Magnetic pressure and tension

The magnetic force (per unit volume) in the equation for fluid motion () may be re-expressed as

$${\bf J \times B} = \frac{1}{\mu_{0}}({\mbox{\boldmath$\nabla ... ...{1}{\mu_{0}}({\mbox{\boldmath$B \cdot \nabla$ }}) {\bf B} \, .$$ (3.46)

The first term corresponds to the magnetic pressure, with $p_{B} = B^{2}/(2 \mu_{0})$. An important diagnostic of a plasma is the plasma beta, defined as the ratio of plasma thermal pressure to the magnetic pressure:

$$\beta = \frac{p}{B^{2}/2\mu_{0}} \, .$$ (3.47)

The second term can be further decomposed into two terms:

$$\frac{1}{\mu_{0}}({\mbox{\boldmath$B \cdot \nabla$ }}) {\bf B... ...right) + \frac{B^{2}}{\mu_{0}} \frac{\bf {\hat n}}{R_{c}} \, ,$$ (3.48)

where ${\bf {\hat b}}$ is a unit vector in the direction of ${\bf B}$and ${\bf {\hat n}}$ is the normal pointing towards the centre of curvature, defined by ( ${\hat b} \cdot \nabla) {\hat b}$ $= {\bf {\hat n}}/R_{c}$, where R_c is the radius of curvature of the field line. The first term cancels out the magnetic pressure gradient term in () in the direction along the field lines. This implies that the magnetic pressure force is not isotropic; only perpendicular components of $\nabla p_{B}$ exert force on the plasma. The second term in () corresponds to the magnetic tension force which is directed towards the centre of curvature of the field lines and thus acts to straighten out the field lines. A suitable analogy is the tension force transferred to an arrow by the stretched string of a bow. In this case the tension force pushes the plasma in the direction that will reduce the length of the field lines.