Diffusion on poloidal and toroidal vector fields | Geowanderer

In this note I shall review the the effect of diffusion operator operated on poloidal and toroidal vector fields. In particular, I will investigate a scheme that is relevant to Earth and planetary fluid dynamics: the viscous/magnetic diffusion with radially varying diffusivity, and how the poloidal and toroidal fields are converted under these operators.

Diffusion operators

The viscous diffusion term in Navier Stokes equation reads

\[D_\nu = \frac{1}{\rho} \nabla\cdot (\mu \nabla \mathbf{v})\]

where \(\mu\) is the dynamic viscosity, and \(\mathbf{v}\) the velocity. In this note I shall assume incompressibility of the fluid, such that \(\nabla\cdot \mathbf{v} = 0\), and there is a toroidal-poloidal decomposition of the solenoidal field (see e.g. [[Vector Representation]]). When assuming uniform viscosity, this term is just scaled Laplacian of the velocity field

\[D_\nu = \frac{\mu}{\rho} \nabla^2 \mathbf{v}\]

The magnetic diffusion term in the induction equation reads

\[D_\eta = -\frac{1}{\mu_0} \nabla\times \left(\frac{1}{\sigma}\nabla\times \mathbf{B}\right) = -\nabla\times \left(\frac{1}{\mu_0 \sigma} \nabla\times \mathbf{B}\right) = -\nabla\times (\eta\nabla\times \mathbf{B})\]

where \(\sigma\) is the electrical conductivity, \(\mu_0\) the magnetic permeability, and \(\eta\) the magnetic diffusivity. When assuming uniform diffusivity (as is very commonly done in geodynamo), this term is just

\[D_\eta = -\eta \nabla\times(\nabla\times \mathbf{B}) = \eta \nabla^2 \mathbf{B}\]

The solenoidal property of \(\mathbf{B}\) always holds, and is already used here. In many cases, especially for giant planets or stars, diffusivity can span several orders of magnitudes at different radius, and a uniform diffusivity is no longer a valid assumption. This motivates the so-called anelastic approximation for the fluid flow, in contrast to the incompressible approximation typically used in geodynamo.

Magnetic diffusion operator

Preposition: The magnetic diffusion operator with radially varying diffusivity, defined as \(\mathcal{L}_\eta: \mathbf{A} \mapsto -\nabla\times (\eta(r) \nabla\times \mathbf{A})\)maps a toroidal field to a toroidal field, and a poloidal field to a poloidal field. That is, introducing the space of toroidal vector fields and polidal vector fields, \(\mathcal{T}\) and \(\mathcal{P}\) respectively, we have

\[\mathcal{L}_\eta \mathbf{A}\in \mathcal{T}\quad (\forall \mathbf{A}\in \mathcal{T}),\qquad \mathcal{L}_\eta \mathbf{A}\in \mathcal{P}\quad (\forall \mathbf{A}\in \mathcal{P})\]

In other words, the subspaces \(\mathcal{T}\) and \(\mathcal{P}\) are invariant subspaces under transform \(\mathcal{L}_\eta\).

Proof:

First, consider a poloidal field \(\mathbf{A}\in \mathcal{P}\). Hence \(\exists P(\mathbf{r})\) such that \(\mathbf{A} = \nabla\times (\nabla\times P(\mathbf{r})\mathbf{r})\). The (magnetic) diffusion of a poloidal field then takes the form

\[\mathcal{L}_\eta \mathbf{A} = \nabla\times (\eta \nabla\times \nabla\times \nabla\times P\mathbf{r})\]

Using the property that the curl of a poloidal field is a toroidal field (see [[Vector Representation]]), i.e.

\[\nabla\times \nabla\times \nabla \times P\mathbf{r} = \hat{\mathbf{r}}\times \nabla_s (\nabla^2 P)\]

and using the commutative property of \(\eta(r)\) with \(\hat{\mathbf{r}}\times \nabla_s\), we arrive at the expression

\[\mathcal{L}_\eta \mathbf{A} = \nabla \times (\hat{\mathbf{r}}\times \nabla_s(\eta \nabla^2 P )) = \nabla\times \nabla\times (-\eta \nabla^2 P \, \mathbf{r}) \in \mathcal{P}\]

Hence the resulting diffusion term is a poloidal field, with modified poloidal scalar \(-\eta \nabla^2 P\).

If, on the other hand, \(\mathbf{A} \in \mathcal{T}\), i.e. \(\exists \, T(\mathbf{r})\) such that \(\mathbf{A} = \nabla\times (T(\mathbf{r}) \mathbf{r})\), the magnetic diffusion

\[\begin{aligned} L_\eta \mathbf{A} &= \nabla\times (\eta \nabla\times \nabla\times T\mathbf{r}) \\ &= \frac{d\eta}{dr} \hat{\mathbf{r}}\times (\nabla\times \nabla\times T\mathbf{r}) + \eta \nabla\times (\nabla\times \nabla\times T\mathbf{r}) \end{aligned}\]

Using again the double curls and triple curls of \(f\mathbf{r}\) (see [[Vector Representation]]), we have

\[\begin{aligned} L_\eta \mathbf{A} &= \frac{d\eta}{dr}\hat{\mathbf{r}}\times \left[-\hat{\mathbf{r}} \nabla_s^2 \frac{T}{r} + \nabla_s\left(\frac{1}{r}\frac{\partial (rT)}{\partial r}\right)\right] + \eta \nabla\times (-\mathbf{r}\nabla^2T) \\ &= \frac{d\eta}{dr}\hat{\mathbf{r}} \times \nabla_s \left(\frac{1}{r}\frac{\partial (rT)}{\partial r}\right) + \eta \hat{\mathbf{r}}\times \nabla_s (\nabla^2 T) \end{aligned}\]

Again applying the commutative property and moving \(\eta\) and \(d\eta/dr\) inside the surface operators

\[L_\eta \mathbf{A} = \hat{\mathbf{r}}\times \nabla_s \left[\frac{1}{r}\frac{d\eta}{dr}\frac{\partial (rT)}{\partial r} + \eta \nabla^2 T\right] \in \mathcal{T}\]

Hence the resulting diffusion term is a toroidal field, with toroidal scalar \(-\eta \nabla^2 T - \frac{1}{r}\frac{d\eta}{dr}\frac{\partial (rT)}{\partial r}\). In summary, \(\mathcal{L}_\eta \mathbf{A} \in \mathcal{T}\) forall \(\mathbf{A}\in \mathcal{T}\), and \(\mathcal{L}_\eta \mathbf{A} \in \mathcal{P}\) forall \(\mathbf{A}\in \mathcal{P}\). \(\blacksquare\) Q.E.D.