Originally Published In: Journal of Geophysical Research, 101, 7677-7678, 1996.
Copyright © 1996 American Geophysical Union
S. M. Petrinec
STELAB, Nagoya University, Toyokawa 442, JAPAN
In their recent paper Cairns et al. [1995a] conclude that when the magnetosonic Mach number is low there can be substantial changes in the standoff distance and/or shock shape. The purpose of this note is not to disagree with that statement which is in accord with our earlier work [Russell and Zhang, 1992] but to suggest a method whereby these two changes can be separated from each other so that their analysis can be made less ambiguous.
Over the forward part of the magnetosheath the flow is subsonic and much of the shock remains in communication with much of the obstacle. Thus the shape of the obstacle plays an important role in determining the shape of the shock and hence the forward part of the terrestrial shock is found to be elliptical in contrast to the hyperbolic shape it is expected to have far behind the Earth. The shape of the obstacle also plays a role in determining the standoff distance. A pointed obstacle would have a much smaller standoff distance than a sphere. A sphere would have a smaller standoff distance than an ellipse (with the Earth at the focus) etc. Physically, the standoff distance depends on the radius of curvature of the obstacle at the stagnation point in the flow and thus is insensitive to the choice of the origin of the coordinate system. Spreiter et al. [1966] gave a standoff formula as a function of Mach number for a specific shape, an elliptical, magnetosphere-like obstacle, with distance measured from the center of the Earth. All later work including that of Cairns et al. [1995a] has followed this procedure of measuring the standoff distance from the center of the Earth, rather than comparing with the radius of curvature of the obstacle thus making the standoff ratio coordinate-system dependent.
The reason the shock front sits anywhere is to allow all the shocked plasma to pass between the shock front and the obstacle. If the plasma is compressed less as when the Mach number drops to low values, then the shock must move away from the obstacle. The failing of the Spreiter et al. [1966] formula is that it does not predict this motion correctly at very low Mach numbers. Since the shock should move to infinity when there is an infinitesimal compression of the flow in order that the (slightly) shocked flow can move around the obstacle, and since this statement is true for a gas or a plasma, we need not look to an MHD effect to explain the failing of Spreiter et al.'s formula. Farris and Russell [1994] proposed an alternate formula equally valid for gas and magnetohydrodynamics that appears to redress this failing. More tests of this formula are certainly in order. However, in order to carry out such tests it is important to separate dynamic pressure effects from Mach number effects, and shape changes from size changes. The methodology of Cairns et al. [1995a] does not do this adequately.
The equation of a conic section is:
where R is the distance from the origin to the
curve; K is the semilatus rectum, the point on the
curve at right angles to the line joining the origin (at the
focus) and the "nose" of the conic section;
is the eccentricity
of the ellipse; and
is the angle between the
line joining the nose and the origin and the line joining the
origin and any point on the curve. The parameters R
and K have dimensions of length;
has dimensions of angle;
and
is dimensionless
as befits a shape parameter. The eccentricity,
, is greater than 1
for a hyperbola; equals 1 for a parabola; and is less than 1
for an ellipse. Cairns et al.
[1995a] assume
=1, despite the
evidence from previous work that
is significantly
smaller than 1. [Holzer et al.,
1966; Slavin and Holzer, 1981;
Farris et al. 1991]. This formula
may be rewritten for
=1:
If we equate this formula with that of Cairns et al. [1995a] with its
focus at the Earth we find that as =
K/2 and bs = 1/(2K), i.e.,
that as/bs =
K2 and bs =
1/(4as). Thus, if the focus of the shock
remains fixed, as and bs
are related and are not independent variables. Since we
expect that the size of the shock should vary with the
dynamic pressure because the obstacle size varies with
dynamic pressure, it is not surprising that the parameter
bs is found to vary too. Furthermore, if
as depends on Mach number, then
bs must depend on Mach number too, contrary
to the authors' conjecture. If one relaxes the assumption
that the focus of the parabola is at the Earth, then the
parabola may be fit to the data by translation along the
solar wind flow direction. That solution is tried by the
authors in Figure 10, curve 
, but as can be seen by inspection the
"shape" of the parabola does not change in this process even
though as and bs vary as
the focus is moved. In fact it is trivial to prove by
substituting x1 = (x-
xo) in their equation for
as that if bs is
constant, the act of changing as is
equivalent to translation of the paraboloid. We note that
throughout their paper Cairns et al.
[1995a] solve for only one parameter assuming the
value of the other. Nowhere do they solve for 2 or as they
now advocate 3 parameters.
In order to separate the effects of shape changes, we
recommend the use of the eccentricity as used by many
previous authors [Holzer et al.,
1966; Slavin and Holzer, 1981;
Russell, 1985]. Since it is
dimensionless it does not vary as the scale size as the
parameter bs does, and will be much less
sensitive to changes in dynamic pressure than
bs. We urge the authors to repeat their
analysis to examine this shape parameter,
. We do not
recommend keeping the eccentricity fixed at unity as Cairns et al. [1995a] do and
changing the location of the focus of the parabola, since it
is clear the eccentricity of the shock is not unity and can
vary with Mach number. Only then will we be able to tell how
dependent is the shape of the forward part of the shock on
Mach number.
In the following reply Cairns et al. [1995b] present new simulation results [Cairns and Lyon, 1995]. We note first that the shock location moves to very large standoff distances at low Mach numbers as expected [Russell and Petrinec, 1995] showing that the treatment of Cairns and Grabbe [1994] is an incorrect approach. Cairns et al. [1995b] plot the shock position versus Alfvenic Mach number and not the magnetosonic Mach number. When the more physical comparison is made with the magnetosonic Mach number, the simulation result is qualitatively the same as Farris and Russell's model but still on the high side. In contrast the low Mach number gas dynamic simulation of Spreiter and Stahara [1995] is qualitatively similar to the Farris and Russell model but on the low side. Since the new MHD model does not approach the asymptotic, and non controversial, limit expected at high Mach numbers, we think the problem may be with the MHD model. Inspection of Figure 2 of Cairns and Lyon [1995] suggests the problem is in the simulation itself. For a perpendicular shock, the fractional jump in density, in magnetic field strength, and the drop in velocity should all be equal, but they are not. The density has the correct jump for the stated conditions but the magnetic field jump and velocity drop are incorrect. On a more minor note, we point out that the reason Farris et al. [1991] did not consider shape changes is that they restricted their attention to Mms > 4.5.
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