Background magnetic field

With magnetic = 1, Kore solves for a small magnetic field perturbation \(\mathbf{b}\) on top of a steady, imposed background field \(\mathbf{B}_0\). The background field is axisymmetric and poloidal:

\[ \mathbf{B}_0 = c_{norm}\, \nabla\times\nabla\times\left( h(r)\, Y_l^0(\theta)\, \mathbf{r} \right), \]

where \(\mathbf{r}\) is the position vector, \(Y_l^0\) is a Schmidt semi-normalised spherical harmonic (\(Y_1^0 = \cos\theta\)), and \(c_{norm}\) is set by cnorm. The radial component is \(B_{0r} = c_{norm}\, l(l+1)\, h\, Y_l^0 / r\). The field enters the Lorentz force and the induction term of the linearised equations, the magnetic torques on the boundaries, and spin_doctor's diagnostics.

Field shapes

B0 selects the radial function \(h(r)\):

B0	\(l\)	\(h(r)\)	Notes
'axial'	1	\(r/2\)	Uniform field along the rotation axis, \(\mathbf{B}_0 = c_{norm}\,\hat{\mathbf{z}}\). Imposed from outside the core. It has no cylindrical-radial component, so it supports no torsional waves. The default.
'dipole'	1	\(1/(2r^2)\)	Point dipole at the centre, current-free in the fluid. Needs an inner core (ricb > 0).
'G21 dipole'	1	\(r/6 - r^3/10\)	Dipole made by currents in the fluid (Gerick et al. 2021, GJI).
'Luo_S1'	1	\((5 - 3r^2)\, r\)	Luo & Jackson (2022, Proc. R. Soc. A) field S1. Same shape as 'G21 dipole' (30 times larger).
'Luo_S2'	2	\((157 - 296 r^2 + 143 r^4)\, r^2\)	Luo & Jackson (2022) quadrupolar field S2.
'FDM'	B0_l	spherical Bessel functions, see below	Poloidal free magnetic decay mode (Zhang & Fearn 1994, 1995; Schmitt 2012).

The free decay mode is \(h = j_l(\beta r)\) in a full sphere (Gubbins & Roberts 1987), and \(h = j_l(\beta r)\, y_{l-1}(\beta) - j_{l-1}(\beta)\, y_l(\beta r)\) in a shell (Zhang & Fearn 1995), with \(j_l\) and \(y_l\) the spherical Bessel functions. \(\beta\) is the decay-mode wavenumber, found by a root search started from beta (default 3); the value found is written to params.dat as beta_actual. B0_l (default 1) is the degree of the mode.

The fields with odd \(l\) are equatorially antisymmetric and 'Luo_S2' is symmetric. The symmetry of \(\mathbf{b}\) is that of the flow (symm) times that of \(\mathbf{B}_0\).

Normalisation

cnorm sets the amplitude \(c_{norm}\), and with it what "a field of unit strength" means in the Lehnert and Elsasser numbers:

cnorm	Meaning
a number	\(c_{norm}\) itself (default 1).
'rms_cmb'	The rms of \(B_{0r}\) over the CMB is 1.
'mag_energy'	\(\int B_0^2\, dV = 1\) over the fluid shell, as in Luo & Jackson (2022).
'Schmitt2012'	\(\int B_0^2\, dV = 4\) over the fluid shell, as in Schmitt (2012).

The comments in params_default.py list numeric values that reproduce published normalisations:

Value	Field	Equivalent to
3.863752890	FDM, \(l = 1\), ricb = 0.35 (Schmitt 2012, G101)	'Schmitt2012'
4.067144	FDM, \(l = 1\), ricb = 0.35 (Zhang & Fearn 1994)	not checked
\(15\sqrt{21/(46\pi)}\)	'G21 dipole', full sphere	'Schmitt2012'
1.094357234	FDM, \(l = 1\), full sphere	'Schmitt2012'
3.438024656	FDM, \(l = 1\), ricb = 0.001	'Schmitt2012'
0.09530063707257978	'Luo_S1', full sphere	'mag_energy'
0.6972166887783963	'Luo_S1', full sphere	volume rms of \(B_{0s}\) = 1
0.005061567359972097	'Luo_S2', full sphere	'mag_energy'
0.0158567582314039	'Luo_S2', full sphere	volume rms of \(B_{0s}\) = 1

With an inner core, 'mag_energy' normalises over the shell, so the field comes out stronger than in a full sphere. To keep exactly the currents of the full-sphere field in the fluid, use its numeric full-sphere value instead.

The field inside the inner core: B0_ic

'G21 dipole', 'Luo_S1' and 'Luo_S2' are defined for a full sphere. With an inner core (ricb > 0) they do not match a potential field at the ICB, so part of the electric current that makes them would have to flow inside the inner core, or in a current sheet on the ICB. Nothing keeps such a current steady in a solid inner core of finite conductivity: there is no flow there, hence no electromotive force, and an axisymmetric azimuthal current decays on the inner core's magnetic diffusion time.

B0_ic	Effect
'potential' (default)	Removes that part of the current. \(h\) becomes \(h + \beta_{ic}\, r^{-(l+1)}\), with \(\beta_{ic} = r_i^{l+1}\,(r_i h' - l h)/(2l+1)\) evaluated at the ICB. The added term carries no current and keeps the CMB matching, so the currents in the fluid are unchanged, and \(\mathbf{B}_0\) continues into the inner core as the potential field \(h(r_i)\,(r/r_i)^l\). This is the self-consistent steady field for an insulating inner core and for any finite inner-core conductivity.
'full sphere'	\(h\) unchanged; its currents extend into the inner core. Self-consistent only for a perfectly conducting inner core, or as a snapshot of currents frozen in the inner core.

B0_ic has no effect for a full sphere, nor for 'axial' and 'FDM' (they already match a potential field at the ICB) or 'dipole' (a point source at the centre). For 'Luo_S2' with ricb = 0.35 the change is 10% of the field at the ICB and 1.7% at the CMB, and it lowers the frequency of the S2 torsional mode by about 1% (insulating inner core).

A thin conducting wall at the CMB: B0_cmb

With a thin conducting layer at the base of the mantle (mantle = 'TWA', see Inner core and mantle), the background field must satisfy the wall's matching condition too. A steady \(\mathbf{B}_0\) drives no current in the wall, so only the wall's permeability term c_cmb acts on it, not its conductance c1_cmb. The condition at the CMB is

\[ \left(\frac{1}{\mu} + l\,c_{cmb}\right) (r h)' + l\,h = 0 . \]

B0_cmb	Effect
'wall' (default)	Adds \(\alpha\, r^l\) to \(h\), with \(\alpha\) chosen to satisfy the condition. The added term carries no current and keeps the ICB matching of B0_ic, so the currents in the fluid are unchanged. For \(l = 1\) it is a uniform axial field.
'plain'	\(h\) unchanged. The field then meets the wall condition only for c_cmb = 0 and mu = 1.

B0_cmb has no effect with an insulating mantle, with c_cmb = 0 and mu = 1, or for 'axial'. For the S2 torsional mode with c1_cmb = 0.015 it raises the frequency by 0.27% at c_cmb = 0.01 and by 5.8% at c_cmb = 0.3.