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831 | class Neoclassics:
@property
def no_roots(self):
return neoclassics_variables.roots.shape[0]
def init_neoclassics(self, r_effin, eps_effin, iotain):
"""Constructor of the neoclassics object from the effective radius,
epsilon effective and iota only.
Parameters
----------
r_effin :
eps_effin :
iotain :
"""
(
neoclassics_variables.densities,
neoclassics_variables.temperatures,
neoclassics_variables.dr_densities,
neoclassics_variables.dr_temperatures,
) = self.init_profile_values_from_PROCESS(r_effin)
neoclassics_variables.roots = np.array([
4.740718054080526184e-2,
2.499239167531593919e-1,
6.148334543927683749e-1,
1.143195825666101451,
1.836454554622572344,
2.696521874557216147,
3.725814507779509288,
4.927293765849881879,
6.304515590965073635,
7.861693293370260349,
9.603775985479263255,
1.153654659795613924e1,
1.366674469306423489e1,
1.600222118898106771e1,
1.855213484014315029e1,
2.132720432178312819e1,
2.434003576453269346e1,
2.760555479678096091e1,
3.114158670111123683e1,
3.496965200824907072e1,
3.911608494906788991e1,
4.361365290848483056e1,
4.850398616380419980e1,
5.384138540650750571e1,
5.969912185923549686e1,
6.618061779443848991e1,
7.344123859555988076e1,
8.173681050672767867e1,
9.155646652253683726e1,
1.041575244310588886e2,
])
neoclassics_variables.weights = np.array([
1.160440860204388913e-1,
2.208511247506771413e-1,
2.413998275878537214e-1,
1.946367684464170855e-1,
1.237284159668764899e-1,
6.367878036898660943e-2,
2.686047527337972682e-2,
9.338070881603925677e-3,
2.680696891336819664e-3,
6.351291219408556439e-4,
1.239074599068830081e-4,
1.982878843895233056e-5,
2.589350929131392509e-6,
2.740942840536013206e-7,
2.332831165025738197e-8,
1.580745574778327984e-9,
8.427479123056716393e-11,
3.485161234907855443e-12,
1.099018059753451500e-13,
2.588312664959080167e-15,
4.437838059840028968e-17,
5.365918308212045344e-19,
4.393946892291604451e-21,
2.311409794388543236e-23,
7.274588498292248063e-26,
1.239149701448267877e-28,
9.832375083105887477e-32,
2.842323553402700938e-35,
1.878608031749515392e-39,
8.745980440465011553e-45,
])
neoclassics_variables.kt = self.neoclassics_calc_KT()
neoclassics_variables.nu = self.neoclassics_calc_nu()
neoclassics_variables.nu_star = self.neoclassics_calc_nu_star()
neoclassics_variables.nu_star_averaged = self.neoclassics_calc_nu_star_fromT(
iotain
)
neoclassics_variables.vd = self.neoclassics_calc_vd()
neoclassics_variables.d11_plateau = self.neoclassics_calc_D11_plateau()
neoclassics_variables.d11_mono = self.neoclassics_calc_d11_mono(
eps_effin
) # for using epseff
neoclassics_variables.d111 = self.calc_integrated_radial_transport_coeffs(
index=1
)
neoclassics_variables.d112 = self.calc_integrated_radial_transport_coeffs(
index=2
)
neoclassics_variables.d113 = self.calc_integrated_radial_transport_coeffs(
index=3
)
neoclassics_variables.gamma_flux = self.neoclassics_calc_gamma_flux(
neoclassics_variables.densities,
neoclassics_variables.temperatures,
neoclassics_variables.dr_densities,
neoclassics_variables.dr_temperatures,
)
neoclassics_variables.q_flux = self.neoclassics_calc_q_flux()
def init_profile_values_from_PROCESS(self, rho):
"""Initialises the profile_values object from PROCESS' parabolic profiles
Parameters
----------
rho :
"""
tempe = (
physics_variables.temp_plasma_electron_on_axis_kev
* (1 - rho**2) ** physics_variables.alphat
* KEV
)
tempT = (
physics_variables.temp_plasma_ion_on_axis_kev
* (1 - rho**2) ** physics_variables.alphat
* KEV
)
tempD = (
physics_variables.temp_plasma_ion_on_axis_kev
* (1 - rho**2) ** physics_variables.alphat
* KEV
)
tempa = (
physics_variables.temp_plasma_ion_on_axis_kev
* (1 - rho**2) ** physics_variables.alphat
* KEV
)
dense = (
physics_variables.nd_plasma_electron_on_axis
* (1 - rho**2) ** physics_variables.alphan
)
densT = (
(1 - physics_variables.f_plasma_fuel_deuterium)
* physics_variables.nd_plasma_ions_on_axis
* (1 - rho**2) ** physics_variables.alphan
)
densD = (
physics_variables.f_plasma_fuel_deuterium
* physics_variables.nd_plasma_ions_on_axis
* (1 - rho**2) ** physics_variables.alphan
)
densa = (
physics_variables.nd_plasma_alphas_vol_avg
* (1 + physics_variables.alphan)
* (1 - rho**2) ** physics_variables.alphan
)
# Derivatives in real space
dr_tempe = (
-2.0
* 1.0
/ physics_variables.rminor
* physics_variables.temp_plasma_electron_on_axis_kev
* rho
* (1.0 - rho**2) ** (physics_variables.alphat - 1.0)
* physics_variables.alphat
* KEV
)
dr_tempT = (
-2.0
* 1.0
/ physics_variables.rminor
* physics_variables.temp_plasma_ion_on_axis_kev
* rho
* (1.0 - rho**2) ** (physics_variables.alphat - 1.0)
* physics_variables.alphat
* KEV
)
dr_tempD = (
-2.0
* 1.0
/ physics_variables.rminor
* physics_variables.temp_plasma_ion_on_axis_kev
* rho
* (1.0 - rho**2) ** (physics_variables.alphat - 1.0)
* physics_variables.alphat
* KEV
)
dr_tempa = (
-2.0
* 1.0
/ physics_variables.rminor
* physics_variables.temp_plasma_ion_on_axis_kev
* rho
* (1.0 - rho**2) ** (physics_variables.alphat - 1.0)
* physics_variables.alphat
* KEV
)
dr_dense = (
-2.0
* 1.0
/ physics_variables.rminor
* rho
* physics_variables.nd_plasma_electron_on_axis
* (1.0 - rho**2) ** (physics_variables.alphan - 1.0)
* physics_variables.alphan
)
dr_densT = (
-2.0
* 1.0
/ physics_variables.rminor
* rho
* (1 - physics_variables.f_plasma_fuel_deuterium)
* physics_variables.nd_plasma_ions_on_axis
* (1.0 - rho**2) ** (physics_variables.alphan - 1.0)
* physics_variables.alphan
)
dr_densD = (
-2.0
* 1.0
/ physics_variables.rminor
* rho
* physics_variables.f_plasma_fuel_deuterium
* physics_variables.nd_plasma_ions_on_axis
* (1.0 - rho**2) ** (physics_variables.alphan - 1.0)
* physics_variables.alphan
)
dr_densa = (
-2.0
* 1.0
/ physics_variables.rminor
* rho
* physics_variables.nd_plasma_alphas_vol_avg
* (1 + physics_variables.alphan)
* (1.0 - rho**2) ** (physics_variables.alphan - 1.0)
* physics_variables.alphan
)
dens = np.array([dense, densD, densT, densa])
temp = np.array([tempe, tempD, tempT, tempa])
dr_dens = np.array([dr_dense, dr_densD, dr_densT, dr_densa])
dr_temp = np.array([dr_tempe, dr_tempD, dr_tempT, dr_tempa])
return dens, temp, dr_dens, dr_temp
def calc_neoclassics(self):
if stellarator_configuration.stella_config_epseff < 0:
logger.error(
f"epseff value lower than 0: {stellarator_configuration.stella_config_epseff}"
)
self.init_neoclassics(
0.6,
stellarator_configuration.stella_config_epseff,
stellarator_variables.iotabar,
)
q_PROCESS = (
(
physics_variables.f_p_alpha_plasma_deposited
* physics_variables.pden_alpha_total_mw
- physics_variables.pden_plasma_core_rad_mw
)
* physics_variables.vol_plasma
/ physics_variables.a_plasma_surface
* impurity_radiation_module.radius_plasma_core_norm
)
q_PROCESS_r1 = (
(
physics_variables.f_p_alpha_plasma_deposited
* physics_variables.pden_alpha_total_mw
- physics_variables.pden_plasma_core_rad_mw
)
* physics_variables.vol_plasma
/ physics_variables.a_plasma_surface
)
q_neo = sum(neoclassics_variables.q_flux * 1e-6)
gamma_neo = sum(
neoclassics_variables.gamma_flux * neoclassics_variables.temperatures * 1e-6
)
total_q_neo = sum(
neoclassics_variables.q_flux * 1e-6
+ neoclassics_variables.gamma_flux
* neoclassics_variables.temperatures
* 1e-6
)
total_q_neo_e = (
2
* 2
* (
neoclassics_variables.q_flux[0] * 1e-6
+ neoclassics_variables.gamma_flux[0]
* neoclassics_variables.temperatures[0]
* 1e-6
)
)
q_neo_e = neoclassics_variables.q_flux[0] * 1e-6
q_neo_D = neoclassics_variables.q_flux[1] * 1e-6
q_neo_a = neoclassics_variables.q_flux[3] * 1e-6
q_neo_T = neoclassics_variables.q_flux[2] * 1e-6
g_neo_e = (
neoclassics_variables.gamma_flux[0]
* 1e-6
* neoclassics_variables.temperatures[0]
)
g_neo_D = (
neoclassics_variables.gamma_flux[1]
* 1e-6
* neoclassics_variables.temperatures[1]
)
g_neo_a = (
neoclassics_variables.gamma_flux[3]
* 1e-6
* neoclassics_variables.temperatures[3]
)
g_neo_T = (
neoclassics_variables.gamma_flux[2]
* 1e-6
* neoclassics_variables.temperatures[2]
)
dndt_neo_e = neoclassics_variables.gamma_flux[0]
dndt_neo_D = neoclassics_variables.gamma_flux[1]
dndt_neo_a = neoclassics_variables.gamma_flux[3]
dndt_neo_T = neoclassics_variables.gamma_flux[2]
dndt_neo_fuel = (
(dndt_neo_D + dndt_neo_T)
* physics_variables.a_plasma_surface
* impurity_radiation_module.radius_plasma_core_norm
)
dmdt_neo_fuel = (
dndt_neo_fuel * physics_variables.m_fuel_amu * constants.PROTON_MASS * 1.0e6
) # mg
dmdt_neo_fuel_from_e = (
4
* dndt_neo_e
* physics_variables.a_plasma_surface
* impurity_radiation_module.radius_plasma_core_norm
* physics_variables.m_fuel_amu
* constants.PROTON_MASS
* 1.0e6
) # kg
chi_neo_e = -(
neoclassics_variables.q_flux[0]
+ neoclassics_variables.gamma_flux[0] * neoclassics_variables.temperatures[0]
) / (
neoclassics_variables.densities[0] * neoclassics_variables.dr_temperatures[0]
+ neoclassics_variables.temperatures[0]
* neoclassics_variables.dr_densities[0]
)
chi_PROCESS_e = self.st_calc_eff_chi()
nu_star_e = neoclassics_variables.nu_star_averaged[0]
nu_star_d = neoclassics_variables.nu_star_averaged[1]
nu_star_T = neoclassics_variables.nu_star_averaged[2]
nu_star_He = neoclassics_variables.nu_star_averaged[3]
return (
q_PROCESS,
q_PROCESS_r1,
q_neo,
gamma_neo,
total_q_neo,
total_q_neo_e,
q_neo_e,
q_neo_D,
q_neo_a,
q_neo_T,
g_neo_e,
g_neo_D,
g_neo_a,
g_neo_T,
dndt_neo_e,
dndt_neo_D,
dndt_neo_a,
dndt_neo_T,
dndt_neo_fuel,
dmdt_neo_fuel,
dmdt_neo_fuel_from_e,
chi_neo_e,
chi_PROCESS_e,
nu_star_e,
nu_star_d,
nu_star_T,
nu_star_He,
)
def neoclassics_calc_KT(self):
"""Calculates the energy on the given grid
which is given by the gauss laguerre roots.
"""
k = np.repeat((neoclassics_variables.roots / KEV)[:, np.newaxis], 4, axis=1)
return (k * neoclassics_variables.temperatures).T
def neoclassics_calc_nu(self):
"""Calculates the collision frequency"""
mass = np.array([
constants.ELECTRON_MASS,
constants.PROTON_MASS * 2.0,
constants.PROTON_MASS * 3.0,
constants.PROTON_MASS * 4.0,
])
z = np.array([-1.0, 1.0, 1.0, 2.0]) * constants.ELECTRON_CHARGE
# transform the temperature back in eV
# Formula from L. Spitzer.Physics of fully ionized gases. Interscience, New York, 1962
lnlambda = (
32.2
- 1.15 * np.log10(neoclassics_variables.densities[0])
+ 2.3
* np.log10(neoclassics_variables.temperatures[0] / constants.ELECTRON_CHARGE)
)
neoclassics_calc_nu = np.zeros((4, self.no_roots), order="F")
for j in range(4):
for i in range(self.no_roots):
x = neoclassics_variables.roots[i]
for k in range(4):
xk = (
(mass[k] / mass[j])
* (
neoclassics_variables.temperatures[j]
/ neoclassics_variables.temperatures[k]
)
* x
)
expxk = np.exp(-xk)
t = 1.0 / (1.0 + 0.3275911 * np.sqrt(xk))
erfn = (
1.0
- t
* (
0.254829592
+ t
* (
-0.284496736
+ t
* (1.421413741 + t * (-1.453152027 + t * 1.061405429))
)
)
* expxk
)
phixmgx = (1.0 - 0.5 / xk) * erfn + expxk / np.sqrt(np.pi * xk)
v = np.sqrt(
2.0 * x * neoclassics_variables.temperatures[j] / mass[j]
)
neoclassics_calc_nu[j, i] = neoclassics_calc_nu[
j, i
] + neoclassics_variables.densities[k] * (
z[j] * z[k]
) ** 2 * lnlambda * phixmgx / (
4.0 * np.pi * constants.EPSILON0**2 * mass[j] ** 2 * v**3
)
return neoclassics_calc_nu
def neoclassics_calc_nu_star(self):
"""Calculates the normalized collision frequency"""
k = np.repeat(neoclassics_variables.roots[:, np.newaxis], 4, axis=1)
kk = (k * neoclassics_variables.temperatures).T
mass = np.array([
constants.ELECTRON_MASS,
constants.PROTON_MASS * 2.0,
constants.PROTON_MASS * 3.0,
constants.PROTON_MASS * 4.0,
])
v = np.empty((4, self.no_roots))
v[0, :] = constants.SPEED_LIGHT * np.sqrt(
1.0 - (kk[0, :] / (mass[0] * constants.SPEED_LIGHT**2) + 1) ** (-1)
)
v[1, :] = constants.SPEED_LIGHT * np.sqrt(
1.0 - (kk[1, :] / (mass[1] * constants.SPEED_LIGHT**2) + 1) ** (-1)
)
v[2, :] = constants.SPEED_LIGHT * np.sqrt(
1.0 - (kk[2, :] / (mass[2] * constants.SPEED_LIGHT**2) + 1) ** (-1)
)
v[3, :] = constants.SPEED_LIGHT * np.sqrt(
1.0 - (kk[3, :] / (mass[3] * constants.SPEED_LIGHT**2) + 1) ** (-1)
)
return (
physics_variables.rmajor
* neoclassics_variables.nu
/ (neoclassics_variables.iota * v)
)
def neoclassics_calc_nu_star_fromT(self, iota):
"""Calculates the collision frequency
Parameters
----------
iota :
"""
temp = (
np.array([
physics_variables.temp_plasma_electron_vol_avg_kev,
physics_variables.temp_plasma_ion_vol_avg_kev,
physics_variables.temp_plasma_ion_vol_avg_kev,
physics_variables.temp_plasma_ion_vol_avg_kev,
])
* KEV
)
density = np.array([
physics_variables.nd_plasma_electrons_vol_avg,
physics_variables.nd_plasma_fuel_ions_vol_avg
* physics_variables.f_plasma_fuel_deuterium,
physics_variables.nd_plasma_fuel_ions_vol_avg
* (1 - physics_variables.f_plasma_fuel_deuterium),
physics_variables.nd_plasma_alphas_vol_avg,
])
mass = np.array([
constants.ELECTRON_MASS,
constants.PROTON_MASS * 2.0,
constants.PROTON_MASS * 3.0,
constants.PROTON_MASS * 4.0,
])
z = np.array([-1.0, 1.0, 1.0, 2.0]) * constants.ELECTRON_CHARGE
# transform the temperature back in eV
# Formula from L. Spitzer.Physics of fully ionized gases. Interscience, New York, 1962
lnlambda = (
32.2
- 1.15 * np.log10(density[0])
+ 2.3 * np.log10(temp[0] / constants.ELECTRON_CHARGE)
)
neoclassics_calc_nu_star_fromT = np.zeros((4,))
for j in range(4):
v = np.sqrt(2.0 * temp[j] / mass[j])
for k in range(4):
xk = (mass[k] / mass[j]) * (temp[j] / temp[k])
expxk = 0.0
if xk < 200.0:
expxk = np.exp(-xk)
t = 1.0 / (1.0 + 0.3275911 * np.sqrt(xk))
erfn = (
1.0
- t
* (
0.254829592
+ t
* (
-0.284496736
+ t * (1.421413741 + t * (-1.453152027 + t * 1.061405429))
)
)
* expxk
)
phixmgx = (1.0 - 0.5 / xk) * erfn + expxk / np.sqrt(np.pi * xk)
neoclassics_calc_nu_star_fromT[j] = (
neoclassics_calc_nu_star_fromT[j]
+ density[k]
* (z[j] * z[k]) ** 2
* lnlambda
* phixmgx
/ (4.0 * np.pi * constants.EPSILON0**2 * mass[j] ** 2 * v**4)
* physics_variables.rmajor
/ iota
)
return neoclassics_calc_nu_star_fromT
def neoclassics_calc_vd(self):
vde = (
neoclassics_variables.roots
* neoclassics_variables.temperatures[0]
/ (
constants.ELECTRON_CHARGE
* physics_variables.rmajor
* physics_variables.b_plasma_toroidal_on_axis
)
)
vdD = (
neoclassics_variables.roots
* neoclassics_variables.temperatures[1]
/ (
constants.ELECTRON_CHARGE
* physics_variables.rmajor
* physics_variables.b_plasma_toroidal_on_axis
)
)
vdT = (
neoclassics_variables.roots
* neoclassics_variables.temperatures[2]
/ (
constants.ELECTRON_CHARGE
* physics_variables.rmajor
* physics_variables.b_plasma_toroidal_on_axis
)
)
vda = (
neoclassics_variables.roots
* neoclassics_variables.temperatures[3]
/ (
2.0
* constants.ELECTRON_CHARGE
* physics_variables.rmajor
* physics_variables.b_plasma_toroidal_on_axis
)
)
vd = np.empty((4, self.no_roots))
vd[0, :] = vde
vd[1, :] = vdD
vd[2, :] = vdT
vd[3, :] = vda
return vd
def neoclassics_calc_D11_plateau(self):
"""Calculates the plateau transport coefficients (D11_star sometimes)"""
mass = np.array([
constants.ELECTRON_MASS,
constants.PROTON_MASS * 2.0,
constants.PROTON_MASS * 3.0,
constants.PROTON_MASS * 4.0,
])
v = np.empty((4, self.no_roots))
v[0, :] = constants.SPEED_LIGHT * np.sqrt(
1.0
- (neoclassics_variables.kt[0, :] / (mass[0] * constants.SPEED_LIGHT**2) + 1)
** (-1)
)
v[1, :] = constants.SPEED_LIGHT * np.sqrt(
1.0
- (neoclassics_variables.kt[1, :] / (mass[1] * constants.SPEED_LIGHT**2) + 1)
** (-1)
)
v[2, :] = constants.SPEED_LIGHT * np.sqrt(
1.0
- (neoclassics_variables.kt[2, :] / (mass[2] * constants.SPEED_LIGHT**2) + 1)
** (-1)
)
v[3, :] = constants.SPEED_LIGHT * np.sqrt(
1.0
- (neoclassics_variables.kt[3, :] / (mass[3] * constants.SPEED_LIGHT**2) + 1)
** (-1)
)
return (
np.pi
/ 4.0
* neoclassics_variables.vd**2
* physics_variables.rmajor
/ neoclassics_variables.iota
/ v
)
def neoclassics_calc_d11_mono(self, eps_eff):
"""Calculates the monoenergetic radial transport coefficients
using epsilon effective
Parameters
----------
eps_eff :
"""
return (
4.0
/ (9.0 * np.pi)
* (2.0 * eps_eff) ** (3.0 / 2.0)
* neoclassics_variables.vd**2
/ neoclassics_variables.nu
)
def calc_integrated_radial_transport_coeffs(self, index: int):
"""Calculates the integrated radial transport coefficients (index `index`)
It uses Gauss laguerre integration
https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
Parameters
----------
index:
"""
return np.sum(
2.0
/ np.sqrt(np.pi)
* neoclassics_variables.d11_mono
* neoclassics_variables.roots ** (index - 0.5)
* neoclassics_variables.weights,
axis=1,
)
def neoclassics_calc_gamma_flux(
self, densities, temperatures, dr_densities, dr_temperatures
):
"""Calculates the Energy flux by neoclassical particle transport
Parameters
----------
densities :
temperatures :
dr_densities :
dr_temperatures :
"""
z = np.array([-1.0, 1.0, 1.0, 2.0])
return (
-densities
* neoclassics_variables.d111
* (
(dr_densities / densities - z * neoclassics_variables.er / temperatures)
+ (neoclassics_variables.d112 / neoclassics_variables.d111 - 3.0 / 2.0)
* dr_temperatures
/ temperatures
)
)
def neoclassics_calc_q_flux(self):
"""Calculates the Energy flux by neoclassicsal energy transport"""
z = np.array([-1.0, 1.0, 1.0, 2.0])
return (
-neoclassics_variables.densities
* neoclassics_variables.temperatures
* neoclassics_variables.d112
* (
(
neoclassics_variables.dr_densities / neoclassics_variables.densities
- z * neoclassics_variables.er / neoclassics_variables.temperatures
)
+ (neoclassics_variables.d113 / neoclassics_variables.d112 - 3.0 / 2.0)
* neoclassics_variables.dr_temperatures
/ neoclassics_variables.temperatures
)
)
def st_calc_eff_chi(self):
volscaling = (
physics_variables.vol_plasma
* stellarator_variables.f_st_rmajor
* (
impurity_radiation_module.radius_plasma_core_norm
* physics_variables.rminor
/ stellarator_configuration.stella_config_rminor_ref
)
** 2
)
surfacescaling = (
physics_variables.a_plasma_surface
* stellarator_variables.f_st_rmajor
* (
impurity_radiation_module.radius_plasma_core_norm
* physics_variables.rminor
/ stellarator_configuration.stella_config_rminor_ref
)
)
nominator = (
physics_variables.f_p_alpha_plasma_deposited
* physics_variables.pden_alpha_total_mw
- physics_variables.pden_plasma_core_rad_mw
) * volscaling
# in fortran there was a 0*alphan term which I have removed for obvious reasons
# the following comment seems to describe this?
# "include alphan if chi should be incorporate density gradients too"
# but the history can be consulted if required (23/11/22 TN)
denominator = (
(
3
* physics_variables.nd_plasma_electron_on_axis
* constants.ELECTRON_CHARGE
* physics_variables.temp_plasma_electron_on_axis_kev
* 1e3
* physics_variables.alphat
* impurity_radiation_module.radius_plasma_core_norm
* (1 - impurity_radiation_module.radius_plasma_core_norm**2)
** (physics_variables.alphan + physics_variables.alphat - 1)
)
* surfacescaling
* 1e-6
)
return nominator / denominator
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