Tomographic Reconstruction of the Beltrami Fields

Abstract

We introduce and study a tomography method and have developed a numerical algorithm for the reconstruction of the linear Beltrami fields $\nabla \times \mathbf{B}=\varkappa \mathbf{B}$ in a bounded domain of $\mathbf{R}^3$ by using a known ray transform. This method is based on the expansion of a vector field over the special basis of vector functions. The results of computer simulation are given.

Appendix A

1. Vector spherical harmonics and Hansen’s vectors.

The set of vector spherical harmonics used in this paper is introduced in [22,23,24].

$$\begin{aligned}&{} \mathbf{P}_l^m({\hat{\mathbf{r}}})={\hat{\mathbf{r}}} Y_{lm}({\hat{\mathbf{r}}}), \end{aligned}$$

(A.1)

$$\begin{aligned}&{} \mathbf{C}_l^m({\hat{\mathbf{r}}})=\frac{-1}{\sqrt{l(l+1)}}({\hat{\mathbf{r}}} \times \nabla _{{\hat{\mathbf{r}}}})Y_{l m}({\hat{\mathbf{r}}})\nonumber \\&\quad = \frac{1}{\sqrt{l(l+1)}} \Bigr ( \frac{1}{\sin \theta }\frac{\partial }{\partial \varphi }{\hat{\varvec{\theta }}}- \frac{\partial }{\partial \theta }\hat{\varvec{\varphi }} \Bigl )Y_{lm}({\hat{\mathbf{r}}}), \end{aligned}$$

(A.2)

$$\begin{aligned}&{} \mathbf{B}_l^m({\hat{\mathbf{r}}})=\frac{1}{\sqrt{l(l+1)}} \nabla _{{\hat{\mathbf{r}}}}Y_{l m}({\hat{\mathbf{r}}})\nonumber \\&\quad = \frac{1}{\sqrt{l(l+1)}} \Bigr ( \frac{\partial }{\partial \theta }{\hat{\varvec{\theta }}}+ \frac{1}{\sin \theta }\frac{\partial }{\partial \varphi }\hat{\varvec{\varphi }} \Bigl )Y_{lm}({\hat{\mathbf{r}}}), \end{aligned}$$

(A.3)

where $Y_{lm}({\hat{\mathbf{r}}})$ are scalar spherical harmonics, and operators $\nabla _\mathbf{r}, \nabla _{{\hat{\mathbf{r}}}}$ have the form

$$\begin{aligned} \nabla _\mathbf{r} = {\hat{\mathbf{r}}}\frac{\partial }{\partial r} + \frac{1}{r}\nabla _{{\hat{\mathbf{r}}}}, \ \ \nabla _{{\hat{\mathbf{r}}}}= {\hat{\varvec{\theta }}}\frac{\partial }{\partial \theta }+\frac{\hat{\varvec{\varphi }}}{\sin \theta }\frac{\partial }{\partial \varphi }. \end{aligned}$$

The vector spherical harmonics are pointwise perpendicular,

$$\begin{aligned} \mathbf{P}_l^m({\hat{\mathbf{r}}}) \cdot \mathbf{C}_l^m({\hat{\mathbf{r}}})= \mathbf{C}_l^m({\hat{\mathbf{r}}}) \cdot \mathbf{B}_l^m({\hat{\mathbf{r}}})= \mathbf{B}_l^m({\hat{\mathbf{r}}}) \cdot \mathbf{P}_l^m({\hat{\mathbf{r}}})=0, \end{aligned}$$

and they satisfy the orthogonality conditions

$$\begin{aligned}&\int \limits _{S^2}{} \mathbf{P}_l^m({\hat{\mathbf{r}}}) \cdot [\mathbf{P}_{l^\prime }^{m^\prime }({\hat{\mathbf{r}}})]^*\hbox {d}\varOmega ({\hat{\mathbf{r}}})= \int \limits _{S^2}{} \mathbf{C}_l^m({\hat{\mathbf{r}}}) \cdot [\mathbf{C}_{l^\prime }^{m^\prime }({\hat{\mathbf{r}}})]^*\hbox {d}\varOmega ({\hat{\mathbf{r}}}) \nonumber \\&\quad =\int \limits _{S^2}{} \mathbf{B}_l^m({\hat{\mathbf{r}}}) \cdot [\mathbf{B}_{l^\prime }^{m^\prime }({\hat{\mathbf{r}}})]^*\hbox {d}\varOmega ({\hat{\mathbf{r}}}) =\frac{4\pi }{2l+1}\frac{(l+|m|)!}{(l-|m|)!}\delta _{ll^\prime }\delta _{mm^\prime },\nonumber \\ \end{aligned}$$

(A.4)

for any $l=0,1,\ldots$ and $|m|\leqslant l$, $*$ means complex conjugation.

There is the following relation of $\mathbf{P}_l^m, \mathbf{B}_l^m, \mathbf{C}_l^m$ to some other vector spherical harmonics $\mathbf{Y}_{lm}^{(-1)}, \mathbf{Y}_{lm}^{(0)}, \mathbf{Y}_{lm}^{(1)}$ introduced in [31], by $\mathbf{P}_l^m = \mathbf{Y}_{lm}^{(-1)}$, $\mathbf{C}_l^m =-i \mathbf{Y}_{lm}^{(0)}$, $\mathbf{B}_l^m = \mathbf{Y}_{lm}^{(1)}$. Under rotation of the frame of reference defined by Euler angles, the vectors $\mathbf{P}_l^m, \mathbf{C}_l^m, \mathbf{B}_l^m$ are transformed in the same manner as vectors $\mathbf{Y}_{lm}^{(-1)}$, $\mathbf{Y}_{lm}^{(0)}$, $\mathbf{Y}_{lm}^{(1)}$.

$$\begin{aligned} \mathbf{Y}_{lm^{\prime }}^{(\lambda )}({\hat{\mathbf{r}}}^{\prime })=\sum \limits _{m=-l}^l \mathbf{Y}_{lm}^{(\lambda )}({\hat{\mathbf{r}}}) D_{mm^{\prime }}^l(\alpha ,\beta ,\gamma ), \quad \lambda =(-1,0,1). \end{aligned}$$

(A.5)

The set of vector spherical harmonics forms a complete and orthonormal set of functions in $(L^2(S^2))^3$.

Hansen’s vectors are defined as in [22, 25, 26]

$$\begin{aligned}&{} \mathbf{M}_l^m(\mathbf{r})= \nabla _\mathbf{r}\times [\mathbf{r}\, z_l(\varkappa r) Y_{lm}({\hat{\mathbf{r}}})]\nonumber \\&\quad = \sqrt{l(l+1)}\, z_l(\varkappa r)\,\mathbf{C}_l^m({\hat{\mathbf{r}}}), \end{aligned}$$

(A.6)

$$\begin{aligned}&{} \mathbf{N}_l^m(\mathbf{r})=\frac{1}{\varkappa }\nabla _\mathbf{r}\times \mathbf{M}_l^m(\mathbf{r})\nonumber \\&\quad =\frac{1}{\varkappa }\nabla _\mathbf{r}\times \nabla _\mathbf{r}\times [\mathbf{r}\, z_l(\varkappa r) Y_{lm}({\hat{\mathbf{r}}})]\nonumber \\&\quad =l(l+1)\frac{z_l(\varkappa r)}{\varkappa r }\, \mathbf{P}_l^m({\hat{\mathbf{r}}})\nonumber \\&\qquad + \sqrt{l(l+1)}\frac{1}{\varkappa r}\Bigl [\frac{\hbox {d}}{\hbox {d} \varkappa r}\bigl (\varkappa r \ z_l(\varkappa r)\bigr )\Bigr ]\mathbf{B}_l^m({\hat{\mathbf{r}}}) \end{aligned}$$

(A.7)

$$\begin{aligned}&{} \mathbf{L}_l^m(\mathbf{r})= \frac{1}{\varkappa }\nabla _\mathbf{r}[ z_l(\varkappa r)Y_{lm}({\hat{\mathbf{r}}})] \nonumber \\&\quad = \sqrt{l(l+1)}\frac{z_l(\varkappa r)}{\varkappa r }\, \mathbf{B}_l^m({\hat{\mathbf{r}}})+\frac{\hbox {d} z_l(\varkappa r)}{\hbox {d}(\varkappa r)} \,\mathbf{P}_l^m({\hat{\mathbf{r}}}), \end{aligned}$$

(A.8)

where $z_l(\varkappa r)= j_l(\varkappa r), y_l(\varkappa r), h_l^{(1)}(\varkappa r), h_l^{(2)}(\varkappa r)$ designate any spherical Bessel function of the first, second and third kinds, respectively.

The definitions (A.6)–(A.8) imply that $M_{lm}$ and $N_{lm}$ have no divergence, and also $L_{lm}$ have no curl,

$$\begin{aligned} \nabla \cdot \mathbf{M}_l^m = 0, \quad \nabla \cdot \mathbf{N}_l^m = 0,\quad \nabla \times \mathbf{L}_l^m = 0. \end{aligned}$$

(A.9)

During the rotation, they are transformed in the same way as the vector spherical harmonics (A.5).

An inner product for vectors $\mathbf{M}_l^m(\varkappa ,\mathbf{r})$ and $\mathbf{N}_l^m(\varkappa ,\mathbf{r})$ is defined as follows

$$\begin{aligned}&\int \limits _0^{R}\!r^2 dr\! \int \limits \mathbf{M}_l^m(\varkappa ,\mathbf{r})\cdot [\mathbf{M}_l^m(\varkappa ,\mathbf{r})]^*d\varOmega (\hat{\mathbf{r}}) = N_{lm} I_l, \end{aligned}$$

(A.10)

$$\begin{aligned}&\int \limits _0^R r^2 \hbox {d}r \int \mathbf{N}_l^m(\varkappa ,\mathbf{r})\cdot [\mathbf{N}_l^m(\varkappa ,\mathbf{r})]^*\hbox {d}\varOmega ({\hat{\mathbf{r}}})\nonumber \\&\quad = N_{lm} \bigl [(l+1)I_{l-1} +l I_{l+1}\bigr ], \end{aligned}$$

(A.11)

where $*$ means complex conjugation, $N_{lm}=\displaystyle \frac{4\pi l(l+1)}{2l+1}\frac{(l+m)!}{(l-m)!}$, $I_l$ is equals [28, p.134]

$$\begin{aligned} I_l=\int \limits _0^R [j_l(\varkappa r)]^2 r^2 \hbox {d}r=\frac{R^3}{2}\bigl ([j_l(\varkappa R)]^2 - j_{l-1}(\varkappa R)j_{l+1}(\varkappa R)\bigr ). \end{aligned}$$

(A.12)

Furthermore, $\mathbf{M},\mathbf{N}$ and $\mathbf{L}$ are mutually orthogonal; that is,

$$\begin{aligned}&\int \limits _0^{2\pi }\hbox {d}\phi \int \limits _0^{\pi } \sin (\theta )\hbox {d}\theta \ \mathbf{M}_l^m(\varkappa ,\mathbf{r})\cdot [\mathbf{N}_{l_1}^{m_1}(\varkappa ,\mathbf{r})]^*=0, \end{aligned}$$

(A.13)

$$\begin{aligned}&\int \limits _0^{2\pi }\hbox {d}\phi \int \limits _0^{\pi } \sin (\theta )\hbox {d}\theta \ \mathbf{M}_l^m(\varkappa ,\mathbf{r})\cdot [\mathbf{L}_{l_1}^{m_1}(\varkappa ,\mathbf{r})]^*=0, \end{aligned}$$

(A.14)

Obviously, $\mathbf{M}_l^m, \mathbf{N}_l^m$ may be used to represent solenoidal fields, and $\mathbf{L}_l^m$ to represent a potential field. Usually, in problems related to plasma diagnostics, both the velocity and the magnetic fields satisfy the solenoidal condition.

2. An auxiliary formula for numerical evaluation of$\mathbf{M}_l^m, \mathbf{N}_l^m$vector wave functions.

From the definitions (A.2), (A.3) and (A.6), (A.7) it can be seen that to evaluate the vectors $\mathbf{M}_l^m, \mathbf{N}_l^m$, some caution is required for calculation the following quantities.

$$\begin{aligned}&\displaystyle \frac{\partial Y_{lm}}{\partial \theta }=\mathop {\mathrm e}\nolimits ^{im\phi }\frac{\hbox {d} P_l^m(\cos \theta )}{\hbox {d}\theta }, \quad \nonumber \\&\quad \frac{1}{\sin \theta }\frac{\partial Y_{lm}}{\partial \phi }=i m \mathop {\mathrm e}\nolimits ^{im\phi }\frac{P_l^m(\cos \theta )}{\sin \theta } \end{aligned}$$

(A.15)

The following recurrence formulas can be used for this purpose [25, p. 402]

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d} \theta }P_l^m(\cos \theta )\\&\quad =\frac{1}{2} \bigl [(l-m+1)(l+m)P_l^{m-1}(\cos \theta )-P_l^{m+1}(\cos \theta )\bigr ] \end{aligned}$$

For evaluation of the $\theta$-component of $Y_{lm}^{(0)}$ and $\phi$-component of $Y_{lm}^{(1)}$ at the points $\theta =0,\pi$, the following formula can be used [25, p. 401]

$$\begin{aligned} \frac{P_l^m(\cos \theta )}{\sin \theta }|_{\theta =0,\pi }= & {} \frac{\cos \theta }{2 m} \bigl [(n-m+1)(n+m)P_l^{m-1}(\cos \theta )\\&\quad + P_l^{m+1}(\cos \theta )\bigl ],\\ \frac{P_l^m(\cos \theta )}{\sin \theta }= & {} \frac{-1}{2m}\bigl [(l-m+2)(l-m+1) P_{l+1}^{m-1}(\cos \theta )\\&\quad +P_{l+1}^{m+1}(\cos \theta )\bigr ], m > 0. \end{aligned}$$

For the quantities $\displaystyle \frac{1}{\varkappa r}\frac{\hbox {d}}{\hbox {d}\varkappa r}(\varkappa r z_l(\varkappa r))$, the following recurrence formula is used [27, p. 439(10.1.20)]

$$\begin{aligned} \frac{\hbox {d}z_l(\varkappa r)}{\hbox {d}(\varkappa r)}=\frac{1}{(2l+1)}\bigl [l z_{l-1}(\varkappa r)-(l+1) z_{l+1}(\varkappa r)\bigr ]. \end{aligned}$$

3. Remarks to Lemma1.

Let the following operators be defined: ${\hat{\mathbf{L}}}_\mathbf{r}=(\mathbf{r} \times \nabla _\mathbf{r})$ and ${\hat{\mathbf{L}}}_\mathbf{k}=(\mathbf{k} \times \nabla _\mathbf{k})$. Then the relations below hold.

$$\begin{aligned}&{\hat{\mathbf{L}}}_\mathbf{r} \mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}=(\mathbf{r} \times \nabla _\mathbf{r}) \mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}=(\mathbf{r} \times i\mathbf{k})\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}= -i(\mathbf{k} \times \mathbf{r})\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}} \nonumber \\&\quad =-i[\mathbf{k} \times (-i\nabla _\mathbf{k})]\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}= -(\mathbf{k} \times \nabla _\mathbf{k})\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}=- {\hat{\mathbf{L}}}_\mathbf{k} \mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}. \nonumber \\&{\hat{\mathbf{L}}}_\mathbf{k}\int \mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}\varPsi (\mathbf{r})\hbox {d}\mathbf{r}= \int {\hat{\mathbf{L}}}_\mathbf{k}\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}\varPsi (\mathbf{r})\hbox {d}{} \mathbf{r}\nonumber \\&= -\int {\hat{\mathbf{L}}}_\mathbf{r}\mathop {\mathrm e}\nolimits ^{i\mathbf{k}\cdot \mathbf{r}}\varPsi (\mathbf{r})\hbox {d}{} \mathbf{r}.\!\!\!\nonumber \\ \end{aligned}$$

(A.16)