## For non-relativistic plasma flows, the electric energy density

1 w_{E }2 =

_{0}E

2

(70)

can be neglected. To see this, we consider the ratio w_{E}/

w_{B }

= µ_{0 0 }

E^{2}/B^{2 }= (E/B)^{2}/c^{2}. How-

ever, from E + v × B = 0, we get E/B ≈ v, where v is the plasma flow speed. Hence we have w_{E}/w_{B }≈ (v/c)^{2}, so that the energy in the electric field usually is negligible in comparison to the magnetic energy.

# 4.7 The interplanetary magnetic field

The relation between the energy densities in a plasma can give us clues to the behaviour of the plasma. For instance, in the solar wind, the kinetic energy density w_{K }is much higher than w_{T }and w_{B }(compare Table 1 on p 94 in Kivelson-Russell). It is therefore reasonable to assume that the solar wind can be described in terms of normal gas flow, without consideration of magnetic effect. Since the magnetic field contains much less energy than the plasma flow, the field can be expected to follow the flow without changing it very much. Why so? Well, think about it this way. Assume w_{K }= 100w_{B }That the flow and the B-field interact mean that they exchange energy (and momentum). If the energy density is higher in the flow than in the B-field, a 1 % loss of energy from the flow is a 100 % gain of energy for the field. This means that just a little change of the flow can completely change the field. On the other hand, even a complete annihilation of the B-field will only change the energy in the flow by 1 %. This means that it is the flow that dominates the dynamics, in this case. In some other situation, where w_{B }>> w_{K }, the opposite would be true.

Thus the solar wind flow determines what the frozen-in magnetic field has to do. As the magnetic field is frozen into the expanding solar wind plasma, magnetic flux is transfered from the sun out into interplanetary space. This is why there is an interplanetary magnetic field with a strength of typically a few nT at Earth orbit. If space was a vacuum, the influence of the solar magnetic should be only through the magnetic dipole field, decaying as 1/r^{3 }(equation (41), thereby being completely negligible at Earth orbit.

# 4.8 Magnetohydrodynamics

In section 3.5, we considered the plasma as consisting of an electron fluid and an ion fluid (or several ion fluids). There is an even simpler description of the plasma, in terms of one conducting fluid. This model is known as magnetohydrodynamics (MHD). Adding the equations of motion for

ions and electrons (33), we get the MHD equation of motion

ρ_{m }

dv = j × B − ∇p dt

(71)

where ρ_{m }is the mass density (equation (13)), p = p_{i }+ p_{e }is the total pressure, and v is the weighted

mean velocity of electrons and ions,

( m i + m e ) v = m i v i + m e v e .

(72)

The magnetohydrodynamical description of a plasma is valid for slow processes on a large scale, so that it is reasonable to assume that the electron and ion fluids have the same number density.

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