scotty.hamiltonian module

scotty.hamiltonian module#

class scotty.hamiltonian.DielectricTensor(electron_density, angular_frequency, B_total, temperature=None)#

Bases: object

Calculates the components of the cold plasma dielectric tensor for a wave with angular frequency \(\Omega\):

\[\begin{split}\epsilon = \begin{pmatrix} \epsilon_{11} & -i\epsilon_{12} & 0 \\ i\epsilon_{12} & \epsilon_{11} & 0 \\ 0 & 0 & \epsilon_{bb} \\ \end{pmatrix}\end{split}\]

where:

\[\begin{split}\begin{align} \epsilon_{11} &= 1 - \frac{\Omega_{pe}^2}{\Omega^2 - \Omega_{ce}^2} \\ \epsilon_{12} &= 1 - \frac{\Omega_{pe}^2\Omega_{ce}}{\Omega(\Omega^2 - \Omega_{ce}^2)} \\ \epsilon_{bb} &= 1 - \frac{\Omega_{pe}^2}{\Omega^2} \\ \end{align}\end{split}\]

The components of the dielectric tensor are calculated in the \((\hat{\mathbf{u}}_1, \hat{\mathbf{u}}_2, \hat{\mathbf{b}})\) basis. Hence, \(\epsilon_{11}\), \(\epsilon_{12}\), and \(\epsilon_{bb}\) correspond to the S, D, and P variables in Stix, respectively. The notation used in this code is chosen to be consistent with Hall-Chen, Parra, Hillesheim, PPCF 2022.

Parameters:

  • electron_density (float | ndarray[tuple[int, ...], dtype[float64]]) – Electron number density

  • angular_frequency (float) – Angular frequency of the beam

  • B_total (float | ndarray[tuple[int, ...], dtype[float64]]) – Magnitude of the magnetic field

  • temperature (float | ndarray[tuple[int, ...], dtype[float64]] | None) – Temperature profile [optional]. Used to calculate relativistic corrections to electron mass, which affects \(\Omega_{pe}\) and :math: Omega_{ce}.

property e_11#

The \(\epsilon_{11}\) component

property e_12#

The \(\epsilon_{12}\) component

property e_bb#

The \(\epsilon_{bb}\) component

class scotty.hamiltonian.Hamiltonian(field, launch_angular_frequency, mode_flag, density_fit, delta_R, delta_Z, delta_K_R, delta_K_zeta, delta_K_Z, temperature_fit=None)#

Bases: object

Functor to evaluate derivatives of the Hamiltonian, \(H\), at a given set of points.

Scotty calculates derivatives using a grid-free finite difference approach. The Hamiltonian is evaluated at, essentially, an arbitrary set of points around the location we wish to get the derivatives at. In practice we define stencils as relative offsets from a central point, and the evaluation points are the product of the spacing in a given direction with the stencil offsets. By carefully choosing our stencils and evaluating all of the derivatives at once, we can reuse evaluations of \(H\) between derivatives, saving a lot of computation.

The stencils are defined as a dict with a tuple of offsets as keys and float weights as values. For example, the CFD1_stencil:

{(1,): 0.5, (-1,): -0.5}

defines the second-order first central-difference:

\[f' = \frac{f(x + \delta_x) - f(x - \delta_x)}{2\delta_x}\]

The keys are tuples so that we can iterate over the offsets for the mixed second derivatives.

The stencils have been chosen to maximise the reuse of Hamiltonian evaluations without sacrificing accuracy.

Parameters:

  • field (MagneticField) – An object describing the magnetic field of the plasma

  • launch_angular_frequency (float) – Angular frequency of the beam

  • mode_flag (int) – Either +/-1, used to determine which mode branch to use

  • density_fit (Callable[[float | ndarray[tuple[int, ...], dtype[float64]]], float | ndarray[tuple[int, ...], dtype[float64]]]) – Function or Callable parameterising the density

  • delta_R (float) – Finite difference spacing in the R direction

  • delta_Z (float) – Finite difference spacing in the Z direction

  • delta_K_R (float) – Finite difference spacing in the K_R direction

  • delta_K_zeta (float) – Finite difference spacing in the K_zeta direction

  • delta_K_Z (float) – Finite difference spacing in the K_Z direction

  • temperature_fit (Callable[[float | ndarray[tuple[int, ...], dtype[float64]]], float | ndarray[tuple[int, ...], dtype[float64]]] | None)

derivatives(q_R, q_Z, K_R, K_zeta, K_Z, second_order=False)#

Evaluate the first-order derivative in all directions at the given point(s), and optionally the second-order ones too

Parameters:

  • q_R (ArrayLike)

  • q_Z (ArrayLike)

  • K_R (ArrayLike)

  • K_zeta (ArrayLike)

  • K_Z (ArrayLike) – Coordinates to evaluate the derivatives at

  • second_order (bool) – If True, also evaluate the second derivatives

Return type:

Dict[str, float | ndarray[tuple[int, …], dtype[float64]]]

scotty.hamiltonian.hessians(dH)#

Compute the elements of the Hessian of the Hamiltonian:

\[\begin{split}\begin{gather} \nabla \nabla H \\ \nabla_K \nabla H \\ \nabla_K \nabla_K H \\ \end{gather}\end{split}\]

given a dict containing the second derivatives of the Hamiltonian (as returned from Hamiltonian.derivatives with second_order=True)

Parameters:

dH (dict)

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