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Sq currents in a three-dimensional ionosphere
PImel. space sci. 1969, Vol. 17. pp. 471 lo 482. F’aemmabs.
RhtedinNortkmhland
Sq CURRENTS IN A THREE-DIMENSIONAL
IONOSPHERE
A. C. COCKS aad A. T. PRICR Department of Mathematics, University of Exeter, Exeter, England AhstrW-Calculations are made of Sq currents on the assumption that they are caused by electromotive forces produced by tidal winds at the base of an ionosphere that extends upwards to considerable heights, and has some of the features of the conductivity distribution of the actual ionosphere. It is shown that the current flow is nearly horizontal throughout the region 04 importance and that the ‘layer equations’ are therefore approximately true. The strictly horizontal part of the current flow is not, however, non-divergent and it is not possible to defhre a current function for it. The calculated currents are divided into two parts, one of which has a field below the ionosphere, while the other does not. The signifkance of this fact is considered in relation to the interpretation of analyses of observations of q variations.
1.INT.RODUCTION
The analysis of the magnetic daily variations observed at a world-wide distribution of stations indicates that the variations proceed mainly according to local time, and they can therefore be regarded as arising from a field which is stationary relative to the Sun, though this field may vary somewhat in intensity and distribution with the angular position of the Earth as it rotates relative to the Sun-Earth line (Price and Wilkins, 1963). The field is generally assumed to be produced by current systems flowing in a thin ionospheric shell, like those shown in Figs. 1 and 2 for the equinoctial and Northern Summer months, respectively. These current systems remain in a tied position relative to the Sun. If they remained entirely steady they would give rise to a strictly local time field, but there are significant deviations from this. These deviations are accounted for by changes in the distribution and intensity of the currents due to the fact that the distribution of ionospheric conductivity is subject to considerable control by the main geomagnetic field, and also by changes in the induced Barth currents, which contribute to the field, and are affected by the geographic distribution of land and sea. The basic idea in the dynamo theory of Sq variations is that the ionospheric currents arise from electromotive forces produced by atmospheric tides carrying conducting air across the lines of force of the main geomagnetic field. The detailed mathematical development of the theory, and the theoretical discussion of various associated phenomena, have been largely based on the assumption that the currents flow horizontally in a thin conducting layer in the E-region of the ionosphere,-the so-called ‘dynamo layer,’ and horizontal currents of the right order of magnitude to account for the variations have indeed been detected at about 100 km altitude by suitably instrumented rockets. (Burrows and Hall, 1964; Davis et al., 1965.) However, the treatment of the dynamo layer in theoretical discussions as a thin conducting sheet or shell is hardly satisfactory, because it implies that the medium above the layer, as well as that below it, is non-conducting. In the actual physical situation the medium above is likely to be a very good conductor, particularly along the lines of force of the main geomagnetic field. Hence it would be more satisfactory to treat the Sq currents as flowing (in the main) near the base of a non-isotropic conducting ionosphere that extends upwards to considerable heights. The upper part of this ionosphere merges into the magnetosphere 471
A. C. COCKS and A. T. PRICE
412
night FIG@.
&PREsENTATIvB
CURRENT
SYSTEM FOR t&J FOR EQtJINoCTIALMONTHS.
and may also have considerable mobility, but we limit ourselves in the present discussion to considering the effects of high conductivity. The fact that the air below the ionosphere is effectively non-conducting ensures that the currents at the lower boundary flow horizontally. Also for points sufficiently near to this lower boundary, any vertical flow of current will be inhibited by the electric field of the surface charge that will accumulate on it. Hence there will be a layer of certain thickness in which the current flow is nearly horizontal, and consequently the well-known ‘layer equations’ (Chapman, 1956), can be used to express approximately the relations between the horizontal components of current and electric field. There does not seem, however, to be any simple way of estimating the thickness of this layer. It cannot of course extend upwards indefinitely, and it has indeed been suspected that there is a significant flow of charge from the top of the dynamo layer along the lines of force in the magnetosphere. Some writers have attempted to investigate this idea theoretically by first treating the layer as a thin sheet or shell, using the layer equations, together with the condition that the horizontal currents are non-divergent, to calculate the corresponding space charge distribution. They have then proceeded to discuss the leakage of this space charge through the conducting medium above. But this procedure appears to the present writers to be highly unsatisfactory. The method of calculation of the space charge implicitly assumes that there is an insulator above the layer, and contradicts the later assumption of an upward (and appreciable) leakage of space charge. Actually, the space charge which accumulates because the three-dimensional current density is non-divergent is profoundly affected by the presence or absence of an insulating layer above. This method of calculation also implies that there is an accumulation of charge on the upper surface of the layer (as well as on the lower surface), but if there is no actual insulating layer between the
Fm. 2. REPR~SENTATWE
CURRENT
SYSTBM FOR
Sq FOR NORTHERN
SUMMER.
&j CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
473
dynamo layer and the region above, there will be no stiace on which charges can accumulate. The only way to get a reliable estimate of the thickness of the layer in which the Sq currents are practically horizontal, and to obtain some idea of the possible effects of the
leakage of charge at the higher levels is to study some illustrative three-dimensional problems in which currents are produced in an extended ionosphere by el~~omotive forces produced by atmospheric tides near the base. A very simple problem of this kind has already been considered by one of us (Price, 1968), but the simplifications there introduced were too drastic to take into account some of the significant features of the problem, We here discuss a slightly more general problem. 2. THE .4PPLICABILlTY OF THE LAYER EQUATIONS
The fundamental relation between the current density j and the electric field intensity E in an ionised gas such as the ionosphere is given by j = k&l + k& + kah A E, where Err and Er are the components of E parallel and perpendicular to the magnetic field, and h is a unit vector along that field; k,, ki and kg are the direct, Pederson and Hall conductivities, respectively. The electric field E will consist of the applied electromotive force a due to atmospheric tidal motions, together with the electrostatic field of the space charge dis~bution, which is set up if the current flow is non-divergent. Hence, if 5 is the potential of the electrostatic field, we have E = II - grad S, (2) and using Cartesian axes with OXSouthwards, 0~ Eastwards and Ozvertically upwards, we have (3) If we now assume that there is a layer of thickness C&say, in which the ratio of the vertical component of current to the horizontal component is everywhere less than some agreed small value 6, say, then, in this layer, the relation between the horizontal components of current and electric field can be obtained by writing down the cartesian components of Equation (1), taking j, zero and eliminating E, between the three resulting equations. This leads to the well-know ‘layer equations’ (4) - = --I&Es JU
+ K,E;
(5)
which will have a degree of accuracy depending on the value of 8. The coefficients J&,, &, & are of course functions of k,, kl and ka which in the actual ionosphere are themselves functions of many parameters including geographic position, altitude, local time, season and time reached within a solar cycle. The results of our calculations below suggest that the layer Equations (4) and (5) will be applicable with reasonable accuracy over most of the region in which appreciable Sq currents flow. It is important to note, however, that though this implies that, for this particular purpose, j# can be ignored, it does not imply that ~j*~~zcan be ignored in
474
A. C. COCKS
and A. T. PRICE
Equation (3). In fact this term is found to be comparable in magnitude with each of the other terms in Equation (3). It follows that the horizontal components taken alone, do not satisfy the non-divergence condition and therefore cannot be derived from a stream function as in earlier theories based on treating the ionosphere as a thin sheet. 3.THEMATHEMA
TICAL PROBLEM
For direct application to the ionosphere, the mathematical problem ought ideally to be expressed in spherical polar coordinates. A problem in this form has been described by one of us (Price, 1968) and we are both at present engaged in finding a method of solving it. In the meantime, two simpler problems, expressed in terms of Cartesian coordinates, have been solved. The solution of one of them is given in the paper just quoted. The solution of the other more general one will now be described. Using cartesian axes with Ox Southwards, Oy Eastwards and Oz vertically upwards, we consider the ionosphere as occupying the half space z > 0. To simulate some of the conditions of the global problem we take all the field quantities to be periodic in x and y with wave lengths 2ra, where a is the Earth’s radius. We take the square bounded by x = -na/2, x = wa/2, y = -ra/2, y = ral2 to correspond to the daylight hemisphere. The electromotive forces arising from the atmospheric motions are assumed to be of the same form as those derived by Price (1968), namely % = CV,e+sin
2ax cos ay
(6)
eV= CV,e-~Z(l - cos 2ax) sin ay
(7)
e, = &CVOe-flZ(l+ cos 2ax) cos ay (8) where a = l/a. The horizontal components of these electromotive forces are shown in Fig. 3. In a thin conducting shell this distribution of e.m.f.s would give rise to current vortices in the Northern and Southern Hemispheres somewhat similar to those shown in Figs. 1 and 2. There would also be present of course the corresponding space charge distribution required to make the horizontal current flow non-divergent. We now consider the currents that would be produced by these e.m.f.s in a threedimensional ionosphere. 4. THE CONDUCTMTY
AND RESISTANCR TENSORS
The conductivity tensor for the Cartesian problem, that would most closely simulate the effect of the Earth’s main dipole field on the ionospheric conductivity for low and mid latitudes, is of the form (Price, 1968)
(k,-kJcos*ax+k A2
c=
-2k, 1
sin ax A
2k2 sin ax A
--Go - kJ sin 2ax A2
-(k,
- kl) sin 2ax A2 k,cosax A
-k2 cos ax A
where A* = 1 + 3 sin2 OX.
4(k,, - kJ sin2 ax + kj A2
li$ CURRENTS
IN A THREE-DIMENSIONAL
IONOSPHERE
r-------------/_A-----.
\
\
\ \
\ \
.
415
l
.
.
.
.
t
.
.
1
I
I I
/ +
--5uk -
-263
-
d.&.--‘__j-
I/ -u/3
-r/6
Flt3. 3. HORIZONTALCXWPONENTS OF ELEC~~~OMU~XVE PORCBS PRODUCED BY THE A.WJMED WlND SYSTEMAT BASE OF IONOSPHERE.
This gives a variation with ax of the conductivity tensor elements which features to some extent the actual variation with latitude, though there is of course some distortion due to the representation of a hemispherical surface by a plane square; the variation of k,, kl and k2 with the Sun’s zenith angle. is also ignored. Nevertheless this tensor probably represents, in a crude fashion, some of the actual features of the conductivity distribution. It will be noted that the tensor (9) has a pair of symmetric elements and two pairs of antisymmetric elements. It is thus not completely symmetric or antisymmetric. The conductivities k,, kl and k2 vary considerably with height in the ionosphere (Kim and Kim, 1962). They are generally negligibly small at about 80 km. The direct conductivity k, rises rapidly from about 3 x MF” e.m.u. at 80 km to 2 x UP at 90 km and 6 x 10-m at 100 km; kl is much smaller and its rise is less remarkable; it is estimated as 4 x 10-l’ e.m.u. at 80 km, 6 x 10-r’ at 90 km and 2 x lo-l6 at 100 km; k, rises from 10-16 e.m.u. at 80 km to 10-16 at 90 km and 5 x 10-r5 at 100 km. Since there is evidence that a considerable part of the Sq current is flowing at about 90-100 km height, it seems that a simple model of the ionosphere which it would be useful to consider is a semi-itinite conductor with its base (z = 0) at height 90 km above the Earth, with conductivities of the above order. Actually the form of the calculated current systems (as distinct from their intensity) is found to depend only on the ratios of the conductivities. At 90 km height k,,/kl is about 350 and k$k, about 20; at 100 km height
476
A. C. COCKS and A. T. PRICE
k,/kI is about 3000 and kzfkI about 23. For ratios of these orders, it has been found from a number of exploratory calculations (A. C. Cocks, Ph.D. Thesis, 1968) that kI can be taken as zero without seriously affecting the general character of the results. Hence in the tensor (9), we take kI as zero. It is found more convenient in the later calculations to use the dynamo equation in the form e--gradS=Rj (11) where R is the resistance tensor. This is the inverse of the conductivity tensor (9) and is given (when kl is zero) by co2 ax
2 sin ccx W
k, -2 sin Qx
R=
-sin 2~ k0 -cos
0
ax
112)
W
&A -sin 2ouc
cos ax
4 sin* ax
It is now found from calculations using increasing values of the ratio k0/k2 that, once it has reached a value of about 10, any further increase in its value does not significantly alter the results. Since ko/k2 is greater than 10 throughout almost all the dynamo layer we take the limit of R as k --+ 00, and obtain the greatly simplified resistance tensor 0
R* =j;-:
-2sin 2
i
0
2 sin ax ax
0
0
-cos ax
cos ax
0,
(13) 1
which it will be noted is antisymmetric. It will be seen from (10) that the common factor l/A in the dements of R* varies from 1 to 4, and corresponds to the variation of the total magni~de of a dipole field from pole to equator. Since the other factor in each element of the tensor varies between 0 and 1, the factor l/A does not greatly alter the general character of the tensor. Its presence, however, greatly increases the complexity of a formal solution of the problem, and it is therefore omitted in the calculations. It should perhaps be emphasized that although only the Hall condu~tivi~ k2 appears explicitly in the resistance tensor (13), the direct conductivity k0 has not been ignored. It has in fact been taken as infinite. On the other hand the Pedersen conductivity kI has been taken as zero. 5. THE FISCAL SOLUTION The Cartesian components of the dynamo Equation (11) can now be written as (14) k,
= -2 sin axj* - cos a.xj*
(19
k2
= cos ax j,.
(16)
Sq CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
47-l
These equations, together with Equation (3) and the boundary conditions at z = 0 and z -+ co are sul&ient to determine S and j. Eliminatingj, between (14) and (16), we obtain (17) It is worth noting that this equation, with the right hand side zero, is obtained for any E corresponding to the dynamo term v A H, if v is derivable from a potential. Taking into account the expressions (6), (7) and (8) for the components of a, the relevant expression for S which wiIl satisfy (3), (15) and (17) is of the form S = cos tcy$ fin cos 2naX r=O where fsn(n= 0, 1,2 . . . ) are functions of z, which satisfy the recurrence relations 2Df, - (D - l)f = 0 (I) + n>fsn- tD - n - l)f,,,
= 0, n > 1
where D denotes the operator (l/a)/(d/dz). From (19) and (20) we have by continued substitution the differential equation 2D(D + 1). . . (D + n - l)fo = (D - l)(D - 2). . . (D - n)J& Now S is a function which is finite and continuous for all x and y and finite and continuous for all positive values of z. Hence the series (18) for S must be convergent for all positive z, andf,, must therefore tend to zero as n tends to infinity for all positive z. We may therefore take f.as the limit as m -+ co of the differential equation D(D + 1) . . . (D + m)f = 0.
(22)
Moreover 5’3 0 as z - co; hence every coefficientf& in (18) must tend to zero as z + co. Hence the required expression for f.is of the form (23) where the coefficients A,,, axe determined by the non-divergence condition on j, together with the boundary condition thatj, = 0 at z = 0. The lotions (19) and (20) can now be used to evaluate all the functions fsn of z, in terms of the coefficients A,. Eliminating S between Equations (15) and (16) and substituting for aj,l& from the zero divergence condition (3), we obtain an equation forjsp in the form (24) where gyand E, are given in (7) and (8). A particular integral of this equation is j, = C&e-@ sin ay 5 C,,, sin (2n + 1)~
(25)
n-0
where the coefficients C,,,, are determined by the known function on the right of (24). The compIementary solution to be added to (25) is a series similar to that obtained for
478
A. C. COCKS and A.. T. PRICE
5’ from (17). However, on substitution of this series into (26) and comparing coefficients of sin nax sin ay we see that all the coefficients of this series are zero. Substituting from (25) into (15) and applying the condition j, = 0 at z = 0 gives
as 3 a=0
(>
= (ay + 2 sin axjZ)Z,o
Substituting from (18), (7) and (25) in (26) then gives a set of equations to determine the A,,,%. The current component j, can now be obtained from (15) and j, from (14). (Further details of the calculations are given by Cocks, 1968.) 6. THE NUMERICAL
CALCULATIONS
The numerical calculations involved in the solution are very elaborate, and some compromise has had to be made between the best values of the parameters to represent the physical problem and the values which make it possible to obtain a numerical solution with the computing facilities available. In his discussion of the simpler problem, in which the conductivity tensor was taken as uniform, Price took @to be O-05* which corresponded to the atmospheric velocities being reduced to one-tenth of their value at a height of 46 km from the base of the ionosphere, and the corresponding value of the ratio #a was about 320. But in the present problem the numerical work becomes prohibitive for such high values of this ratio, and considerably smaller values have been used. Also it is found that the calculations are simplified if #a is taken to be half an odd integer because the series for S and for the components of j become finite, the number of terms being slightly greater than the value of this integer. It is then necessary to solve certain matrix equations of about the same order. The results given in this paper are for p = 205a, which corresponds to the el~tromotive force dropping to one-tenth of its value at z = 0 in about 800 km. This height is undoubtediy much too great, but some further calculations have been partially completed for the case when B = 40*5a, corresponding to a height of about 400 km, and the general nature of the results does not appear to be appr~iably altered. 7. DISCUSSION OF THE RESULTS
The results of the present calculations confirm two of the important conclusions reached by Price (1968), namely, (1) throughout the region where the current density is large, the flow is almost entirely horizontal, and therefore the layer equations can be used for obtaining a good approximation to the relation between the horizontal components of current and electric field. (2) the vahte of a@z is of the same order of magnitude as that of Z&/ax or &/lay at all heights. Hence we cannot deduce from (3) that the horizontal flow of current at any height has zero divergence. Consequently this flow cannot be derived from a stream function, as has been assumed in most theoretical discussions of Sq. There are several additional features of the present results that are of interest, and it is useful to compare them in greater detail with those of the previous calculation. The horizontal flow at the base of the ionosphere, when it is assumed uniform (but non-isotropic) as in the earlier calculation, is shown in Fig. 4. It is noticeable that the pattern of the horizontal current flow is quite similar to the horizontal part of the applied * Note this is incornxtly print& as 05 in Price’s paper.
Sq CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
r-------------
419
\ \ \ t .
I
1 / / FIG. 4. CURRENT SYSlZMAT
BASR OF IONC6PIiBRE
PRODUCED
IN
A
UNIFORM
NON4SOTROPIC
IONOSPHERE.
electromotive forces, and differs considerably from the non-divergent flow that wouldihave been produced by the same electromotive forces in a thin shell. This implies that the horizontal component of the electric field of the space charge distribution, which accumulates in the three-dimensional ionosphere because the (total) current density is nondivergent, is much smaller than that arising in the case of a thin shell. In the latter case the field is large enough to overcome the primary electromotive forces along the dipole equator and turns back the horizontal currents to form closed loops. This does not occur in the threedimensional case because much of the current is able to escape vertically. Since analyses of the actual Sq field have indicated a very considerable return current is very important to flow along the magnetic dip equator- the equatorial electrojet-it see whether the new calculations for a non-uniform non-isotropic ionosphere lead to such a current. It might be expected that this will be so because it has been found that the layer equations are applicable near the base of the ionosphere, and the layer conductivities rise to sharp maxima near the equator, but it is not immediately obvious because the essential part of the total electric field is dependent on the space charge distribution set up. However, the calculations do in fact give a very large return current along the equator, as shown in Fig. 5. This horizontal current system at the base of the ionosphere does not
A. C. COCKS and A. T. PRICE
480
satisfy the condition of zero divergence, but there is present a very considerable concentration of current along the equator. 8. iUU’RJ3tENTA~
AND ACTUAL .+&CWRRENTSYSW
As already pointed out by Price (1968), the representative current system, derived from the analysis of observations of the Sq variations, corresponds to one part only of the actual ionospheric currents. The other part, which includes vertical as well as horizontal currents, cannot be found from surface observations of the magnetic geld. It may, however, be possible to deduce it from calculations similar to the present one, if a sufficiently accurate knowledge of the various parameters is available, and improved numerical methods for solving the elaborate numerical problems involved can be devised. For the present model ionosphere it is possible to separate the current system into two parts, one which has a field beneath the ionosphere and one which does not, It may be shown that any non-divergent current system in the three-dimensional ionosphere can be expressed in the form j = curl klp + curl curl kQ (27) where P and Q are functions of (x, y, z) and k is a unit vector in the z-direction.
i l-l -7&Frtr.
5.
-
CURRENT
ii._
-r/3
SYSTEM
AT
RAW
XSOTROPXZ IONOSPEIERE
BY
OR IOMMPHERB THB
SAW3
PRODUCED
ELECTROMOTWE
M
A
NON-UNIFORM
FORCES
AS IN
FIG
AND
4.
NOW
j--,-
L’
1--L-L---L_
-17-/2
-
t-
-
1 +-T/3
T
-
’
-)
-
-
-E%
-i_
/
n/6
..L QY
-
7
-
-Ilw2
-L_t-L lCf3 '?
2.p
d2
d/2 4;o
Scale
FIG. 6. HORIZONTAL
COMPONENT OF THE CURRENT SYSTEM OF FIG. 5 THAT HAS A MAGNETIC FIELD BELOW THE IONOSPHERE.
--
I
-C
\
f
,
rL_
\
f
I
/-
-
I
t 1
\
I I-
n/3
\
la
I
-
-
-
iota,
l
I
I /
-
/
L/6
t
I I I t
/
-
f
t
f
t
! I
\ .C
f J_-
u/6
-%--
T-
-I
%3-0
-w/3.
I
-I
-lv2
d/2 2,o 40
Scale
FIG: 7. HORIZONTAL
~~~0~~0~~ CURRENT SYSTEM OF FIG. 5 TIZATHASNO FIELD BELOW THE IONOSPHERE. 481
MAGNETIC
A. C. COCKS and A. T. PRICE
482
We then have
asQ axaz
(28)
ju=-aP+asQ
ax ayaz
(29)
jz=-($+C$).
(30)
aP
jZ=T+-
Eliminating Q from (15) and (16), we have
av
asp
aj,
Z2+3jZYj
--.
ai,
ax
(31)
Using the values of jpDand jV obtained in the present calculations, we can solve (30) for P and thus calculate the part of j that has a field in the region z < 0. This current system is shown for z = 0 in Fig. 6, and the horizontal part of the system not having a field in z < 0 is shown in Fig. 7. The horizontal currents in Fig. 6 are non-divergent and may be obtained from a current function. There is a strong return current along the equator. The currents shown in Fig. 7 are not derivable from a current function and there are significant vertical currents associated with them. The results suggest that this part of the ionospheric current system may be comparable with the part usually associated with Sq. REFERElNcEs Bmows, K. and HALL.,S. H. (1964). Nume, Lmd. 204, 721. CHAPMAN,S. (1956). Nuovo Cim., Suppl. 4,1385. Cows, A. C. (1968). Ph.D. Thesis, University of Exeter. DAM, T. N., STOURIK, J. D. and HE~PNER,J. P. (1965), J. geophys. Res. 70, 5883. KIM, J. S. and Km, H. Y. (1962). Nuture, Lmd. 196, 630. PRICB,A. T. (1968). Geophys. J. R. astr. Sot. 15, 93. PRICE,A. T. and WILKINS,G. A. (1963). Phil. Trans. A256,31.
RhtedinNortkmhland
Sq CURRENTS IN A THREE-DIMENSIONAL
IONOSPHERE
A. C. COCKS aad A. T. PRICR Department of Mathematics, University of Exeter, Exeter, England AhstrW-Calculations are made of Sq currents on the assumption that they are caused by electromotive forces produced by tidal winds at the base of an ionosphere that extends upwards to considerable heights, and has some of the features of the conductivity distribution of the actual ionosphere. It is shown that the current flow is nearly horizontal throughout the region 04 importance and that the ‘layer equations’ are therefore approximately true. The strictly horizontal part of the current flow is not, however, non-divergent and it is not possible to defhre a current function for it. The calculated currents are divided into two parts, one of which has a field below the ionosphere, while the other does not. The signifkance of this fact is considered in relation to the interpretation of analyses of observations of q variations.
1.INT.RODUCTION
The analysis of the magnetic daily variations observed at a world-wide distribution of stations indicates that the variations proceed mainly according to local time, and they can therefore be regarded as arising from a field which is stationary relative to the Sun, though this field may vary somewhat in intensity and distribution with the angular position of the Earth as it rotates relative to the Sun-Earth line (Price and Wilkins, 1963). The field is generally assumed to be produced by current systems flowing in a thin ionospheric shell, like those shown in Figs. 1 and 2 for the equinoctial and Northern Summer months, respectively. These current systems remain in a tied position relative to the Sun. If they remained entirely steady they would give rise to a strictly local time field, but there are significant deviations from this. These deviations are accounted for by changes in the distribution and intensity of the currents due to the fact that the distribution of ionospheric conductivity is subject to considerable control by the main geomagnetic field, and also by changes in the induced Barth currents, which contribute to the field, and are affected by the geographic distribution of land and sea. The basic idea in the dynamo theory of Sq variations is that the ionospheric currents arise from electromotive forces produced by atmospheric tides carrying conducting air across the lines of force of the main geomagnetic field. The detailed mathematical development of the theory, and the theoretical discussion of various associated phenomena, have been largely based on the assumption that the currents flow horizontally in a thin conducting layer in the E-region of the ionosphere,-the so-called ‘dynamo layer,’ and horizontal currents of the right order of magnitude to account for the variations have indeed been detected at about 100 km altitude by suitably instrumented rockets. (Burrows and Hall, 1964; Davis et al., 1965.) However, the treatment of the dynamo layer in theoretical discussions as a thin conducting sheet or shell is hardly satisfactory, because it implies that the medium above the layer, as well as that below it, is non-conducting. In the actual physical situation the medium above is likely to be a very good conductor, particularly along the lines of force of the main geomagnetic field. Hence it would be more satisfactory to treat the Sq currents as flowing (in the main) near the base of a non-isotropic conducting ionosphere that extends upwards to considerable heights. The upper part of this ionosphere merges into the magnetosphere 471
A. C. COCKS and A. T. PRICE
412
night FIG@.
&PREsENTATIvB
CURRENT
SYSTEM FOR t&J FOR EQtJINoCTIALMONTHS.
and may also have considerable mobility, but we limit ourselves in the present discussion to considering the effects of high conductivity. The fact that the air below the ionosphere is effectively non-conducting ensures that the currents at the lower boundary flow horizontally. Also for points sufficiently near to this lower boundary, any vertical flow of current will be inhibited by the electric field of the surface charge that will accumulate on it. Hence there will be a layer of certain thickness in which the current flow is nearly horizontal, and consequently the well-known ‘layer equations’ (Chapman, 1956), can be used to express approximately the relations between the horizontal components of current and electric field. There does not seem, however, to be any simple way of estimating the thickness of this layer. It cannot of course extend upwards indefinitely, and it has indeed been suspected that there is a significant flow of charge from the top of the dynamo layer along the lines of force in the magnetosphere. Some writers have attempted to investigate this idea theoretically by first treating the layer as a thin sheet or shell, using the layer equations, together with the condition that the horizontal currents are non-divergent, to calculate the corresponding space charge distribution. They have then proceeded to discuss the leakage of this space charge through the conducting medium above. But this procedure appears to the present writers to be highly unsatisfactory. The method of calculation of the space charge implicitly assumes that there is an insulator above the layer, and contradicts the later assumption of an upward (and appreciable) leakage of space charge. Actually, the space charge which accumulates because the three-dimensional current density is non-divergent is profoundly affected by the presence or absence of an insulating layer above. This method of calculation also implies that there is an accumulation of charge on the upper surface of the layer (as well as on the lower surface), but if there is no actual insulating layer between the
Fm. 2. REPR~SENTATWE
CURRENT
SYSTBM FOR
Sq FOR NORTHERN
SUMMER.
&j CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
473
dynamo layer and the region above, there will be no stiace on which charges can accumulate. The only way to get a reliable estimate of the thickness of the layer in which the Sq currents are practically horizontal, and to obtain some idea of the possible effects of the
leakage of charge at the higher levels is to study some illustrative three-dimensional problems in which currents are produced in an extended ionosphere by el~~omotive forces produced by atmospheric tides near the base. A very simple problem of this kind has already been considered by one of us (Price, 1968), but the simplifications there introduced were too drastic to take into account some of the significant features of the problem, We here discuss a slightly more general problem. 2. THE .4PPLICABILlTY OF THE LAYER EQUATIONS
The fundamental relation between the current density j and the electric field intensity E in an ionised gas such as the ionosphere is given by j = k&l + k& + kah A E, where Err and Er are the components of E parallel and perpendicular to the magnetic field, and h is a unit vector along that field; k,, ki and kg are the direct, Pederson and Hall conductivities, respectively. The electric field E will consist of the applied electromotive force a due to atmospheric tidal motions, together with the electrostatic field of the space charge dis~bution, which is set up if the current flow is non-divergent. Hence, if 5 is the potential of the electrostatic field, we have E = II - grad S, (2) and using Cartesian axes with OXSouthwards, 0~ Eastwards and Ozvertically upwards, we have (3) If we now assume that there is a layer of thickness C&say, in which the ratio of the vertical component of current to the horizontal component is everywhere less than some agreed small value 6, say, then, in this layer, the relation between the horizontal components of current and electric field can be obtained by writing down the cartesian components of Equation (1), taking j, zero and eliminating E, between the three resulting equations. This leads to the well-know ‘layer equations’ (4) - = --I&Es JU
+ K,E;
(5)
which will have a degree of accuracy depending on the value of 8. The coefficients J&,, &, & are of course functions of k,, kl and ka which in the actual ionosphere are themselves functions of many parameters including geographic position, altitude, local time, season and time reached within a solar cycle. The results of our calculations below suggest that the layer Equations (4) and (5) will be applicable with reasonable accuracy over most of the region in which appreciable Sq currents flow. It is important to note, however, that though this implies that, for this particular purpose, j# can be ignored, it does not imply that ~j*~~zcan be ignored in
474
A. C. COCKS
and A. T. PRICE
Equation (3). In fact this term is found to be comparable in magnitude with each of the other terms in Equation (3). It follows that the horizontal components taken alone, do not satisfy the non-divergence condition and therefore cannot be derived from a stream function as in earlier theories based on treating the ionosphere as a thin sheet. 3.THEMATHEMA
TICAL PROBLEM
For direct application to the ionosphere, the mathematical problem ought ideally to be expressed in spherical polar coordinates. A problem in this form has been described by one of us (Price, 1968) and we are both at present engaged in finding a method of solving it. In the meantime, two simpler problems, expressed in terms of Cartesian coordinates, have been solved. The solution of one of them is given in the paper just quoted. The solution of the other more general one will now be described. Using cartesian axes with Ox Southwards, Oy Eastwards and Oz vertically upwards, we consider the ionosphere as occupying the half space z > 0. To simulate some of the conditions of the global problem we take all the field quantities to be periodic in x and y with wave lengths 2ra, where a is the Earth’s radius. We take the square bounded by x = -na/2, x = wa/2, y = -ra/2, y = ral2 to correspond to the daylight hemisphere. The electromotive forces arising from the atmospheric motions are assumed to be of the same form as those derived by Price (1968), namely % = CV,e+sin
2ax cos ay
(6)
eV= CV,e-~Z(l - cos 2ax) sin ay
(7)
e, = &CVOe-flZ(l+ cos 2ax) cos ay (8) where a = l/a. The horizontal components of these electromotive forces are shown in Fig. 3. In a thin conducting shell this distribution of e.m.f.s would give rise to current vortices in the Northern and Southern Hemispheres somewhat similar to those shown in Figs. 1 and 2. There would also be present of course the corresponding space charge distribution required to make the horizontal current flow non-divergent. We now consider the currents that would be produced by these e.m.f.s in a threedimensional ionosphere. 4. THE CONDUCTMTY
AND RESISTANCR TENSORS
The conductivity tensor for the Cartesian problem, that would most closely simulate the effect of the Earth’s main dipole field on the ionospheric conductivity for low and mid latitudes, is of the form (Price, 1968)
(k,-kJcos*ax+k A2
c=
-2k, 1
sin ax A
2k2 sin ax A
--Go - kJ sin 2ax A2
-(k,
- kl) sin 2ax A2 k,cosax A
-k2 cos ax A
where A* = 1 + 3 sin2 OX.
4(k,, - kJ sin2 ax + kj A2
li$ CURRENTS
IN A THREE-DIMENSIONAL
IONOSPHERE
r-------------/_A-----.
\
\
\ \
\ \
.
415
l
.
.
.
.
t
.
.
1
I
I I
/ +
--5uk -
-263
-
d.&.--‘__j-
I/ -u/3
-r/6
Flt3. 3. HORIZONTALCXWPONENTS OF ELEC~~~OMU~XVE PORCBS PRODUCED BY THE A.WJMED WlND SYSTEMAT BASE OF IONOSPHERE.
This gives a variation with ax of the conductivity tensor elements which features to some extent the actual variation with latitude, though there is of course some distortion due to the representation of a hemispherical surface by a plane square; the variation of k,, kl and k2 with the Sun’s zenith angle. is also ignored. Nevertheless this tensor probably represents, in a crude fashion, some of the actual features of the conductivity distribution. It will be noted that the tensor (9) has a pair of symmetric elements and two pairs of antisymmetric elements. It is thus not completely symmetric or antisymmetric. The conductivities k,, kl and k2 vary considerably with height in the ionosphere (Kim and Kim, 1962). They are generally negligibly small at about 80 km. The direct conductivity k, rises rapidly from about 3 x MF” e.m.u. at 80 km to 2 x UP at 90 km and 6 x 10-m at 100 km; kl is much smaller and its rise is less remarkable; it is estimated as 4 x 10-l’ e.m.u. at 80 km, 6 x 10-r’ at 90 km and 2 x lo-l6 at 100 km; k, rises from 10-16 e.m.u. at 80 km to 10-16 at 90 km and 5 x 10-r5 at 100 km. Since there is evidence that a considerable part of the Sq current is flowing at about 90-100 km height, it seems that a simple model of the ionosphere which it would be useful to consider is a semi-itinite conductor with its base (z = 0) at height 90 km above the Earth, with conductivities of the above order. Actually the form of the calculated current systems (as distinct from their intensity) is found to depend only on the ratios of the conductivities. At 90 km height k,,/kl is about 350 and k$k, about 20; at 100 km height
476
A. C. COCKS and A. T. PRICE
k,/kI is about 3000 and kzfkI about 23. For ratios of these orders, it has been found from a number of exploratory calculations (A. C. Cocks, Ph.D. Thesis, 1968) that kI can be taken as zero without seriously affecting the general character of the results. Hence in the tensor (9), we take kI as zero. It is found more convenient in the later calculations to use the dynamo equation in the form e--gradS=Rj (11) where R is the resistance tensor. This is the inverse of the conductivity tensor (9) and is given (when kl is zero) by co2 ax
2 sin ccx W
k, -2 sin Qx
R=
-sin 2~ k0 -cos
0
ax
112)
W
&A -sin 2ouc
cos ax
4 sin* ax
It is now found from calculations using increasing values of the ratio k0/k2 that, once it has reached a value of about 10, any further increase in its value does not significantly alter the results. Since ko/k2 is greater than 10 throughout almost all the dynamo layer we take the limit of R as k --+ 00, and obtain the greatly simplified resistance tensor 0
R* =j;-:
-2sin 2
i
0
2 sin ax ax
0
0
-cos ax
cos ax
0,
(13) 1
which it will be noted is antisymmetric. It will be seen from (10) that the common factor l/A in the dements of R* varies from 1 to 4, and corresponds to the variation of the total magni~de of a dipole field from pole to equator. Since the other factor in each element of the tensor varies between 0 and 1, the factor l/A does not greatly alter the general character of the tensor. Its presence, however, greatly increases the complexity of a formal solution of the problem, and it is therefore omitted in the calculations. It should perhaps be emphasized that although only the Hall condu~tivi~ k2 appears explicitly in the resistance tensor (13), the direct conductivity k0 has not been ignored. It has in fact been taken as infinite. On the other hand the Pedersen conductivity kI has been taken as zero. 5. THE FISCAL SOLUTION The Cartesian components of the dynamo Equation (11) can now be written as (14) k,
= -2 sin axj* - cos a.xj*
(19
k2
= cos ax j,.
(16)
Sq CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
47-l
These equations, together with Equation (3) and the boundary conditions at z = 0 and z -+ co are sul&ient to determine S and j. Eliminatingj, between (14) and (16), we obtain (17) It is worth noting that this equation, with the right hand side zero, is obtained for any E corresponding to the dynamo term v A H, if v is derivable from a potential. Taking into account the expressions (6), (7) and (8) for the components of a, the relevant expression for S which wiIl satisfy (3), (15) and (17) is of the form S = cos tcy$ fin cos 2naX r=O where fsn(n= 0, 1,2 . . . ) are functions of z, which satisfy the recurrence relations 2Df, - (D - l)f = 0 (I) + n>fsn- tD - n - l)f,,,
= 0, n > 1
where D denotes the operator (l/a)/(d/dz). From (19) and (20) we have by continued substitution the differential equation 2D(D + 1). . . (D + n - l)fo = (D - l)(D - 2). . . (D - n)J& Now S is a function which is finite and continuous for all x and y and finite and continuous for all positive values of z. Hence the series (18) for S must be convergent for all positive z, andf,, must therefore tend to zero as n tends to infinity for all positive z. We may therefore take f.as the limit as m -+ co of the differential equation D(D + 1) . . . (D + m)f = 0.
(22)
Moreover 5’3 0 as z - co; hence every coefficientf& in (18) must tend to zero as z + co. Hence the required expression for f.is of the form (23) where the coefficients A,,, axe determined by the non-divergence condition on j, together with the boundary condition thatj, = 0 at z = 0. The lotions (19) and (20) can now be used to evaluate all the functions fsn of z, in terms of the coefficients A,. Eliminating S between Equations (15) and (16) and substituting for aj,l& from the zero divergence condition (3), we obtain an equation forjsp in the form (24) where gyand E, are given in (7) and (8). A particular integral of this equation is j, = C&e-@ sin ay 5 C,,, sin (2n + 1)~
(25)
n-0
where the coefficients C,,,, are determined by the known function on the right of (24). The compIementary solution to be added to (25) is a series similar to that obtained for
478
A. C. COCKS and A.. T. PRICE
5’ from (17). However, on substitution of this series into (26) and comparing coefficients of sin nax sin ay we see that all the coefficients of this series are zero. Substituting from (25) into (15) and applying the condition j, = 0 at z = 0 gives
as 3 a=0
(>
= (ay + 2 sin axjZ)Z,o
Substituting from (18), (7) and (25) in (26) then gives a set of equations to determine the A,,,%. The current component j, can now be obtained from (15) and j, from (14). (Further details of the calculations are given by Cocks, 1968.) 6. THE NUMERICAL
CALCULATIONS
The numerical calculations involved in the solution are very elaborate, and some compromise has had to be made between the best values of the parameters to represent the physical problem and the values which make it possible to obtain a numerical solution with the computing facilities available. In his discussion of the simpler problem, in which the conductivity tensor was taken as uniform, Price took @to be O-05* which corresponded to the atmospheric velocities being reduced to one-tenth of their value at a height of 46 km from the base of the ionosphere, and the corresponding value of the ratio #a was about 320. But in the present problem the numerical work becomes prohibitive for such high values of this ratio, and considerably smaller values have been used. Also it is found that the calculations are simplified if #a is taken to be half an odd integer because the series for S and for the components of j become finite, the number of terms being slightly greater than the value of this integer. It is then necessary to solve certain matrix equations of about the same order. The results given in this paper are for p = 205a, which corresponds to the el~tromotive force dropping to one-tenth of its value at z = 0 in about 800 km. This height is undoubtediy much too great, but some further calculations have been partially completed for the case when B = 40*5a, corresponding to a height of about 400 km, and the general nature of the results does not appear to be appr~iably altered. 7. DISCUSSION OF THE RESULTS
The results of the present calculations confirm two of the important conclusions reached by Price (1968), namely, (1) throughout the region where the current density is large, the flow is almost entirely horizontal, and therefore the layer equations can be used for obtaining a good approximation to the relation between the horizontal components of current and electric field. (2) the vahte of a@z is of the same order of magnitude as that of Z&/ax or &/lay at all heights. Hence we cannot deduce from (3) that the horizontal flow of current at any height has zero divergence. Consequently this flow cannot be derived from a stream function, as has been assumed in most theoretical discussions of Sq. There are several additional features of the present results that are of interest, and it is useful to compare them in greater detail with those of the previous calculation. The horizontal flow at the base of the ionosphere, when it is assumed uniform (but non-isotropic) as in the earlier calculation, is shown in Fig. 4. It is noticeable that the pattern of the horizontal current flow is quite similar to the horizontal part of the applied * Note this is incornxtly print& as 05 in Price’s paper.
Sq CURRENTS IN A THREE-DIMENSIONAL IONOSPHERE
r-------------
419
\ \ \ t .
I
1 / / FIG. 4. CURRENT SYSlZMAT
BASR OF IONC6PIiBRE
PRODUCED
IN
A
UNIFORM
NON4SOTROPIC
IONOSPHERE.
electromotive forces, and differs considerably from the non-divergent flow that wouldihave been produced by the same electromotive forces in a thin shell. This implies that the horizontal component of the electric field of the space charge distribution, which accumulates in the three-dimensional ionosphere because the (total) current density is nondivergent, is much smaller than that arising in the case of a thin shell. In the latter case the field is large enough to overcome the primary electromotive forces along the dipole equator and turns back the horizontal currents to form closed loops. This does not occur in the threedimensional case because much of the current is able to escape vertically. Since analyses of the actual Sq field have indicated a very considerable return current is very important to flow along the magnetic dip equator- the equatorial electrojet-it see whether the new calculations for a non-uniform non-isotropic ionosphere lead to such a current. It might be expected that this will be so because it has been found that the layer equations are applicable near the base of the ionosphere, and the layer conductivities rise to sharp maxima near the equator, but it is not immediately obvious because the essential part of the total electric field is dependent on the space charge distribution set up. However, the calculations do in fact give a very large return current along the equator, as shown in Fig. 5. This horizontal current system at the base of the ionosphere does not
A. C. COCKS and A. T. PRICE
480
satisfy the condition of zero divergence, but there is present a very considerable concentration of current along the equator. 8. iUU’RJ3tENTA~
AND ACTUAL .+&CWRRENTSYSW
As already pointed out by Price (1968), the representative current system, derived from the analysis of observations of the Sq variations, corresponds to one part only of the actual ionospheric currents. The other part, which includes vertical as well as horizontal currents, cannot be found from surface observations of the magnetic geld. It may, however, be possible to deduce it from calculations similar to the present one, if a sufficiently accurate knowledge of the various parameters is available, and improved numerical methods for solving the elaborate numerical problems involved can be devised. For the present model ionosphere it is possible to separate the current system into two parts, one which has a field beneath the ionosphere and one which does not, It may be shown that any non-divergent current system in the three-dimensional ionosphere can be expressed in the form j = curl klp + curl curl kQ (27) where P and Q are functions of (x, y, z) and k is a unit vector in the z-direction.
i l-l -7&Frtr.
5.
-
CURRENT
ii._
-r/3
SYSTEM
AT
RAW
XSOTROPXZ IONOSPEIERE
BY
OR IOMMPHERB THB
SAW3
PRODUCED
ELECTROMOTWE
M
A
NON-UNIFORM
FORCES
AS IN
FIG
AND
4.
NOW
j--,-
L’
1--L-L---L_
-17-/2
-
t-
-
1 +-T/3
T
-
’
-)
-
-
-E%
-i_
/
n/6
..L QY
-
7
-
-Ilw2
-L_t-L lCf3 '?
2.p
d2
d/2 4;o
Scale
FIG. 6. HORIZONTAL
COMPONENT OF THE CURRENT SYSTEM OF FIG. 5 THAT HAS A MAGNETIC FIELD BELOW THE IONOSPHERE.
--
I
-C
\
f
,
rL_
\
f
I
/-
-
I
t 1
\
I I-
n/3
\
la
I
-
-
-
iota,
l
I
I /
-
/
L/6
t
I I I t
/
-
f
t
f
t
! I
\ .C
f J_-
u/6
-%--
T-
-I
%3-0
-w/3.
I
-I
-lv2
d/2 2,o 40
Scale
FIG: 7. HORIZONTAL
~~~0~~0~~ CURRENT SYSTEM OF FIG. 5 TIZATHASNO FIELD BELOW THE IONOSPHERE. 481
MAGNETIC
A. C. COCKS and A. T. PRICE
482
We then have
asQ axaz
(28)
ju=-aP+asQ
ax ayaz
(29)
jz=-($+C$).
(30)
aP
jZ=T+-
Eliminating Q from (15) and (16), we have
av
asp
aj,
Z2+3jZYj
--.
ai,
ax
(31)
Using the values of jpDand jV obtained in the present calculations, we can solve (30) for P and thus calculate the part of j that has a field in the region z < 0. This current system is shown for z = 0 in Fig. 6, and the horizontal part of the system not having a field in z < 0 is shown in Fig. 7. The horizontal currents in Fig. 6 are non-divergent and may be obtained from a current function. There is a strong return current along the equator. The currents shown in Fig. 7 are not derivable from a current function and there are significant vertical currents associated with them. The results suggest that this part of the ionospheric current system may be comparable with the part usually associated with Sq. REFERElNcEs Bmows, K. and HALL.,S. H. (1964). Nume, Lmd. 204, 721. CHAPMAN,S. (1956). Nuovo Cim., Suppl. 4,1385. Cows, A. C. (1968). Ph.D. Thesis, University of Exeter. DAM, T. N., STOURIK, J. D. and HE~PNER,J. P. (1965), J. geophys. Res. 70, 5883. KIM, J. S. and Km, H. Y. (1962). Nuture, Lmd. 196, 630. PRICB,A. T. (1968). Geophys. J. R. astr. Sot. 15, 93. PRICE,A. T. and WILKINS,G. A. (1963). Phil. Trans. A256,31.