Spherical Coordinates

In spherical coordinates, the drift velocity

\[\begin{split}\begin{aligned} & \boldsymbol{V}_d = \frac{pcw}{3q}\nabla\times\left(\frac{\boldsymbol{B}}{B^2}\right) = \frac{1}{3q}\frac{p^2c^3}{\sqrt{p^2c^2+m^2c^4}} \left(\frac{1}{B^2}\nabla\times\boldsymbol{B} - \frac{2}{B^3}\nabla B\times\boldsymbol{B}\right) \\ & (\nabla\times\boldsymbol{B})_r = \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta B_\phi) - \frac{1}{r\sin\theta}\frac{\partial B_\theta}{\partial\phi} = \frac{1}{r}\frac{\partial B_\phi}{\partial\theta} + \frac{\cos\theta}{r\sin\theta}B_\phi - \frac{1}{r\sin\theta}\frac{\partial B_\theta}{\partial\phi} \\ & (\nabla\times\boldsymbol{B})_\theta = \frac{1}{r\sin\theta}\frac{\partial B_r}{\partial\phi} - \frac{1}{r}\frac{\partial}{\partial r}(rB_\phi) = \frac{1}{r\sin\theta}\frac{\partial B_r}{\partial\phi} -\frac{\partial B_\phi}{\partial r} - \frac{B_\phi}{r} \\ & (\nabla\times\boldsymbol{B})_\phi = \frac{1}{r}\frac{\partial}{\partial r}(rB_\theta) - \frac{1}{r}\frac{\partial B_r}{\partial\theta} = \frac{\partial B_\theta}{\partial r} + \frac{B_\theta}{r} - \frac{1}{r}\frac{\partial B_r}{\partial\theta} \\ & (\nabla B)_r=\frac{\partial B}{\partial r};\quad (\nabla B)_\theta=\frac{1}{r}\frac{\partial B}{\partial\theta};\quad (\nabla B)_\phi=\frac{1}{r\sin\theta}\frac{\partial B}{\partial\phi} \\ & (\nabla B\times\boldsymbol{B})_r = (\nabla B)_\theta B_\phi - (\nabla B)_\phi B_\theta \\ & (\nabla B\times\boldsymbol{B})_\theta = (\nabla B)_\phi B_r - (\nabla B)_r B_\phi \\ & (\nabla B\times\boldsymbol{B})_\phi = (\nabla B)_r B_\theta - (\nabla B)_\theta B_r \end{aligned}\end{split}\]

The spatial diffusion coefficient is in the same form.

\[\begin{split}\begin{aligned} & \kappa = \begin{bmatrix} \kappa_{rr} & \kappa_{r\theta} & \kappa_{r\phi} \\ \kappa_{r\theta} & \kappa_{\theta\theta} & \kappa_{\theta\phi} \\ \kappa_{r\phi} & \kappa_{\theta\phi} & \kappa_{\phi\phi} \end{bmatrix} % & \kappa_{rr} = \kappa_\perp - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_r^2\\ % & \kappa_{\theta\theta} = \kappa_\perp - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_\theta^2\\ % & \kappa_{\phi\phi} = \kappa_\perp - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_\phi^2\\ % & \kappa_{r\theta} = - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_rB_\theta\\ % & \kappa_{r\phi} = - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_rB_\phi\\ % & \kappa_{\theta\phi} = - \frac{\kappa_\perp-\kappa_\parallel}{B^2}B_\theta B_\phi \end{aligned}\end{split}\]

where \(\kappa_{ij}=\kappa_\perp\delta_{ij} - (\kappa_\perp - \kappa_\parallel)b_ib_j\), and \(i,j\) are \(r,\theta,\phi\).

Since \(\kappa\) is of the same form as that in the Cartesian coordinates, the gradients of \(\kappa\) are

\[\begin{split}\begin{aligned} \partial_r\kappa_{rr} & = \partial_r\kappa_\perp + \partial_r(\kappa_\parallel-\kappa_\perp)b_r^2 + 2(\kappa_\parallel-\kappa_\perp)(B_rB\partial_rB_r - B_r^2\partial_r B)/B^3, \\ \partial_\theta\kappa_{\theta\theta} & = \partial_\theta\kappa_\perp + \partial_\theta(\kappa_\parallel-\kappa_\perp)b_\theta^2 + 2(\kappa_\parallel-\kappa_\perp)(B_\theta B\partial_\theta B_\theta - B_\theta^2\partial_\theta B)/B^3, \\ \partial_\phi\kappa_{\phi\phi} & = \partial_\phi\kappa_\perp + \partial_\phi(\kappa_\parallel-\kappa_\perp)b_\phi^2 + 2(\kappa_\parallel-\kappa_\perp)(B_\phi B\partial_\phi B_\phi - B_\phi^2\partial_\phi B)/B^3, \\ \partial_r\kappa_{r\theta} & = \partial_r(\kappa_\parallel-\kappa_\perp)b_rb_\theta + (\kappa_\parallel-\kappa_\perp)[(B_\theta\partial_rB_r + B_r\partial_rB_\theta)B - 2B_rB_\theta\partial_rB] / B^3, \\ \partial_\theta\kappa_{r\theta} & = \partial_\theta(\kappa_\parallel-\kappa_\perp)b_rb_\theta + (\kappa_\parallel-\kappa_\perp)[(B_\theta\partial_\theta B_r + B_r\partial_\theta B_\theta)B - 2B_rB_\theta\partial_\theta B] / B^3, \\ \partial_r\kappa_{r\phi} & = \partial_r(\kappa_\parallel-\kappa_\perp)b_rb_\phi + (\kappa_\parallel-\kappa_\perp)[(B_\phi\partial_rB_r + B_r\partial_rB_\phi)B - 2B_rB_\phi\partial_rB] / B^3, \\ \partial_\phi\kappa_{r\phi} & = \partial_\phi(\kappa_\parallel-\kappa_\perp)b_rb_\phi + (\kappa_\parallel-\kappa_\perp)[(B_\phi\partial_\phi B_r + B_r\partial_\phi B_\phi)B - 2B_rB_\phi\partial_\phi B] / B^3, \\ \partial_\theta\kappa_{\theta\phi} & = \partial_\theta(\kappa_\parallel-\kappa_\perp)b_\theta b_\phi + (\kappa_\parallel-\kappa_\perp)[(B_\phi\partial_\theta B_\theta + B_\theta\partial_\theta B_\phi)B - 2B_\theta B_\phi\partial_\theta B] / B^3, \\ \partial_\phi\kappa_{\theta\phi} & = \partial_\phi(\kappa_\parallel-\kappa_\perp)b_\theta b_\phi + (\kappa_\parallel-\kappa_\perp)[(B_\phi\partial_\phi B_\theta + B_\theta\partial_\phi B_\phi)B - 2B_\theta B_\phi\partial_\phi B] / B^3. \end{aligned}\end{split}\]

We then need to transfer the Parker transport equation to the spherical coordinates. Since we don’t have cross-diffusion terms (spatial and momentum), we can ignore the momentum diffusion for now.

Note

For a more complete equation, the cross-diffusion terms should be included.

\[\begin{aligned} \frac{\partial F}{\partial t} & = -(\boldsymbol{V}+\boldsymbol{V}_d)\cdot\nabla F - (\nabla\cdot\boldsymbol{V})F + \frac{\partial}{\partial p}\left[\frac{p}{3}(\nabla\cdot\boldsymbol{V})F\right] + \nabla\cdot(\boldsymbol{\kappa}\cdot\nabla F) \end{aligned}\]

where \(F=fp^2\). Since \(\nabla\cdot\boldsymbol{V}_d=0\), we can add one more term \(-(\nabla\cdot\boldsymbol{V}_d)F\) to the right. Then,

\[\begin{aligned} \frac{\partial F}{\partial t} & = -\nabla\cdot((\boldsymbol{V}+\boldsymbol{V}_d)F) + \frac{\partial}{\partial p}\left[\frac{p}{3}(\nabla\cdot\boldsymbol{V})F\right] + \nabla\cdot(\boldsymbol{\kappa}\cdot\nabla F) \end{aligned}\]

Taking \(\boldsymbol{V}+\boldsymbol{V}_d\rightarrow\boldsymbol{V}\),

\[\begin{split}\begin{aligned} \nabla\cdot(\boldsymbol{V}F) & = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2V_rF) +\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta V_\theta F) +\frac{1}{r\sin\theta}\frac{\partial}{\partial\phi}(V_\phi F) \\ & = \frac{\partial(V_rF)}{\partial r} + \frac{2}{r}V_rF +\frac{\partial}{\partial\theta}\left(\frac{V_\theta F}{r}\right) +\frac{\cos\theta}{r\sin\theta}V_\theta F +\frac{\partial}{\partial\phi}\left(\frac{V_\phi F}{r\sin\theta}\right) \end{aligned}\end{split}\]

so there is 2 additional terms (2nd and 4th) if we want to write the equation Fokker–Planck form. It turns out that we need to change \(F\) to \(F_1=F\sin\theta r^2\) [Jokipii77] [Pei10]. Multiplying the above equation by \(r^2\sin\theta\), we get

\[\begin{aligned} r^2\sin\theta\nabla\cdot(\boldsymbol{V}F) & = \frac{\partial(V_rF_1)}{\partial r} +\frac{\partial}{\partial\theta}\left(\frac{V_\theta F_1}{r}\right) +\frac{\partial}{\partial\phi}\left(\frac{V_\phi F_1}{r\sin\theta}\right) \end{aligned}\]

For the diffusion term,

\[\begin{split}\begin{aligned} \boldsymbol{\kappa}\cdot\nabla F = & \left(\kappa_{rr}\frac{\partial F}{\partial r} + \kappa_{r\theta}\frac{1}{r}\frac{\partial F}{\partial\theta} + \kappa_{r\phi}\frac{1}{r\sin\theta}\frac{\partial F}{\partial\phi} \right)\hat{e}_r + \\\nonumber & \left(\kappa_{r\theta}\frac{\partial F}{\partial r} + \kappa_{\theta\theta}\frac{1}{r}\frac{\partial F}{\partial\theta} + \kappa_{\theta\phi}\frac{1}{r\sin\theta}\frac{\partial F}{\partial\phi} \right)\hat{e}_\theta + \\\nonumber & \left(\kappa_{r\phi}\frac{\partial F}{\partial r} + \kappa_{\theta\phi}\frac{1}{r}\frac{\partial F}{\partial\theta} + \kappa_{\phi\phi}\frac{1}{r\sin\theta}\frac{\partial F}{\partial\phi} \right)\hat{e}_\phi \end{aligned}\end{split}\]

Taking \(\boldsymbol{A}=\boldsymbol{\kappa}\cdot\nabla F\),

\[\begin{aligned} r^2\sin\theta\nabla\cdot\boldsymbol{A} = \frac{\partial(r^2\sin\theta A_r)}{\partial r} + \frac{\partial(r\sin\theta A_\theta)}{\partial\theta} + \frac{\partial(rA_\phi)}{\partial\phi} \end{aligned}\]

The 1st term on the right is expanded to

\[\begin{split}\begin{aligned} & \frac{\partial^2}{\partial r^2}(\kappa_{rr}F_1) + \frac{\partial^2}{\partial r\partial\theta}\left(\frac{\kappa_{r\theta}}{r}F_1\right) + \frac{\partial^2}{\partial r\partial\phi}\left(\frac{\kappa_{r\phi}}{r\sin\theta}F_1\right) \\ \nonumber & -\frac{\partial}{\partial r}\left[\left(\frac{1}{r^2} \frac{\partial(r^2\kappa_{rr})}{\partial r} + \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{r\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{r\phi}}{\partial\phi}\right)F_1\right] \end{aligned}\end{split}\]

The 2nd term one the right is expanded to

\[\begin{split}\begin{aligned} & \frac{\partial^2}{\partial r\partial\theta}\left(\frac{\kappa_{r\theta}}{r}F_1\right) + \frac{\partial^2}{\partial\theta^2}\left(\frac{\kappa_{\theta\theta}}{r^2}F_1\right) + \frac{\partial^2}{\partial\theta\partial\phi}\left(\frac{\kappa_{\theta\phi}}{r^2\sin\theta}F_1\right) \\ \nonumber & -\frac{\partial}{\partial\theta}\left[\left(\frac{1}{r^2} \frac{\partial(r\kappa_{r\theta})}{\partial r}+ \frac{1}{r^2\sin\theta}\frac{\partial(\sin\theta\kappa_{\theta\theta})}{\partial\theta}+ \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\theta\phi}}{\partial\phi}\right)F_1\right] \end{aligned}\end{split}\]

The 3rd term one the right is expanded to

\[\begin{split}\begin{aligned} & \frac{\partial^2}{\partial r\partial\phi}\left(\frac{\kappa_{r\phi}}{r\sin\theta}F_1\right) + \frac{\partial^2}{\partial\theta\partial\phi}\left(\frac{\kappa_{\theta\phi}}{r^2\sin\theta}F_1\right) + \frac{\partial^2}{\partial\phi^2}\left(\frac{\kappa_{\phi\phi}}{r^2\sin^2\theta}F_1\right) \\ \nonumber & -\frac{\partial}{\partial\phi}\left[\left(\frac{1}{r^2\sin\theta} \frac{\partial(r\kappa_{r\phi})}{\partial r}+ \frac{1}{r^2\sin\theta}\frac{\partial(\kappa_{\theta\phi})}{\partial\theta}+ \frac{1}{r^2\sin^2\theta}\frac{\partial\kappa_{\phi\phi}}{\partial\phi}\right)F_1\right] \end{aligned}\end{split}\]

The final transferred version of Parker transport equation is

\[\begin{split}\begin{aligned} \frac{\partial F_1}{\partial t} = & -\frac{\partial}{\partial r}\left[\left(v_r+v_{dr} +\frac{1}{r^2}\frac{\partial(r^2\kappa_{rr})}{\partial r} + \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{r\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{r\phi}}{\partial\phi} \right)F_1\right] \\\nonumber & -\frac{\partial}{\partial\theta}\left[\left(\frac{v_\theta+v_{d\theta}}{r} +\frac{1}{r^2}\frac{\partial(r\kappa_{r\theta})}{\partial r}+ \frac{1}{r^2\sin\theta}\frac{\partial(\sin\theta\kappa_{\theta\theta})}{\partial\theta}+ \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\theta\phi}}{\partial\phi} \right)F_1\right]\\\nonumber & -\frac{\partial}{\partial\phi}\left[\left(\frac{v_\phi+v_{d\phi}}{r\sin\theta} +\frac{1}{r^2\sin\theta}\frac{\partial(r\kappa_{r\phi})}{\partial r}+ \frac{1}{r^2\sin\theta}\frac{\partial(\kappa_{\theta\phi})}{\partial\theta}+ \frac{1}{r^2\sin^2\theta}\frac{\partial\kappa_{\phi\phi}}{\partial\phi} \right)F_1\right]\\\nonumber & +\frac{\partial}{\partial p}\left(\frac{p}{3} \left(\frac{1}{r^2}\frac{\partial(r^2v_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial(\sin\theta v_\theta)}{\partial\theta} + \frac{1}{r\sin\theta}\frac{\partial v_\phi}{\partial\phi}\right)F_1\right)\\\nonumber & +\frac{\partial^2}{\partial r^2}(\kappa_{rr}F_1) + \frac{\partial^2}{\partial r\partial\theta}\left(\frac{\kappa_{r\theta}}{r}F_1\right) + \frac{\partial^2}{\partial r\partial\phi}\left(\frac{\kappa_{r\phi}}{r\sin\theta}F_1\right)\\\nonumber & +\frac{\partial^2}{\partial r\partial\theta}\left(\frac{\kappa_{r\theta}}{r}F_1\right) + \frac{\partial^2}{\partial\theta^2}\left(\frac{\kappa_{\theta\theta}}{r^2}F_1\right) + \frac{\partial^2}{\partial\theta\partial\phi}\left(\frac{\kappa_{\theta\phi}}{r^2\sin\theta}F_1\right)\\\nonumber & +\frac{\partial^2}{\partial r\partial\phi}\left(\frac{\kappa_{r\phi}}{r\sin\theta}F_1\right) + \frac{\partial^2}{\partial\theta\partial\phi}\left(\frac{\kappa_{\theta\phi}}{r^2\sin\theta}F_1\right) + \frac{\partial^2}{\partial\phi^2}\left(\frac{\kappa_{\phi\phi}}{r^2\sin^2\theta}F_1\right)\nonumber \end{aligned}\end{split}\]

This corresponds to a set of SDEs.

\[\begin{split}\begin{aligned} dr & = \left(v_r+v_{dr} + \frac{\partial\kappa_{rr}}{\partial r} + \frac{2}{r}\kappa_{rr}+ \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{r\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{r\phi}}{\partial\phi} \right)dt + [P.dW_t]_r \\ d\theta & = \left(\frac{v_\theta+v_{d\theta}}{r} + \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial r} + \frac{\kappa_{r\theta}}{r^2}+ \frac{1}{r^2}\frac{\partial\kappa_{\theta\theta}}{\partial\theta}+ \frac{\cos\theta}{r^2\sin\theta}\kappa_{\theta\theta}+ \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\theta\phi}}{\partial\phi} \right)dt + [P.dW_t]_\theta \\ d\phi & = \left(\frac{v_\phi+v_{d\phi}}{r\sin\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{r\phi}}{\partial r} + \frac{\kappa_{r\phi}}{r^2\sin\theta} + \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\theta\phi}}{\partial\theta}+ \frac{1}{r^2\sin^2\theta}\frac{\partial\kappa_{\phi\phi}}{\partial\phi} \right)dt + [P.dW_t]_\phi\\ dp & = -\frac{p}{3}\left(\frac{\partial v_r}{\partial r}+\frac{2v_r}{r}+ \frac{1}{r}\frac{\partial v_\theta}{\partial\theta} + \frac{\cos\theta}{r\sin\theta}v_\theta + \frac{1}{r\sin\theta}\frac{\partial v_\phi}{\partial\phi}\right) \end{aligned}\end{split}\]

where

\[\begin{split}\begin{aligned} & PP^T = \begin{bmatrix} 2\kappa_{rr} & \dfrac{2\kappa_{r\theta}}{r} & \dfrac{2\kappa_{r\phi}}{r\sin\theta} \\ \dfrac{2\kappa_{r\theta}}{r} & \dfrac{2\kappa_{\theta\theta}}{r^2} & \dfrac{2\kappa_{\theta\phi}}{r^2\sin\theta} \\ \dfrac{2\kappa_{r\phi}}{r\sin\theta} & \dfrac{2\kappa_{\theta\phi}}{r^2\sin\theta} & \dfrac{2\kappa_{\phi\phi}}{r^2\sin^2\theta} \end{bmatrix} \end{aligned}\end{split}\]

According to [Pei10], one possibility for \(P\) is

\[\begin{split}\begin{aligned} \begin{bmatrix} \sqrt{\dfrac{\kappa_{rr}\kappa_{\theta\phi}^2+\kappa_{\theta\theta}\kappa_{r\phi}^2 +\kappa_{\phi\phi}\kappa_{r\theta}^2-2\kappa_{r\phi}\kappa_{r\theta}\kappa_{\theta\phi} -\kappa_{rr}\kappa_{\theta\theta}\kappa_{\phi\phi}} {0.5(\kappa_{\theta\phi}^2 - \kappa_{\theta\theta}\kappa_{\phi\phi})}} & \dfrac{\kappa_{r\phi}\kappa_{\theta\phi}-\kappa_{r\theta}\kappa_{\phi\phi}} {\kappa_{\theta\phi}^2 - \kappa_{\theta\theta}\kappa_{\phi\phi}} \sqrt{2\kappa_{\theta\theta}-\dfrac{2\kappa_{\theta\phi}^2}{\kappa_{\phi\phi}}} & \dfrac{\sqrt{2}\kappa_{r\phi}}{\sqrt{\kappa_{\phi\phi}}} \\ 0 & \dfrac{\sqrt{2\left(\kappa_{\theta\theta}-\kappa_{\theta\phi}^2/\kappa_{\phi\phi}\right)}}{r} & \dfrac{\kappa_{\theta\phi}}{r}\sqrt{\dfrac{2}{\kappa_{\phi\phi}}} \\ 0 & 0 & \dfrac{\sqrt{2\kappa_{\phi\phi}}}{r\sin\theta} \end{bmatrix} \end{aligned}\end{split}\]

For 1D probelms, \(F_1=fp^2r^2\), and the corresponding SDEs are

\[\begin{split}\begin{aligned} dr & = \left(v_r + \frac{\partial\kappa_{rr}}{\partial r} + \frac{2}{r}\kappa_{rr}\right)dt + \sqrt{2\kappa_{rr}}dW_t \\ dp & = -\frac{p}{3}\left(\frac{\partial v_r}{\partial r}+\frac{2v_r}{r}\right) \end{aligned}\end{split}\]

For 2D problems, \(F_1=fp^2r^2\sin\theta\), and the corresponding SDEs are

\[\begin{split}\begin{aligned} dr & = \left(v_r + \frac{\partial\kappa_{rr}}{\partial r} + \frac{2}{r}\kappa_{rr}+ \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{r\theta} \right)dt + [P.dW_t]_r \\ d\theta & = \left(\frac{v_\theta}{r} + \frac{1}{r}\frac{\partial\kappa_{r\theta}}{\partial r} + \frac{\kappa_{r\theta}}{r^2}+ \frac{1}{r^2}\frac{\partial\kappa_{\theta\theta}}{\partial\theta}+ \frac{\cos\theta}{r^2\sin\theta}\kappa_{\theta\theta} \right)dt + [P.dW_t]_\theta \\ dp & = -\frac{p}{3}\left(\frac{\partial v_r}{\partial r}+\frac{2v_r}{r}+ \frac{1}{r}\frac{\partial v_\theta}{\partial\theta} + \frac{\cos\theta}{r\sin\theta}v_\theta\right) \end{aligned}\end{split}\]

where

\[\begin{split}\begin{aligned} & PP^T = \begin{bmatrix} 2\kappa_{rr} & \dfrac{2\kappa_{r\theta}}{r} \\ \dfrac{2\kappa_{r\theta}}{r} & \dfrac{2\kappa_{\theta\theta}}{r^2} \end{bmatrix} \end{aligned}\end{split}\]

One possibility for \(P\) is

\[\begin{split}\begin{aligned} & \begin{bmatrix} -\dfrac{Q_{--}\sqrt{-Q_{-+}}}{\sqrt{Q_{--}^2+4b^2}} & \dfrac{Q_{+-}\sqrt{Q_{++}}}{\sqrt{Q_{+-}^2+4b^2}} \\ \dfrac{2b\sqrt{-Q_{-+}}}{\sqrt{Q_{--}^2+4b^2}} & \dfrac{2b\sqrt{Q_{++}}}{\sqrt{Q_{+-}^2+4b^2}} \end{bmatrix} \end{aligned}\end{split}\]

where

\[\begin{split}\begin{aligned} Q_{++} &=\sqrt{(a-c)^2+4b^2} + (a + c) \\ Q_{-+} &=\sqrt{(a-c)^2+4b^2} - (a + c) \\ Q_{+-} &=\sqrt{(a-c)^2+4b^2} + (a - c) \\ Q_{--} &=\sqrt{(a-c)^2+4b^2} - (a - c) \end{aligned}\end{split}\]

where \(a=\kappa_{rr}\), \(b=\kappa_{r\theta}/r\), \(c=\kappa_{\theta\theta}/r^2\).

[Jokipii77]

Jokipii, J.R. and Levy, E.H., 1977. Effects of particle drifts on the solar modulation of galactic cosmic rays. The Astrophysical Journal, 213, pp.L85-L88.