Radial diffusion models

The dll provides radial diffusion models.

Models:

  • Brautigam & Albert, (2000)

  • Ozeke et al., (2014)

  • Liu et al., (2016)

  • Ali et al., (2016)

A2016(L, kp[, dll_type])

Calculate radial diffusion coefficient following Ali et al. [2016] model.

BA2000(L, kp[, mu, dll_type])

Calculate electromagnetic and electrostatic radial diffusion coefficient following Brautigam & Albert [2000] model, eq.

L2016(L, kp, mu[, dll_type])

Calculate radial diffusion coefficient following Liu et al. [2016] model.

O2014(L, kp[, dll_type])

Calculate radial diffusion coefficient following Ozeke et al. [2014] model.

Functions

rbamlib.models.dll.BA2000(L, kp, mu=None, dll_type='M')

Calculate electromagnetic and electrostatic radial diffusion coefficient following Brautigam & Albert [2000] model, eq. (6) and eq. (4, 5).

Parameters:

  • kp (ndarray) – Kp-index, vector of geomagnetic activity indices corresponding to the times.

  • L (ndarray) – Array of L values, representing the radial distance in Earth radii where magnetic field lines cross the magnetic equator.

  • mu (ndarray, default=None) – Array of mu values, vector of corresponding first adiabatic invariants required for calculation electrostatic radial diffusion coefficient, in MeV/G. mu is not used in calculation of electromagnetic radial diffusion coefficient.

  • dll_type (str, default='M') – A string, indicating which diffusion coefficient to calculate. ‘M’ stands for electromagnetic radial diffusion coefficient (eq. 6). ‘E’ stands for electrostatic diffusion coefficient (eq. 4, 5). ‘ME’, ‘both’, or ‘’ computes both coefficients and returns a tuple (dllm, dlle).

Returns:

  • dllm (numpy.ndarray) – Electromagnetic radial diffusion coefficient, in 1/days (if dll_type=’M’).

  • dlle (numpy.ndarray) – Electrostatic radial diffusion coefficient, in 1/days (if dll_type=’E’).

  • dllm, dlle (tuple of numpy.ndarray) – Both electromagnetic and electrostatic diffusion coefficients (dllm, dlle) if dll_type=’ME’ or ‘’.

Notes

The electromagnetic radial diffusion coefficient is calculated as:

\[D^{M}_{LL} = 10^{(0.506 \cdot Kp - 9.325)} \cdot L^{10}\]

The electrostatic radial diffusion coefficient is calculated as:

\[D^{E}_{LL} = \frac{1}{4} \left(\frac{c E_{ms}}{B_0}\right)^2 \frac{T}{1 + (\omega_D T / 2)^2} L^6\]

where:

\[\omega_D = \frac{3 \mu c}{e L^2 R_E^2} \left(1 + \frac{2 \mu B}{E_0}\right)^{-1/2}\]
\[E_{ms} = 0.26 (Kp - 1) + 0.1\]
rbamlib.models.dll.O2014(L, kp, dll_type='E')

Calculate radial diffusion coefficient following Ozeke et al. [2014] model.

Parameters:

  • kp (ndarray) – Kp-index, vector of geomagnetic activity indices corresponding to the times.

  • L (ndarray) – Array of L values, representing the radial distance in Earth radii where magnetic field lines cross the magnetic equator.

  • dll_type (str, default='E') – A string, indicating which diffusion coefficient to calculate. ‘M’ or ‘B’: stands for magnetic radial diffusion coefficient (eq. 20). ‘E’ stands for electric diffusion coefficient (eq. 23). ‘ME’, ‘BE’, ‘both’, or ‘’ computes both coefficients and returns a tuple (dllb, dlle).

Returns:

  • dllb (numpy.ndarray) – Magnetic radial diffusion coefficient, in 1/days (if dll_type=’B’).

  • dlle (numpy.ndarray) – Electric radial diffusion coefficient, in 1/days (if dll_type=’E’).

  • dllb, dlle (tuple of numpy.ndarray) – Both electromagnetic and electrostatic diffusion coefficients (dllb, dlle) if dll_type=’ME’ or ‘’.

Notes

The electromagnetic radial diffusion coefficient is calculated as:

\[D^{B}_{LL} = 6.62 \times 10^{-13} \cdot L^8 10^{-0.0327 L^2 + 0.625 L - 0.0108 Kp^2 + 0.499 Kp}\]

The electrostatic radial diffusion coefficient is calculated as:

\[D^{E}_{LL} = 2.16 \times 10^{-8} \cdot L^6 \cdot 10^{0.217 L + 0.461 Kp}\]
rbamlib.models.dll.L2016(L, kp, mu, dll_type='E')

Calculate radial diffusion coefficient following Liu et al. [2016] model.

Parameters:

  • L (ndarray) – Array of L values, representing the radial distance in Earth radii where magnetic field lines cross the magnetic equator.

  • kp (ndarray) – Kp-index, vector of geomagnetic activity indices corresponding to the times.

  • mu (ndarray) – First adiabatic invariant in MeV/G.

  • dll_type (str, default='E') – This function only calculates the electric radial diffusion coefficient (‘E’).

Returns:

dlle – Electric radial diffusion coefficient, in 1/days.

Return type:

numpy.ndarray

Notes

The electric radial diffusion coefficient is calculated as:

\[D^{E}_{LL} = 1.115 \times 10^{-6} \cdot 10^{0.281 \cdot Kp} \cdot L^{8.184} \cdot \mu^{-0.608}\]
rbamlib.models.dll.A2016(L, kp, dll_type='M')

Calculate radial diffusion coefficient following Ali et al. [2016] model.

Parameters:

  • L (ndarray) – Array of L values, representing the radial distance in Earth radii where magnetic field lines cross the magnetic equator.

  • kp (ndarray) – Kp-index, vector of geomagnetic activity indices corresponding to the times.

  • dll_type (str, default='E') – A string, indicating which diffusion coefficient to calculate. ‘M’ or ‘B’ stands for magnetic radial diffusion coefficient (eq. 14). ‘E’ stands for electric diffusion coefficient (eq. 15). ‘ME’, ‘BE’, ‘both’, or ‘’ computes both coefficients and returns a tuple (dllb, dlle).

Returns:

  • dllb (numpy.ndarray) – Magnetic radial diffusion coefficient, in 1/days (if dll_type=’M’, ‘B’).

  • dlle (numpy.ndarray) – Electric radial diffusion coefficient, in 1/days (if dll_type=’E’).

  • dllb, dlle (tuple of numpy.ndarray) – Both magnetic and electric diffusion coefficients (dllb, dlle) if dll_type=’ME’, ‘BE’, ‘both’, or ‘’.

Notes

The magnetic radial diffusion coefficient is calculated as:

\[D^{B}_{LL} = \exp \left( a_1 + b_1 \cdot Kp \cdot L^* + L^* \right)\]

The electric radial diffusion coefficient is calculated as:

\[D^{E}_{LL} = \exp \left( a_2 + b_2 \cdot Kp \cdot L^* + c_2 \cdot L^* \right)\]

where the constants are given by:

\[a_1 = -16.253, \quad b_1 = 0.224, \quad a_2 = -16.951, \quad b_2 = 0.181, \quad c_2 = 1.982\]