Space Research Institute of Russian Academy of Sciences Terrestrial bow shock analytical global...

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.



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.



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


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


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


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).


-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

Space Research Institute of Russian Academy of Sciences Terrestrial bow shock analytical global...

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