*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* [1994] 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*
[1981]. 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* [1994] 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.*
[1966], 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* [1994] 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* [1994] 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* [1981] 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* [1994] 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* [1994]. = 1 degree is also included.

*Cairns and Grabbe* [1994] 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. *D _{BS}*/

In the introduction we mentioned that the conjecture of *Farris and Russell* [1994] 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
*D _{BS}*/

Fig. 3. *D _{BS}*/

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.* [1966]
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 (*M _{ms}*), , plasma ,
and polytropic index ().

where

, and

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