Parallel force balance

Consider the force balance in the direction of $\mathbf {B}$. Dotting the equilibrium equation (43) by $\mathbf {B}$, we obtain

$$0 =\mathbf{B} \cdot \nabla P,$$ (44)

which implies that $P$ is constant along a magnetic field line. Since $\Psi$ is also constant along a magnetic field line, $P$ can be expressed in terms of only $\Psi$ on a single magnetic line. Note that this does not necessarily mean $P$ is a single-valued function of $\Psi$, i.e. $P = P (\Psi)$. For instance, $P$ can take different value on different magnetic field lines with the same value of $\Psi$ while still satisfying $\mathbf{B} \cdot \nabla P = 0$. However, for an asymmetric pressure, it is ready to prove that $P$ is indeed a single-valued function of $\Psi$ on a flux surface since $\Psi$ is a constant on a magnetic surface (however, on different magnetic surfaces with the same value of $\Psi$, $P$ can be different, refer to Sec. 13.8). On the other hand, if $P = P (\Psi)$, then we obtain

$$\mathbf{B} \cdot \nabla P = \frac{d P}{d \Psi} \mathbf{B} \cdot \nabla \Psi = 0,$$

i.e., Eq. (44) is satisfied. Therefore $P = P (\Psi)$ is equivalent to the force balance equation in the parallel (to the magnetic field) direction.