7 - A unified theory of steady states  pp. 123-132

A unified theory of steady states

By Karl Schindler

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Earlier in this part we have encountered several different forms of equations describing steady states. The simplest case is magnetohydrostatics, physically richer versions include an anisotropic pressure tensor, directed flow or gravity.

In this chapter we consider a generalized formulation for a class of steady states that includes such generalizations. The method reduces the original fluid equations to two field equations, which can be understood as generalizations of the MHS equations (5.21) and (5.22).

In the first part the effect of an external gravity force is ignored; however, it will be incorporated in the second part. In all cases we assume that the magnetic field possesses Euler potentials. Again, it should be kept in mind that in space physics environments Euler potentials have considerable applicability (Section 5.1.2).

Remarkably, it turns out that in each of the cases that we consider the steady state problem can be reduced to solving the field equations for the Euler potentials. The form of these equations is uniquely determined by a single scalar function, which also serves as a Lagrangian generating the field equations.

However, it should be kept in mind that this is not a general theory of steady states. Besides the existence of Euler potentials this procedure excludes bulk flow perpendicular to the magnetic field. Is is only for a particular symmetric configuration that we show how a perpendicular flow component can be included (Section 7.3.2).