The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 22

to Eq. ( 1.10 ) and making formally " ! 0 yields

ŒV n D 0; (1.41)

where square brackets denote the jump of the corresponding value and V n D V z
stands for the normal component of the fluid velocity. The implication here is that
the flux of mass is continuous across the shock surface.
Similarly, applying the operator ( 1.40 )toEq.( 1.39 ) and taking into account of
Eq. ( 1.12 ) gives the boundary conditions
V n V C P C n

B 2
2 0

B n B
0

D 0; (1.42)

where n is the unit vector perpendicular to the shock surface. Integrating of Eq. ( 1.3 )
across the discontinuity surface yields

ŒB n D 0: (1.43)

One more boundary equation describes the energy balance at the bow shock. We
refer the reader to Akasofu and Chapman ( 1972 ) and Landau and Lifshitz ( 1982 )
for a more complete treatise on the boundary conditions on the bow shock.
To study the magnetic field at the bow shock in a little more detail, it is necessary
at this point to consider the electric current flowing in the vicinity of the shock.
Taking the cross product of Eq. ( 1.36 ) with B , the current density perpendicular to
B is given by

B 2 B dt C B r P :

1 d V

j ? D (1.44)

As illustrated in Fig. 1.8 , because of the deceleration of the solar wind in the
bow shock, the eastward current arises in the region between the bow shock and
magnetopause. The current flowing along the magnetopause is shown with an arrow
in Fig. 1.8 , which shows a view looking down onto the ecliptic plane. Notice that this
current, called the Chapman-Ferraro/magnetopause current, exists in a thin plane
sheet/magnetosheath extending also out of the paper. This surface current is due to
both the solar wind ions and electrons since the planetary magnetic field deflects the
solar wind ions to the east and electrons to the west as they move to the Earth. The
current flowing around the magnetopause results in the generation of the secondary
magnetic field. As is seen from Fig. 1.8 , this field is parallel to the Earth's field on the
earthward side of the current sheet. Conversely, on the sunward side of the current
the secondary magnetic field is directed in opposition to the Earth's field. To the first
order the secondary magnetic field cancels nearly the Earth's field in the solar wind
whereas the value of magnetic field is approximately doubled on the earthward side
of the current sheet.