Spherical Coordinates

The spatial diffusion coefficient is in the same form.

\[\begin{split}\begin{aligned} & \boldsymbol{\kappa}_\perp = \begin{bmatrix} \kappa_{\perp rr} & \kappa_{\perp r\theta} & \kappa_{\perp r\phi} \\ \kappa_{\perp r\theta} & \kappa_{\perp\theta\theta} & \kappa_{\perp\theta\phi} \\ \kappa_{\perp r\phi} & \kappa_{\perp\theta\phi} & \kappa_{\perp\phi\phi} \end{bmatrix} \end{aligned}\end{split}\]

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

\[\begin{aligned} \partial_i\kappa_{\perp ij} & = \delta_{ij}\partial_i\kappa_\perp -b_ib_j\partial_i\kappa_\perp - \frac{\kappa_\perp}{B}(b_j\partial_iB_i + b_i\partial_iB_j - 2b_ib_j\partial_iB) \end{aligned}\]

We then need to transfer the focused 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.

\[\begin{split}\begin{aligned} \frac{\partial F}{\partial t} = & \nabla\cdot(\boldsymbol{\kappa}_\perp\cdot\nabla F) - \nabla\cdot\left[(v\mu\boldsymbol{b} + \boldsymbol{V} + \boldsymbol{V}_d)F\right] \nonumber\\ & + \frac{\partial}{\partial\mu^2}\left(D_{\mu\mu}F\right) - \frac{\partial}{\partial\mu}\left[\left(\frac{d\mu}{dt}+\frac{\partial D_{\mu\mu}}{\partial\mu}\right)F\right] - \frac{\partial}{\partial p}\left(\frac{dp}{dt}F\right) \end{aligned}\end{split}\]

The rest is essentially the same as before except for \(\boldsymbol{\kappa}_\perp\), \(v\mu\boldsymbol{b}\), and the pitch-angle diffusion terms. We will first need to change \(F\) to \(F_1=F\sin\theta r^2\) [Jokipii77] [Pei10]. The resulting SDEs are

\[\begin{split}\begin{aligned} dr & = \frac{dr}{dt}dt + [P.dW_t]_r \\ d\theta & = \frac{d\theta}{dt}dt + [P.dW_t]_\theta \\ d\phi & = \frac{d\phi}{dt}dt + [P.dW_t]_\phi\\ dp & = \frac{dp}{dt}dt \\ d\mu & = \left(\frac{d\mu}{dt}+\frac{\partial D_{\mu\mu}}{\partial\mu}\right)dt + \sqrt{2D_{\mu\mu}}dW_\mu(t) \end{aligned}\end{split}\]

where

\[\begin{split}\begin{aligned} \frac{dr}{dt} & = v\mu b_r + V_r+V_{dr} + \frac{\partial\kappa_{\perp rr}}{\partial r} + \frac{2}{r}\kappa_{\perp rr}+ \frac{1}{r}\frac{\partial\kappa_{\perp r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{\perp r\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{\perp r\phi}}{\partial\phi} \\ \frac{d\theta}{dt} & = \frac{v\mu b_\theta + V_\theta+V_{d\theta}}{r} + \frac{1}{r}\frac{\partial\kappa_{\perp r\theta}}{\partial r} + \frac{\kappa_{\perp r\theta}}{r^2}+ \frac{1}{r^2}\frac{\partial\kappa_{\perp \theta\theta}}{\partial\theta}+ \frac{\cos\theta}{r^2\sin\theta}\kappa_{\perp \theta\theta}+ \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\perp \theta\phi}}{\partial\phi} \\ \frac{d\phi}{dt} & = \frac{v\mu b_\phi + V_\phi+V_{d\phi}}{r\sin\theta}+ \frac{1}{r\sin\theta}\frac{\partial\kappa_{\perp r\phi}}{\partial r} + \frac{\kappa_{\perp r\phi}}{r^2\sin\theta} + \frac{1}{r^2\sin\theta}\frac{\partial\kappa_{\perp \theta\phi}}{\partial\theta}+ \frac{1}{r^2\sin^2\theta}\frac{\partial\kappa_{\perp \phi\phi}}{\partial\phi} \end{aligned}\end{split}\]

To calculate \(d\mu/dt\) and \(dp/dt\), we will need

\[\begin{split}\begin{aligned} -\boldsymbol{b}\cdot\nabla\ln B & = -\frac{1}{B}\left(b_r\partial_r B + \frac{b_\theta}{r}\partial_\theta B + \frac{b_\phi}{r\sin\theta}\partial_\phi B\right) \\ \nabla\cdot\boldsymbol{V} & = \partial_r V_r +\frac{2V_r}{r}+ \frac{1}{r}\partial_\theta V_\theta + \frac{\cos\theta}{r\sin\theta}V_\theta + \frac{1}{r\sin\theta}\partial_\phi V_\phi \end{aligned}\end{split}\]

\(b_ib_j\frac{\partial V_i}{\partial x_j}\) is more complicated.

\[\begin{split}\begin{aligned} b_ib_j\frac{\partial V_i}{\partial x_j} = & b_r^2\partial_r V_r + \frac{b_rb_\theta}{r}\partial_\theta V_r + \frac{b_rb_\phi}{r\sin\theta}\partial_\phi V_r - \frac{b_rb_\theta V_\theta + b_rb_\phi V_\phi}{r} + \nonumber\\ & b_rb_\theta\partial_r V_\theta + \frac{b_\theta^2}{r}\partial_\theta V_\theta + \frac{b_\theta b_\phi}{r\sin\theta}\partial_\phi V_\theta + \frac{b_\theta^2V_r}{r} - \frac{\cot\theta b_\theta b_\phi V_\phi}{r} + \nonumber\\ & b_rb_\phi\partial_r V_\phi + \frac{b_\theta b_\phi}{r}\partial_\theta V_\phi + \frac{b_\phi^2}{r\sin\theta}\partial_\phi V_\phi + \frac{b_\phi^2V_r}{r} + \frac{\cot\theta b_\phi^2V_\theta}{r} \end{aligned}\end{split}\]

Similar for \(b_iV_j\frac{\partial V_i}{\partial x_j}\),

\[\begin{split}\begin{aligned} b_iV_j\frac{\partial V_i}{\partial x_j} = & b_rV_r\partial_r V_r + \frac{b_rV_\theta}{r}\partial_\theta V_r + \frac{b_rV_\phi}{r\sin\theta}\partial_\phi V_r - \frac{b_r(V_\theta^2+V_\phi^2)}{r} + \nonumber\\ & b_\theta V_r\partial_r V_\theta + \frac{b_\theta V_\theta}{r}\partial_\theta V_\theta + \frac{b_\theta V_\phi}{r\sin\theta}\partial_\phi V_\theta + \frac{b_\theta V_\theta V_r}{r} - \frac{\cot\theta b_\theta V_\phi^2}{r} + \nonumber\\ & b_\phi V_r\partial_r V_\phi + \frac{b_\phi V_\theta}{r}\partial_\theta V_\phi + \frac{b_\phi V_\phi}{r\sin\theta}\partial_\phi V_\phi + \frac{b_\phi V_\phi V_r}{r} + \frac{\cot\theta b_\phi V_\phi V_\theta}{r} \end{aligned}\end{split}\]

For 1D probelms, \(F_1=fp^2r^2\), and the corresponding SDE for \(r\) is

\[\begin{aligned} dr = & \left(v\mu b_r + V_r + \frac{\partial\kappa_{\perp rr}}{\partial r} + \frac{2}{r}\kappa_{\perp rr}\right)dt + \sqrt{2\kappa_{\perp rr}}dW_t \end{aligned}\]

For \(d\mu\) and \(dp\), we will need

\[\begin{split}\begin{aligned} -\boldsymbol{b}\cdot\nabla\ln B = & -\frac{1}{B}b_r\partial_r B \\ \nabla\cdot\boldsymbol{V} = & \partial_r V_r +\frac{2V_r}{r} \\ b_ib_j\frac{\partial V_i}{\partial x_j} = & b_r^2\partial_r V_r - \frac{b_rb_\theta V_\theta + b_rb_\phi V_\phi}{r} + \nonumber \\ & b_rb_\theta\partial_r V_\theta + \frac{b_\theta^2V_r}{r} - \frac{\cot\theta b_\theta b_\phi V_\phi}{r} + \nonumber \\ & b_rb_\phi\partial_r V_\phi + \frac{b_\phi^2V_r}{r} + \frac{\cot\theta b_\phi^2V_\theta}{r} \\ b_iV_j\frac{\partial V_i}{\partial x_j} = & b_rV_r\partial_r V_r - \frac{b_r(V_\theta^2+V_\phi^2)}{r} + \nonumber \\ & b_\theta V_r\partial_r V_\theta + \frac{b_\theta V_\theta V_r}{r} - \frac{\cot\theta b_\theta V_\phi^2}{r} + \nonumber \\ & b_\phi V_r\partial_r V_\phi + \frac{b_\phi V_\phi V_r}{r} + \frac{\cot\theta b_\phi V_\phi V_\theta}{r} \end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned} dr & = \left(v\mu b_r + V_r+ \frac{\partial\kappa_{\perp rr}}{\partial r} + \frac{2}{r}\kappa_{\perp rr}+ \frac{1}{r}\frac{\partial\kappa_{\perp r\theta}}{\partial\theta}+ \frac{\cos\theta}{r\sin\theta}\kappa_{\perp r\theta}\right)dt + [P.dW_t]_r \\ d\theta & = \left(\frac{v\mu b_\theta + V_\theta}{r} + \frac{1}{r}\frac{\partial\kappa_{\perp r\theta}}{\partial r} + \frac{\kappa_{\perp r\theta}}{r^2}+ \frac{1}{r^2}\frac{\partial\kappa_{\perp \theta\theta}}{\partial\theta}+ \frac{\cos\theta}{r^2\sin\theta}\kappa_{\perp \theta\theta}\right)dt + [P.dW_t]_\theta \end{aligned}\end{split}\]

where \(P\) has the same form as the one in Parker transport. For \(d\mu\) and \(dp\), we will need

\[\begin{split}\begin{aligned} -\boldsymbol{b}\cdot\nabla\ln B = & -\frac{1}{B}\left(b_r\partial_r B + \frac{b_\theta}{r}\partial_\theta B\right) \\ \nabla\cdot\boldsymbol{V} = & \partial_r V_r +\frac{2V_r}{r}+ \frac{1}{r}\partial_\theta V_\theta + \frac{\cos\theta}{r\sin\theta}V_\theta \\ b_ib_j\frac{\partial V_i}{\partial x_j} = & b_r^2\partial_r V_r + \frac{b_rb_\theta}{r}\partial_\theta V_r - \frac{b_rb_\theta V_\theta + b_rb_\phi V_\phi}{r} + \nonumber\\ & b_rb_\theta\partial_r V_\theta + \frac{b_\theta^2}{r}\partial_\theta V_\theta + \frac{b_\theta^2V_r}{r} - \frac{\cot\theta b_\theta b_\phi V_\phi}{r} + \nonumber\\ & b_rb_\phi\partial_r V_\phi + \frac{b_\theta b_\phi}{r}\partial_\theta V_\phi + \frac{b_\phi^2V_r}{r} + \frac{\cot\theta b_\phi^2V_\theta}{r} \\ b_iV_j\frac{\partial V_i}{\partial x_j} = & b_rV_r\partial_r V_r + \frac{b_rV_\theta}{r}\partial_\theta V_r - \frac{b_r(V_\theta^2+V_\phi^2)}{r} + \nonumber\\ & b_\theta V_r\partial_r V_\theta + \frac{b_\theta V_\theta}{r}\partial_\theta V_\theta + \frac{b_\theta V_\theta V_r}{r} - \frac{\cot\theta b_\theta V_\phi^2}{r} + \nonumber\\ & b_\phi V_r\partial_r V_\phi + \frac{b_\phi V_\theta}{r}\partial_\theta V_\phi + \frac{b_\phi V_\phi V_r}{r} + \frac{\cot\theta b_\phi V_\phi V_\theta}{r} \end{aligned}\end{split}\]

The drift velocity

\[\begin{aligned} & \boldsymbol{V}_d=\frac{cpv}{qB}\left\{\frac{1-\mu^2}{2}\frac{\boldsymbol{B}\times\nabla B}{B^2}+\mu^2\frac{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]}{B^3}+\frac{1-\mu^2}{2}\frac{\boldsymbol{B}(\boldsymbol{B}\cdot\nabla\times\boldsymbol{B})}{B^3}\right\} \end{aligned}\]

Calculations needed for the first term (gradient drift):

\[\begin{split}\begin{aligned} & (\nabla B)_r=\partial_r B;\quad (\nabla B)_\theta=\frac{\partial_\theta B}{r};\quad (\nabla B)_\phi=\frac{\partial_\phi B}{r\sin\theta} \\ & (\boldsymbol{B}\times\nabla B)_r = B_\theta(\nabla B)_\phi - B_\phi(\nabla B)_\theta \\ & (\boldsymbol{B}\times\nabla B)_\theta = B_\phi(\nabla B)_r - B_r(\nabla B)_\phi \\ & (\boldsymbol{B}\times\nabla B)_\phi = B_r(\nabla B)_\theta - B_\theta(\nabla B)_r \end{aligned}\end{split}\]

Calculations needed for the second term (curvature drift):

\[\begin{split}\begin{aligned} & \boldsymbol{C} = (\boldsymbol{B}\cdot\nabla)\boldsymbol{B} \\ & C_r = B_r\partial_r B_r + \frac{B_\theta}{r}\partial_\theta B_r + \frac{B_\phi}{r\sin\theta}\partial_\phi B_r - \frac{B_\theta^2+B_\phi^2}{r} \\ & C_\theta = B_r\partial_r B_\theta + \frac{B_\theta}{r}\partial_\theta B_\theta + \frac{B_\phi}{r\sin\theta}\partial_\phi B_\theta + \frac{B_r B_\theta}{r} - \frac{\cot\theta B_\phi^2}{r} \\ & C_\phi = B_r\partial_r B_\phi + \frac{B_\theta}{r}\partial_\theta B_\phi + \frac{B_\phi}{r\sin\theta}\partial_\phi B_\phi + \frac{B_r B_\phi}{r} + \frac{\cot\theta B_\theta B_\phi}{r} \\ & (\boldsymbol{B}\times\boldsymbol{C})_r = B_\theta C_\phi - B_\phi C_\theta \\ & (\boldsymbol{B}\times\boldsymbol{C})_\theta = B_\phi C_r - B_r C_\phi \\ & (\boldsymbol{B}\times\boldsymbol{C})_\phi = B_r C_\theta - B_\theta C_r \end{aligned}\end{split}\]

Calculations needed for the third term (parallel drift):

\[\begin{split}\begin{aligned} & (\nabla\times\boldsymbol{B})_r = \frac{\partial_\theta(\sin\theta B_\phi)}{r\sin\theta} - \frac{\partial_\phi B_\theta}{r\sin\theta} = \frac{\partial_\theta B_\phi}{r} + \frac{\cos\theta}{r\sin\theta}B_\phi - \frac{\partial_\phi B_\theta}{r\sin\theta} \\ & (\nabla\times\boldsymbol{B})_\theta = \frac{\partial_\phi B_r}{r\sin\theta} - \frac{\partial_r(rB_\phi)}{r} = \frac{\partial_\phi B_r}{r\sin\theta} -\partial_r B_\phi - \frac{B_\phi}{r} \\ & (\nabla\times\boldsymbol{B})_\phi = \frac{\partial_r(rB_\theta)}{r} - \frac{\partial_\theta B_r}{r} = \partial_r B_\theta + \frac{B_\theta}{r} - \frac{\partial_\theta B_r}{r} \end{aligned}\end{split}\]