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[1]:

%matplotlib inline

Braginskii coefficients

A short example of how to calculate classical transport coefficients from Bragiński’s theory.

[2]:

import astropy.units as u

from plasmapy.formulary import ClassicalTransport

We’ll use some sample ITER data, without much regard for whether the regime is even fit for classical transport theory:

[3]:

thermal_energy_per_electron = 8.8 * u.keV
electron_concentration = 10.1e19 / u.m**3

thermal_energy_per_ion = 8.0 * u.keV
ion_concentration = electron_concentration
ion = "D+"  # a crude approximation

We now make the default ClassicalTransport object:

[4]:

braginskii = ClassicalTransport(
    thermal_energy_per_electron,
    electron_concentration,
    thermal_energy_per_ion,
    ion_concentration,
    ion,
)

/home/docs/checkouts/readthedocs.org/user_builds/plasmapy/envs/latest/lib/python3.12/site-packages/plasmapy/utils/decorators/checks.py:1405: RelativityWarning: thermal_speed is yielding a velocity that is 18.561% of the speed of light. Relativistic effects may be important.
  warnings.warn(
/home/docs/checkouts/readthedocs.org/user_builds/plasmapy/envs/latest/lib/python3.12/site-packages/plasmapy/utils/decorators/checks.py:1405: RelativityWarning: V is yielding a velocity that is 18.561% of the speed of light. Relativistic effects may be important.
  warnings.warn(
/home/docs/checkouts/readthedocs.org/user_builds/plasmapy/envs/latest/lib/python3.12/site-packages/plasmapy/utils/decorators/checks.py:1405: RelativityWarning: thermal_speed is yielding a velocity that is 18.559% of the speed of light. Relativistic effects may be important.
  warnings.warn(
/home/docs/checkouts/readthedocs.org/user_builds/plasmapy/envs/latest/lib/python3.12/site-packages/plasmapy/utils/decorators/checks.py:1405: RelativityWarning: V is yielding a velocity that is 18.559% of the speed of light. Relativistic effects may be important.
  warnings.warn(

These variables are calculated during initialization and can be referred to straight away:

[5]:

print(braginskii.coulomb_log_ei)
print(braginskii.coulomb_log_ii)
print(braginskii.hall_e)
print(braginskii.hall_i)

18.015542112815666
20.41557520752423
0.0
0.0

These quantities are not calculated during initialization and can be referred to via methods. To signify the need to calculate them, we call them via ().

[6]:

print(braginskii.resistivity)
print(braginskii.thermoelectric_conductivity)
print(braginskii.electron_thermal_conductivity)
print(braginskii.ion_thermal_conductivity)

1.1541382845331304e-09 Ohm m
0.7110839986207994
1065176076.6192665 W / (K m)
21360464.46376672 W / (K m)

They also change with magnetization:

[7]:

mag_braginskii = ClassicalTransport(
    thermal_energy_per_electron,
    electron_concentration,
    thermal_energy_per_ion,
    ion_concentration,
    ion,
    B=0.1 * u.T,
)

print(mag_braginskii.resistivity)
print(mag_braginskii.thermoelectric_conductivity)
print(mag_braginskii.electron_thermal_conductivity)
print(mag_braginskii.ion_thermal_conductivity)

1.1541382845331304e-09 Ohm m
0.7110839986207994
1065176076.6192665 W / (K m)

21360464.46376672 W / (K m)

They also change with direction with respect to the magnetic field. Here, we choose to print out, as arrays, the (parallel, perpendicular, and cross) directions. Take a look at the docs to ClassicalTransport for more information on these.

[8]:

all_direction_braginskii = ClassicalTransport(
    thermal_energy_per_electron,
    electron_concentration,
    thermal_energy_per_ion,
    ion_concentration,
    ion,
    B=0.1 * u.T,
    field_orientation="all",
)

print(all_direction_braginskii.resistivity)
print(all_direction_braginskii.thermoelectric_conductivity)
print(all_direction_braginskii.electron_thermal_conductivity)
print(all_direction_braginskii.ion_thermal_conductivity)

[1.15413828e-09 2.25078755e-09 1.39690568e-15] Ohm m
[7.11083999e-01 6.76676822e-13 5.46328992e-07]
[1.06517608e+09 2.08451764e-04 3.06777888e+02] W / (K m)
[2.13604645e+07 4.24754851e-03 2.69422221e+02] W / (K m)

The viscosities return arrays:

[9]:

print(braginskii.electron_viscosity)
print(mag_braginskii.electron_viscosity)
print(braginskii.ion_viscosity)
print(mag_braginskii.ion_viscosity)

[16.29411376 16.28874805 16.28874805  0.          0.        ] Pa s
[1.62941138e+01 2.00480711e-25 8.01922844e-25 1.47442522e-12
 2.94885044e-12] Pa s

[1271.38945503 1267.52435833 1267.52435833    0.            0.        ] Pa s

[1.27138946e+03 5.99222933e-17 2.39689173e-16 2.57162285e-07
 5.14324570e-07] Pa s