S. Mühlbachler,^{1} C. J. Farrugia,^{2} H. K. Biernat,^{3} D. F. Vogl,^{3} V. S. Semenov,^{4} P. Aber,^{5} J. M. Quinn,^{2} N. V. Erkaev,^{6} K. W. Ogilvie,^{7} R. P. Lepping,^{7} S. Kokubun,^{8} and T. Mukai^{9}
^{1}Space Research Institute, Austrian Academy of Sciences,
Graz, Austria
^{2}Institute of the Study of Earth, Oceans, and
Space, University of New Hampshire, Durham, New Hampshire, USA
^{3}Space
Research Institute, Austrian Academy of Sciences,
Graz, Austria
^{4}Institute of Physics,
St. Petersburg State
University, St. Petersburg, Russia
^{5}Faculty of Arts and Sciences, Harvard University, Cambridge,
Massachusets, USA
^{6}Institute of
Computational Modelling, Russian Academy of Sciences,
Krasnojarsk, Russia
^{7}NASA
Goddard Space Flight Center,
Greenbelt, Maryland, USA
^{8}Solar-Terrestrial
Environment Laboratory, Nagoya University, Nagoya, Japan
^{9}Institute of Space and
Astronautical Science, Kanagawa, Japan
Received 3 May 2001, published online 24 August 2001
Figure 1 |
Magnetic clouds are very useful for investigating the interaction between the solar wind and the magnetosphere because of their special properties, which allow them to couple energy and momentum to the magnetosphere, thus driving storms and substorms. Interplanetary magnetic clouds are characterized by (1) strong magnetic field strengths relative to ambient values, (2) low proton b and proton temperature, and (3) large and smooth rotation of magnetic field direction [Burlaga et al., 1981; Lepping et al., 1990].
Their passage at Earth typically lasts about 1-2 days, and their dimension at AU is ~0.25 AU. Furthermore, magnetic clouds are often a dramatic source of long-lasting, strong, negative B_{z} of interplanetary magnetic field, which is an optimum condition for reconnection at the dayside magnetopause. Ahead of fast magnetic clouds, interplanetary shocks are often observed [Burlaga, 1995].
Figure 2 |
The key parameter data we examine are as follows. Plasma data are from the SWE (Wind) and from the LEP instruments (Geotail). Magnetic field data are obtained from MFI and MGF instruments [e.g., Kokubun et al. 1992; Lepping et al. 1995; Mukai et al., 1992; Ogilvie et al., 1995].
Figure 3 |
The magnetic cloud arrived at Wind at approximately 1900 UT on 18 October 1995, preceded by an interplanetary shock at ~1040 UT. The magnetic field turned abruptly and strongly southward when Wind entered the magnetic cloud, and it rotated gradually to a northward orientation during the next ~24 hours. The magnetic field strength in the cloud was large (20-30 nT) and relatively constant. Note the relatively constant bulk speed in the cloud. The magnetosonic Mach number in the cloud is very low (between 2 and 4), which is ideal to check the position of the bow shock because this is precisely the range where in MHD theories the standoff distance starts to increase.
Solar wind dynamic pressure is high in the cloud's sheath and very low inside the cloud with a gradual increase from ~1 nPa up to ~10 nPa. This increase is mainly due to the interaction with a faster trailing stream [Farrugia et al., 1998].
Most of the time p_{dyn} is below the historical average of 2.2 nPa. The interplanetary parameters provide an ideal situation to examine the bow shock position as a function of low magnetosonic Mach number and under a wide range of dynamic pressure from 0.2
dyn<10 nPa in the cloud.
Figure 4 |
In a statistical analysis, Farris et al. [1991] studied 351 independent bow shock crossings and 233 independent magnetopause crossings made by the ISEE 1 spacecraft from 1977 to 1980 to determine the average positions and shapes of the bow shock and the magnetopause. They represented the bow shock as a paraboloid and obtained statistically X=a_{s}-b_{s}(Y^{2}+Z^{2}) and a_{s}=13.7 0.2R_{E} and b_{s}=0.0223 0.0003R_{E}^{-1} for the subsolar standoff distance and the shape parameters, respectively.
Specifically for low Alfvén Mach numbers, Farrugia et al. [1995] derived a quasi-linear connection between the thickness of the magnetosheath D_{ms} normalized to the subsolar radius of the magnetopause a_{mp} ( D_{ms} (a_{s}-a_{mp})/a_{ mp} ) and the inverse square of the Alfvén Mach number, 1/M_{A}^{2}, as it is in our study.
Therefore, ignoring the motion of the bow shock, we fit the crossings to the Farris et al. [1991] formula to a functional form which brings out the 1/M_{A}^{2} dependence explicitly. Instead of two parameters, a_{s} and b_{s}, in the Farris formula, a four-parameter formula is employed:
(1) |
Least squares fitting yields a_{1}=13.37, a_{2}=12.97, a_{3}=0.005, and a_{4}=0.036.
Figure 5 |
We employ two different methods of calculating the bow shock normals: (1) from the shape of the Farris et al. bow shock and (2) from the coplanarity theorem [after Abraham-Shrauner and Yun, 1976].
For method (1) we know the position vector r of the boundary
(2) |
Thus the shock normal vector at any point at the curve can be derived from vector analysis
(3) |
For the shock normal derived from the coplanarity theorem we compute upstream and downstream values of the magnetic field and obtain
(4) |
where subscripts 1 and 2 refer to upstream and downstream values
of
Figure 6 |
Figure 7 |
In Figure 7 we plot for each interval the angle l between the derived shock normals and the subsolar line, also for each method. The observed normal directions have large scatter, which however decreases in groups 2 and 3, i.e., as Geotail approaches the subsolar line. The large scatter of the coplanarity normals in group 1 (at the flanks of the bow shock) may be due to localized disturbances on the shock and hint to a more fluttery shock shape at the flanks. The last group, where the scatter is small still has Dl= 4.6^{o}. This may indicate that the actual bow shock shape departs from an axisymmetrical shape, what may be due to the large B_{y} component of the cloud field at this time.
The angles q between the shock normals and the IMF B_{n} at each bow shock crossing are all q>45^{o}, and thus all shock crossings are perpendicular shocks.
Now we use the coplanarity normals to derive the bow shock velocity after Burgess [1995]
(5) |
Figure 8 |
The magnetosonic Mach number is very low at times of shock in and out motions, between 1.2 and 3. The trend for large sunward displacement for decreasing M_{ms} is evident here. It has been shown in previous studies [e.g., Cairns and Grabbe, 1994; Cairns and Lyon, 1995, 1996; Cairns et al., 1995; Fairfield, 1971; Farris et al., 1991; Formisano et al., 1971; Grabbe, 1997; Peredo et al., 1995] that at very low Alfvén and magnetosonic Mach numbers the subsolar distance could increase up to 30 or more R_{E}. Note, however, that we never observe a static bow shock but one moving either earthward or sunward.
We now discuss the dynamic pressure. For an increasing dynamic pressure, the magnetopause standoff distance moves inward, as does the bow shock. We assume here that this is the primary effect of dynamic pressure. We shall therefore not study changes of the shape of the magnetosphere (blunt to more pointed), which rapid and large dynamic pressure changes may be expected to occasion; that is, we shall consider in first approximation only changes in dynamic pressure, which are slow, i.e., which affect the whole magnetosphere. The crossings are obviously correlated with changes in dynamic pressure. When the dynamic pressure is low and <1 nPa, there are no crossings at all; that is, the M_{ms} and the p_{dyn} effects on the bow shock position act in the same direction.
Figure 9 |
Much work has been done on the bow shock standoff distance as a function of interplanetary parameters [see e.g. Grabbe and Cairns, 1995, and references therein]. In recent years there is renewed interest on this issue for cases when the Alfvén Mach number is low [Cairns and Grabbe, 1994; Cairns et al., 1995; Russell and Petrinec, 1996a, 1996b]. In their paper, Grabbe and Cairns [1995] present an analytical MHD formula for the density jump r_{2}/r_{1}=X
Because of the perturbation technique used to derive this formula, it is valid only for values of q60^{o}. In our case, where the average value of q75^{o}, one has to take a simplified solution also presented by Grabbe and Cairns [1995]
(6) |
with
(7) |
An empirical relation between the bow shock standoff distance ( a_{s} ), the magnetopause nose ( a_{mp} ), and X takes the following form [Cairns and Lyon, 1995; Farris and Russell, 1994; Seiff, 1962; Spreiter et al., 1966]
(8) |
For the gas dynamic empirical relation found by Seiff [1962] and further developed by Spreiter et al. [1966], j=1 and k=1.1, where the value of k depends on the obstacle shape. In the model presented by Farris and Russell [1994] the value for k is modified at lower Mach numbers by k=1.1M_{ms}^{2}/(M_{ms}^{2}-1), while j stays at 1. In the model developed from MHD simulations by Cairns and Lyon [1995], j=0.4 and k=3.4 for quasi-perpendicular flows with M_{S}8 and M_{A}>1.5. These values are appropriate for our problem, and so we calculate the ratio
(9) |
using (6) and (7) for X.
Figure 10 |
Figure 11 |
Figure 11 shows six panels where the first one contains the predicted a_{s} from (9) (solid line) and the given position dependent on M_{A}, keeping the dynamic pressure at its average value for the first group of crossings. In the second panel we keep the Mach number at its average value for the group of crossings and check the effects of p_{dyn} through parameter a_{mp} in (9). The other four panels repeat this procedure for the other two groups of crossings. The most impressive thing which can be made out of this figure is that it seems that especially for the large upstream excursions of the bow shock, M_{A} influences the bow shock motion most. Of course, when looking at the solar wind parameters, this is an unexpected result, because of fairly constant values of B and the proportionality of p_{dyn} and M_{A} via the solar wind density and bulk speed ( M_{A}^{2}=m_{0}rv_{sw}^{2}/B^{2}=m_{0}p_{dyn}/B^{2} ). Two considerations have to be taken into account when analyzing this figure: (1) a_{s} is calculated in subsolar distance, and our crossings are not subsolar; (2) to derive the nose of the magnetopause, we have used the formula for pressure balance, which might not give the most realistic behavior of the magnetopause for this event. The very large, negative B_{z}
Figure 12 |
Figure 13 |
From the figure we can see that for the first period, where we have very negative B_{z}, the Cairns and Lyon formula and the Shue et al. formula fit quite well; and in the third period with positive B_{z}, the Farrugia et al. magnetopause leads to rather good agreement with Grabbe and Cairns.
1. We examined 26 repeated crossings of the bow shock on 18-19 October 1995, made by Geotail.
2. The period studied corresponded to an Earth passage of an interplanetary magnetic cloud.
3. We related these crossings to interplanetary parameters,
the solar wind dynamic pressure, and the solar wind Alfvén
and
magnetosonic Mach numbers. For the interval studied,
the ranges of these parameters were
1 dyn<10nPa,
respectively. Thus
we expect large sunward displacements of the bow shock.
4. Compared to the model bow shock of Farris et al., we find a net average sunward displacement of 1.85 R_{E} due to the low Alfvén Mach number.
5. We calculated the bow shock normals in two different ways and found that the coplanarity normals agree with the Farris et al. shape normals except near the flanks, where a wide scatter in the derived normals is observed.
6. All bow shock crossings were quasi-perpendicular, q_{av} 75^{o}.
7. Small density jumps at bow shock occurred in association with low bow shock speed (of the order of 20 km s ^{-1} ).
8. We examine a delay in the response time of the bow shock between M_{ms} and P_{dyn} changes at Geotail and the bow shock crossings. This delay was of the order of ~10-20 min.
9. Our results are in fair agreement with the simulations of Cairns and Lyon on the standoff bow shock position in relation to M_{ms}.
10. We compare the position of the magnetopause and bow shock as predicted by various models and offered reasons for discrepancies between them.
11. The drawing of any conclusions due to the extreme conditions of the interplanetary magnetic field should also have been part of bow shock observations in this special magnetic cloud event. As seen from the data plots, there was strong negative B_{z} for a long period then rotating to the northward direction, also a strong eastward component rotating to strong westward values. However, reconnection might occur, the magnetopause could be eroded, and asymmetries in the Earth magnetosphere could play a nonnegligible role. This we point out in Figure 13, comparing the a_{s}/a_{mp} values in the first panel when B_{z} was less than zero. In this panel the result of the Cairns and Lyon model agrees rather well with the Shue et al. formula, which takes into account the direction of B_{z}. The pressure balance results show much lower values. Vice versa in the third panel with B_{z}>0, the a_{s}/a_{mp} derived from the actual SW characteristics fit better with pressure balance than with Shue et al. Further work will be reported elsewhere [Farrugia et al., 2001].
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