The Equation Solved

Warning

This part is still in early development. Please use it with caution.

Note

It is better to read the section on Parker’s transport equation first since some of the details are skipped here.

The focused transport equation is [Zank14]

\[\begin{split}\begin{aligned} & \frac{\partial f}{\partial t} + (U_i + c\mu b_i)\frac{\partial f}{\partial x_i} + \frac{dc}{dt}\frac{\partial f}{\partial c} + \frac{d\mu}{dt}\frac{\partial f}{\partial\mu} = \left<\left.\frac{\delta f}{\delta t}\right\vert_s\right> \\ & \frac{dc}{dt} = \left[\frac{1-3\mu^2}{2}b_ib_j\frac{\partial U_i}{\partial x_j}-\frac{1-\mu^2}{2}\nabla\cdot\boldsymbol{U}-\frac{\mu b_i}{c}\left(\frac{\partial U_i}{\partial t} + U_j\frac{\partial U_i}{\partial x_j}\right)\right]c \\ & \frac{d\mu}{dt} = \frac{1-\mu^2}{2}\left[c\nabla\cdot\boldsymbol{b}+\mu\nabla\cdot\boldsymbol{U} - 3\mu b_ib_j\frac{\partial U_i}{\partial x_j}-\frac{2b_i}{c}\left(\frac{\partial U_i}{\partial t} + U_j\frac{\partial U_i}{\partial x_j}\right)\right] \end{aligned}\end{split}\]

where \(\boldsymbol{c}\) is the particle velocity in the flow frame (\(\boldsymbol{v}=\boldsymbol{c} + \boldsymbol{U}\)), \(\mu\equiv\cos\theta=\boldsymbol{c}\cdot\boldsymbol{b}/c\) is the particle pitch angle, \(\boldsymbol{b}\equiv\boldsymbol{B}/B\) is the unit vector along the magnetic field. The following focused transport equation is often used [Zhang09] [Zuo13] [Zhang17] [Kong22],

(1)\[\begin{aligned} \frac{\partial f}{\partial t} = \nabla\cdot(\boldsymbol{\kappa}_\perp\nabla f) - (v\mu\boldsymbol{b} + \boldsymbol{V} + \boldsymbol{V}_d)\cdot\nabla f + \frac{\partial}{\partial\mu}\left(D_{\mu\mu}\frac{\partial f}{\partial\mu}\right) - \frac{d\mu}{dt}\frac{\partial f}{\partial\mu} - \frac{dp}{dt}\frac{\partial f}{\partial p} \end{aligned}\]

where the terms on the right-hand side are cross-field spatial diffusion with a tensor \(\boldsymbol{\kappa}_\perp=\kappa_\perp\left(\overline{\overline{\mathbf{I}}}-\boldsymbol{b}\boldsymbol{b}\right)\) [Zhang09], streaming along the ambient or average magnetic field direction \(\boldsymbol{b}\) with particle speed \(v\) and pitch-angle cosine \(\mu\), convection with the background plasma \(\boldsymbol{V}\), partial gradient/curvature drift \(\boldsymbol{V}_d\), pitch-angle diffusion with a coefficient \(D_{\mu\mu}\), focusing \(d\mu/dt\), and adiabatic heating/cooling \(dp/dt\). In the adiabatic approximation, the drift velocity, focusing rate, and cooling rate may be calculated from the ambient magnetic field \(\boldsymbol{B}=B\boldsymbol{b}\) and plasma velocity \(\boldsymbol{V}\) through

\[\begin{split}\begin{aligned} & \boldsymbol{V}_d=\frac{cpv}{qB}\left\{\frac{1-\mu^2}{2}\frac{\boldsymbol{B}\times\nabla B}{B^2}+\mu^2\frac{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]}{B^3}+\frac{1-\mu^2}{2}\frac{\boldsymbol{B}(\boldsymbol{B}\cdot\nabla\times\boldsymbol{B})}{B^3}\right\}\\ & \frac{d\mu}{dt} = \frac{1-\mu^2}{2}\left[-v\boldsymbol{b}\cdot\nabla\ln B+\mu\nabla\cdot\boldsymbol{V} - 3\mu b_ib_j\frac{\partial V_i}{\partial x_j}-\frac{2b_i}{v}\frac{dV_i}{dt}\right] \\ & \frac{dp}{dt} = -p\left[\frac{1-\mu^2}{2}\left(\nabla\cdot\boldsymbol{V}-b_ib_j\frac{\partial V_i}{\partial x_j}\right)+\mu^2b_ib_j\frac{\partial V_i}{\partial x_j}+\frac{\mu b_i}{v}\frac{dV_i}{dt}\right] \end{aligned}\end{split}\]

where \(\boldsymbol{V}_d\) includes gradient, curvature, and parallel drifts. We may ignore \(\partial V_i/\partial t\) in \(dV_i/dt\) if the flow is not dramatically evolving.

In the quasi-linear theory, resonant interaction between the particle and the turbulent magnetic field can be related by the pitch-angle diffusion coefficient \(D_{\mu\mu}\) [Jokipii77] [Kong22]

\[D_{\mu\mu} = \frac{\pi}{4}\Omega_0(1-\mu^2)\frac{k_\text{res}P(k_\text{res})}{B_0^2}\]

where \(\Omega_0\) is particle gyrofrequency, \(k_\text{res}=|\Omega_0/v\mu|\) is the resonant wavenumber. The above equation is strictly applicable only for the case of 1D turbulence in which the wavevectors are aligned with the mean field. For anisotropic turbulence (e.g., 2D + slab), only the slab turbulence (about 20% of all turbulent fluctuations [Bieber96]) affect particle parallel transport [Florinski03]. The turbulence power is usually expressed as

\[P(k) = \frac{\left<\delta B^2\right>}{1+(kL_c)^\gamma}\left[\int_{k_\text{min}}^{k_\text{max}}\frac{dk}{1+(kL_c)^\gamma}\right]^{-1}\]

where \(k_\text{min}\) and \(k_\text{max}\) are the smallest and largest wavenumbers in the system, \(L_c\) is the turbulence correlation length, and \(\gamma\) is the turbulence spectrum index (e.g., 5/3). For \(k_\text{min}\ll 1/L_c \ll k_\text{max}\), the integral in the above equation can be taken from 0 to \(\infty\). From the table of integral, \(\int_0^\infty x^{\mu-1} dx / (1+x^\nu) = \pi\csc(\mu\pi/\nu)/\nu\). Then,

\[P(k) = \frac{\left<\delta B^2\right>L_c}{1+(kL_c)^\gamma}A_0\]

where \(A_0=\left[(\pi/\gamma)\csc(\pi/\gamma)\right]^{-1}\). In the nonrelativistic limit, we take the pitch-angle diffusion coefficient in the form of [Kong22]

\[\begin{aligned} D_{\mu\mu}=D_{\mu\mu0}\left(\frac{p}{p_0}\right)^{\gamma-1}(1-\mu^2)(|\mu|^{\gamma-1} + h_0) \end{aligned}\]

where \(D_{\mu\mu0}=\frac{\pi}{4}A_0\sigma^2\Omega_0^{2-\gamma}L_c^{1-\gamma}v_0^{\gamma-1}\), and \(p_0\)(\(v_0\)) is the initial particle momentum (velocity) at the injection energy. The parameter \(h_0\) is added to describe the scattering through \(\mu=0\), and we set \(h_0=0.2\) [Zhang17] [Kong22].

\[\begin{aligned} \frac{\partial D_{\mu\mu}}{\partial\mu} = D_{\mu\mu0}\left(\frac{p}{p_0}\right)^{\gamma-1}\left[-2\mu(|\mu|^{\gamma-1} + h_0) + (1-\mu^2)\text{sign}(\mu)|\mu|^{\gamma-2}\right] \end{aligned}\]

[Kong22] (1,2,3,4)

Kong, X., Chen, B., Guo, F., Shen, C., Li, X., Ye, J., Zhao, L., Jiang, Z., Yu, S., Chen, Y. and Giacalone, J., 2022. Numerical modeling of energetic electron acceleration, transport, and emission in solar flares: connecting loop-top and footpoint hard X-ray sources. The Astrophysical Journal Letters, 941(2), p.L22.

[Zank14]

Zank, G.P., 2014. Transport processes in space physics and astrophysics (Vol. 877, p. 185). Berlin: Springer.

[Zhang09] (1,2)

Zhang, M., Qin, G. and Rassoul, H., 2009. Propagation of solar energetic particles in three-dimensional interplanetary magnetic fields. The Astrophysical Journal, 692(1), p.109.

[Zhang17] (1,2)

Zhang, M. and Zhao, L., 2017. Precipitation and release of solar energetic particles from the solar coronal magnetic field. The Astrophysical Journal, 846(2), p.107.

[Zuo13]

Zuo, P., Zhang, M. and Rassoul, H.K., 2013. The role of cross-shock potential on pickup ion shock acceleration in the framework of focused transport theory. The Astrophysical Journal, 776(2), p.93.