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Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for the use with vector fields.

Definition

Several conventions have been used to define the VSH. We follow that of Barrera et al. Given a scalar spherical harmonic

Y_{lm}(theta,varphi)

we define three VSH:

$mathbf{Y}_{lm} = Y_{lm}hat{mathbf{r}}$

$mathbf{Psi}_{lm} = rnabla Y_{lm}$

$mathbf{Phi}_{lm} = mathbf{r}timesnabla Y_{lm}$

being $hat{mathbf{r}}$ the unitary vector along the radial direction. The radial factors are included to guarantee that the dimensions of the VSH are the same as the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.

The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical ordinates, so that a vector field admits a multipole expansion

mathbf{E}=sum_{l=0}^inftysum_{m=-l}^lleft(E^r_{lm}(r)mathbf{Y}_{lm}+E^{(1)}_{lm}(r)mathbf{Psi}_{lm}+E^{(2)}_{lm}(r)mathbf{Phi}_{lm}right)

The labels on the components reflect that $E^r_{lm}$ is the radial component of the vector field, while $E^{(1)}_{lm}$ and $E^{(2)}_{lm}$ are transverse components.

Main Properties

Symmetry

Like the scalar spherical harmonics, the VSH verify

mathbf{Y}_{l,-m} = (-1)^m mathbf{Y}^*_{lm}qquadmathbf{Psi}_{l,-m} = (-1)^m mathbf{Psi}^*_{lm}qquadmathbf{Phi}_{l,-m} = (-1)^m mathbf{Phi}^*_{lm}

Orthogonality

The VSH are orthogonal in the usual three-dimensional way

mathbf{Y}_{lm}cdotmathbf{Psi}_{lm}=0qquadmathbf{Y}_{lm}cdotmathbf{Phi}_{lm}=0qquadmathbf{Psi}_{lm}cdotmathbf{Phi}_{lm}=0

but also in the Hilbert space

intmathbf{Y}_{lm}cdot mathbf{Y}^*_{l'm'},mathrm{d}Omega = delta_{ll'}delta_{mm'}

intmathbf{Psi}_{lm}cdot mathbf{Psi}^*_{l'm'},mathrm{d}Omega = l(l+1)delta_{ll'}delta_{mm'}

intmathbf{Phi}_{lm}cdot mathbf{Phi}^*_{l'm'},mathrm{d}Omega = l(l+1)delta_{ll'}delta_{mm'}

intmathbf{Y}_{lm}cdot mathbf{Psi}^*_{l'm'},mathrm{d}Omega = 0

intmathbf{Y}_{lm}cdot mathbf{Phi}^*_{l'm'},mathrm{d}Omega = 0

intmathbf{Psi}_{lm}cdot mathbf{Phi}^*_{l'm'},mathrm{d}Omega = 0

Vector multipole moments

The orthogonality relations allow to compute the spherical multipole moments of a vector field as

E^r_{lm} = int mathbf{E}cdot mathbf{Y}^*_{lm},mathrm{d}Omega

E^{(1)}_{lm} = frac{1}{l(l+1)}int mathbf{E}cdot mathbf{Psi}^*_{lm},mathrm{d}Omega

E^{(2)}_{lm} = frac{1}{l(l+1)}int mathbf{E}cdot mathbf{Phi}^*_{lm},mathrm{d}Omega

The gradient of a scalar field

Given the multipole expansion of a scalar field

phi = sum_{l=0}^infty sum_{m=-l}^l phi_{lm}(r) Y_{lm}(theta,phi)

we can express its gradient in terms of the VSH as

nablaphi = sum_{l=0}^infty sum_{m=-l}^lleft(frac{mathrm{d}phi_{lm}}{mathrm{d}r} mathbf{Y}_{lm}+

frac{phi_{lm}}{r}mathbf{Psi}_{lm}right)

Divergence

For any multipole field we have

nablacdotleft(f(r)mathbf{Y}_{lm}right) = left(frac{mathrm{d}f}{mathrm{d}r}+frac{2}{r}fright)Y_{lm}

nablacdotleft(f(r)mathbf{Psi}_{lm}right) = -frac{l(l+1)}{r}fY_{lm}

nablacdotleft(f(r)mathbf{Phi}_{lm}right) = 0

By superposition we obtain the divergence of any vector field

nablacdotmathbf{E} = sum_{l=0}^infty sum_{m=-l}^l left(frac{mathrm{d}E^r_{lm}}{mathrm{d}r}+frac{2}{r}E^r_{lm}-frac{l(l+1)}{r}E^{(1)}_{lm}right)Y_{lm}

we see that the component on $mathbf{Phi}_{lm}$ is always solenoidal.

Curl

For any multipole field we have

nablatimesleft(f(r)mathbf{Y}_{lm}right) =-frac{1}{r}fmathbf{Phi}_{lm}

nablatimesleft(f(r)mathbf{Psi}_{lm}right) = left(frac{mathrm{d}f}{mathrm{d}r}+frac{1}{r}fright)mathbf{Phi}_{lm}

nablatimesleft(f(r)mathbf{Phi}_{lm}right) = -frac{l(l+1)}{r}fmathbf{Y}_{lm}-left(frac{mathrm{d}f}{mathrm{d}r}+frac{1}{r}fright)mathbf{Psi}_{lm}

By superposition we obtain the curl of any vector field

nablatimesmathbf{E} = sum_{l=0}^infty sum_{m=-l}^l

left(-frac{l(l+1)}{r}E^{(2)}_{lm}mathbf{Y}_{lm}-left(frac{mathrm{d}E^{(2)}_{lm}}{mathrm{d}r}+ frac{1}{r}E^{(2)}_{lm}right)mathbf{Psi}_{lm}+ left(-frac{1}{r}E^r_{lm}+frac{mathrm{d}E^{(1)}_{lm}}{mathrm{d}r}+frac{1}{r}E^{(1)}_{lm}right)mathbf{Phi}_{lm}right)

Examples

Firsts vector spherical harmonics

$l=0,$

mathbf{Y}_{00}= sqrt{frac{1}{4pi}}hat{mathbf{r}}

mathbf{Psi}_{00}= mathbf{0}

mathbf{Phi}_{00}= mathbf{0}

$l=1,$

mathbf{Y}_{10}= sqrt{frac{3}{4pi}}costheta,hat{mathbf{r}}

mathbf{Y}_{11}= -sqrt{frac{3}{8pi}}mathrm{e}^{mathrm{i}varphi}sintheta,hat{mathbf{r}}

mathbf{Psi}_{10}= -sqrt{frac{3}{4pi}}sintheta,hat{mathbf{theta}}

mathbf{Psi}_{11}= -sqrt{frac{3}{8pi}}mathrm{e}^{mathrm{i}varphi}left(costheta,hat{mathbf{theta}}+mathrm{i}hat{mathbf{varphi}}right)

mathbf{Phi}_{10}= -sqrt{frac{3}{4pi}}sintheta,hat{mathbf{varphi}}

mathbf{Phi}_{11}= sqrt{frac{3}{8pi}}mathrm{e}^{mathrm{i}varphi}left(mathrm{i},hat{mathbf{theta}}-costheta,hat{mathbf{varphi}}right)

The expression for negative values of m are obtained applying the symmetry relations.

Application to electrodynamics

The VSH are especially useful in the study of multipole radiation fields. For instance, a magnetic multipole is due to an oscillating current with angular frequency

omega,

and complex amplitude

hat{mathbf{J}}= J(r)mathbf{Phi}_{lm}

and the corresponding electric and magnetic fields can be written as

hat{mathbf{E}}= E(r)mathbf{Phi}_{lm}

hat{mathbf{B}}= B^r(r)mathbf{Y}_{lm}+B^{(1)}(r)mathbf{Psi}_{lm}

Substituting into Maxwell equations, Gauss' law is automathically satisfied

nablacdothat{mathbf{E}}=0

while Faraday's law decouples in

nablatimeshat{mathbf{E}}=-mathrm{i}omegahat{mathbf{B}}qquadRightarrowqquad

left{begin{array}{l}displaystyle frac{l(l+1)}{r}E = mathrm{i}omega B^r displaystylefrac{mathrm{d}E}{mathrm{d}r} +frac{E}{r}= mathrm{i}omega B^{(1)}end{array}right.

Gauss' law for the magnetic field implies

nablacdothat{mathbf{B}} = 0quadRightarrow quadfrac{mathrm{d}B^r}{r}+frac{2}{r}B^r - frac{l(l+1)}{r}B^{(1)}=0

and Ampère-Maxwell's equation gives

nablatimeshat{mathbf{B}} = mu_0 hat{mathbf{J}} + mathrm{i}mu_0varepsilon_0omega hat{mathbf{E}} quadRightarrowquad -frac{B^r}{r}+frac{mathrm{d}B^{(1)}}{mathrm{d}r}+frac{B^{(1)}}{r} = mu_0J+mathrm{i}omegamu_0varepsilon_0E

In this way, the partial differential equations have been transformed in a set of ordinary differential equations.

Application to fluid dynamics

In the calculation of the Stokes' law for the drag than a viscous fluid exerts on a small spherical particle, the velocity distribution obeys Navier-Stokes equations neglecting inertia, i.e.

nablacdot mathbf{v} = 0

mathbf{0} = -nabla p + eta nabla^2mathbf{v}

with the boundary conditions

mathbf{v} = mathbf{0}quad (r=a)

mathbf{v} = -mathbf{U}_0quad (rtoinfty)

being $mathbf{U},$ the relative velocity of the particle to the fluid far from the particle. In spherical coordinates this velocity at infinity can be written as