# Flux expansion module¶

PLEQUE provides set of functions for mapping of upstream heat fluxes.

## API Reference¶

pleque.utils.flux_expansions.effective_poloidal_heat_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Effective poloidal heat flux expansion coefficient

Definition:

$f_\mathrm{pol, heat, eff} = \frac{B_\theta^\mathrm{u}}{B_\theta^\mathrm{t}} \frac{1}{\sin \beta} = \frac{f_\mathrm{pol}}{\sin \beta}$

Where $$\beta$$ is inclination angle of the poloidal magnetic field and the target plane.

Typical usage:

Effective poloidal heat flux expansion coefficient is typically used scale upstream poloidal heat flux to the target plane.

$q_\perp^\mathrm{t} = \frac{q_\theta^\mathrm{u}}{f_{\mathrm{pol, heat, eff}}}$
Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.
pleque.utils.flux_expansions.effective_poloidal_mag_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Effective poloidal magnetic flux expansion coefficient

Definition:

$f_\mathrm{pol, eff} = \frac{B_\theta^\mathrm{u} R^\mathrm{u}}{B_\theta^\mathrm{t} R^\mathrm{t}} \frac{1}{\sin \beta} = \frac{f_\mathrm{pol}}{\sin \beta}$

Where $$\beta$$ is inclination angle of the poloidal magnetic field and the target plane.

Typical usage:

Effective magnetic flux expansion coefficient is typically used for $$\lambda$$ scaling of the target $$\lambda$$ with respect to the upstream value.

$\lambda^\mathrm{t} = \lambda_q^\mathrm{u} f_{\mathrm{pol, eff}}$

This coefficient can be also used to calculate peak target heat flux from the total power through LCFS if the perpendicular diffusion is neglected. Then for the peak value stays

$q_{\perp, \mathrm{peak}} = \frac{P_\mathrm{div}}{2 \pi R^\mathrm{t} \lambda_q^\mathrm{u}} \frac{1}{f_\mathrm{pol, eff}}$

Where $$P_\mathrm{div}$$ is total power to outer strike point and $lambda_q^mathrm{u}$ is e-folding length on the outer midplane.

Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.
pleque.utils.flux_expansions.impact_angle_cos_pol_projection(coords: pleque.core.coordinates.Coordinates)[source]

Impact angle calculation - dot product of PFC norm and local magnetic field direction poloidal projection only. Internally uses incidence_angle_sin function where vecs are replaced by the vector of the poloidal magnetic field (Bphi = 0).

Returns: array of impact angles
pleque.utils.flux_expansions.impact_angle_sin(coords: pleque.core.coordinates.Coordinates)[source]

Impact angle calculation - dot product of PFC norm and local magnetic field direction. Internally uses incidence_angle_sin function where vecs are replaced by the vector of the magnetic field.

Returns: array of impact angles cosines
pleque.utils.flux_expansions.incidence_angle_sin(coords: pleque.core.coordinates.Coordinates, vecs)[source]
Parameters: coords – Coordinate object (of length N_vecs)of a line in the space on which the incidence angle is evaluated. vecs – array (3, N_vecs) vectors in (R, Z, phi) space. array of sines of angles of incidence. I.e. cosine of the angle between the normal to the line (in the poloidal plane) and the corresponding vector.
pleque.utils.flux_expansions.parallel_heat_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Parallel heat flux expansion coefficient

Definition:

$f_\parallel= \frac{B^\mathrm{u}}{B^\mathrm{t}}$

Typical usage:

Parallel heat flux expansion coefficient is typically used to scale total upstream heat flux parallel to the magnetic field along the magnetic field lines.

$q_\parallel^\mathrm{t} = \frac{q_\parallel^\mathrm{u}}{f_\parallel}$
Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.
pleque.utils.flux_expansions.poloidal_heat_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Poloidal heat flux expansion coefficient

Definition:

$f_\mathrm{pol, heat} = \frac{B_\theta^\mathrm{u}}{B_\theta^\mathrm{t}}$

Typical usage: Poloidal heat flux expansion coefficient is typically used to scale poloidal heat flux (heat flux projected along poloidal magnetic field) along the magnetic field line.

$q_\theta^\mathrm{t} = \frac{q_\theta^\mathrm{u}}{f_{\mathrm{pol, heat}}}$
Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.
pleque.utils.flux_expansions.poloidal_mag_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Poloidal magnetic flux expansion coefficient.

Definition:

$f_\mathrm{pol} = \frac{\Delta r^\mathrm{t}}{\Delta r^\mathrm{u}} = \frac{B_\theta^\mathrm{u} R^\mathrm{u}}{B_\theta^\mathrm{t} R^\mathrm{t}}$

Typical usage:

Poloidal magnetic flux expansion coefficient is typically used for $$\lambda$$ scaling in plane perpendicular to the poloidal component of the magnetic field.

Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.
pleque.utils.flux_expansions.total_heat_flux_exp_coef(equilibrium: pleque.core.equilibrium.Equilibrium, coords: pleque.core.coordinates.Coordinates)[source]

Total heat flux expansion coefficient

Definition:

$f_\mathrm{tot} = \frac{B^\mathrm{u}}{B^\mathrm{t}} \frac{1}{\sin \alpha} = \frac{f_\parallel}{\sin \alpha}$

Where $$\alpha$$ is an inclination angle of the total magnetic field and the target plane.

Important

$$\alpha$$ is an inclination angle of the total magnetic field to the target plate. Whereas $$\beta$$ is an inclination of poloidal components of the magnetic field to the target plate.

Typical usage:

Total heat flux expansion coefficient is typically used to project total upstream heat flux parallel to the magnetic field to the target plane.

$q_\perp^\mathrm{t} = \frac{q_\parallel^\mathrm{u}}{f_{\mathrm{tot}}}$
Parameters: equilibrium – Instance of Equilibrium. coords – Coordinates where the coefficient is evaluated.

## References¶

Theiler, C., et al.: Results from recent detachment experiments in alternativee divertor cofigurations on TCV, Nucl. Fusion 57 (2017) 072008 16pp

Vondracek, P.: Plasma Heat Flux to Solid Structures in Tokamaks, PhD thesis, Prague 2019