Some personal notes PT decomposition

Here are notes on PT decomposition.

The PT decomposition is defined as follows, $\mathbf{B}=\mathrm{\nabla }\times (\mathrm{\nabla }\eta \times \mathrm{\nabla }\mathrm{\Phi })+\mathrm{\nabla }\eta \times \mathrm{\nabla }\mathrm{\Psi },$

$\eta$ is a coordinate parameter in Cartesian or Spherical coordinates, it can be chosen as $z$ and $r$, respectively. The electric current is defined as follows, here we ignore the vacuum permeability.

$$\begin{array}{r}\mathbf{J}=\mathrm{\nabla }\times \mathbf{B}\end{array}$$

Thus we will have the following equations

$$\begin{array}{r}{\mathrm{\nabla }}_{||}^2\mathrm{\Phi }=B_n,\\ {\mathrm{\nabla }}_{||}^2\mathrm{\Psi }=J_n,\end{array}$$

where the subscript $||$ indicates the operator only works in the directions parallel to the plane, which is perpendicular to $\mathrm{\nabla }\eta$. As the gradient operator along $\eta$ commutates with the parallel Laplace operator, $[{\partial }_{\eta },{\mathrm{\nabla }}_{||}^2]=0$. We will have,

$$\begin{array}{}\text{(1)}& {\mathrm{\nabla }}_{||}^2{\partial }_{\eta }\mathrm{\Phi }={\partial }_{\eta }{B}_n,\text{(2)}& {\mathrm{\nabla }}_{||}^2{\partial }_{\eta }\mathrm{\Psi }={\partial }_{\eta }{J}_n.\end{array}$$

Adopting the divergence-free and charge-free conditions, we can obtain,

$$\begin{array}{}\text{(3)}& {\mathrm{\nabla }}_{||}^2{\partial }_{\eta }\mathrm{\Phi }=-{\mathrm{\nabla }}_{||}\cdot {\mathbf{B}}_{||},\text{(4)}& {\mathrm{\nabla }}_{||}^2{\partial }_{\eta }\mathrm{\Psi }=-{\mathrm{\nabla }}_{||}\cdot {\mathbf{J}}_{||}.\end{array}$$

The parallel current components can be obtained using the force-free condition $\mathbf{J}=\alpha \mathbf{B}$.

Thus in each layer, we can obtain the variable $(\mathrm{\Phi },\mathrm{\Psi }{)}^{\mathrm{\top }}$ and their $\eta$ derivative $({\partial }_{\eta }\mathrm{\Phi },{\partial }_{\eta }\mathrm{\Psi }{)}^{\mathrm{\top }}$. Therefore, we can make an integral upward along $\eta$, just like the integration of an ordinary equation.

We can obtain the physical variable on a staggered grid, described in the following section.

Cartesian Case

We choose $\eta =z$, and the above equations become,

$$\begin{array}{}\text{(5)}& {\mathrm{\nabla }}_{x,y}^2\mathrm{\Phi }& =B_z,\text{(6)}& {\mathrm{\nabla }}_{x,y}^2\mathrm{\Psi }& =J_z,\text{(7)}& {\mathrm{\nabla }}_{x,y}^2{\partial }_z\mathrm{\Phi }& =-{\partial }_x{B}_x-{\partial }_y{B}_y,\text{(8)}& {\mathrm{\nabla }}_{x,y}^2{\partial }_z\mathrm{\Psi }& =-{\partial }_x{J}_x-{\partial }_y{J}_y.\end{array}$$

Therefore, the $\mathbf{B}$ can be obtained directly from observations, and $\mathbf{J}$ can be obtained by combining the force-free condition $\mathbf{J}=\alpha \mathbf{B}$, where $\alpha =J_z/B_z$. In the following discussion, we assume that $\mathbf{B}$, $J_z$, and $\alpha$ have already been provided on the bottom boundary.

Combining the Eq. 1, we can express the transverse components as,

$$\begin{array}{}\text{(9)}& {\mathbf{B}}_{x,y}& ={\hat{e}}_x({\partial }_x{\partial }_z\mathrm{\Phi })+{\hat{e}}_y({\partial }_y{\partial }_z\mathrm{\Phi })\text{(10)}& & +{\hat{e}}_x({\partial }_y\mathrm{\Psi })+{\hat{e}}_y(-{\partial }_x\mathrm{\Psi }).\end{array}$$

In the $\hat{z}$ direction, the transverse components of $\mathbf{B}$ only contain the first-order derivative.

Here, we design a numerical scheme for the upward integral along the $\hat{z}$ direction. Suppose that we already have $(\mathrm{\Phi },\mathrm{\Psi }{)}^{\mathrm{\top }}$ and $(\dot{\mathrm{\Phi }},\dot{\mathrm{\Psi }}{)}^{\mathrm{\top }}$ on the location of $z_i$, where $(\dot{})$ over the variable indicates the derivative along $\hat{z}$, hereafter we denote them as $({\mathrm{\Phi }}_i,{\mathrm{\Psi }}_i{)}^{\mathrm{\top }}$ and $({\dot{\mathrm{\Phi }}}_i,{\dot{\mathrm{\Psi }}}_i{)}^{\mathrm{\top }}$. With $({\dot{\mathrm{\Phi }}}_i,{\dot{\mathrm{\Psi }}}_i{)}^{\mathrm{\top }}$, we can deal it like the integral of an ODE.

Lame parameters