Zabusky et al. (1973, 197.5, 1976), Doles III et al. (1976) and Pudovkin et al. (1987). In these models the magnetic field lines were considered to be equipotential and the diffusion terms in the transport equation were neglected, so that 3-D equations were reduced to the 2-D ones for the densities integrated along magnetic field B. Such reduction is only possible for inhomogeneities with the transverse scales a, 2 3-5 km (see below). For inhomogeneities with smaller scales a complete 3-D approach is necessary. The analytical solution for linear perturbation on the homogeneous background was obtained by Gurevich and Tsedilina (1967), and Rozhansky and Tsendin (1988). For strong ~rturbation only selfsimilar analytical solutions and results for special
limiting cases were obtained ; for further details see Rozhansky and Tsendin (1988). The direct computer simulation of the complete transport equation system is a rather difficult problem due to the drastic difference between longitudinal and transverse transport coefficients. In the ionosphere their ratio is ofthe order lo’--106. For model transport coefficients a 3-D simulation was performed by Voskoboynikov et al. (1987) for the case of a homogeneous background plasma. In the recent work of Drake et al. (1988) 3-D calculation of perturbed potential was performed for an initial density profile with a large spatial scale ratio along and across B. Computer simulation presented in this work is based on the reduced 3-D transport equation system suggested by Rozhansky (1985) where the main problem of drastic transport coefficient difference is overcome. For the first time simulation is performed for the inhomogeneities of arbitrary form and nonlinearity in the real ionosphere. It is shown that only for rather large-scale clouds of aI 2 3-5 km, magnetic field lines are close to equipotential ones. For al 5 1 km cloud evolution is essentially 3-D. Small-scale cloud deformation with a, 5 100 m is determined by diffusion and occurs in the same way as in homogeneous ambient plasma. For scales in the interval 100 m < a, < 1 km the strong inhomogeneity lifetime is governed by non-linear deformation in elf-insistent electric fields and ambipolar diffusion along magnetic field lines. The analytical formulae for maximal density time dependence and cloud velocity for arbitrary non-linearity are suggested and tested by comparison with the numerical calculations. The simulation results are in good agreement with the barium cloud experiments “Spolokh” data ; see Andreeva et al.
1375
1376
V. A.
ROZIUNSKY
(1984), and Dzubenko ef al. (1983). For a, > 1 km the calculations obtained are consistent with the 2-D model results. We consider the slow diffusion-type evolution of plasma cloud. For the barium clouds created by the explosion this means that the initial rapid expansion phase is over and the pressure gradient of the injected plasma is balanced by the neutral gas and magnetic pressure. For the experiments of the “Spolokh” type, for example, the duration of the initial phase is l-2 s at a height of 170 km. For the barium plasma clouds created by photoionization of neutral clouds the diffusion-type evolution begins after 20 s when the photoionization process is over. There are also many other different methods of producing strong plasma inhomogeneities-electron and ion beams, meteors, rocket traces, etc. In each case one should analyse the initial phase and obtain the criteria of validity of the diffusion-type approach.
et al.
cp(X,Y--f a) = 0; n,(x,y-,
co) = 0;
h is the height in the ionosphere. The injected density maximum is situated at z = 0 (h = h,). At h < 100 km (h = h J and at h = h2 > ho the transverse ionospheric conductivity is negligible compared with the ambient conductivity at z = 0 (h = ho) inside the cloud. So the boundary condition at h = h,, h2 corresponds to the condition on the dielectric surface where ion fluxes along B are equal to the electron ones : ren = ri,, + rll, . Neglecting longitudinal ion mobility with respect to the electron one (we consider at h 2 100 km b,,, >>bi,,ll, be1 <
l-e,,Ih-h,,h,= -D.,,z+b,“n$
Three-dimensional evolution of plasma cloud created by injection into the ionosphere of ions with different natures is described by the continuity equation system
V@,Vn, T &n,Vrp f &n,Eo)+L -R.,
(1)
where n, = particle densities, cpis the perturbed potential, o!= e, i, I, where subscript e corresponds to electrons, and subscripts i, I to the background and injected ions, respectively, E. is the ambient electric field in the neutral wind frame, 6. and d, are. the height-dependent mobility and diffusion tensors, and I,, &are ionization and recombination terms. Particle temperatures are considered to be constant and the magnetic field B vertical. The z axis is parallel to the magnetic field B, the y axis to the ambient electric field, and the x axis coincides with the E, x B (Hall) direction. In these coordinates mobility and diffusion tensors have the form
6, =
i
b
7b.H
0
kzH
b,,
0
0
bull I
0
,
TD.H
0
D,,
0
0
=O.
(3)
so
n, = ni+n,.
2. MODEL
at
_ h-h,.hl
We assume quasineutrality,
an e=
(2)
RCx,Y + 03) = nO(h)= nO(z)~
.
DC41
The boundary conditions at x, y + cc are
(4)
The injected ions’ initial profile was taken in the form M,Y,
z*
4 = h40> exp - q [
The longitudinal small compared ionosphere. According to by Rozhansky potential can be
(x"+y'> 2
aA
1
.
cloud scale ui is considered to be with the global vertical size of the the analytical and numerical results and Tsendin (1988) the perturbed taken in the form
rp = :
ln (+J
+ WX,Y, 4,
where Y is a function with large longitudinal I,, >>a,, . We assume also alI <
(5) scale
(6)
For non-boundary homogeneous plasma the characteristic scale I,, at t = 0 is simply a,,/b,,lb, and then increases with time. In the real ionosphere the I, value can also be determined by the vertical scale of no(z) and b,(z). If (6) is fulfilled the potential form (5) is justified for the homogeneous plasma but the algorithm presented revealed the Y longitudinal scale to be large enough in the real ionosphere also. According to equation (10) with the boundary conditions (2), (1 I), (12) only two scales for Y exist-u,dmand the vertical scale of the ionosphere. The ambient density perturbation can be obtained from (15) and has the same scale. So form (51 . I is correct.
1377
Three-dimensional computer simulation of plasma cloud evolution
We consider the strong density perturbation A >>1 so that near the injected cloud n, = n,. From (1) and (4) we have the equation for particle fluxes div I?, = div Fi + div I?, .
(7)
For the ambient ions at ]z] > a,, assuming n, = ni we obtain
(8) Let us introduce dimensionless variables
where hi,(O), b,,,(O) are the mobilities at z = 0 at the injected cloud height. Neglecting the longitudinal ambient ion flux in the ambient plasma with respect to the electron one we obtain the perturbed potential equation for the given density profile (Rozhansky, 1985) :
Equations (10) and (11) are obtained for the arbitrary profile of ni(z) if @‘““I >>ni. Relation (11) must be considered as a mixed boundat-y condition for equation (10) at 4 = 0. The potential Y at [ = 0 is a continuous function but the longitudinal electric field at [ -+ + 0 differs from the one at [ + -0. The boundary condition (3) has the form
Plasma density can be obtained from ion equation (1). After integrating the injected ions’ continuity equation over z up to a,, < ]zD]CCI, with 1, = RI = 0 we obtain (in the variables <, q, [)
+&fi,(V~‘I’--EoaJ
1 .
(13)
The equation is a 2-D one. The ni profile can be reconstructed in a practically important case when T, 3 Inn1 -auf az I-0 7-x----~
T, call
So the ~zi temporal evolution is determined by the Boltzmann part of potential (5) and corresponds to the 1-D ambipolar diffusion (c = Ti) (Rozhansky and Tsendin, 1988) :
f W) where
Ni exp {--2*/[4(1+
Kll = bell@)lbell Co); bL,El= &,H = Dd..li/biJ.(o>;
ii,
baLJitz)lbil
Co);
&~/~iJ(o)*
=
We assume mcV,i<<&vii+(m,, h are the electron and ion masses, and v,i,N the electron-ion and electronneutral collision frequencies) so that D,,, = (T,/e)b,,, . Inserting (5) into (7), integrating (7) over z up to a,, < Izoj <
nr =
T,l~:,)~,,,~+~~ll
J;;[a:+4(1+T,/~)D,,,t]‘/2
’
04)
This treatment is not valid in the transition region (n, < n, and ni N ni) so the far tails of the profile (14) are missing. The ambient density longitudinal perturbation scale is I, >>aW. Neglecting ambient ion shift along B 4Q,,r <
lie1 1(=+0-f&-0
= -V,
-V,[(6;+6:)~*,@,ul-Eoa1)]
(&--&)-I-
++&I)
1
v,&
+&&(3)@,Y_E,a,)
>I r=o’
(11)
where re,, =
ayl f&x
cu i%$=
s -03
dz
n,bJ2(0) iv,bi’!z(O) ~~(O)~~~‘(O)~~= a~~~(O)~~~‘(O).
1 +(&-Ri)al.
(15)
Equation (15) is also 2-D, and [-dependence is a parametric one. The equation system (lo), (13), (15) with the boundary conditions (2), (11) and (12) was simulated numerically. In contrast to the initial equation (1) our basic system (IO), (13), (15) reveals only transport coefficients’ height dependence and the density equations are 2-D. This system describes both plasma evolution in the strong ambient electric field and transverse diffusion processes.
V. A. ROZHANSKY ef al.
1378 3.PERTURBED
POTENTIALSIMULATION
OF A STRONG
AMBIENT
INTHE
ELECTRIC
CASE
FIELD
For large-scale inhomogeneities the diffusion terms in (IO), (1 I), (13) and (15)can be neglected for smooth density profiles. At t = 0 when the background plasma is undisturbed the potential equation is reduced to the Laplace-type one
with the coefficients depending on 1 only c,(C) = &n*,;
c&)
= blln”Q.
While deriving (16) we neglect the transverse electron diffusion coefficients. We also consider (in this section only) the Hall current at c = 0 to be zero. The perturbed potential distribution for the given density profile is described by 116) with the boundary condition (11) where the diffusion terms are neglected. According to (10) and (11) the potential Y is determined by the injected particles’ integrated density only
and depends on parameter A”= J&4. In the calculations presented the chosen values were : 1 km < al < 5 km, the cloud height h,, 160 km < ho < 250 km, the density non-linearity parameter A - 1 C A 6 100 for a, = a, = a. For at # a, the corresponding value of A”is A”= Aa, /aI. The lower boundary of the ionosphere was taken at 90 km, the upper one at h = 2SO300 km. The simulation parameters correspond to night and day aurora1 ionosphere ; see Gurevich et al. (1973). To solve equation (16) the finite-difference method was used. The arising linear algebraic equations system for Y was solved by the technique described in Voskoboynikov ef al. (1987). This technique includes the implicit method of minimal residuals in combination with a cyclic reduction algorithm. The typical equipotential lines Y = const. in the plane pe~ndicular to the magnetic field at I = 0 are presented in Fig. 1. The perturbed electric field EI height dependence is shown in Fig. 2. According to Fig. 2 E; is practically constant along magnetic field lines only for rather large-scale inhomogeneities a, k 5 km. In this case the perturbed potential value inside the cloud E,t is given by (17) obtained for the two-layer model at t = 0 for 6, = biH (Lloyd and Haerendel, 1973 ; see also Rozhansky and Tsendin, 1988)
-h---E; _ -&
1
1 +e NIbII12&’
FIG. 1. EQLJIPOTENTUL LINES'P = CoNSI'.IN THE 5Oq PLANE
WHERE THEINITIAL CLOUD IS SITUATED. Cloud height ho = 160 km, al = 1 km, A = 10. The parameters chosen correspond to the day ionosphere.
where xi, is the height-integrated total Pedersen conductivity of the ionosphere. According to (17) for A = 1, ho = 160 km, aA = 5 km and in the day-time E$E, = 2.75. 10m2. The value obtained in the 3-D numerical simulation is 2.52 * 10A2. For a, = 1 km the difference increases. For example in the day ionosphere for A = 10, ho = 160 km, from (17) we have E;/Eo = 5.4. 10e2, while the numerical simulation gives EJEo = 7.5 - lo- *. So for a, < 5 km the equipotential field lines approximation (17) underestimates the perturbed electric field value. This fact is
z? ”
-ii
5
i ‘W
loo
140 i60 180 200 220 240 h (km)
FIG.2. PERTURBED HEIGHT
(17)
120
ELECTMC
DEPENDENCE
FIELD IN THE VICINITY FOR NIGHT
[ = q =
0
IONOSPHERE.
Cloud height ho = 160 km, A = IO. (1) ai = 0.1 km; (2) a, =0.3km;(3)a,=1km;(4)a,=2.5km;(5)a,=5km.
1379
Three-dimensional computer simulation of plasma cloud evolution
connected with the Ei decrease in the low ionosphere where Pedersen conductivity is maximal. Accordingly the maximal Pedersen conductivity zone is “switched off” from the “influence region” where the ambient ion short-circuiting current is formed. The E; height asymmetry is clearly seen in Fig. 2. At h > 200 km the perturbed electric field remains practically constant due to the sharp Pedersen conductivity decrease ; so the simulation results are unsensitive to the upper boundary position. The perturbed electric field noticeably decreases under the cloud. For a, < 100 m the Ei distribution becomes symmetrical in the z direction, so we can use the formula obtained by Rozhansky (1985); see also Rozhansky and Tsendin (1988) for unboundary homogeneous plasma (E; = 0)
/I Co
h (km)
FIG. 3. PERTURBED ELECTRIC FIELD INSIDE THE CLOUD DIFFERENT INITIAL CJBUD HEIGHTS, A = 1, a,=2.5 NIGHT
(1)ho =
EI_ Eo -
FOR
km,
IONOSPHERE.
160 km; (2) h, = 200 km; (3) h, = 250 km.
-1
>.
(18)
For example, in the day ionosphere for A = 10, a, = 100 m, h,, = 160 km the simulation value is EJEo = 5.1. lo-‘, while from (18) we have EJEo = 6.8. 10m2. The equipotential magnetic field lines model (17) gives us E//E0 = 5.7. 10m3 which is 10 times less than (18). For a given non-linearity parameter A the perturbed electric field decreases with the cloud height ho. For the small-scale inhomogeneities with a, < 100 m this fact is connected with mobility height dependence, so that the “influence region” of the ambient ion shortcircuiting current with the scale al,/m increases. For the large-scale inhomogeneities a, > 5 km the transverse conductivity inside the cloud decreases with height while the region of the ion short-circuiting current remains practically constant. The perturbed electric field dependence on the cloud height ho for the intermediate case is presented in Fig. 3. In our calculations the magnetospheric inertial currents were neglected. It is justified for the inhomogeneities with a, 5 300 m because the perturbed electric field is localized in the ionosphere and practically vanishes at magnetospheric heights. For the large-scale inhomogeneities the presented model is valid if the wave magnetospheric conductivity c, = c2/47rvA (vA is the Alfven velocity) is less than the Pedersen integrated conductivity of the ionosphere. In this case the conductivity of the conjuction ionosphere must be taken into account also. The effects of the finite c,, were discussed by Rozhansky (1986) and Pudovkin et al. (1986). 4. THREEDIMENSIONAL
250
150
DENSITY EVOLUTION
The equation system (lo), (13), (15) with the boundary conditions (2), (1 1), (12) and initial condition
n,(x,~,z)l,=o = AdO) x exp
was solved numerically
by it splitting according to physical processes. The equations (lo), (13), (15) at every time step are considered as linear ones. The above-mentioned method was used for solving the elliptical potential equation. The parabolic density equations were solved by splitting according to the physical processes and directions. The method reduced the problem to the 1-D equations. For their solution the Godunov method (Godunov et al., 1973, 1976) was used and also the method of CrankNicolson. The diffusion equations were integrated by the explicit Euler method. The calculations were performed for a, = 0.1, 0.3, parameter and 1 km, a,, = (I~, the non-linearity A = 10,100,500, and the ambient electric field E, = 1 and 2 mV m- I. Ionization and recombination terms were taken into account for uI = 0.3 km. The initial cloud was situated at ho = 160 km, the low boundary height h, = 100 km, and the upper one h, = 190 km. It can be seen from numerical results that the ambient density perturbation is small enough at h > 190-200 km. So the h, variation causes no significant effects. The part of integrated Pedersen conductivity could be easily taken into account by renormalization of A (A becomes smaller). Such a procedure is necessary for a, 2 1 km only. The spatial step in the z direction was 5 km. It is smaller than both the characteristic scale of the perturbed electric field in the vertical direction and
1380
V.
PUENTIALPRDFlI&5 FIG. 4. PERTURBED
A.
ROZHANSKY
(MINUS
ALTO
et
al.
TRRM)
IN THE PiXXBSN
($I
AND
HALL (t)
DIRECTIONS.
Cloud height ho = 160 km, r = rb,&/a,. (a) A = 10, al = 0.1 km, T = 0, t = 0. (1) h = 160 km ; (2) h = 170 km; (3) h = 180 km; (4) h = 150 km; (5) h = 140 km. (b) A = 10, aI = 0.1 km, h = 160 km (1) ‘c= 0.0, q = 0.0;(2)x = 0.6, q = 0.5. (c)A = 500, a, = 0.3 km, h = 160 km, T = 0, 5 = 0.
the ambient density scale (the smallest vertical scale I,, for a, = 0.1 km was 13.3 km). In the XOY plane the grid with the spatial step 0.2%~~ was used covering the region (8 x @a,. The grid was moving in the Pedersen direction with the injected ions’ maximum density velocity. The calculations were performed for the day ionosphere and T, = q. (a) a, = 0.1 km The Y profiles (the perturbed potential minus Boltzmann part) are presented in Fig. 4. For al = 0.1 km ‘P decreases along z with the spatial scale I,, >>al (Rozhanslcy and Tsendin, 1988) lr = aLtber (0)lk!_("ll"'*
e EoaL
cp([ = 0) = T,/e(lnA-~2-tz)+Y
> 0.
When 5 = 0, q= 5 = 0 (more accurate when }zl >>a,,) the Boltzmann part turns zero and 50= Y c 0. So the potential is of quadrupole type; see Rozhansky and Tsendin (1988). The estimate for Y could be easily obtained from (7). The transverse ion flux divergence and the electron longitudinal one are of the same order, so we have
(19)
For the parameters taken I, = 133~~. The diffusion Diln,Jai and convective bilEgo ffuxes are of the same order when a, = 0.1 km. The corresponding parameter is
r, ff=------.
this case is of diffusion type. From (5) we see that at < = 0, t = 0 the total perturbed potential
(20)
For a,=O.l km and Eo=l mV m-’ a=0.77. According to Fig. 4a-4c the perturbed potential in
For A = 10 and #“@ = 0.13,9 = Y/(E,a,) is of the order of 0.1 according to estimate (21) in agreement with results presented in Fig. 4a4c. The dipole electric field which partially screens the ambient one is added to the diffusion-type quadrupole field; see Fig. 4a. The perturbed electric field value in the cloud centre for o! < 1 and bIH = beH is given by (18). To estimate
Three-dimensional computer simulation of plasma cloud evolution
1381
O.& T 0
I
1
0. I
0.2
I 0.3
I
I
0.4
0.5
I 06
I 0.5
t 1.0
I 1.5
tE,
FIG. 5. MAXI~LDENSITY
~~DE~ENDEN~~
= tbeHE&,.
Cloud height h, = 160 km. (a) ai = 0.1 km. (1) Calculation according to equation (22), 2,3-sim~la~on results for A = 10, and 100, correspondingly. (b) a, = 0.3 km, A = 500. Dotted line-calculation by (22), straight lines-simulation results. (1) IT,,= 1 mV m- ’ ; (2) E. = 2 mV m- ‘. (c) al = 1 km, A = 100. Dotted line-calculation according to (22).
for a, = 1 km we subtract the diffusion part of electric field (Y profile along 5) from the total perturbed electric field (Y profile along q). The EJE, calculated value even in the case G(w 1 agreed with (18) for A = 10 and 100 with an accuracy less than 30% (the same agreement was observed for the dipole E’ component
the calculations in Section 3). The potential distribution is similar to that obtained by Drake et al. (1988). It is shown in Rozhansky and Tsendin (1988) that the plasma inhomogeneity evolution is governed by shop-cir~iting currents in the background plasma.
1382
V.
FIG. 6. R,(q)
A.
HALF-WIDTH TIME DEPENDENCE, a, = 0.1
ROZHANSKY
km,
A = 10.
Crosseenumerical simulation ; straight line-calculation according to (22) : A: = (1 + S&l/a:).
Such a mechanism is effective if the relative density perturbation in the depletion regions h < 1. When @-’ << 1, h N @“-j cc 1. In this case the perturbed electric field is small, so the initial cloud drifts with the injected ions velocity in the ambient field v,, = &E,. Such a shift was observed for A = 10. Cloud decay is determined by injected ions’ unipolar diffusion
N,(5,v,t) =
NJ(0,0,0) 1+ 80,, t/a:
1
-(x-b*“Eet)2-(Y-bllE,t)2 .
x exp [
a:+8D*,t
(22)
The simulation results are compared with (22) in Figs 5 and 6. For A = 10 the agreement is good enough and the deviation from the straight line in Fig. 6 is caused by mesh finiteness. When A = 100 and RimaX)= 1.33, density perturbation in the depletion region becomes comparable with the ambient one (Fig. 7a), so cloud decay decelerates according to Rozhansky and Tsendin (1988) ; see Figs 5a and 9a. The perturbed electric field E; = 0.37E,, in agreement with (18) becomes of order E,, and the cloud velocity correspondingly decreases. The @i profile remains almost symmetrical while the background density profile presented in Fig. 7a is noticeably asymmetric. The negative ambient density perturbation at q = 2 exceeds the one at q = -2 due to the ambient ions’ shift in the (-E,) direction under the perturbed electric field influence. The difference in z direction is caused by transport coefficients’ height dependence.
et
al.
(b) a, = 0.3 km In this case the parameter a = 0.26 so the transverse diffusion term’s role is smaller than for a, = 0.1 km. Accordingly the Y character in Fig. 4c is closer to dipole-type than in Fig. 4a. As for al = 0.1 km, the perturbed potential longitudinal scale is given by (19) but the potential profile becomes asymmetric in the z direction-it is more sharp below the cloud. The perturbed electric field inside the cloud is given by (18) with accuracy less than 10 % . For A = 10 the electric field is slightly disturbed so the injected cloud moves practically with the velocity v0 = 61E,, and its decay is controlled by diffusion in accordance with (22). For A = 100 when h _ film@ N 1 cloud velocity and decay rate decrease. The most interesting case is fi$max)> 1 when depletion regions are deep enough. The parameters chosen (A = 500, a, = 0.3 km, h,, = 160 km) correspond to the barium cloud experiment “Spolokh” data ; seeAndreeva et al. (1984) and Dzubenko et al. (1983). In agreement with (18) the ambient electric field is almost completely screened inside the cloud (Fig. lc) and the cloud velocity strongly decreases (Figs 9b and 10a). Indeed at t = 3 the injected ion maximum shift in the [ direction is < = 1 instead of r = 3 for the unperturbed electric field. It is interesting to note that the cloud does not move in the E, direction while for the Hall to Pedersen mobility ratio bIH/bII = 3.5 the corresponding shift should be noticeable (-0.28 at 7 = 3). This fact is connected with perturbed electric field x-component E: > 0. For 6 = (bcH-blH)/bll << 1 (for parameters chosen at h,, = 160 km, 6 = 0.29) and #{maX)>> 1 according to Rozhansky and Tsendin (1988), E:/(E,- EJ = 6. Such perturbed field was obtained in numerical simulation. Taking into account E: we obtain the cloud velocity in they direction uv = - bIHE: + b,, (E, -E;) = (E, - E; )bn, (-6+
n>.
(23)
Since 6 - bI,/blH N 5 * IO- 3 the u,, is practically zero. The injected ion outflow occurs at the cloud side where & 5 1 in the 5 direction as can be seen in Fig. 10a. The front side of the cloud in the [E, x B] direction becomes more sloping while at the back steepening takes place. Such evolution is presented in Figs 9a and 10a. Equidensities with & < 1 elongate in the direction forming an angle with the r axis. Similar elongation was observed in the two-layer model numerical simulation of Scannapieco et al. (1974), Pudovkin et al. f 1987). etc. It is caused bv the electron
Three-dimensional computer simulation of plasma cloud evolution
‘30
_:
1,
-:
0
1383
II
I2
’ 3
130
& ,90
-3
-2
-I
0
-
II30
150-
140 -
130-
? Cc)
FIG. 7. AMBIENT ION EQUIDENSITTES IN THE 2 Y I’LANJZ WHERE THE fl, MAXIMUM IS SITUA~. Figures correspond to 64 = (q-~(z))/+,(z). (a ) nl = 0.1, A = 100, z = 2.0, C = 1.5; (b) a, = 0.3, A=500,s=1.0,~=0.25;(c)al=1.0;A=100,r=2.0,~=1.
and ion Hall mobilities difference ; the corresponding mechanism was discussed by Rozhansky and Tsendin (1988). Ambient ion density profiles presented in Figs 7b and 8 essentially differ from the case a, = 0.1 km, Fig.
7a. The depletion region is situated mainly on the right side where q > 0 due to ambient ions’ redistribution caused by the perturbed field E;. Positive % perturbation is displaced in the negative q direction. Cloud lifetime is one of its most important charac-
1384
V.
A. ROZHAN~KY ef al.
:t~3~ -3
-2
-I
0
FIG.8. AMBIENT IONEQUIDENSITJJB INTHE
teristics. If we assume the injected ions flow in the 5 direction with density #, = 1 and velocity v0 = CEO/B
then we obtain the linear density decrease law (see also Rozhanksy and Tsendin, 1988) ~rnEX)(~)= urns)
- KZ,
(24)
where 1cis a numerical coefficient. Such linear decay was obtained in our simulation, Fig. 5b. For E. = 2 mV m- ‘, K = 0.3. For E0 = 1 mV m-’ the decay time is smaller due to the relative diffusion influence increase. Naturally for I,, of order (19) the injected ion numbers flowing out due to diffusion is given by the estimate
E (bl FIG.
9. INJECTED IONS’ DENSITYEVOLUTION IN THE HALL DECECTION.
The parameter Sa~i~~~=~ is equal to 0.54 for & = 1 mV m- ’ and 0.27 for .!&,= 2 mV m- ‘. So in the second case & decreases more slowly-Fig. 5b. It should be noted that fir time decay is larger than that due to ion unipolar diffusion (22) corresponding to the case fl,(m”) -C l-the dotted line in Fig. 5b. So the cloud lifetime for A = 500, uI = 300 m is essentially determined by the non-linear deformation in the Hall direction and the density decrease law is given by (24). The simulation results are consistent with “Spolokh” experimental data-Andreeva et al. (19841, Dzubenko et al. (1983 )---where the bulk of the cloud was practically at rest and the injected particle numbers flowing from the main cloud into the tail was increasing linearly with time in accordance with (24). The equipotential magnetic field lines model (17) predicts for t = 0, beH = bin and A = 500 the cloud
Profites presented
include maximum & data (the corresponding tf was chosen). {a) al = 0.1 km, A = 100. (1) ~=0.0;~=0.0;(2)~=0.6;~=0.5;(3)~=2.0;~=1.5; (b)ua,=0.3km,A==500.(1)~=0.0;(2)~=2.0;~=3.0.
velocity in the c direction to be 1.5 times greater than obtained in 3-D simulation. (c) a, = 1 km
This case is an intermediate one so it reveals both features of homogeneous ambient plasma and equipotential magnetic field line models. Simulation was performed for E,, = 1 mV m- ‘. The non-linear parameter chosen, A = 100, lies between the strong nonlinear case when the depletion regions are deep enough
Three-dimensional
computer
simulation
of plasma -3
1385
cloud evolution -2
-I
0
I
2
3
-I
-I
0
0 I
I
6
I -3
I -2
-I
I<
0
I I
I 2
I 3
2
2
3
3
4
4
5
5
6 -3
-2
(al
-I
0
I
2
3
tb) FIG. 10. INJECTED
ION EQUIDENSmIE.5
IN THE @fj
PLANE.
(a)a,=0.3km,A=5OO,r=3.0;(b)a,=1km,A=lOO,r=2.O.
observed, similar to the two-layer model resultsand the perturbed electric field is of order E. and the situation when the perturbed electric field is small Fig. lob. compared with Eo. The cloud velocity in the Hall direction, as can be 5. CONCLUSIONS seen from Fig. lob, is smaller than for the unperturbed (1) For the first time 3-D computer simulation has electric field. It corresponds to the electric field perbeen performed for the inhomogeneities of arbitrary turbation calculated by both (17) and (18). The maxinon-linearity and form in the ionosphere. mal density linear decrease presented in Fig. 5c is in (2) Small-scale (al 6 100 m) cloud evolution is accordance with (24) with the coefficient JC= 0.13. It governed by diffusion ; it occurs in the same way as in is smaller than the one for A = 500 because the density homogeneous ambient plasma. maximum moves with noticeable velocity - 0.6cE,,/B, (3) Inhomogeneities with intermediate transverse sothe relative outflow velocity in the Hall direction is scales a, - 0.3-l km (for the ambient electric field of of order O.rlcE,,/B and K is two times less than in order l-2 mV m- ‘) are in the medium position the previous case. The injected cloud transverse scale between the homogeneous background plasma case increases faster than due to the unipolar diffusion and the equipotential magnetic field lines model. For because parameter CIis three times smaller than for the very strong inhomogeneities (i?, > 1), when the a, = 0.3 km. Equidensities with IjI(“‘@ 5 1 elongate depletion regions in the background plasma are deep in the direction forming an angle with the < axis. enough, maximal cloud density decreases with time This angle is smaller than in the two-layer model of according to the linear law. Cloud evolution in this Scannapieco et al. (1974) or in calculations performed case is determined by the non-linear deformation in by Pudovkin et al. (1987). Such an effect is produced the self-consistent electric field. by the 3-D character of our simulation so that in (4) Perturbed electric field and injected cloud velthe low ionosphere at heights of order 120 km the ocity for inhomogeneities with a, < 1 km are conmagnetic field lines are essentially non-equipotential sistent with the analytical formula (18) obtained for -SO the Hall current perturbation responsible for homogeneous non-boundary ambient plasma. the elongation character is smaller than in equipoten(5) Large-scale inhomogeneities’ evolution with tial magnetic field line models. (21 N 3-5 km could be described by the equipotential Depletion regions are more powerful than for magnetic field lines model. small-scale inhomogeneities a, = 100 m (Fig. 7a,c). Most of it is situated at q > 0 on the right side of Fig. 7c. Such a picture corresponds to perturbed potential dipole character while the quadrupole part is practically absent. Back-side cloud steepening at t; < 0 is
REFERENCES
Andreeva, L. A., Ivchenko, I. S., Milinevsky, G. P., Rozhansky, V. A., Rnzhin, Yu. Ya., Skomarovsky, V. S.
1386
V. A. ROVIANSKYel al.
and Tsendin, L. D. (1984) Dynamics of artificial plasma clouds in “Spolokh” experiments. Cloud deformation. Planet. Space Sci. 32, 1045. Doles, J. H., III, Zabusky, N. J. and Perkins, F. W. (1976) Deformations and striations of plasma clouds in the ionosphere. 3. Numerical simulation of a multilevel model with recombination chemistry. J. geophys. Res. 81,5987. Drake, J. F., Mulbrandon, M. and Huba, J. D. (1988) Threedimensional equilibrium and stability of ionospheric plasma clouds. Phys. Fluids 31, 3412.; Dzubenko, N. I., Zhilinsky, A. P., Zhuiin, I. A., Ivchenko, I. S., Molotai, A. A., Rozhansky, V. A., R&in, Yu. Ya., Skomorovsky, V. S. and Tsendin, L. D. (1983) Dynamics of artificial plasma clouds in “Spolokh” experiments : movement pattern. Planet. Space Sci. 31,849. Godunov, S. K. and Ryabenky, V. S. (1973) Dl@rence Scheme. Nauka, Moskow. Godunov, S. K., Zabrodin, A. V., Ivanov, M. Ya., Krajko, A. N. and Prokopov, G. P. (1976) Gas Dynamics Mullidimensional Problems Numerical ;Simulation. Nauka, Moskow. _$ Gurevich, A. V., Fishuk, D. I. and Tsedilina, E. E. (1973) Three-dimensional model of electron density distribution in calm ionosphere. Geomagn. Aeronomia 13, 31. Gurevich, A. V. and Tsedilina, E. E. (1967) Motion and diffusion of plasma inhomogeneities. Sou. Phys. Usp. 91, 609.
Haerendel, G., Lust, R. and Rieger, E. (1967) Motion of artificial ion clouds in the upper atmosphere. Planet. Space sci. 15, 1. Lloyd, K. H. and Haerendel, G. (1973) Numerical modelling of plasma clouds in the ionosphere. J. geophys. Res. 78, 7389. Perkins, F. W., Zabusky, N. J. and Doles, J. H., 111(1973) Deformation and striation of plasma clouds in the ionosphere. J. geophys. Res. 78, 697. Pudovkin, M. I., Golovchanskaya, I. V., Nikolaev, N. V. and Klimov, M. P. (1987) Two-layer model validity for
strong inhomogeneity Aeronomia
evolution description.
Geomagn.
27, 560.
Pudovkin, M. I., Lyatskaya, A. M. and Golovchanskaya, I. V. (1986) Longitudinal current structure over the bank of increased conductivity in the ionosphere. Geomagn. Aeronomia 26,767.
Rozhansky, V. A. (1985) Perturbed electric fields and evolution of ionosuheric inhomogeneities. Geomaqn. Aeronomia 25,906.
_
Rozhansky, V. A. (1986) Magnetospheric effects caused by the ionospheric inhomogeneities. Geomagn. Aeronomia 26, 780.
-
Rozhansky, V. A. and Tsendin, L. D. (1988) Collisional Transport in Partially Ionized Plasma. Energoatomizdat, Moskow. Scannapieco, A. J., Ossakov, S. L., Book, D. L., McDonald, B. E. and Goldman, S. R. (1974) Conductivity ratio effect on the drift and deformation of F-region barium clouds coupled to E-region ionosphere. J. geophys. Res. 79,2913. Voskobovnikov. S. P. and Rozhanskv. V. A. (1987) Threedimensional simulation of perturdkh elect& fields produced by the ionospheric inhomogeneities. Geomagn. Aeronomia 26,161.
Voskoboynikov, S. P., Rozhansky, V. A. and Tsendin, L. D. (1987) Numerical modelling of three-dimensional plasma cloud evolution in crossed E x B fields. Planet. Space Sci. 35, 835.
Zabusky, N. J. and Doles, J. H., III (1975) The effect of artificially close boundaries on numerical simulations of the instabilitv and evolution of barium clouds. J. _ aeophys. _. Res. 80, 1846.
Zabusky, N. J., Doles, J. H., III and Perkins, F. W. (1973) Deformation and striation of plasma clouds in the ionosphere. 2. Numerical simulation of a nonlinear two-dimensional model. J. ueophvs. Rex 78, 711. Zabusky, N. J., Doles; i H., III and Perkins, F. W. (1976) Deformation and striation of plasma clouds in the ionosphere. J. geophys. Res. 81,5987.