Magnetohydrostatics

Abstract

A comparison of the sizes of terms in the equation of motion

$$ \rho \frac{{Dv}}{{Dt}} = - \nabla p + k \times B + \rho g $$

(2.21)

shows that the inertial terms on the left-hand side may be neglected when the flow speed is much smaller than both the sound speed (γp ₀/ρ ₀)^1/2, the Alfvén speed B ₀/(μρ ₀)^1/2 and the gravitational free-fall speed (2gl ₀)^l/2 for a vertical scale-length l ₀. The result is a magnetohydrostatic balance

$$ O = - \nabla p + j \times B + \rho g $$

(3.1)

between the pressure gradient, the Lorentz force and the gravitational force. The object of the present chapter is to solve Equation (3.1) together with the subsidiary equations

$$ j = curl\,B/\mu $$

(3.2)

$$ div\,B = 0 $$

(3.3)

$$ \rho = \frac{{mp}} {{{k_B}T}} $$

(3.4)

And (in general) an energy equation for the temperature.