Geophysical Research Letters, 101, 7677-7678, 1996.
Copyright © 1996 American Geophysical Union
C. T. Russell
Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA
S. M. Petrinec
Solar-Terrestrial Environment Laboratory, Nagoya University, Toyokawa 442, JAPAN
In their recent article Cairns and Grabbe  suggest that the major differences be- tween the observed shock location and the theoretical shock location is due to MHD effects at low Mach numbers and then repeat work done as part of a larger study of the shock and magnetosheath by Zhuang and Russell . In the end they leave the problem unsolved, but make several discoveries such as that "gasdynamics and phenomenological models ... generally provide poor approximations to the MHD results"; and that "changes in field orientation ... cause factor of ~4 changes" in the standoff distance of the shock. The purpose of this note is to point out that the solution to the problem of the distant location of the low Mach number bow shock is independent of whether the fluid is MHD or gasdynamic; that over most of pa- rameter space gasdynamics is a good approximation to the MHD solution with the simple substitution of "magnetosonic" for the gasdynamic sonic Mach number; and that Cairns and Grabbe  have made a significant mistake in their analysis.
The standoff position of the shock has long been thought to obey the postulate of Spreiter et al. , that the ratio of the thickness of the magnetosheath at the nose over the radial distance to the nose of the shock from the center of the Earth was equal to , where the subscripts 1 and 2 refer to upstream and downstream of the shock respectively and the factor 1.1 is shape dependent. The reason for this postulate is that it produces a good fit at high Mach numbers, but there is no physical basis for this postulate. The familiar formula for the standoff distance follows from this postulate by substituting the formula for the compression of the density as a function of Mach number and ratio of specific heats. Farris and Russell  point out that a slightly different postulate gives the same asymptotic magnetosheath thickness for high Mach numbers and the expected retreat of the bow shock to infinity at low Mach numbers. This result is not dependent on whether the fluid is gasdynamic or MHD, and one would not expect it to be because intuitively the shock should move to infinity as the Mach number approaches unity in all situations. In short, a solution exists to the problem that Cairns and Grabbe  say motivates them.
The second point we wish to make is that the Earth's bow shock is three-dimensional. This was clearly recognized by Zhuang and Russell  in their classic analytic study of the MHD Rankine-Hugoniot jumps across the forward part of the shock. In that study, as in the case of the real world, the Mach number and angle between the magnetic field and the shock normal () vary over the shock surface. Thus, there is no one value of that is appropriate over the whole surface, and it is a red herring to investigate oddly behaved, singular regions of parameter space. It is common practice to use the magnetosonic Mach number for perpendicular magnetic field orientation ( = 90 degrees) in quoting the Mach number for the overall interaction. Figure 5 of Cairns and Grabbe illustrates the wisdom of that approach as this curve (the dotted line) lies very close to the curves for all other angles.
Now let us examine the error that Cairns and Grabbe  make in their 1-D solution. Figure 1 repeats their Figure 2 with the addition of a solution for = 1 degree. We see that there is a singular solution at = 0 degrees, but for a magnetic field only one degree away from the shock normal there is a solution that emulates the solution at much higher angles. The constant distance obtained for the 0 degree solution appears to be a singular point, but it turns out not to be a physical solution at all. In their analysis Cairns and Grabbe drop the "unphysical" switch-on shocks. However, the low Mach number quasi parallel shock is a switch-on shock (Farris et al., 1994). Figure 2 shows the 'switch-on' solution at = 0 degrees for fixed low beta. It is remarkably like the 1 degree solution over the range we expect the shock to be a "switch-on" shock! Thus in fact there is no singular position of the shock in nature. It asymptotes to the same location for low Mach numbers at 0 degrees as it does at 90 degrees.
Fig. 1. Adapted from Figure 2 of Cairns and Grabbe . = 1 degree is also included.
Cairns and Grabbe  also do not appreciate the role of plasma beta in their solutions. Because they use the Alfven and sonic Mach numbers and not magnetosonic Mach number and beta in ordering their analysis they overlook an important attribute of the solutions. In the bottom panel of Figure 2 we repeat the analysis for high beta. The curves all overlap one another. The small differences between the solutions at low Mach number are only a low beta phenomenon.
Fig. 2. DBS/DOB solutions as a function of magnetosonic Mach number, extrapolating the formula of Spreiter et al.  to very low Mach numbers. a) Plasma = 0.01, = 5/3. b) Plasma = 100, = 5/3.
In the introduction we mentioned that the conjecture of Farris and Russell  had provided a solution to the low Mach number shock location problem. Our Figures 1 and 2 clearly illustrate the naivete of using the Spreiter et al. Mach number relation when the Mach number reaches very low values. Although the curves differ slightly from one another for different (and for low plasma , in Figure 2), they are all constrained, for no physical reason, to meet at DBS/DOB = 2.1 (DBS is the bow shock standoff distance; DOB is the obstacle standoff distance) when the Mach number approaches unity. Thus we should no longer use the Spreiter et al.  postulate in drawing Figure 2. This figure was only shown to contrast with the results of Cairns and Grabbe . We repeat the analysis in Figure 3 using the Farris and Russell  conjecture (see Appendix for explicit formula used). It is these curves that should be used in determining if there is still a mystery about the location of the bow shock at low Mach numbers. They move to infinity as the Mach number approaches unity as expected (and are in close agreement with recent low Mach number gasdynamic calculations by Spreiter and Stahara ).
Fig. 3. DBS/DOB solutions as a function of magnetosonic Mach number, using the recent formulation of Farris and Russell . a) Plasma = 0.01, = 5/3. b) Plasma = 100, = 5/3.
In summary we feel there is not an outstanding problem in determining where the bow shock should stand at low Mach numbers. The problem lay in attempting to use the original postulate of Spreiter et al.  in the low Mach number regime. It is not a question of the appropriateness of gasdynamics versus MHD. Under most circumstances the magnetosonic Mach number can be used as the Mach number that is analogous to the gasdynamic Mach number. Finally, we conclude that the singular results found by Cairns and Grabbe at low beta and Mach number at = 0 degrees do not occur in nature.
The following is the general solution of the Farris and Russell modification of the Spreiter et al. formula of the relative distance of the terrestrial bow shock to magnetopause standoff distance, as a function of magnetosonic Mach number (Mms), , plasma , and polytropic index ().
Cairns, I. H., and C. L. Grabbe, Towards an MHD theory for the standoff distance of Earth's bow shock, Geophys. Res. Lett., 21, 2781-2784, 1994.
Farris, M. H., and C. T. Russell, Determining the standoff distance of the bow shock: Mach number dependence and use of models, J. Geophys. Res., 99, 17,681-17,689, 1994.
Farris, M. H., C. T. Russell, R. J.
Fitzenreiter, and K.W. Ogilvie, The subcritical, quasi-
parallel, switch-on shock, Geophys. Res. Lett., 21,
Spreiter, J. R., A. L. Summers, and A. Y.
Alksne, Hydromagnetic flow around the magnetosphere,
Planet. Space Sci., 14, 223-253, 1966.
Spreiter, J. R., and S. S. Stahara, The
location of planetary bow shocks: A critical overview of
theory and observations, Adv. Space Res., 15,
Zhuang, H. C., and C. T. Russell, An
analytic treatment of the structure of the bow shock and
magnetosheath, J. Geophys. Res., 86, 2191-2205,
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Spreiter, J. R., A. L. Summers, and A. Y. Alksne, Hydromagnetic flow around the magnetosphere, Planet. Space Sci., 14, 223-253, 1966.
Spreiter, J. R., and S. S. Stahara, The location of planetary bow shocks: A critical overview of theory and observations, Adv. Space Res., 15, (8/9)433-(8/9)449, 1995.
Zhuang, H. C., and C. T. Russell, An analytic treatment of the structure of the bow shock and magnetosheath, J. Geophys. Res., 86, 2191-2205, 1981.