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An analytic model of the neutral cloud evolution in the Earth's atmosphere
Pher. Space Sci., Vol. 40, No. 8, pp. 1139-l 152. 1992 Printed in Great Britain.
0
00324633/92 $5.L-KJ+O.O0 1992 Pergamon Press Ltd
AN ANALYTIC MODEL OF THE NEUTRAL CLOUD EVOLUTION IN THE EARTH’S ATMOSPHERE L. G. BRUSKIN*
Irkutsk
Polytechnical
Institute,
Department
of Mathematics
83 Lermontov
St, Irkutsk,
Russia
and G. V. KHAZANOV_F Altai State University, 66 Dimitrov St, Bamaul, (Received
Russia
in final form 6 January 1992)
Abstract-We discuss an analytic model for the expansion of a chemically reacting gas in the upper atmosphere. The spatial-temporal distribution of released particles in the initial regime is described on the basis of self-similar solutions of Euler’s equations. In the case of transition to the diffusion regime we perform an approximate solution of the kinetic equation with the collision integral in the form of BGK. Gravitation and the atmospheric inhomogeneity are taken into account. The diffusion regime is described by an analytic solution of the diffusion equation in an exponential atmosphere taking account of possible losses of the gas due to chemical reactions. We discuss some peculiarities of the expansion of various gases as well as the possibility of applying the model for describing ionospheric “holes”. 1. INTRODUCTION
making active experiments and launching spacecraft at heights of the ionospheric F-layer, agents are released which are able to enter into chemical reactions with ionospheric ions. This substantially modifies the chemical processes occurring in the ionospheric plasma. Gases such as HZ, CO* and HI0 transform atomic O+ ions to molecular ions, and the rate of this transformation is three orders of magnitude higher than the loss rate of O+ in an undisturbed ionosphere. Resulting molecular ions rapidly recombine with electrons, thus leading to a significant decrease in plasma density of the disturbed region : an ionospheric hole is produced (Mendillo, 1981). Interest in these unusual phenomena is caused by a large number of factors (Bernhardt, 1982 ; Mendillo, 1988). Such experiments provide valuable data on the chemical and diffusion processes occurring in the upper atmosphere and on the airglow emission (Mendillo and Baumgardner, 1982). In the injection zone, there is a change in radio wave propagation conditions, which can lead to the formation of ducts in the plasmasphere (Bemhardt and Park, 1977), to radio wave focusing by an ionospheric “lens” (Bemhardt and da Rosa, 1977), and to reception of radio signals of extraterrestrial origin with a frequency below a critical frequency of the F-layer; the injected gas can both increase and decrease the amplitude of natural When
*On leave of absence at the Plasma Research Center, University of Tsukuba, l-l-l, Tenno-dai 305, Japan. t Now at the Space Physics Research Laboratory, University of Michigan, Ann Arbor, MI, U.S.A.
inhomogeneities, and in the case of injections in the equatorial region the “hole” can become unstable, which would lead to the formation of an ionospheric “bubble” (Anderson and Bernhardt, 1978). When making model simulations of active experiments, it is necessary to investigate two main processes occurring due to injection, namely the evolution of the cloud of released particles and the modification of the background atmosphere. In the early period of expansion of the neutral gas the free path length of injected molecules is significantly smaller than the size of the cloud itself, and its density exceeds the background density. The expansion has a continuum character and can be accompanied by the formation of shock waves. In the case of injections at high altitudes the rarefied background atmosphere has little influence on the injected gas, and no shock waves are produced on the interface. Gases such as H,O and CO* in the initial stage of expansion can condense, which reduces the effectiveness of the ion-molecule reaction (Bernhardt et al., 1982 ; Sjolander and Szuszcewicz, 1979). The continuum flow was investigated both theoretically (Mirels and Mullen, 1963 ; Hamel and Willis, 1966 ; Brode and Enstrom, 1972 ; Aristov and Shakhov, 1985 ; Bernhardt et al., 1988b) and experimentally (Muntz et al., 1970). In the subsequent expansion, internal collisions cease to play the decisive role. Also, in the case of injection at high altitudes the gas moves during a certain period in a collisionless regime before it is scattered due to atmospheric molecules. The subsequent evolution of the cloud is ever increasingly affected by collisions with background particles, and 1139
1140
L.
G.
BRUSKIN
and
the regime of expansion changes from collisionless to diffusive. The process of transition was investigated by Baum (1973), Bernhardt (1979a) and Bruskin et al. (1987) for the case of injection of a small mass of gas into the rarefied atmosphere. The last diffusion stage of evolution is the longest. In order to describe it correctly, it is necessary to take into account the entire set of chemical reactions, the influence of gravitation, the atmospheric inhomogeneities and the neutral wind (Mendillo and Forbes, 1978; Bernhardt, 1979b; Bernhardt et al., 1975; Yu and Klein, 1964; Bruskin et nl., 1985, 1988). Despite the large number of publications on the study of separate evolutionary stages of the cloud, at present there does not exist a model to determine parameters of the gas in all stages of its expansion. Care should be taken when constructing numerical models because the complexity of the model can make its use difficult when calculating the ionospheric plasma disturbance. In this paper we develop an analytical model of an injected cloud which is convenient for practical purposes. The model is useful for determining the density of the released gas in the initial, transition and diffusion regimes of expansion. We investigate two methods of injection : an instantaneous point release and continuous injection by a moving source. The model can be used for calculating ionospheric holes resulting from releases of chemically active gases into the rarefied atmosphere.
2. CONTINUUM
EXPANSION
In the case of injections, in the rarefied atmosphere in the.initial period of expansion the collision frequency of injected particles between themselves significantly exceeds that between background molecules. The cloud in this case evolves similarly to the escape of the gas into a vacuum. One-dimensional flows (plane, or with spherical or cylindrical symmetry) were investigated by Mirels and Mullen (1963) on the basis of solving the hydrodynamical equations for a perfect gas :
aP @PU) -++(v-l).$=O
at+ar
G.
V.
KHAZANOV
the gas, respectively; y = cp/cv is the adiabatic index ; and v = 1, 2, 3, for plane, cylindrical and spherical flow, respectively. These equations do not involve any dimensional constants. Therefore, if more than two constants with independent dimensions are not contained in the additional conditions, the flow will be a self-similar one.
3. POINT
Let us consider a spherically symmetric flow (v = 3). The initial radius of the cloud at rest a and its internal energy E will be the two constants with independent dimensions. The self-similar solution obtained by Mirels and Mullen (1963) has the form : *(l-‘I*)‘w
R’(t)
(1)
R=
(n2+$*)lil,
where c1= (l/MoR ‘) *lr*p dIf; for our density distribution, the factor c( = -3(y- 1)/(5y-3). Let us estimate the influence of the initial condition R(0) = a. The value of tl is about unity. At the initial temperature of _ 1000 K and a radius a of = 1 m the augend is comparable with the addend only at t < lo-’ s, and at t > lo-’ s it is vanishingly small, and the solution has the form
For other values of y, this conclusion obviously remains valid. Time t - low3 s is the interval during which the system with any initial density distribution “forgets” its initial conditions and reaches an asymptotic self-similar mode.
HYPERSONIC
JET
_‘ap P ar
$(Pp-y)
’
where B(a,b) is a beta function, q = r/R(t), and MO is the mass of gas. The motion of an expanding front R(t) is defined by the formula
4. AXISYMMETRICAL
$+u$
INJECTION
= 0.
The flow is assumed to be adiabatic; p, v and P are the density, hydrodynamic velocity and pressure of
A steady-state hypersonic jet, i.e. a flux of particles with a hydrodynamical velocity which very much exceeds the thermal velocity, is investigated by analysing a system of hydrodynamical equations in cylindrical coordinates. Let u be the velocity of the gas along the direction
Analytic model of the neutral cloud evolution in Earth’s atmosphere of the jet x, and let D be the transverse velocity component (along rJ. The equation of continuity in the case of azimuthal symmetry has the form :
and the equation of motion projected onto rl is
au
au +vdr
z
I
+ug =-cyp’-2$ 1
where c = const. In the case of a steady-state motion ap/at = 0 and au/at= 0. The relationships v CCuO, u x u,, hold true in the hypersonic approximation (uO is the initial value of velocity). The hydrodynamical equations then take the form
5. THE TRANSITION
ap+apv+pL’=o
Oax
ar,
2
MOY
,=L
7Wo(~-1) t’
af
(
af
= 4)W,
From this, one can conclude that the transverse expansion of a steady-state hypersonic jet is described by the same relationships as the non-steady-state 1-D flow of an axisymmetrical gas cloud into a vacuum. The only difference is that this equation involves r = x/u0 rather than t. Therefore the self-similar solution is written similarly to the above case of a 1-D flow with spherical symmetry. Finally, for the gas density in the jet and the transverse velocity, we obtain (Mirels and Mullen, 1963) the expressions
=
OF
at+“ar+%
au au ap ““ax +vaY, = -cypy-‘dr,.
p
REGIMES
Following a continual expansion at some distance from the injector, let the free path length of released molecules increase to the size of a gas cloud. Then, when investigating the subsequent expansion, eigencollisions can be neglected. Let us also suppose that the background atmosphere is disturbed only on small spatial scales such that injected energetic particles move in an undisturbed medium. The evolution of the distribution function is described by the kinetic Boltzmann’s equation where the scattering of the gas due to atmospheric molecules is taken into account by the collision integral in the form of BGK :
rl
l-
AND DIFFUSION EXPANSION
af u
1141
I/y- 1
%
0) R2
(2)
where MO is the mass of gas injected for 1 s, and R is the cross-sectional radius of the expanding jet. The dependence R(t) :
where E is the thermal energy of the molecules injected during 1 s, and T and M are the temperature and mass of a molecule, respectively. For this density distribution, the factor CI= (y - 1)/(2y - 1). Thus, the formulae obtained, together with the adiabatic equation of an ideal gas p = cpv, define macroscopic parameters of the injected gas in the cases of instantaneous or continuous injection.
-f)
+ Qo(OWv)dr,
0 -L
Here g is gravitational acceleration, Q. is the source intensity ; in the case of an instantaneous injection Qo(t) = No e(t), cp(r, t) is a function that characterizes the location and shape of the source at some distance from the injector (in the case of a continual injection) or a certain time to after the injection (in the case of an instantaneous release), starting from and onwards from which the eigen-collisions can be neglected ; n = 5f dv is the gas density, and w(r) is the collision frequency with background particles. For the law of conservation of the number of particles to be satisfied, in the BGK model the collision frequency is considered independent of the velocity of the colliding molecules, OS,.= (m/2kT,,J 3/2 exp (- (m/2kT,,,) (v-II,,+)‘) is the Maxwellian distribution function of the source and the atmosphere respectively, and L is the term that represents possible losses of the gas in chemical reactions. It can be of importance only for large times (comparable with the lifetime of the released gas in the most rapid chemical reaction) and can be neglected before the onset of the diffusion stage. The initial conditions are assumed to be zero. In the absence of the gravitational force and of the atmospheric inhomogeneity in the case of a point injection [cp= a(r)] by a continuous source, the Boltzmann’s equation in the injector-associated frame of reference has the form : :+v:=
o(n(D.-f)+Qo(t)a(r)~,(v).
(3)
This equation was solved by Baum (1973), and the following results obtained. The spatial-temporal distribution of particles is described by the following expression :
1142
L. G. BRUSKIN and G. V. KHAZANOV
where
ns(r, t) =
’dt’ s dr’n,(r’, t’)wG(r -r’,
s0
t-t’).
(4~)
ma1 equilibrium with the atmosphere, and the subsequent expansion of the cloud has a purely diffusive character. Let us show further that, for Boltzmann’s equation, in view of the gravity and inhomogeneity of the medium, the gas density also consists of two parts similar to equations (4b) and (4c). By replacing the variables 2
Such a representation clarifies the physical meaning of the functions ncrns and G. First of all, n, is the density of molecules which did not undergo collisions with the atmosphere. Their distribution function differs from the solution of Boltzmann’s equations without collisions n,,(r,t)
r’ = r-vt+
v’ = v-gt
f,
and integrating over t and v, the gas density will be determined from the equation
= lw(&r* n(r,t) =.,(r,t)+r$jdr’exp{-l’o(r(l-?) xexp(
- &(;
-uJ)df’
(5)
by a factor exp (--at) that represents the scattering of part of the molecules. The function n,(r, t) is the density of all other molecules, i.e. of particles scattered by the atmosphere. wn, molecules are scattered per unit time in the atmosphere and are the source of the scattered mode. The function G, as follows from equation (4c), is Green’s function of the scattered mode. The asymptotic expressions for G, when t--t’ c co-’ or when t-t’ >>a-‘, obtained by Baum (1973), have the form :
+r’:+
F(t2-t,)
dt, >
co(r’) Ii (7a)
xQI,t$-F)n(r’,t-t,),
where n,(r,t) =ldt,[dvexp{-l’a(r-vt2+$)dt2} 2 x
r-vt,
Q,(t-t,)@(v-gt,)cp
+ $f,
t-t,
> (64
(6b) where t, = t-t’ and r. = r-r’, and 1 = (2kTa/m)‘/2/w is the free path length. By comparing equation (5) with expression (6a), one can conclude that for small times the function G coincides with the solution of the collisionless Boltzmann’s equation for an instantaneous source. The only difference is that in the expression (6a), instead of the released gas parameters (2kTJm) I/*,the atmospheric parameters lo appear. The representation for G for large times coincides with the Green’s function of the classical diffusion operator in a homogeneous medium with the diffusion coefficients D = kT/mw. This is not an unexpected result because at t >>co-’ particles of the gas have enough time to undergo collisions and to reach ther-
U’b)
The above relationship is an inhomogeneous integral Volterra-Fredholm equation of the second kind ; therefore, its solution in the general case can be represented as n(r, t) = n,(r, t) +
dr’n,(r’, t ,)R(r, r’, t, t ,),
‘dt ,
s0
s (8)
where R is a certain resolvent kernel of equation (7a). On comparing equation (4) with (8), one can conclude that for Boltzmann’s equations also, in view of the gravity and inhomogeneity of the atmosphere, the density proves to consist of two modes. The augend corresponds to collisionless molecules, and their distribution function falls off exponentially. The addend corresponds to a scattered mode, and without loss of generality, the resolvent R(r, r’, t, t ,) can be written, similarly to equation (4c), as o(r’)G(r, r’, t, t J. G will then have the sense of a point source function for the scattered mode.
Analytic model of the neutral cloud evolution in Earth’s atmosphere The collisionless summand very much simplifies for a point source. Thus, for an exponential model of the atmosphere w = w. exp ( - z/H,), the collisionless term can be represented as
It,
1143
=
--
2
gtt m L
-w-t,>
11 /J .
(94
When the source moves with a constant velocity a, its trajectory S(t) = at, and the collisionless term is n, =
From here onwards, T means the injected gas temperature T,. The axis z is considered to be directed upwards, and x and y are directed horizontally. Of most interest is the case of a horizontal motion of the source because at high altitudes the trajectory of a satellite or a rocket is horizontal. Expression (9a) for this case takes a rather simple form. Most of the gas molecules will be scattered by the atmosphere in the height range less than its typical size. Therefore, the atmospheric inhomogen~ity should be taken into account only for the scattered mode, while for the collisionless mode the collision frequency can be considered constant. In this case in a frame of reference associated with the source of the collisionless mode is defined as
In the virtually important case of an instantaneous injection this expression will be written as
where NOis the amount of the released molecules. In the case of a curvilinear motion of the source along the trajectory S(t) of the injected gas, with velocity u,(t) with respect to the atmosphere, n;is defined by the following integral :
1
where u = ~,--a is the hydrodynamic velocity with respect to the injector. It is impossible to determine the form of the dependence G(r, r’, t, t ,) using rigorous analytic methods ; therefore, in order to establish the asymptotic representation of the Green’s function G, we shall make use of the physical analogy with a simplified problem (3). For equation (3), it was shown that at small times, G is a point source function of the problem of collisionless expansion with a mean velocity kw [expression (6a)]. Therefore, in view of the gravitationai force, the Green’s function at smali times takes the form similar to the collisionless solution in the gravitational field
At large times the function G, as expression (6b) for a homogeneous problem, must correspond to the diffusion of particles. Let us determine the explicit expression for the Green’s function at t >>co- ‘. In this case the gravity, the atmospheric inhomogeneity and losses in chemical reactions introduce significant changes to the overall picture of the diffusion. Therefore, an asymptotic representation of Green’s function at large times must satisfy the diffusion equation which includes all these factors :
L. G. BRUSKIN and G. V. KHAZANOV
1144
with the boundary conditions
where
GL.,,,+*, -+ 9, and the initial condition
N(L ?* a, t) =
m
PIx,,~,;++o3+ 0
is -m
G(x, Y, 2, r) xexp(-i(5xot_rlu,))dxdy
G(t = t’) = 6(x-x’,y--y’,z--z’). The Green’s function G is equal to the ratio of the gas density n to the number of injected molecules No. Here P is the flux of injected particles, H is their height scale, and D(z) is the diffusion coefficient of the gas in the atmosphere. Assuming that in the height range under ~nsideration (z w 2~5~ km) the atmosphere consists mainly of atomic oxygen and is in hydrostatic equilibrium, for the coefficient of reciprocal diffusion we have the expression D(z) =
H (
N = e-(kfolr)e -w2)/p,/w+ r/z+p x(AM(‘,+~-k,1+2Kh)$_BU(‘,+C1-k,1+2~,h)). (13)
>
Here
H,+O-+OH+H.
h-ho H 0
h=
2~o(Btt/~o~"2exp(-(~-~~)/~~~; HO
7 = 2(Do/po)"2;
l2 +H,2(eZ+$). 2
( > N,
P2=
___ 2H
to = t-t’;
M and U are confluent hypergeometrical functions, and k is a constant of separation. Expression (13) will be bounded when z + If: co for any { and TV only with a discrete set of k, eigenvalues such that
In this case the loss term has the form G_ >
j+p--km= -m,
B. is the value of the loss coefficient at the origin of height zo. In the case of injection of H,O or CO1, chemical losses are insignificant for w 10 min after the release, and the loss term can either be neglected (L = 0) as done by Yu and Klein (1964) or taken into account approximately in the form L = c&S, where o[is a certain effective constant of losses. The loss term was taken in such a form by Bernhardt et al. (1975, 1988b). Equation (11) with the exponential loss term was solved by Mendillo and Forbes (1978) using the method of separation of variables. The two-dimensional Fourier-transform in horizontal coordinates brings equation (11) to the form
aiv
It should be noted that when 5 = rl = 0 equation (12) becomes a 1-D equation of continuity for integral density, which was considered in detail by Ginzburg and Kim (1974). The method of separation of variables is applied to find the general solution of the problem :
. 0
Ho is the height scale of the atmosphere and Do is the value of D at the origin of height zO. This equation takes into account losses of the ejected gas due to chemical reactions (the term L). As was pointed out by Mendillo and Forbes (1978), in the case of injection of hydrogen the main loss source is the interaction of H 2 with atomic oxygen :
(
yo = Y-Y'.
z-20
D,exp
L=Boexp
xg = x-x’,
( >
m=0,1,2,....
In this case
W-m,1-+2kbN=
mw +211) m L’*“‘(h)
r(l+2p+mj
U= c-m, 1-t2fi,h) = (-l)~rn!~~)(~}, where L:*)(h) is a generalized Lagerr polynomial. With such a spectrum of k,, the Fourier-transform of the vertical particle flux p at a high altitude (h << 1) also tends to zero at any p. The Fourier-transform of the equation solution is represented as a series in eigenfunctions
z-20
at= -Poexp - ffo N
Expression (14) can be brought to a form convenient for practical purposes. Let us determine A, from the initial condition taking into account the orthogonality of the eigenfunctions : > (12)
A, =
??Z! HJ(1--2~-trn)
e-'h'2)~I12-_(H,/2H)+C~~~~(~),
1145
Analytic model of the neutral cloud evolution in Earth’s atmosphere
where h’ = h(Y) is the coordinate of the injection point. For calculating the sum (14), we shall make use of the formula of summation of the series of Lagerr polynomials (Bateman and Erdalyi, 1974)
m m!L~‘(x)L’“‘(y)z”
c
m=O
(xyz) - “2a = ______ l-z
r@n+a; 1)
xexp (-zz)ZJ+@$,
holds. This function very rapidly decreases when v > e. Therefore, it is important to approximate it correctly when v varies in the interval of order e, and with a further increase of v the accuracy of approximation is unimportant. In the expression for Zcv~+pyw(e),where p/v << 1 and e/v CC1, the denominator and the exponent index can be expanded into a series ; the terms of second order of smallness with respect to p/v and e/v can be neglected :
where Z. is a modified Bessel’s function. The Fouriertransform of the problem solution then assumes the form 1 N=m
h (HO/2H)+1/2 h’ F 0 sinh 2 x exp
h+h’
Z,,(e).
- Tcoth$
Here e = (h/h’) ‘/‘/(sinh (t,/2r)). The Green’s function G will be defined by the inverse Fourier-transform which in the case of azimuthal symmetry coincides with the zeroth-order Hankel transform
With a further increase of p (p > H,/H- l), values of both the function and its approximation are negligibly small, and their contribution to the integral is insignificant. Using equation (17)
H” I I-1
‘,?,‘; = ZjCHOIHj - 1,(4 -
HO --1 H
arcsi-;
i fm G=&
11e)
J NJo(P~Pdp> 0
where
Xexp[-&arcsin;;~l,,ej. p = (r2fq2)“2,
For calculating the integral Int =
(18)
rl = (x;+y:)“2.
cc ZCH,IH-ij+)JO
we shall make use of the approximation which is valid at large values of e (Bernhardt et al., 1975)
The choice of the representation (16) for Bessel’s function is motivated by the fact that when e >> 1 the expression (18) together with its derivative, naturally becomes (15). Therefore, formula (18) can be extended to the entire range of variation of e. Then the solution of the equation has the form h’
G=
.
16H&r sinh
2 0
0
h (Ho/=,)+ 112 ZwbIH)- II
7 h
Then the integral is (Gradstein and Ryzhik, 1971) (sing2r))
Int , = ZIcHoIHj_ ,,(e) . e * exp
(15)
e,, I
*exp At small e the following representation Erdalyi, 1974)
- ycoth
(2))
W*exp ($$
(Bateman and where
1 Z,(e) = (2s)
l/2(v2+e2)
l/2 I
(v2+e2)112-varcsinhE
(16)
w = arcsinh (1%
-:smh
I
($)/&‘)’
IV),
(19)
1146
L. G.
BRUSKIN and G.
In the case of diffusion without losses (Yu and Klein, 1964), in much the same way one can obtain G, =-
r,h’,
V.
KHAZANOV
For large times, it is convenient not to calculate the integrals over x and y in equation (8) because the integral over z cannot be taken analytically at all, but to make use of the following identity for the diffusion Green’s function:
h, (H0/2H)+ l/2
8H;nt0 0 h;
xexp( - i’(h;dh”)W, exp( - @ WI),(20) where
I I
-.
HO u - 1 hm~ ,&a (I I
>
HO -H
w, =
arcsinh
dr’G(r’,r,,t)G(r,r’,
t) = G(r,ro, t+z).
(22)
Equation (22) reflects the obvious circumstance that the system with the initial condition corresponding to a point injection at time t = --z at the point r. will subsequently diffuse in the same way as does the point source 6(r,) 6(--t). Using equation (22) one can roughly calculate the integral over the space in the expression that defines the scattered mode :
1
n, =
Expression (19) is the Green’s function of the problem (11) for the case of an exponential dependence of the loss term on the height, and formula (20) applies to the case L = 0. When t <<7 both relationships coincide with the solution reported by Bernhardt et al. (1975).
Using the asymptotic representations of equations (10) and (19) of the Green’s function one can perform integration over spatial variables in expression (8). Thus, for the case of a continuous injection when t
I’ s 0
dt,
dr’n,(r’, t,)oG(r,
r’, t- t2).
The main contribution to the integral is made by small times tZ [(not larger than a few collision times) because the function n, decreases very rapidly with increasing t (as t; 3exp (-0x2)) for an instantaneous injection] ; but at small values of time, as has already been pointed out, the Green’s function G differs little from the Gaussian curve
G= (4nDt)’
v2
exp(-w),
where D is the diffusion coefficient at the height of injection; therefore, n,(r, t) can be represented as G(r, r’, z). For a continuous injection, from equation (9b) it follows that n,(r, 0 =
s0
dt,Qdt-t,)
xexp(--wt,)G
(21) where c = u+atJt
,, and a is
the source velocity
with
Hence, calculating
the scattered mode implies inte-
grating only over time
respect to the atmosphere (A;2
Inkgy)t,
n, =
~~d~~~2d~,Qo~~4
&(r-ct,)‘+
r,at,-ut,+$,t-t,+-
:
kT 2mD (2W
.
In the case of an instantaneous
injection
Analytic model of the neutral cloud evolution in Earth’s atmosphere
s
n, = CON,
0
dt,exp(-wt,)
(23b) Thus, the above expressions (9), (21) and (23) are useful for determining the space-time distribution of injected particles in the second transition regime of expansion. Since when t >>co-’ the scattered mode is calculated by using the Green’s function for the diffusion of a chemicafly reactive gas in an exponential atmosphere, the asymptotic representations of n(t, t) obtained for large times also prove to be valid for both the transition and diffusion regimes of expansion. It is no longer necessary to match the solutions for these two regimes because they are matched in a natural manner.
6. MATCHING OF SOLUTIONS FOR THREE REGIMES OF EXPANSION
In Section 2 we have developed a solution to describe the expansion of the gas in a continual regime. It remains valid as long as the size of the cloud is less than the free path length. At the moment of time when this condition ceases to be satisfied, it is necessary to calculate the density by formulae corresponding to the transition regime of expansion. With such an approach, however, in order to calculate the density of particles scattered by the atmosphere, one would have to calculate the integral of order four, and this is difficult to do, even using a computer. Therefore, we shall not match expressions (1) and (2) with the free-molecular regime, but the distributions close to them, which will allow us to simplify the calculation of the integrals in equation (8). The expression for a collisionless expansion of the gas that is instantaneously injected by a point source has the form
(This formula is written in a coordinate system associated with the centre of mass of the cloud, and gravity on small times is neglected.) The velocity of the molecules is r/t, and temperature tends to zero, Equation (24) has much in common with the expression for the density in the continual regime (1). Indeed, in both cases the density decreases with time according to a cubic law, and the size of the cloud [the half-width in equation (24)] grows linearly. Therefore, one can specify the initial condition for the
1147
collisionless stage not in the form of a corresponding hydrodynamic profile (1) but in the form n&to), where to is defined such that profiles (1) and (24) be close to each other. This can be achieved by requiring the equality of absolute values of density at the cloud centre : to = R(&y2(2sB(;,
-&))“‘.
The hydrodynamic velocity and temperature in both cases also behave in the same way (a = r/t, T + 0) so that the initial condition for the transition regime can be approximated in the form nCo(to). On substituting the expression for R into the last formula, we find t0 = t,($.-.-“(2~B(;,
-5)>“‘,
(25)
where t, is the time of transition to the free-molecular solution. Thus, the collisionless mode in the second stage of expansion of a spherically symmetric cloud can be described by the following expression (the scattering due to atmospheric particles is not taken into account here) : n = ?z,o(t-tt,+t*), where nCOis the collisionless solution corresponding to a point source, t is the time from the beginning of the first stage of expansion, and I, is the duration of the first stage. The time of transition to a free-molecular flow can be determined from the conditions I = R: A = (nu) -I, where CT is the collision cross-section (-2-4 x lo-” cm’) I!2
R=
We now investigate the question of the matching of two first stages in the case of an expansion of a hypersonic jet. For a continual steady-state flow, expression (2) was obtained. In the case of a frec-
L. G. BRUSKIN and G. V. KHAZANOV
1148
molecular expansion in the steady-state the velocity a0 along the direction x
regime with
Thus, the free-molecular flow of a steady-state hypersonic jet (i.e. the downstream flow from the point x, on the axis) obeys the expression :
n&, = ~d~~~(~~~ n = nl.o(x+x,-xt), xex~(-~((~-~lr+~~)l)). Through substitution of l/t’ = y this integral is calculated via the error function 4 :
where n&, corresponds to a free expansion of the jet from the point source, and x is counted from the place of injection. The distance x,, where the flow reaches a collisionless regime, is chosen from considerations R=I=(ncs)-‘:
It= ~~~~*) 0 which corresponds to the distance from the injector
(29
+
When u. >>(ZkT/m)“* (the hypersonic approximation) the augend is much larger than the addend, and the expression in parentheses equals two. Then
40
Qouo M =
nzz
mug r: --( 2kTx*+r; x2(1+r:/x2)3’2
exp
> ’
In the hypersonic approximation the distance from the source to an arbitrary cross-section is much larger than the cross-sectional radius ; therefore
‘ _
n,, Expression (2) can be approximated using equation (27) because the dependence of the density and of the typical width of the profile on x is the same in both cases, and the dependence n(r) within the half-width differs little. We now equalize the densities specified by the expressions (2) and (27) on the axis of the jet at the matching point X, :
The point x, corresponds to the distance from the source, below which the flow is considered freemolecular. The hydrodynamic velocity in the case of a continual expansion u I r,uo/x. ~al~uIating the azimuthal velocity for uO B (ZkT/m) r/Zfor the collisionless distribution function leads to the same result.
It should be noted that the time elapsed from the beginning of injector operation must be sufhcientiy long such that at distance x, the jet is in steady state. From expressions (2.5) and (28) it follows that in the case of both a spherical expansion and a jet the inequality to > t, (or x0 > xi) is satsified, i.e. in the continuum regime the expansion proceeds faster. When calculating the density of particles which underwent collisions with the atmosphere, it should be assumed that the molecules started to experience a scattering in the second stage of expansion, i.e. at time t, or at the distance x, from the injector. Therefore, in view of the scattering of particles and gravity, the collisionless mode for an instantaneous injection will have the form n&r, 8) = gr2 n,, r-W-----, t--tl +f, e-“fr-t~) 2 ( >
t>
5,
f< I,
i0
(304
and for a continuous
I0
injection :
X
rl = yj+zk.
Wb)
Analytic model of the neutral
cloud evolution
The scattered mode is defined through Green’s function as
Expression (31a) corresponds injection. For a hypersonic jet ‘dt”
dr’n6(r’,t”)oG(r,r’,t-t”),
(31b)
where n, is defined by the piece~se-continuous function (30b). The integral (31a) does not present any particular problems and is calculated in the same way as in Section 3 when inv~t~gating the transition regime for an instantaneous point injection. Thus, when t > o- ’ n,
= Jf
dtfe-““‘-‘,‘G
CON,
1149
atmosphere
z (km)
to an instantaneous
J0 J
n,=
in Earth’s
-20
t-
FIG. i. THE COLLISIONL~ APPROXIMATION
SOLUTtON
BY A CONTINUOUS
(SOLID
FUNCTION
LINE)
AND
(DASHED
ITS
LINE).
g1’2
r,uf+
t__t’
2 ’
0
2 r,at,+ul,+
x
+&(“+t,
+to)’
>
.
(32)
However, expression (31 b) cannot be integrated in this way because n, is defined differently for x c xi and for x 2 x1. In order to integrate (3 1b) over spatial variables, we approximate the piecewise-continuous function (30b) by another function which, though continuous for any r, differs little from (30b) :
(x+x,--x,
+
-a,t--u,r’)*
( -EC-_af2 r
J.
2
r >>> 3 (33)
where xa again is the distance from a fictitious point source of collisionless particles. As is apparent from Fig. 1, the functions (33) and (30b) differ only in a small region near X, and, outside this region, are virtually the same. This means that when using expression (33), matching of the solutions occurs not at the point x, but in its certain vicinity. It should also be added that equation (33), and all the approximations used, do not violate the law of conservation of the number of particles. The integral (31 b) is representable as
9
-xo+xl,t-*2+
(
$&tf
>
. (34)
In summarizing the results obtained, we can say the following. An expansion of the gas when t > ? , occurs in such a fashion as if at time - (to-t ,), slightly before the instantaneous point source, a fictitious source of collisionless particles operated. For a continuous injection, the flow at x > x, is described by the solution for a fictitious collisionless source located behind the real source at the distance x , -xw Matching of the solutions for the transition and diffusion expansions has been performed in Section 3 ; or more exactly, it was shown that for large times the solution describing the transition regime naturally reaches the diffusion. Thus, expressions (30a), (3Ob), (32) and (34) specify the space-time distribution of released particles in the transition and diffusion stages. When t --ct, and x -C x, (regime of a continuum flow), the density of particles for an instantaneous and continuous injection is defined by expressions (1) and (2).
7.
DISCUSSION
The relationships obtained in the preceding sections are useful for studying the influence of the various factors on the gas evolution in the upper atmosphere. In the early stage (when I < w- ‘1, collisions with the atmosphere play a secondary role. Only a small part of the molecules that are instantaneously injected
1150
L. G. BRUSKIN and G. V. KHAZANOV
z (km)
= (km) 80 40
k
la1
80
$
(a)
-
,O 40
-
-60
-
40
80
120
160
200
24
104 5 x lo4 cm”
80
FIG. 2. POINTINJECTION OF1O25 H2 MOLECULES AT THEORIGIN OF COORDINATES;THE INITIALVELOCITYOF THE CLOUD IS 8kms-‘,ANDTHETlME1S28s(a). Continuum injection by a source moving with the velocity of 8 km s- ’ ; the gas velocity with respect to the injector is -4 km s- ‘, and the source intensity is 10” molecules s- ’ (b).
at the point x = y = z = 0 (Fig. 2a) form a tail of the cloud (the scattered mode). The influence of the gravitational force is still not manifested. In the case of a continuous injection by a rapidly moving source (Fig. 2b) the tail also consists mainly of collisionless particles up to a distance -u/w from the injector. The evolution when t Q u.- ’ is determined by the temperature and the hydrodynamic velocity of the gas itself and depends weakly on atmospheric parameters. In a later stage (Fig. 3a,b) the entire cloud consists of scattered particles, and its expansion is controlled by the diffusion. The gravity effect at small times manifests itself to a small extent. Thus, from Fig. 2b it is evident that the gas tail behind the source slightly decreases. When t >> w - ’ the density maximum decreases by 20-25 km, and the entire cloud deforms considerably. Further, in the case of injection of a heavy gas (water vapours, Fig. 3b) the cloud flattens, and in the case of injection of light-weight matter (hydrogen, Fig. 3a) the effects of decrease and deformation are less significant due to the fast diffusion. Obviously, if the cloud initially has a hydrodynamic velocity, then it becomes elongated aong the direction of motion; and with the same conditions of injection
t
(b)
FIG. 3. HYDROGENDENSITYCONTOURS600 s AFTERTHEPOINT INJECTION
OF 1oz6 MOLECULES
AT THE ORIGIN OF COORDINATES.
The initial velocity of the cloud is 4 km s- ’ (a). The concentration of water for the same injection conditions (b).
(u, = 4 km s-‘, z+, = U, = 0 and T = 1000 K), the heavy gas has enough time to traverse a longer distance as compared with the light gas which is rapidly dissipated and decelerated in the atmosphere. At long times the atmospheric inhomogeneity also has some influence. In the upper part of the cloud the diffusive flows are more intensive, and losses due to chemical reactions are less significant than those at lower heights. Therefore, in the lower part of the cloud density gradients are large, and in upper parts constant density lines are located at greater distances from each other. The spatial-temporal distribution of particles is defined by the expressions from the preceding section ; however, a large number of typical features of the cloud’s evolution in the ionosphere can be understood by analysing only the diffusion Green’s function (19). Since in a certain time t > w - ’ the released cloud expands diffusively, its behaviour retains the features inherent to the diffusive point source. By analysing the Green’s function, one is led to the following conclusions. At small times (r <
Analytic model of the neutral
cloud evolution
diffusion are small, and expression (19) coincides with a Gaussoid, i.e. the fundamental solution of the diffusion equation in a homogeneous medium. The density of particles in the cloud substantially depends on the height of injection and on the ratio of the height scales of the ambient atmosphere to the gas, Ho/H. When H < Ho, the distribution of particles at high altitudes is close to a barometric one : n-exp(-y)/r$).
The shape of the cloud differs from spherical. For the release of hydrogen, 300 s after the injection the Green’s function has the form shown in Fig. 4. Due to the combined effect of the processes of losses and diffusion, the region of steep gradients is located in the lower part of the cloud. The typical scale of density variation in the horizontal plane is given by the expression
1151
varies with time. When t <<22, the height of maximum is defined by the relationship
The velocity of maximum is directed downward and is
dz,_ dt - -
For the release of a light gas, for example hydrogen, H > H,, and at higher altitudes the density varies with the height scale of the atmosphere :
&(2JLJh’)’
During the process of downward motion the velocity of the cloud decreases. Due to intensive losses in lower parts of the cloud and a rapid diffusion in upper parts, the gravitational downward motion is replaced by a rise of the cloud. In a sufficiently long time interval the height of maximum stabilizes and is located near the point
I I
h,=g+l++
For the injection of a light gas, the position of maximum will be independent of its height scale : =lll = z,-H,ln(&@y2).
Hr = H,
X
The height at which the particle density is a maximum
I
I -200
1 -120
I 40
I
I
0
40
I
I
120
200
c
RN0
FIG. 4.
in Earth’s atmosphere
GREEN’S
FUNCTION
INHOMOGENEOUS
OF THE DIFFUSION
ATMOSPHERE
OPERATOR
(LINES OF EQUAL
IN AN
LEVEL).
The height of maximum is stabilized in a time roughly equal to 7 (40-50 min). It should be noted that the last expressions reflect only qualitatively the picture of motion of the cloud because below 180-200 km the initial diffusion equation in an exponential atmosphere is no longer valid. The results obtained are valid for the following restriction. The absolute value of all typical sizes of the problem in the initial stage of expansion should not exceed the free path length of atmospheric particles ; otherwise, it would not be justifiable to neglect collisions with background particles in the first stage. This imposes constraints on the specified amount of gas. The maximum allowable mass of injection can be estimated by setting the total number of injected particles equal to the number of atmospheric molecules contained in a sphere of radius A, z (cm,)-‘, where cr is the collision cross-section. Then the total N,,=n,+rA;= number of injected particles $rn, 2a-3 where n, is the density of atmospheric molecules. For the F-layer conditions (h zz 300 km, n, x lo9 cm-‘, and a x 2x lo-l5 cm’) N, is about 1 kmole. For the case accepted by Bernhardt et al. (1988a), (n, N 1.3 x 10’ cm-‘) an estimate of maximum gas No N 50 kmoles. If the amount of injected gas significantly exceeds
1152
L. G. BRUSKIN and
NO then after the first self-collision-governed stage both types of collisions should be taken into account, in which case the background atmosphere disturbance must be investigated using the kinetic Boltzmann equation. It was noticed by Bernhardt et al. (1988a) that the coupled, gas-kinetic equations for injected and ambient gases seem to be too complex, not only for analytical implementation but also for numerical implementation. The assumption N < N, provides the validity of the unperturbed atmosphere approximation. We can estimate the spatial scale I of atmospheric disturbance in the first stage of gas evolution. Ambient molecules will be scattered by the cloud while injected particle density exceeds unperturbed ambient concentration : I N (N/(&n,)) ‘I’ . When N approaches the upper limit NO the perturbation scale I approaches the mean free path 1,. While I < 1, the atmosphere perturbation effect may not be taken into account when modelling gas cloud expansion. The validity of fluid and kinetic models is discussed by Bernhardt et al. (1988a). In terms of this constraint the analytic model presented here makes it possible to determine the spatialtemporal distribution of injected particles in the continuum, transitional and diffusive regimes of expansion. Acknowledgements-This work was supported by the NASA Space Physics Theory Program under grant number NAGW2162. We are also grateful to Mr V. G. Mikhalkovsky (SibIZMIR) for his assistance in preparing the English version of the manuscript and typing the text. REFERENCES Anderson, D. N. and Bemhardt, P. A. (1978) Modeling the effects of an H, gas release on the equatorial ionosphere. J. geophys. Res. 83,4777. Aristov, V. V. and Shakhov, E. M. (1985) Motion of rarefied gas caused by finite-mass point release. JVMMF 25, 1066. Bateman, H. and Erdalyi, A. (1974) Higher Transcendental Functions, p. 230. Nauka, Moscow. Baum, H. R. (1973) The interaction of a transient exhaust plume with rarefied atmosphere. J. Fluid Mech. 58, 795. Bernhardt, P. A. (1982) Environmental effect of plasma depletion experiments. Adv. Space Rex 2, 129. Bernhardt, P. A. (1979a) High-altitude gas releases : transition from collisionless flow to diffusive flow in a nonuniform atmosphere. J. geophys. Res. 84,434l. Bemhardt, P. A. (1979b) Three-dimensional time-dependent modeling of neutral gas diffusion in a nonuniform, chemically reactive atmosphere. J. geophys. Res. 84, 793.
G. V. KHAZANOV Bernhardt, P. A. and da Rosa,A. V. (1977) A refracting radio telescope. Radio Sci. 12, 327. Bernhardt, P. A. and Park, C. G. (1977) Protonosphericionospheric modeling of VLF ducts. J. geophys. Res. 82, 5222. Bernhard& P. A., Park, C. G. and Banks, P. M. (1975) Depletion of the F2 region ionosuhere and the nrotonosphere by the release of molecular hydrogen. Geophys. Res. Lett. 2. 341. Bemhardt, P. A., Kashiwa, B. A., Tepley, C. A. and Noble, S. T. (1988a) Spacelab 2 upper atmospheric modification experiment over Arecibo, 1. Neutral gas dynamics. Astron. Lett. Commun. 27, 169. Bernhard& P. A., Swartz, W. E., Kelley, M. C., Sulzer, M. P. and Noble, S. T. (1988b) Soacelab 2 uDDer atmosoheric modification experiment over Arecibo, 2. Plasma dynamics. Astron. Letl. Commun. 27, 183. Bernhardt, P. A., Zinn, J., Mendillo, M. and Baumgardner, J. (1982) Modification of the upper atmosphere with chemical found in rocket exhaust. AIAA Paper 145, 1. Brode, H. L. and Enstrum, J. E. (1972) Analysis of gas expansion in a rarefied atmosphere. Phys. Fluids 25, 1913. Bruskin, L. G., Koen, M. A. and Khazanov, G. V. (1985) Diffusion of chemically reactive gas cloud in an exponential atmosphere. Geomagn. Aeronomiya 25, 794. Bruskin, L. G., Koen, M. A. and Khazanov, G. V. (1987) Transition regime of neutral gas expansion at ionospheric heights. Geomagn. Aeronomiya 21, 790. Bruskin, L. G., Koen, M. A. and Sidorov, I. M. (1988) Modeling of neutral gas releases into the Earth’s ionosphere. Pure appl. Geoihys. 127,415. Ginzbure. E. I. and Kim. V. F. (1974) An analvtic model for diffusildn processes in’the upper atmosphe;e. Izu. vurou. Radiofiz. 17, 1175. Gradstein, I. S. and Ryzhik, I. M. (1971) Tables of Integrals, p. 1100. Nauka, Moscow. Hamel, B. B. and Willis, D. R. (1966) Kinetic theory of source flow expansion with application to the free jet. Phys. Fluids 9, 829. Mendillo, M. (1981) The effect of rocket launches on the ionosphere. Adv. Space Res. 1,275. Mendillo, M. (1988) Ionospheric holes: a review of theory and recent experiments. Adv. Space Res. 8, 5 1. Mendillo, M. and Baumgardner, J. (1982) Optical signature of an ionospheric hole. Geophys. Res. Lett. 9, 215. Mendillo, M. and Forbes, J. M. (1978) Artificially created holes in the ionosphere. J. geophys. Res. 83, 15 1: Mirels, M. and Mullen, J. F. (1963) Exnansion of aas clouds and’hypersonic jets.bounded by a bacuum. A>AA J. 1, 596. Muntz, E. P., Hamel, B. B. and Maguire, B. L. (1970) Some characteristics of exhaust plume rarefaction. AIAA J. 9, 1651. Sjolander, G. W. and Szuszcewicz, E. P. (1979) Chemically depleted F2 ion composition : measurements and theory. J. geophys. Res. 84,4393. Yu, K. and Klein, M. M. (1964) Diffusion of small particles in a nonuniform atmosphere. Phys. Flu& 7, 65 1. .A
1
0
00324633/92 $5.L-KJ+O.O0 1992 Pergamon Press Ltd
AN ANALYTIC MODEL OF THE NEUTRAL CLOUD EVOLUTION IN THE EARTH’S ATMOSPHERE L. G. BRUSKIN*
Irkutsk
Polytechnical
Institute,
Department
of Mathematics
83 Lermontov
St, Irkutsk,
Russia
and G. V. KHAZANOV_F Altai State University, 66 Dimitrov St, Bamaul, (Received
Russia
in final form 6 January 1992)
Abstract-We discuss an analytic model for the expansion of a chemically reacting gas in the upper atmosphere. The spatial-temporal distribution of released particles in the initial regime is described on the basis of self-similar solutions of Euler’s equations. In the case of transition to the diffusion regime we perform an approximate solution of the kinetic equation with the collision integral in the form of BGK. Gravitation and the atmospheric inhomogeneity are taken into account. The diffusion regime is described by an analytic solution of the diffusion equation in an exponential atmosphere taking account of possible losses of the gas due to chemical reactions. We discuss some peculiarities of the expansion of various gases as well as the possibility of applying the model for describing ionospheric “holes”. 1. INTRODUCTION
making active experiments and launching spacecraft at heights of the ionospheric F-layer, agents are released which are able to enter into chemical reactions with ionospheric ions. This substantially modifies the chemical processes occurring in the ionospheric plasma. Gases such as HZ, CO* and HI0 transform atomic O+ ions to molecular ions, and the rate of this transformation is three orders of magnitude higher than the loss rate of O+ in an undisturbed ionosphere. Resulting molecular ions rapidly recombine with electrons, thus leading to a significant decrease in plasma density of the disturbed region : an ionospheric hole is produced (Mendillo, 1981). Interest in these unusual phenomena is caused by a large number of factors (Bernhardt, 1982 ; Mendillo, 1988). Such experiments provide valuable data on the chemical and diffusion processes occurring in the upper atmosphere and on the airglow emission (Mendillo and Baumgardner, 1982). In the injection zone, there is a change in radio wave propagation conditions, which can lead to the formation of ducts in the plasmasphere (Bemhardt and Park, 1977), to radio wave focusing by an ionospheric “lens” (Bemhardt and da Rosa, 1977), and to reception of radio signals of extraterrestrial origin with a frequency below a critical frequency of the F-layer; the injected gas can both increase and decrease the amplitude of natural When
*On leave of absence at the Plasma Research Center, University of Tsukuba, l-l-l, Tenno-dai 305, Japan. t Now at the Space Physics Research Laboratory, University of Michigan, Ann Arbor, MI, U.S.A.
inhomogeneities, and in the case of injections in the equatorial region the “hole” can become unstable, which would lead to the formation of an ionospheric “bubble” (Anderson and Bernhardt, 1978). When making model simulations of active experiments, it is necessary to investigate two main processes occurring due to injection, namely the evolution of the cloud of released particles and the modification of the background atmosphere. In the early period of expansion of the neutral gas the free path length of injected molecules is significantly smaller than the size of the cloud itself, and its density exceeds the background density. The expansion has a continuum character and can be accompanied by the formation of shock waves. In the case of injections at high altitudes the rarefied background atmosphere has little influence on the injected gas, and no shock waves are produced on the interface. Gases such as H,O and CO* in the initial stage of expansion can condense, which reduces the effectiveness of the ion-molecule reaction (Bernhardt et al., 1982 ; Sjolander and Szuszcewicz, 1979). The continuum flow was investigated both theoretically (Mirels and Mullen, 1963 ; Hamel and Willis, 1966 ; Brode and Enstrom, 1972 ; Aristov and Shakhov, 1985 ; Bernhardt et al., 1988b) and experimentally (Muntz et al., 1970). In the subsequent expansion, internal collisions cease to play the decisive role. Also, in the case of injection at high altitudes the gas moves during a certain period in a collisionless regime before it is scattered due to atmospheric molecules. The subsequent evolution of the cloud is ever increasingly affected by collisions with background particles, and 1139
1140
L.
G.
BRUSKIN
and
the regime of expansion changes from collisionless to diffusive. The process of transition was investigated by Baum (1973), Bernhardt (1979a) and Bruskin et al. (1987) for the case of injection of a small mass of gas into the rarefied atmosphere. The last diffusion stage of evolution is the longest. In order to describe it correctly, it is necessary to take into account the entire set of chemical reactions, the influence of gravitation, the atmospheric inhomogeneities and the neutral wind (Mendillo and Forbes, 1978; Bernhardt, 1979b; Bernhardt et al., 1975; Yu and Klein, 1964; Bruskin et nl., 1985, 1988). Despite the large number of publications on the study of separate evolutionary stages of the cloud, at present there does not exist a model to determine parameters of the gas in all stages of its expansion. Care should be taken when constructing numerical models because the complexity of the model can make its use difficult when calculating the ionospheric plasma disturbance. In this paper we develop an analytical model of an injected cloud which is convenient for practical purposes. The model is useful for determining the density of the released gas in the initial, transition and diffusion regimes of expansion. We investigate two methods of injection : an instantaneous point release and continuous injection by a moving source. The model can be used for calculating ionospheric holes resulting from releases of chemically active gases into the rarefied atmosphere.
2. CONTINUUM
EXPANSION
In the case of injections, in the rarefied atmosphere in the.initial period of expansion the collision frequency of injected particles between themselves significantly exceeds that between background molecules. The cloud in this case evolves similarly to the escape of the gas into a vacuum. One-dimensional flows (plane, or with spherical or cylindrical symmetry) were investigated by Mirels and Mullen (1963) on the basis of solving the hydrodynamical equations for a perfect gas :
aP @PU) -++(v-l).$=O
at+ar
G.
V.
KHAZANOV
the gas, respectively; y = cp/cv is the adiabatic index ; and v = 1, 2, 3, for plane, cylindrical and spherical flow, respectively. These equations do not involve any dimensional constants. Therefore, if more than two constants with independent dimensions are not contained in the additional conditions, the flow will be a self-similar one.
3. POINT
Let us consider a spherically symmetric flow (v = 3). The initial radius of the cloud at rest a and its internal energy E will be the two constants with independent dimensions. The self-similar solution obtained by Mirels and Mullen (1963) has the form : *(l-‘I*)‘w
R’(t)
(1)
R=
(n2+$*)lil,
where c1= (l/MoR ‘) *lr*p dIf; for our density distribution, the factor c( = -3(y- 1)/(5y-3). Let us estimate the influence of the initial condition R(0) = a. The value of tl is about unity. At the initial temperature of _ 1000 K and a radius a of = 1 m the augend is comparable with the addend only at t < lo-’ s, and at t > lo-’ s it is vanishingly small, and the solution has the form
For other values of y, this conclusion obviously remains valid. Time t - low3 s is the interval during which the system with any initial density distribution “forgets” its initial conditions and reaches an asymptotic self-similar mode.
HYPERSONIC
JET
_‘ap P ar
$(Pp-y)
’
where B(a,b) is a beta function, q = r/R(t), and MO is the mass of gas. The motion of an expanding front R(t) is defined by the formula
4. AXISYMMETRICAL
$+u$
INJECTION
= 0.
The flow is assumed to be adiabatic; p, v and P are the density, hydrodynamic velocity and pressure of
A steady-state hypersonic jet, i.e. a flux of particles with a hydrodynamical velocity which very much exceeds the thermal velocity, is investigated by analysing a system of hydrodynamical equations in cylindrical coordinates. Let u be the velocity of the gas along the direction
Analytic model of the neutral cloud evolution in Earth’s atmosphere of the jet x, and let D be the transverse velocity component (along rJ. The equation of continuity in the case of azimuthal symmetry has the form :
and the equation of motion projected onto rl is
au
au +vdr
z
I
+ug =-cyp’-2$ 1
where c = const. In the case of a steady-state motion ap/at = 0 and au/at= 0. The relationships v CCuO, u x u,, hold true in the hypersonic approximation (uO is the initial value of velocity). The hydrodynamical equations then take the form
5. THE TRANSITION
ap+apv+pL’=o
Oax
ar,
2
MOY
,=L
7Wo(~-1) t’
af
(
af
= 4)W,
From this, one can conclude that the transverse expansion of a steady-state hypersonic jet is described by the same relationships as the non-steady-state 1-D flow of an axisymmetrical gas cloud into a vacuum. The only difference is that this equation involves r = x/u0 rather than t. Therefore the self-similar solution is written similarly to the above case of a 1-D flow with spherical symmetry. Finally, for the gas density in the jet and the transverse velocity, we obtain (Mirels and Mullen, 1963) the expressions
=
OF
at+“ar+%
au au ap ““ax +vaY, = -cypy-‘dr,.
p
REGIMES
Following a continual expansion at some distance from the injector, let the free path length of released molecules increase to the size of a gas cloud. Then, when investigating the subsequent expansion, eigencollisions can be neglected. Let us also suppose that the background atmosphere is disturbed only on small spatial scales such that injected energetic particles move in an undisturbed medium. The evolution of the distribution function is described by the kinetic Boltzmann’s equation where the scattering of the gas due to atmospheric molecules is taken into account by the collision integral in the form of BGK :
rl
l-
AND DIFFUSION EXPANSION
af u
1141
I/y- 1
%
0) R2
(2)
where MO is the mass of gas injected for 1 s, and R is the cross-sectional radius of the expanding jet. The dependence R(t) :
where E is the thermal energy of the molecules injected during 1 s, and T and M are the temperature and mass of a molecule, respectively. For this density distribution, the factor CI= (y - 1)/(2y - 1). Thus, the formulae obtained, together with the adiabatic equation of an ideal gas p = cpv, define macroscopic parameters of the injected gas in the cases of instantaneous or continuous injection.
-f)
+ Qo(OWv)dr,
0 -L
Here g is gravitational acceleration, Q. is the source intensity ; in the case of an instantaneous injection Qo(t) = No e(t), cp(r, t) is a function that characterizes the location and shape of the source at some distance from the injector (in the case of a continual injection) or a certain time to after the injection (in the case of an instantaneous release), starting from and onwards from which the eigen-collisions can be neglected ; n = 5f dv is the gas density, and w(r) is the collision frequency with background particles. For the law of conservation of the number of particles to be satisfied, in the BGK model the collision frequency is considered independent of the velocity of the colliding molecules, OS,.= (m/2kT,,J 3/2 exp (- (m/2kT,,,) (v-II,,+)‘) is the Maxwellian distribution function of the source and the atmosphere respectively, and L is the term that represents possible losses of the gas in chemical reactions. It can be of importance only for large times (comparable with the lifetime of the released gas in the most rapid chemical reaction) and can be neglected before the onset of the diffusion stage. The initial conditions are assumed to be zero. In the absence of the gravitational force and of the atmospheric inhomogeneity in the case of a point injection [cp= a(r)] by a continuous source, the Boltzmann’s equation in the injector-associated frame of reference has the form : :+v:=
o(n(D.-f)+Qo(t)a(r)~,(v).
(3)
This equation was solved by Baum (1973), and the following results obtained. The spatial-temporal distribution of particles is described by the following expression :
1142
L. G. BRUSKIN and G. V. KHAZANOV
where
ns(r, t) =
’dt’ s dr’n,(r’, t’)wG(r -r’,
s0
t-t’).
(4~)
ma1 equilibrium with the atmosphere, and the subsequent expansion of the cloud has a purely diffusive character. Let us show further that, for Boltzmann’s equation, in view of the gravity and inhomogeneity of the medium, the gas density also consists of two parts similar to equations (4b) and (4c). By replacing the variables 2
Such a representation clarifies the physical meaning of the functions ncrns and G. First of all, n, is the density of molecules which did not undergo collisions with the atmosphere. Their distribution function differs from the solution of Boltzmann’s equations without collisions n,,(r,t)
r’ = r-vt+
v’ = v-gt
f,
and integrating over t and v, the gas density will be determined from the equation
= lw(&r* n(r,t) =.,(r,t)+r$jdr’exp{-l’o(r(l-?) xexp(
- &(;
-uJ)df’
(5)
by a factor exp (--at) that represents the scattering of part of the molecules. The function n,(r, t) is the density of all other molecules, i.e. of particles scattered by the atmosphere. wn, molecules are scattered per unit time in the atmosphere and are the source of the scattered mode. The function G, as follows from equation (4c), is Green’s function of the scattered mode. The asymptotic expressions for G, when t--t’ c co-’ or when t-t’ >>a-‘, obtained by Baum (1973), have the form :
+r’:+
F(t2-t,)
dt, >
co(r’) Ii (7a)
xQI,t$-F)n(r’,t-t,),
where n,(r,t) =ldt,[dvexp{-l’a(r-vt2+$)dt2} 2 x
r-vt,
Q,(t-t,)@(v-gt,)cp
+ $f,
t-t,
> (64
(6b) where t, = t-t’ and r. = r-r’, and 1 = (2kTa/m)‘/2/w is the free path length. By comparing equation (5) with expression (6a), one can conclude that for small times the function G coincides with the solution of the collisionless Boltzmann’s equation for an instantaneous source. The only difference is that in the expression (6a), instead of the released gas parameters (2kTJm) I/*,the atmospheric parameters lo appear. The representation for G for large times coincides with the Green’s function of the classical diffusion operator in a homogeneous medium with the diffusion coefficients D = kT/mw. This is not an unexpected result because at t >>co-’ particles of the gas have enough time to undergo collisions and to reach ther-
U’b)
The above relationship is an inhomogeneous integral Volterra-Fredholm equation of the second kind ; therefore, its solution in the general case can be represented as n(r, t) = n,(r, t) +
dr’n,(r’, t ,)R(r, r’, t, t ,),
‘dt ,
s0
s (8)
where R is a certain resolvent kernel of equation (7a). On comparing equation (4) with (8), one can conclude that for Boltzmann’s equations also, in view of the gravity and inhomogeneity of the atmosphere, the density proves to consist of two modes. The augend corresponds to collisionless molecules, and their distribution function falls off exponentially. The addend corresponds to a scattered mode, and without loss of generality, the resolvent R(r, r’, t, t ,) can be written, similarly to equation (4c), as o(r’)G(r, r’, t, t J. G will then have the sense of a point source function for the scattered mode.
Analytic model of the neutral cloud evolution in Earth’s atmosphere The collisionless summand very much simplifies for a point source. Thus, for an exponential model of the atmosphere w = w. exp ( - z/H,), the collisionless term can be represented as
It,
1143
=
--
2
gtt m L
-w-t,>
11 /J .
(94
When the source moves with a constant velocity a, its trajectory S(t) = at, and the collisionless term is n, =
From here onwards, T means the injected gas temperature T,. The axis z is considered to be directed upwards, and x and y are directed horizontally. Of most interest is the case of a horizontal motion of the source because at high altitudes the trajectory of a satellite or a rocket is horizontal. Expression (9a) for this case takes a rather simple form. Most of the gas molecules will be scattered by the atmosphere in the height range less than its typical size. Therefore, the atmospheric inhomogen~ity should be taken into account only for the scattered mode, while for the collisionless mode the collision frequency can be considered constant. In this case in a frame of reference associated with the source of the collisionless mode is defined as
In the virtually important case of an instantaneous injection this expression will be written as
where NOis the amount of the released molecules. In the case of a curvilinear motion of the source along the trajectory S(t) of the injected gas, with velocity u,(t) with respect to the atmosphere, n;is defined by the following integral :
1
where u = ~,--a is the hydrodynamic velocity with respect to the injector. It is impossible to determine the form of the dependence G(r, r’, t, t ,) using rigorous analytic methods ; therefore, in order to establish the asymptotic representation of the Green’s function G, we shall make use of the physical analogy with a simplified problem (3). For equation (3), it was shown that at small times, G is a point source function of the problem of collisionless expansion with a mean velocity kw [expression (6a)]. Therefore, in view of the gravitationai force, the Green’s function at smali times takes the form similar to the collisionless solution in the gravitational field
At large times the function G, as expression (6b) for a homogeneous problem, must correspond to the diffusion of particles. Let us determine the explicit expression for the Green’s function at t >>co- ‘. In this case the gravity, the atmospheric inhomogeneity and losses in chemical reactions introduce significant changes to the overall picture of the diffusion. Therefore, an asymptotic representation of Green’s function at large times must satisfy the diffusion equation which includes all these factors :
L. G. BRUSKIN and G. V. KHAZANOV
1144
with the boundary conditions
where
GL.,,,+*, -+ 9, and the initial condition
N(L ?* a, t) =
m
PIx,,~,;++o3+ 0
is -m
G(x, Y, 2, r) xexp(-i(5xot_rlu,))dxdy
G(t = t’) = 6(x-x’,y--y’,z--z’). The Green’s function G is equal to the ratio of the gas density n to the number of injected molecules No. Here P is the flux of injected particles, H is their height scale, and D(z) is the diffusion coefficient of the gas in the atmosphere. Assuming that in the height range under ~nsideration (z w 2~5~ km) the atmosphere consists mainly of atomic oxygen and is in hydrostatic equilibrium, for the coefficient of reciprocal diffusion we have the expression D(z) =
H (
N = e-(kfolr)e -w2)/p,/w+ r/z+p x(AM(‘,+~-k,1+2Kh)$_BU(‘,+C1-k,1+2~,h)). (13)
>
Here
H,+O-+OH+H.
h-ho H 0
h=
2~o(Btt/~o~"2exp(-(~-~~)/~~~; HO
7 = 2(Do/po)"2;
l2 +H,2(eZ+$). 2
( > N,
P2=
___ 2H
to = t-t’;
M and U are confluent hypergeometrical functions, and k is a constant of separation. Expression (13) will be bounded when z + If: co for any { and TV only with a discrete set of k, eigenvalues such that
In this case the loss term has the form G_ >
j+p--km= -m,
B. is the value of the loss coefficient at the origin of height zo. In the case of injection of H,O or CO1, chemical losses are insignificant for w 10 min after the release, and the loss term can either be neglected (L = 0) as done by Yu and Klein (1964) or taken into account approximately in the form L = c&S, where o[is a certain effective constant of losses. The loss term was taken in such a form by Bernhardt et al. (1975, 1988b). Equation (11) with the exponential loss term was solved by Mendillo and Forbes (1978) using the method of separation of variables. The two-dimensional Fourier-transform in horizontal coordinates brings equation (11) to the form
aiv
It should be noted that when 5 = rl = 0 equation (12) becomes a 1-D equation of continuity for integral density, which was considered in detail by Ginzburg and Kim (1974). The method of separation of variables is applied to find the general solution of the problem :
. 0
Ho is the height scale of the atmosphere and Do is the value of D at the origin of height zO. This equation takes into account losses of the ejected gas due to chemical reactions (the term L). As was pointed out by Mendillo and Forbes (1978), in the case of injection of hydrogen the main loss source is the interaction of H 2 with atomic oxygen :
(
yo = Y-Y'.
z-20
D,exp
L=Boexp
xg = x-x’,
( >
m=0,1,2,....
In this case
W-m,1-+2kbN=
mw +211) m L’*“‘(h)
r(l+2p+mj
U= c-m, 1-t2fi,h) = (-l)~rn!~~)(~}, where L:*)(h) is a generalized Lagerr polynomial. With such a spectrum of k,, the Fourier-transform of the vertical particle flux p at a high altitude (h << 1) also tends to zero at any p. The Fourier-transform of the equation solution is represented as a series in eigenfunctions
z-20
at= -Poexp - ffo N
Expression (14) can be brought to a form convenient for practical purposes. Let us determine A, from the initial condition taking into account the orthogonality of the eigenfunctions : > (12)
A, =
??Z! HJ(1--2~-trn)
e-'h'2)~I12-_(H,/2H)+C~~~~(~),
1145
Analytic model of the neutral cloud evolution in Earth’s atmosphere
where h’ = h(Y) is the coordinate of the injection point. For calculating the sum (14), we shall make use of the formula of summation of the series of Lagerr polynomials (Bateman and Erdalyi, 1974)
m m!L~‘(x)L’“‘(y)z”
c
m=O
(xyz) - “2a = ______ l-z
r@n+a; 1)
xexp (-zz)ZJ+@$,
holds. This function very rapidly decreases when v > e. Therefore, it is important to approximate it correctly when v varies in the interval of order e, and with a further increase of v the accuracy of approximation is unimportant. In the expression for Zcv~+pyw(e),where p/v << 1 and e/v CC1, the denominator and the exponent index can be expanded into a series ; the terms of second order of smallness with respect to p/v and e/v can be neglected :
where Z. is a modified Bessel’s function. The Fouriertransform of the problem solution then assumes the form 1 N=m
h (HO/2H)+1/2 h’ F 0 sinh 2 x exp
h+h’
Z,,(e).
- Tcoth$
Here e = (h/h’) ‘/‘/(sinh (t,/2r)). The Green’s function G will be defined by the inverse Fourier-transform which in the case of azimuthal symmetry coincides with the zeroth-order Hankel transform
With a further increase of p (p > H,/H- l), values of both the function and its approximation are negligibly small, and their contribution to the integral is insignificant. Using equation (17)
H” I I-1
‘,?,‘; = ZjCHOIHj - 1,(4 -
HO --1 H
arcsi-;
i fm G=&
11e)
J NJo(P~Pdp> 0
where
Xexp[-&arcsin;;~l,,ej. p = (r2fq2)“2,
For calculating the integral Int =
(18)
rl = (x;+y:)“2.
cc ZCH,IH-ij+)JO
we shall make use of the approximation which is valid at large values of e (Bernhardt et al., 1975)
The choice of the representation (16) for Bessel’s function is motivated by the fact that when e >> 1 the expression (18) together with its derivative, naturally becomes (15). Therefore, formula (18) can be extended to the entire range of variation of e. Then the solution of the equation has the form h’
G=
.
16H&r sinh
2 0
0
h (Ho/=,)+ 112 ZwbIH)- II
7 h
Then the integral is (Gradstein and Ryzhik, 1971) (sing2r))
Int , = ZIcHoIHj_ ,,(e) . e * exp
(15)
e,, I
*exp At small e the following representation Erdalyi, 1974)
- ycoth
(2))
W*exp ($$
(Bateman and where
1 Z,(e) = (2s)
l/2(v2+e2)
l/2 I
(v2+e2)112-varcsinhE
(16)
w = arcsinh (1%
-:smh
I
($)/&‘)’
IV),
(19)
1146
L. G.
BRUSKIN and G.
In the case of diffusion without losses (Yu and Klein, 1964), in much the same way one can obtain G, =-
r,h’,
V.
KHAZANOV
For large times, it is convenient not to calculate the integrals over x and y in equation (8) because the integral over z cannot be taken analytically at all, but to make use of the following identity for the diffusion Green’s function:
h, (H0/2H)+ l/2
8H;nt0 0 h;
xexp( - i’(h;dh”)W, exp( - @ WI),(20) where
I I
-.
HO u - 1 hm~ ,&a (I I
>
HO -H
w, =
arcsinh
dr’G(r’,r,,t)G(r,r’,
t) = G(r,ro, t+z).
(22)
Equation (22) reflects the obvious circumstance that the system with the initial condition corresponding to a point injection at time t = --z at the point r. will subsequently diffuse in the same way as does the point source 6(r,) 6(--t). Using equation (22) one can roughly calculate the integral over the space in the expression that defines the scattered mode :
1
n, =
Expression (19) is the Green’s function of the problem (11) for the case of an exponential dependence of the loss term on the height, and formula (20) applies to the case L = 0. When t <<7 both relationships coincide with the solution reported by Bernhardt et al. (1975).
Using the asymptotic representations of equations (10) and (19) of the Green’s function one can perform integration over spatial variables in expression (8). Thus, for the case of a continuous injection when t
I’ s 0
dt,
dr’n,(r’, t,)oG(r,
r’, t- t2).
The main contribution to the integral is made by small times tZ [(not larger than a few collision times) because the function n, decreases very rapidly with increasing t (as t; 3exp (-0x2)) for an instantaneous injection] ; but at small values of time, as has already been pointed out, the Green’s function G differs little from the Gaussian curve
G= (4nDt)’
v2
exp(-w),
where D is the diffusion coefficient at the height of injection; therefore, n,(r, t) can be represented as G(r, r’, z). For a continuous injection, from equation (9b) it follows that n,(r, 0 =
s0
dt,Qdt-t,)
xexp(--wt,)G
(21) where c = u+atJt
,, and a is
the source velocity
with
Hence, calculating
the scattered mode implies inte-
grating only over time
respect to the atmosphere (A;2
Inkgy)t,
n, =
~~d~~~2d~,Qo~~4
&(r-ct,)‘+
r,at,-ut,+$,t-t,+-
:
kT 2mD (2W
.
In the case of an instantaneous
injection
Analytic model of the neutral cloud evolution in Earth’s atmosphere
s
n, = CON,
0
dt,exp(-wt,)
(23b) Thus, the above expressions (9), (21) and (23) are useful for determining the space-time distribution of injected particles in the second transition regime of expansion. Since when t >>co-’ the scattered mode is calculated by using the Green’s function for the diffusion of a chemicafly reactive gas in an exponential atmosphere, the asymptotic representations of n(t, t) obtained for large times also prove to be valid for both the transition and diffusion regimes of expansion. It is no longer necessary to match the solutions for these two regimes because they are matched in a natural manner.
6. MATCHING OF SOLUTIONS FOR THREE REGIMES OF EXPANSION
In Section 2 we have developed a solution to describe the expansion of the gas in a continual regime. It remains valid as long as the size of the cloud is less than the free path length. At the moment of time when this condition ceases to be satisfied, it is necessary to calculate the density by formulae corresponding to the transition regime of expansion. With such an approach, however, in order to calculate the density of particles scattered by the atmosphere, one would have to calculate the integral of order four, and this is difficult to do, even using a computer. Therefore, we shall not match expressions (1) and (2) with the free-molecular regime, but the distributions close to them, which will allow us to simplify the calculation of the integrals in equation (8). The expression for a collisionless expansion of the gas that is instantaneously injected by a point source has the form
(This formula is written in a coordinate system associated with the centre of mass of the cloud, and gravity on small times is neglected.) The velocity of the molecules is r/t, and temperature tends to zero, Equation (24) has much in common with the expression for the density in the continual regime (1). Indeed, in both cases the density decreases with time according to a cubic law, and the size of the cloud [the half-width in equation (24)] grows linearly. Therefore, one can specify the initial condition for the
1147
collisionless stage not in the form of a corresponding hydrodynamic profile (1) but in the form n&to), where to is defined such that profiles (1) and (24) be close to each other. This can be achieved by requiring the equality of absolute values of density at the cloud centre : to = R(&y2(2sB(;,
-&))“‘.
The hydrodynamic velocity and temperature in both cases also behave in the same way (a = r/t, T + 0) so that the initial condition for the transition regime can be approximated in the form nCo(to). On substituting the expression for R into the last formula, we find t0 = t,($.-.-“(2~B(;,
-5)>“‘,
(25)
where t, is the time of transition to the free-molecular solution. Thus, the collisionless mode in the second stage of expansion of a spherically symmetric cloud can be described by the following expression (the scattering due to atmospheric particles is not taken into account here) : n = ?z,o(t-tt,+t*), where nCOis the collisionless solution corresponding to a point source, t is the time from the beginning of the first stage of expansion, and I, is the duration of the first stage. The time of transition to a free-molecular flow can be determined from the conditions I = R: A = (nu) -I, where CT is the collision cross-section (-2-4 x lo-” cm’) I!2
R=
We now investigate the question of the matching of two first stages in the case of an expansion of a hypersonic jet. For a continual steady-state flow, expression (2) was obtained. In the case of a frec-
L. G. BRUSKIN and G. V. KHAZANOV
1148
molecular expansion in the steady-state the velocity a0 along the direction x
regime with
Thus, the free-molecular flow of a steady-state hypersonic jet (i.e. the downstream flow from the point x, on the axis) obeys the expression :
n&, = ~d~~~(~~~ n = nl.o(x+x,-xt), xex~(-~((~-~lr+~~)l)). Through substitution of l/t’ = y this integral is calculated via the error function 4 :
where n&, corresponds to a free expansion of the jet from the point source, and x is counted from the place of injection. The distance x,, where the flow reaches a collisionless regime, is chosen from considerations R=I=(ncs)-‘:
It= ~~~~*) 0 which corresponds to the distance from the injector
(29
+
When u. >>(ZkT/m)“* (the hypersonic approximation) the augend is much larger than the addend, and the expression in parentheses equals two. Then
40
Qouo M =
nzz
mug r: --( 2kTx*+r; x2(1+r:/x2)3’2
exp
> ’
In the hypersonic approximation the distance from the source to an arbitrary cross-section is much larger than the cross-sectional radius ; therefore
‘ _
n,, Expression (2) can be approximated using equation (27) because the dependence of the density and of the typical width of the profile on x is the same in both cases, and the dependence n(r) within the half-width differs little. We now equalize the densities specified by the expressions (2) and (27) on the axis of the jet at the matching point X, :
The point x, corresponds to the distance from the source, below which the flow is considered freemolecular. The hydrodynamic velocity in the case of a continual expansion u I r,uo/x. ~al~uIating the azimuthal velocity for uO B (ZkT/m) r/Zfor the collisionless distribution function leads to the same result.
It should be noted that the time elapsed from the beginning of injector operation must be sufhcientiy long such that at distance x, the jet is in steady state. From expressions (2.5) and (28) it follows that in the case of both a spherical expansion and a jet the inequality to > t, (or x0 > xi) is satsified, i.e. in the continuum regime the expansion proceeds faster. When calculating the density of particles which underwent collisions with the atmosphere, it should be assumed that the molecules started to experience a scattering in the second stage of expansion, i.e. at time t, or at the distance x, from the injector. Therefore, in view of the scattering of particles and gravity, the collisionless mode for an instantaneous injection will have the form n&r, 8) = gr2 n,, r-W-----, t--tl +f, e-“fr-t~) 2 ( >
t>
5,
f< I,
i0
(304
and for a continuous
I0
injection :
X
rl = yj+zk.
Wb)
Analytic model of the neutral
cloud evolution
The scattered mode is defined through Green’s function as
Expression (31a) corresponds injection. For a hypersonic jet ‘dt”
dr’n6(r’,t”)oG(r,r’,t-t”),
(31b)
where n, is defined by the piece~se-continuous function (30b). The integral (31a) does not present any particular problems and is calculated in the same way as in Section 3 when inv~t~gating the transition regime for an instantaneous point injection. Thus, when t > o- ’ n,
= Jf
dtfe-““‘-‘,‘G
CON,
1149
atmosphere
z (km)
to an instantaneous
J0 J
n,=
in Earth’s
-20
t-
FIG. i. THE COLLISIONL~ APPROXIMATION
SOLUTtON
BY A CONTINUOUS
(SOLID
FUNCTION
LINE)
AND
(DASHED
ITS
LINE).
g1’2
r,uf+
t__t’
2 ’
0
2 r,at,+ul,+
x
+&(“+t,
+to)’
>
.
(32)
However, expression (31 b) cannot be integrated in this way because n, is defined differently for x c xi and for x 2 x1. In order to integrate (3 1b) over spatial variables, we approximate the piecewise-continuous function (30b) by another function which, though continuous for any r, differs little from (30b) :
(x+x,--x,
+
-a,t--u,r’)*
( -EC-_af2 r
J.
2
r >>> 3 (33)
where xa again is the distance from a fictitious point source of collisionless particles. As is apparent from Fig. 1, the functions (33) and (30b) differ only in a small region near X, and, outside this region, are virtually the same. This means that when using expression (33), matching of the solutions occurs not at the point x, but in its certain vicinity. It should also be added that equation (33), and all the approximations used, do not violate the law of conservation of the number of particles. The integral (31 b) is representable as
9
-xo+xl,t-*2+
(
$&tf
>
. (34)
In summarizing the results obtained, we can say the following. An expansion of the gas when t > ? , occurs in such a fashion as if at time - (to-t ,), slightly before the instantaneous point source, a fictitious source of collisionless particles operated. For a continuous injection, the flow at x > x, is described by the solution for a fictitious collisionless source located behind the real source at the distance x , -xw Matching of the solutions for the transition and diffusion expansions has been performed in Section 3 ; or more exactly, it was shown that for large times the solution describing the transition regime naturally reaches the diffusion. Thus, expressions (30a), (3Ob), (32) and (34) specify the space-time distribution of released particles in the transition and diffusion stages. When t --ct, and x -C x, (regime of a continuum flow), the density of particles for an instantaneous and continuous injection is defined by expressions (1) and (2).
7.
DISCUSSION
The relationships obtained in the preceding sections are useful for studying the influence of the various factors on the gas evolution in the upper atmosphere. In the early stage (when I < w- ‘1, collisions with the atmosphere play a secondary role. Only a small part of the molecules that are instantaneously injected
1150
L. G. BRUSKIN and G. V. KHAZANOV
z (km)
= (km) 80 40
k
la1
80
$
(a)
-
,O 40
-
-60
-
40
80
120
160
200
24
104 5 x lo4 cm”
80
FIG. 2. POINTINJECTION OF1O25 H2 MOLECULES AT THEORIGIN OF COORDINATES;THE INITIALVELOCITYOF THE CLOUD IS 8kms-‘,ANDTHETlME1S28s(a). Continuum injection by a source moving with the velocity of 8 km s- ’ ; the gas velocity with respect to the injector is -4 km s- ‘, and the source intensity is 10” molecules s- ’ (b).
at the point x = y = z = 0 (Fig. 2a) form a tail of the cloud (the scattered mode). The influence of the gravitational force is still not manifested. In the case of a continuous injection by a rapidly moving source (Fig. 2b) the tail also consists mainly of collisionless particles up to a distance -u/w from the injector. The evolution when t Q u.- ’ is determined by the temperature and the hydrodynamic velocity of the gas itself and depends weakly on atmospheric parameters. In a later stage (Fig. 3a,b) the entire cloud consists of scattered particles, and its expansion is controlled by the diffusion. The gravity effect at small times manifests itself to a small extent. Thus, from Fig. 2b it is evident that the gas tail behind the source slightly decreases. When t >> w - ’ the density maximum decreases by 20-25 km, and the entire cloud deforms considerably. Further, in the case of injection of a heavy gas (water vapours, Fig. 3b) the cloud flattens, and in the case of injection of light-weight matter (hydrogen, Fig. 3a) the effects of decrease and deformation are less significant due to the fast diffusion. Obviously, if the cloud initially has a hydrodynamic velocity, then it becomes elongated aong the direction of motion; and with the same conditions of injection
t
(b)
FIG. 3. HYDROGENDENSITYCONTOURS600 s AFTERTHEPOINT INJECTION
OF 1oz6 MOLECULES
AT THE ORIGIN OF COORDINATES.
The initial velocity of the cloud is 4 km s- ’ (a). The concentration of water for the same injection conditions (b).
(u, = 4 km s-‘, z+, = U, = 0 and T = 1000 K), the heavy gas has enough time to traverse a longer distance as compared with the light gas which is rapidly dissipated and decelerated in the atmosphere. At long times the atmospheric inhomogeneity also has some influence. In the upper part of the cloud the diffusive flows are more intensive, and losses due to chemical reactions are less significant than those at lower heights. Therefore, in the lower part of the cloud density gradients are large, and in upper parts constant density lines are located at greater distances from each other. The spatial-temporal distribution of particles is defined by the expressions from the preceding section ; however, a large number of typical features of the cloud’s evolution in the ionosphere can be understood by analysing only the diffusion Green’s function (19). Since in a certain time t > w - ’ the released cloud expands diffusively, its behaviour retains the features inherent to the diffusive point source. By analysing the Green’s function, one is led to the following conclusions. At small times (r <
Analytic model of the neutral
cloud evolution
diffusion are small, and expression (19) coincides with a Gaussoid, i.e. the fundamental solution of the diffusion equation in a homogeneous medium. The density of particles in the cloud substantially depends on the height of injection and on the ratio of the height scales of the ambient atmosphere to the gas, Ho/H. When H < Ho, the distribution of particles at high altitudes is close to a barometric one : n-exp(-y)/r$).
The shape of the cloud differs from spherical. For the release of hydrogen, 300 s after the injection the Green’s function has the form shown in Fig. 4. Due to the combined effect of the processes of losses and diffusion, the region of steep gradients is located in the lower part of the cloud. The typical scale of density variation in the horizontal plane is given by the expression
1151
varies with time. When t <<22, the height of maximum is defined by the relationship
The velocity of maximum is directed downward and is
dz,_ dt - -
For the release of a light gas, for example hydrogen, H > H,, and at higher altitudes the density varies with the height scale of the atmosphere :
&(2JLJh’)’
During the process of downward motion the velocity of the cloud decreases. Due to intensive losses in lower parts of the cloud and a rapid diffusion in upper parts, the gravitational downward motion is replaced by a rise of the cloud. In a sufficiently long time interval the height of maximum stabilizes and is located near the point
I I
h,=g+l++
For the injection of a light gas, the position of maximum will be independent of its height scale : =lll = z,-H,ln(&@y2).
Hr = H,
X
The height at which the particle density is a maximum
I
I -200
1 -120
I 40
I
I
0
40
I
I
120
200
c
RN0
FIG. 4.
in Earth’s atmosphere
GREEN’S
FUNCTION
INHOMOGENEOUS
OF THE DIFFUSION
ATMOSPHERE
OPERATOR
(LINES OF EQUAL
IN AN
LEVEL).
The height of maximum is stabilized in a time roughly equal to 7 (40-50 min). It should be noted that the last expressions reflect only qualitatively the picture of motion of the cloud because below 180-200 km the initial diffusion equation in an exponential atmosphere is no longer valid. The results obtained are valid for the following restriction. The absolute value of all typical sizes of the problem in the initial stage of expansion should not exceed the free path length of atmospheric particles ; otherwise, it would not be justifiable to neglect collisions with background particles in the first stage. This imposes constraints on the specified amount of gas. The maximum allowable mass of injection can be estimated by setting the total number of injected particles equal to the number of atmospheric molecules contained in a sphere of radius A, z (cm,)-‘, where cr is the collision cross-section. Then the total N,,=n,+rA;= number of injected particles $rn, 2a-3 where n, is the density of atmospheric molecules. For the F-layer conditions (h zz 300 km, n, x lo9 cm-‘, and a x 2x lo-l5 cm’) N, is about 1 kmole. For the case accepted by Bernhardt et al. (1988a), (n, N 1.3 x 10’ cm-‘) an estimate of maximum gas No N 50 kmoles. If the amount of injected gas significantly exceeds
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L. G. BRUSKIN and
NO then after the first self-collision-governed stage both types of collisions should be taken into account, in which case the background atmosphere disturbance must be investigated using the kinetic Boltzmann equation. It was noticed by Bernhardt et al. (1988a) that the coupled, gas-kinetic equations for injected and ambient gases seem to be too complex, not only for analytical implementation but also for numerical implementation. The assumption N < N, provides the validity of the unperturbed atmosphere approximation. We can estimate the spatial scale I of atmospheric disturbance in the first stage of gas evolution. Ambient molecules will be scattered by the cloud while injected particle density exceeds unperturbed ambient concentration : I N (N/(&n,)) ‘I’ . When N approaches the upper limit NO the perturbation scale I approaches the mean free path 1,. While I < 1, the atmosphere perturbation effect may not be taken into account when modelling gas cloud expansion. The validity of fluid and kinetic models is discussed by Bernhardt et al. (1988a). In terms of this constraint the analytic model presented here makes it possible to determine the spatialtemporal distribution of injected particles in the continuum, transitional and diffusive regimes of expansion. Acknowledgements-This work was supported by the NASA Space Physics Theory Program under grant number NAGW2162. We are also grateful to Mr V. G. Mikhalkovsky (SibIZMIR) for his assistance in preparing the English version of the manuscript and typing the text. REFERENCES Anderson, D. N. and Bemhardt, P. A. (1978) Modeling the effects of an H, gas release on the equatorial ionosphere. J. geophys. Res. 83,4777. Aristov, V. V. and Shakhov, E. M. (1985) Motion of rarefied gas caused by finite-mass point release. JVMMF 25, 1066. Bateman, H. and Erdalyi, A. (1974) Higher Transcendental Functions, p. 230. Nauka, Moscow. Baum, H. R. (1973) The interaction of a transient exhaust plume with rarefied atmosphere. J. Fluid Mech. 58, 795. Bernhardt, P. A. (1982) Environmental effect of plasma depletion experiments. Adv. Space Rex 2, 129. Bernhardt, P. A. (1979a) High-altitude gas releases : transition from collisionless flow to diffusive flow in a nonuniform atmosphere. J. geophys. Res. 84,434l. Bemhardt, P. A. (1979b) Three-dimensional time-dependent modeling of neutral gas diffusion in a nonuniform, chemically reactive atmosphere. J. geophys. Res. 84, 793.
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