JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A5, 1174, doi:10.1029/2002JA009569, 2003

Three-dimensional modeling of Earth’s bow shock: Shock shape as a function of Alfve´n Mach number J. F. Chapman and Iver H. Cairns School of Physics, University of Sydney, Sydney, New South Wales, Australia Received 3 July 2002; revised 8 January 2003; accepted 31 January 2003; published 8 May 2003.

[1] Earth’s bow shock changes its three-dimensional (3-D) location in response to

changes in the solar wind ram pressure Pram, Alfve´n Mach number MA, magnetic field orientation, fast mode Mach number Mms, and sonic Mach number MS. Using shock locations from global 3-D ideal MHD simulations [Cairns and Lyon, 1995], empirical models are derived for the 3-D shape and location of Earth’s bow shock in the near-Earth regime as a function of solar wind conditions. Multiple simulations with different MA and Pram but two orientations of the interplanetary magnetic field BIMF are analyzed: qIMF = 45 and 90 with respect to the solar wind direction vsw. Models for the (paraboloid) flaring parameter bs as a function of MA, azimuthal angle f, and qIMF = 45 or 90, show bs decreasing with MA, corresponding to the shock becoming blunter and less swept back (with a larger cross section), as expected. Together with models for the shock’s standoff distance (which increases with decreasing MA) the models for bs(MA, f) predict the shock’s 3-D location. Variations of bs with f represent eccentricities in the shock’s cross section (i.e., a departure from circularity), with the shock extending further perpendicular to vms (the fast mode speed) than parallel, as MA ! 1. An additional effect is observed in which the shock shape is ‘‘skewed’’ for qIMF = 45 (but not for qIMF = 90) in the plane containing BIMF and vsw. These latter two effects are consistent with the fast INDEX TERMS: 2154 mode velocity varying with propagation direction relative to BIMF. Interplanetary Physics: Planetary bow shocks; 7851 Space Plasma Physics: Shock waves; 2784 Magnetospheric Physics: Solar wind/magnetosphere interactions; 2199 Interplanetary Physics: General or miscellaneous; KEYWORDS: bow shock, Earth, Mach number, magnetohydrodynamics, magnetopause Citation: Chapman, J. F., and I. H. Cairns, Three-dimensional modeling of Earth’s bow shock: Shock shape as a function of Alfve´n Mach number, J. Geophys. Res., 108(A5), 1174, doi:10.1029/2002JA009569, 2003.

1. Introduction [2] Earth’s bow shock results from the interaction of the supersonic, superAlfe´nic solar wind with Earth’s magnetopause: fast mode magnetohydrodynamic (MHD) waves created near the magnetopause obstacle travel back upstream, combining and steepening to form the shock. The position and shape of Earth’s bow shock has been studied for over 40 years, since early predictions in the 1960s [Kellogg, 1962] and its discovery in 1964 [Ness et al., 1964]. It has long been predicted and observed that the three dimensional (3-D) location and shape of the shock depend upon the prevailing solar wind conditions and the size and shape of the magnetopause obstacle [e.g., Spreiter et al., 1966; Spreiter and Stahara, 1985; Fairfield, 1971; Farris et al., 1991; Russell and Zhang, 1992; Cairns and Lyon, 1995; Cairns et al., 1995; Peredo et al., 1995; Bennett et al., 1997; Verigin et al., 2000; Fairfield et al., 2001]. Solar wind 2 , conditions are described by the ram pressure Pram = rswvsw Copyright 2003 by the American Geophysical Union. 0148-0227/03/2002JA009569$09.00

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Alfve´n Mach number MA = vsw/vA, sonic Mach number MS = vsw/cS, qIMF (the angle between vsw and BIMF), and the fast magnetosonic Mach number Mms = vsw/vms (dependent upon MA, MS and qIMF). Here vsw, rsw, vA, cS, and BIMF denote the solar wind velocity, mass density, Alfve´n speed, sound speed and interplanetary magnetic field vector, respectively. [3] At the present time accurate theoretical and observational models do not exist for the bow shock’s 3-D location and shape as a function of the above solar wind parameters. This paper presents and analyzes 3-D MHD simulations [Cairns and Lyon, 1995] of solar wind flow onto an imposed magnetopause obstacle (that varies only with Pram) in order to study intrinsic effects on the shock’s 3-D shape and location, including its flaring away from the solar wind direction and cross-sectional asymmetries, as functions of MA, Mms, and Pram. This is done for two IMF orientations, the Parker spiral angle of qIMF = 45 and the case qIMF = 90 when vsw is perpendicular to BIMF, when MS effects are unimportant (MS = 7.6). Motivations include: (1) the development of theoretical models for the shock’s 3-D location and shape under these conditions that can be compared with, and used to organize and understand, observational

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CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

data, (2) identifying and understanding changes in the shock’s intrinsic shape, symmetries, and location with MA and qIMF, and (3) parameterizing changes in the shock’s 3-D shock shape that are intrinsic to the shock physics (e.g., Mach cone and compression effects), so that future studies can addresses changes in the shock’s 3-D location that are caused by changes in magnetopause location and shape. [4] The speed vms of fast mode waves that form the shock is dependent upon the local angle qBn between the shock normal (proportional to the wave vector k) and the upstream BIMF [e.g., Spreiter et al., 1966], with v2ms ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 2 c2S þ v2A 4c2S v2A cos2 qBn : cS þ v2A  2

ð1Þ

In the asymptotic limit the bow shock lies on the magnetosonic Mach cone with angular half width qms defined by [e.g., Michel, 1965] qms ¼ sin1

    vms 1 ¼ sin1 : vsw Mms

ð2Þ

Since qms for a given qIMF explicitly depends upon the Mach numbers MA and MS, a decrease in MA and/or MS, and/or Mms(= vsw/vms), corresponds to an increase in qms: this should cause the shock’s extent in the direction perpendicular to vsw to increase, with the shock becoming increasingly blunt or planar, as inferred observationally [e.g., Cairns et al., 1995; Bennett et al., 1997; Fairfield et al., 2001]. In addition, fast mode waves propagating perpendicular to BIMF (qBn = 90) travel faster then those traveling parallel to BIMF (qBn = 0), so that the shock should be found further from Earth in the direction perpendicular to BIMF than in the direction parallel to BIMF. Accordingly, qms? > qms > qmsk. [5] Effects due to vms depending on qBn have been both predicted and observed [e.g., Romanov, 1977; Romanov et al., 1978; Russell et al., 1988; Bennett et al., 1997]. The most common effect studied is the departure from circularity of the shock’s cross section; e.g., Peredo et al.’s [1995] analysis of 1392 crossings of Earth’s bow shock (for 60RE < X < 20RE) showed that the shock’s extent in the north-south (XZ in GSE) direction was typically 2 –7% larger than in east-west (XY in GSE) direction. [6] Previous models for the shock’s 3-D location and shape use paraboloidal [e.g., Filbert and Kellogg, 1979; Cairns et al., 1995] and hyperboloidal [e.g., Fairfield, 1971; Slavin et al., 1984; Bennett et al., 1997] shapes for the shock, with a hyperboloidal model being more desirable for large distances downstream of Earth since the shock should eventually asymptote to the fast mode Mach cone. Most models assume that the shock is symmetric about the tail axis (along vsw) with a circular cross section [e.g., Fairfield, 1971; Farris et al., 1991]. Paraboloid models [e.g., Filbert and Kellogg, 1979; Cairns et al., 1995] are viable sufficiently close to the Earth that the shock has not yet asymptoted to the Mach cone, with the form   x ¼ as  bs y2 þ z2 :

ð3Þ

Here vsw is parallel to the x axis, x, y, and z are equivalent to GSE coordinates, Earth is located at (0, 0, 0), as is the

shock’s standoff distance from Earth (measured along the x axis), and bs is a flaring parameter which describes the extent of the shock in the y  z plane. The terminator distance of the shock, measured in the plane x = 0, is then L ¼ ðas =bs Þ1=2 :

ð4Þ

Unfortunately, at present, models do not exist for bs, L, and the shock cross section as functions of MA, MS and qIMF. Moreover, the dependence of vms on qIMF suggest the cross section is not always symmetric about vsw, as shown in the simulations below. This paper addresses these deficiencies. [7] For a high MA solar wind the Rankine-Hugoniot conditions imply a large density jump across the subsolar shock, so that the transmitted plasma is highly compressed and fits into a much smaller magnetosheath volume. Thus for high MA the shock is found further earthward than for low MA [e.g., Spreiter et al., 1966; Fairfield, 1971]. Accordingly, as should decrease with increasing MA. Similarly, pressure balance dictates that 1/6 (e.g., equation (5)), as does the as varies with Pram subsolar distance to the magnetopause amp [e.g., Binsack and Vasyliunas, 1968; Fairfield, 1971]. Assuming that all distances within the shock-magnetopause system vary 1/6 . Spreiter et al.’s self-similarly with Pram, then L / Pram [1966] gas dynamic model for as is   ðg  1ÞM 2 þ 2 as / ðPram Þ 1 þ 1:1 ; ðg þ 1ÞM 2 16

ð5Þ

where M refers to the sonic Mach number MS and g is the adiabatic index. Cairns and Lyon’s [1995, 1996] empirical relationship for as, based upon 3-D global ideal MHD simulations for qIMF  45, is as ¼ 3:4 cðMA ; qIMF ; MS Þ þ 0:4; amp

ð6Þ

where c = rsw/rd is the MHD density jump at the subsolar bow shock, and rd is the mass density of the downstream (shocked) solar wind. Here c1 monotonically increases with MA. However, as decreases with increasing MA for field-aligned flow (qIMF = 0) [e.g., Spreiter and Rizzi, 1974; Cairns and Lyon, 1996]. Outside the scope of this paper, magnetic reconnection should cause amp and as to decrease for southward IMF (qIMF = 90) [e.g., Farris et al., 1991; Sibeck et al., 1991; Shue et al., 1998]. [8] The flaring parameter bs in (3) varies self-similarly with changes in as induced by variations in Pram [Slavin and Holzer, 1981]. Then, since as and L are proportional 1/6 1/6 , (4) implies that bs is proportional to Pram . Cairns to Pram et al.’s [1995] self-similar model, normalized using the observations of Farris et al. [1991], is  bs ðPram Þ ¼ 0:0223

 Pram 1=6 1 RE : 1:8nPa

ð7Þ

The asymptotic approach of the shock to the Mach cone, however, implies that the shock must be blunter, with larger L, for smaller MA and Mms (see (2)). That is, bs should also decrease with MA and/or Mms. Except for Cairns et al.

CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

[1995], these Mach number effects have not been quantified previously for Earth’s bow shock, nor have variations of bs with qIMF and f, the latter corresponding to cross-sectional asymmetries. [9] There are three main results in this paper. Firstly, we present new models for the flaring parameter bs as functions of MA, Pram, and f for two IMF orientations, qIMF = 45 and 90, with Ms = 7.6 (typical for the solar wind at 1AU). We also present simple empirical models for as(MA, qIMF). These models give the shock’s 3-D location and shape as a function of the solar wind conditions in the near-Earth regime. They are derived from the shock locations obtained in the global 3-D ideal MHD simulations of Cairns and Lyon [1995], which were previously analyzed only for as(MA, qIMF). The new models for the shock’s location are in a suitable form to test with observational data (section 5.3 discusses this in more detail). The second major result is the departure from circularity of the shock’s cross section (in the y  z plane). This is modeled via the dependence of bs on an azimuthal angle f, and interpreted in terms of vms varying with qBn. Thirdly, for the qIMF = 45 simulations, we show that there is a skewing of the shock’s shape in the x  z plane due to the dependence of vms on qBn, and we model this using tilted paraboloids. This skewing is shown to be qualitatively consistent with some previous predictions [e.g., Walters, 1964; Hundhausen et al., 1969; Formisano, 1979], and is consistent with a fast mode Mach cone effect. [10] The structure of this paper is as follows. In section 2 the 3-D shock locations in Cairns and Lyon’s [1995] simulation runs are described as a function of the solar wind conditions, together with the fitting functions. Changes in the shock’s shape are described and modeled in section 3, with the shock’s cross section modeled in section 4. The paper’s results are discussed in section 5, with conclusions given in section 6.

2. Simulation Results and Fitting Equations [11] In this section and below new shock models are derived from shock locations given by global 3-D ideal MHD simulations [Cairns and Lyon, 1995] for IMF orientations of qIMF = 45 and 90. The simulations involved solving the ideal MHD equations in conservative form on a deformed spherical (r, q, j) grid with 20 24 24 cells. The grid is distorted in (r, q) space so that 14 of the 24 cells in the q coordinate are located on the day-side, giving higher resolution of the bow shock there. The IMF field lies in the x  z plane, with j = 90 with respect to the simulation domain. These simulations use an impermeable infinitely conducting magnetopause obstacle, whose location is given by Farris et al.’s [1991] 3-D magnetopause model (averaged over IMF orientation), as the inner boundary for the calculations. That is, the magnetopause is an ellipsoid with an eccentricity of 0.43, focus at the Earth (0, 0, 0), and standoff and terminator distances of 10.3RE and 14.7RE, respectively. Accurate locations for the shock as a function of x, y, and z were obtained by studying profiles of the plasma density, flow speed, and magnetic field strength along rays parallel to vsw with constant y and z, as described by Cairns and Lyon [1995]. [12] In the simulations vsw is parallel to the x axis, BIMF lies in the x  z plane, g = 5/3, vsw = 400 km/s, MS = 7.6,

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and 1.3 < MA < 10. In the qIMF = 45 simulations BIMF = 5 nT but nsw is varied, causing changes in MA, Mms, Pram, amp, and the shock location. Conversely, the qIMF = 90 runs have constant amp = 10.2RE, nsw, and Pram = 1.87 nPa but varying BIMF, MA and so Mms. For these analyses shock locations were found in the x  y, x  z, y = z and y = z planes. [13] In the time independent limit, the MHD equations can be written in forms dependent upon qIMF and MA, so that self-similar scalings of Pram cancel in the ratio as/amp. This means that variations in Pram for the qIMF = 45 simulations can be scaled out, allowing direct comparisons between the two sets of Cairns and Lyon’s [1995] simulation runs for qIMF = 45 and 90. [14] The shock positions for the six qIMF = 90 simulations, and associated uncertainties due to the resolution of the simulation grid, are shown in Figure 1 for the x  y and x  z planes. Solid lines represent parabolic fits in these planes. Clear differences exist in the shape and location of the shock as MA is varied. The parabolic fits are a generalized form of (3), with x ¼ as  bsf rf2 ;

ð8Þ

where bsf and rf denote the flaring parameter and distance to the shock from the origin, as a function of azimuthal angle f (measured from the z axis) in the y  z plane; i.e., r0 = z and r90 = y. The shock shapes and fits in Figure 1 show that the shock becomes increasingly blunt (smaller bsf) as MA ! 1. Also, differences between the x  y and x  z planes are apparent in Figure 1 with the shock extending further in the y direction (perpendicular to BIMF) than in the z direction (parallel to BIMF) for a given MA. Both of these characteristics are qualitatively consistent with the anisotropic nature of vms, as shown in more detail below. [15] For qIMF = 45 we expect that the shock’s shape, in planes of constant y (i.e., planes containing BIMF and vsw), should reflect the anisotropic nature of vms. Figure 2 shows density contours in the x  z plane for MA = 1.9, allowing the shock’s position and shape to be identified. Here the contours clearly show that the shock is not symmetric about the x axis (in contrast to the qIMF = 90 simulations in Figure 1), but that the shock is ‘skewed’ or flattened in the +x and +z quadrant. These results are consistent with vms(qBn), i.e., the shock is found further from Earth in directions perpendicular to BIMF than parallel to BIMF. Also note, that the shock was found to be symmetric about the x axis in the x  y plane, which is also expected since vms does not vary in this plane. [16] Figure 3 shows the shock positions for the qIMF = 45 simulation runs. The skewing of the shock in the x  z plane is modeled geometrically as a tilting of the whole shock in the x  z plane. This model was chosen, firstly, to allow for a comparison with the paraboloid model developed for qIMF = 90, and secondly, to describe the skewing effects as simply as possible; the fits in Figure 3 show that this model is appropriate. [17] The solid curves in Figure 3a represent rotated parabola fits where the tilt angle yz is measured clockwise from the x axis in the x  z plane. In Figure 3a, yz can be

1-4

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CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

Figure 1. Shock locations in the a) xz, r = r0 = z and b) x-y, r = r90 = y planes for the simulations with qIMF = 90 (circle symbols with error bars), with associated parabolic fits (solid lines). The lowest curves have MA = 9.7, with MA decreasing (4.9, 3.2, 1.9, 1.6, and 1.4) for the higher curves.

identified as the angle between the symmetry axes (solid straight line from the origin) and the x axis. The fits agree well with the data points, supporting modeling of the skewing as a rotated parabola. The x  y curves are of the same form as (8) since we expect that the shock is symmetric in this plane. [18] A full 3-D description of the data is given by rotating the paraboloid x45 ¼ a0s  b0sf0 ¼90 y2  b0sf0 ¼0 z2 ;

ð9Þ

0

where f is the azimuthal angle (see equation (8)), about the angle yz in the x  z plane. The shock then has coordinates 2 0 3 2 3 3 xs x45 as  b0s90 y2  b0s0 z2 5; 4 ys 5 ¼ A4 y 5 ¼ A4 y z zs z

[20] We seek models for the paraboloid parameters as, 0 0 0, and/or Lf and Lf0 in (8), including tilts when as0 , bsf, bsf appropriate using (10), as functions of MA, Pram, and f. Previous models for as include dependences upon MA2 [e.g., (5) and (6)], suggesting similar dependences for Lf and L0f0. Figure 4 shows that as, as0 , Lf, and Lf0 0 all increase monotonically as MA ! 1, as expected for a MA2 dependence. The solid curves in Figures 4a and 4b, obtained by least squares minimizations, are the fitted functions as ¼

1=6   a1 Pram a0 þ 2 RE ; 1:87nPa MA

ð12Þ

2

ð10Þ

where 2

cos yz A¼4 0 sin yz

3 0  sin yz 5: 1 0 0 cos yz

ð11Þ

3. New Models for the Shock’s 3-D Shape [19] Figure 4 shows the standoff distances as and as0 (Figures 4a and 4b, respectively) and the terminator distances L and L0 (Figures 4c and 4d, respectively) calculated from the parabola fits in Figures 1 and 3, for 1/6 scaling qIMF = 90 and 45, respectively. Here the Pram discussed near (5) and (7) has been used to remove the effects of variations in Pram on as0 and L0 for the qIMF = 45 simulation runs. The dependences of L and L0, upon f and f0, respectively, are clearly seen; i.e., Lf=90 > Lf=45 > Lf=0 and Lf0 0=90 > Lf0 0=0.

Figure 2. Density contours in the xz plane, for a simulation run with qIMF = 45 and MA = 1.9. Note: the upper horizontal contours (near x = 20RE) represent the edge of the simulation grid.

CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

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Figure 3. Bow shock positions for the qIMF = 45 simulation runs in the a) x  z, r = r0 = z and b) x  y, r = r90 = y. Solid lines in (3a) show tilted parabolic fits with increasing as0 for decreasing MA (MA = 9.6, 4.8, 2.4, 1.9, and 1.6). The new symmetry axes (solid lines from the origin to the shock) are rotated from the x-axis by the angles yz. Similarly, (b) shows parabolic fits, but with zero tilt. where the fitted parameters are (a0, a1) = (13.9, 32.2) and (13.9, 29.0), for as and as0 , respectively. These curves agree very well with the simulation points, explicitly showing strong dependencies of as and as0 on MA2. Equation (12)

agrees with the more general model (6), being essentially indistinguishable when plotted on analogs of Figure 4. Note, however, that as0 is measured along a new symmetry axis rotated through an angle yz in the x  z plane.

Figure 4. The standoff distances (4a and 4b), terminator distances (4c and 4d) and flaring parameters (4e and 4f) for the qIMF = 90 (left panels) and qIMF = 45 (right panels) simulation runs. In 4c, d, e, and f the 8, 4, and , symbols represent simulation results in the x  y (f = 90), y = z (f = 45), and x  z (f = 0) planes, respectively. Note, however, the data points for the qIMF = 45 runs (4b, d, and f) are measured in the tilted coordinate system (x0, y0, z0) described by (10). The solid curves in 4a and 4b correspond to the models (12) with the curves in 4c and 4d described by (13), for f = 0 (dotted), 45 (solid) and 90 (dot dash). The curves in 4e and 4f are the resulting ratios (16).

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CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

Table 1. Fit Parameters for Equation (13) Quantity qIMF Lf L0f0

i

j

k

l

m

n

90 25.42 0.82 0.98 79.45 3.96 9.55 45 24.55 0.1341 0.1341 80.147 8.5087 8.5087

[21] A least-squares fit of the Lf values was performed using the 18 available data points (6 runs for each MA times 3 values of f), to the function  Lf ¼

Af þ

Bf MA2



1=6 Pram RE ; 1:87nPa

ð13Þ

where Af and Bf are second order polynomials in cos f, Af ¼ i þ jjcos fj þ k cos f2 ;

ð14Þ

and 2

Bf ¼ l þ mjcos fj þ n cos f :

ð15Þ

These choices are motivated by the dependence of vms on qBn in (1). The fitted parameters i, j, k, l, m, and n are shown in Table 1, and the associated analytic predictions (13) are plotted over Lf for qIMF = 90 in Figure 4c for f = 0, 45, and 90. This model agrees very well with the simulation data. [22] The associated ratio bsf 1 ¼ 2 R2 ; as Lf E

ð16Þ

1/6 is plotted in Figure 4e. In using the model (16) the Pram scaling in both as and Lf, e.g., using (12) and (13), must be 1/6 , as included to give the required scaling of bsf with Pram discussed near (7). The only significant deviation between the model (16) and the ratio bs/as from the simulations in Figure 4 is for MA = 9.7: the simulations show no differences in L (and thus bs) with f for MA = 9.7, whereas the model predicts cross-sectional asymmetries (L changing with f) for high MA. We interpret the simulation results in terms of the magnetopause shape, rather than Mach cone effects, dominating the bow shock’s shape at high MA, resulting in a circular cross-sectional shock shape that

Figure 5. Tilt angle yz (in degrees) for the qIMF = 45 model. The solid line is described by (17).

reflects the obstacle shape. Physically, this is due to the shock being blown back onto the obstacle in such a way that the predicted Mach cone would be interior to the magnetopause. [23] The same methods were applied to the qIMF = 45 simulation data. The corresponding fit parameters are shown in Table 1 for equations (13) – (16). The associated 0 0 are plotted in Figures 4d and model curves for Lf0 0 and bsf 4f, respectively, showing very good agreement between the models and simulation data. The accurate representation of the simulation cross sections by these models, shown below in section 4, also provides strong support for this fitting procedure. [24] Plotted in Figure 5 are the tilt angles extracted from the fits of the qIMF = 45 simulations using (10). The solid line in Figure 5 shows the linear fit yz ¼ ð7:0  0:74MA Þ :

ð17Þ

[25] The shock’s 3-D shape can thus be modeled using (10) and (8) for qIMF = 45 and 90, respectively, with the 0 , and yz, given by equations (12) – quantities as, as0 , bsf, bsf (17) and Table 1. The skewing of the shock (or tilting as we have modeled it) is interpreted in terms of vms varying with propagation direction (and so qBn) in section 5.1.

4. Shock Cross Sections [26] The cross sections in the terminator plane (x = 0) predicted for the qIMF = 90 paraboloid model (equations (8) and (12) – (16)) are shown in Figure 6 for each MA. The terminator locations, with associated error bars, for the simulation runs are also shown for each MA. Very good agreement is apparent, supporting the validity of the fitting

Figure 6. Cross sections in the terminator plane for qIMF = 90. The solid curves are ellipses with semi-major and minor axes Ly and Lz respectively, predicted by the paraboloidal model (13). The plotted points (with error bars) are the terminator distances shown in Figure 4c, for f = 0, 45, and 90 with additional distances measured for f = 135, 180, 215, 270, and 315. The extent of the cross section increases as MA ! 1: the innermost ellipses correspond to MA = 9.7 and the outermost ones to MA = 1.4.

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[28] The solid lines in Figure 8 show the cross sections given by (20) using equations (12) – (17). These predictions agree well with the data points calculated from the original fits (section 2) to the simulation data. The centers (0, cz) of the ellipses shown in Figure 8 are increasingly translated along the z axis as MA ! 1. The semi-minor and semimajor axes for the translated ellipses increase for MA ! 1, as also observed for the un-translated cross sections in the qIMF = 90 simulations. Also plotted in Figure 7 are the eccentricities (18) for the qIMF = 45 tilted parabola fits. The trends are similar to those as found for the qIMF = 90 simulations. Figure 7. Eccentricities of the cross sections in the terminator plane calculated from (18), for the qIMF = 90 and qIMF = 45 simulations, using symbols 4 and 8, respectively. methods in sections 2 and 3. The cross sections of the qIMF = 90 paraboloid model are well described by ellipses, with centers at (y, z) = (0, 0), and with semi-major and semiminor axes L90 = Ly and L0 = Lz, respectively. The eccentricity of these ellipses, for all x, are given by sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi bsy L2  ¼ 1  z2 ¼ 1  ; Ly bsz

ð18Þ

which is plotted in Figure 7 for each of the simulation runs. The eccentricity first increases as MA decreases from 9.7, reaching a maximum at MA = 1.9, before sharply decreasing as MA ! 1. That is, for both high MA and low MA, the eccentricities tend to 0, corresponding to the shock cross section becoming more closely circular. This is consistent with (1), which predicts that vms becomes essentially independent of qBn in these limits. [27] Figure 8 shows the distances to the shock in the terminator plane, calculated from the tilted parabola fits of section 2 for the qIMF = 45 simulations, as the data points with error bars. These cross sections emphasize the ‘skewing’ effects observed in the x  z plane. In the terminator plane, the x coordinate of the tilted paraboloid model is zero [xs = 0 in (10)], imposing the requirement 0 ¼ x45 cos yz  z sin yz ;

5. Discussion [29] The theoretical models for as and bsf given by equations (12) – (16), combined with equations (8) – (11), and (17) for the shock’s functional form, are the first empirical models for the bow shock’s 3-D location and shape as a function of MA and f for the specific cases qIMF = 45 and 90. These models are next discussed critically. 5.1. Shape and Cross-Section Effects [30] The models in section 2 predict that the shock becomes increasingly flared (bs decreases) as MA ! 1, as predicted qualitatively from the Mach cone (2). That is, as MA ! 1, qms should increase, implying that the extent of the shock in the x  y and x  z planes should also increase. This prediction is qualitatively consistent with previous observations [e.g., Cairns et al., 1995; Peredo et al., 1995]. Moreover the model results (12) – (16) represent the first quantitative models for bs as a function of MA and f. Note, however, that for MA ^ 8 the shock’s shape appears to be primarily determined by the magnetopause obstacle rather than by MA or qIMF effects (i.e., see Figure 4 and the discussion in section 3). [31] An analytic model was developed to describe the variations of bsf with MA. This model is based upon the assumption that the shock responds self-similarly to

ð19Þ

0 0 where x45 = as0  bs90 y2  bs0 z2. Substituting for x45 and rearranging gives the equation of an ellipse centered (y, z) ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pat b=b0s90 = (0,pc z) with semi-major and semi-minor axes ffiffiffiffiffiffiffiffiffiffiffiffi and b=b0s0 , respectively, described by

y2 ð z þ cz Þ2 þ ¼ 1; 0 b=bs90 b=b0s0

ð20Þ

where cz ¼

tan yz ; 2b0s0

ð21Þ

and b ¼ a0s þ

tan2 yz : 4bs0

ð22Þ

Figure 8. Cross sections in the terminator plane for the qIMF = 45 simulations. Points with error bars are the terminator distances from the simulation runs. The solid lines are the elliptical cross sections, predicted by (20). The inner most ellipse corresponds to the MA = 9.6 run, with the cross sections increasing in size as MA decreases.

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CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

changes in both MA and Pram, rather than to Pram alone as in (7). Then as = aLf, where a is a proportionality constant, whence bsf = as/Lf2 = a/Lf. Then the ratio of two values of bsf is bsf ðMA1 Þ Lf ðMA2 Þ as ðMA2 Þ ¼ ¼ : bsf ðMA2 Þ Lf ðMA1 Þ as ðMA1 Þ

ð23Þ

[32] Figure 9 shows the values of bs predicted via this self-similar method, normalizing with as(MA = 9.7) = 13.95 RE and bs = 0.0223RE1. This model agrees well with the simulations for large MA (4.8 < MA < 10) but deviates increasingly for lower MA. Moreover, the model cannot explain the dependence of bsf on f, since as is independent of f. Conversely, normalizing with as(MA = 1.4) = 30.8 RE and bs = 0.0069 RE1, the model predictions agree with the simulation data for 1.4 < MA < 3 but differ increasingly for 3 < MA < 10. Thus while this model shows the correct qualitative variation of bsf with MA it cannot quantitatively predict bsf. [33] We attribute the main quantitative failure of this model to the shock’s shape being dependent upon the obstacle shape for high MA > 8, so that normalizing with the high MA, results in an over prediction for low MA, and vice versa. More simply put, the shock’s shape does not vary fully self-similarly in response to changes in both MA and Pram, since we cannot smoothly connect the two regimes MA ^ 4 and 1 < MA < 4. More work needs to be done to develop an analytic model to accurately describe the variations in bsf with MA, and the variations of bs with f. [34] Also plotted in Figure 9 (using a star symbol) is a low MA value for bs derived by Cairns et al. [1995] using 19 observed shock crossings by ISEE 1 and IMP 8. Here bs = (0.0134 ± 0.0049) RE1 for MA = 2.5 (see Figure 9 of Cairns et al. [1995]). This data point agrees well with the new shock models (plotted curves). Accordingly, the shock models can be used with reasonable confidence for the shock’s bs parameter in future comparisons with observational data. [35] The cross sections in Figures 1, 3, 6, and 8 show that the shock extends further from Earth in directions perpendicular to BIMF than parallel to BIMF, qualitatively consistent with the non-isotropic nature of the fast mode speed, i.e., vms? > vmsk. In particular, for the qIMF = 90 simulations, y is perpendicular to BIMF and z is parallel to BIMF, so that the shock’s shape should have a 180 rotational symmetry about vsw with a larger extent in y than in z. [36] The simulation coordinates x  y  z are equivalent to the GSE coordinates X  Z  Y for Parker spiral BIMF (which holds well on average), so that the asymmetries shown in Figure 7 are consistent with those of Peredo et al. [1995], who measured north-south (XZ in GSE) versus east-west (X-Y in GSE) asymmetries of the shock, finding that the extent of the shock in the north-south direction was typically 2 – 7% larger than in the east-west direction. They also found this effect to be more evident for low Mach numbers (MA ! 2). In the discussion presented in section 4, a cross-sectional eccentricity of par = 0.2 implies an asymmetry (= 1  Lz/Ly) of 2% from (18). This corresponds to a high Mach number 5 < MA < 10 in Figure 7. Similarly, the high eccentricity of par = 0.4, for the simulations with

Figure 9. Flaring parameters bsf for the qIMF = 90 simulation runs predicted by (12), (13), and (16): curves dot, solid and dot-dash show bs in the x  z (f = 0), y = z (f = 45), and x  y (f = 90) planes, respectively. The * and . symbols are for the theoretical model (23), normalized by the values of bs for MA = 9.7 (*) and MA = 1.4 (.), respectively. The star symbol represents the observational value for bs found by Cairns et al. [1995]. MA  2, gives an asymmetry of 8%, consistent with the higher asymmetries measured by Peredo et al. [1995] under these conditions. 5.2. Skewing of the Shock’s Shape [37] The ‘skewing’ of the overall shock shape observed and modeled in the qIMF = 45 simulations is an important result. The shape deformations in the x  z plane, are qualitatively consistent with those predicted in the earlier literature [e.g., Walters, 1964; Hundhausen et al., 1969; Formisano, 1979; Shen, 1972]. Also, more recently Verigin et al. [2000] analyzed bow shock crossings and found asymmetric cross sections not centered on the origin (0, 0, 0) of the GSE coordinate system. We interpret Verigin et al.’s [2000] translated elliptical shock cross sections in terms of skewing of the shock in the x  z plane, as shown in the cross sections for qIMF = 45 in Figure 8. [38] These deformations, with the shock extending out further perpendicular to BIMF than parallel to BIMF, are in agreement with the non-isotropic nature of fast mode waves. That is, the distance to the shock found in the qIMF = 45 simulations and models is smaller for +z, where the shock normals are quasi-parallel to BIMF and the fast mode speed is smaller, than for z, where the shock normals are closer to quasi-perpendicular to BIMF and the fast mode speed is larger. The simplest explanation then is that the shock’s skewing in the x  z plane is a consequence of the fast mode speed depending on the IMF direction and the shock’s local propagation direction (which is parallel to the shock normal). 5.3. Limitations and Further Work [39] The empirical models developed above describe the 3-D shape of Earth’s bow shock in the near-Earth regime (20RE < x < 35RE) as a function of the solar wind conditions MA and Pram, with an additional dependence upon the azimuthal angle f. These models are restricted to IMF orientations qIMF = 45 and 90, which are clearly not always appropriate in the variable solar wind (although the 45 models correspond to the average Parker spiral angle at 1AU), and are predicated on the simple mean magnetopause

CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

model of Farris et al. [1991] which assumes no magnetic reconnection or azimuthal asymmetries. Other models are likely to be more representative of the magnetopause’s shape, including asymmetries in the cross section and magnetic reconnection [Farris et al., 1991; Sibeck et al., 1991; Shue et al., 1998]. The new models also cannot describe the shock’s response to changes in MS; however, changes in MA produce considerably larger effects on the shock than changes in MS [e.g., Peredo et al., 1995] and typical values of MS in the solar wind are 7 – 8 [e.g., Fairfield and Feldman, 1975; Slavin and Holzer, 1981]. Thus, although these new shock models do not include functional dependencies upon qIMF, MS, and the magnetopause shape, they do describe the expected variations of the shock’s shape and 3-D location with MA and Pram (based upon the ideal MHD simulations) when qIMF = 45 or 90, 1.3 ] MA ] 10, MS is large enough for MS effects to be ignorable, and the magnetopause obstacle can be well approximated by Farris et al.’s [1991] model. [40] Further simulations are required to determine accurately how the shock’s shape and location vary with the magnetopause model. However, the relation as / amp found in the simulations [Cairns and Lyon, 1995, 1996], the definition bs = as/L2 in (4), and the assumption L / Lmp suggest the following plausible approximations: a00s ¼ as

b00s

¼ bs

a00mp amp

Lmp L00mp

!2

ð24Þ

a00mp amp

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underway. It appears that these models do well in predicting the shock’s location in the near-Earth regime.

6. Conclusions [44] Shock locations obtained from Cairns and Lyon’s [1995] global ideal MHD simulations have been analyzed and the 3-D shape and location of Earth’s bow shock modeled using paraboloidal functions. Analytic expressions for the fit parameters contained in these models now exist as functions of the solar wind parameters MA and Pram for qIMF = 45 and 90. The complete set of model equations are (8) – (17). They have additional dependences upon an azimuthal angle f, which introduces eccentricities to the shock’s cross section, and a tilt angle yz, which allows for ‘skewing’ of the shock in planes containing BIMF and vsw. Both of these effects, departure of circularity in the cross section and skewing of the shock, are consistent with the propagation speed of fast mode waves varying with direction, being larger in directions perpendicular to BIMF than parallel to BIMF. The new models yield the shock’s 3-D shape and location as a function of the solar wind parameters for qIMF = 45 and 90 in forms suitable for testing against observational data. [45] Acknowledgments. We thank John Lyon for performing these simulations in 1994 and 1995 on our behalf. We gratefully acknowledge funding support from the Australian Research Council and NASA grant NAG5-6369. [46] Shadia Rifai Habbal thanks Raymond J. Walker and Jih Kwin Chao for their assistance in evaluating this paper.

References :

ð25Þ

Here symbols without superscripts refer to the predictions of (12) – (16) and Table 1, based on simulations for Farris et al.’s [1994] mean magnetopause, while symbols with double-dashed superscripts are plausible estimates for a different magnetopause model with standoff distance a00mp and terminator distance L00mp. [41] While the simulations analyzed here only have qIMF = 45 and 90, intermediate ‘skewing’ effects in the planes containing BIMF and vsw should occur for 0 < qIMF < 45 and 45 < qIMF < 90 since the fast mode speed is also nonisotropic for these field orientations. In particular, symmetry about the x axis is expected for qIMF = 0 and 90, with expected skewing of the shock in planes containing BIMF and vsw for 0 < qIMF < 45 and 45 < qIMF < 90, tending to zero as qIMF ! 0 or 90. New simulations which support this statement have been performed and will be presented elsewhere. [42] While these skewing effects should be robust, since they depend on the non-isotropic variation of the fast mode speed with propagation direction relative to BIMF, they may be modified by allowing the magnetopause boundary to vary its shape. In Figure 2 note that the density distribution is not uniform near the magnetopause. This suggests that the magnetopause might deform in a fully self-consistent MHD simulation, perhaps reducing the skewing effect on the shock. Such simulations should be performed. [43] Comparisons of the new shock models with spacecraft data close to the nose region of the shock are currently

Bennett, L., M. G. Kivelson, K. K. Khurana, L. A. Frankand, and W. R. Paterson, A model of the Earth’s distant bow shock, J. Geophys. Res., 102, 26,927 – 26,941, 1997. Binsack, J. H., and V. M. Vasyliunas, Simultaneous Imp-2 and Ogo-1 observations of bow shock compression, J. Geophys. Res., 73, 429, 1968. Cairns, I. H., and J. G. Lyon, MHD simulations of Earth’s bow shock at low Mach numbers: Standoff distances, J. Geophys. Res., 100, 17,173 – 17,180, 1995. Cairns, I. H., and J. G. Lyon, Magnetic field orientation effects on the standoff distance of the Earth’s bow shock, Geophys. Res. Lett., 23, 2883 – 2886, 1996. Cairns, I. H., D. H. Fairfield, R. R. Anderson, V. E. H. Carlton, K. I. Paularena, and A. J. Lazarus, Unusual locations of the Earth’s bow shock on September 24 – 25, 1987: Mach number effects, J. Geophys. Res., 100, 47 – 62, 1995. Fairfield, D. H., Average and unusual locations of the Earth’s magnetopause and bow shock, J. Geophys. Res., 76, 6700 – 6716, 1971. Fairfield, D. H., and W. C. Feldman, Standing waves at low Mach number laminar bow shocks, J. Geophys. Res., 80, 515 – 522, 1975. Fairfield, D. H., I. H. Cairns, M. D. Desch, A. Szabo, A. J. Lazarus, and M. R. Aellig, The location of low Mach number bow shocks at Earth, J. Geophys. Res., 106, 25,361 – 25,376, 2001. Farris, M. H., S. M. Petrinec, and C. T. Russell, The thickness of the magnetosheath: Constraints on the polytropic index, Geophys. Res. Lett., 18, 1821 – 1824, 1991. Filbert, P. C., and P. J. Kellogg, Electrostatic noise at the plasma frequency upstream of the Earth’s bow shock, J. Geophys. Res., 84, 1369 – 1381, 1979. Formisano, V., Orientation and shape of the Earth’s bow shock in three dimensions, Planet. Space Sci., 27, 1151 – 1161, 1979. Hundhausen, A. J., S. J. Bame, and J. R. Asbridge, Plasma flow in the Earth’s magnetosheath, J. Geophys. Res., 11, 2799 – 2806, 1969. Kellogg, P. J., Flow of plasma around the Earth, J. Geophys. Res., 67, 3805, 1962. Michel, F. C., Detectability of disturbances in the solar wind, J. Geophys. Res., 70, 1 – 7, 1965. Ness, N. F., C. S. Scearce, and J. B. Seek, Initial results of the IMP 1 magnetic field experiment, J. Geophys. Res., 69, 3531, 1964. Peredo, M., J. A. Slavin, E. Mazur, and C. A. Curtis, Three-dimensional position and shape of the bow shock and their variation with Alfve´nic,

SSH

1 - 10

CHAPMAN AND CAIRNS: 3-D SHAPE OF EARTH’S BOW SHOCK

sonic and magnetosonic Mach numbers and the interplanetary magnetic field orientation, J. Geophys. Res., 100, 7907 – 7916, 1995. Romanov, S. A., Asymmetry of the region of interaction of the solar wind with Venus based on data of the automatic interplanetary stations Venera 9 and Venera 10, Cosmic Res., 16, 318 – 319, 1977. Romanov, S. A., V. N. Smirnoff, and O. L. Vaisburg, Interaction of the solar wind with Venus, Cosmic Res., 16, 746 – 756, 1978. Russell, C. T., and T. L. Zhang, Unusually distant bow shock encounters at Venus, Geophys. Res. Lett, 19, 833 – 836, 1992. Russell, C. T., E. Chou, and J. G. Luhmann, Solar and interplanetary control of the location of the Venus bow shock, J. Geophys. Res., 93, 5461 – 5469, 1988. Shen, W. W., The Earth’s bow shock in an oblique interplanetary field, Cosmic Electrodyn., 2, 381 – 395, 1972. Shue, J. H., P. Song, C. T. Russell, J. T. Steinberg, J. K. Chao, G. Zastenker, O. L. Vaisberg, S. Kokubun, H. J. Singer, T. R. Detman, and H. Kawano, Magnetopause location under extreme solar wind conditions, J. Geophys. Res., 103, 17,691 – 17,700, 1998. Sibeck, D. G., R. E. Lopez, and E. C. Roelof, Solar wind control of the magnetopause shape, location and motion, J.Geophys. Res., 96, 5489 – 5495, 1991. Slavin, J. A., and R. E. Holzer, Solar wind flow about the terrestrial planets: 1. Modelling bow shock position and shape, J. Geophys. Res., 86, 11,401 – 11,418, 1981.

Slavin, J. A., R. E. Holzer, J. R. Spreiter, and S. S. Stahara, Planetary Mach cones: Theory and observation, J. Geophys. Res., 89, 2708, 1984. Spreiter, J. R., and A. W. Rizzi, Aligned magnetohydrodynamic solution for solar wind flow past the Earth’s magnetosphere, Acta Astron., 1, 15 – 35, 1974. Spreiter, J. R., and S. S. Stahara, Magnetohydrodynamic and gas dynamic theories for planetary bow waves, in Collisionless Shocks in the Magnetopause, Geophys. Monogr. Ser., vol. 35, pp. 85 – 107, AGU, Washington, D. C., 1985. Spreiter, J. R., A. L. Summers, and A. Y. Alksne, Hydromagnetic flow around the magnetosphere, Planet. Space Sci., 14, 223 – 253, 1966. Verigin, M., et al., On the location and asymmetry of the terrestrial bow shock, paper presented at International Symposium, From Solar Corona Through Interplanetary Space, Into Earth’s Magnetosphere and Ionosphere: Interball, ISTP Satellites, and Ground-Based Observations, Natl. Space Agency of Ukraine, Kyiv, Ukraine, Feb. 1 – 4 2000. Walters, G. K., Effect of oblique interplanetary magnetic field on shape and behavior of the magnetosphere, J. Geophys. Res., 69, 1769 – 1783, 1964. 

I. H. Cairns, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia. ([email protected]) J. F. Chapman, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia. ( [email protected])