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Space Research Institute of Russian Academy of SciencesSpace Research Institute of Russian Academy of Sciences
Terrestrial bow shockTerrestrial bow shock
analytical global modelinganalytical global modeling
M. VeriginM. Verigin
Talk to workshop:
Solar - Terrestrial Interactions from Microscale to Global Models
Sinaia, Romania, September 6-10, 2005
Микиженское озеро
Terrestrial bow shock analytical global modelingTerrestrial bow shock analytical global modeling
IntroductionIntroduction
A large number of observations of shocks and shock-related phenomena at different planets require robust and convenient BS model for data classification and analysis, e.g., • Exploration of electron and ion foreshock formation• Studies of plasma wave generation by specularly and diffusely reflected ions • Investigations of mechanisms of the solar wind deceleration in the shock foot and shock itself• Studies of the plasma wave propagation along shock surfaces• Analyses of the processes of shock overshoot formation and plasma turbulence generation inside the magnetosheath• Investigation of micropulsations generated at the bow shock and magnetopause, etc.
Huge uniform databases recently appeared with Earth’s BS crossings by:IMP-8 (1977-01-01 - 2000-06-09) Geotail (1995-01-01 - 1997-11-02)
Magion-4 (1996-02-01 - 1997-07-31)Cluster (2001-02-02 - 2002-05-20)
http://nssdc.gsfc.nasa.gov/ftphelper/bowshock.htmhttp://spdf.gsfc.nasa.gov/bowshock/
Theoretical BS models provide a reliable baseline for bow shock related studies. These are:
• gasdynamic (GD), or • magnetohydrodynamic (MHD), or
• semikinetic (hybrid) particle simulations
Theoretical approaches, especially the last two, • require sophisticated, non-transparent numerical codes and • plenty of processor time for even a single run on modern supercomputers.
e.g., Spreiter et al., 1966; Cairns & Lyon, 1995; Tanaka, 1995; Kabin et al., 2000; Brecht, 1997…
They are presently inconvenient for wide use, and practically impossible to be used for the bow shock position and shape tracing with high time resolution.
Terrestrial bow shock analytical global modelingTerrestrial bow shock analytical global modeling
IntroductionIntroduction
Empirical models are used for most of shock related studies:
Though providing general description of the average shape and position of the bow shock, these models generally• have uncertain limits of applicability• fail for unusual solar wind conditions • use conic sections for description of bow shock shape • do not provide correct asymptotic behavior and• do not consider the influence of specific heat ratio on the planetary bow shock position
e.g., Fairfield, 1971; Formisano, 1979Slavin and Holzer, 1981
Nĕmeček & Šafránková, 1991 Peredo et al., 1995; Chao et al., 2002
Chapman and Cairns, 2003
Maximal use of exact analytical theoretical relations is highly welcomed for reasonable parameterization of accurate numerical BS modeling – so called semi-empirical modeling approach.
Terrestrial bow shock analytical global modelingTerrestrial bow shock analytical global modeling
IntroductionIntroduction
MagnetopauseMagnetopause shape parameterizationshape parameterization
ro
rs
Ro
Rs
obstacle
bow shock
bo
bs
X
Y
VV1
vn
V
Different equations for magnetopause shapes
20000
2 )()(2)( xrbxrRxy e.g. detailed GD tables by Lyubimov & Rusanov [1970]
r0 – obstacle nose positionR0 – nose radius of curvature b0 – obstacle ‘bluntness’
Obstacle shape: b0 < -1 - blunt elliptic b0 = -1 - spherical -1 < b0 < 0 - elongated elliptic b0 = 0 - parabolic b0 > 0 - hyperbolic
Holzer & Slavin [1978 r = l/(1+cos) r0 = l/(1+) R0 = l b0 = 2-1
Roelof & Sibeck [1993] y2+ax2+bx+c=0 ar02+br0+c=0 R0 = ar0+b/2 b0 = -a
Petrinec &Russell [1996] r = r0(1+)/(1+cos) r0 R0 = r0(1+) b0 = 2-1
Kawano al. [1998],
parametric, - parameter
x = (r0-x0)(1+)cos/(1+cos)+x0
y = (r0-x0)(1+)sin/(1+cos) r0 R0 = (r0-x0)(1+) b0 = 2-1
Kuznetsov & Yushkov [2000] x = r0 – gy2 r0 R0 = 1/(2g) b0 = 0
Shue et al. [2000] r = r0(2/(1+cos)) r0 R0 =2r0/(2-) b0 =(6-8)(-1)/(-2)3
Spreiter et al. [1966] pressure balanced p(r) = p(r0)(r0/r)6 r0
Spreiter et al. [1970] pressure balanced
p(r) = p(r0)exp((r0-r)/H) r0
Verigin et al. [1997]
pressure balanced
p(r) = p(r0)[(1- ) (r0/r)6
+ exp((r0-r)/H)]
r0
6/)213(00 rR
2/)/811( 000 rHrR
Hr
rpprR
/)1(6
))(/1(811
2 0
0000
30/)2119(0 b
)/8/()/21(21 000 HRrHb
H
R
rpp
rH
r
H
rH
rH
rH
rpp
b0
00
0
00
20
2
0
00
0
))(/1(2
/)1(64
1/)1(6
/)1(42
/)1(6
))(/1(2
1
Basic relations for non-dissipative Basic relations for non-dissipative perfect compressible GD flowperfect compressible GD flow
0)div( V
p VV ),(
012
div2
VpV
0)( VS
pSSV )( 2
012
2
pV
VS
0
p
012
2
pV
2
2
)1(
2)1(
s
s
M
M
2
2
)1(
tan2
s
vnn
M
vnnn VV cos1 vnt VV sin1n /1
)1()1(
)1()1(1
n
npp
Rankine-Hugoniot relations Solution of R-H relations
0][ nV 0tV
0][ 2 pVn 012
2
pV
e.g. Landau & Lifshitz [1970]
Semi-empirical modeling GD approachSemi-empirical modeling GD approach
1)(1121
1,sMitimlhypersonics dx
Vd
VRdx
dS
S
Rankine-Hugoniot relation for the expansion rate of flow tube cross-section S
e.g. Biermann et al. [1967] and Wallis [1973]
1
1*
/1
/
1
reasonable to search and Rs as functions of
3/2*3/1)1(
229.1*
c
3/5*3/4)1(
3*
cRs
a) High Mach number limit
b) Low Mach number limit
Standoff distance
Nose curvature radiusShugaev [1965]
Empiric relations for BS stand off distance Empiric relations for BS stand off distance
constsM
1
30/)2119(0 b
Reference Relation Specific feature Comment
Serbin [1958] b0 = -1 , but with incorrect power
Ambrosio & Wortman [1962] b0 = -1 , but with incorrect power
Seiff [1962] b0 = -1 , unrealistic
Hida [1955], Shugaev [1964] b0 - unspecified correct power of
Spreiter et al. [1966] , unrealistic
Farris & Russell [1994] , but with incorrect power
Minailos [1973]
b0 = -1 , unrealistic, but wide region of applicability
See below the table*) - < b0 < -0.25same advantages and disadvan –tages, but describes dependency on bo also
Verigin et al. [1997] b0 - unspecified , with correct power
*03
2 R
1sM
*052.0 R
078.0 R constsM
1
3/2*3/21
~)1(~ sM
Ms
1sM
30/)2119(0 b 00 0.872/)321(1.1 RR
)1/(0.87 220 ss MMR
1sM
1sM
with,)05.176.0( 20
R2/07.0 sM
constsM
1
3/25/3**0 ))/86.087.1/(( R
.25.078.2,0
,78.2,/29.0/38.017.005.1
18.0982.0584.1
0
000
0
02
000
0
b
bbb
b
R
bbb
R *)
1sM
Comparison of the GD simulations with empirical Comparison of the GD simulations with empirical relations for standoff distance relations for standoff distance
1 2 3 4 5so n ic Ma ch n u mb e r
0
1
2
3
/r 0
- Spreiter & S tahara, 1995, H D sim ula tion- Spreiter et a l., 1966- M inailos, 1973 (R 0 /r0 = 1 .26, b 0 = -0 .786)- Farriss & R ussell, 1994- Verig in et a l., 2003 (R 0 /r0 = 1.26, b 0 = -0.786)
Empiric relations for BS Empiric relations for BS nose curvature radius nose curvature radius RR00
1sMsR
1sMsR
Reference Relation Specific feature Comment
Hayes & Probstein [1966] b0 - unspecified , but with incorrect power
Shugaev [1965] b0 - unspecified correct power of
Verigin et al. [1999] b0 - unspecified correct power of
1sMsR
*)
3
81sR
3/43/5*1
)1/(~ SMsR
Empiric relations for BS shapeEmpiric relations for BS shape Reference Relation Specific feature Comment
Van Dyke [1958] conic section
Maslennikov [1967] bS is a function of MS mainly b0 = - 1; = 5/3, 7/5, 1.15
Verigin et al. [1999] See below the table*) b0 - unspecified
22 )()(2)( xrbxrRxy ssss
)1(4
)1(
1
)1(1)1()1()1(
2
1)1(
222
22
2
2220
ssss
sssssMR
y
MR
yMRyMMRrx
)1(2.3 sM
,)067.1/)058.1(( 3/5*0 RRs
Gasdynamic flow around sphere
Maslennikov et al., 1967
• Shape of BS nose is elongated ellipsoid (bs < 0) for Ms > 2.5 • bs > 0 should be for correct downstream slope
Bow shock cannot be approximated by a simple conic section from its nose to far downstream
Is conic section good approximation Is conic section good approximation for BS surface?for BS surface?
Conic section is widely used in Earth’s BS modeling2
002 )()(2)( xrbxrRxy ss
– standoff distanceRs – radius of BS nose curvature
bs – BS nose bluntness & tan2 downstream slope
e.g., Fairfield, JGR, 6700, 1971Formisano, PSS, 1151,1979
Slavin & Holzer, JGR, 11401, 1981Nĕmeček & Šafránková, JATP, 1049, 1991
Peredo et al., JGR, 7907, 1995
1 2 3 4 5 6 7 8 9 10 11 12M ach num ber, M s
-1 .5
-1
-0.5
0
0.5
1
Bo
w s
ho
ck b
lun
tnes
s, b
s -
-
parabolo id
blunt e llip tic
spherical
e longated e llip tic
hyperbo lic
BS nose shape
- C F 4 - = 1 .15
- A r - = 5 /3
- A ir - = 7 /5
W ind tunnel experim ent
GD bow shock modeling approachGD bow shock modeling approach
a) rational function for shock shape
ss
ss
ss Rxrd
Mb
M
xrxrRxy
/)(1
11
1
)()(2)(
0
2
2
20
02
with convexity condition:
1)1(22
1
1)1(2 22
2
sss
sss Md
M
Mdb
),( 0* R
),( 0
* RRs – standoff distance determines the nose position itself; – radius of curvature determines the shock shape close to the nose;bs(Ms) – bluntness determines the shock shape further from the
nose than Rs;
ds – provides the transition from the bluntness influenced region
to the asymptotic regime;Ms – determines the asymptotic bow shock slope
Verigin et al., JGR, 2003
GD bow shock modeling approachGD bow shock modeling approach
b) empiric expressions for parameters of BS equation
6/1*
03/23/1
03/2*
00
* ),(1
)50/)1(1()1(
)(229.1),(
bbRbc
R ,
)(*
03/53/40
3/5*00
*
0
),(
)50/)1(1()1(
1)(3),(
bds
baRbcRR
,
4
2
0
002
02
1
30/),(231
4/79/),(1417/),(21),(
1
1)(
s
s
sss
M
M
be
bebebe
MMb
,
11/77/1100
0 13
)21/4(81
13
)21/4(8
68
371
29
107exp)(
bbbds
,84
97
)5/12(
1
)1(
1
10
33
)|16/7|1(
16/71
)5/12(
1
)1(
1
10
33
84
97
25
52
2
1),(
413413
8333380
04134130
b
bba
,)5/7(
11
18
13
13
24
)|10/3|20/119(
10/31
)5/7(
11
18
13
13
24
)5/12(
1
)1(
1
3
43
35
23
2
1),(
1357135720
0
13571357136813680
b
b
bb
4 2
02
5/35/3000
)9/26(
52/41201712017
5
6)(
bbbbc
,
29
15
)|33/)70/39(19|1(
33/)70/39(191
29
15
47
85
2
1)(
56650
00
b
bbd ,
39
1318
39
1318
)5/7(
1140
17
1042
)11/))5/7((16061/841()18/809(
11/))5/7((16061/8411
2
1),(
415415
25/165/160
2
5/165/160
0
b
bbe
.
Verigin et al., JGR, 2003
Basic dependencies of Basic dependencies of , , RRss, , bbss, , ddss
on on MMss and and bb00 for for = 7/5 = 7/5
Reference b0 Ms Tables by Lyubimov & Rusanov [1970]
7/5
-1 -0.25
0 0.031 0.217 0.490
1.5, 2, 3, 4, 5, 6, 7, 8, 10, 20, 2, 4, 6, 20, 2, 4, 6, 8, 10, 20, 2, 4, 6, 20, 2, 4, 6, 20, 4, 6
0 1 2
0
1
2
3
4
5
y/R
0
M s = 1.25
1.5
2
- 3 - 2 - 1 0 1
1
2
3
4
5
6
20
M s = 2
4
3
- 1 0 1x /R 0
1
2
3
4
y/R
0
20
M s = 2
4
- 3 - 2 - 1 0 1x /R 0
1
2
3
4
y /R 0
6 M s = 24
a) b)
c) d)
Comparison of the bow shocks calculated by GD code by Lyubimov & Rusanov [1970] (points) with those evaluated by relations suggested in the present paper for flows around
a) sphere (b0 = -1), b) ellipsoid (b0 = -0.25), c) paraboloid (b0 = 0), d) hyperboloid (b0 = 0.217).
Standard deviation: ~ 8·10-3R0
Expansion of the basic dependencies Expansion of the basic dependencies into 1.15 < into 1.15 < < 2 region < 2 region
Reference
b0 Ms
Maslennikov [1967] GD experiments 1.15 7/5 5/3 1.15 7/5 5/3
-
-1
1.67, 2.3, 3.24, 3.8, 6.14, 7.43, 9.17 1.84, 2.03, 2.91, 3.98, 4.35, 5.92 1.54, 1.95, 2.47, 2.78, 3.63, 5.1, 5.65 2.03, 3.06, 3.43, 4.0, 5.1, 8.9, 11.8 1.5, 2.09, 3.28, 4.2, 6.15 1.4, 2.12, 2.84, 4.1, 6.0
- 1 0 1x-b 0/R 0
1
2
3 = 7/5
4.35 2.91 2.03 1.84 M s
- 1 0 1x-b 0/R 0
1
2
3
y-b
0/R
0
= 1.15
7.43 3.8
- 1 0 1x-b 0/R 0
1
2
3 = 5/3
5.65 3.63 2.47 1.951.67 M s
- 1 0 1 2x /R 0
1
2
y/R
0
8.9 5.1 3.43 2.03 M s
- 1 0 1 2x /R 0
1
2
0 1 2x /R 0
1
2
2.3
6.15 4.2 3.28 2.09 M s 6.0 4.1 2.84 1.4
b 0=-
b 0=-1 Standard deviation: ~ 2.3·10-2R0
Comparison of the model bow shocks (smooth curves) calculated by present relations with results of GD experiments (dots) of Maslennikov et al. [1967]
Expansion of the basic dependencies Expansion of the basic dependencies into 1.15 < into 1.15 < < 2 region < 2 region
Reference b0 Ms Spreiter & Stahara [1995], dipole pressure balanced obstacle
5/3 2
-0.786 2, 4, 8 2, 4, 8
Spreiter et al. [1970], ionosphere pressure balance obstacle
5/3 -0.981, -0.878, -0.777, -0.644, -0.520, -0.400
8
Stahara et al. [1989] 2 -0.51 10, 12
Comparison of the bow shocks calculated by GD codes (points) by Spreiter et al. [1970], Stahara et al. [1989], Spreiter & Stahara [1992] for obstacles of different shapes and = 5/3, 2 with present model bow shocks (smooth curves).
Standard deviation: ~ 1.8·10-2R0
- 1 0 1 2x /r0
1
2
3
y /r 0
- 1 0 1 2x /r0
1
2
3
y /r 0
- 3 - 2 - 1 0 1 2x /r0
1
2
3
= 5/3
= 2
0 1 2x /r0
1
2
3
M s = 12
10
R 0 1.264 b 0
- 0.51
R 0 1.96
1.44
b 0
- 0.786
M s = 2
4
8
M s = 2
4
8
H /r0 = 0 .010.1
0.25
0.5
0.75
1.0
M s = 8
R 0 1.264b 0
- 0.786
• The only relation from Friedrichs I diagram consideration is:
There is no (x,y,z) variables in it and, hence, there is no idea on space position of the point where shock normal direction fits above relation.
Usual approaches to MHD Mach cone Usual approaches to MHD Mach cone slope determinationslope determination
)/1(sin)/(sin 11ms
phms MVV
vn
BS
n
BS
?v
bv
X
Y
V m s
V /2 s w +
• Usual assumption is invalid, that MHD asymptotic shock surface is asymmetric cone with a half width of
e.g., Khurana et al., 1994Bennet et al., 1997
Perinec & Russell, 1997Chapman & Cairns, 2003
• Incorrectness of usual approach comes from implicit neglecting of azimuthal component of shock normal to asymmetric cone
)cos1()cos1(sin 22222vnsvnabn MM
• Approach of waves from a point disturbance (Friedrichs II diagram) require unrealized complicated construction to obtain tangent surface to parametric Friedrichs II diagram
Usual approaches to MHD Mach cone Usual approaches to MHD Mach cone slope determinationslope determination
vn vn bv
X
Y
V m s
V s w
n
n
BS
BS e.g. Spreiter et al., PSS,14223,1966
• This approach was completed for VB solar wind flow, only (McKenzie et al., JGR, 9201, 1993).
Differential equation for MHD Mach Differential equation for MHD Mach cone determinationcone determination
0)(),,(
axxFx
• asymptotic shock is a ’cone’ surface
• cone slope at =const
adx
dcons
1)tan(
• cone cross section at -x=const
)()()(
a
const
a
xasymptotic surface equation:
22
22222
1)cossin)cossin((
aa
MMMMaa sa
sabvbv
cos))sin)((cos(
sinsin)sin)((cos
cos
)()(
2222
222
bvbvsa
bvbvbvsa
bMM
bMMba
implicit solution:
(b) b() a() ()
b
bMMbMM
bbMM
MM
b
bvbvsa
bvbvbvsa
bvbvsa
sa
2222
22
2222
22
))sin(cos(sin)sin(cos
1)sin(cos
arctan)(
Verigin et al., EPS, 33, 2003
Exact analytic solution for Exact analytic solution for distant BS cross-sectiondistant BS cross-section
-1.5 -0.5 0.5 1.5
-1.5
-0.5
0.5
1.5
M s = 6
M a = 4
bv= 60o
tan().min ( M a2, M s
2)-1
Y
Z
Fast shock Slow shocks
Alfven wings
Cross-section of the fast (smooth curve), slow (triangle-like features) shocks,and of Alfven wings (asterisks) discontinuities far downstream of the obstacle. Dashed circle - minimal possible slope of the fast shock Dot-dashed circle - maximal possible slope of the fast shock
Exact analytic solution for Exact analytic solution for distant BS cross-sectiondistant BS cross-section
M a=10
M s=10
bv=90o1
-1
-1 1
M a=7
M s=10
bv=90o1
-1
-1 1
M a=20
M s=10
bv=90o1
-1
-1 1
M a=5
M s=10
bv=90o1
-1
-1 1
M a=10
M s= 5
bv=90o1
-1
-1 1
M a=5
M s=5
bv=90o1
-1
-1 1
M a=4
M s=5
bv=90o1
-1
-1 1
M a=2
M s=5
bv=90o1
-1
-1 1
M a=2
M s=2
bv=90o1
-1
-1 1
M a=1.3
M s=2
bv=90o1
-1
-1 1
M a=4
M s=2
bv=90o1
-1
-1 1
M a=1.5
M s=2
bv=90o1
-1
-1 1
VB solar wind flow – ‘kindersurprise’ cross-section
Exact analytic solution for Exact analytic solution for distant BS cross-sectiondistant BS cross-section
Ma = Ms = M, bv =acos(1/M) solar wind flow – ‘chopped’’ bow shock
M a= M s = 2
bv= 60o
ch= 62.2o
1
-1
-1 1
M a= M s = 1.5
bv= 48.2o
ch= 73.4o
1
-1
-1 1
M a= M s = 5
bv= 78.5o
ch= 54.3o
1
-1
-1 1
M a= M s = 10
bv= 84.3o
ch= 53.4o
1
-1
-1 1
Chopping angle:
54arctan2
2M
Mch
1),min(
1)tan(
22
aa
consMM
‘ 1 ‘ ==>
Approach for generalization Approach for generalization of GD shock model to MHDof GD shock model to MHD flowflow
1. Use Ms = 1/sin() everywhere in GD expressions, with downstream slope being taken from exact analytic MHD solution described above
2. Use compression ratio everywhere in GD expressions taken from solution of following MHD equation:
2
2
4
22
22
23 2
1)1()1(
2
)1(
)2(
1
1
s
bv
a
bv
sa
bv
M
Cos
M
Cos
MM
Cos
,041
2)1()1(
122
222
sabvbv
a MMCosCos
M
2. Use b-2/5(b and b-2/5R0(b) everywhere in GD expressions instead of () and R0() with MHD factor .
22 /cos1 abv Mb
Verigin et al., EPS, 1001, 2001
10 12 14 16 18 20 22 24 26
0
10
20
30
B, n
T
10 12 14 16 18 20 22 24 26
10
20
30
40
R,
Re
10 12 14 16 18 20 22 24 26
1
10
Mac
h nu
mbe
r
A lfvenic
son ic
10 12 14 16 18 20 22 24 26U T, hours
0.1
1.0
10.0
ram
pre
ssur
e, n
Pa
M ay 5 - 6 , 1999 bow shock crossings
m ode l
spacecraft
a
b
c
d
12 14 16 18 20 22 24 26
0
10
20
30
B, n
T
12 14 16 18 20 22 24 26
10
20
30
40
R,
Re
12 14 16 18 20 22 24 26
1
10
Mac
h nu
mbe
r
A lfvenic
son ic
12 14 16 18 20 22 24 26U T, hours
0.1
1.0
10.0
ram
pre
ssur
e, n
Pa
D ecem ber 22 - 23, 1995bow shock crossings
spacecraft
m odel
a
b
c
d
Examples of variations of modeled BS Examples of variations of modeled BS relative to Wind positionrelative to Wind position
Verigin et al., Adv. Space Res., 28, No. 6, 857, 2001.
10 10020 30 50
W in d o b se rve d d ista n ce to th e b o w sh o ck, R e
10
100
20
30
50
Mo
de
l dis
tan
ce to
the
bo
w s
ho
ck,
Re
- BS inbound crossings
10 10020 30 50
W in d o b se rve d d ista n ce to th e b o w sh o ck, R e
10
100
20
30
50
Mo
de
l dis
tan
ce to
the
bo
w s
ho
ck,
Re
- BS outbound crossings
rmodel = (1.010.02)·robs rmodel = (0.970.02)·robs
Small difference between the coefficients 1.01 obtained for the inbound crossings and 0.97 for outbound crossings as a consequence of motion of the bow shock.
Scatter of modeled relative observed BS positions Scatter of modeled relative observed BS positions during its inbound/outbound crossings by Windduring its inbound/outbound crossings by Wind
Kotova et al., Adv. Space Res., 2005 (accepted).
Bow shock is located closer to the obstacle for small bv and farther – for bv ~ 90 for specified Ma, Ms.
The increase of bow shock subsolar distance with increasing bv is particularly well seen for small Ma values. 0 20 40 60 80
bv, degree
12
14
16
18
2010 < M
s < 20
3.2 < Ma
< 4.8
model
Nor
mal
ized
nos
e B
S p
ositi
on, R
e
Dependence of the terrestrial BS position on the IMF Dependence of the terrestrial BS position on the IMF direction: direction: Prognoz, Prognoz 2,4,5,6,9, Interball 1,Prognoz, Prognoz 2,4,5,6,9, Interball 1,
Magion 4 dataMagion 4 data
4 8 12 16 20 24
10
12
14
16
18
4 8 12 16 20 24
10
12
14
16
18
4 8 12 16 20 24
10
12
14
16
18
Bo
w s
ho
ck s
ub
sola
r d
ista
nce
, Re
M a M aM a
0 o < bv< 10 o
M s = 11±1.540 o < bv< 50 o
M s = 12±180 o < bv< 90 o
M s = 13±2
Bow shock approach to Earth with Bow shock approach to Earth with decreasing of Mdecreasing of Maa in field aligned flows according to in field aligned flows according to
INTERBALL/MagionINTERBALL/Magion 4 observations 4 observations
With decreasing of Alvenic Mach number standoff distance of the bow shock rapidly gets away from the Earth for VB flow of the solar wind, gets away from the planet slower for VB flow, and tends to approach to Earth for V B flow.
Verigin et al., Adv. Space Res., 28, No. 6, 857, 2001.
Earth’s bow shock terminator anisotropy as seen Earth’s bow shock terminator anisotropy as seen from Wind observations analysisfrom Wind observations analysis
Ygipm
Zgipm Zgipm
0
90
180
270
0 10 20
0o < bv < 30o
0
90
180
270
0 10 20
1.06 < (VV )m s< 1 .19 1.24 < (VV )m s< 1 .34
0
90
180
270
0 10 20
0
90
180
270
0 10 20
1.04 < (VV )m s< 1 .15 1.3 < (VV )m s< 1.41
0
90
180
270
0 10 20
0
90
180
270
0 10 20
1.03 < (VV )m s< 1 .17 1.33 < (VV )m s< 1 .41
30o < bv < 60o
60o < bv < 90o
Ygipm
Ygipm
Terminator cross-section of the BS is elongated in Zgipm direction for the solar wind flow with highly anisotropic Friedrichs diagram, and the center of the approximating ellipsoid shifts in Ygipm direction (for non field aligned and non quasi-perpendicular flow).
Verigin et al., EPS, 1001, 2001
-70 -50 -30 -10 10 30
Xse, Re
-50
-30
-10
10
30
50
May 11, 1999
Apr. 5, 1999
Ys/c plane, Re
Most distant and high latitude BS crossings Most distant and high latitude BS crossings by Wind observationsby Wind observations
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