Solenoidal fields

The fundamental lemma is due to Chadwick and Trowbridge (1967).

Lemma 1 (Chadwick-Trowbridge)   Let $\mathbf{v}$ be a vector field possessing partial derivatives of order up to two which are Hölder continuous on $S=\{(r,\theta,\phi): 0<r_1\leq r\leq r_2<\infty, 0\leq\theta\leq\pi, 0\leq\phi\leq 2\pi\}$. Then if, in addition, $\mbox{\boldmath$\nabla$}\cdot \mathbf{v}=0$ on $S$, there are scalars $\mbox{$\mathcal P$}[\mathbf{v}]$ and $\mbox{$\mathcal T$}[\mathbf{v}]$ such that $\mathbf{v}=(\mbox{\boldmath$\nabla$}\times)^2(\mbox{$\mathcal P$}[\mathbf{v}]\m... ...r}) +\mbox{\boldmath$\nabla$}\times(\mbox{$\mathcal T$}[\mathbf{v}]\mathbf{r})$ on $S$.

Vectors of the form $(\mbox{\boldmath$\nabla$}\times)^2\mbox{$\mathcal P$}\mathbf{r}$ and $\mbox{\boldmath$\nabla$}\times\mbox{$\mathcal T$}\mathbf{r}$ are called poloidal and toroidal, respectively. It has been shown by Backus (1986) that the lemma can be extended from spherical annuli to spheres. The scalars can be obtained as the regular solutions of

$${\mathcal L}^2\mbox{$\mathcal P$}[\mathbf{u}] = -\mathbf{r}\cdot\mathbf{u}$$ (B.1)

$${\mathcal L}^2\mbox{$\mathcal T$}[\mathbf{u}] = -\mathbf{r}\cdot(\mbox{\boldmath$\nabla$}\times\mathbf{u}),$$ (B.2)

where

$${\mathcal L}^2\equiv r^2\nabla^2 - \frac{\partial }{\partial... ...} + \frac{1}{\sin^2\theta}\frac{\partial^2 }{\partial \phi^2}$$ (B.3)

(Moffatt 1978, p. 18).

The operator ${\mathcal L}^2$ is the surface Laplacian for the unit sphere (Aris 1989, pp. 196-7, 222). Its eigenfunctions are the spherical harmonics:

$${\mathcal L}^2S_n = -n(n+1) S_n$$ (B.4)

$$S_n \equiv P_n^m (\cos\theta) \left\{ \sin m\phi \atop \cos m\phi \right\}$$ (B.5)

(Lamb 1932, pp. 112-117). The $P_n^m(\cos\theta)$ are given by

$$P_n^m(\cos\theta) \equiv \frac{\sin^m\theta}{2^n n!} \frac... ...\cos\theta)^{(m+n)}} (-\sin^2\theta)^n, \quad (0\leq m\leq n)$$ (B.6)

for integer $m$ and $n$ (Lamb 1932, pp. 114-7). The first few $P_n^m$ are listed in table B.1.

Table B.1: The first few $P_n^m(c)$. Note that $s^2=1-c^2$.

	$m$
$n$	0	1	2	3	4
0	1	--	--	--	--
1	$c$	$s$	--	--	--
2	$\mbox{$\frac{1}{2}$}(3c^2-1)$	$3cs$	$3s^2$	--	--
3	$\mbox{$\frac{1}{2}$}(5c^3-3c)$	$\frac{3}{2}s(5c^2-1)$	$15s^2c$	$15s^3$	--
4	$\frac{1}{8}(35c^4-30c^2+3)$	$\frac{5}{2}s(7c^3-3c)$	$\frac{15}{2}s^2(7c^2-1)$	$105s^3c$	$105s^4$

An immediate and extremely useful consequence of (B.4) is

$$\nabla^2 r^m S_n = [m(m+1)-n(n+1)]r^{m-2}S_n.$$ (B.7)

The spherical components of poloidal and toroidal fields are:

$$(\mbox{\boldmath$\nabla$}\times)^2\mbox{$\mathcal P$}\mathbf{r} = -\frac{1}{r}{\mathcal L}^2\mbox{$\mathcal P$}\mbox{\boldmath$\hat... ...al^2(r\mbox{$\mathcal P$})}{\partial r\partial\phi}\mbox{\boldmath$\hat{\phi}$}$$ (B.8)

$$\mbox{\boldmath$\nabla$}\times\mbox{$\mathcal T$}\mathbf{r} = \frac{1}{\sin\theta}\frac{\partial \mbox{$\mathcal T$}}{\partial ... ...rac{\partial \mbox{$\mathcal T$}}{\partial \theta}\mbox{\boldmath$\hat{\phi}$}.$$ (B.9)

Clearly, arbitrary functions of $r$ may be added to $\mbox{$\mathcal P$}[\mathbf{v}]$ or $\mbox{$\mathcal T$}[\mathbf{v}]$ without affecting $\mathbf{v}^{(P)}$ or $\mathbf{v}^{(T)}$ (Moffatt 1978, p. 19).