Magnetic configuration with concentric-circular magnetic surfaces

Assume magnetic surfaces of a magnetic configuration are given by

$$R (r, \theta) = R_0 + r \cos \theta,$$ (333)

and

$$Z (r, \theta) = r \sin \theta,$$ (334)

where $r$ is the label of magnetic surfaces (i.e., $\partial \Psi / \partial \theta \vert _r = 0$). The above parametric equations specify a series of concentric-circular magnetic surfaces with $r$ being the minor radius.

Assume the poloidal plasma current is zero, i.e., $g = R B_{\phi}$ is a constant (this constant is denoted by $g_0$ in the following). In this case the toroidal magnetic field is determined by $B_{\phi} = g_0 / R$. Assume the $q$ profile is given. Next, let us determine the poloidal magnetic field $\mathbf{B}_p$, which is given by

$$\mathbf{B}_p = \nabla \Psi \times \nabla \phi = \frac{1}{2 \pi} \nabla \Psi_p \times \nabla \phi,$$ (335)

which involves the poloidal magnetic flux $\Psi _p$. Our task is to express the poloidal magnetic flux $\Psi _p$ in terms of $q$ and $g_0$. Using $q (r) = d \Psi_t / d \Psi_p$, we obtain

$$d \Psi_p = \frac{1}{q} d \Psi_t,$$

Integrate the above equation over $r$, we obtain

$$\int_0^r d \Psi_p = \int_0^r \frac{1}{q} d \Psi_t,$$ (336)

which an be written as

$$\Psi_p (r) - \Psi_p (0) = \int_0^r \frac{1}{q (r)} d \left( \int_0^r \int_0^{2 \pi} B_t r d r d \theta \right),$$ (337)

where use has been made of $\Psi_t = \int_0^r \int_0^{2 \pi} B_t r d r d \theta$. Using $B_t = g_0 / R$ and $R = R_0 + r \cos \theta$, the above equation is written

$$\Psi_p (r) - \Psi_p (0) = \int_0^r \frac{1}{q (r)} d \left( \int_0^r \int_0^{2 \pi} \frac{g_0}{R_0 + r \cos \theta} r d r d \theta \right)$$ (338)

Using maxima (an open-source computer algebra system), the above integration over $\theta$ can be performed analytically, giving

$$\int_0^{2 \pi} \frac{g_0}{R_0 + r \cos \theta} d \theta = \frac{2 \pi g_0 r}{\sqrt{R_0^2 - r^2}} .$$ (339)

Using this, equation (338) is written as

$$\Psi_p (r) - \Psi_p (0) = \int_0^r \frac{1}{q (r)} d \left( \int_0^r \frac{2 \pi g_0 r}{\sqrt{R_0^2 - r^2}} d r \right),$$ (340)

which can be simplified as

$$\Psi_p (r) - \Psi_p (0) = \int_0^r \frac{1}{q (r)} \frac{2 \pi g_0 r}{\sqrt{R_0^2 - r^2}} d r.$$ (341)

Using this, the poloidal magnetic field in Eq. (335) is written as

$$\mathbf{B}_p = \frac{\nabla r \times \nabla \phi}{2 \pi} \frac{\partial}{\partia... ...ft( \int_0^r \frac{1}{q (r)} \frac{2 \pi g_0 r}{\sqrt{R_0^2 - r^2}} d r \right)$$

$$= \nabla r \times \nabla \phi \frac{1}{q (r)} \frac{g_0 r}{\sqrt{R_0^2 - r^2}} .$$ (342)

[Using the formulas $\nabla r = - \frac{R}{\mathcal{J}} (Z_{\theta} \hat{\mathbf{R}} - R_{\theta} \hat{\mathbf{Z}})$ and $\mathcal{J} = R (R_{\theta} Z_r - R_r Z_{\theta})$, we obtain $\mathcal{J} = - R r$ and $\nabla r = \cos \theta \hat{\mathbf{R}} + \sin \theta \hat{\mathbf{Z}}$, $\nabla \phi = \hat{\ensuremath{\boldsymbol{\phi}}} / R$. Then Eq. (342) is written as

$$\mathbf{B}_p = \frac{\cos \theta \hat{\mathbf{Z}} - \sin \theta \hat{\mathbf{R}}}{R} \frac{1}{q (r)} \frac{g_0 r}{\sqrt{R_0^2 - r^2}}$$ (343)

This is the formula for calculating the poloidal magnetic field. The magnitude of $\mathbf{B}_p$ is written as

$$B_p = \frac{1}{(R_0 + r \cos \theta)} \frac{1}{q (r)} \frac{g_0 r}{\sqrt{R_0^2 - r^2}}$$ (344)

Note that both $B_p$ and $B_{\phi}$ depend on the poloidal angle $\theta$.]

I use Eq. (341) to compute the 2D data of $\Psi$ ( $\Psi = \Psi_p / 2 \pi$) on the poloidal plane when creating a numerical G-eqdsk file for the above equilibrium.