pyrokinetics.local_geometry.mxh.LocalGeometryMXH

class pyrokinetics.local_geometry.mxh.LocalGeometryMXH(*args, **kwargs)

Bases: LocalGeometry

Local equilibrium representation defined as in: PPCF 63 (2021) 012001 (5pp)

Miller eXtended Harmonic (MXH)

\[\begin{split}\begin{align} R(r, \theta) &= R_{major}(r) + r \cos(\theta_R) \\ Z(r, \theta) &= Z_0(r) + r \kappa(r) \sin(\theta) \\ \theta_R &= \theta + c_0(r) + \sum_{n=1}^N [c_n(r) \cos(n \theta) + s_n(r) \sin(n \theta)] \\ r &= (\max(R) - \min(R)) / 2 \end{align}\end{split}\]

Data stored in a ordered dictionary

psi_n

Normalised Psi

Type:

Float

rho

r/a

Type:

Float

a_minor

Minor radius of LCFS [m]

Type:

Float

Rmaj

Normalised Major radius (Rmajor/a_minor)

Type:

Float

Z0

Normalised vertical position of midpoint (Zmid / a_minor)

Type:

Float

f_psi

Torodial field function

Type:

Float

B0

Toroidal field at major radius (Fpsi / Rmajor) [T]

Type:

Float

bunit_over_b0

Ratio of GACODE normalising field = \(q/r \partial \psi/\partial r\) [T] to B0

Type:

Float

dpsidr

\(\frac{\partial \psi}{\partial r}\)

Type:

Float

q

Safety factor

Type:

Float

shat

Magnetic shear \(r/q \partial q/ \partial r\)

Type:

Float

beta_prime

\(\beta = 2 \mu_0 \partial p \partial \rho 1/B0^2\)

Type:

Float

kappa

Elongation

Type:

Float

s_kappa

Shear in Elongation \(r/\kappa \partial \kappa/\partial r\)

Type:

Float

shift

Shafranov shift

Type:

Float

dZ0dr

Shear in midplane elevation

Type:

Float

thetaR

thetaR values at theta

Type:

ArrayLike

dthetaR_dtheta

Derivative of thetaR w.r.t theta at theta

Type:

ArrayLike

dthetaR_dr

Derivative of thetaR w.r.t r at theta

Type:

ArrayLike

cn

cosine moments of thetaR

Type:

ArrayLike

sn

sine moments of thetaR

Type:

ArrayLike

dcndr

Shear in cosine moments \(\partial c_n/\partial r\)

Type:

ArrayLike

dsndr

Shear in sine moments \(\partial s_n/\partial r\)

Type:

ArrayLike

R_eq

Equilibrium R data used for fitting

Type:

Array

Z_eq

Equilibrium Z data used for fitting

Type:

Array

b_poloidal_eq

Equilibrium B_poloidal data used for fitting

Type:

Array

theta_eq

theta values for equilibrium data

Type:

Float

R

Fitted R data

Type:

Array

Z

Fitted Z data

Type:

Array

b_poloidal

Fitted B_poloidal data

Type:

Array

theta

Fitted theta data

Type:

Float

dRdtheta

Derivative of fitted \(R\) w.r.t \(\theta\)

Type:

Array

dRdr

Derivative of fitted \(R\) w.r.t \(r\)

Type:

Array

dZdtheta

Derivative of fitted \(Z\) w.r.t \(\theta\)

Type:

Array

dZdr

Derivative of fitted \(Z\) w.r.t \(r\)

Type:

Array

d2Rdtheta2

Second derivative of fitted \(R\) w.r.t \(\theta\)

Type:

Array

d2Rdrdtheta

Derivative of fitted \(R\) w.r.t \(r\) and \(\theta\)

Type:

Array

d2Zdtheta2

Second derivative of fitted \(Z\) w.r.t \(\theta\)

Type:

Array

d2Zdrdtheta

Derivative of fitted \(Z\) w.r.t \(r\) and \(\theta\)

Type:

Array

__init__(*args, **kwargs)

Methods

__init__(*args, **kwargs)
convert_physical_units(norms)	Convert physical-unit attributes to generic simulation units of norms.
default()	Default parameters for geometry Same as GA-STD case
from_gk_data(params)	Initialise from data gathered from GKCode object, and additionally set bunit_over_b0
from_global_eq(eq, psi_n, norms[, show_fit, ...])	Loads LocalGeometry object from an Equilibrium Object
from_local_geometry(local_geometry[, ...])	Loads LocalGeometry object of one type from a LocalGeometry Object of a different type
get_RZ_derivatives(theta[, params])	Calculates the derivatives of \(R(r, heta)\) and \(Z(r, heta)\) w.r.t \(r\) and :math:` heta`, used in B_poloidal calc
get_RZ_second_derivatives(theta)	Calculates the second derivatives of \(R(r, heta)\) and \(Z(r, heta)\) w.r.t \(r\) and :math:` heta`, used in geometry terms
get_b_poloidal(theta[, params])	Returns Miller prediction for get_b_poloidal given flux surface parameters
get_bunit_over_b0()	Get Bunit/B0 using q and loop integral of Bp
get_d2Rdrdtheta(thetaR, dthetaR_dr, ...)	Calculate the second derivative of \(R(r, heta)\) w.r.t \(r\) and :math:` heta`
get_d2Rdtheta2(thetaR, dthetaR_dtheta, ...)	Calculates the second derivative of \(R(r, heta)\) w.r.t :math:` heta`
get_d2Zdrdtheta(theta, s_kappa)	Calculates the second derivative of \(Z(r, \theta)\) w.r.t \(r\) and \(\theta\)
get_d2Zdtheta2(theta)	Calculates the second derivative of \(Z(r, theta)\) w.r.t :math:` heta`
get_d2thetaR_drdtheta(theta, dcndr, dsndr)
get_d2thetaR_dtheta2(theta)
get_dLdtheta(theta)	Returns dLdtheta used in loop integrals
get_dRdr(shift, thetaR, dthetaR_dr)	Calculates the derivatives of \(R(r, \theta)\) w.r.t \(r\)
get_dRdtheta(thetaR, dthetaR_dtheta)	Calculates the derivatives of \(R(r, heta)\) w.r.t :math:` heta`
get_dZdr(theta, dZ0dr, s_kappa)	Calculates the derivatives of \(Z(r, \theta)\) w.r.t \(r\)
get_dZdtheta(theta)	Calculates the derivatives of \(Z(r, theta)\) w.r.t \(\theta\)
get_dthetaR_dr(theta, dcndr, dsndr)
get_dthetaR_dtheta(theta)
get_f_prime([ntheta])	Calculate F' from and other geometry terms
get_f_psi()	Calculate safety factor from b poloidal field, R, Z and q \(f = \frac{2\pi q}{\oint \frac{dl}{R^2 B_{\theta}}}\)
get_flux_surface(theta)	Generates (R,Z) of a flux surface given a set of MXH fits
get_flux_surface_area_volume()	Calculate the poloidal and toroidal area of the flux surface and the toroidal volume in units of lref
get_flux_surface_area_volume_derivatives()	Calculate the derivative of the poloidal and toroidal area of the flux surface and the toroidal volume with respect to r
get_grad_r(theta[, params])	MXH definition of grad r from MXH, R.
get_s_hat([Fprime, ntheta])	Calculate magnetic shear from F' and other geometry terms
get_thetaR(theta)
keys()
minimise_b_poloidal(params[, even_space_theta])	Function for least squares minimisation of poloidal field
normalise(norms[, context])	Convert LocalGeometry Parameters to current NormalisationConvention Note this creates the attribute unit_mapping which is used to apply units to the LocalGeometry object :type norms: :param norms: Normalisation convention to convert to :type norms: SimulationNormalisation
plot_equilibrium_to_local_geometry_fit([...])
test_safety_factor()	Calculate safety factor from LocalGeometry object b poloidal field \(q = \frac{1}{2\pi} \oint \frac{f dl}{R^2 B_{\theta}}\)
to(norms[, context])	Thin wrapper for normalise

Attributes

delta
n
n_moments
s_delta
s_zeta
zeta

default()

Default parameters for geometry Same as GA-STD case

property delta

from_local_geometry(local_geometry, verbose=False, show_fit=False, **kwargs)

Loads LocalGeometry object of one type from a LocalGeometry Object of a different type

Miller is a special case which is a subset of MXH so we can directly set values :type local_geometry: :param local_geometry: LocalGeometry object :type local_geometry: LocalGeometry :type verbose: :param verbose: Controls verbosity :type verbose: Boolean

get_RZ_derivatives(theta, params=None)

Calculates the derivatives of \(R(r, heta)\) and \(Z(r, heta)\) w.r.t \(r\) and :math:` heta`, used in B_poloidal calc

Parameters:

• theta (ArrayLike) – Array of theta points to evaluate grad_r on

• params (Array [Optional]) – If given then will use params = [shift, s_kappa, dZ0dr, cn[nmoments], sn[nmoments] ] when calculating derivatives, otherwise will use object attributes

Return type:

ndarray

Returns:

• dRdtheta (Array) – Derivative of \(R\) w.r.t :math:` heta`

• dRdr (Array) – Derivative of \(R\) w.r.t \(r\)

• dZdtheta (Array) – Derivative of \(Z\) w.r.t :math:` heta`

• dZdr (Array) – Derivative of \(Z\) w.r.t \(r\)

get_RZ_second_derivatives(theta)

Calculates the second derivatives of \(R(r, heta)\) and \(Z(r, heta)\) w.r.t \(r\) and :math:` heta`, used in geometry terms

Parameters:

theta (ArrayLike) – Array of theta points to evaluate grad_r on

Return type:

ndarray

Returns:

• d2Rdtheta2 (Array) – Second derivative of \(R\) w.r.t :math:` heta`

• d2Rdrdtheta (Array) – Second derivative of \(R\) w.r.t \(r\) and :math:` heta`

• d2Zdtheta2 (Array) – Second derivative of \(Z\) w.r.t :math:` heta`

• d2Zdrdtheta (Array) – Second derivative of \(Z\) w.r.t \(r\) and :math:` heta`

get_d2Rdrdtheta(thetaR, dthetaR_dr, dthetaR_dtheta, d2thetaR_drdtheta)

Calculate the second derivative of \(R(r, heta)\) w.r.t \(r\) and :math:` heta`

Parameters:

• theta (ArrayLike) – Array of theta points to evaluate dRdr on

• thetaR (ArrayLike) – Array of thetaR points to evaluate dRdtheta on

• dthetaR_dr (ArrayLike) – Radial derivative of thetaR

• dthetaR_dtheta (ArrayLike) – Theta derivative of thetaR

• d2thetaR_drdtheta (ArrayLike) – Second derivative of thetaR w.r.t \(r\) and :math:` heta`

Returns:

d2Rdrdtheta – Second derivative of R w.r.t \(r\) and :math:` heta`

Return type:

Array

get_d2Rdtheta2(thetaR, dthetaR_dtheta, d2thetaR_dtheta2)

Calculates the second derivative of \(R(r, heta)\) w.r.t :math:` heta`

Parameters:

• thetaR (ArrayLike) – Array of thetaR points to evaluate dRdtheta on

• dthetaR_dtheta (ArrayLike) – Theta derivative of thetaR

• d2thetaR_dtheta2 (ArrayLike) – Second theta derivative of thetaR

• -------

• d2Rdtheta2 (Array) – Second derivative of \(R\) w.r.t :math:` heta`

get_d2Zdrdtheta(theta, s_kappa)

Calculates the second derivative of \(Z(r, \theta)\) w.r.t \(r\) and \(\theta\)

Parameters:

• theta (ArrayLike) – Array of theta points to evaluate dZdr on

• s_kappa (Float) – Shear in Elongation \(r/\kappa \partial \kappa/\partial r\)

Returns:

d2Zdrdtheta – Second derivative of \(Z\) w.r.t \(r\) and \(\theta\)

Return type:

Array

get_d2Zdtheta2(theta)

Calculates the second derivative of \(Z(r, theta)\) w.r.t :math:` heta`

Parameters:

theta (ArrayLike) – Array of theta points to evaluate dZdtheta on

Returns:

d2Zdtheta2 – Second derivative of \(Z\) w.r.t :math:` heta`

Return type:

Array

get_d2thetaR_drdtheta(theta, dcndr, dsndr)

Parameters:

• theta (Array) – theta angles

• dcndr (Array) – Asymmetric coefficients in thetaR

• dsndr (Array) – Symmetric coefficients in thetaR

get_d2thetaR_dtheta2(theta)

Parameters:

theta

Returns:

d^2thetaR/dtheta^2 – second theta derivative of poloidal angle used in R

Return type:

Array

get_dRdr(shift, thetaR, dthetaR_dr)

Calculates the derivatives of \(R(r, \theta)\) w.r.t \(r\)

Parameters:

• theta (ArrayLike) – Array of theta points to evaluate dRdr on

• shift (Float) – Shafranov shift

• thetaR (ArrayLike) – Array of thetaR points to evaluate dRdtheta on

• dthetaR_dr (ArrayLike) – Radial derivative of thetaR

Returns:

dRdr – Derivative of \(R\) w.r.t \(r\)

Return type:

Array

get_dRdtheta(thetaR, dthetaR_dtheta)

Calculates the derivatives of \(R(r, heta)\) w.r.t :math:` heta`

Parameters:

• thetaR (ArrayLike) – Array of thetaR points to evaluate dRdtheta on

• dthetaR_dtheta (ArrayLike) – Theta derivative of thetaR

• -------

• dRdtheta (Array) – Derivative of \(R\) w.r.t :math:` heta`

get_dZdr(theta, dZ0dr, s_kappa)

Calculates the derivatives of \(Z(r, \theta)\) w.r.t \(r\)

Parameters:

• theta (ArrayLike) – Array of theta points to evaluate dZdr on

• dZ0dr (Float) – Derivative in midplane elevation

• s_kappa (Float) – Shear in Elongation \(r/\kappa \partial \kappa/\partial r\)

Returns:

dZdr – Derivative of \(Z\) w.r.t \(r\)

Return type:

Array

get_dZdtheta(theta)

Calculates the derivatives of \(Z(r, theta)\) w.r.t \(\theta\)

Parameters:

theta (ArrayLike) – Array of theta points to evaluate dZdtheta on

Returns:

dZdtheta – Derivative of \(Z\) w.r.t \(\theta\)

Return type:

Array

get_dthetaR_dr(theta, dcndr, dsndr)

Parameters:

• theta (Array) – theta angles

• dcndr (Array) – Asymmetric coefficients in thetaR

• dsndr (Array) – Symmetric coefficients in thetaR

get_dthetaR_dtheta(theta)

Parameters:

theta

Returns:

dthetaR/dtheta – theta derivative of poloidal angle used in R

Return type:

Array

get_flux_surface(theta)

Generates (R,Z) of a flux surface given a set of MXH fits

Parameters:

theta (Array) – Values of theta to evaluate flux surface

Return type:

Tuple[ndarray, ndarray]

Returns:

• R (Array) – R values for this flux surface (if not normalised then in [m])

• Z (Array) – Z Values for this flux surface (if not normalised then in [m])

get_thetaR(theta)

Parameters:

theta (Array)

Returns:

thetaR – Poloidal angle used in definition of R

Return type:

Array

property n

property n_moments

property s_delta

property s_zeta

property zeta