Poly

Polynomial functions and conveniences that are needed for the spectral or spatial discretization.

Polynomials

Limace.Poly.ylm — Function
ylm(ℓ::Int64, m::Int64, θ, φ) -> Any

Spherical harmonic $Y_l^m$ in full norm, i.e.

\[\int Y_l^mY_i^j\, \sin(\theta)\,\mathrm{d}\theta\mathrm{d}\phi = \delta_{li}\delta_{mj}\]

where $\theta$ is the colatitude and $\phi$ the azimuthal angle.

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Integral functions

Integrals over the spherical surfaces are reduced to two specific integrals, known as the Adam-Gaunt and Elsasser integrals (James, 1973).

Limace.Poly.adamgaunt — Function
adamgaunt(la,lb,lc,ma,mb,mc)

Adam-Gaunt integral (James, 1973), given by

\[A_{abc} = \oint\int Y_aY_bY_c\sin\theta\,\mathrm{d}\theta\mathrm{d}\phi,\]

where the spherical harmonics are abbreviated, so that $Y_a = Y_{l_a}^{m_a}$.

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Limace.Poly.elsasser — Function
elsasser(la,lb,lc,ma,mb,mc)

Elsasser integral (James, 1973), given by

\[E_{abc} = \oint\int Y_c\left( \frac{\partial Y_a}{\partial \theta} \frac{\partial Y_b}{\partial \phi} - \frac{\partial Y_a}{\partial \phi}\frac{\partial Y_b}{\partial \theta} \right)\,\mathrm{d}\theta\mathrm{d}\phi,\]

where the spherical harmonics are abbreviated, so that $Y_a = Y_{l_a}^{m_a}$.

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The radial part of the projection of poloidal and toroidal vectors can be described by two scalar functions, Limace.Poly.inners and Limace.Poly.innert.

Limace.Poly.inners — Function
inners(s, s2, l, r) -> Any

\[r\rightarrow \frac{l(l+1)}{r^2}\left( l(l+1)s(r) s_2(r) + \frac{\partial r s(r)}{\partial r}\frac{\partial r s_2(r)}{\partial r}\right)\]

Radial function to be integrated in radius when computing the inner product of two poloidal vectors.

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Limace.Poly.innert — Function
innert(t, t2, l::Int64, r) -> Any

\[r\rightarrow l(l+1) t(r) t_2(r)\]

Radial function to be integrated in radius when computing the inner product of two toroidal vectors.

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These functions are used for the discrete integration in radius. The derivatives are computed using automatic differentiation (based on ForwardDiff.jl).

Limace.Quadrature.rquad — Function
rquad(nr, a, b)
rquad(nr, V::Volume)

Computes nr Gauss-Legendre quadrature points and weights on a radial grid. 

r0 = 0.3
r1 = 1.0

r, wr = Limace.Quadrature.rquad(50, r0, r1)

V = SphericalShell(r0,r1)

r, wr = Limace.Quadrature.rquad(50, V)
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The radial integrals of the projections are all integrated using

Limace.Quadrature.∫dr — Function
∫dr(f, r::Vector{T}, wr::Vector{T})::Complex{T} where T

Computes the integral of a function f

\[\int_{r_0}^{r_1} r^2 f(r)\,\mathrm{d}r,\]

with $r_0<r<r_1$, using quadrature points r and weights wr computed by Limace.Quadrature.rquad. This function returns a complex value by default for easier type stability.

Type \int<TAB> to write ∫.

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using Limace.Quadrature: rquad, ∫dr
r,wr = rquad(10, 0.0, 1.0)
f = r->r^2
∫dr(f,r,wr) ≈ 1/5
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Miscellaneous functions

The functions listed here are mostly for internal convenience. Nevertheless, they may be useful for further implementations.

Limace.Poly._∂ll — Function
_∂ll(f, l, l1, r) -> Any

\[\partial_l^{l_1} = \begin{cases} \frac{\partial f}{\partial r} + \frac{l+1}{r}f \quad \mathrm{if}\, l_1 = l-1\\ \frac{\partial f}{\partial r} - \frac{l}{r}f \quad \mathrm{if}\, l_1 = l+1 \end{cases}\]

Equation (25) in Ivers and Phillips (2008).

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