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Convective diffusion in the equatorial F region
Journal of Atmospheric and Terrestrial Physics, 1956, Yol. 9, 1)P. 304 to 310.
Pergamon Press Ltd., London
Convective diffusion in the equatorial
F
region
J . W . DU?~GEr The Cavendish Laboratory,
Cambridge
(Received 11 July 1 9 5 6 )
Abstract--This paper discusses the convective difl\lsion which JotI~so_~ and HULB~-aTshowe(l eouhl take place across the geomagnetic field in the F region near tile equator. It is show1/that the convective diffusion will increase any irregularities which may be present. The usual formula for the conductivity is found to be inappropriate when diffusion is involved. The convective motion is regarded as that of a gravity-driven d3~mrno and its speed is controlled by the euri'ellt flowing along the lines of for('e inlo lower levels of the ionosphere. The speed is found to be inversely proportio,~al to the ('ast-w~sl scale of the irregularities, and for a scale of 100 metres may be a few metres/see.
I. INTRODUCTION IX the F region, because the collision frequency for positive ions is much snmller than their gyrofrequency, diffusion of the ions across the magnetic field is very slow. Then, since the forces acting on the ionization, namely gravity and the partial pressure gradient, are nearly vertical, it is at first sight expected that near the geomagnetic equator diffusion will be unimportant; the ionization should be supported by the electromagnetic force. It was realised by JoHNso~.,~ and HVLBURT (1950) and JOI~NSON (1950), however, that this situation is unstable and that a convective motion of the ionization must occur. They suppose that some irregularities exist in the ionization density and show that the currents required to support them against gravity will tend to build up space charge. The resulting electric field causes a drift of the ionization, which is downwards in the region of excess density. Two properties of the motion follow from this: (i) Since the mean gradient of the ionization density is upwards in the observable part of a layer, a downward motion increases the density at a fixed point, and so the irregularities are amplified. (ii) The motion produces a nett downward transport of ionization. HULBURT and JOBNSON point out that the current required to balance a pressure gradient does not set up space charge, so that diffusion nnder a pressure gradient is completely prevented by the magnetic field. Although they say that diffusion under gravity is unaffected by the field, it is of course modified by having to take a convectional form. It will be shown here that the velocity of diffusion is controlled b y t h e flow o f e l e c t r i c c u r r e n t do~31 t h e l i n e s of force i n t o l o w e r l e v e l s o f t h e i o n o s p h e r e w h e r e t h e c o l l i s i o n f r e q u e n c y is g r e a t e r . T h i s side o f t h e p r o b l e m w a s n o t d i s c u s s e d b y HVLBVt~T a n d J o n ~ ' s o ~ ", a n d is t h e m a i n s u b j e c t of t h i s p a p e r , b u t a l i t t l e f u r t h e r c l a r i f i c a t i o n o f t h e i r d i s c u s s i o n will b e g i v e n first. 2. HITLBURT AND JOHNSON'S PHENOMENON I n t h e F r e g i o n t h e c o n d u c t i v i t y is l a r g e a n d it is well k n o w n f r o m t h e s t u d y o f c o s m i c e l e e t r o d y n a m i e s t h a t t h e n t h e m a g n e t i c field m a y b e r e g a r d e d as m o v i n g w i t h t h e i o n i z a t i o n . I n t h e i o n o s p h e r e , f u r t h e r m o r e , t h e field is s t r o n g i n t h e s e n s e t h a t t h e m a g n e t i c p r e s s u r e H2/Srr is m u c h l a r g e r t h a n t h e k i n e t i c p r e s s m ' e o f t h e i o n i z a t i o n ; it t h e n follows t h a t t h e d i s t u r b a n c e i n t h e field m u s t be s m a l l c o m p a r e d 3()4
Convective diffusion in the equatoriM 1'~ region
with the unperturbed field. In particular, the lines of force cannot be appreciably bent. In the present problem the motion is perpendicular to the field and it then follows that the velocity is constant on a line of force, if the field is taken to be uniform. Then, although the small initial irregularities in the density of ionization m a y have any form, the irregularities generated by the motion are also constant on a line of force. Thus, with I-ILTL~U~T and Jott~-so~-'s approximations the problem is twodimensional, all quantities being constant on a line of force. For the purpose of a simple demonstration of the instability in physical terms, consider the simplest motion which will not disturb the field: consider a circular cylinder with its axis parallel to the field, and suppose that the only motion
0 Fig. 1. A convective motion.
allowed is a uniform rotation of the cylinder. It is well known that such a system is ,rustable if the centre of gravity of the cylinder is above its axis; here the centre of gravity refers to the charged particles only, and they are assumed to be free to move independently of the neutral particles. Instability therefore occurs for some cylinder, whenever the electron density increases upwards. The form of the actual motion will be more complicated, lint. is probably of the convective type shown in Fig. 1, the regions of excess density being those of downward motion. Since the magnetic field moves with the velocity U of the charged particles and is taken to be uniform, the condition that the field should not change requires that the area enclosed b y any curve moving with velocity U should not change or div U = 0
(1)
Then. if the loss of ionization is omitted, the rate of change of the electron density N,. is given b y ~\'~/~t =- --(U. V) N,. (2) which implies that, if we follow a lump of ionization through the motion, N~ remains constant. The regions of high density tend to fall and so reduce the mean value of the upward component of VN~, b u t the rate of loss of ionization probably decreases with height (IIATCLIF~E, SCHMERLING, SETTY, and THOMAS, 1956), in which case the loss tends to re-establish the upward gradient. 3. TIIIIElgDIMENSIONAL ~OIIMULATION
It has been pointed out that the twodimensional model is not really adequate and that the flow of current down the lines of force to lower levels is important. 305
J . W. DU~=GE¥
For readers who are familiar with the idea of the motion of the field, it m a y be said t h a t the motion of the field and ionization at low levels is resisted by collisions with neutral particles and this bends the field slightly. The bending opposes the motion at the high level. The same result can be seen by considering the system as an electrical machine. Consider again a cylinder at the high level, with its centre of gravity above the axis. I t m a y be regarded as a gravity-driven unipolar inductor, generating an e.m.f, between its axis and its outer surface. This drives a current down the lines of force, which are analogous to a coaxial cable, as shown in Fig. 2. With increasing depth in the atmosphere the resistivity parallel to the field increases while t h a t
Fig. 2. The flow of c u r r e n t along a n d across the lines of force.
across the field decreases, so t h a t the performance of the cable deteriorates. The effective resistance of this "cable" determines the electric field and this determines the speed of the motion. This approach is developed mathematically here, but it is first necessary to discuss the values of the conductivity appropriate to various conditions; although the theory of the conductivity is well known, it is not obvious what value to use for the internal resistance of the " d y n a m o . " 4. THE CONDUCTIVITY The basic equations determining the conductivity of a weakly ionized gas of one constituent are mvvv(Vv - - U.) = e(E -~ Up A H/c)
(3)
m~ve(U~ - - U,,) ~- - - e ( E -~ Ue ~ H/c)
(4)
and where the suffices n, p, and e refer to neutral molecules, positive ions and electrons and r~ and ~ are the frequencies of collisions with neutral molecules. The result of eliminating Un from (3) and (4) m a y be written
where a 0 is the conductivity in the absence of a magnetic field and the densities of positive ions and electrons are taken to be equal and are denoted by Ne. The second term is the Hall electric field. The last term is a field induced by the motion of the charged particles in which the velocities of the electrons and positive ions are weighted with their collision frequencies as well as their masses. It is usual, 306
Convective diffusion in the equatorial F region
however, to define the induced field as Un A H/c, and in order to obtain an expression for the conductivity consistent with this definition use is made of the equation obtained b y eliminating E from (3) and (4): m~,v~,Uv + m~v~U~
j AH
m~v~ + m~v~
Nec(m~G +
When this is substituted into (5) a term proportional to H A j +, H results, and, i f j is perpendicular to H, this inereases the component of E parallel to j and hence the work done b y the electric field. The physical reason for this is that the force density j A H/c causes the ionization to move relative to the neutral molecules and the motion is resisted b y collisions, which dissipate energy. When the equation is solved f o r j the solution takes the we|l-known form involving % and ~2 (CowLi.X(;, 1945). Here it is necessary to consider what modifications occur, when the pressure gradient and gravity are included in equations (3) and (4). The terms containing the pressure gradients which then occur in (5) represent a generalized form of the thermoelectric field and are negligible; the gravity term is also negligible (,JoH~-SO~T and HULBI;RT, 1950). Equation (6) becomes.
=¥gm#~,u,, + m#~G) = ~'~(m~,,, + m~e)U,, + j ,', HIc
Vp ÷ /,g
((~')
which is just the equation determining diffusion. Now it has already been seen that in the F region at low geomagnetic latitudes diffusion in the ordinary sense is prevented b y the magnetic field, although it would be quite fast in the absence of a magnetic field. This means that the magnetic foree j A H/c just balances the forces of gravity and the pressure gradient, and there is hardly any motion of the ionization relative to the neutral molecules. Then (6) is quite incorrect an(1 the usual formula for the conductivity must also be wrong. When nmtion of the ionization occurs, the full equation (6') should be used in eonjunetion with (5). When the collision frequencies are small, however, the collision terms in (6') m a y be omitted, and in the following this approximation will be made in the F region. Equation (6') then determines the current and equation (5) determines the motion. I f this is to be expressed physically in terms of internal resistance, the internal resistance corresponds to the resistive term j / % in (5), so that % is the appropriate internal conductivity. 5. TIIE SIMPLIFIED MODEL
The purpose here is to estimate the velocity of the convective motion and a model will be used in which only the essential features are retained. The damping of the motion depends on currents flowing along lines of force into the lower ionosphere. The model will be straightened out, however, so that the magnetic field is uniform, and the density of neutral molecules increases with the distance from the geomagnetic equator. Cartesian axes will be used with the x axis pointing north, the y axis west and the z axis upwards; the geomagnetic equatorial plane will be x = 0 and the model will be symmetrical with respect to this plane. The model will be divided into two regions: i n t h e " d y n a m o region," defined b y x < a, collisions will be completely neglected and in the "load region," defined b y x > a, 307
J. W. Du~rG~r
the forces of gravity and the pressure gradient will be neglected. This separation is artificial, but greatly simplifies the work and should not introduce a serious error into the calculated velocity. The value of a is determined roughly by the geomagnetic latitude at which a line of force has fallen say 200 km below its height at the geomagnetic equator; this is about 10 ° and hence a ~--1000 kin. The problem is now t h a t of a dynamo driven by a known force and feeding into a known load. The force determines the current, the current and load determine the voltage and the voltage deternfines the speed of the dynamo. In the dynamo region, since collisions are neglected, (6') is approximated by
j
,,.,,
u/c = v p
-/~g
(7)
where ff is the mass density of the ionization; it has also been assumed t h a t the acceleration of the motion is negligible. I t follows t h a t
and, using d i v j ~ 0 and the s y m m e t r y with respect to the plane x
- -
0.
qc tTM 9 u ---
~"
d:c
(S)
[['his determines the current flowing from the dynamo region into the load region. Next, the electric field is eMeulated as a multiple ofj~ from the properties of the load region. 6. TuE LoA[) I-Ir,:G,OX The condition t h a t the field is not appreciably distorted requires curl E ~- 0 and we put E -- --V~. In the load region, since gravity and the pressure gradient are neglected, (6) and the usual formulae tbr the conductivities are correct. We have .L, = - a o O ¢ / ~ z
(:!)
aud hence
cx
, ~
by
O" I
u/
~-
.
.
.
.
~cz , al bz ,
^
~ by
--
~.-
~
.
c y c z ) ~'~
(10)
This equation will now be simplified by ruthless approximations, which are partially justified by the final result. This shows t h a t the speed of the motion is inversely proportional to its scale in the east-west direction. There must of course be some lower limit to this scale; for instance it cannot be smaller t h a n the radius of the orbits of the positive ions, but this is only ~ 1 0 metres. Consequently, the east-west scale of the convection is likely to be small compared with the vertical scale, and it m a y not be unreasonable to assume t h a t the import a n t term on the right hand side of (10) is al~2c}/by 2. The problem is then linear 308
Convective diffusionin the equatorial F region and the Fourier component corresponding to ¢ ~ cosly will now be discussed. Then (10) reduces to C'd2
= --~1/2¢
(l |)
which nlust be solved in conjunction with (9). The variation of ~o and ~1 with x depends on the variation of the densities ~\'~ of electrons and N,~ of neutral molecules with x, according to the well-known formulae a o = Y,,e21mY<, (12) and
where ~ denotes the gyrofrequency. Now when :\~,, is small enough, so t h a t r,, is substantially less t h a n fl~,, (13) can be approximated by (14)
a 1 ~ N,,e2vJm~f~, 2
h i this approximation the product a0a 1 is independent of N,~ and, if it is further assmned t h a t N~ is independent of x, a0a 1 is independent of x, and the solution of (9) and (ll) is A exp
I
1
1 1
~1 d x
L J \o'0/
-~ Bex])
J
[ /i o) ] --l
' a l ~dx
(15)
wh(~t'e A and B are arbitrary constants. I f this solution were valid for the whole of the load region, the b o u n d a r y condition t h a t ~ -~ 0 as x -+ 09 would require A - - 0 , and this condition will still be approximately correct, if the exponents in (15) reach large enough values before the approximation fails. Now v~ ~ 10~,p and for atomic oxygen ions m~, ~-~ 3. lO%n(,, whence the approximation requires al/aO to t)e substantially less t h a n 3. 10 4. The exponents in (15) can then be large only if 1 1 is considerably less t h a n 1 kni. Consequently the use of (15) with A := () is probably rather inaccurate, but nevertheless we proceed to estimate the velocity of the convective motion on this basis; since a~ is overestiniated here, the velocity is underestimated. 7. ESTISIATE OF THE VELOCITY
Substituting (15) with A = 0 into (9) gives j~c = l(O'0G'l)~¢
(16)
which together with (8) determines ¢ at the boundary between the dynamo and load regions. In the d y n a m o region the variation of ¢ with x is neglected, because ~0 is large, and hence ¢ is known everywhere. The velocity of the ionization m a y now be determined from (5) in which only the last term is not known. The term J/ao is negligible and the Hall field contributes a velocity of order g / ~ v , which is only a few cm/sec and will also be neglected. The speed of the motion is therefore 309
J. W. DU=NGEY
c E / H a n d t h e m a g n i t u d e of its v e r t i c a l c o m p o n e n t is cl¢/H. Using (16), (8), (12), a n d (14) it is t h e n
w h e r e ~N~ r e p r e s e n t s t h e v a r i a t i o n of N~ in t h e d i s t u r b a n c e in t h e d y n a m o region; it will be a s s u m e d t h a t ~N~/N~ is n o t v e r y small. F o r a t o m i c o x y g e n , g/£2, ~ 4 c m / sec a n d m~u~/m~v~ ---- 3 . 10 -a. I f 1-1 ~ 100 m e t r e s a n d a --~ 1000 k m , the speed is t h e n 7 5N~/N~ m / s or 25 &V~/N~ k m / h r a n d m i g h t be a p p r e c i a b l e . This c a l c u l a t i o n requires quite a lot of r e f i n e m e n t , b u t we m a y conclude t.hat t h e p h e n o m e n o n p r o p o s e d b y HULBURT a n d JOH~'SO~= is likely to be i m p o r t a n t a n d t h a t a m o r e refined t h e o r e t i c a l s t u d y w o u l d be w o r t h while. O b s e r v a t i o n s will n o t be discussed here, b u t t h e r e should be no difficulty in o b s e r v i n g t h e irregularities; t h e effect on t h e a v e r a g e p r o p e r t i e s of t h e l a y e r is c o m p l i c a t e d b y other movements.
Acknowledgements--I a m i n d e b t e d to Mr. J. A. I~ATCLIFFE for suggesting this p r o b l e m to m e a n d for his v e r y helpful c o m m e n t s d u r i n g t h e course of the work. T h i s w o r k was carried out d u r i n g t h e t e n u r e of an I.C.I. Fellowship.
(I!O~,VLIN(~ ' T. G. .IO~NSO-',~M. H.
REFERENCES 1945 Proc. Roy. ~'~'oc.A183, 453. 1950 Proc. Conf. I<~no.~'pb. l'l~ys., I)."l~. >~'t<~t~,
P~per J. ,JOHNSON ~'~. H. and HULBURT E. (). ]~ATCLIFFE J. A., SCHMERLINGE. R., SETTY C. •. G. K., and THO)IAS J. O.
] 950 1956
310
l'l~/s. Rev. 79, S02.
t'hil Tr~lns. A248, 621.
Pergamon Press Ltd., London
Convective diffusion in the equatorial
F
region
J . W . DU?~GEr The Cavendish Laboratory,
Cambridge
(Received 11 July 1 9 5 6 )
Abstract--This paper discusses the convective difl\lsion which JotI~so_~ and HULB~-aTshowe(l eouhl take place across the geomagnetic field in the F region near tile equator. It is show1/that the convective diffusion will increase any irregularities which may be present. The usual formula for the conductivity is found to be inappropriate when diffusion is involved. The convective motion is regarded as that of a gravity-driven d3~mrno and its speed is controlled by the euri'ellt flowing along the lines of for('e inlo lower levels of the ionosphere. The speed is found to be inversely proportio,~al to the ('ast-w~sl scale of the irregularities, and for a scale of 100 metres may be a few metres/see.
I. INTRODUCTION IX the F region, because the collision frequency for positive ions is much snmller than their gyrofrequency, diffusion of the ions across the magnetic field is very slow. Then, since the forces acting on the ionization, namely gravity and the partial pressure gradient, are nearly vertical, it is at first sight expected that near the geomagnetic equator diffusion will be unimportant; the ionization should be supported by the electromagnetic force. It was realised by JoHNso~.,~ and HVLBURT (1950) and JOI~NSON (1950), however, that this situation is unstable and that a convective motion of the ionization must occur. They suppose that some irregularities exist in the ionization density and show that the currents required to support them against gravity will tend to build up space charge. The resulting electric field causes a drift of the ionization, which is downwards in the region of excess density. Two properties of the motion follow from this: (i) Since the mean gradient of the ionization density is upwards in the observable part of a layer, a downward motion increases the density at a fixed point, and so the irregularities are amplified. (ii) The motion produces a nett downward transport of ionization. HULBURT and JOBNSON point out that the current required to balance a pressure gradient does not set up space charge, so that diffusion nnder a pressure gradient is completely prevented by the magnetic field. Although they say that diffusion under gravity is unaffected by the field, it is of course modified by having to take a convectional form. It will be shown here that the velocity of diffusion is controlled b y t h e flow o f e l e c t r i c c u r r e n t do~31 t h e l i n e s of force i n t o l o w e r l e v e l s o f t h e i o n o s p h e r e w h e r e t h e c o l l i s i o n f r e q u e n c y is g r e a t e r . T h i s side o f t h e p r o b l e m w a s n o t d i s c u s s e d b y HVLBVt~T a n d J o n ~ ' s o ~ ", a n d is t h e m a i n s u b j e c t of t h i s p a p e r , b u t a l i t t l e f u r t h e r c l a r i f i c a t i o n o f t h e i r d i s c u s s i o n will b e g i v e n first. 2. HITLBURT AND JOHNSON'S PHENOMENON I n t h e F r e g i o n t h e c o n d u c t i v i t y is l a r g e a n d it is well k n o w n f r o m t h e s t u d y o f c o s m i c e l e e t r o d y n a m i e s t h a t t h e n t h e m a g n e t i c field m a y b e r e g a r d e d as m o v i n g w i t h t h e i o n i z a t i o n . I n t h e i o n o s p h e r e , f u r t h e r m o r e , t h e field is s t r o n g i n t h e s e n s e t h a t t h e m a g n e t i c p r e s s u r e H2/Srr is m u c h l a r g e r t h a n t h e k i n e t i c p r e s s m ' e o f t h e i o n i z a t i o n ; it t h e n follows t h a t t h e d i s t u r b a n c e i n t h e field m u s t be s m a l l c o m p a r e d 3()4
Convective diffusion in the equatoriM 1'~ region
with the unperturbed field. In particular, the lines of force cannot be appreciably bent. In the present problem the motion is perpendicular to the field and it then follows that the velocity is constant on a line of force, if the field is taken to be uniform. Then, although the small initial irregularities in the density of ionization m a y have any form, the irregularities generated by the motion are also constant on a line of force. Thus, with I-ILTL~U~T and Jott~-so~-'s approximations the problem is twodimensional, all quantities being constant on a line of force. For the purpose of a simple demonstration of the instability in physical terms, consider the simplest motion which will not disturb the field: consider a circular cylinder with its axis parallel to the field, and suppose that the only motion
0 Fig. 1. A convective motion.
allowed is a uniform rotation of the cylinder. It is well known that such a system is ,rustable if the centre of gravity of the cylinder is above its axis; here the centre of gravity refers to the charged particles only, and they are assumed to be free to move independently of the neutral particles. Instability therefore occurs for some cylinder, whenever the electron density increases upwards. The form of the actual motion will be more complicated, lint. is probably of the convective type shown in Fig. 1, the regions of excess density being those of downward motion. Since the magnetic field moves with the velocity U of the charged particles and is taken to be uniform, the condition that the field should not change requires that the area enclosed b y any curve moving with velocity U should not change or div U = 0
(1)
Then. if the loss of ionization is omitted, the rate of change of the electron density N,. is given b y ~\'~/~t =- --(U. V) N,. (2) which implies that, if we follow a lump of ionization through the motion, N~ remains constant. The regions of high density tend to fall and so reduce the mean value of the upward component of VN~, b u t the rate of loss of ionization probably decreases with height (IIATCLIF~E, SCHMERLING, SETTY, and THOMAS, 1956), in which case the loss tends to re-establish the upward gradient. 3. TIIIIElgDIMENSIONAL ~OIIMULATION
It has been pointed out that the twodimensional model is not really adequate and that the flow of current down the lines of force to lower levels is important. 305
J . W. DU~=GE¥
For readers who are familiar with the idea of the motion of the field, it m a y be said t h a t the motion of the field and ionization at low levels is resisted by collisions with neutral particles and this bends the field slightly. The bending opposes the motion at the high level. The same result can be seen by considering the system as an electrical machine. Consider again a cylinder at the high level, with its centre of gravity above the axis. I t m a y be regarded as a gravity-driven unipolar inductor, generating an e.m.f, between its axis and its outer surface. This drives a current down the lines of force, which are analogous to a coaxial cable, as shown in Fig. 2. With increasing depth in the atmosphere the resistivity parallel to the field increases while t h a t
Fig. 2. The flow of c u r r e n t along a n d across the lines of force.
across the field decreases, so t h a t the performance of the cable deteriorates. The effective resistance of this "cable" determines the electric field and this determines the speed of the motion. This approach is developed mathematically here, but it is first necessary to discuss the values of the conductivity appropriate to various conditions; although the theory of the conductivity is well known, it is not obvious what value to use for the internal resistance of the " d y n a m o . " 4. THE CONDUCTIVITY The basic equations determining the conductivity of a weakly ionized gas of one constituent are mvvv(Vv - - U.) = e(E -~ Up A H/c)
(3)
m~ve(U~ - - U,,) ~- - - e ( E -~ Ue ~ H/c)
(4)
and where the suffices n, p, and e refer to neutral molecules, positive ions and electrons and r~ and ~ are the frequencies of collisions with neutral molecules. The result of eliminating Un from (3) and (4) m a y be written
where a 0 is the conductivity in the absence of a magnetic field and the densities of positive ions and electrons are taken to be equal and are denoted by Ne. The second term is the Hall electric field. The last term is a field induced by the motion of the charged particles in which the velocities of the electrons and positive ions are weighted with their collision frequencies as well as their masses. It is usual, 306
Convective diffusion in the equatorial F region
however, to define the induced field as Un A H/c, and in order to obtain an expression for the conductivity consistent with this definition use is made of the equation obtained b y eliminating E from (3) and (4): m~,v~,Uv + m~v~U~
j AH
m~v~ + m~v~
Nec(m~G +
When this is substituted into (5) a term proportional to H A j +, H results, and, i f j is perpendicular to H, this inereases the component of E parallel to j and hence the work done b y the electric field. The physical reason for this is that the force density j A H/c causes the ionization to move relative to the neutral molecules and the motion is resisted b y collisions, which dissipate energy. When the equation is solved f o r j the solution takes the we|l-known form involving % and ~2 (CowLi.X(;, 1945). Here it is necessary to consider what modifications occur, when the pressure gradient and gravity are included in equations (3) and (4). The terms containing the pressure gradients which then occur in (5) represent a generalized form of the thermoelectric field and are negligible; the gravity term is also negligible (,JoH~-SO~T and HULBI;RT, 1950). Equation (6) becomes.
=¥gm#~,u,, + m#~G) = ~'~(m~,,, + m~e)U,, + j ,', HIc
Vp ÷ /,g
((~')
which is just the equation determining diffusion. Now it has already been seen that in the F region at low geomagnetic latitudes diffusion in the ordinary sense is prevented b y the magnetic field, although it would be quite fast in the absence of a magnetic field. This means that the magnetic foree j A H/c just balances the forces of gravity and the pressure gradient, and there is hardly any motion of the ionization relative to the neutral molecules. Then (6) is quite incorrect an(1 the usual formula for the conductivity must also be wrong. When nmtion of the ionization occurs, the full equation (6') should be used in eonjunetion with (5). When the collision frequencies are small, however, the collision terms in (6') m a y be omitted, and in the following this approximation will be made in the F region. Equation (6') then determines the current and equation (5) determines the motion. I f this is to be expressed physically in terms of internal resistance, the internal resistance corresponds to the resistive term j / % in (5), so that % is the appropriate internal conductivity. 5. TIIE SIMPLIFIED MODEL
The purpose here is to estimate the velocity of the convective motion and a model will be used in which only the essential features are retained. The damping of the motion depends on currents flowing along lines of force into the lower ionosphere. The model will be straightened out, however, so that the magnetic field is uniform, and the density of neutral molecules increases with the distance from the geomagnetic equator. Cartesian axes will be used with the x axis pointing north, the y axis west and the z axis upwards; the geomagnetic equatorial plane will be x = 0 and the model will be symmetrical with respect to this plane. The model will be divided into two regions: i n t h e " d y n a m o region," defined b y x < a, collisions will be completely neglected and in the "load region," defined b y x > a, 307
J. W. Du~rG~r
the forces of gravity and the pressure gradient will be neglected. This separation is artificial, but greatly simplifies the work and should not introduce a serious error into the calculated velocity. The value of a is determined roughly by the geomagnetic latitude at which a line of force has fallen say 200 km below its height at the geomagnetic equator; this is about 10 ° and hence a ~--1000 kin. The problem is now t h a t of a dynamo driven by a known force and feeding into a known load. The force determines the current, the current and load determine the voltage and the voltage deternfines the speed of the dynamo. In the dynamo region, since collisions are neglected, (6') is approximated by
j
,,.,,
u/c = v p
-/~g
(7)
where ff is the mass density of the ionization; it has also been assumed t h a t the acceleration of the motion is negligible. I t follows t h a t
and, using d i v j ~ 0 and the s y m m e t r y with respect to the plane x
- -
0.
qc tTM 9 u ---
~"
d:c
(S)
[['his determines the current flowing from the dynamo region into the load region. Next, the electric field is eMeulated as a multiple ofj~ from the properties of the load region. 6. TuE LoA[) I-Ir,:G,OX The condition t h a t the field is not appreciably distorted requires curl E ~- 0 and we put E -- --V~. In the load region, since gravity and the pressure gradient are neglected, (6) and the usual formulae tbr the conductivities are correct. We have .L, = - a o O ¢ / ~ z
(:!)
aud hence
cx
, ~
by
O" I
u/
~-
.
.
.
.
~cz , al bz ,
^
~ by
--
~.-
~
.
c y c z ) ~'~
(10)
This equation will now be simplified by ruthless approximations, which are partially justified by the final result. This shows t h a t the speed of the motion is inversely proportional to its scale in the east-west direction. There must of course be some lower limit to this scale; for instance it cannot be smaller t h a n the radius of the orbits of the positive ions, but this is only ~ 1 0 metres. Consequently, the east-west scale of the convection is likely to be small compared with the vertical scale, and it m a y not be unreasonable to assume t h a t the import a n t term on the right hand side of (10) is al~2c}/by 2. The problem is then linear 308
Convective diffusionin the equatorial F region and the Fourier component corresponding to ¢ ~ cosly will now be discussed. Then (10) reduces to C'd2
= --~1/2¢
(l |)
which nlust be solved in conjunction with (9). The variation of ~o and ~1 with x depends on the variation of the densities ~\'~ of electrons and N,~ of neutral molecules with x, according to the well-known formulae a o = Y,,e21mY<, (12) and
where ~ denotes the gyrofrequency. Now when :\~,, is small enough, so t h a t r,, is substantially less t h a n fl~,, (13) can be approximated by (14)
a 1 ~ N,,e2vJm~f~, 2
h i this approximation the product a0a 1 is independent of N,~ and, if it is further assmned t h a t N~ is independent of x, a0a 1 is independent of x, and the solution of (9) and (ll) is A exp
I
1
1 1
~1 d x
L J \o'0/
-~ Bex])
J
[ /i o) ] --l
' a l ~dx
(15)
wh(~t'e A and B are arbitrary constants. I f this solution were valid for the whole of the load region, the b o u n d a r y condition t h a t ~ -~ 0 as x -+ 09 would require A - - 0 , and this condition will still be approximately correct, if the exponents in (15) reach large enough values before the approximation fails. Now v~ ~ 10~,p and for atomic oxygen ions m~, ~-~ 3. lO%n(,, whence the approximation requires al/aO to t)e substantially less t h a n 3. 10 4. The exponents in (15) can then be large only if 1 1 is considerably less t h a n 1 kni. Consequently the use of (15) with A := () is probably rather inaccurate, but nevertheless we proceed to estimate the velocity of the convective motion on this basis; since a~ is overestiniated here, the velocity is underestimated. 7. ESTISIATE OF THE VELOCITY
Substituting (15) with A = 0 into (9) gives j~c = l(O'0G'l)~¢
(16)
which together with (8) determines ¢ at the boundary between the dynamo and load regions. In the d y n a m o region the variation of ¢ with x is neglected, because ~0 is large, and hence ¢ is known everywhere. The velocity of the ionization m a y now be determined from (5) in which only the last term is not known. The term J/ao is negligible and the Hall field contributes a velocity of order g / ~ v , which is only a few cm/sec and will also be neglected. The speed of the motion is therefore 309
J. W. DU=NGEY
c E / H a n d t h e m a g n i t u d e of its v e r t i c a l c o m p o n e n t is cl¢/H. Using (16), (8), (12), a n d (14) it is t h e n
w h e r e ~N~ r e p r e s e n t s t h e v a r i a t i o n of N~ in t h e d i s t u r b a n c e in t h e d y n a m o region; it will be a s s u m e d t h a t ~N~/N~ is n o t v e r y small. F o r a t o m i c o x y g e n , g/£2, ~ 4 c m / sec a n d m~u~/m~v~ ---- 3 . 10 -a. I f 1-1 ~ 100 m e t r e s a n d a --~ 1000 k m , the speed is t h e n 7 5N~/N~ m / s or 25 &V~/N~ k m / h r a n d m i g h t be a p p r e c i a b l e . This c a l c u l a t i o n requires quite a lot of r e f i n e m e n t , b u t we m a y conclude t.hat t h e p h e n o m e n o n p r o p o s e d b y HULBURT a n d JOH~'SO~= is likely to be i m p o r t a n t a n d t h a t a m o r e refined t h e o r e t i c a l s t u d y w o u l d be w o r t h while. O b s e r v a t i o n s will n o t be discussed here, b u t t h e r e should be no difficulty in o b s e r v i n g t h e irregularities; t h e effect on t h e a v e r a g e p r o p e r t i e s of t h e l a y e r is c o m p l i c a t e d b y other movements.
Acknowledgements--I a m i n d e b t e d to Mr. J. A. I~ATCLIFFE for suggesting this p r o b l e m to m e a n d for his v e r y helpful c o m m e n t s d u r i n g t h e course of the work. T h i s w o r k was carried out d u r i n g t h e t e n u r e of an I.C.I. Fellowship.
(I!O~,VLIN(~ ' T. G. .IO~NSO-',~M. H.
REFERENCES 1945 Proc. Roy. ~'~'oc.A183, 453. 1950 Proc. Conf. I<~no.~'pb. l'l~ys., I)."l~. >~'t<~t~,
P~per J. ,JOHNSON ~'~. H. and HULBURT E. (). ]~ATCLIFFE J. A., SCHMERLINGE. R., SETTY C. •. G. K., and THO)IAS J. O.
] 950 1956
310
l'l~/s. Rev. 79, S02.
t'hil Tr~lns. A248, 621.