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# MHD Equations

In MHD, the magnetic induction and the plasma velocity are determined by the equation of motion (4.1)

and the ideal induction equation (4.2)

In addition, there is the equation of mass continuity (4.3)

and the remainder of Maxwell's equations (4.4) (4.5)

plus (4.6)

and an energy equation. This may, for example, be taken as an isothermal plasma.

Note,

1. In the equation of motion we have an extra magnetic force, (the Lorentz force), that is normal to both the magnetic field and the current. On using (4.5), this force may be written in the form (4.7)

The second term on the right hand side represents a magnetic pressure force, , acting from regions of high magnetic pressure to low magnetic pressure. The first term is interpreted as a magnetic tension force which acts when the field lines are curved. Magnetic tension is similar to the force exerted by a stretched elastic band. Interpreting the Lorentz force in terms of these two basic quantities is important since it is possible to see how the plasma will respond to a given magnetic field configuration.

2. In the Maxwell equation, which is called Ampère's Law, the last term is negligible, except for extremely rapid time variations. Dropping this term gives (4.5). In order of magnitude, the Maxwell equation, which is called Faraday's Induction Law, can be approximated by , where and are typical time and length variations of the plasma. Comparing the left hand side with the last term (the displacement current) in Ampère's law, we find that Thus, where we have used for the speed of light and the typical plasma velocity is . Hence, (4.5) is valid provided the plasma flows are small compared with the speed of light.

3. A plasma moving with a velocity experiences an electric field in addition to so that Ohm's Law is (4.8)

Normally, the left hand side is negligible so that (4.8) reduces to (4.9)

Using (4.9) to eliminate the Maxwell equation reduces to (4.2).

Subsections   Next: Effect of on Up: Magnetohydrodynamics MHD Previous: Solar Observations
Prof. Alan Hood
2000-11-06