chaosmagpy.model_utils.power_spectrum

chaosmagpy.model_utils.power_spectrum(coeffs, radius=None, *, nmax=None, source=None, axis=None)

Compute the spatial power spectrum.

Parameters:

coeffsndarray, shape (…, N*(N+2))

Spherical harmonic coefficients for degree N.

radiusfloat, optional

Radius in kilometers (defaults to mean Earth’s surface radius). It has no effect for source='toroidal'.

nmaxint, optional

Maximum spherical degree (defaults to N).

source{‘internal’, ‘external’, ‘toroidal’}

Source of the field model (defaults to internal).

axisint, optional

Axis of coeffs along which to compute the spatial power spectrum (defaults to -1, last dimension of coeffs).

Returns:

W_nndarray, shape (…, nmax)

Power spectrum for degrees up to degree nmax

Notes

The spatial power spectrum for a potential field is defined as

\[\begin{split}W_n(r) &= \langle|\mathbf{B}|^2\rangle = \frac{1}{4\pi}\iint_\Omega |\mathbf{B}|^2 \mathrm{d} \Omega \\ &= W_n^\mathrm{i}(r) + W_n^\mathrm{e}(r) + W_n^\mathrm{T}\end{split}\]

where the internal \(W_n^\mathrm{i}\), external \(W_n^\mathrm{e}\) and the non-potential toroidal \(W_n^\mathrm{T}\) spatial power spectra are

\[\begin{split}W_n^\mathrm{i}(r) &= (n+1)\left(\frac{a}{r}\right)^{2n+4} \sum_{m=0}^n [(g_n^m)^2 + (h_n^m)^2] \\ W_n^\mathrm{e}(r) &= n\left(\frac{r}{a}\right)^{2n-2}\sum_{m=0}^n [(q_n^m)^2 + (s_n^m)^2] \\ W_n^\mathrm{T} &= \frac{n(n+1)}{2n+1}\sum_{m=0}^n [(T_n^{m,c})^2 + (T_n^{m,s})^2]\end{split}\]

References

Sabaka, T. J.; Hulot, G. & Olsen, N., “Mathematical properties relevant to geomagnetic field modeling”, Handbook of geomathematics, Springer, 2010, 503-538