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Plasma.py
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Plasma.py
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# Plasma Class - Create a plasma and calculate the relative parameters
import numpy as np
from scipy.interpolate import CubicSpline, interp1d
import scipy.constants as cons
from MagPy.Ionisation import IaeaTable, IaeaTableMod
class Plasma:
def __init__(self, A, ANum, ne, Te, Ti, V, B, Z = None, gamma = None):
'''
Initialise a Plasma Object given the following parameters:
Args:
example:
---------------------------------------------------------------------------------------
al_flow = {'A':27, 'ANum': 14, 'ne':1e18, 'Te': Te, 'Ti': Ti, 'V':4e6, 'B': 5, 'Z': Z, gamma: 5/3}
al=Plasma(**al_flow)
---------------------------------------------------------------------------------------
A: ion mass in nucleon masses
ANum: Atomic Number
ne: Electron Density in cm^-3
Te electron temperature in eV
V: velocity in cm/s
B: Magnetic Field [Tesla]
Z: if not provided use z_model based on Te and Ne (specify 'lte' or 'ss' to use the custom tables)
gamma: polytropic gamma value, if not specified gamma = 1
'''
self.A = A # Atomic mass weight [gr/mol]
self.ANum = ANum # Atomic Number
self.ne = ne # Electron Density [cm^-3]
self.Te = Te # Electron Temperature
self.Ti = Ti # Ion Temperature [eV]
self.V = V # Bulk Velocity [cm/s]
self.B = B # Magnetic Field [Tesla T]
if gamma is None:
self.gamma = 1
else:
self.gamma = gamma
if Z is None:
# Estimate Ionisation Charge State - Z - from Tabled Values
Z_mod = IaeaTable(self.ANum)
if np.isscalar(self.Te) == True:
self.Z = Z_mod.model(self.Te, self.ne) # Charge State for a given Te
else:
self.Z = []
for i in range(len(self.Te)):
z = Z_mod.model(self.Te[i], self.ne[i])
self.Z.extend(z)
self.Z = np.array(self.Z)
elif Z == 'lte':
Z_mod = IaeaTableMod(self.ANum, 'lte')
if np.isscalar(self.Te) == True:
self.Z = Z_mod.model(self.Te, self.ne) # Charge State for a given Te
else:
self.Z = []
for i in range(len(self.Te)):
z = Z_mod.model(self.Te[i], self.ne[i])
self.Z.extend(z)
self.Z = np.array(self.Z)
elif Z == 'ss':
Z_mod = IaeaTableMod(self.ANum, 'ss')
if np.isscalar(self.Te) == True:
self.Z = Z_mod.model(self.Te, self.ne) # Charge State for a given Te
else:
self.Z = []
for i in range(len(self.Te)):
z = Z_mod.model(self.Te[i], self.ne[i])
self.Z.extend(z)
self.Z = np.array(self.Z)
else:
self.Z = Z
# -----------------------------------------------------------------
# Density
self.density = self.ne * self.A * cons.m_p * 1e3 / self.Z # Mass Density [gr/cm3]
# Ion density
self.ni = self.ne/self.Z # Ion Density [cm^-3]
# Calculate Coulomb Log
self.CoulombLog()
# Parameters
self.speed()
self.frequency()
self.lengthScale()
self.viscosity()
self.resistivity()
self.pressure()
self.timing()
self.dimensionless()
self.thermal_conductivityEH()
# self.thermalconductivity()
def CoulombLog(self):
"""
method to calculate Coulomb Log:
- Formulas taken from NRL formulary pg 34
"""
m_e = cons.m_e # Electron Mass [Kg]
m_i = self.A*cons.m_u
X = self.Ti * (m_e/m_i)
Y = 10*self.Z**2
if np.isscalar(X) == False:
log_ei = []
for i in range(len(X)):
if ((self.Te[i] > X[i]) and (self.Te[i] < Y[i])):
collog = 23-np.log(self.ne[i]**0.5*self.Z[i]*self.Te[i]**-1.5)
log_ei.append(collog)
elif (X[i] < Y[i]) and (Y[i] < self.Te[i]):
collog = 24-np.log(self.ne[i]**0.5*self.Te[i]**-1)
log_ei.append(collog)
elif (self.Te[i] < X[i]*self.Z[i]):
collog = 30-np.log(self.ni[i]**0.5*self.Z[i]**2*self.Te[i]**-1.5/self.A[i])
log_ei.append(collog)
self.col_log_ei = np.array(log_ei)
else:
if ((self.Te > X) and (self.Te < Y)):
self.col_log_ei = 23-np.log(self.ne**0.5*self.Z*self.Te**-1.5) # see NRL formulary pg 34
elif (X < Y) and (Y < self.Te):
self.col_log_ei = 24-np.log(self.ne**0.5*self.Te**-1)
elif (self.Te < X*self.Z):
self.col_log_ei = 30-np.log(self.ni**0.5*self.Z**2*self.Te**-1.5/self.A)
def speed(self):
"""
Method to calculate main Speeds:
- Electron Thermal speed
- Ions Thermal speed
- Sound Speed
- Alfven Speed
[Using SI units, Kelvin, m, Kg, s]
"""
# Scientific Constant
m_e = cons.m_e # Electron Mass [Kg]
m_i = self.A*cons.m_u # Ion Mass [Kg]
e = cons.e # Elemental Charge [C]
mu_0 = cons.mu_0 # Vacuum Permeability [N A^-2]
epsilon_0 = cons.epsilon_0 # Vacuum Permittivity [F m^-1]
kb = cons.k # Boltzmann Constant [J K^-1]
c = cons.c # Light Speed [m s^-1]
T_e = self.Te*e/kb # Electron Temperature [K]
T_i = self.Ti*e/kb # Ion Temperature [K]
n_e = self.ne * 1e6 # Electron Density, SI [m^-3]
n_i = n_e/self.Z # Ion Density, SI [m^-3]
self.V_te = np.sqrt(kb*T_e/m_e) # Electron Thermal Speed [m s^-1]
self.V_ti = np.sqrt(kb*T_i/m_i) # Ion Thermal Speed [m s^-1]
self.V_S = np.sqrt(self.gamma * kb*(self.Z*T_e+T_i)/m_i) # Sound Speed [m s^-1]
self.V_A = np.sqrt(self.B**2/(mu_0*n_i*m_i)) # Alfven Speed [m s^-1]
def frequency(self):
"""
Method to calculate main plasma frequencies
[Using SI units, Kelvin, m, Kg, s]
"""
m_e = cons.m_e # Electron Mass [Kg]
m_i = self.A*cons.m_u # Ion Mass [Kg]
e = cons.e # Elemental Charge [C]
epsilon_0 = cons.epsilon_0 # Vacuum Permittivity [F m^-1]
n_e = self.ne * 1e6 # Electron Density, SI [m^-3]
n_i = n_e/self.Z # Ion Density, SI [m^-3]
self.om_ce = e*self.B/m_e # Electron Cyclotron frequency [rad s^-1]
self.om_ci = self.Z*e*self.B/m_i # Ion Cyclotron frequency [rad s^-1]
self.om_pe = np.sqrt(e**2*n_e/(epsilon_0*m_e)) # Electron Plasma Frequency [rad s^-1]
self.om_pi = np.sqrt(self.Z**2*e**2*n_i/(epsilon_0*m_i)) # Ion Plasma Frequency [rad s^-1]
# Collision Rate
"""Using CGS Units, eV, cm, g, s"""
self.nu_ei = 2.91e-6*self.Z*self.ne*self.col_log_ei*self.Te**-1.5 # Collision Frequency: Electrons - Ions [1/s] ref. NRL FUNDAMENTAL PLASMA PARAMETERS chapter does not include Z - refer to Braginskii
self.nu_ie = 4.80e-8*self.Z**4*self.A**-0.5*self.ni*self.col_log_ei*self.Ti**-1.5 # Collision Frequency: Ions - Electrons [1/s] taken from near Maxwellian formulas
def lengthScale(self):
"""
Method to calculate main lenght scales
[Using SI units Kelvin, m, Kg, s]
"""
e = cons.e # Elemental Charge [C]
epsilon_0 = cons.epsilon_0 # Vacuum Permittivity [F m^-1]
kb = cons.k # Boltzmann Constant [J K^-1]
c = cons.c # Light Speed [m s^-1]
T_e = self.Te*e/kb # electron Temperature, kelvin [K]
T_i = self.Ti*e/kb # ion Temperature, kelvin [K]
n_e = self.ne * 1e6 # Electron Density, SI [m^-3]
self.la_de = np.sqrt(epsilon_0*kb*T_e/(n_e*e**2)) # Debye length [m]
self.delta_i = c/self.om_pi # ion inertial length (ion skin depth) [m]
self.delta_e = c/self.om_pe # electron inertial length (electron skin depth) [m]
if np.nonzero(self.B):
self.rho_i = self.V_ti/self.om_ci # Ion Larmor Radius [m]
self.rho_e = self.V_te/self.om_ce # Electron Larmor Radius [m]
self.rho_e *= 1e2
self.rho_i *= 1e2
else:
self.rho_e = np.nan
self.rho_i = np.nan
self.mfp_e = self.V_te/self.nu_ei # thermal electron mean-free-path [m]
self.mfp_i = self.V_ti/self.nu_ie # thermal Ion mean-free-path
self.mfp_ii = self.V * 1e-2/self.nu_ie # velocity ion-ion mean-free-path (shock thickness)
self.mfp_ee = self.V * 1e-2/self.nu_ei # velocity electron mean-free-path (shock thickness)
""" Convert to CGS units """
self.la_de *= 1e2 # [m] --> [cm]
self.delta_e *= 1e2 # [m] --> [cm]
self.delta_i *= 1e2 # [m] --> [cm]
self.mfp_i *= 1e2 # [m] --> [cm]
self.mfp_e *= 1e2 # [m] --> [cm]
self.mfp_ii *= 1e2 # [m] --> [cm]
self.mfp_ee *= 1e2 # [m] --> [cm]
def viscosity(self):
# Viscosity
"""
Method to calculate Plasma Viscosity (Ryutov 99)
Using CGS Units, eV, cm, g, s
"""
self.visc = 2e19*(self.Ti**2.5)/(self.col_log_ei*self.A**0.5*self.Z**3*self.ne) # [cm^2 s^-1]
# self.visc = 5e-6 * self.A**0.5 * self.Ti**2.5 / (self.Z**4 * self.density) # [cm^2 s^-1] Rayleigh-Taylor in finaly structured medium, Ryuton 1996
self.Lvisc = self.visc/self.V # Viscous Length Scale [cm]
def resistivity(self):
"""
Method to calculate plasma resistivity and relative resistive scale
[Using SI units Kelvin, m, Kg, s]
"""
m_e = cons.m_e # Electron Mass [Kg]
e = cons.e # Elemental Charge [C]
mu_0 = cons.mu_0 # Vacuum Permeability [N A^-2]
n_e = self.ne * 1e6 # Electron Density [m-3]
self.sigma = n_e*e**2/(m_e*self.nu_ei) # Electric Conductivity [s kg^-3 m^-3 C^-3]
self.Dm = 1/(self.sigma*mu_0) # Magnetic Diffusivity [m^2 s^-1]
self.eta = self.Dm*mu_0 # Electric Resistivity [s kg^-3 m^-3 C^-3]^-1
""" Convert to CGS units """
self.Dm = self.Dm*1e4 # [m^2 s^-1] --> [cm^2 s^-1]
self.Leta = self.Dm / self.V # Electric Resistive scale [cm]
def pressure(self):
"""
Method to calculate Pressures:
- Magnetic Pressure
- Thermal Pressure
- Ram Pressure
[Using SI units Kelvin, m, Kg, s]
"""
kb = cons.k # Boltzmann Constant [J K^-1]
mu_0 = cons.mu_0 # Vacuum Permeability [N A^-2]
m_i = self.A*cons.m_u # Ion Mass [Kg]
e = cons.e # Elemental Charge [C]
n_e = self.ne * 1e6 # Electron Density [m^-3]
n_i = n_e/self.Z # Ion Density, SI [m^-3]
V = self.V*1e-2 # Bulk Speed [cm/s] --> [m/s]
T_e = self.Te*e/kb # Electron Temperature [K]
T_i = self.Ti*e/kb # Ion Temperature [K]
self.P_B = self.B**2/(2*mu_0) # Magnetic Pressure [N m^-2]
self.P_th = n_i*kb*(self.Z*T_e+T_i) # Thermal Pressure [N m^-2]
self.P_ram = n_i*m_i*V**2 # Ram Pressure [N m^-2]
def dimensionless(self, l = 1):
"""
Calculate main dimensionless parameters given a characteristic spatial length
input:
- l: Characteristic Spatial Length
[CGS]
"""
self.l = l # Length [cm]
self.HallNumber = self.delta_i / (self.l*1e-2) # Hall Number
self.Re = self.l*self.V / self.visc # Reynolds Number
self.Re_m = self.l*self.V / self.Dm # Magnetic Reynolds Number
if np.nonzero(self.B):
self.beta_th = self.P_th / self.P_B # Thermal Beta
self.beta_ram = self.P_ram / self.P_B # Dynamic Beta
self.M_A = self.V*1e-2 / self.V_A # Alvenic Mach Number
else:
self.M_A = np.nan
self.beta_ram = np.nan
self.beta_th = np.nan
self.M_S = self.V*1e-2 / self.V_S # Sonic Mach Number
self.M_SA = self.V*1e-2 / np.sqrt(self.V_A**2 + self.V_S**2) # Magnitosonic Mach Number
self.omega_t_e = self.om_ce / self.nu_ei # omega tau electron
self.omega_t_i = self.om_ci / self.nu_ie # omega tau ions
def timing(self):
m_e = cons.m_e * 1e3 # Electron Mass [g]
m_i = self.A*cons.m_u *1e3 # Ion Mass [g]
e = cons.e # Elemental Charge [C]
# equilibration time
self.ni_ei = 1.8e-19 * (m_e * m_i)**0.5*self.Z**2*self.ne*self.col_log_ei / (m_e*self.Ti + m_i*self.Te)**1.5 # [s^-1]
self.ni_ie = 1.8e-19 * (m_i * m_e)**0.5*self.Z**2*self.ni*self.col_log_ei / (m_i*self.Te + m_e*self.Ti)**1.5 # [s^-1]
self.tau_eq_ei = 1/self.ni_ei # Second [s]
self.tau_eq_ie = 1/self.ni_ie # Second [s]
#collisional time ions and electrons
self.tau_ei = 1/self.nu_ei
self.tau_ie = 1/self.nu_ie
def thermal_conductivityEH(self):
# Thermal conductivity - Epperlein_Haines 1985 (More accurate) - only electrons
# Coefficients in the following table were computed assuming fully ionised plasma so (Atomic Number == Z_bar)
# NB In our case, it might be more appropriate to use z_bar instead of atomic number!
def near_ANum(Anum, Anum_arr):
idx = np.argmin(np.abs(Anum_arr - Anum))
# print('Calculating transport using Anum = {}'.format(Anum_arr[idx]))
return idx
## Heat transport
Anum_arr = np.array([1,2,3,4,5,6,7,8,10,12,14,60, 100])
g0 = np.array([3.2, 4.93, 6.12, 7.00, 7.68,8.23, 8.69, 9.07, 9.67, 10.1, 10.5, 12.7, 13.58])
gp0 = np.array([6.2, 9.3, 10.2, 9.1, 8.6, 8.6, 8.8, 7.9, 7.4, 7.3, 7.1, 6.4, 6.21])
gp1 = np.array([4.7, 4.0, 3.7, 3.6, 3.5, 3.5, 3.5, 3.4, 3.4, 3.4, 3.4, 3.27, 3.25])
cp0 = np.array([1.9, 1.9, 1.7, 1.3, 1.1, 1.0, 1.0, 0.9, 0.7, 0.7, 0.7, 0.5, 0.5])
cp1 = np.array([2.3, 3.8, 4.8, 4.6, 4.6, 4.8, 5.2, 4.7, 4.6, 4.7, 4.6, 4.7, 4.8])
cp2 = np.array([5.4, 7.8, 8.9, 8.8, 8.8, 9.0, 9.2, 8.8, 8.7, 8.7, 8.7, 8.5, 8.5])
def kc_par(Anum, Anum_arr):
idx = near_ANum(Anum, Anum_arr)
return g0[idx]
def kc_perp(chi, Anum, Anum_arr):
idx = near_ANum(Anum, Anum_arr)
return (gp1[idx] * chi + gp0[idx])/(chi**3 + cp2[idx] * chi**2 + cp1[idx] * chi +cp0[idx])
"""
This function is in MKS.
"""
m_e = cons.m_e # Electron Mass [Kg]
e = cons.e # Elemental Charge [C]
kb = cons.k # Boltzmann Constant [J K^-1]
T_e = self.Te*e/kb # Electron Temperature [K]
n_e = self.ne * 1e6 # Electron Density [m^-3]
Anum = self.ANum # Atomic mass
chi = self.omega_t_e # omega tau electron
self.k_conv = kb * n_e * T_e/(m_e * self.nu_ei) # NRL p 37, 1/(m s)
self.kc_perp = kc_perp(chi, self.Z, Anum_arr)
self.kc_par = kc_par(self.Z, Anum_arr)
self.par_to_perp = self.kc_par/self.kc_perp
self.k_perp = self.kc_perp * self.k_conv
self.k_par = self.kc_par * self.k_conv
self.C_p = 5/2*(n_e * (1 + 1/self.Z)) # heat capacity of electrons and ions
self.Dth_perp = self.k_perp/self.C_p # m^2/s
self.Dth_par = self.k_par/self.C_p # m^2/s
def thermalconductivity(self):
# Coefficient Braginskii
m_e = cons.m_e * 1e3 # Electron Mass [g]
m_i = self.A*cons.m_u *1e3 # Ion Mass [g]
e = cons.e # Elemental Charge [C]
def ThermalCoefficient(Z):
## function retrieved from spreadsheet Thermal Conductivity
par = 3.2132*Z**0.5697
perp = 4.6211*Z**-0.191
return par, perp
a_par, a_perp = ThermalCoefficient(self.Z)
self.xi_i_par = 1.6e-12 * a_par * (self.ni * self.Ti * self.tau_ie / m_i)
self.Dth_i_par = self.xi_i_par / self.ne
if np.nonzero(self.B) is False:
self.xi_i_perp = np.nan
self.Dth_i_perp = np.nan
else:
self.xi_i_perp = 1.6e-12 * a_perp * (self.ni * self.Ti / ( self.om_ci**2 * self.tau_ie * m_i))
self.Dth_i_perp = self.xi_i_perp / self.ne
self.xi_e_par = 1.6e-12 * a_par * (self.ne * self.Te * self.tau_ei / m_e)
self.Dth_e_par = self.xi_e_par / self.ne
if np.nonzero(self.B) is False:
self.xi_e_perp = np.nan
self.Dth_e_perp = np.nan
else:
self.xi_e_perp = 1.6e-12 * a_perp * (self.ne * self.Te / ( self.om_ce**2 * self.tau_ei * m_e * 1e3 ))
self.Dth_e_perp = self.xi_e_perp / self.ne
def params(self):
#useful function tht really should be built in....rounds to n sig figs
round_to_n = lambda x, n: round(x, -int(np.floor(np.log10(np.abs(x)))) + (n - 1))
# Create print list
electrondensity = 'Electron Density = ' + str(np.format_float_scientific(self.ne, precision = 1, exp_digits=2)) + ' [cm^-3]'
ioninertiallength = 'Ion Inertial Length = ' + str(np.format_float_scientific(self.delta_i, precision = 1, exp_digits=2)) + ' [cm]'
ionlarmorradius = 'Ion Larmor Radius = ' + str(np.format_float_scientific(self.rho_i, precision = 1, exp_digits=2)) + ' [cm]'
ionplasmafrequency = 'Ion Plasma Frequency = ' + str(np.format_float_scientific(self.om_pi, precision = 1, exp_digits=2)) + ' [rad s^-1]'
ionmeanfreepath = 'Ion Mean-Free-Path = ' + str(np.format_float_scientific(self.mfp_i, precision = 1, exp_digits=2)) + ' [cm]'
elemeanfreepath = 'Electron Mean-Free-Path = ' + str(np.format_float_scientific(self.mfp_e, precision = 1, exp_digits=2)) + ' [cm]'
collog = 'Coulomb Logaritm = ' + str(np.format_float_scientific(self.col_log_ei, precision = 1, exp_digits=2))
magneticdiff = 'Magnetic Diffusivity = ' + str(np.format_float_scientific(self.Dm, precision = 1, exp_digits=2)) + ' [cm^2/s]'
resistivescale = 'Resistive Scale = ' + str(np.format_float_scientific(self.Leta, precision = 1, exp_digits=2)) + ' [cm]'
sonicmachnumber = 'Sonic Mach Number = ' + str(np.format_float_scientific(self.M_S, precision = 1, exp_digits=2))
magneticmachnumber = 'Alfven Mach Number = ' + str(np.format_float_scientific(self.M_A, precision = 1, exp_digits=2))
msonicmachnumber = 'Magnetosonic Mach Number = ' + str(np.format_float_scientific(self.M_SA, precision = 1, exp_digits=2))
omegatau_i = 'Magnatisation Ions (omega_tau_i) = ' + str(np.format_float_scientific(self.omega_t_i, precision = 1, exp_digits=2))
omegatau_e = 'Magnatisation Electrons (omega_tau_e) = ' + str(np.format_float_scientific(self.omega_t_e, precision = 1, exp_digits=2))
viscositykinem = 'kinematic viscosity = ' + str(np.format_float_scientific(self.visc, precision = 1, exp_digits=2))
reynoldsnumber = 'Reynolds Number = ' + str(np.format_float_scientific(self.Re, precision = 1, exp_digits=2))
mareynoldsnumber = 'Magnetic Reynolds Number = ' + str(np.format_float_scientific(self.Re_m, precision = 1, exp_digits=2))
mabeta = 'Thermal Beta = ' + str(np.format_float_scientific(self.beta_th, precision = 1, exp_digits=2))
rambeta = 'Dynamic Beta = ' + str(np.format_float_scientific(self.beta_ram, precision = 1, exp_digits=2))
electrontemperature = 'Electron Temperature = ' + str(np.format_float_scientific(self.Te, precision = 1, exp_digits=2)) + ' [eV]'
iontemperature = 'Ion Temperature = ' + str(np.format_float_scientific(self.Ti, precision = 1, exp_digits=2)) + ' [eV]'
chargestate = 'Charge State - Z = ' + str(self.Z)
txtstr = electrondensity + '\n' + ioninertiallength + '\n' + ionlarmorradius + '\n' + ionmeanfreepath + '\n' + elemeanfreepath + '\n' + collog + '\n' + ionplasmafrequency + '\n' + resistivescale + '\n' + sonicmachnumber + '\n' + magneticmachnumber + '\n' + msonicmachnumber + '\n' + viscositykinem + '\n' + omegatau_i + '\n' + omegatau_e + '\n' + reynoldsnumber + '\n' + mareynoldsnumber + '\n' + mabeta + '\n' + rambeta + '\n' + electrontemperature + '\n' + iontemperature + '\n' + chargestate + '\n' + magneticdiff
print(txtstr)