Analytic Fields

Arnold-Beltrami-Childress (ABC) field in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \big( a \, \sin(z) + c \, \cos(y) , \, b \, \sin(x) + a \, \cos(z) , \, c \, \sin(y) + b \, \cos(x) \big)^T\]

resulting in the magnetic field $B(x,y,z) = A(x,y,z)$.

Parameters: a, b, c

Axisymmetric tokamak equilibrium in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \frac{1}{2} \frac{B_0}{q_0} \, \bigg( \frac{q_0 R_0 x z - r^2 y}{R^2} , \, \frac{q_0 R_0 y z + r^2 x}{R^2} , \, - q_0 R_0 \, \ln \bigg( \frac{R}{R_0} \bigg) \bigg)^T ,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = \frac{B_0}{q_0} \, \bigg( - \frac{q_0 R_0 y + x z}{R^2} , \, \frac{q_0 R_0 x - y z}{R^2} , \, \frac{R - R_0}{R} \bigg)^T ,\]

where $R = \sqrt{ x^2 + y^2 }$ and $r = \sqrt{ (R - R_0)^2 + z^2 }$.

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• q₀: safety factor at magnetic axis

Axisymmetric tokamak equilibrium in (R,Z,ϕ) coordinates with covariant components of the vector potential given by

\[A (R, Z, \phi) = \frac{B_0}{2} \, \bigg( R_0 \, \frac{Z}{R} , \, - R_0 \, \ln \bigg( \frac{R}{R_0} \bigg) , \, - \frac{r^2}{q_0} \bigg)^T ,\]

resulting in the magnetic field with covariant components

\[B (R, Z, \phi) = \frac{B_0}{q_0} \, \bigg( - \frac{Z}{R} , \, \frac{R - R_0}{R} , \, - q_0 R_0 \bigg)^T ,\]

where $r = \sqrt{ (R - R_0)^2 + Z^2 }$.

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• q₀: safety factor at magnetic axis

Axisymmetric tokamak equilibrium in (r,θ,ϕ) coordinates with covariant components of the vector potential given by

\[A (r, \theta, \phi) = B_0 \, \bigg( 0 , \, \frac{r R_0}{\cos (\theta)} - \bigg( \frac{R_0}{\cos (\theta)} \bigg)^2 \, \ln \bigg( \frac{R}{R_0} \bigg) , \, - \frac{r^2}{2 q_0} \bigg)^T ,\]

resulting in the magnetic field with covariant components

\[B (r, \theta, \phi) = \frac{B_0}{q_0} \, \bigg( 0 , \, \frac{r^2}{R}, \, q_0 R_0 \bigg)^T ,\]

where $R = R_0 + r \cos \theta$.

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• q₀: safety factor at magnetic axis

Penning trap with uniform magnetic field in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 \, ( 0, x, 0)^T ,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 0, 0, 1)^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

• B₀: B-field strength
• E₀: E-field strength

Penning trap with magnetic bottle in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 / 2 \, ( -y , x, 0)^T - B_1 \, (xz, yz, (x^2 + y^2)/2 - z^2)^T,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 0, 0, 1)^T - B_1 \, ( yz^2 - y^3 / 6, x^3 / 6, xyz )^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

• B₀: B-field strength
• Bₚ: B-field perturbation strength
• E₀: E-field strength

Penning trap with asymmetric magnetic field in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 / 2 \, ( -y , x-z/6, y / 6)^T + B_1 / 2 \, (z^2 - y^2, z^2 - x^2, y^2 - x^2)^T,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 1/3, 0, 1)^T + B_1 \, ( y-z, x+z, y-x )^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

• B₀: B-field strength
• Bₚ: B-field perturbation strength
• E₀: E-field strength

Singular magnetic field in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \frac{B_0}{\sqrt{(x^2 + y^2)}^3} \big( y , \, - x , \, 0 \big)^T\]

resulting in the magnetic field with covariant components

\[B(x,y,z) = B_0 \, \begin{pmatrix} 0 \\ 0 \\ (x^2 + y^2)^{-3/2} \\ \end{pmatrix}\]

Parameters: B₀

Symmetric quadratic mangetic field in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \frac{B_0}{4} \big( - y \, (2 + x^2 + y^2) , \, x \, (2 + x^2 + y^2) , \, 0 \big)^T\]

resulting in the magnetic field with covariant components

\[B(x,y,z) = B_0 \, \begin{pmatrix} 0 \\ 0 \\ 1 + x^2 + y^2 \\ \end{pmatrix}\]

Parameters: B₀

Symmetric Solov'ev equilibrium in (R,Z,phi) coordinates. Based on McCarthy, Physics of Plasmas 6, 3554, 1999.

The covariant components of the vector potential are given by

\[A (x, y) = \frac{B_0}{2} \, \bigg( 0 , \, 0 , \, - \frac{\alpha}{4} (R_0 + x)^4 - \beta y^2 \bigg)^T ,\]

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• α, β: free constants

Axisymmetric Solov'ev equilibra in (R/R₀,Z/R₀,ϕ) coordinates. Based on Cerfon & Freidberg, Physics of Plasmas 17, 032502, 2010, and Freidberg, Ideal Magnetohydrodynamics, 2014.

The covariant components of the vector potential are given by

\[A (x, y, \phi) = \left( \frac{B_0 R_0}{2} \, \frac{y}{x} , \, - \frac{B_0 R_0}{2} \, \ln x , \, \psi(x,y) \right)^T ,\]

with $x = R/R_0$ and $y = Z/R_0$. The normalised poloidal flux $\psi$ is given by

\[\psi (x,y) = \psi_0 + \sum \limits_{i=1}^{7} c_i \psi_i (x,y) ,\]

with

\[\begin{aligned} \psi_{0} &= \frac{x^4}{8} + \alpha \left( \frac{1}{2} x^2 \, \ln x - \frac{x^4}{8} \right) , \\ \psi_{1} &= 1 , \\ \psi_{2} &= x^2 , \\ \psi_{3} &= y^2 - x^2 \, \ln x , \\ \psi_{4} &= x^4 - 4 x^2 y^2 , \\ \psi_{5} &= 2 y^4 9 y^2 x^2 + 3 x^4 \, \ln x - 12 x^2 y^2 \, \ln x , \\ \psi_{6} &= x^6 - 12 x^4 y^2 + 8 x^2 y^4 , \\ \psi_{7} &= 8 y^6 - 140 y^4 x^2 + 75 y^2 x^4 - 15 x^6 \, \ln x + 180 x^4 y^2 \, \ln x - 120 x^2 y^4 \, \ln x . \end{aligned}\]

This formula describes exact solutions of the Grad-Shafranov equation with up-down symmetry. The constants $c_i$ are determined from boundary constraints on $\psi$, that are derived from the following analytic model for a smooth, elongated "D" shaped cross section:

\[\begin{aligned} x &= 1 + \epsilon \, \cos (\tau + \delta_0 \, \sin \tau) , \\ y &= \epsilon \kappa \, \sin (\tau) , \end{aligned}\]

where $0 \leq \tau < 2 \pi$, $\epsilon = a / R_0$ is the inverse aspect ratio, $\kappa$ the elongation, and $\sin \delta_0 = \delta$ is the triangularity.

Defining three test points, namely

• the high point $(1 - \delta \epsilon, \kappa \epsilon)$,
• the inner equatorial point $(1 - \epsilon, 0)$,
• and the outer equatorial point $(1 + \epsilon, 0)$,

the following geometric constraints can be posed on the solution:

\[\begin{aligned} \psi (1 + \epsilon, 0) &= 0 , \\ \psi (1 - \epsilon, 0) &= 0 , \\ \psi (1 - \delta \epsilon, \kappa \epsilon) &= 0 , \\ \psi_{x} (1 - \delta \epsilon, \kappa \epsilon) &= 0 , \\ \psi_{yy} (1 + \epsilon, 0) &= - N_1 \psi_{x} (1 + \epsilon, 0) , \\ \psi_{yy} (1 - \epsilon, 0) &= - N_2 \psi_{x} (1 - \epsilon, 0) , \\ \psi_{xx} (1 - \delta \epsilon, \kappa \epsilon) &= - N_3 \psi_y (1 - \delta \epsilon, \kappa \epsilon) . \end{aligned}\]

The first three equations define the three test points, the fourth equations enforces the high point to be a maximum, and the last three equations define the curvature at the test points.

The coefficients $N_j$ can be found from the analytic model cross section as

\[\begin{aligned} N_1 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = 0} = - \frac{(1 + \delta_0)^2}{\epsilon \kappa^2} , \\ N_2 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = \pi} = \hphantom{-} \frac{(1 - \delta_0)^2}{\epsilon \kappa^2} , \\ N_3 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = \pi/2} = - \frac{\kappa}{\epsilon \, \cos^2 \delta_0} . \end{aligned}\]

For a given value of the constant $a$ above conditions reduce to a set of seven linear inhomogeneous algebraic equations for the unknown $c_i$, which can easily be solved.

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• ϵ: inverse aspect ratio
• κ: elongation
• δ: triangularity
• α: free constant, determined to match a given beta value

Axisymmetric Solov'ev equilibra with X-point in (R/R₀,Z/R₀,phi) coordinates. Based on Cerfon & Freidberg, Physics of Plasmas 17, 032502, 2010, and Freidberg, Ideal Magnetohydrodynamics, 2014.

The covariant components of the vector potential are given by

\[A (x, y, \phi) = \left( \frac{B_0 R_0}{2} \, \frac{y}{x} , \, - \frac{B_0 R_0}{2} \, \ln x , \, \psi(x,y) \right)^T ,\]

with $x = R/R_0$ and $y = Z/R_0$. The normalised poloidal flux $\psi$ is given by

\[\psi (x,y) = \psi_0 + \sum \limits_{i=1}^{12} c_i \psi_i (x,y) ,\]

with

\[\begin{aligned} \psi_{0} &= \frac{x^4}{8} + \alpha \left( \frac{1}{2} x^2 \, \ln x - \frac{x^4}{8} \right) , \\ \psi_{1} &= 1 , \\ \psi_{2} &= x^2 , \\ \psi_{3} &= y^2 - x^2 \, \ln x , \\ \psi_{4} &= x^4 - 4 x^2 y^2 , \\ \psi_{5} &= 2 y^4 9 y^2 x^2 + 3 x^4 \, \ln x - 12 x^2 y^2 \, \ln x , \\ \psi_{6} &= x^6 - 12 x^4 y^2 + 8 x^2 y^4 , \\ \psi_{7} &= 8 y^6 - 140 y^4 x^2 + 75 y^2 x^4 - 15 x^6 \, \ln x + 180 x^4 y^2 \, \ln x - 120 x^2 y^4 \, \ln x , \\ \psi_{8} &= y , \\ \psi_{9} &= y x^2 , \\ \psi_{10} &= y^3 - 3 y x^2 \, \ln x , \\ \psi_{11} &= 3 y x^4 - 4 y^3 x^2 , \\ \psi_{12} &= 8 y^5 - 45 y x^4 - 80 y^3 x^2 \, \ln x + 60 y x^4 \, \ln x . \end{aligned}\]

This formula describes exact solutions of the Grad-Shafranov equation with up-down asymmetry. The constants $c_i$ are determined from boundary constraints on $\psi$, that are derived from the following analytic model for a smooth, elongated "D" shaped cross section:

\[\begin{aligned} x &= 1 + \epsilon \, \cos (\tau + \arcsin \delta \, \sin \tau) , \\ y &= \epsilon \kappa \, \sin (\tau) , \end{aligned}\]

where $0 \leq \tau < 2 \pi$, $\epsilon = a / R_0$ is the inverse aspect ratio, $\kappa$ the elongation, and $\sin \delta_0 = \delta$ is the triangularity.

Defining four test points, namely

• the high point $(1 - \delta \epsilon, \kappa \epsilon)$,
• the inner equatorial point $(1 - \epsilon, 0)$,
• and the outer equatorial point $(1 + \epsilon, 0)$,
• the position of the X-point $(x_{\mathrm{sep}}, y_{\mathrm{sep}})$,

the following geometric constraints can be posed on the solution:

\[\begin{aligned} \psi (1 + \epsilon, 0) &= 0 , \\ \psi (1 - \epsilon, 0) &= 0 , \\ \psi (1 - \delta \epsilon, \kappa \epsilon) &= 0 , \\ \psi (x_{\mathrm{sep}}, y_{\mathrm{sep}}) &= 0 , \\ \psi_{y} (1 + \epsilon, 0) &= 0 , \\ \psi_{y} (1 - \epsilon, 0) &= 0 , \\ \psi_{x} (1 - \delta \epsilon, \kappa \epsilon) &= 0 , \\ \psi_{x} (x_{\mathrm{sep}}, y_{\mathrm{sep}}) &= 0 , \\ \psi_{y} (x_{\mathrm{sep}}, y_{\mathrm{sep}}) &= 0 , \\ \psi_{yy} (1 + \epsilon, 0) &= - N_1 \psi_{x} (1 + \epsilon, 0) , \\ \psi_{yy} (1 - \epsilon, 0) &= - N_2 \psi_{x} (1 - \epsilon, 0) , \\ \psi_{xx} (1 - \delta \epsilon, \kappa \epsilon) &= - N_3 \psi_y (1 - \delta \epsilon, \kappa \epsilon) . \end{aligned}\]

The first four equations define the four test points, the fifth and sixth equations define the up-down symmetry, the seventh equations enforces the high point to be a maximum, the eighth and ninth eqaution set the $x$- and $y$-components of the magnetic field at the X-point to zero, and the last three equations define the curvature at the first three test points.

The coefficients $N_j$ can be found from the analytic model cross section as

\[\begin{aligned} N_1 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = 0} = - \frac{(1 + \delta_0)^2}{\epsilon \kappa^2} , \\ N_2 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = \pi} = \hphantom{-} \frac{(1 - \delta_0)^2}{\epsilon \kappa^2} , \\ N_3 &= \left[ \frac{d^2 x}{dy^2} \right]_{\tau = \pi/2} = - \frac{\kappa}{\epsilon \, \cos^2 \delta_0} . \end{aligned}\]

For a given value of the constant $a$ above conditions reduce to a set of seven linear inhomogeneous algebraic equations for the unknown $c_i$, which can easily be solved.

Parameters:

• R₀: position of magnetic axis
• B₀: B-field at magnetic axis
• ϵ: inverse aspect ratio
• κ: elongation
• δ: triangularity
• α: free constant, determined to match a given beta value
• xsep: x position of the X point
• ysep: y position of the X point

θ-pinch equilibrium in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \frac{B_0}{2} \big( - y , \, x , \, 0 \big)^T\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = \big( 0 , \, 0 , \, B_0 \big)^T .\]

Parameters: B₀: B-field at magnetic axis