between the solar wind and a strongly active comet show that a bilobed magnetic
field configuration arises. The International Cometary Explorer (ICE) spacecraft flew
~7800 km down the plasma tail of 21P/Giacobini-Zinner making magnetic field
observations. It confirmed the presence of a well developed magnetotail (Slavin et al.
1986). The tail was indeed composed of two lobes of opposite magnetic polarity
(note the direction of the arrows on the magnetic field lines in Fig. 5.13, right)
separated by a 1500 km thick plasma sheet.
In Fig. 5.13 (right), we can see the diamagnetic cavity and the high field strengths
surrounding it in the stagnation region.
The figures illustrate how MHD can be used to treat the mutual interaction
between flows of electrically conducting, non-magnetic fluids and magnetic fields.
We have already seen the Navier-Stokes equations for fluid flow. The basic MHD
equation incorporates the Lorentz force into the Navier-Stokes equations. Previously
we saw the equation
∂ρ p u p
∂t
þ u p Á ∇
À
Á ρ p u p þ ρ p u p ∇ Á u p
À
Á þ ∇p p À υρ p ∇
2 u p ¼ ρ p F
ð5:29Þ
and incorporating the Lorentz force, we obtain
∂ρ p u p
∂t
þ u p Á ∇
À
Á ρ p u p þ ρ p u p ∇ Á u p
À
Á þ ∇p p À υρ p ∇
2 u p þ J Â B
ð
Þ
¼ ρ p F
ð5:30Þ
By using Ampere’s law, the Lorentz force can be re-written (see Davidson 2001)
so that
J Â B = B Á ∇
ð
Þ
B
μ 0
À ∇
B
2
2μ 0
ð5:31Þ
from which we can again recognize the magnetic pressure as the second term on the
right hand side. The momentum equation still contains the ρ p F term on the right side
which can be considered as a means to include additional, potentially minor, forces
(e.g. gravity).
Figure 5.12 is a result from a multifluid simulation and follows work by Najib
et al. (2011) in which there are separate mass, momentum, and energy equations for
each ion species in the total fluid. These are coupled through ensuring charge
neutrality by defining the electron density as
n e ¼
X
k T
q k T n k T
ð5:32Þ
where q kT is the charge of species, k T , and n kT is the number density of that species
and through source terms which include ion-ion elastic collisions that couple the ion
5.3 Dynamics of the Interaction
417
field configuration arises. The International Cometary Explorer (ICE) spacecraft flew
~7800 km down the plasma tail of 21P/Giacobini-Zinner making magnetic field
observations. It confirmed the presence of a well developed magnetotail (Slavin et al.
1986). The tail was indeed composed of two lobes of opposite magnetic polarity
(note the direction of the arrows on the magnetic field lines in Fig. 5.13, right)
separated by a 1500 km thick plasma sheet.
In Fig. 5.13 (right), we can see the diamagnetic cavity and the high field strengths
surrounding it in the stagnation region.
The figures illustrate how MHD can be used to treat the mutual interaction
between flows of electrically conducting, non-magnetic fluids and magnetic fields.
We have already seen the Navier-Stokes equations for fluid flow. The basic MHD
equation incorporates the Lorentz force into the Navier-Stokes equations. Previously
we saw the equation
∂ρ p u p
∂t
þ u p Á ∇
À
Á ρ p u p þ ρ p u p ∇ Á u p
À
Á þ ∇p p À υρ p ∇
2 u p ¼ ρ p F
ð5:29Þ
and incorporating the Lorentz force, we obtain
∂ρ p u p
∂t
þ u p Á ∇
À
Á ρ p u p þ ρ p u p ∇ Á u p
À
Á þ ∇p p À υρ p ∇
2 u p þ J Â B
ð
Þ
¼ ρ p F
ð5:30Þ
By using Ampere’s law, the Lorentz force can be re-written (see Davidson 2001)
so that
J Â B = B Á ∇
ð
Þ
B
μ 0
À ∇
B
2
2μ 0
ð5:31Þ
from which we can again recognize the magnetic pressure as the second term on the
right hand side. The momentum equation still contains the ρ p F term on the right side
which can be considered as a means to include additional, potentially minor, forces
(e.g. gravity).
Figure 5.12 is a result from a multifluid simulation and follows work by Najib
et al. (2011) in which there are separate mass, momentum, and energy equations for
each ion species in the total fluid. These are coupled through ensuring charge
neutrality by defining the electron density as
n e ¼
X
k T
q k T n k T
ð5:32Þ
where q kT is the charge of species, k T , and n kT is the number density of that species
and through source terms which include ion-ion elastic collisions that couple the ion
5.3 Dynamics of the Interaction
417