pyoculus.toybox.cylindrical_toybox

This module provides functions to calculate equilibrium and perturbation fields in cylindrical coordinates.

The fields are calculated from the vector potential \(A\) as \(B = \nabla \times A\).

authors:

Functions

A_circularcurrentloop(rr, R, Z)

Contravariant component of the vector potential \(A\) for the circular current loop perturbation.

A_gaussian(rr, R, Z, mu, sigma, m, n[, ...])

Holonomous component of the vector potential \(A\) for the Gaussian distributed perturbation.

A_maxwellboltzmann(rr, R, Z, d, m, n[, ...])

Holonomous component of the vector potential \(A\) for the Maxwell-Boltzmann distributed perturbation.

A_r_squared(rr, R, Z, sf, shear)

Vector potential \(A_r\) component (giving the poloidal flux \(F\)) for the squared circle equilibrium field.

A_squared(rr, R, Z, sf, shear)

Holonomous component of the vector potential \(A\) for the squared circle equilibrium field.

ellpe(m)

Complete elliptic integral of the second kind.

ellpk(m)

Complete elliptic integral of the first kind.

psi_circularcurrentloop(rr, R, Z)

\(\psi\) flux function generated at \((r, \phi, z)\) by a circular current loop located at position \((R, Z)\).

psi_gaussian(rr, R, Z, mu, sigma, m, n[, ...])

Gaussian distributed \(\psi\) flux function.

psi_maxwellboltzmann(rr, R, Z, d, m, n[, ...])

Maxwell-Boltzmann distributed \(\psi\) flux function.

psi_squared(rr, R, Z)

\(\psi\) flux function for the squared circle equilibrium field.

psitob(f)

Decorator to calculate the contribution of a \(\psi\) function to a \(B\) field using the relation \(B = \nabla \times A\), with \(A_\phi \textbf{e}_\phi = \psi / r \textbf{e}_\phi\), where \(g(\textbf{e}_\phi, \textbf{e}_\phi) = 1\).

rot(f[, from_holonomous])

Decorator that calculate the curl in cylindrical coordinates, useful to get the \(B\) field from a vector potential \(A\) using the relation \(B = \nabla \times A\).
pyoculus.toybox.cylindrical_toybox.psitob(f)

Decorator to calculate the contribution of a \(\psi\) function to a \(B\) field using the relation \(B = \nabla \times A\), with \(A_\phi \textbf{e}_\phi = \psi / r \textbf{e}_\phi\), where \(g(\textbf{e}_\phi, \textbf{e}_\phi) = 1\).

Parameters:

f (function) – Function that returns the \(\psi\) field. The function must take the position vector \(\textbf{r}\) as first argument.

Returns:

Function that returns the \(B\) field.

Return type:

function

pyoculus.toybox.cylindrical_toybox.rot(f, from_holonomous=True)

Decorator that calculate the curl in cylindrical coordinates, useful to get the \(B\) field from a vector potential \(A\) using the relation \(B = \nabla \times A\).

Parameters:

  • f (function) – Function that returns the vector potential \(A\) field. The function must take the position vector \(\textbf{r}\) as first argument.

  • from_holonomous (bool) – If True, the input of the function is in holonomous coordinates \(\{\partial_r, \partial_\phi, \partial_z\}\). If False, the input is in non-holonomous coordinates \(\{\textbf{e}_r, \textbf{e}_\phi, \textbf{e}_z\}\).

Returns:

Function that returns the \(B\) field.

Return type:

function

pyoculus.toybox.cylindrical_toybox.psi_squared(rr: array, R: float, Z: float) → float64

\(\psi\) flux function for the squared circle equilibrium field. The squared circle equilibrium field is defined as:

\[\psi(r, z) = (r - R)^2 + (z - Z)^2.\]
Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the squared circle.

  • Z (float) – Z coordinate of the center of the squared circle.

Returns:

\(\psi\) flux at position rr.

Return type:

float

pyoculus.toybox.cylindrical_toybox.A_r_squared(rr: array, R: float, Z: float, sf: float, shear: float) → float64

Vector potential \(A_r\) component (giving the poloidal flux \(F\)) for the squared circle equilibrium field. Using it with the psi_squared function makes sense of sf and shear as the \(q\)-factor profile becomes:

\[q(r, z) = sf + shear / 2 * ((r - R)^2 + (z - Z)^2).\]
Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the squared circle.

  • Z (float) – Z coordinate of the center of the squared circle.

  • sf (float) – Safety factor at \(\rho = 0\).

  • shear (float) – Shear of the \(q\)-profile.

Returns:

\(A_r\) component at position rr.

Return type:

float

pyoculus.toybox.cylindrical_toybox.A_squared(rr: array, R: float, Z: float, sf: float, shear: float) → Array

Holonomous component of the vector potential \(A\) for the squared circle equilibrium field.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the squared circle.

  • Z (float) – Z coordinate of the center of the squared circle.

  • sf (float) – Safety factor at \(\rho = 0\).

  • shear (float) – Shear of the \(q\)-profile.

Returns:

\(A\) field at position rr.

Return type:

array

pyoculus.toybox.cylindrical_toybox.psi_maxwellboltzmann(rr: array, R: float, Z: float, d: float, m: int, n: int, phase_poloidal: float = 0.0, phase_toroidal: float = 0.0, A: float = 1.0, B: float = 1.0) → float64

Maxwell-Boltzmann distributed \(\psi\) flux function.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the Maxwell-Boltzmann distribution.

  • Z (float) – Z coordinate of the center of the Maxwell-Boltzmann distribution.

  • d (float) – Standard deviation of the Maxwell-Boltzmann distribution.

  • m (int) – Poloidal mode number.

  • n (int) – Toroidal mode number.

  • phase_poloidal (float) – Poloidal phase of the perturbation.

  • phase_toroidal (float) – Toroidal phase of the perturbation.

  • A (float) – Scaling factor for the R coordinate.

  • B (float) – Scaling factor for the Z coordinate.

Returns:

\(\psi\) flux at position rr.

Return type:

float

pyoculus.toybox.cylindrical_toybox.A_maxwellboltzmann(rr: array, R: float, Z: float, d: float, m: int, n: int, phase_poloidal: float = 0.0, phase_toroidal: float = 0.0, A: float = 1.0, B: float = 1.0) → Array

Holonomous component of the vector potential \(A\) for the Maxwell-Boltzmann distributed perturbation.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the Maxwell-Boltzmann distribution.

  • Z (float) – Z coordinate of the center of the Maxwell-Boltzmann distribution.

  • d (float) – Standard deviation of the Maxwell-Boltzmann distribution.

  • m (int) – Poloidal mode number.

  • n (int) – Toroidal mode number.

  • phase_poloidal (float) – Poloidal phase of the perturbation.

  • phase_toroidal (float) – Toroidal phase of the perturbation.

  • A (float) – Scaling factor for the R coordinate.

  • B (float) – Scaling factor for the Z coordinate.

Returns:

\(A\) field at position rr.

Return type:

array

pyoculus.toybox.cylindrical_toybox.psi_gaussian(rr: array, R: float, Z: float, mu: float, sigma: float, m: int, n: int, phase_poloidal: float = 0.0, phase_toroidal: float = 0.0, A: float = 1.0, B: float = 1.0) → float64

Gaussian distributed \(\psi\) flux function.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the Gaussian distribution.

  • Z (float) – Z coordinate of the center of the Gaussian distribution.

  • mu (float) – Mean of the Gaussian distribution.

  • sigma (float) – Standard deviation of the Gaussian distribution.

  • m (int) – Poloidal mode number.

  • n (int) – Toroidal mode number.

  • phase_poloidal (float) – Poloidal phase of the perturbation.

  • phase_toroidal (float) – Toroidal phase of the perturbation.

  • A (float) – Scaling factor for the R coordinate.

  • B (float) – Scaling factor for the Z coordinate.

Returns:

\(\psi\) flux at position rr.

Return type:

float

pyoculus.toybox.cylindrical_toybox.A_gaussian(rr: array, R: float, Z: float, mu: float, sigma: float, m: int, n: int, phase_poloidal: float = 0.0, phase_toroidal: float = 0.0, A: float = 1.0, B: float = 1.0) → Array

Holonomous component of the vector potential \(A\) for the Gaussian distributed perturbation.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the Gaussian distribution.

  • Z (float) – Z coordinate of the center of the Gaussian distribution.

  • mu (float) – Mean of the Gaussian distribution.

  • sigma (float) – Standard deviation of the Gaussian distribution.

  • m (int) – Poloidal mode number.

  • n (int) – Toroidal mode number.

  • phase_poloidal (float) – Poloidal phase of the perturbation.

  • phase_toroidal (float) – Toroidal phase of the perturbation.

  • A (float) – Scaling factor for the R coordinate.

  • B (float) – Scaling factor for the Z coordinate.

Returns:

\(A\) field at position rr.

Return type:

array

pyoculus.toybox.cylindrical_toybox.ellpe(m)

Complete elliptic integral of the second kind.

pyoculus.toybox.cylindrical_toybox.ellpk(m)

Complete elliptic integral of the first kind.

pyoculus.toybox.cylindrical_toybox.psi_circularcurrentloop(rr: array, R: float, Z: float) → float64

\(\psi\) flux function generated at \((r, \phi, z)\) by a circular current loop located at position \((R, Z)\).

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the circular current loop.

  • Z (float) – Z coordinate of the center of the circular current loop.

Returns:

\(\psi\) flux at position rr.

Return type:

float

pyoculus.toybox.cylindrical_toybox.A_circularcurrentloop(rr: array, R: float, Z: float) → Array

Contravariant component of the vector potential \(A\) for the circular current loop perturbation.

Parameters:

  • rr (array) – Position vector in cylindrical coordinates.

  • R (float) – R coordinate of the center of the circular current loop.

  • Z (float) – Z coordinate of the center of the circular current loop.

Returns:

\(A\) field at position rr.

Return type:

array