Axisymmetric magnetic field

Due to the divergence-free nature of magnetic field, i.e., $\nabla \cdot \mathbf{B}= 0$, magnetic field can be generally expressed as the curl of a vector potential,

$$\mathbf{B}= \nabla \times \mathbf{A}.$$ (1)

We consider axisymmetric magnetic field. The axial symmetry means that, when expressed in the cylindrical coordinate system $(R, \phi, Z)$, the components of $\mathbf {B}$, namely $B_R$, $B_Z$, and $B_{\phi}$, are all independent of $\phi$. For this case, it can be proved that an axisymmetric vector potential $\mathbf{A}$ suffices for expressing the magnetic field, i.e., all the components of the vector potential $\mathbf{A}$ can also be taken independent of $\phi$. Using this, Eq. (1) is written

$$\mathbf{B}= - \frac{\partial A_{\phi}}{\partial Z} \hat{\mathbf{R... ... \frac{\partial A_Z}{\partial R} \right) \hat{\ensuremath{\boldsymbol{\phi}}} .$$ (2)

In tokamak literature, $\hat{\ensuremath{\boldsymbol{\phi}}}$ direction is called the toroidal direction and $(R, Z)$ plane is called the poloidal plane. Equation (2) indicates that the two poloidal components of $\mathbf {B}$, namely $B_R$ and $B_Z$, are determined by a single component of $\mathbf{A}$, namely $A_{\phi}$. This motivates us to define a function $\Psi (R, Z)$ by

$$\Psi (R, Z) \equiv R A_{\phi} (R, Z) .$$ (3)

Then Eq. (2) implies the poloidal components, $B_R$ and $B_Z$, can be written as

$$B_R = - \frac{1}{R} \frac{\partial \Psi}{\partial Z}$$ (4)

and

$$B_Z = \frac{1}{R} \frac{\partial \Psi}{\partial R} .$$ (5)

It is obvious that it is the property of being axisymmetric and divergence-free that enables the two components of $\mathbf {B}$, $B_R$ and $B_Z$, to be expressed in terms of a single function $\Psi (R, Z)$. (The function $\Psi$ is usually called the ``poloidal flux function'' in tokamak literature. The reason for this nomenclature will become clear when we discuss the physical meaning of $\Psi$ in Sec. 1.4.) Furthermore, it is ready to verify $\Psi$ is constant along a magnetic field line, i.e. $\mathbf{B} \cdot \triangledown \Psi = 0$. [Proof:

$$\mathbf{B} \cdot \triangledown \Psi = \mathbf{B} \cdot \left( \frac{\partial \Psi}{\partial R} \hat{\mathbf{R}} + \frac{\partial \Psi}{\partial Z} \hat{\mathbf{Z}} \right)$$

$$= - \frac{1}{R} \frac{\partial \Psi}{\partial Z} \frac{\partial \Ps... ...+ \frac{1}{R} \frac{\partial \Psi}{\partial R} \frac{\partial \Psi}{\partial Z}$$

$$= 0.$$ (6)

] For later use, define $g \equiv R B_{\phi} (R, Z)$, then the toroidal component of the magnetic field is written

$$\mathbf{B}_{\phi} = B_{\phi} \hat{\ensuremath{\boldsymbol{\phi}}} = \frac{g}{R} \hat{\ensuremath{\boldsymbol{\phi}}} = g \nabla \phi .$$ (7)

Using Eqs. (4), (5), and (7), a general axisymmetric magnetic field can be written as

$$\mathbf{B} = B_R \hat{\mathbf{R}} + B_Z \hat{\mathbf{Z}} + B_{\phi} \hat{\ensuremath{\boldsymbol{\phi}}}$$

$$= - \frac{1}{R} \frac{\partial \Psi}{\partial Z} \hat{\mathbf{R}} + \frac{1}{R} \frac{\partial \Psi}{\partial R} \hat{\mathbf{Z}} + g \nabla \phi$$

$$= \frac{1}{R} \nabla \Psi \times \hat{\ensuremath{\boldsymbol{\phi}}} + g \nabla \phi$$

$$= \nabla \Psi \times \nabla \phi + g \nabla \phi .$$ (8)