Nonsolenoidal fields

While the velocity fields of interest here are all solenoidal, the inertia and buoyancy forces can have nonzero divergence. A general vector field (of sufficient smoothness) can be decomposed as follows. The treatment is specific to the domain $\{\mathbf{r}: 0\leq r<1/2\}$, but it could easily be generalized to other spheres.

Lemma 2 (Decomposition of vector fields)   Any vector field, $\mathbf{v}$, with Hölder continuous second order partial derivatives in a sphere can be expressed:

$$\mathbf{v} = \mbox{\boldmath$\nabla$}\mbox{$\mathcal S$}[\ma... ...x{$\mathcal T$}[\mathbf{v}-\mathbf{v}^{(S)}]\mathbf{r}\right).$$ (B.10)

Proof: Define $\mbox{$\mathcal S$}[\mathbf{v}]$ as the solution of the problem

$$\nabla^2 \mbox{$\mathcal S$}[\mathbf{v}] = \mbox{\boldmath$\nabla$}\cdot \mathbf{v},\qquad(0\leq r<\mbox{$\frac{1}{2}$})$$ (B.11)

$$\mbox{\boldmath$\hat{r}$}\cdot\mbox{\boldmath$\nabla$}\mbox{$\mathcal S$}[\mathbf{v}] = \mbox{\boldmath$\hat{r}$}\cdot\mathbf{v}, \qquad(r=\mbox{$\frac{1}{2}$}),$$ (B.12)

and let

$$\mathbf{v}^{(S)} = \mbox{\boldmath$\nabla$}\mbox{$\mathcal S$}[\mathbf{v}].$$ (B.13)

Then, $\mathbf{v}-\mathbf{v}^{(S)}$ is solenoidal and can be decomposed into poloidal and toroidal parts as before. $\Box$

Following Elsasser (1946), $\mathbf{v}^{(S)}$ is called the scaloidal part of $\mathbf{v}$.

Note that if $\mathbf{v}$ is solenoidal but has a nonvanishing normal component at the surface of the sphere; i.e. is solenoidal but not purely toroidal; the poloidal part obtained by the present procedure,

$$\mathbf{v}^{(P)} \equiv \mbox{\boldmath$\nabla$}\times \mbox... ...ox{$\mathcal P$}[\mathbf{v}-\mathbf{v}^{(S)}]\mathbf{r}\right)$$ (B.14)

will not necessarily equal that from (B.1). This is because some poloidal vector fields are irrotational and so can be alternatively represented as gradients. Only null toroidal fields are irrotational (Backus 1986).

It is advantageous here to apply the present procedure to all vector fields, whether or not they are solenoidal. This is because if a field is both irrotational and solenoidal, so that it has both scaloidal and poloidal representations, the scaloidal representation is usually easier to work with. In particular, in a momentum equation it can be combined with the pressure gradient term. The classic example of this is the integration of the Euler equations when the flow is both irrotational and incompressible (Lamb 1932, p. 19).

The toroidal part of a general vector field is then here defined as

$$\mathbf{v}^{(T)} \equiv \mbox{\boldmath$\nabla$}\times \lef... ...x{$\mathcal T$}[\mathbf{v}-\mathbf{v}^{(S)}]\mathbf{r}\right).$$ (B.15)

Note that the normal component at the spherical surface $r=1/2$ of the poloidal part of a vector field (defined by Lemma 2) vanishes, since

$$\mbox{\boldmath$\hat{r}$}\cdot\mathbf{v}^{(P)} = \mbox{\boldmath$\hat{r}$}\cdot\mathbf{v} - \mbox{\boldmath$\hat{r... ... - \mbox{\boldmath$\hat{r}$}\cdot\mathbf{v}^{(T)},\quad(r=\mbox{$\frac{1}{2}$})$$

$$= 0$$ (B.16)

by (B.12) and (B.9).