- Email: [email protected]
A new theoretical approach to the modelling of toroidal Pc5 pulsations
Planet. Space Sci. Vol. 29, No. II, Printed in Great Britain.
pp. 1159-1168, 1981
0032W33/8l/l1ll59-10S02.00/0 0 1981 Pergaman Press Ltd.
A NEW THEORETICAL MODELLING
APPROACH
OF TOROIDAL
TO THE
Pc5 PULSATIONS
D. R. MCDIARMID
Herzberg
Institute
of Astrophysics,
National Research Council of Canada, Ottawa, Ontario KIA 0R6, Canada (Receioed IS April 1981)
Abstract-A model is developed to represent a toroidal mode of Pc5 geomagnetic pulsations. It is shown that this model is consistent in its predictions, such as the latitude profiles of amplitude and phase and their dependence on the height integrated Pedersen conductivity, Z,, with those of Walker’s (1980) theory. It is also shown that this theory is relatively easily capable of accommodating (i) a variety of field line plasma mass density distributions, (ii) a variety of external excitation schemes, (iii) unequal Z?‘s at each end of the field lines and (iv) non-dipolar geomagnetic fields. The theory yields the transient as well as the steady state response, an important feature permitting application to short-lived events or to those for which the generator is amplitude modulated. It is shown, for instance, that the amplitude-latitude profile varies during the transient. It is also shown that the steady state latitude profiles of amplitude and phase are the dual of those observed as a function of frequency when the excitation frequency is scanned through a resonance. A more realistic steady state energy flow from a generator along the field lines to the ionosphere is inherent in this theory compared with that from the mode to the ionosphere which is inherent in Walker’s theory.
1. INTRODUCTION
Of the recent developments of PCS pulsation theory which include the consequence of ionospheric Pedersen conductivity (Newton et al., 1978; Allan and Knox, 1979a, b; Walker, 1980), that by Walker is particularly useful because it produces the phase and amplitude variation in the meridian plane through the resonance latitude. These properties have been measured by the STARE radar system (Walker at af., 1979) and their theoretical prediction is important in the ordering and understanding of future observational data. There are, however, several limitations to Walker’s theory. The energy dissipated in the ionosphere is supplied from that stored in the (damped) standing wavC whereas in reality it must be supplied from some source even if temporarily stored in the mode. Also, his work was confined to a description of the steady state whereas, given the common occurrence of either short wavetrains or of amplitude modulated (burst-like) wavetrains, an understanding of the transient response is equally important. It is one of the purposes of this work to diminish these limitations while retaining the strong points of the Walker theory. The theoretical formulation to be described below is a logical extension of the lumped circuit
description of McDiarmid (1980a, b) (which are denoted as I and II respectively below) to a full distributed transmission line picture. Here we will confine ourselves to the formulation and analysis of a longitudinally limited toroidal mode which will be seen to have properties very similar to Walker’s model. This formulation permits the investigation of the consequences of (a) variations in the plasma mass distribution both along and perpendicular (in the meridian plane) to the magnetic field lines, (b) the non-equality of northern and southern hemisphere height integrated Pedersen conductivities and (c) the possible loading by the generation circuit. Although a dipole geomagnetic field is assumed throughout, the model can be adapted to non-dipolar fields. 2. THE MODEL
The distributed non-uniform transmission line which, will represent the resonant flux tubes and their plasma can be described by its equivalent series inductance and shunt capacitance per unit length. An easy way to visualize the geometry of the transmission line is to imagine a pure toroidal mode with infinitely conducting ionospheric boundaries. Here, at the resonant latitude, the geomagnetic field experiences an oscillatory disturbance in the East-West direction whereas the 1159
D. R. MCDIARMID
1160
field lines just poleward of equatorward (i.e. outside the resonance) experience no disturbance. These latter field lines are isolated from the resonant lines by two sheets of oscillatory fieldaligned current flow, one poleward and the other equatorward of the resonance. These sheets can be thought of as plates of a parallel plate transmission line. If the ionosphere is not perfectly conducting, the resonance region has extended dimension in the meridian plane as shown by Walker (1980) and, as a consequence, the current sheets also have thickness in the meridian plane. Even though our transmission line model consists of infinitely thin plates, it will become evident below that the actual behaviour can be synthesized from our model. The distributed parameters of a parallel plate transmission line are L= po$
H/m
(1)
expressions for L, and C, there are reduced to their per unit length values (along the field line) by removing 2/1, the expressions (1) above result. The capacitance C, contains the kinetic energy of the oscillating plasma and the inductance L, contains the stored magnetic energy of the field-aligned pulsation current. We must define the functional forms of p and B. For p we adopt the form used by Walker for the purpose of being able to compare results. p(B) = p(0) cos-*‘e
where 0 is the dipole latitude representing the position along the field line and p(0) is the plasma mass density at the magnetic equat-or. Once the L value of the field line is chosen, 0 adequately defines the position in space since, over the longitudinal extent of the disturbance, axial symmetry is assumed. The magnetic field being dipolar,
and B(e)
C=er
where b is the distance between the plates (current sheets) and w is the longitudinal extent of the sheets. b/w reflects of the ratio of the meridional to longitudinal scale sizes of the pulsation. As such it determines the type of transmission line and hence the functional form of the characteristic (wave) impedance required in the model. The use of a parallel plate transmission line here corresponds to a resonance of limited yet significant longitudinal extent consistent with observation. The phase velocity everywhere along the transmission line must equal the AlfvCn velocity at that point on the geomagnetic field line.
or
= &)
‘c4 - 3 cos2e, co?e
B(O)
(4)
where B(0) is the magnitude of the field at the magnetic equator. Finally, some algebra gives us
b/w co.1
vph = V(L)
=
F/m
provided
(3)
___ = L$)
(2)
.
where Bi is the value of 0 at the E-region foot of the magnetic field line and similarly for (w/b)i. The wave impedance of the line is of paramount importance since it tells us what the internal and terminal wave reflections are. These reflections determine the response of the resonant system. The wave impedance is
z(e) =
In addition to the variation of B and p along the field lines, b and w also vary as defined by the geomagnetic model used. The transmission line, then, is clearly non-uniform. The connection between this model and that for circuit 1 of paper I can now be seen. If the
6)
where Z(0) = B(0) &$)
e=--_i. BP
J(S)=,,~~~~,, d/(4-
3 co? e)(i),
I
Clearly it too varies with position along the field line. In order to complete the description of the resonant circuit, the ionosphere boundaries must be characterized. These boundaries behave as a resistor which corresponds to the finite height integrated Pedersen conductivity. Following the
A new theoretical approach to the modelling of toroidal argument of paper I, the total Pedersen current flow between the plates in response to an electric field is determined. Ohm’s law then yields
1
b
0
rt= w ic,
cash
where 2, is the height integrated Pedersen conductivity and A is the dip angle of the magnetic field at the ionosphere. Thus 1
tan A=2tan&’ The cos A factor is included for the purpose comparison with Walker’s (1980) results.
(8) of
3. THE GENERATOR
With the resonant circuit now defined, we will examine the method of excitation. In this we differ from Walker’s approach which was to supply the ionosphere energy dissipation from the stored energy of the resonance. Here the source of the dissipated energy is explicitly an equatorial
PCS
pulsations
1161
generator which can be given whatever characteristics seem appropriate. These can, in principle, be devised to correspond to any physical mechanism though to be relevant such as the Kelvin-Helmholtz instability at the magnetopause. The circuit itself, as shown in Fig. l(a), consists of either a voltage or current generator feeding the equatorial point of the transmission line through an appropriate impedance. In the case of a KelvinHelmholtz generator, the disturbance of which propagates evanescently into the magnetosphere from the magnetopause, a combination of a voltage generator and series inductance seems appropriate by analogy to an evanescent TE mode in a parallel plate waveguide. Perhaps a shunt resistance across the generator to represent energy dissipation at the magnetopause should be included. However, as will be discussed later, various generator circuits used numerically did not yield any differences in the nature of the steady state solutions. Thus for the bulk of the presentation, the generator circuit consisted, for simplicity, of a current generator feeding the resonant circuit
FIG. 1. THE CONCEPTUAL STRUCTURE OF THE
PCS PULSATION MODEL.
D. R. MCDIARMID
1162
directly. However, the transient solution did change with alterations to the generator circuit. Since this response may be a significant contributor to the observed pulsation, more attention must ultimately be given to the nature of the excitation. This is a topic for future work. As in paper I, it is important to realize that the temporal output of the generator can be anything we deem suitable. When more is known about the growth and decay of a Kelvin-Helmholtz instability at the magnetopause, the voltage generator can be made to mirror this variation.
where W -=
cos8 ( cos
Wi
3 ei >
and b &=
d
4-3cosz 4-3cosze
ei
w >( Gi >.
Standard scattering analysis yields the following results. With reference to Fig. l(d), we obtain
4. ANALYSIS AND RESULTS
The analysis procedure is that outlined in Mallinckrodt and Carlson (1978), namely the transmission line is divided into a number of segments for which the wave propagation times are all equal. In our formulation, sixty segments from hemisphere to hemisphere are used. Each of these segments is represented by its mean wave impedance. Since the line is non-uniform, partial reflection of any incident wave will occur at all segment boundaries as well as at the ionosphere terminations. The former are the previously noted internal partial reflections. The sequence of mean wave impedances as illustrated in Fig. 1 is obtained from the expression
(9)
where 0,_, and 19. are the end points of the nth equal time and dl= LR, v/(4segment 3 cos’ 0) cos 0 d0 is a differential of length along the field line. The 0”‘s were determined by suitably subdividing the time of flight curve for the resonant field line. The values obtained are normalized with respect to p(0) but are of necessity dependent on the value of p. Thus a set of 2”‘s must be computed for each value of p desired. We have used p = 3 and 4. The analysis is in terms of voltage and current so as to avoid the scaling factors which must be used if electric field and current density are the computation parameters. The conversion after solution is straightforward. E(e)
v(e) = b(e)
Vi-Y(n) c-Y(n) = p/s ” ’
Zy+z) = ZJy(n)
W(n) = p/s II IF(n) =m
VFxn)
K,“” = reflection coefficient
and N/S is used to delineate between segments north of the equator and those south of the equator as shown in Fig. l(a). These equations describe the internal partial reflections of the line. In the specific formulation used herein, the N/S designation is not required because of the symmetry of the line about the equatorial plane. However, we carry the distinction to permit the extension to non-symmetrical geomagnetic field models. Behaviour at the ionosphere terminations is similarly described. With reference to Fig. l(c), we obtain Zy
=
(I-
K
fJ’S)zy(N
ZzS(N + 1) =
K~J’~Z:“(N
Vz’(N
K r””
+ 1) =
V f”” = (1 + where
(IO) j(0) = $$
where
_-
+
+ 1)
1)
V$‘:l”(N + 1) KIN/~)V;“(N
(12) +
1)
A new theoretical approach to the modelling of toroidal PCS pulsations
and the use of N instead of n is to denote the maximum value of n which, in our case, is 30. Finally we must describe the scattering at the generator input. For a simple current generator input (no input impedance) with the current direction that defined by Z, in Fig. l(b), we obtain Z;(l)
=
K’I,I;(l)+
v;(l)
=
K’v;(l)
tS(If$(l) +
-0.5&)
tNV;&l)-
.??,I,
(13)
and similarly for IS,(l) and V?,(l). Here
K
s_ zs-zr: -
.z;+.zy p=
-KN
(14)
and z = -2% ’ Z’;‘+Zf. In the event that there is an inductive input impedance, we must integrate the differential equation
dZ v, - v, - L, $ = 0 over one transmission time, At, to obtain
line segment
propagation
&(m + 1) = [V,(m) - V,(m)] g + &(m) 8
(15)
where m represents the time via t = mAt. This value of Z, is then used in equation (13). The method of solution in iterative. For any given iteration, the following transformation must first be performed: Z:‘:/“(n) = ZZ’:‘“(n- 1) ZE’;l”(n)= ZSS(n + 1) VP(n)
= VX”(n - 1)
of Z, is calculated. The computation commences with the initial value of Z, and all the response parameters set to zero. Thus the transient part of the response is inherent. Clearly then, it is easy to change the generator function. The time sequence of whatever solution terms are desired is retained in a separate file for either direct output or automatic plotting. Alternatively, the output of all the solution terms for set times into a separate file is possible. Response along the resonant field line
=
l+KS=Z-fN
1163
(16)
VXS(n) = VZS(n + 1) where the left-hand side terms represent incident waves at time m. These are equal to the right-hand side reflected terms which were generated at time m - 1 and have propagated to the next segment boundary. The scattered waves are then computed with the use of equations (1 l)-(14) or (15) as the case may be. At this stage, of course, a new value
The latter output scheme was used to determine the spatial response at two times one quarter period apart after the steady state had been reached. The input parameters were those listed in the next section and the output data are plotted in Fig. 2. The times were keyed to that of maximum ionosphere electric field. The voltage and current solutions were converted to electric field and current density to facilitate direct comparison with Walker’s (1980) Fig. 1. Comparison shows that Walker’s curves l(a), (b) and (c) have shapes very close to our counterparts. This is not true of the remaining curve. The reason for this discrepancy becomes apparent when one examines the energy flow in each model. In Walker’s model, the “energy is being lost from the mode to the ionosphere” (Walker, 1979). In our model, the energy is supplied by the generator and flows down the field line, along which the energy flow is constant, to be dissipated in the two ionospheres. Such is a more realistic scheme during the steady state. In addition, our model also naturally accommodates the other energy flow patterns which exist during the transient phase including the loss of energy from the mode to the ionospheres which occurs during the decay transient. A more accurate picture reflecting distributed excitation is possible in our mode1 through appropriate input at a number of segment boundaries. To return to Fig. 2, we note that the equatorial response is 90” out of phase with the ionosphere response as also predicted by Walker. However, by 0 = 10” this is no longer true by a wide margin in our j response. Response
us frequency
at fixed latitude
Equation (9) above pertains to a specific latitude (0,) and for our base model, 13,= 67” was chosen. It will be shown below how to alter the model to obtain the response at neighbouring latitudes. However, examination of the variation of the res-
D. R. MCDIARMID
1164
,::r 20"
30" 8 40"
t’ \
50"
60"
A’ __-- _-
I
1
I
I
I
I
I
0204060810
2
x (lo-? FIG.
2.
THE
STEADY
STATE
x
ELECTRIC
FUNCTION
FIELD
OF POSITION
(D3)
AND
FIELD-ALIGNED
ALONG
x ( 0’1
x ( lo-4) CURRENT
A RESONANT
FIELD
PULSATION
RESPONSE
AS
A
LINE.
The response amplitudes are displayed to facilitate intercomparison between curves and the current is toward the ionosphere on the high voltage side.
ponse at fixed latitude as a function of frequency is illuminating. Equations (6) and (7) show that with a simple current generator input, the factor (b/ w)i drops out of the K’S; it only remains in 2, of equations (13) which describe the excitation. The computed response then pertains to 67” only, and by varying the frequency, we obtain something like a Q curve. Such a plot has been produced using the following input data and the curve of the ionosphere response is shown in Fig. 3:
Walker’s results is the following. The points on the amplitude curve for periods progressively less than 402 s correspond to resonances occurring on field lines at successively lower geomagnetic latitudes. Thus this portion of our curve is a kind of mirror image of the portion of Walker’s curve .O
0
p(O) = 9.5( IO-*‘) kg mm3 2,” = 2,” = 6.51 a-’ p=3 Geomognetlc
(b/ w)i = 0.072.
The ratio (b/w), corresponds roughly to the ratio of scale sizes in Walker’s (1980) results. The time of flight between hemispheres under these conditions is 300s and the resonant period obtained was 402 s. The solid lines in Fig. 3 pertain to the steady state response and the dashed lines to the transient. They will be examined later. A salient feature of the steady state curves is their similarity of Walker’s (1980) ionosphere response vs. latitude curves (e.g. his Fig. 4(b)) if the period axis is reversed to increase to the left instead of to the right as shown. This is no coincidence. Also the resulting phase variation of something approaching 180” as the period varies through the resonance is an inherent property of a resonance. The reason for the similarity between Fig. 3 and
686 0 -100" -
675 II
I
latitude
670 II
665 I
-200"
-260"
Period,
set
FIG. 3. THE STEADY STATE AMPLITUDE AND PHASE PROFILES WITH
PERIOD
(LATITUDE)
ARE SHOWN
AS SOLID
LINES.
The dashed lines pertain to the transient which was determined as a function of A. The input parameters were p = 3,X,, = 6.5 1 0-l and p(O) = 9.5( 10m20) kg m-j. Also p(0) was assumed constant with L.
1165
A new theoretical approach to the modelling of toroidal Pc5 pulsations 1740 6
3046
3916.L
0 695
-2
1;
p-3
= C;=651
mho
1
320
6700” 6725” 67 50” 67 75”
p,,,=9
5
( IO-'")kg mm3
T :
402 s
-t
13001
Time.
set
670
, 006 E Ef ; 0
I 342 0
290
-0322
-1650 4334
FIG.~.
8663
THEPULSATIONRESPONSEASAFUNCTIONOFI.ATITUDEWITHTHEI.ATITUDESSHOWNATTHECENTER
The resonant latitude is 67”. The vertical dashed lines denote equal times and thus show the phase dependence of the response on latitude.
of the resonance. A similar correspondence holds for points on our curve for periods greater than 402 s. In other words, our response vs period curve at fixed latitude is a kind of dual of the response vs latitude curve for fixed frequency. As such, it is clear that the phase variation with latitude shown by Walker is as inherent to the resonance as that observed by varying the frequency through a resonance. poleward
Response
us latitude at fixed frequency
The model above can easily be altered to correspond to latitudes in the neighbourhood of 67”. This is possible because, to’ a very high degree of accuracy, the change in the time of flight curves required by the latitude variations is only through a multiplicative factor; the time axis is either stretched or contracted while the curve is unaltered. Thus the line segment boundaries, &, remain fixed with the exception, obviously, of the final one, ON, the new geomagnetic latitude. As a consequence, all the 2’s except 2: are altered only by a common multiplicative constant which accounts for (a) the change in the equatorial value of the magnetic field, (b) the change, if any, in the equatorial plasma mass density and (c) the change
in @.,(@,).However, the change in &., means that 2:“” requires a separate correction. Consequently, all the reflection coefficients except K:” and KY” are unaltered; only these two reflection coefficients and At, the segment propagation time, need be changed. The required change in At is straightforward but that for 2:/s is a little more complicated. The upper limit of the integrals in equation (9) is changed. Fortunately it is relatively easy to express ZK’“(&) as f(67”- 0N)zE’S(67”) where the function f was determined by matching to calculated values of the mean impedance. This done, the response at any nearby latitude can be calculated by first altering the required parameters and then by proceeding as described above. This has been done for the input data in the preceding section and the result appears as the latitude axis in the center of Fig. 3. It is consistent with the duality discussion above. We also note that the transient profiles in Fig. 3 are spatial, i.e. relative to latitude. The transient response
Figure 4 shows the response at the ionosphere for latitudes 67.0”, 67.25”, 67.50” and 67.75”, as indicated at the center of the figure, for the same
1166
D. R. MCDIARMID
parameters as used for Fig. 3. The input circuit consisted of a current generator feeding the line directly (2, = 0) with an output of sin ot. The character of the transient response varies as the observation point is moved poleward from the latitude of the resonance. Similar behaviour is obtained if the observation point is shifted equatorward from the resonance. First of all a delay, corresponding to the time required for the disturbance to propagate down the field lines, is observed in every response. Not surprisingly, the response grows monotonically to the steady state value at the resonant latitude (0, = 67”). During this time there is a build-up of stored energy in the resonant flux tubes. As we depart from this latitude, the response begins to show overshoot. Thus the amplitude vs latitude profile varies during the transient phase as shown by the dashed lines in Fig. 3 which are for times of 667 and 1500s. These profiles differ significantly from the solid line. Hence care must be taken in the interpretation of measured profiles, such as determined from radar observations (Walker et al., 1979) for instance, for events which are either short lived or show significant amplitude modulation. It is intended to investigate this problem in future using more suitable generator functions and input impedances. Both of these parameters contribute to the character of the transient response. Figure 4 also shows the phase variation with latitude of the steady state solution as indicated by the vertical dashed lines which denote a common time. A phase variation approaching 90” is clearly evident. The other 90” of phase shift occurs equatorward of 67”. Response as a function of 8, and p The C,‘s corresponding to the three values of k,/k, shown in Walker’s (1980) Fig. 8 were determined using the appropriate expressions in Walker (1979). They are C,” = 2,’ = 6.51, 4.65 and 3.62 R-‘. Response curves were computed for each using p(O) = 9.5(10m2”) kg mm3 and p equal each of 3 and 4. The period, T,, and the latitudinal distance between half amplitude points, Al\, for each solution are shown in Table 1 along with Ah’s scaled from Walker’s Fig. 8. Comparison with Walker’s values is good for C, = 6.51 a-’ but the results diverge as 2, is lowered. This might in part be a consequence of the coupling to a poloidal type field which is inherent in his work. If this coupling leads to an increase in the ratio of stored energy to power dissipation, the discrepancy follows. Another, and possibly more important
TABLET. THERESONANTPER~ODANDLATITLJDE HALF-WIDTH AS A FUNCTIONOF&, ANDP 2,
p=3
R-’
Walker
6.51
1.03 1.42 1.82
4.65 3.62
The parameter culations.
T,(S)
Ah(‘)
1.01 1.48 1.98
p=4
p=3
1.14 1.69 2.32
402 403 407
p=4 409 411 416
wi was fixed for the cal-
source of the divergence is a slight difference in the stored energies inherent in the two theories resulting from the two excitation schemes. As noted above, the power flow patterns differ and it thus is natural to expect the stored energy distribution to differ as well. In so far as this is the cause of the divergence, our results are probably more realistic since our power flow pattern is more realistic. A third and probably the most important source of the divergence is the fact that in Walker’s calculations, the ratio of latitudinal to longitudinal scale sizes is > 0.1 for XP = 4.65 and 3.62R-‘. Since our impedance expression is only valid for ratios I 0.1, our .?s are slightly in error for both these C,‘s with the larger error associated with 3.62 0-l. This corresponds well with the increase in the divergence of our results as C,, is decreased. However, the other two sources of discrepancy may also contribute. (Note that the ratio (b/w), on p. 1164 is somewhat lower than that of Walker as a result of a computation error. However, as noted there, the ratio, if I 0.1, only affects the excitation. It will be seen below that, as such, the steady state solution is unaltered.) The consequence of changing p from 3 to 4, the other parameters remaining fixed, is also shown in Fig. 5 as well as in Table 1. This change is to a , 60
-1 t
AA
Frc.5.
THEPULSATION
LATITUDEHALF-WIDTHASA
FUNCTIONOFP
ANDOF&,.
1167
A new theoretical approach to the modelling of toroidal PCS pulsations
more uniform transmission line and, as a consequence, the line is more heavily loaded. This is because (K,( is decreased as a result of a decrease in the Z’s, particularly at the ionosphere ends of the field line. Also the ratio of the resonant period to the time of flight is changed as previously noted by Warner and Orr (1979). The time of flight periods are 300 s for p = 3 and 338 s for p = 4; they are thus approx. 25 and 17% less, respectively, than the computed periods. The interesting thing to note is that for fixed equatorial plasma mass density, the resonant period is almost unchanged for the above change in p.
2,”
z C,?
Unequal Northern and Southern hemisphere &,‘s, which are the norm in reality, present no problem in our formulation. During the investigation of the nature of such solutions, it was observed that &,” = 6.51 R-’ and 2,’ = 3.62 R-’ yield the same shape of steady state response vs frequency (latitude) curve as did C,” = Xi,’ = 4.65 0-l. In other words, 4.65 R-’ can be used as an equivalent conductivity in theories which require 2,” = C,“. A general expression for calculating the equivalent conductivity, Z,“, is
This is the same as requiring the series addition of the two mean ionosphere termination resistances to equal the series addition of the actual termination resistances. Such is the requirement resulting from specifying that the product of all the scattering coefficients be invariant. A cursory examination of this criterion via a lumped circuit analysis indicates that it is only valid when 2, ti r, which is virtually always the case in any real event (Wallis and Budzinski, 1981).
Consequence
of generator impedance
While it may not represent the most appropriate description of the excitation process, a generator circuit consisting of a resistance, an inductance, and a voltage generator, all in series, was tested. Two sets of calculations were conducted in which r, 6 ZL~’ and wL, = 0.3Zf“/“. In both cases the shape of the steady state response vs frequency curves were the same as those obtained using the simple current generator. However, not surprisingly, the transient solutions varied.
5. DISCUSSIONAND CONCLUSIONS
The question may naturally arise as to the extent of the error which occurs when one matches the theory to observed results with an incorrect assumption of the plasma mass distribution along the field line (i.e. incorrect choice of p). For this discussion let p = 4, C, = 6.51 R-’ and p(O) = 9.5 (10~“) kgmm3 be the correct parameters. We then match the theory to observation after incorrectly assuming p = 3 by varying Zp and p(0) until the theory yields the observed values of rr and AA. The difference between these values and those above give an indication of the possible error in using this matching to measure C, and p(O). A crude estimate of the error may be determined as follows using the data in Table 1 and Fig. 5. To increase AA from 1.01” to 1.14” when p = 3 requires a change in Xp from 6.51 to 5.94 R-l. This leads to no significant change in TV.Now to increase TVfrom 402 to 409 s requires an increase in p(0) of about 3.5%. Such a change reduces the Z’s by about 1.7%. With respect to the above change in C, of about 8.8%, the subsequent change in 2, does not significantly further alter Kt. Thus further iteration of the above process will probably converge rapidly and the resultant error in 2, and p(0) are probably of the order of 10% and 5% respectively. These are acceptable if they are indicative of the errors obtained using other equally appropriate plasma distributions. Allan and Knox (1979b) pointed out that it is theoretically possible to have a h/4 mode if there is very low Pedersen conductivity at one end of the resonant field line. To be specific, r, must be greater than ZN. In our two cases (p = 3 and 4), this implies z
P
< 0.60(10? q(p(o)) 1 1.2 (l@? d\/(P(W
P = 3 P = 4
at f&= 67”. If p(0) = 9.5 (lo-*“) km rnm3, we obtain Z, < 0.18 and 0.37 C’ respectively. It is very unlikely that such a condition will be satisfied at aurora1 latitudes (Wallis and Budzinski, 1981). Increasing these limits on Z, by an order of magnitude requires either an increase in 0, to about 75” or of p(0) by two orders of magnitude. In the former case, the times of flight are increased by a factor of approx. 25 if one assumes the field lines to be still dipolar. In the latter case, the times of flight are increased by an order of magnitude. Thus if the conditions are altered such that a A/4 mode approaches the realm of possibility, particularly without severe damping, the period is then too
1168
D. R. MCDIARMID
large for the pulsation to be called PCS. Thus if our parameter values are realistic, it is unlikely that a A/4 Pc5 mode will occur. Before discussing the particular accomplishments of our theory which were obtained above, we note again that each individual solution we have produced is valid for its latitude only. Also, since (w/b), cancels out of all the K’S, it is a factor only in the excitation through Z,, in equations (13). The entire or synthesized solution is obtained through the calculation of a sequence of individual solutions. This has been done above in Fig. 3 for the ionosphere and the same could have been done for any point in the magnetosphere. Clearly, the end result cannot be faulted because it is not spatially distributed as a consequence of being generated by a “hard wire” circuit. The synthesized solution is spatially distributed. In fact, we see that for each solution the model probes the pulsation response at the latitude which it describes. The results presented in this paper show that our longitudinally limited toroidal mode model predicts behaviour consistent with that predicted by Walker’s model. It can also relatively easily accomodate (a) changes in the plasma density distribution along the field line as represented by p, (b) the insertion of a generator circuit which permits, in principle, a proper description of the excitation process, (c) the condition C,“# &,‘, and (d) the alteration of the magnetic field to non-dipolar form. The transient response is an integral part of the solution and the importance of understanding its contribution is now clear since latitudinal variation of the pulsation differs from the steady state profile. It is hoped to obtain further insights into the consequences of this behaviour in the course of future work since not all pulsations give evidence of steady state behaviour. For the parameters chosen in our calculations, we have seen that the generator impedance did not alter the shape of the steady state latitudinal response profile. However, it is not known if these
parameters are realistic; thus only a tentative conclusion is possible at best. Further work would be useful here. In the discussion on the latitudinal variation of the pulsation response it was shown that because of the duality with the frequency response, the phase shift of - 180” across the resonance region is as inherent as that which occurs when the frequency of excitation of a resonant structure is varied through its resonance. REFERENCES Allan, W. and Knox, F. B. (1979a). A dipole field model for axisymmetric AlfvCn waves with finite ionosphere conductivities. Planet. Space Sci. 27, 79. Allan, W. and Knox, F. B. (1979b). The effect of finite ionosphere conductivities on axisymmetric toroidal Alfv& wave resonances. Planet. Space Sci. 27,939. Mallinckrodt. A. J. and Carlson. C. W. (1978). Relations between transverse electric fields and field-aligned currents. J. geophys. Res. 83, 1426. McDiarmid. D. R. (1980a). Toward some new directions in PCS pulsation theory-l. Consequences of _ ionosphere Hall current. Planet. Space Sci. 28, 713. McDiarmid, D. R. (1980b). Toward some new directions in PCS pulsation theory-2. Extension of the analysis to two hemispheres and determination of the model’s uolarization characteristics. Planet. Space Sci. 28. 727. Newton, R. S., Southwood, D. J. and Hughes, W. J. (1978). Damping of geomagnetic pulsations by the ionosphere. Planet. Space Sci. 26,201. Walker, A. D. M. (1979). Modelling of Pc5 pulsation structure in the Magnetosphere-i. Toroidat motion and associated fields. Research Renort R1/79. Dent. of Physics, University of Natal, Durban. Walker, A. D. M. (1980). Modelling of PCS pulsation structure in the magnetosphere. Planet. Space Sci. 28, 213.
Walker, A. D. M., Greenwald, R. A., Stuart, W. F. and Green, C. A. (1979). STARE aurora1 radar observations of Pc5 geomagnetic pulsations. J. geophys. Res. 84, 3373. Wallis, D. D. and Budzinski, E. E. (1981). Empirical models of height integrated conductivities. J. geophys. Res. 86, 125. Warner, M. R. and Orr, D. (1979). Time of flight calculations for high latitude geomagnetic pulsations. Planet. Space Sci. 27, 679.
pp. 1159-1168, 1981
0032W33/8l/l1ll59-10S02.00/0 0 1981 Pergaman Press Ltd.
A NEW THEORETICAL MODELLING
APPROACH
OF TOROIDAL
TO THE
Pc5 PULSATIONS
D. R. MCDIARMID
Herzberg
Institute
of Astrophysics,
National Research Council of Canada, Ottawa, Ontario KIA 0R6, Canada (Receioed IS April 1981)
Abstract-A model is developed to represent a toroidal mode of Pc5 geomagnetic pulsations. It is shown that this model is consistent in its predictions, such as the latitude profiles of amplitude and phase and their dependence on the height integrated Pedersen conductivity, Z,, with those of Walker’s (1980) theory. It is also shown that this theory is relatively easily capable of accommodating (i) a variety of field line plasma mass density distributions, (ii) a variety of external excitation schemes, (iii) unequal Z?‘s at each end of the field lines and (iv) non-dipolar geomagnetic fields. The theory yields the transient as well as the steady state response, an important feature permitting application to short-lived events or to those for which the generator is amplitude modulated. It is shown, for instance, that the amplitude-latitude profile varies during the transient. It is also shown that the steady state latitude profiles of amplitude and phase are the dual of those observed as a function of frequency when the excitation frequency is scanned through a resonance. A more realistic steady state energy flow from a generator along the field lines to the ionosphere is inherent in this theory compared with that from the mode to the ionosphere which is inherent in Walker’s theory.
1. INTRODUCTION
Of the recent developments of PCS pulsation theory which include the consequence of ionospheric Pedersen conductivity (Newton et al., 1978; Allan and Knox, 1979a, b; Walker, 1980), that by Walker is particularly useful because it produces the phase and amplitude variation in the meridian plane through the resonance latitude. These properties have been measured by the STARE radar system (Walker at af., 1979) and their theoretical prediction is important in the ordering and understanding of future observational data. There are, however, several limitations to Walker’s theory. The energy dissipated in the ionosphere is supplied from that stored in the (damped) standing wavC whereas in reality it must be supplied from some source even if temporarily stored in the mode. Also, his work was confined to a description of the steady state whereas, given the common occurrence of either short wavetrains or of amplitude modulated (burst-like) wavetrains, an understanding of the transient response is equally important. It is one of the purposes of this work to diminish these limitations while retaining the strong points of the Walker theory. The theoretical formulation to be described below is a logical extension of the lumped circuit
description of McDiarmid (1980a, b) (which are denoted as I and II respectively below) to a full distributed transmission line picture. Here we will confine ourselves to the formulation and analysis of a longitudinally limited toroidal mode which will be seen to have properties very similar to Walker’s model. This formulation permits the investigation of the consequences of (a) variations in the plasma mass distribution both along and perpendicular (in the meridian plane) to the magnetic field lines, (b) the non-equality of northern and southern hemisphere height integrated Pedersen conductivities and (c) the possible loading by the generation circuit. Although a dipole geomagnetic field is assumed throughout, the model can be adapted to non-dipolar fields. 2. THE MODEL
The distributed non-uniform transmission line which, will represent the resonant flux tubes and their plasma can be described by its equivalent series inductance and shunt capacitance per unit length. An easy way to visualize the geometry of the transmission line is to imagine a pure toroidal mode with infinitely conducting ionospheric boundaries. Here, at the resonant latitude, the geomagnetic field experiences an oscillatory disturbance in the East-West direction whereas the 1159
D. R. MCDIARMID
1160
field lines just poleward of equatorward (i.e. outside the resonance) experience no disturbance. These latter field lines are isolated from the resonant lines by two sheets of oscillatory fieldaligned current flow, one poleward and the other equatorward of the resonance. These sheets can be thought of as plates of a parallel plate transmission line. If the ionosphere is not perfectly conducting, the resonance region has extended dimension in the meridian plane as shown by Walker (1980) and, as a consequence, the current sheets also have thickness in the meridian plane. Even though our transmission line model consists of infinitely thin plates, it will become evident below that the actual behaviour can be synthesized from our model. The distributed parameters of a parallel plate transmission line are L= po$
H/m
(1)
expressions for L, and C, there are reduced to their per unit length values (along the field line) by removing 2/1, the expressions (1) above result. The capacitance C, contains the kinetic energy of the oscillating plasma and the inductance L, contains the stored magnetic energy of the field-aligned pulsation current. We must define the functional forms of p and B. For p we adopt the form used by Walker for the purpose of being able to compare results. p(B) = p(0) cos-*‘e
where 0 is the dipole latitude representing the position along the field line and p(0) is the plasma mass density at the magnetic equat-or. Once the L value of the field line is chosen, 0 adequately defines the position in space since, over the longitudinal extent of the disturbance, axial symmetry is assumed. The magnetic field being dipolar,
and B(e)
C=er
where b is the distance between the plates (current sheets) and w is the longitudinal extent of the sheets. b/w reflects of the ratio of the meridional to longitudinal scale sizes of the pulsation. As such it determines the type of transmission line and hence the functional form of the characteristic (wave) impedance required in the model. The use of a parallel plate transmission line here corresponds to a resonance of limited yet significant longitudinal extent consistent with observation. The phase velocity everywhere along the transmission line must equal the AlfvCn velocity at that point on the geomagnetic field line.
or
= &)
‘c4 - 3 cos2e, co?e
B(O)
(4)
where B(0) is the magnitude of the field at the magnetic equator. Finally, some algebra gives us
b/w co.1
vph = V(L)
=
F/m
provided
(3)
___ = L$)
(2)
.
where Bi is the value of 0 at the E-region foot of the magnetic field line and similarly for (w/b)i. The wave impedance of the line is of paramount importance since it tells us what the internal and terminal wave reflections are. These reflections determine the response of the resonant system. The wave impedance is
z(e) =
In addition to the variation of B and p along the field lines, b and w also vary as defined by the geomagnetic model used. The transmission line, then, is clearly non-uniform. The connection between this model and that for circuit 1 of paper I can now be seen. If the
6)
where Z(0) = B(0) &$)
e=--_i. BP
J(S)=,,~~~~,, d/(4-
3 co? e)(i),
I
Clearly it too varies with position along the field line. In order to complete the description of the resonant circuit, the ionosphere boundaries must be characterized. These boundaries behave as a resistor which corresponds to the finite height integrated Pedersen conductivity. Following the
A new theoretical approach to the modelling of toroidal argument of paper I, the total Pedersen current flow between the plates in response to an electric field is determined. Ohm’s law then yields
1
b
0
rt= w ic,
cash
where 2, is the height integrated Pedersen conductivity and A is the dip angle of the magnetic field at the ionosphere. Thus 1
tan A=2tan&’ The cos A factor is included for the purpose comparison with Walker’s (1980) results.
(8) of
3. THE GENERATOR
With the resonant circuit now defined, we will examine the method of excitation. In this we differ from Walker’s approach which was to supply the ionosphere energy dissipation from the stored energy of the resonance. Here the source of the dissipated energy is explicitly an equatorial
PCS
pulsations
1161
generator which can be given whatever characteristics seem appropriate. These can, in principle, be devised to correspond to any physical mechanism though to be relevant such as the Kelvin-Helmholtz instability at the magnetopause. The circuit itself, as shown in Fig. l(a), consists of either a voltage or current generator feeding the equatorial point of the transmission line through an appropriate impedance. In the case of a KelvinHelmholtz generator, the disturbance of which propagates evanescently into the magnetosphere from the magnetopause, a combination of a voltage generator and series inductance seems appropriate by analogy to an evanescent TE mode in a parallel plate waveguide. Perhaps a shunt resistance across the generator to represent energy dissipation at the magnetopause should be included. However, as will be discussed later, various generator circuits used numerically did not yield any differences in the nature of the steady state solutions. Thus for the bulk of the presentation, the generator circuit consisted, for simplicity, of a current generator feeding the resonant circuit
FIG. 1. THE CONCEPTUAL STRUCTURE OF THE
PCS PULSATION MODEL.
D. R. MCDIARMID
1162
directly. However, the transient solution did change with alterations to the generator circuit. Since this response may be a significant contributor to the observed pulsation, more attention must ultimately be given to the nature of the excitation. This is a topic for future work. As in paper I, it is important to realize that the temporal output of the generator can be anything we deem suitable. When more is known about the growth and decay of a Kelvin-Helmholtz instability at the magnetopause, the voltage generator can be made to mirror this variation.
where W -=
cos8 ( cos
Wi
3 ei >
and b &=
d
4-3cosz 4-3cosze
ei
w >( Gi >.
Standard scattering analysis yields the following results. With reference to Fig. l(d), we obtain
4. ANALYSIS AND RESULTS
The analysis procedure is that outlined in Mallinckrodt and Carlson (1978), namely the transmission line is divided into a number of segments for which the wave propagation times are all equal. In our formulation, sixty segments from hemisphere to hemisphere are used. Each of these segments is represented by its mean wave impedance. Since the line is non-uniform, partial reflection of any incident wave will occur at all segment boundaries as well as at the ionosphere terminations. The former are the previously noted internal partial reflections. The sequence of mean wave impedances as illustrated in Fig. 1 is obtained from the expression
(9)
where 0,_, and 19. are the end points of the nth equal time and dl= LR, v/(4segment 3 cos’ 0) cos 0 d0 is a differential of length along the field line. The 0”‘s were determined by suitably subdividing the time of flight curve for the resonant field line. The values obtained are normalized with respect to p(0) but are of necessity dependent on the value of p. Thus a set of 2”‘s must be computed for each value of p desired. We have used p = 3 and 4. The analysis is in terms of voltage and current so as to avoid the scaling factors which must be used if electric field and current density are the computation parameters. The conversion after solution is straightforward. E(e)
v(e) = b(e)
Vi-Y(n) c-Y(n) = p/s ” ’
Zy+z) = ZJy(n)
W(n) = p/s II IF(n) =m
VFxn)
K,“” = reflection coefficient
and N/S is used to delineate between segments north of the equator and those south of the equator as shown in Fig. l(a). These equations describe the internal partial reflections of the line. In the specific formulation used herein, the N/S designation is not required because of the symmetry of the line about the equatorial plane. However, we carry the distinction to permit the extension to non-symmetrical geomagnetic field models. Behaviour at the ionosphere terminations is similarly described. With reference to Fig. l(c), we obtain Zy
=
(I-
K
fJ’S)zy(N
ZzS(N + 1) =
K~J’~Z:“(N
Vz’(N
K r””
+ 1) =
V f”” = (1 + where
(IO) j(0) = $$
where
_-
+
+ 1)
1)
V$‘:l”(N + 1) KIN/~)V;“(N
(12) +
1)
A new theoretical approach to the modelling of toroidal PCS pulsations
and the use of N instead of n is to denote the maximum value of n which, in our case, is 30. Finally we must describe the scattering at the generator input. For a simple current generator input (no input impedance) with the current direction that defined by Z, in Fig. l(b), we obtain Z;(l)
=
K’I,I;(l)+
v;(l)
=
K’v;(l)
tS(If$(l) +
-0.5&)
tNV;&l)-
.??,I,
(13)
and similarly for IS,(l) and V?,(l). Here
K
s_ zs-zr: -
.z;+.zy p=
-KN
(14)
and z = -2% ’ Z’;‘+Zf. In the event that there is an inductive input impedance, we must integrate the differential equation
dZ v, - v, - L, $ = 0 over one transmission time, At, to obtain
line segment
propagation
&(m + 1) = [V,(m) - V,(m)] g + &(m) 8
(15)
where m represents the time via t = mAt. This value of Z, is then used in equation (13). The method of solution in iterative. For any given iteration, the following transformation must first be performed: Z:‘:/“(n) = ZZ’:‘“(n- 1) ZE’;l”(n)= ZSS(n + 1) VP(n)
= VX”(n - 1)
of Z, is calculated. The computation commences with the initial value of Z, and all the response parameters set to zero. Thus the transient part of the response is inherent. Clearly then, it is easy to change the generator function. The time sequence of whatever solution terms are desired is retained in a separate file for either direct output or automatic plotting. Alternatively, the output of all the solution terms for set times into a separate file is possible. Response along the resonant field line
=
l+KS=Z-fN
1163
(16)
VXS(n) = VZS(n + 1) where the left-hand side terms represent incident waves at time m. These are equal to the right-hand side reflected terms which were generated at time m - 1 and have propagated to the next segment boundary. The scattered waves are then computed with the use of equations (1 l)-(14) or (15) as the case may be. At this stage, of course, a new value
The latter output scheme was used to determine the spatial response at two times one quarter period apart after the steady state had been reached. The input parameters were those listed in the next section and the output data are plotted in Fig. 2. The times were keyed to that of maximum ionosphere electric field. The voltage and current solutions were converted to electric field and current density to facilitate direct comparison with Walker’s (1980) Fig. 1. Comparison shows that Walker’s curves l(a), (b) and (c) have shapes very close to our counterparts. This is not true of the remaining curve. The reason for this discrepancy becomes apparent when one examines the energy flow in each model. In Walker’s model, the “energy is being lost from the mode to the ionosphere” (Walker, 1979). In our model, the energy is supplied by the generator and flows down the field line, along which the energy flow is constant, to be dissipated in the two ionospheres. Such is a more realistic scheme during the steady state. In addition, our model also naturally accommodates the other energy flow patterns which exist during the transient phase including the loss of energy from the mode to the ionospheres which occurs during the decay transient. A more accurate picture reflecting distributed excitation is possible in our mode1 through appropriate input at a number of segment boundaries. To return to Fig. 2, we note that the equatorial response is 90” out of phase with the ionosphere response as also predicted by Walker. However, by 0 = 10” this is no longer true by a wide margin in our j response. Response
us frequency
at fixed latitude
Equation (9) above pertains to a specific latitude (0,) and for our base model, 13,= 67” was chosen. It will be shown below how to alter the model to obtain the response at neighbouring latitudes. However, examination of the variation of the res-
D. R. MCDIARMID
1164
,::r 20"
30" 8 40"
t’ \
50"
60"
A’ __-- _-
I
1
I
I
I
I
I
0204060810
2
x (lo-? FIG.
2.
THE
STEADY
STATE
x
ELECTRIC
FUNCTION
FIELD
OF POSITION
(D3)
AND
FIELD-ALIGNED
ALONG
x ( 0’1
x ( lo-4) CURRENT
A RESONANT
FIELD
PULSATION
RESPONSE
AS
A
LINE.
The response amplitudes are displayed to facilitate intercomparison between curves and the current is toward the ionosphere on the high voltage side.
ponse at fixed latitude as a function of frequency is illuminating. Equations (6) and (7) show that with a simple current generator input, the factor (b/ w)i drops out of the K’S; it only remains in 2, of equations (13) which describe the excitation. The computed response then pertains to 67” only, and by varying the frequency, we obtain something like a Q curve. Such a plot has been produced using the following input data and the curve of the ionosphere response is shown in Fig. 3:
Walker’s results is the following. The points on the amplitude curve for periods progressively less than 402 s correspond to resonances occurring on field lines at successively lower geomagnetic latitudes. Thus this portion of our curve is a kind of mirror image of the portion of Walker’s curve .O
0
p(O) = 9.5( IO-*‘) kg mm3 2,” = 2,” = 6.51 a-’ p=3 Geomognetlc
(b/ w)i = 0.072.
The ratio (b/w), corresponds roughly to the ratio of scale sizes in Walker’s (1980) results. The time of flight between hemispheres under these conditions is 300s and the resonant period obtained was 402 s. The solid lines in Fig. 3 pertain to the steady state response and the dashed lines to the transient. They will be examined later. A salient feature of the steady state curves is their similarity of Walker’s (1980) ionosphere response vs. latitude curves (e.g. his Fig. 4(b)) if the period axis is reversed to increase to the left instead of to the right as shown. This is no coincidence. Also the resulting phase variation of something approaching 180” as the period varies through the resonance is an inherent property of a resonance. The reason for the similarity between Fig. 3 and
686 0 -100" -
675 II
I
latitude
670 II
665 I
-200"
-260"
Period,
set
FIG. 3. THE STEADY STATE AMPLITUDE AND PHASE PROFILES WITH
PERIOD
(LATITUDE)
ARE SHOWN
AS SOLID
LINES.
The dashed lines pertain to the transient which was determined as a function of A. The input parameters were p = 3,X,, = 6.5 1 0-l and p(O) = 9.5( 10m20) kg m-j. Also p(0) was assumed constant with L.
1165
A new theoretical approach to the modelling of toroidal Pc5 pulsations 1740 6
3046
3916.L
0 695
-2
1;
p-3
= C;=651
mho
1
320
6700” 6725” 67 50” 67 75”
p,,,=9
5
( IO-'")kg mm3
T :
402 s
-t
13001
Time.
set
670
, 006 E Ef ; 0
I 342 0
290
-0322
-1650 4334
FIG.~.
8663
THEPULSATIONRESPONSEASAFUNCTIONOFI.ATITUDEWITHTHEI.ATITUDESSHOWNATTHECENTER
The resonant latitude is 67”. The vertical dashed lines denote equal times and thus show the phase dependence of the response on latitude.
of the resonance. A similar correspondence holds for points on our curve for periods greater than 402 s. In other words, our response vs period curve at fixed latitude is a kind of dual of the response vs latitude curve for fixed frequency. As such, it is clear that the phase variation with latitude shown by Walker is as inherent to the resonance as that observed by varying the frequency through a resonance. poleward
Response
us latitude at fixed frequency
The model above can easily be altered to correspond to latitudes in the neighbourhood of 67”. This is possible because, to’ a very high degree of accuracy, the change in the time of flight curves required by the latitude variations is only through a multiplicative factor; the time axis is either stretched or contracted while the curve is unaltered. Thus the line segment boundaries, &, remain fixed with the exception, obviously, of the final one, ON, the new geomagnetic latitude. As a consequence, all the 2’s except 2: are altered only by a common multiplicative constant which accounts for (a) the change in the equatorial value of the magnetic field, (b) the change, if any, in the equatorial plasma mass density and (c) the change
in @.,(@,).However, the change in &., means that 2:“” requires a separate correction. Consequently, all the reflection coefficients except K:” and KY” are unaltered; only these two reflection coefficients and At, the segment propagation time, need be changed. The required change in At is straightforward but that for 2:/s is a little more complicated. The upper limit of the integrals in equation (9) is changed. Fortunately it is relatively easy to express ZK’“(&) as f(67”- 0N)zE’S(67”) where the function f was determined by matching to calculated values of the mean impedance. This done, the response at any nearby latitude can be calculated by first altering the required parameters and then by proceeding as described above. This has been done for the input data in the preceding section and the result appears as the latitude axis in the center of Fig. 3. It is consistent with the duality discussion above. We also note that the transient profiles in Fig. 3 are spatial, i.e. relative to latitude. The transient response
Figure 4 shows the response at the ionosphere for latitudes 67.0”, 67.25”, 67.50” and 67.75”, as indicated at the center of the figure, for the same
1166
D. R. MCDIARMID
parameters as used for Fig. 3. The input circuit consisted of a current generator feeding the line directly (2, = 0) with an output of sin ot. The character of the transient response varies as the observation point is moved poleward from the latitude of the resonance. Similar behaviour is obtained if the observation point is shifted equatorward from the resonance. First of all a delay, corresponding to the time required for the disturbance to propagate down the field lines, is observed in every response. Not surprisingly, the response grows monotonically to the steady state value at the resonant latitude (0, = 67”). During this time there is a build-up of stored energy in the resonant flux tubes. As we depart from this latitude, the response begins to show overshoot. Thus the amplitude vs latitude profile varies during the transient phase as shown by the dashed lines in Fig. 3 which are for times of 667 and 1500s. These profiles differ significantly from the solid line. Hence care must be taken in the interpretation of measured profiles, such as determined from radar observations (Walker et al., 1979) for instance, for events which are either short lived or show significant amplitude modulation. It is intended to investigate this problem in future using more suitable generator functions and input impedances. Both of these parameters contribute to the character of the transient response. Figure 4 also shows the phase variation with latitude of the steady state solution as indicated by the vertical dashed lines which denote a common time. A phase variation approaching 90” is clearly evident. The other 90” of phase shift occurs equatorward of 67”. Response as a function of 8, and p The C,‘s corresponding to the three values of k,/k, shown in Walker’s (1980) Fig. 8 were determined using the appropriate expressions in Walker (1979). They are C,” = 2,’ = 6.51, 4.65 and 3.62 R-‘. Response curves were computed for each using p(O) = 9.5(10m2”) kg mm3 and p equal each of 3 and 4. The period, T,, and the latitudinal distance between half amplitude points, Al\, for each solution are shown in Table 1 along with Ah’s scaled from Walker’s Fig. 8. Comparison with Walker’s values is good for C, = 6.51 a-’ but the results diverge as 2, is lowered. This might in part be a consequence of the coupling to a poloidal type field which is inherent in his work. If this coupling leads to an increase in the ratio of stored energy to power dissipation, the discrepancy follows. Another, and possibly more important
TABLET. THERESONANTPER~ODANDLATITLJDE HALF-WIDTH AS A FUNCTIONOF&, ANDP 2,
p=3
R-’
Walker
6.51
1.03 1.42 1.82
4.65 3.62
The parameter culations.
T,(S)
Ah(‘)
1.01 1.48 1.98
p=4
p=3
1.14 1.69 2.32
402 403 407
p=4 409 411 416
wi was fixed for the cal-
source of the divergence is a slight difference in the stored energies inherent in the two theories resulting from the two excitation schemes. As noted above, the power flow patterns differ and it thus is natural to expect the stored energy distribution to differ as well. In so far as this is the cause of the divergence, our results are probably more realistic since our power flow pattern is more realistic. A third and probably the most important source of the divergence is the fact that in Walker’s calculations, the ratio of latitudinal to longitudinal scale sizes is > 0.1 for XP = 4.65 and 3.62R-‘. Since our impedance expression is only valid for ratios I 0.1, our .?s are slightly in error for both these C,‘s with the larger error associated with 3.62 0-l. This corresponds well with the increase in the divergence of our results as C,, is decreased. However, the other two sources of discrepancy may also contribute. (Note that the ratio (b/w), on p. 1164 is somewhat lower than that of Walker as a result of a computation error. However, as noted there, the ratio, if I 0.1, only affects the excitation. It will be seen below that, as such, the steady state solution is unaltered.) The consequence of changing p from 3 to 4, the other parameters remaining fixed, is also shown in Fig. 5 as well as in Table 1. This change is to a , 60
-1 t
AA
Frc.5.
THEPULSATION
LATITUDEHALF-WIDTHASA
FUNCTIONOFP
ANDOF&,.
1167
A new theoretical approach to the modelling of toroidal PCS pulsations
more uniform transmission line and, as a consequence, the line is more heavily loaded. This is because (K,( is decreased as a result of a decrease in the Z’s, particularly at the ionosphere ends of the field line. Also the ratio of the resonant period to the time of flight is changed as previously noted by Warner and Orr (1979). The time of flight periods are 300 s for p = 3 and 338 s for p = 4; they are thus approx. 25 and 17% less, respectively, than the computed periods. The interesting thing to note is that for fixed equatorial plasma mass density, the resonant period is almost unchanged for the above change in p.
2,”
z C,?
Unequal Northern and Southern hemisphere &,‘s, which are the norm in reality, present no problem in our formulation. During the investigation of the nature of such solutions, it was observed that &,” = 6.51 R-’ and 2,’ = 3.62 R-’ yield the same shape of steady state response vs frequency (latitude) curve as did C,” = Xi,’ = 4.65 0-l. In other words, 4.65 R-’ can be used as an equivalent conductivity in theories which require 2,” = C,“. A general expression for calculating the equivalent conductivity, Z,“, is
This is the same as requiring the series addition of the two mean ionosphere termination resistances to equal the series addition of the actual termination resistances. Such is the requirement resulting from specifying that the product of all the scattering coefficients be invariant. A cursory examination of this criterion via a lumped circuit analysis indicates that it is only valid when 2, ti r, which is virtually always the case in any real event (Wallis and Budzinski, 1981).
Consequence
of generator impedance
While it may not represent the most appropriate description of the excitation process, a generator circuit consisting of a resistance, an inductance, and a voltage generator, all in series, was tested. Two sets of calculations were conducted in which r, 6 ZL~’ and wL, = 0.3Zf“/“. In both cases the shape of the steady state response vs frequency curves were the same as those obtained using the simple current generator. However, not surprisingly, the transient solutions varied.
5. DISCUSSIONAND CONCLUSIONS
The question may naturally arise as to the extent of the error which occurs when one matches the theory to observed results with an incorrect assumption of the plasma mass distribution along the field line (i.e. incorrect choice of p). For this discussion let p = 4, C, = 6.51 R-’ and p(O) = 9.5 (10~“) kgmm3 be the correct parameters. We then match the theory to observation after incorrectly assuming p = 3 by varying Zp and p(0) until the theory yields the observed values of rr and AA. The difference between these values and those above give an indication of the possible error in using this matching to measure C, and p(O). A crude estimate of the error may be determined as follows using the data in Table 1 and Fig. 5. To increase AA from 1.01” to 1.14” when p = 3 requires a change in Xp from 6.51 to 5.94 R-l. This leads to no significant change in TV.Now to increase TVfrom 402 to 409 s requires an increase in p(0) of about 3.5%. Such a change reduces the Z’s by about 1.7%. With respect to the above change in C, of about 8.8%, the subsequent change in 2, does not significantly further alter Kt. Thus further iteration of the above process will probably converge rapidly and the resultant error in 2, and p(0) are probably of the order of 10% and 5% respectively. These are acceptable if they are indicative of the errors obtained using other equally appropriate plasma distributions. Allan and Knox (1979b) pointed out that it is theoretically possible to have a h/4 mode if there is very low Pedersen conductivity at one end of the resonant field line. To be specific, r, must be greater than ZN. In our two cases (p = 3 and 4), this implies z
P
< 0.60(10? q(p(o)) 1 1.2 (l@? d\/(P(W
P = 3 P = 4
at f&= 67”. If p(0) = 9.5 (lo-*“) km rnm3, we obtain Z, < 0.18 and 0.37 C’ respectively. It is very unlikely that such a condition will be satisfied at aurora1 latitudes (Wallis and Budzinski, 1981). Increasing these limits on Z, by an order of magnitude requires either an increase in 0, to about 75” or of p(0) by two orders of magnitude. In the former case, the times of flight are increased by a factor of approx. 25 if one assumes the field lines to be still dipolar. In the latter case, the times of flight are increased by an order of magnitude. Thus if the conditions are altered such that a A/4 mode approaches the realm of possibility, particularly without severe damping, the period is then too
1168
D. R. MCDIARMID
large for the pulsation to be called PCS. Thus if our parameter values are realistic, it is unlikely that a A/4 Pc5 mode will occur. Before discussing the particular accomplishments of our theory which were obtained above, we note again that each individual solution we have produced is valid for its latitude only. Also, since (w/b), cancels out of all the K’S, it is a factor only in the excitation through Z,, in equations (13). The entire or synthesized solution is obtained through the calculation of a sequence of individual solutions. This has been done above in Fig. 3 for the ionosphere and the same could have been done for any point in the magnetosphere. Clearly, the end result cannot be faulted because it is not spatially distributed as a consequence of being generated by a “hard wire” circuit. The synthesized solution is spatially distributed. In fact, we see that for each solution the model probes the pulsation response at the latitude which it describes. The results presented in this paper show that our longitudinally limited toroidal mode model predicts behaviour consistent with that predicted by Walker’s model. It can also relatively easily accomodate (a) changes in the plasma density distribution along the field line as represented by p, (b) the insertion of a generator circuit which permits, in principle, a proper description of the excitation process, (c) the condition C,“# &,‘, and (d) the alteration of the magnetic field to non-dipolar form. The transient response is an integral part of the solution and the importance of understanding its contribution is now clear since latitudinal variation of the pulsation differs from the steady state profile. It is hoped to obtain further insights into the consequences of this behaviour in the course of future work since not all pulsations give evidence of steady state behaviour. For the parameters chosen in our calculations, we have seen that the generator impedance did not alter the shape of the steady state latitudinal response profile. However, it is not known if these
parameters are realistic; thus only a tentative conclusion is possible at best. Further work would be useful here. In the discussion on the latitudinal variation of the pulsation response it was shown that because of the duality with the frequency response, the phase shift of - 180” across the resonance region is as inherent as that which occurs when the frequency of excitation of a resonant structure is varied through its resonance. REFERENCES Allan, W. and Knox, F. B. (1979a). A dipole field model for axisymmetric AlfvCn waves with finite ionosphere conductivities. Planet. Space Sci. 27, 79. Allan, W. and Knox, F. B. (1979b). The effect of finite ionosphere conductivities on axisymmetric toroidal Alfv& wave resonances. Planet. Space Sci. 27,939. Mallinckrodt. A. J. and Carlson. C. W. (1978). Relations between transverse electric fields and field-aligned currents. J. geophys. Res. 83, 1426. McDiarmid. D. R. (1980a). Toward some new directions in PCS pulsation theory-l. Consequences of _ ionosphere Hall current. Planet. Space Sci. 28, 713. McDiarmid, D. R. (1980b). Toward some new directions in PCS pulsation theory-2. Extension of the analysis to two hemispheres and determination of the model’s uolarization characteristics. Planet. Space Sci. 28. 727. Newton, R. S., Southwood, D. J. and Hughes, W. J. (1978). Damping of geomagnetic pulsations by the ionosphere. Planet. Space Sci. 26,201. Walker, A. D. M. (1979). Modelling of Pc5 pulsation structure in the Magnetosphere-i. Toroidat motion and associated fields. Research Renort R1/79. Dent. of Physics, University of Natal, Durban. Walker, A. D. M. (1980). Modelling of PCS pulsation structure in the magnetosphere. Planet. Space Sci. 28, 213.
Walker, A. D. M., Greenwald, R. A., Stuart, W. F. and Green, C. A. (1979). STARE aurora1 radar observations of Pc5 geomagnetic pulsations. J. geophys. Res. 84, 3373. Wallis, D. D. and Budzinski, E. E. (1981). Empirical models of height integrated conductivities. J. geophys. Res. 86, 125. Warner, M. R. and Orr, D. (1979). Time of flight calculations for high latitude geomagnetic pulsations. Planet. Space Sci. 27, 679.