A magnetohydrodynamic (MHD) theory is developed for the standoff distance a(s) of the bow shock and the thickness Delta(ms) of the magnetosheath, using the empirical Spreiter et al. relation Delta(ms) = kX and the MHD density ratio X across the shock. The theory includes as special cases the well-known gasdynamic theory and associated phenomenological MHD-like models for Delta(ms) and As. In general, however, MHD effects produce major differences from previous models, especially at low Alfev (Ma) and Sonic (Ms) Mach numbers. The magnetic field orientation Ma, Ms and the ratio of specific heats gamma are all important variables of the theory. In contrast, the fast mode Mach number need play no direct role. Three principle conclusions are reached. First the gasdynamic and phenomenological models miss important dependences of field orientation and Ms generally provide poor approximations to the MHD results. Second, changes in field orientation and Ms are predicted to cause factor of approximately 4 changes in Delta(ms) at low Ma. These effects should be important when predicting the shock's location or calculating gramma from observations. Third, using Spreiter et al.'s value for k in the MHD theory leads to maxima a(s) values at low Ma and nominal Ms that are much smaller than observations and MHD simulations require. Resolving this problem requires either the modified Spreiter-like relation and larger k found in recent MHD simulations and/or a breakdown in the Spreiter-like relation at very low Ma

Carins, Iver H.

Grabbe, Crockett L.

English

NASA Technical Reports Server

GEOPHYSICAL RESEARCH LETTERS, VOL. 21, NO. 25, PAGES 2781-2784, DECEMBER 15, 1994
f
Towards an MHD theory for the standoff distance of
Earth's bow shock
Iver H. Cairns and Crockett L. Grabbe
Department of Physics and Astronomy, University of Iowa
NASA-CR-202¥39
Abstract. An MHD theory is developed for the stand-
off distance as of the bow shock and the thickness A,_s
of the magnetosheath, using the empirical Spreiter et
al. relation Ares = kX and the MHD density ratio X
across the shock. The theory includes as special cases
the well-known gasdynamic theory and associated phe-
nomenological MHD-like models for A,n, and as. In
general, however, MHD effects produce major differ-
ences from previous models, especially at low Alfven
(MA) and sonic (Ms) Mach numbers. The magnetic
field orientation, Ma, Ms, and tlle ratio of specific heats
7 are all important variables of the theory. In contrast,
the fast mode Mach number need play no direct role.
Three principal conclusions are reached. First, the gas-
dynamic and phenomenological models miss important
dependances on field orientation and Ms and generally
provide poor approximations to the MHD results. Sec-
ond, changes in field orientation and Ms are predicted
to cause factor of-_ 4 changes in Ares at low MA. These
effects should be important when predicting the shock's
location or calculating 3' from observations. Third, us-
ing Spreiter et al.'s value for k in the MHD theory leads
to maximum as values at low A'IA and nominal Ms that
are much smaller than observations and MHD simula-
tions require. Resolving this problem requires either
the modified Spreiter-like relation and larger k found
ira recent MHD simulations and/or a breakdown in the
Spreiter-like relation at very low MA.
1. Introduction
The location of Earth's bow shock has been actively
researched since its prediction and discovery. Subjects
of particular interest include the shock's farthest ex-
tent sunwards (known as the standoff distance as) and
the thickness Ares of the magnetosheath region separat-
ing the shock from the rnagnetopause, due to their im-
portance in understanding foreshock observations, so-
lar wind-magnetosphere interactions, and the ratio of
specific heats 7 for the plasma. Figure 1 defines as,
A,,_,, and the magnetopause standoff distance a,,_p in
the X-Y-Z coordinate system formed by rotating the
GSE system so that the shock is symmetric about the
solar wind's velocity vector relative to Earth. Clearly
as = amp +A,,s. Balancing the solar wind ram pressure
P = pswV_w and the magnetostatic pressure of Earth's
Copyright 1994 by the American Geophysical Union.
Paper number 94GL02551
0094-8534/94/94GL-02551 $03.00
dipole magnetic field requires amp = KP -x/6, where K
is a slowly varying function of the ¿MF B_ component,
the ring current, the magnetopause current system and
drag effects for the solar wind-magnetosphere system
[Formisano et al., 1971; Slavin and Hoher, 1978; Far-
ris et al., 1991; Sibeck et al., 1991]. Spreiter et al. [1966,
and references therein] developed the first detailed the-
oretical model for a_ and Ares,
a, = I,'P 1+ 1.1 (11
The ratio Ares/am p is given by the entire second term,
with the number 1.1 depending on the obstacle's shape.
This equation follows from the empirical linear relation
-- = kX = k (2)
amp Pd
obtained from gasdynamic simulations for M = Ms > 5
[Spreiter et al., 1966] with the ratio p,w/pd specified
subsequently by the jump conditions for a gasdynamic
shock. (Here Pd is the mass density downstream from
the shock.) An analytic explanation for (2) remains
unavailable. Spreiter et al. generally identified the
(sonic) Mach number M with the Alfven Mach num-
ber MA = Vsw/Va for arbitrary magnetic field orien-
tations and MA_ >> M_ >> 1 (a pseudo Mach number
was defined instead for aligned flows with v_ II Bs,_).
This theory is therefore intrinsically gasdynamic with
the subsequent phenomenological replacement of the
sonic Mach number Ms = vsw/cs by MA. Spreiter
et al. emphasized the theory's expected limitations at
low Mach numbers. More recently Russell [1985] sug-
gested that the proper replacement for M in (1) is the
fast magnetosonic Mach number M,,_s, since the bow
shock is a fast mode shock. Again, however, this is a
phenomenological replacement in a _asdynamic result.
Earth's bow shock is indeed a fast 'mode shock, not a
gasdynamic shock, and so MHD theory is a priori more
appropriate. Spreiter et al. 's gasdynamic' equation and
its phenomenological variants therefore need to be re-
considered and a MIlD version of (1) derived more rig-
orously. Other motivations for studying the bow shock's
location include the finding that (1) with M replaced
by MA or Mms predicts as values that are too small for
AlIA & 3_rns "_ ] -- 3 [Russell and Zhang, 1992; Cairns et
al., 1994] and the scattered values for 7 extracted from
the measured A,_s via (1) [Fairfield, 1971; Zhuan9 and
Russell, 1981; Farris et al., 1991]. Lastly, MHD predic-
tions for as and Ares are needed now that global MHD
simulation codes are available [e.g., Cairns and Lyon,
1994] for studying the Earth-solar wind interaction.
2781
https://ntrs.nasa.gov/search.jsp?R=19970007044 2020-06-16T02:38:53+00:00Z
BOW SHOCK Vsw I_' Bsw
/ 7 -2 ..... °
Y E
Figure 1. Definitions of as, amp, Ares, the angle 0,
and the X-Y-Z coordinate system.
This paper addresses the basis of previous gasdy-
namic and phenomenological MHD-like models for as
and Ares, the importance of MHD effects, and current
attempts to explain unusually distant shock locations
and the plasma's ratio of specific heats 3'. The simplest
MHD theory for as and A,ns is constructed: Spreiter et
al. 's empirical relation between A,n, and X is retained
and MHD theory is used to specify the density ratio
X across the shock. This MHD theory (Sections 2 and
3) includes the gasdynamic and related phenomenolog-
ical theories as special cases but in general shows differ-
ent theoretical dependances. In fact, the MHD theory
predicts that A,_s and as should depend strongly on
the magnetic field orientation, /'1/I a and Ms (and 3').
Zhuang and Russell [1981] previously developed an ap-
proximate but not self-consistent MHD expression for
X and an unrelated calculation for A ..... when MA
Ms >> 1. This paper's theory extends Zhuang and Rus-
sell's work on X, by retaining all contributions to X
and not assuming high 1_ A _5 Ms flows, and merges it
with Spreiter et al. 's empirical approach. Quantitative
comparisons are made between tile various theories in
Section 3. A discussion and the conclusions are pre-
sented in Sections 4 and 5, respectively.
2. Analytic theory
Spreiter et al. [1966] used gasdynamic simulations to
show that tile linear relation (2) between Am s and X
holds for Ms > 5. The jump conditions for a gasdy-
namic shock lead to a quadratic equation for X whence
x = (3 -` 1)M + 2
(3`+ 1)i '
thereby leading to (1) via (2). The transition to the
magnetised solar wind was then attempted by phe-
nomenologically replacing Ms by Mn [Spreiter el al.,
1966] or Mms [Russell, 1985]. The limiting values
of Ares and as for 3' = 5/3 are then: Ams/arn p "+
1.1/4 and a_/amp --+ 1.275 as _a and Mm, --_ oo;
Ares/am p -+ 1.1 and as/amp -+ 2.1 as i_Ia and A4rns --+
1. Note that the gasdynamic and phenomenological re-
sults have no explicit dependences on magnetic field ori-
entation and Ms except through Alms.
The theory developed here assumes that (2) remains
valid, as supported by Cairns and Lyon's [1994] MHD
simulations for MA _ 1.5, and uses MHD theory to
specify the density jump X. Factoring out the solu-
tion X = 1, the MHD jump conditions lead to a cubic
equation for X [e.g., Zhuang and Russell, 1981]:
AX a + BX 2 + CX + D = 0 (4)
as A = (3` + 1)MA6
[ -B = (7-1)M_+(3`+2)cos_OM]+(3`+3`/3)M4a
¢ C = (3`- 2 + 7c°s_ O)M] + (3` + l + 27/3) cos 20M]
-D = (3`- 1)cos20M_ +3`flcos40 •
The new variables introduced are 0 C [0,900], the angle
between vsw and Bsw (the shock normal is antiparal-
lel to vs,_), and the upstream plasma/3 is defined by
7_ = 2Cs/VA2 _ = 2M2t/M 2. The natural Mach numbers
in the theory are then Ma and Ms; M, ns need play
no role. In comparison, neither 0 nor /3 play roles in
the gasdynamic or phenomenological theories (except
through Mms). The standard cubic analysis provides
general solutions to (4), although these typically pro-
vide little insight. Consider, however, the special cases
of parallel (0 = 0°) and perpendicular (0 = 90 °) flows.
The cubic is easily factored when B,_ is parallel to
the shock normal (cos 0 = 1). Ignoring the two 'switch-
on' shock solutions X = MA 2, (2) and (4) yield
a---L-"= l+k (3`- 1)MA_ +3`/3- l+k(3'-l)Ms 2+2
amp (3' + 1) M2 (3` + 1)Ms 2
(5)
The rightmost form reveals that the gasdynamic ex-
pressions for X and a_ are recovered, cf. (1) and (3),
as expected since the magnetic field essentially drops
out of the problem in the parallel case. This equa-
tion implies two important results for 0 = 0°. First,
phenomenological replacement of Ms by Ma or M,,_s
(= MA for 0 = 0°) in (1), as proposed by Spreiter et
al. and Russell, is incorrect except in the special case
3`/3 = 2. The phenomenological theories are therefore
restricted special cases of the MHD theory. Second,
Ares, as and .¥ are independent of AIA and M ..... and
depend only on 3` and Ms (with the intuitive caveats
that MA & Ms >_ 1). This is a major difference from (l)
with the replacements M -+ MA or Mms, which predict
Ares/am p "-4 k a.s MA --4 1 instead of the correct result
Ares amp _ k/4 for 3`= 5/3 and Ms >> 1.
For perpendicular flows (cos 0 = 0) the cubic equation
(3) for X collapses to a quadratic, whence
ampa-'_8 k(_, )1+2(3`+D A+ A2-4(3`-2)(3`+l)M_ 2
(6)
with A = (7- 1)+7/M2A +2/M 2. For 3` < 2 only
the solution as = as+ is relevant (only X = X+ >
0). In general, (6) is not equivalent to the gasdynamic
solution (1), the Spreiter et al. and Russell variants of
(1) with M -+ ,VIA and Mou, respectively, or (5)'s MIlD
solution for 0 = 0°. However, in the special case 3` = 2
(6) reduces to Russell's form for arbitrary /3 (writing
MA and Ms in terms of M,n, and /3) and to Spreiter
et al.'s form in the limit /3 --+ oo. Furthermore, the
gasdynamic result (1) is recovered in the limit MA -+
oc (or /3 -+ 0) for arbitrary 7. (This is true for all
0.) Equation (6) also implies three corollary results.
-2
o*
First,a,, Am, and X vary strongly with the angle 0.
Second, in general the solution depends intrinsically on
both MA and Ms; while (6) can easily be rewritten in
terms of Mm, and /3, only for 7 = 2 does the explicit
/3 dependance disappear and the solution depend solely
on Mm,. Last, the behavior as MA --+ 1 for 0 = 90 °
differs greatly from the MHD result (5) for 0 = 0: for
7 = 5/3, (5) implies X -+ 1/4 for /3 = 0 (Ms = oc)
while (6) implies X+ -+ 1 for/3 = 0 and X+ _ 1.7 for
3'/3 = 1. Thus, factor of > 4 changes in Ares should
exist at low MA for different 0 and Ms.
3. Numerical Analyses
When the magnetic field is neither parallel nor per-
pendicular to the flow direction and shock normal, (4)'s
solutions are most instructive when presented graphi-
cally. Figure 2 shows the MHD theory's predictions for
the ratio as/amp = 1 + 1.1X, where Spreiter et al.'s
empirical value for k in (2) is used (see Section 4) and
(4) is solved numerically, as a function of MA for 0 ---=0,
45, and 90 ° and Ms = 8. Figure 3 is an analogous plot
for Ms = 2. The strong dependences on 0, MA and Ms
implied by (4) - (6) are clearly evident. Note that the
0 = 0 ° curves are all flat, and so independent of MA
as in (5), with a level that depends on Ms. The curves
for 0 -= 45 ° lie below the 0 = 90 ° curves. Indeed, it
may be shown analytically that the maximum allowed
values of as (and X) for given Ms occur at 0 = 900 and
MA = 1+, with smaller Ms leading to larger X and a,.
Figure 4 shows how X and as depend on 3': for a
given 0 and Ms, a larger 3' leads to a higher curve for
as versus Ma. However, it can be shown analytically
from (4) that the maximum values of X and as are
independent of 7: curves for different 7 therefore all
converge to the same maximum at MA = 1 and 0 = 90 °.
Quantitative comparisons between the MHD theory
developed here, Spreiter et al. 's model given by (1) with
M = MA, and Russell's model are shown for various 0
and MA in Figure 5. For all 0 the gasdynamic theory
with M = Ms coincides with the 0 = 0° MHD solution.
In general significant differences between the MHD pre-
dictions and the phenomenological models are apparent,
except in the limited region MA < 1.5 and 0 > 300 (de-
creasing Ms shrinks this region further). Note that
the phenomenological models have no 0 or (direct) Ms
I I I
8 = 90"
8 = 45"
I , I I
2 4 6
MA
Figure 2. Ratio as  amy as a function of MA and 0 for
3' = 5/3, Ms -- 8 and k = 1.1. The ratio is independent
alMA for 0 = 0° and is maximum at 0 = 90 ° & MA = 1.
E_
J
I I
O = 90 °
/
/ 0 = 45 °
I I , I , I
2 4 6
M A
Figure 3. Similar to Figure 2 but with Ms : 2. Note
that as increases as Ms decreases (for all 0).
dependences, whereas the MHD theory shows these de-
pendences to be very important. The maximum as val-
ues predicted by the MHD and phenomenological mod-
els (MA _ l) are, however, almost identical for large
Ms > 5. When Ms is small the MHD theory predicts
larger as values. However, the maximum difference in
as between the MHD and 'gasdynamie' theories remains
less than 50% for Ms > 1.
4. Discussion
Differences of 50% or more in a, and 400% in Ares,
due to 0 and Ms effects when MA < 5, are easily dis-
cerned in Figures 2 - 5. Differences occur both between
the MHD, gasdynamic and phenomenological theories
and between different parameter sets for the MHD the-
ory. That 0 effects should be very important, as well as
MA and Ms variations, in determining as and Ar, s is
one of this paper's important predictions. Refinements
to the present 'local' MHD theory will undoubtedly oc-
cur when global MHD effects are considered.
The above results argue that values for 7 derived from
measurements of Am, [Fairfield, 1971; Zhuan9 and Rus-
sell, 1981; Farris et al., 1991] should depend on the for-
mulae used for Am, and on whether MA, 0 and Ms
variations are all considered. Finite MA, 0 and Ms ef-
fects may explain the wide scatter in the published 3'
values. Analyses to measure 3' should be redone using
(4)'s explicit MHD solutions (or successors thereof).
Near 1 AU the solar wind speed and electron temper-
ature vary relatively little, whence Ms lies within a fac-
tor of 2 of the nominal value Ms --_7. For MA _ 1 -- 3,
I I I
7,= 5/5
2 y:2
"-x 0 =oo
I I I _ I
2 4 6
MA
Figure 4. Ratio as amp versus MA for 3' = 5/3 (full
lines) and 3' = 2 (dashed), Ms = 8, and 0 = 0 & 90 °.
2784 CAIRNS AND GRABBE: MHD THEORY FOR SttOCK STANDOFF DISTANCES
I F I
2 C-G
SPREITER ET AL.
_ . WITH M -- M A
o = o .........RUSSELL (8=90 ° )
o 1.5
. 0:
_'0 : 0 ° - 1
I , I , t , 1 , /
2 4 6
MA
Figure 5. Comparison between the MttD theory given
by (2) and (4) {full lines) for various O, and the phe-
nomenological eopreiter et al. (dashed) and Russell(dot-
ted for 0 = 90 o) models for Ms = 5 and 7 = 5/3. For
arbitrary 0 the Russell curve lies between the dotted
and dashed curves.
then, Figures 2-5 predict that as will be within 10%
of (1)'s prediction (for M = MA). This paper's MHD
theory, using Spreiter et al.'s empirical value for k in
(2) and MHD theory to determine X, can therefore
not explain unusually distant bow shock observations at
low MA [Russell and Zhang, 1992; Cairns et al., 1994].
Cairns and Lyon's [1994] MHD simulations show, how-
ever, that intrinsically MHD effects modify (2) and in-
crease k by a factor ,-_ 3 over Spreiter et al. 's value.
Incorporating the modifications into this paper's MHD
theory they can explain the large as values at low MA in
the simulations. Comparisons with observational data
still need to be done. It remains possible that the MHD
variant of (2) breaks down at very low MA.
5. Conclusions
The analyses above show that more detailed consid-
eration of MHD effects results in major differences from
the well-known gasdynamic theory, and its phenomeno-
logical variants, for Ares and as. The MHD model de-
pends on MA, Ms, O, and 7, all of which have important
effects; in comparison the gasdynamic and phenomeno-
logical models involve only a single (differing) Mach
number and no explicit 0 effects (save through Mms).
The MHD theory reduces to the gasdynamic theory in
the special cases 0 = 0 ° or MA >> Ms >> 1. The Spre-
iter et al. [1966] and Russell [1985] phenomenological
theories reappear as the special cases of the MHD the-
ory for (i) _ = 0 with 7/? = 2, and (ii) 0 = 900 with
7 = 2 and # = 0 or arbitrary _, respectively. The MHD
theory predicts that Ares and as should depend strongly
on _, MA and Ms when MA and/or Ms _< 5, with the
functional form of the MA and Ms dependence varying
with O.ln particular, varying _ from 0 ° to 90 ° at low MA
should cause Ares to vary by ,-_ 400%. Since MA and
Ms are frequently of this order out to 1 AU, the pre-
dicted MIID effects should have widespread applicabil-
ity; for instance, in calculations of 3' from measurements
of Ares. It is found that the MHD theory with Spre-
iter et al. 's value for k cannot explain observations and
MHD simulations of distant, very low MA bow shocks.
Cairns and Lyon [1994] can explain the simulation re-
suits by inserting the larger, MHD value for k into the
present paper's MHD theory. It remains possible, nev-
ertheless, that a nonlinear model for Ares = A,,s(X) is
necessary at very low MA.
Acknowledgments. Grants NAGW-2040 and ATM-
9312263 supported this research.
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(received February 3, 1994; revised June 14, 1994;
accepted September 16, 1994.)


Towards an MHD Theory for the Standoff Distance of Earth's Bow Shock

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