Episodic cosmic-ray modulation in the heliosphere

0273—1177/89 $0.00 + .50 Copyright © 1989 COSPAR

Adv. Space Res. Vol. 9. No. 4. pp. (4)225—(4)228, 1989

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EPISODIC COSMIC-RAY MODULATION IN THE HELIOSPHERE J. A. Le Roux and M. S. Potgieter Department of Physics, Potchefstroom University, Potchefstroom, South Africa

ABSTRACT A newly developed two—dimensional, time—dependent modulation model is applied to the simula-

tion of propagating Forbush decreases (Fd’s) in the heliosphere. Compared to a one—dimensional model the recovery time of a Fd is found to be significant less with this model. The magnitude as a function of radial distance is determined by (1/Krr — Kj/rKrr). In general, results are in reasonable agreement with observations. With drifts included the recovery rate differs significantly compared with the non—drift model, and depends strongly on the polarity of the solar magnetic field. INTRODUCTION Forbush decreases which are transient changes, in the galactic cosmic—ray intensity (associated with impulsive solar processes), characterized by a fast decrease within - 1 day followed by a more gradual, nearly exponential recovery over a few days, have been observed continually with neutron monitors since the ~95Q~5• With the launching of several spacecraft during the 1970’s a space network of cosmic—ray detectors has been established. This has made it possible to observe Forbush decreases (Fd’s) as far distant as 40 AU, enabling the analysis of the propagation properties of Fd’s. A detailed picture of the characteristics of Fd’s from 1—30 AU were presented by Lockwood etal. /1/ and Webber etal. /2/. For the period 1972—1984 they found the recovery time of about 30 Fd’s at Earth to vary between 2—10 days, averaging 5 days. No significant difference in the average recovery time was observed before and after the IMF polarity reversal in 1980. They also found the recovery time 20 times longer at 30 AU than at 1 AU, while the magnitude of the decrease at 30 AU was a factor of 2 smaller than at Earth. The observance of longer recovery times for individual Fd’s at larger radial distances was disputed by Burlaga et al. /3/. However, despite some disagreement about the actual features of propagating Fd’s, the work by Lockwood and Webber can be used as a guide in the numerical modelling thereof. To gain insight into what is actually determining the features of propagating Fd’swe have developed a two—dimensional (2D), axially—symmetric, time—dependent modulation model. The characteristics of this model are discussed and compared with the mentioned observations. We did not, however, try to do exact fits to experimental data. We also present preliminary results of a 2D drift model. THE MODEL We solved the standard cosmic—ray transport equation numerically, in two spatial dimensions, using the ADI (alternating direction implicit) method /4/, which was adapted to include time—dependence. This equation is Krr

+

~

~

00

+

Ii

2f

1 K

—~

—i

+

~ —

2 (r Krr)

+

rsin8

—

1 ~ (sine ‘~er~”J — 2V)

[-51 r

~

1

ar (rKr0) + r 2 sinO 1 ~~0 (sin Kee)J ~~8

+

~3r2 1~ ar (r

.J~L, ~2nP

where f is the omnidirectional distribution function with respect to radial distance r, polar angle 9, rigidity P and time t — the differential intensity is related to f by iT p2~ — ~ is the radially directed, latitude de~endentsolar wind velocity. The diffusion coefficients are given by Krr K 2i~+Kjsin ~, K 11cos 00 K1 and Kr9 = KBr = KTSifl~, (4)225

J. A. Le Roux and M. S. Potgieter

(4)226

where K

11 and Kj, respectively, are the diffusion coefficients parallel and perpendicular to the IMP, KT = 0P133 represents the effect of gradient and curvature drift in the large 21 cm2/s, corresponds to high solar activity scale IMF given by B; and p is the angle between the which radial and IMF direction. We assumed since= many of the Fd’s observed in the heliosphere were seen during that time /21. K11 K0PP(0.3/B) = 20K1, with K0deep = 3.4x10 To simulate a Fd we assumed that it was caused by a propagating region of enhanced scattering and diminished drift. Therefore, K 11 , Kj and K.r were disturbed simultaneously by a factor of• 10 decrease in the equatorial plane. The disturbance was assumed 1 AU wide in the radial direction and extended to 15°above the equatorial plane without any spatial evolution. We let the disturbance propagate from the inner boundary, at 0.1 AU, to the Outer modulation boundary, at 50 AU, with the solar wind speed of 400 km/s. RESULTS AND DISCUSSION In Figures 1(a) and (b) we show the 1 GeV proton intensity—time profiles, at Earth and at 20 AU, obtained with the 2D, non—drift model. There is a striking similarity between the simulated Fd of 9%, at 1AU, with its small precursor, fast decrease and almost exponential reset, and the observations. The recovery time to — the time for a 63% recovery — of 4 days compare well with the observations of Lockwood eta].. Ill. This gives credibility to the argument that Fd’s are probably caused by regions of enhanced scattering /5/. The intensity—time profile at 20 AU illustrates how the recovery time increased to 33 days, while the magnitude decreased to 4%. This supports the view of Webber eta].. /2/ of how a Fd evolves with radial distance. The presence of the precursor at 20 AU, although too large, is in agreement with observations reported by McDonald eta].. /6/. 3.7

8.7 1 GeV protons

I 6eV protons

‘F~~ ~

~

~

3.3.

0

82~U

5

10.

15

Time (days)

20

25

70

80

90

100

110

120

130

Time (days)

Fig. 1.(a) Intensity—time profile of a simulated Forbush decrease (Ed) at 1 AU. 2ssr~GeV~. (b) Same The as Time zero is the moment the disturbance starts out from the inner boundary. in (a) at time 20 AU. recovery to = 4 days. Intensity in units of m The magnitude of a propagating Fd, as a function of radial distance, obtained with a ID model (‘~i = KT = 0) and a 20 model without drifts (KT = 0), is displayed in Figure 2 with the data of Webber eta].. /2/. The data and the model results are normalized at 1 AU. Evidently, the radial dependence predicted by both models, for the particular parameter set, differ from that of the observations. The large difference between the model predictions can be understood by a simple linear combination of the coefficients associated with parallel and transverse diffusion. For instance, the magnitude of a Fd with the ID model is proportional to (l/Krr), in agreement with the analytical work of Chih and Lee /71. With the 2D non—drift model it is proportional to (l/Krr — Ki/rKrr), which implies that since particles can also diffuse perpendicular to the IMP direction the Ed is smaller than in the 1D model. The steep decrease in the magnitude of the Fd at small radial distances is caused by the sharp increase of the given expression in that region, and is primarily the consequence of assuming Kj ~ K 11 ~ (1/B). We find that a di~fferent spatial dependence for K11 and Kj (but not necessarilS’ more realistic) disposes of this significant decrease in th& inner heliosphere and gives better agreement with the data. We also find that the extent and the spatial evolution of the disturbance have a major effect on the magnitude of the Fd.

Episodic Cosmic-Ray Modulation

_1.60

(4)227

50

.___.

,

—

U

zI.40

/÷2.3

50~

~::~ ~

t: >.

~1.20.

1.00.,

0 40

0

5

10

15

20

/

..~

.1.~—----

‘—-

—

10

-----——

25

30

~

10

~

0

5

10

15

20

25

30

Radial Distance (Au)

Radial Distance (AU) Fig. 2. Magnitude of a Pd w.r.t. its value at 1 AU. Results of a ID and 2D model are shown with data from Webber eta].. /2/.

Fig. 3. Recovery time of a propagating Fd obtained with ID and 2D non—drift models. Data are from Webber etal:/2/.

In Figure 3 the recovery time of a Fd, as a function of radial distanc8, is shown for the ID and 2D models compared with data /2/. The recovery time is obviously very large for the ID model, but can be improved by assuming a decaying disturbance. For the non—drift model the reset time agrees fairly well with the data, i.e., without a decaying disturbance. The difference between the two models can be understood by considering the way particles stream in the heliosphere. Differential stream calculations show that in the case of the 20 model without drifts the particles Stream effectively downward to the equatorial plane from higher latitudes, particularly in the radial direction. Therefore, the recovery of the intensity behind the disturbance in the 2D model is quicker in the inner heliosphere than further away from the Sun, and much quicker compared to the simpler ID model. 1.30

1.14 1 GeV protons

~dIe

1 6eV protons

IE~~AA~~

Time (days) Fig. 4(a) Same as Fig. 1(a) with full drift; IMF polarity reversed; qA < 0.

Time (days) qA

>

0.

(b)

Same as in (a) with

In Figures 4(a) and (b) we show Pd’s for 1 GeV protons at Earth when full drifts are included in the model. Figure 4(a) depicts the situation with the northern—hemispheric IMP pointing away from the Sun (qA > 0) and Figure 4(b) the situation with the polarity reversed (qA < 0). The striking difference between the two Pd’s is the reset time of 3 days with qA > 0, and 11 days for qA < 0 (4 days for the non—drift model). However, this marked difference seems to disagree with observations /2/. The magnitude is slightly larger at 13%, with qA > 0, against 11% with qA < 0. The inclusion of drifts results into an intensity level associated with low solar activity rather than high solar activity. The difference in

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3. A. Le Roux and M. S. Potgieter

reset times is clearly understood considering the direction particles drift in the helio— sphere. With qA > 0 particles drift equatorward from the poles and enhance the filling in of the depression in the intensity, while with qA < 0, when particles drift poleward, the process is retarded and leads to slower recovery. Although we use a less complex propagating disturbance, our drift results agree with those of Kadokura and Nishida /8/. We note, however, that no significant difference in reset times before and after the reversal of the IMP polarity was calculated with the Jokipii model used by Webber eta].. /2/. The apparent reason is that their disturbance decays exponentially with radial distance, masking the polarity dependence. Unfortunately such a decay diminished the chance of studying Fd’s at large radial distances for preliminary results show that the magnitude of a Pd, as a function of radial distance, decreases more steeply with qA > 0, but less steeply with qA < 0 than with the non—drift model. Compared with the non—drift model the recovery rate of a Fd is faster for qA > 0, but slower for qA < 0. This effect enhances with increasing radial distance. The strong polarity dependence of the reset times may point to a too large drift effect in full drift models. Decreasing drifts give reset times less dependent on IMF polarity and a radial dependence of the magnitude of the Fd closer to that of the non—drift model. CONCLUSIONS Our newly developed 2D, time—dependent modulation model which simulates a propagating Pd in the heliosphere gives results in fair agreement with observations. The magnitude of the Fd, as a function of radial distance, depends on (l/Krr — Kj/rKrr). The recovery rate of the Fd is much faster than with a 10 model because of the downward streaming from higher latitudes caused by transverse diffusion. With the inclusion of drifts the recovery time of the Fd differs significantly depending on the polarity of the IMF. On the whole the drift results point to a too large drift effect when compared with observations /see also 9,10/. This drift model will be studied further. REFERENCES 1.

J.A. Lockwood, W.R. Webber, and J.R. Jokipii, Characteristic Recovery Times of Forbush— Type Decreases in the Cosmic Radiation. 1. Observations at Earth at Different Energies, .3. Geophys. Res. 91, 2851 (1986) 2. W.R. Webber, .I.A. Lockwood, and .J.R. Jokipii, Characteristic Recovery Times of Forbush— Type Decreases in the Cosmic Radiation. 1. Observations at Different Heliocentric Radial Distances, J. Geophys. Res. 91, 4103 (1986) 3. L.F. Burlaga, P.S. McDonald, M.L. Goldstein, and A.J. Lazarus, Cosmic Ray Variations and Turbulent Plow Systems: 0.3—1.0 AU; 1977—1980, 3. Geophys, Res. 90, 12027 (1985) 4. M.S. Potgieter, and H. Moraal, A Drift Model for the Modulation of Galactic Cosmic Rays, Ap. .3. 294, 425 (1985) 5. G. Zhang and L.F. Burlaga, Magnetic Clouds, Geomagnetic Disturbances, and Cosmic Ray Decreases, 3. Geophys. Res. 93, 2511 (1988) 6. P.8. McDonald, J.H. Trainor, and W.R. Webber, Pioneer and Voyager Observations of For— bush Decreases between 6 and 24 AU, Proc. 17th Tnt. Conf. Cos. Rays, 7, 147 (1981) 7. P.P. Chih and M.A. Lee, A Perturbation Approach to Cosmic Ray Transients in Interplanetary Space, J. Geophys. Res. 91, 2903 (1986) 8. A. Kadokura and A. Nishida, Two—Dimensional Numerical Modelling of the Cosmic Ray Storm, J. Geophys. Res. 91, 13 (1986) 9. M.S. Potgieter, J.A. Le Roux, and R.A. Burger, Interplanetary cosmic—ray radial gradients with steady—state modulation models, J. Geophys. Res. 1989, in press 10. M.S. Potgieter, R.A. Burger, and J.A. Le Roux, The modulation of cosmic—ray electrons in drift models, Proc. 20th Int. Conf. Cos. Rays, 3, 295 (1987)