Spectral representation of dissipation in spherical coordinates | Geowanderer

In some applications, we are interested in the quantity

\[\mathscr{D} = \int_{B} |\nabla\times \mathbf{A}|^2\, dV\]

where \(B\) denotes a unit ball in 3-D space, \(\mathbf{A}\) a solenoidal field, i.e. \(\nabla\cdot \mathbf{A} = 0\). The condition that \(\mathbf{A}\) is solenoidal perhaps seems a bit restrictive, but such fields naturally arise when \(\mathbf{A}\) represents the velocity field of a incompressible, uniform density fluid, or a general magnetic field. This integral naturally arise when calculating e. g. the viscous dissipation in incompressible fluid,

\[\mathcal{E}(\mathbf{u}) = \int_B |\nabla\times \mathbf{u}|^2\, dV = \int_B |\nabla \mathbf{u}|^2\, dV\]

which is also called enstrophy, and when calculating Ohmic dissipation,

\[\mathscr{D}_I = \int_B |\nabla\times \mathbf{B}|^2\, dV.\]

Since the field is solenoidal, and is defined within a sphere, we have the toroidal-poloidal representation of such vector field \(\mathbf{A}\),

\[\mathbf{A} = \nabla\times T \mathbf{r} + \nabla\times \nabla\times S\mathbf{r}\]

and the spectral representation of the toroidal and poloidal scalars

\[\begin{aligned} T(\mathbf{r}) &= \sum_{lmn} T_{lmn}\, R_n^{lm}(r) \,Y_l^m(\theta, \phi), \\ S(\mathbf{r}) &= \sum_{lmn} S_{lmn}\, R_n^{lm}(r) \,Y_l^m(\theta, \phi), \end{aligned}\]

where \(\sum_{lmn} = \sum_{l=0}^L \sum_{m=-l}^l \sum_{n=0}^N\), \(R_n^{lm}(r)\) is some radial basis function. A popular choice is a spherical-harmonic-order-independent one-sided Jacobi polynomial, known as the (Jones-)Worland polynomial, \(W_n^l(r) \propto r^l P_n^{(-\frac{1}{2}, l-\frac{1}{2})}(2r^2 - 1)\) (Livermore et al. 2007). What, then, would be the representation of \(\mathscr{D}\) purely in the spectral domain?

Let us first rewrite the curl of \(\mathbf{A}\) in Helmholtz representation, but with surface operators,

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

where \(\nabla_s\) is the surface gradient operator, and we used the properties of toroidal and poloidal fields. This is the Helmholtz representation of field \(\nabla\times \mathbf{A}\), where we decompose it into a surface gradient term (first term), a surface curl term (third term), and a radial component (second term) (see the notes on vector representations on the surface of a sphere). We then substitute the scalars with their respective spectral expansion, and

\[\begin{aligned} \nabla\times \mathbf{A} &= \sum_{lmn} \bigg[T_{lmn} \frac{1}{r} \frac{d(rR_n^{lm})}{d r} \nabla_s Y_l^m - T_{lmn} \frac{R_n^{lm}}{r} \hat{\mathbf{r}} \nabla_s^2 Y_l^m \\ &\qquad \,\,\, - S_{lmn} \nabla\times \left(\frac{1}{r^2} \frac{d}{d r}\left(r^2\frac{dR_n^{lm}}{d r}\right) + \frac{R_n^{lm}}{r^2} \nabla_s^2\right)Y_l^m \mathbf{r}\bigg] \\ &= \sum_{lmn} \bigg[T_{lmn} \frac{1}{r} \frac{d(rR_n^{lm})}{d r} \nabla_s Y_l^m + T_{lmn} \frac{l(l+1)}{r} R_n^{lm} Y_l^m \hat{\mathbf{r}} \\ &\qquad \,\,\, - S_{lmn} \left(\frac{1}{r^2} \frac{d}{d r}\left(r^2\frac{dR_n^{lm}}{d r}\right) - \frac{l(l+1)}{r^2} R_n^{lm}\right) \nabla \times Y_l^m \mathbf{r}\bigg] \\ &= \sum_{lmn} \bigg[T_{lmn} \frac{1}{r} \frac{d(rR_n^{lm})}{d r} \mathbf{B}_l^m + T_{lmn} \frac{l(l+1)}{r} R_n^{lm} \mathbf{P}_l^m \\ &\qquad \,\,\, + S_{lmn} \left(\frac{l(l+1)}{r^2} R_n^{lm} - \frac{1}{r^2} \frac{d}{d r}\left(r^2\frac{dR_n^{lm}}{d r}\right)\right) \mathbf{C}_l^m\bigg] \end{aligned}.\]

where \(\mathbf{P}_l^m\), \(\mathbf{B}_l^m\) and \(\mathbf{C}_l^m\) are the vector spherical harmonics. Using the orthogonality of the vector spherical harmonics, we have

\[\begin{aligned} \mathscr{D} &= \int_B |\nabla\times \mathbf{A}|^2\, dV \\ &= \sum_{lm} \sum_{nn'} T_{lmn'}^* T_{lmn} \langle \mathbf{B}_l^m, \mathbf{B}_l^m \rangle_S \int_0^1 \left(\frac{1}{r} \frac{d(rR_{n'}^{lm})}{d r}\right) \left(\frac{1}{r} \frac{d(rR_n^{lm})}{d r}\right) r^2 dr \\ &+ \sum_{lm} \sum_{nn'} T_{lmn'}^* T_{lmn} \langle \mathbf{P}_l^m, \mathbf{P}_l^m \rangle_S \int_0^1 \left(\frac{l(l+1)}{r} R_{n'}^{lm}\right) \left(\frac{l(l+1)}{r} R_n^{lm}\right) r^2 dr \\ &+ \sum_{lm} \sum_{nn'} S_{lmn'}^* S_{lmn} \langle \mathbf{C}_l^m, \mathbf{C}_l^m \rangle_S \\ &\quad \times \int_0^1 \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_{n'}^{lm} \cdot \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_n^{lm} r^2 dr \\ &= \sum_{lm} \sum_{nn'} T_{lmn'}^* T_{lmn} \left[ l(l+1) \int_0^1 \frac{d(rR_{n'}^{lm})}{d r} \frac{d(rR_n^{lm})}{d r} dr + l^2(l+1)^2 \int_0^1 R_{n'}^{lm} R_n^{lm} dr\right] \\ &+ \sum_{lm} \sum_{nn'} S_{lmn'}^* S_{lmn} l(l+1) \\ &\quad \times \int_0^1 \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_{n'}^{lm} \cdot \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_n^{lm} r^2 dr. \end{aligned}\]

Therefore, defining matrices

\[\begin{aligned} (\mathbf{M}^{lm}_{\mathscr{D}T})_{n'n} &= l(l+1) \int_0^1 \frac{d(rR_{n'}^{lm})}{d r} \frac{d(rR_n^{lm})}{d r} dr + l^2(l+1)^2 \int_0^1 R_{n'}^{lm} R_n^{lm} dr \\ (\mathbf{M}^{lm}_{\mathscr{D}S})_{n'n} &= l(l+1) \int_0^1 \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_{n'}^{lm} \left(\frac{l(l+1)}{r^2} - \frac{1}{r^2} \frac{d}{d r}r^2\frac{d}{d r}\right) R_n^{lm} r^2 dr \\ \end{aligned}\]

the volume integral can be written in the quadratic form

\[\mathscr{D} = \int_B |\nabla\times \mathbf{A}|^2\, dV = \sum_{lm} \left[\mathbf{T}_{lm}^H \mathbf{M}_{\mathscr{D}T}^{lm} \mathbf{T}_{lm} + \mathbf{S}_{lm}^H \mathbf{M}_{\mathscr{D}S}^{lm} \mathbf{S}_{lm}\right].\]

This gives the representation of \(\mathscr{D}\) purely in terms of spectral coefficients of \(\mathbf{A}\). The specific form of the matrix depends on the radial basis of choice, but these matrices are usually dense (e.g. for Worland polynomials).