2.1. Governing equations

We consider a fluid-filled spherical shell of inner and outer radii ${a}_s$ and ${a}_m$ , bounded by inner and outer solid domains (figure 2). The forced problem is formulated in the non-inertial frame attached to the outer solid of radius $a_e$ , with origin $O$ at the outer-sphere centre and axes rotating at ${\boldsymbol{\varOmega }}_o=\varOmega _o\hat {\boldsymbol{z}}$ , so that the outer solid has no translational motion in this frame. The incompressible Newtonian fluid has uniform electrical conductivity $\sigma _{\!f}$ , density $\rho _{\!f}$ , kinematic viscosity $\nu$ and magnetic permeability $\mu _{\!f}$ . The spherical interfaces separate the fluid from inner and outer solid domains with uniform electrical conductivities $\sigma _s$ and $\sigma _m$ , and magnetic permeabilities $\mu _s$ and $\mu _m$ , respectively. The fluid velocity $\boldsymbol{V}$ is governed by

(2.1a,b) \begin{equation} \partial _t {\boldsymbol{V}} + ({\boldsymbol{V}} \boldsymbol{\cdot } \boldsymbol{\nabla }) \, {\boldsymbol{V}} +2 { \boldsymbol{\varOmega }}_o \times {\boldsymbol{V}} = -\boldsymbol{\nabla }{\varPi } + \nu \, \boldsymbol{\nabla }^2 {\boldsymbol{V}} +(\rho _{\!f} \mu _{\!f})^{-1} ({\boldsymbol{B}} \boldsymbol{\cdot }\boldsymbol{\nabla }) {\boldsymbol{B}} , \quad \boldsymbol{\nabla } \boldsymbol{\cdot } {\boldsymbol{V}} = 0, \end{equation}

in the fluid region $\varOmega _{\!f}$ of volume $(4 \pi /3) ({a}_m^3-{a}_s^3)$ , and the magnetic field $\boldsymbol{B}$ is governed by

(2.2a,b) \begin{equation} \partial _t {\boldsymbol{B}} = \boldsymbol{\nabla }\times ({\boldsymbol{V}} \times {\boldsymbol{B}}) + \eta \boldsymbol{\nabla} ^2 {\boldsymbol{B}} , \quad \boldsymbol{\nabla } \boldsymbol{\cdot } {\boldsymbol{B}} = 0 , \end{equation}

where the magnetic diffusivity $\eta$ is $\eta _{\!f}=(\mu _{\!f}\sigma _{\!f})^{-1}$ , $\eta _s=(\mu _s\sigma _s)^{-1}$ and $\eta _m=(\mu _m\sigma _m)^{-1}$ in, respectively, the fluid, the solid inner and outer domains. The reduced pressure is defined as $\varPi =\varPi _h+B^2/(2\rho _{\!f}\mu _{\!f})$ , where $\varPi _h$ includes the centrifugal contribution, such that $\varPi$ contains both hydrodynamic and magnetic pressures.

For a free planetary mode, our frame of reference would additionally translate relative to an inertial centre-of-mass frame, as required by the global linear-momentum balance (Grinfeld & Wisdom Reference Grinfeld and Wisdom2005; Greff-Lefftz & Legros Reference Greff-Lefftz and Legros2007). The resulting spatially uniform inertial acceleration is a gradient and can therefore be absorbed into the reduced pressure; the local equations for prescribed relative motion consequently retain the same form, while the physical pressure force and planetary eigenfrequency must be reconstructed together with the global translational dynamics. Let $\boldsymbol{\xi }_s(t)$ and $\boldsymbol{\xi }_m(t)$ denote the positions of the centres of the inner and outer solid domains, respectively, in an inertial frame. In our working frame of reference, the prescribed mechanical forcing is their relative displacement

(2.3) \begin{equation} {\boldsymbol{\delta }}(t) ={\boldsymbol{\xi }}_s(t)-{\boldsymbol{\xi }}_m(t) =\textit{Re} \big({\boldsymbol{\epsilon }}_s \mathrm{e}^{\textrm{i} {\omega _{\textit{s}}}t}\big ), \end{equation}

with relative velocity ${\boldsymbol{V}}_{\!s}=\mathrm{d}_t{\boldsymbol{\delta }}$ , with the oscillation frequency $\omega _{\textit{s}}$ and the time-independent displacement vector $\boldsymbol{\epsilon }_s$ . We use the Cartesian basis $(\hat {\boldsymbol{x}},\hat {\boldsymbol{y}},\hat {\boldsymbol{z}})$ and spherical-coordinate basis $(\hat {\boldsymbol{r}},\hat {\boldsymbol{\theta }},\hat {\boldsymbol{\phi }})$ , centred on $O$ . The oscillation can be along the rotation axis, with

(2.4) \begin{eqnarray} {\boldsymbol{\delta }}=\textit{Re} \big [ {\epsilon _s} \hat {\boldsymbol{z}} \mathrm{e}^{\textrm{i} {\omega _{\textit{s}}} t} \big ] = \textit{Re} \big [ {\epsilon _s} (\cos \theta \hat {\boldsymbol{r}}-\sin \theta \hat {\boldsymbol{\theta }}) \mathrm{e}^{\textrm{i} {\omega _{\textit{s}}} t} \big ], \end{eqnarray}

which is the so-called polar mode, or along a circular trajectory, with

(2.5) \begin{eqnarray} {\boldsymbol{\delta }}=\textit{Re} \big [{\epsilon _s} (\hat {\boldsymbol{x}}\pm \mathrm{i}\hat {\boldsymbol{y}}) \mathrm{e}^{\textrm{i} {\omega _{\textit{s}}} t} \big ] = \textit{Re} \big [ {\epsilon _s} (\sin \theta \hat {\boldsymbol{r}}+\cos \theta \hat {\boldsymbol{\theta }}\pm \mathrm{i} \hat {\boldsymbol{\phi }}) \mathrm{e}^{\textrm{i} ( {\omega _{\textit{s}}} t \pm \phi )} \big ], \end{eqnarray}

which are the prograde and retrograde equatorial modes (equations (4) and (5) of BG95). The three modes have distinct frequencies $\omega _{\textit{s}}$ and amplitudes $\epsilon _s$ in planetary cores and are analysed independently below (using the generic notation $\omega _{\textit{s}}$ and $\epsilon _s$ ). The amplitude $\epsilon _s$ is assumed to be the smallest length scale, so the inner-boundary position is fixed at leading order. From (2.4) and (2.5), the polar mode corresponds to a rectilinear harmonic translation, whose velocity magnitude oscillates in time. By contrast, the equatorial modes correspond to a uniform circular translation: the Cartesian velocity components oscillate harmonically with a phase shift of $\pi /2$ , but the velocity magnitude remains constant. Although the maximum kinetic energy is identical for all three modes, the time-averaged kinetic energy per unit mass is $\langle (\mathrm{d}_t {\boldsymbol{\delta }})^2/2 \rangle =\mathcal{E}(\epsilon _s\omega _s)^2/2$ , where $\mathcal{E}$ is the kinetic energy factor: $\mathcal{E}=1/2$ for the polar mode and $\mathcal{E}=1$ for equatorial modes (twice as much).

To close the problem, boundary conditions are needed. With outward normal $\hat {\boldsymbol{n}}$ , the velocity of inviscid fluids verifies the non-penetration condition $\hat {\boldsymbol{n}} \boldsymbol{\cdot }{\boldsymbol{V}}=\hat {\boldsymbol{n}} \boldsymbol{\cdot }\mathrm{d}_t{\boldsymbol{\delta }}$ and $\hat {\boldsymbol{n}} \boldsymbol{\cdot }{\boldsymbol{V}}=0$ at the inner and outer boundaries, respectively. For viscous fluids, conditions on tangential velocities are also required: no-slip boundaries impose ${\boldsymbol{V}}=\mathrm{d}_t{\boldsymbol{\delta }}$ and ${\boldsymbol{V}}=\boldsymbol{0}$ at the inner and outer boundaries, respectively, whereas stress-free boundaries impose

(2.6a,b) \begin{equation} ( {\boldsymbol{\tau }}_{\!v} \boldsymbol{\cdot }\hat {\boldsymbol{n}}) \times \hat {\boldsymbol{n}} =\boldsymbol{0} , \quad {\boldsymbol{\tau }}_{\!v}=\rho _{\!f} \nu (\boldsymbol{\nabla }{\boldsymbol{V}}+(\boldsymbol{\nabla }{\boldsymbol{V}})^\top ), \end{equation}

where ${\boldsymbol{\tau }}_{\!v}$ is the viscous-stress tensor. For the magnetic field, (2.2) is solved in the solid domains, using ${\boldsymbol{V}}=\mathrm{d}_t {\boldsymbol{\delta }}$ and ${\boldsymbol{V}}= \boldsymbol{0}$ in the inner and outer domains, respectively. The outer solid domain extends with uniform properties up to external radius ${a}_e$ .

2.2. Forces from pressure, viscous and magnetic stresses

The total fluid force $\boldsymbol{F}$ on the sphere results from the combined action of tangential ( ${\boldsymbol{F}}_{\!v}$ ) and normal ( ${\boldsymbol{F}}_{\!vn}$ ) viscous stresses, pressure forces ${\boldsymbol{F}}_{\!p}$ and Lorentz force $\boldsymbol{F}_{\!\eta} ={\boldsymbol{F}}_{\!mt}+{\boldsymbol{F}}_{\!mp}$ that splits into magnetic-tension ${\boldsymbol{F}}_{\!mt}$ and magnetic-pressure ${\boldsymbol{F}}_{\!mp}$ forces. This gives

(2.7) \begin{eqnarray} {\boldsymbol{F}}={\boldsymbol{F}}_{\!v}+{\boldsymbol{F}}_{\!vn}+{\boldsymbol{F}}_{\!p}+{\boldsymbol{F}}_{\!mt}+{\boldsymbol{F}}_{\!mp} , \end{eqnarray}

with ${\boldsymbol{F}}_{\!v}+{\boldsymbol{F}}_{\!vn}=\int _S {\boldsymbol{\tau }}_{\!v}\boldsymbol{\cdot }\mathrm{d}\boldsymbol{S}$ and ${\boldsymbol{F}}_{\!p}=-\int _S \rho _{\!f}{\varPi }_h\,\mathrm{d}\boldsymbol{S}$ , where $\mathrm{d}\boldsymbol{S}=\hat {\boldsymbol{n}}\mathrm{d}S$ is related to the sphere outward unit normal $\hat {\boldsymbol{n}}=\hat {\boldsymbol{r}}$ . Writing the Lorentz stress as

(2.8) \begin{eqnarray} \mu _{\!f}^{-1}(\boldsymbol{\nabla }\times \boldsymbol{B})\times \boldsymbol{B}=-\mu _{\!f}^{-1}\boldsymbol{\nabla }(\boldsymbol{B}^2/2)+\mu _{\!f}^{-1}(\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{B} \end{eqnarray}

gives $\boldsymbol{F}_{\!\eta} =\mu _{\!f}^{-1}\int _V(\boldsymbol{\nabla }\times \boldsymbol{B})\times \boldsymbol{B}\,\mathrm{d}V$ as the sum of ${\boldsymbol{F}}_{\!mt}=\mu _{\!f}^{-1}\int _S\boldsymbol{B}(\boldsymbol{B}\boldsymbol{\cdot }\mathrm{d}\boldsymbol{S})$ and ${\boldsymbol{F}}_{\!mp}=-(2\mu _{\!f})^{-1}\int _S\boldsymbol{B}^2\,\mathrm{d}\boldsymbol{S}$ . As viscous effects modify the pressure, and thus ${\boldsymbol{F}}_{\!p}$ , we calculate the viscous contribution $\boldsymbol{F}_\nu$ as $\boldsymbol{F}_\nu =\boldsymbol{F}_{\!v}+{\boldsymbol{F}}_{\!n}-\boldsymbol{F}_{\!n}(\nu =0)$ , where ${\boldsymbol{F}}_{\!n}={\boldsymbol{F}}_{\!p}+{\boldsymbol{F}}_{\!vn}$ is the total normal force and $\boldsymbol{F}_{\!n}(\nu =0)=\boldsymbol{F}_{\!p}(\nu =0)$ is the inviscid-fluid force. Hence, $\boldsymbol{F}_\nu$ , unlike $\boldsymbol{F}_{\!v}$ , can remain non-zero for stress-free viscous boundaries (and both vanish for inviscid flows).

Neglecting magnetic and rotational effects, the oscillating no-slip sphere was analysed by Stokes (Reference Stokes1851), who showed that ${\boldsymbol{F}}_{\!n}+{\boldsymbol{F}}_{\!v}$ is the sum of a viscous drag proportional to ${\boldsymbol{V}}_{\!s}$ (vanishing as $\nu \to 0$ ) and an inertial contribution. Noting that the fluid mass $M_{\!fs}=4\pi a_s^3 \rho _{\!f}/3$ is displaced by the inner sphere, this latter inertial contribution is proportional to $\mathrm{d}_t{\boldsymbol{V}}_{\!s}$ , written $-C_a M_{\!fs} \mathrm{d}_t{\boldsymbol{V}}_{\!s}$ , where $C_a\to C_a^i\neq 0$ in the inviscid limit owing to ${\boldsymbol{F}}_{\!p}$ . Although the steady result ${\boldsymbol{F}}_{\!n}+{\boldsymbol{F}}_{\!v}=-6\pi \rho _{\!f}\nu a_s{\boldsymbol{V}}_{\!s}$ for $\omega _{\textit{s}}\to 0$ is well known, the relative roles of pressure and viscous stresses, and the rapidly oscillating regime, have received much less attention. Moreover, Stokes (Reference Stokes1851) also considered an outer spherical boundary, deriving implicit bounded-flow solutions and an explicit force on the sphere. Revisiting these results, we derive an explicit velocity field (Appendix A.1) and asymptotic expansions of the force for slow and rapid oscillations, quantifying confinement effects (Appendix A.2).

For stress-free boundary conditions, the force differs but remains relevant to the no-slip configuration, as it directly reflects viscous dissipation in the fluid bulk. This contribution becomes significant when viscosity is not negligible; we therefore recall the corresponding stress-free bulk-force expressions (Appendix A.3).

In the presence of rotation or magnetic effects, the inviscid-fluid force does not necessarily retain the form ${\boldsymbol{F}}_{\!p}=-C_a^i M_{\!fs} \mathrm{d}_t {\boldsymbol{V}}_{\!s}$ ; in general, ${\boldsymbol{F}}_{\!p}$ is not even opposite to the acceleration (B74). However, for the three Slichter modes, B74 showed that this property persists when rotation is included (without magnetic or viscous effects). In this inviscid rotating limit, the pressure force is parallel to $\hat {\boldsymbol{z}}$ for the polar mode and antiparallel to the cylindrical-radius unit vector $\hat {\boldsymbol{s}}$ for equatorial modes.

2.3. Ohmic and viscous dissipation

In the fluid, the Ohmic and viscous dissipations are

(2.9) \begin{eqnarray} {\mathcal{D}}_\eta = \eta _{\!f} \mu _{\!f}^{-1} \int _{\varOmega _{\!f}} (\boldsymbol{\nabla }\times {\boldsymbol{B}})^2\mathrm{d}V,\quad \quad {\mathcal{D}}_\nu =\int _{\varOmega _{\!f}} {\boldsymbol{\tau }}_{\!v} \boldsymbol{:} \boldsymbol{\nabla }{\boldsymbol{V}} \mathrm{d}V , \end{eqnarray}

with ${\boldsymbol{\tau }}_{\!v}=\rho _{\!f} \nu [\boldsymbol{\nabla }{\boldsymbol{V}}+(\boldsymbol{\nabla }{\boldsymbol{V}})^\top ]$ . Ohmic dissipation in solids follows from analogous expressions with the corresponding volumes and material properties. Although viscous dissipation is often expressed using the vorticity $\boldsymbol{\mathcal{W}}=\boldsymbol{\nabla }\times \boldsymbol{V}$ alone, the oscillating inner boundary prevents this simplification. Using the Bobyleff–Forsyth identity (Buresti Reference Buresti2009),

(2.10) \begin{eqnarray} {\mathcal{D}}_\nu =\int _{\varOmega _{\!f}} {\boldsymbol{\tau }}_{\!v} \boldsymbol{:} \boldsymbol{\nabla }{\boldsymbol{V}} \mathrm{d}V = \rho _{\!f} \nu \int _{\varOmega _{\!f}} \boldsymbol{\mathcal{W}}^2\mathrm{d}V + 2 \rho _{\!f} \nu \int _{ {\varOmega _{\!f}}} \boldsymbol{\nabla }\boldsymbol{\cdot }(\mathrm{D}_t \boldsymbol{V} ) \mathrm{d}V, \end{eqnarray}

with $\mathrm{D}_t \boldsymbol{V}=\partial _t \boldsymbol{V}+(\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{V}$ , an additional acceleration term must be retained. Since $\boldsymbol{\nabla }\boldsymbol{\cdot }\partial _t\boldsymbol{V}=0$ ,

(2.11) \begin{equation} \int _{ {\varOmega _{\!f}}} \boldsymbol{\nabla }\boldsymbol{\cdot }(\mathrm{D}_t \boldsymbol{V} ) \mathrm{d}V = \int _{\partial {\varOmega _{\!f}}} \hat {\boldsymbol{n}} \boldsymbol{\cdot }(\mathrm{D}_t \boldsymbol{V}) \mathrm{d}S = \int _{\partial {\varOmega _{\!f}}} \hat {\boldsymbol{n}} \boldsymbol{\cdot }\big( \boldsymbol{\mathcal{W}} \times \boldsymbol{V} + \boldsymbol{\nabla }( \boldsymbol{V}^2/2) \big ) \mathrm{d}S , \end{equation}

where $\boldsymbol{\mathcal{W}}\times \boldsymbol{V}$ is the Lamb vector and

(2.12) \begin{eqnarray} \int _{\partial {\varOmega _{\!f}}} \hat {\boldsymbol{n}}\boldsymbol{\cdot }(\mathrm{D}_t \boldsymbol{V})\mathrm{d}S=\int _{r=a_m} (\mathrm{D}_t {V}_r) \mathrm{d}S-\int _{r=a_s} (\mathrm{D}_t {V}_r) \mathrm{d}S . \end{eqnarray}

The surface stress work rate (power input from the boundary) is then (Wu et al. Reference Wu, Zhou and Fan1999)

(2.13) \begin{eqnarray} \rho _{\!f} \nu \int _{\varOmega _{\!f}} \boldsymbol{V} \boldsymbol{\cdot }(\boldsymbol{\nabla} ^2 \boldsymbol{V}) \mathrm{d}V+ {\mathcal{D}}_\nu =\rho _{\!f} \nu \int _{\partial \varOmega _{\!f}} \hat {\boldsymbol{n}} \boldsymbol{\cdot }\big(\boldsymbol{\mathcal{W}} \times \boldsymbol{V} + \boldsymbol{\nabla }\boldsymbol{V}^2 \big ) \mathrm{d}S . \end{eqnarray}

3.1. Numerical modelling

Equations (3.1)−(3.2) are solved using the finite-difference/pseudo-spectral code XSHELLS (Schaeffer et al. Reference Schaeffer, Jault, Nataf and Fournier2017) or the finite-element code COMSOL $\textrm {Multiphysics}^{\circledR}$ . The XSHELLS code employs a poloidal–toroidal decomposition of the velocity field in spherical harmonics with degree $l\le l_{\textit{max}}$ and azimuthal wavenumber $m\le m_{\textit{max}}$ via the shtns library (Schaeffer Reference Schaeffer2013), combined with second-order finite differences on $N_r$ radial points. Its spectral formulation enables access to extreme viscosity and magnetic-field regimes and has been extensively validated (e.g. Marti et al. Reference Marti2014). In the weak-forcing limit $\epsilon \ll 1$ , the nonlinear equations are integrated in a fixed geometry, with translational forcing imposed through the boundary velocity $\mathrm{d}_t\boldsymbol{\delta }$ at the inner boundary (as in Rieutord Reference Rieutord2002). Only diffusion, discarding any motion, is solved in the solid regions. The region surrounding the outer solid shell is treated as a perfect electrical insulator, with the magnetic boundary conditions described in Christensen & Wicht (Reference Christensen and Wicht2015). Implementation details are given in P26. Simulations use a semi-implicit third-order backward-difference scheme (SBDF3, Ascher et al. Reference Ascher, Ruuth and Wetton1995). A typical resolution at $L_\nu =10^{-3}$ , $\varLambda =0.5$ and $P_m=10^{-2}$ is $N_r=511$ , $l_{\textit{max}}=127$ and $m_{\textit{max}}=8$ , with sufficient refinement in the nested boundary layers and, when Alfvén waves are present, in the interior. For laminar flows with simple forcing (e.g. axial magnetic field), only a few azimuthal modes are required: the polar mode is axisymmetric ( $m=0$ ), whereas equatorial modes require the single non-zero azimuthal order ( $m=\pm 1$ ). Simulations with XSHELLS are restricted here to no-slip boundaries. The system starts from rest and is allowed to evolve until it reaches a converged regime over a few diffusive time scales. The initial transient is discarded. Dissipation is computed from volume integrals (§ 2.3) and time-averaged in the converged regime. The vorticity formulation avoids solving for the pressure, reducing computational cost but precluding direct evaluation of pressure forces and, therefore, of the added-mass coefficient. For these quantities, COMSOL is used instead (§ S1).

3.2. Perturbation approach and BLT

We restrict attention to the weak-forcing regime, $\epsilon =\epsilon _s/D\ll 1$ . Analytical solutions are derived using BLT, for which the boundary-layer components must decay over small distances compared with the local radii of curvature and the separation between neighbouring boundaries. In the fluid, the asymptotic expansion is therefore organised by $L_\nu /\ell$ and $L_\eta /\ell$ , with $\ell =\min (a_s,D)$ ; as shown below by deriving their thicknesses, the magnetic layers in the inner and outer solids additionally require $L_\eta \ll a_s \breve {\eta }_s^{-1/2}$ and $L_\eta \ll \breve {\eta }_m^{-1/2} \min (a_m,a_e-a_m)$ , respectively. Except for this last condition, the boundary-layer problem is fully independent with respect to $a_e$ . These restrictions do not apply to the exact bounded Stokes hydrodynamic solution of Stokes (Reference Stokes1851) in Appendix A.1, which remains valid for arbitrary $L_\nu /\ell$ within the linear problem.

With the dependence $\mathrm{e}^{\textrm{i}(\omega t+m\phi )}$ , the polar and equatorial modes correspond to $m=0$ and $m=\pm 1$ , respectively. The analysis starts from the imposed magnetic field $\boldsymbol{B}_0$ and the inviscid non-magnetic solution $(\boldsymbol{U}, P)$ . Boundary layers generated at the inner and outer spheres then induce successive magnetic, velocity and pressure corrections, yielding the hierarchy

(3.3) \begin{align} \boldsymbol{B} &= \boldsymbol{B}_0+\epsilon \textrm{Re} \big [(\boldsymbol{b}+\boldsymbol{\mathcal{B}}+\boldsymbol{\mathfrak{b}}+\boldsymbol{b}^{(o)}+\boldsymbol{\mathcal{B}}^{(o)}+\boldsymbol{\mathfrak{b}}^{(o)}) \mathrm{e}^{\textrm{i} ( \omega t + m \phi )} \big ], \\[-12pt]\nonumber \end{align}

(3.4) \begin{align} \boldsymbol{V} &= \epsilon \textrm{Re} \big [(\boldsymbol{U}+ \boldsymbol{u}+\boldsymbol{\mathcal{U}}+\boldsymbol{\mathfrak{u}}+ \boldsymbol{u}^{(o)}+\boldsymbol{\mathcal{U}}^{(o)}+\boldsymbol{\mathfrak{u}}^{(o)}) \mathrm{e}^{\textrm{i} ( \omega t + m \phi )} \big ] \end{align}

and

(3.5) \begin{eqnarray} \varPi &=&\epsilon \textrm{Re} \left [(P+p+\mathcal{P}+\mathfrak{p}+p^{(o)}+\mathcal{P}^{(o)}+\mathfrak{p}^{(o)}) \mathrm{e}^{\textrm{i} ( \omega t + m \phi )} \right ] \!, \end{eqnarray}

where $\boldsymbol{B}_0(r,\theta )$ is an imposed steady magnetic field, whose normal component $B_{0r}$ at the solid boundary primarily controls boundary-layer magnetic effects (e.g. BG95). Figure 3 provides the roadmap of the analytical construction adopted throughout the paper, indicating the successive bulk and boundary-layer corrections and their coupling.

The boundary-layer analysis holds for an arbitrary $\phi$ -independent $\boldsymbol{B}_0(r,\theta )$ , while a uniform axial field is used in DNS. For $r\le a_s$ , the field remains uniform when a uniform external field $\boldsymbol{B}_e$ is applied, and can reach up to three times $\boldsymbol{B}_e$ for large $\mu _s \gg \mu _0$ , or can be strongly reduced for large $\mu _{\!f}$ due to magnetic shielding. The analytical derivation of $\boldsymbol{B}_0$ , including the effective permeability formulation and shielding limits, is given in § S2 of the supplementary material.

Considering the hydrodynamic inviscid solution $(\boldsymbol{U},P)$ for $\boldsymbol{B}_0=\boldsymbol{0}$ , the non-zero $\boldsymbol{B}_0$ and viscous no-slip conditions generate a hierarchy of BLT corrections from inner and outer boundaries, the latter denoted by ${}^{(o)}$ . The perturbation hierarchy relies on the thin boundary-layer assumption, introducing a second small parameter in addition to $\epsilon$ . Starting from the irrotational solution $(\boldsymbol{U}, P)$ , successive boundary-layer and bulk corrections are determined order by order. We first consider the leading-order corrections $(\boldsymbol{b},\boldsymbol{u},p)$ and $(\boldsymbol{b}^{(o)},\boldsymbol{u}^{(o)},p^{(o)})$ to the basic state $(\boldsymbol{B}_0,\boldsymbol{U},P)$ , arising from the inner- and outer-boundary layers, respectively (figure 3). The fields $(\boldsymbol{b},\boldsymbol{u})=(\boldsymbol{b}^{(p)}+\boldsymbol{b}^{(h)},\,\boldsymbol{u}^{(p)}+\boldsymbol{u}^{(h)})$ decompose into a bulk contribution $(\boldsymbol{b}^{(p)},\boldsymbol{u}^{(p)})$ , driven by induction and vanishing in the non-magnetic limit where $^{(p)}$ stands for particular, and a boundary-layer correction $(\boldsymbol{b}^{(h)},\boldsymbol{u}^{(h)})$ enforcing the boundary conditions at $r=a_s$ , where $^{(h)}$ stands for homogeneous. Through higher-order boundary-layer radial flow $\mathfrak{u}_r$ , required to ensure mass conservation, these corrections, in turn, force higher-order bulk fields $(\boldsymbol{\mathcal{B}},\boldsymbol{\mathcal{U}},\mathcal{P})$ and $(\boldsymbol{\mathcal{B}}^{(o)},\boldsymbol{\mathcal{U}}^{(o)},\mathcal{P}^{(o)})$ , smaller by a factor of the order of the thickness of the boundary layer. The resulting secondary bulk fields generally violate the tangential boundary conditions and, therefore, generate a final boundary-layer readjustment $(\boldsymbol{\mathfrak{b}}_\parallel ,\boldsymbol{\mathfrak{u}}_\parallel )$ , and similarly at the outer boundary (see §§ 3.5 and 5.3).

Figure 3.

Hierarchy and spatial distribution of velocity and magnetic base fields and perturbations. All fields are of order $\epsilon$ in this work, except the zeroth-order field $\boldsymbol{B}_0$ , which interacts with the hydrodynamic solution $(\boldsymbol{U},P)$ forced by the inner-sphere velocity $\mathrm{d}_t \boldsymbol{\delta }$ , to generate corrections in the whole space (e.g. $\boldsymbol{u}$ ). At next order in the boundary-layer thickness, ( $\boldsymbol{u}_\parallel ,\boldsymbol{u}^{(o)}_\parallel$ ) drive boundary-layer radial flows $(\mathfrak{u}_r,\mathfrak{u}_r^{(o)})$ , which force secondary bulk fields (e.g. $\boldsymbol{\mathcal{B}}$ ) and their associated boundary-layer corrections (e.g. $\boldsymbol{\mathfrak{b}}_\parallel$ ).

3.3. Calculation of the boundary-layer solution

Boundary-layer homogeneous solutions decay exponentially into the bulk, where diffusion is negligible, so that the magnetic particular solution $\boldsymbol{b}^{(p)}$ approaches the diffusionless limit $\boldsymbol{b}^{(p)}_0$ induced by $\boldsymbol{U}$ . In the weak-Lorentz-force regime, $(\boldsymbol{b}^{(p)},\boldsymbol{u}^{(p)}) \approx (\boldsymbol{b}^{(p)}_0,\boldsymbol{u}^{(p)}_0)$ can be obtained analytically (Appendix B). Outer-boundary corrections (superscript ${}^{(o)}$ ) satisfy the same equations as those at $r=a_s$ and vanish in the unbounded limit $a_m \gg a_s$ ; they are thus deduced from the inner solutions, and only $(\boldsymbol{b},\boldsymbol{u},p)$ is detailed below.

With $\boldsymbol{U}$ satisfying non-penetration, the leading-order boundary-layer perturbation verifies $u_r=0$ . Within BLT, radial derivatives $\partial _r \boldsymbol{u}$ and $\partial _r \boldsymbol{b}$ dominate over the fields and over derivatives in other directions. Radial variations of $\boldsymbol{B}_0$ are neglected relative to those of $\boldsymbol{b}$ , so $\partial _r \boldsymbol{B}_0$ is omitted when $\partial _r \boldsymbol{b}$ is present; this is exact for the uniform axial fields considered below. For $\boldsymbol{B}_0=[B_{0r},B_{0\theta },B_{0\phi }]$ depending only on $\theta$ , the tangential components of $\boldsymbol{u}=(0,\boldsymbol{u}_\parallel )$ and $\boldsymbol{b}=(b_r,\boldsymbol{b}_\parallel )$ satisfy the BLT form of (3.1) and (3.2),

(3.6) \begin{eqnarray} \mathrm{i} \omega \boldsymbol{u}_\parallel +\gamma \omega \hat {\boldsymbol{z}} \times \boldsymbol{u}_\parallel &=& - \boldsymbol{\nabla} _\parallel p+P_m \partial _r^2 \boldsymbol{u}_\parallel + B_{0r}\partial _r \boldsymbol{b}_\parallel , \end{eqnarray}

(3.7) \begin{eqnarray} \mathrm{i} \omega \boldsymbol{b}_\parallel &=& \breve {\eta } \partial _r^2 \boldsymbol{b}_\parallel + B_{0r} \partial _r \boldsymbol{u}_\parallel + \mathcal{T}(\boldsymbol{U}, \boldsymbol{B}_0) , \end{eqnarray}

where $(\boldsymbol{u}_\parallel ,\boldsymbol{b}_\parallel )$ also denote $\boldsymbol{u}_\parallel =[0,u_\theta ,u_\phi ]^\top$ and $\boldsymbol{b}_\parallel =[0,b_\theta ,b_\phi ]^\top$ for simplicity. Combining (3.7) with the curl of (3.6) yields a homogeneous system for $(\boldsymbol{u}_\parallel ,\boldsymbol{b}_\parallel )$ , except for

(3.8) \begin{align} \mathcal{T} &= B_{0r} \partial _r \boldsymbol{U}- \boldsymbol{B}_0 (\partial _r U_r) + r^{-1} (B_{0r} \boldsymbol{U}-U_r \boldsymbol{B}_0)+ r^{-1} \mathcal{F} , \\[-12pt]\nonumber \end{align}

(3.9) \begin{align} \mathcal{F} &= \mathrm{i} m ( \sin \theta )^{-1} (B_{0\phi } U_\theta -B_{0\theta } U_\phi ) \hat {\boldsymbol{\theta }} + \mathcal{F}_\phi \hat {\boldsymbol{\phi }} , \\[-12pt]\nonumber \end{align}

(3.10) \begin{align} \mathcal{F}_\phi &= B_{0\theta } \partial _\theta U_\phi -B_{0\phi } \partial _\theta U_\theta +U_\phi \partial _\theta B_{0\theta }-U_\theta \partial _\theta B_{0\phi } , \end{align}

related to the induction forcing. Neglecting rotation and viscosity recovers equations (20) and (21) of BG95 when $B_{0r}\partial _r\boldsymbol{U}$ is assumed dominant in (3.8); in their subsequent analysis all forcing terms are then neglected, i.e. $\mathcal{T}=0$ . Here, all inhomogeneous contributions in (3.8) are retained, showing that $B_{0r}\partial _r\boldsymbol{U}$ is not generally dominant (see § 5).

Since $b_r$ does not enter (3.6) and (3.7), it is determined independently within the BLT framework. At leading order, the forcing $\boldsymbol{\nabla }\times (\boldsymbol{U}\times \boldsymbol{B}_0)$ leads to

(3.11) \begin{eqnarray} \mathrm{i} \omega b_r&=& \breve {\eta } \partial _r^2 b_r+ r^{-1 } [(U_r B_{0 \theta } - U_\theta B_{0r} ) \cot \theta +B_{0 \theta } \partial _\theta U_r - B_{0r} \partial _\theta U_\theta ] \nonumber \\ & & + \mathrm{i} m(r \sin \theta )^{-1}(B_{0 \phi } U_r - B_{0r} U_\phi )+ r^{-1}(U_r \partial _\theta B_{0\theta }-U_\theta \partial _\theta B_{0r}) , \end{eqnarray}

as detailed in § 5.1. All theoretical evaluations, from basic flows (e.g. from B74) to boundary-layer solutions or surface integrations, are carried out with the arbitrary-precision library mpmath ensuring the high accuracy required.

3.4. Boundary and interface conditions

Boundary conditions at $r=a_s$ or $r=a_m$ require continuity of (i) $b_r$ , (ii) the tangential components $\breve {\mu }^{-1}\boldsymbol{b}_\parallel$ of the $\boldsymbol{H}$ field $\boldsymbol{B}= \mu \boldsymbol{H}$ , and (iii) the tangential electric field $\breve {\eta } \boldsymbol{\nabla }\times \boldsymbol{B}-\boldsymbol{V}\times \boldsymbol{B}$ (e.g. Stratton Reference Stratton2007). In the BLT limit, condition (iii) reduces to

(3.12) \begin{eqnarray} [ \breve {\eta } \partial _r \boldsymbol{b}_\parallel ]_-^+ + B_{0r} [\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel ]_-^+ = \boldsymbol{0}, \end{eqnarray}

as in BG95. Using $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{B}=0$ , conditions (i) and (ii) also imply continuity of $\breve {\mu }^{-1}\partial _r (r^2 b_r)$ . For inviscid fluids, the above boundary conditions are sufficient, as the velocity field is slaved to the magnetic field. Velocity discontinuities may then occur at the boundaries, introducing forcing through the term $B_{0r}[\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel ]_-^+\neq \boldsymbol{0}$ in (3.12). At $r=a_s$ , $[\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel ]_-^+=\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel -\mathrm{d}_t\boldsymbol{\delta }_\parallel$ , whereas at $r=a_m$ , $[\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel ]_-^+=\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel$ since the outer boundary is at rest. This boundary forcing can then generate a non-zero homogeneous solution $(\boldsymbol{b}_\parallel ^{(h)},\boldsymbol{u}_\parallel ^{(h)})$ , even for a negligible particular solution $(\boldsymbol{b}_\parallel ^{(p)},\boldsymbol{u}_\parallel ^{(p)})$ .

For viscous fluids, additional boundary conditions are required. The stress-free condition in the BLT formulation (2.6) imposes $\partial _r(\boldsymbol{U}_\parallel +\boldsymbol{u}_\parallel )=\boldsymbol{0}$ , whereas the no-slip condition enforces $\boldsymbol{U}+\boldsymbol{u}=\mathrm{d}_t\boldsymbol{\delta }$ , reducing (3.12) to $ [\breve {\eta }\partial _r\boldsymbol{b}_\parallel ]_-^+=\boldsymbol{0}$ .

3.5. Higher-order MHD solutions

Equations (3.6) and (3.7) give the boundary-layer solution but do not enforce $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0$ . To do so, the radial flow $\epsilon \text{Re} (\mathfrak{u}_r\mathrm{e}^{\textrm{i} ( \omega t + m \phi )} )$ is needed at the next order (in the small boundary-layer thickness). With the non-penetration condition, the BLT approach gives (Greenspan Reference Greenspan1968)

(3.13) \begin{equation} r \partial _r \mathfrak{u}_r =- \cot (\theta ){u}_{\theta }-\mathrm{i} m (\sin \theta )^{-1} {u}_{\phi }-\partial _{\theta } {u}_{\theta } , \end{equation}

where the factor $r$ on the left-hand side of (3.13) is set constant to $r=a_s$ or $r=a_m$ . The solution $\mathfrak{u}_r$ is then obtained by integrating (3.13) with the condition that $\mathfrak{u}_r$ vanishes away from the boundary and is non-zero at the boundary ( $\mathfrak{u}_r$ being called secondary flow or Ekman pumping in rotating fluids). The non-penetration boundary condition $\mathfrak{u}_r+\mathcal{U}_r=0$ then forces a secondary bulk flow $\boldsymbol{\mathcal{U}}$ (the so-called Ekman circulation for rotating fluids). Neglecting diffusive effects in the bulk, $\boldsymbol{\mathcal{U}}$ and the pressure field $\mathcal{P}$ are governed by

(3.14) \begin{align} \mathrm{i} \omega \boldsymbol{\mathcal{U}} &= - \boldsymbol{\nabla }\mathcal{P}+ \boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{B}}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{B}_0-\gamma \omega \hat {\boldsymbol{z}} \times \boldsymbol{\mathcal{U}} , \\[-12pt]\nonumber \end{align}

(3.15) \begin{align} \mathrm{i} \omega \boldsymbol{\mathcal{B}} &= (\boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{\mathcal{U}} - (\boldsymbol{\mathcal{U}} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{B}_0 , \end{align}

with the secondary bulk field $\boldsymbol{\mathcal{B}}$ . In contrast to the exponentially decaying $\mathfrak{u}_r$ , $\boldsymbol{\mathcal{U}}$ is nearly uniform across the boundary layer, with $\mathcal{U}_r(r=a_s)=-\mathfrak{u}_r(r=a_s)$ , so that the radial flow is $\mathfrak{u}_r+\mathcal{U}_r \approx \mathfrak{u}_r-\mathfrak{u}_r(r=a_s)$ in the boundary layer. This higher-order contribution can be significant (e.g. for recovering the Stokes force in Appendix C).

Although explicit determination of $(\boldsymbol{\mathcal{U}},\boldsymbol{\mathcal{B}})$ is difficult, their contribution to mean dissipation can be obtained indirectly via power arguments (§ 3.6). When used below, $\boldsymbol{\mathcal{U}}$ is approximated by neglecting magnetic and rotational effects in (3.14) and (3.15) and assuming an irrotational form; weak rotational corrections can nevertheless be derived a posteriori for the polar mode (§ 4.1). Once the secondary bulk fields are known, their residual tangential mismatch at the boundaries is removed by the higher-order boundary-layer corrections $(\boldsymbol{\mathfrak{u}}_\parallel ,\boldsymbol{\mathfrak{b}}_\parallel )$ . As illustrated in § 5.3, they are obtained by solving the same homogeneous boundary-layer problem as $(\boldsymbol{u}^{(h)}_\parallel ,\boldsymbol{b}^{(h)}_\parallel )$ with amplitudes fixed by the interface conditions of § 3.4, but where $\boldsymbol{U}$ is replaced by $\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)}$ , and with a stationary inner boundary (as its motion is already accounted for in $\boldsymbol{u}$ ).

3.6. Scaled forces and dissipation from the boundary-layer approach

We consider the forces on the inner solid domain using leading-order corrections to the basic state; those on the outer solid domain are obtained similarly, using the relevant fields at $r=a_m$ . At the order $\epsilon$ , the viscous force $\boldsymbol{F}_{\!v}$ , the hydrodynamic pressure force $\boldsymbol{F}_{\!p}$ and the magnetic tension $\boldsymbol{F}_{\!mt}$ are obtained from $\boldsymbol{F}_{\!j}= \epsilon \mathrm{e}^{\textrm{i} \omega t} \int _S \boldsymbol{f}_{\!j} \, \mathrm{e}^{\textrm{i} m \phi } \mathrm{d}S$ , with $\boldsymbol{f}_{\!v} = P_m \partial _r \boldsymbol{u}_\parallel$ , $\boldsymbol{f}_{\!p} = - (P+p_h) \hat {\boldsymbol{r}}$ and $\boldsymbol{f}_{\!mt} = B_{0r} \boldsymbol{b}+b_r \boldsymbol{B}_0$ , where $p_h$ and $p_m$ are the hydrodynamic and magnetic parts of $p=p_h+p_m$ . In BG95, $\boldsymbol{f}_{\!mt}$ is further simplified to its BLT expression $\boldsymbol{f}_{\!mt} = B_{0r} \boldsymbol{b}_\parallel$ . The magnetic-pressure force $\boldsymbol{F}_{\!mp}$ is similarly obtained from $\boldsymbol{f}_{mp} = {f}_{mp} \hat {\boldsymbol{r}}$ , with ${f}_{mp} = - \boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{b}$ since $B^2 \approx 2 \boldsymbol{B}_0 \boldsymbol{\cdot }\boldsymbol{b}$ at this order. We also note that the torque $\boldsymbol{\varGamma }_{\!j}=\epsilon \mathrm{e}^{\textrm{i} \omega t }\int _S \boldsymbol{r} \times \boldsymbol{f}_{\!j} \, \mathrm{e}^{\textrm{i} m \phi } \mathrm{d}S$ for each local force $\boldsymbol{f}_{\!j}$ . The magnetic and viscous torques originate from the magnetic-tension stress $\boldsymbol{f}_{\!mt}$ and tangential viscous stress $\boldsymbol{f}_{\!v}$ , respectively.

Following BG95, we characterise each force contribution by the scaled force

(3.16) \begin{eqnarray} \frac {\Delta \boldsymbol{\omega }_{\!j}}{\omega } = - \frac {1}{2 M_s \omega ^2} \int _S \boldsymbol{f}_{\!j} \, \mathrm{e}^{\textrm{i} m \phi } \mathrm{d}S = - \frac {1}{2 M_s \omega ^2} \frac {\boldsymbol{F}_{\!j}} {\epsilon \mathrm{e}^{\textrm{i} \omega t }} , \end{eqnarray}

which measures the force relative to the inertial-force amplitude $M_s\epsilon \omega ^2$ . Although introduced here for a forced problem, this notation follows directly from the corresponding fixed-outer-boundary free-oscillation dynamics governed by $M_s \mathrm{d}_t^2 \boldsymbol{\delta }= \boldsymbol{F}$ , where $M_s=(4\pi /3)\rho _s a_s^3$ is the solid-sphere mass and $\boldsymbol{F}$ is the total force (as in the fixed-outer-boundary linear formulation of Rieutord Reference Rieutord2002). Indeed, a weak frequency perturbation gives $M_s\epsilon (\omega +\Delta \omega _{\!j})^2 \simeq M_s\epsilon \omega ^2(1+2\Delta \omega _{\!j}/\omega )$ , so that projection of (3.16) onto the oscillation direction yields the first-order complex frequency correction associated with $\boldsymbol{F}_{\!j}$ for weak forces. In the absence of rotation, the polar-mode forces considered by BG95 are parallel to $\hat {\boldsymbol{z}}$ , such that $\Delta \boldsymbol{\omega }_{\!j}=\Delta \omega _{\!j}\hat {\boldsymbol{z}}$ ; the force response can then be characterised by the scalar $\Delta \omega _{\!j}$ . Within this fixed-outer-boundary free-oscillation interpretation, the real and imaginary parts of the resulting detuning $\Delta \omega _{\!j}/\omega$ quantify, respectively, the eigenfrequency correction and modal damping-rate contribution, while $-2(\rho _s/\rho _{\!f})\text{Re}(\Delta \omega _{\!j}/\omega )$ gives the associated relative correction to the effective inertia of the solid domain. For a finite-mass freely translating outer solid, however, $\boldsymbol{\delta }$ is the relative displacement and $\Delta \boldsymbol{\omega }_{\!j}/\omega$ retains its meaning as a fluid-force response, while its planetary eigenfrequency interpretation requires the global translational dynamics (Grinfeld & Wisdom Reference Grinfeld and Wisdom2005; Greff-Lefftz & Legros Reference Greff-Lefftz and Legros2007).

Decomposing the viscous force $\boldsymbol{F}_\nu$ and the Lorentz force $\boldsymbol{F}_{\!\eta}$ into non-dissipative components $[\boldsymbol{F}_\nu ^{nd},\boldsymbol{F}_{\!\eta} ^{nd}]\propto \mathrm{d}_t^2\boldsymbol{\delta }$ (in phase with acceleration) and dissipative components $[\boldsymbol{F}_\nu ^d,\boldsymbol{F}_{\!\eta} ^d]\propto \mathrm{d}_t\boldsymbol{\delta }$ (in phase with velocity), we can write

(3.17) \begin{eqnarray} \big[\boldsymbol{F}_\nu ^d,\boldsymbol{F}_{\!\eta} ^d\big] = - \frac {\mathrm{d}_t \boldsymbol{\delta }}{\omega \epsilon } \big[F_\nu ^d,F_\eta ^d \big] =-\frac {\mathrm{d}_t\boldsymbol{\delta }}{\omega }\int _S \big[f_\nu ^d,f_\eta ^d \big] \mathrm{e}^{\textrm{i} m\phi } \mathrm{d}S . \end{eqnarray}

Time averages follow by equating the work required to maintain the motion to viscous and Ohmic losses (Motz Reference Motz1966b ; Batchelor Reference Batchelor1967). Projecting onto the velocity $\text{Re}(\mathrm{d}_t\boldsymbol{\delta })$ gives $\omega ^{-1}\langle \text{Re}(\mathrm{d}_t\boldsymbol{\delta })^2\rangle \int _S[f_\nu ^d,f_\eta ^d] \,\mathrm{e}^{\textrm{i} m\phi } \mathrm{d}S =\langle [\mathcal{D}_\nu ,\mathcal{D}_\eta ]\rangle$ , where $\langle \text{Re}(\mathrm{d}_t\boldsymbol{\delta })^2\rangle /(\epsilon \omega )^2=\mathcal{E}$ is the kinetic energy factor (§ 2.1). Using BLT, these forces are given by (e.g. Batchelor Reference Batchelor1967)

(3.18) \begin{align} \int _S f_\nu ^d \mathrm{d}S &= \frac { P_m}{2 \mathcal{E} \omega } \int _{\varOmega _{\!f}} \left (\textrm{Re} \left ( \partial _r \boldsymbol{u}_\parallel \right )\right )^2+\left (\mathrm{Im} \left ( \partial _r \boldsymbol{u}_\parallel \right )\right )^2\mathrm{d}V \implies {f}_\nu ^d = - \frac { P_m {\mathcal{G}_\nu }{(a_s)}}{2 \mathcal{E}\omega } , \\[-12pt]\nonumber\end{align}

(3.19) \begin{align} \int _S f_\eta ^d \mathrm{d}S &= \frac {1}{2 \mathcal{E} \omega } \int _{\varOmega _{\!f}} \left (\textrm{Re} \left ( \partial _r \boldsymbol{b}_\parallel \right )\right )^2+\left (\mathrm{Im} \left ( \partial _r \boldsymbol{b}_\parallel \right )\right )^2 \mathrm{d}V \implies {f}_\eta ^d= - \frac { {\mathcal{G}_\eta }{(a_s)} }{2 \mathcal{E}\omega } \end{align}

for the (thin) inner fluid boundary layer of velocity amplitude $\omega \epsilon$ . Because the boundary-layer perturbations decay exponentially, the thin-layer limit yields $\int _{\varOmega _{\!f}}{\cdots} \,\mathrm dV=\int _S{\cdots} \,\mathrm dS \int _{a_s}^{\infty }\mathrm dr$ , so the radial integration reduces to the opposite of the primitives $(\mathcal{G}_\nu ,\mathcal{G}_\eta )$ evaluated at $r=a_s$ . The same procedure applies to the outer-boundary layer by evaluating the primitive at $r=a_m$ . Ohmic dissipation in solid domains and the associated forces follow from analogous expressions using the appropriate volume, magnetic diffusivity and permeability.

Whereas (3.16) is based on instantaneous-force amplitudes, mean dissipation follows from the power balance. We define

(3.20) \begin{eqnarray} \widehat {\mathcal D}_{\!j} \equiv \frac {\langle \mathcal D_{\!j}\rangle }{M_s\epsilon ^2\omega ^3} = \frac {\mathcal E}{M_s\omega ^2}\int _S f_{\!j}^d\,\mathrm{e}^{\textrm{i} m\phi } \mathrm{d}S = \frac {\mathcal E}{M_s\omega ^2}\frac {F_{\!j}^d}{\epsilon }, \quad j\in \{\nu ,\eta \}. \end{eqnarray}

Since the modal energy is $M_s\mathcal E(\epsilon \omega )^2$ , (3.16) and (3.20) give, for the dissipative component of the same force, $\mathrm{Im}(\Delta \omega _{\!j})/\omega = \widehat {\mathcal D}_{\!j}/(2\mathcal E)$ . Thus, the normalised amplitude-damping rate equals $\widehat {\mathcal D}_{\!j}$ for the polar mode and $\widehat {\mathcal D}_{\!j}/2$ for the equatorial modes.

In (3.16)–(3.20), surface integration over $S$ is generally cumbersome and, when required, performed numerically. Analytical simplifications follow from the BG95 assumptions: (i) retaining only the homogeneous solution, which is accurate for sufficiently thin boundary layers; and (ii) neglecting spatial variations of boundary-layer thickness, i.e. assuming a uniform layer. The latter is exact in specific limits (e.g. no rotation with a monopole field at $r=a_s$ ), and otherwise relies on a characteristic thickness. This approach is hereafter termed the homogeneous uniform boundary layer (huBL).

4.1. Potential flows, added-mass and weak rotational perturbations

For $\gamma \ll 1$ , rotational effects are negligible and the basic flow reduces to an irrotational oscillatory motion. For $\boldsymbol{B}_0=\boldsymbol{0}$ , an irrotational bulk flow $\boldsymbol{U}$ can then be sought (with finite viscosity; see Buresti Reference Buresti2009), so that $\boldsymbol{U}=\boldsymbol{\nabla }\varPhi$ with $\boldsymbol{\nabla} ^2 \varPhi =0$ . In the unbounded limit $a_s \ll a_m$ used by BG95 (or problem 92 in Lamb Reference Lamb1924), the velocity potential $\varPhi$ can be obtained for arbitrary displacements at small velocities. In the frame of the sphere, the flow reduces to dipolar contributions with internal sources of the form $(k_1 x + k_2 y + k_3 z) r^{-3}$ , where the coefficients $k_i$ are determined from the non-penetration condition $(U_r - \mathrm{d}_t \boldsymbol{\delta } \boldsymbol{\cdot }\hat {\boldsymbol{r}})\vert _{r=a_s} = 0$ . Extension to bounded domains follows by superposing dipolar terms with external sources, as in Article 9 of Stokes (Reference Stokes1843) or problem 93 of Lamb (Reference Lamb1924). In the frame of the outer boundary at rest, additional terms ${k}_4 x + {k}_5 y + {k}_6 z$ enforce non-penetration at $r=a_m$ . The flow $\epsilon \boldsymbol{U} \mathrm{e}^{\textrm{i}(\omega t + m \phi )} = \epsilon \boldsymbol{\nabla }[\varPhi \mathrm{e}^{\textrm{i}(\omega t + m \phi )}]$ is given by

(4.1) \begin{equation} [\varPhi _0,\varPhi _\pm ]=-\frac {\mathrm{i}}{2} \left ( \frac {a_s}{r} \right )^2 \omega a_s \frac {1+2(r/a_m)^3}{1-a^3} [ \cos \theta , \sin \theta ] \end{equation}

for the polar and equatorial modes ( $a=a_s/a_m$ ). The associated velocity fields are

(4.2) \begin{equation} \boldsymbol{U}= \mathrm{i} \omega \frac {1}{1-a^3} \frac {[a_s^3-(a r)^3] \cos (\theta )\, \hat {\boldsymbol{r}} +[(a r)^3+a_s^3/2] \sin (\theta )\, \hat {\boldsymbol{\theta }}}{r^3} \end{equation}

for the polar mode, while the velocity field of equatorial modes is

(4.3) \begin{equation} \boldsymbol{U}= \mathrm{i} \omega \frac {1}{1-a^3} \frac {[a_s^3-(a r)^3] \sin (\theta )\, \hat {\boldsymbol{r}} -[(a r)^3+a_s^3/2][\cos (\theta )\, \hat {\boldsymbol{\theta }} + \mathrm{i} m \hat {\boldsymbol{\phi }}]}{r^3}. \end{equation}

These flows are exact solutions of the viscous Navier–Stokes equation (3.1) in the absence of magnetic and rotational effects (as for any potential flow; see Buresti Reference Buresti2009).

Figure 4.

Theoretical basic flow (T) versus XSHELLS (N, black-dotted line for $\gamma =0$ , and cyan for the polar mode at $\gamma =1/3$ ) for polar (a) and prograde equatorial (b) forcing ( $\theta =60^{\circ }$ , $a=0.7$ , $L_\nu =6\times 10^{-3}$ ). Potential flow: unbounded (orange line) and bounded (red line) at $\gamma =0$ (solid, (4.2)–(4.3)), and rotation perturbation of the polar mode (non-zero azimuthal flow (4.5), bottom plot of panel (a)). The B74 solution (pink dashed line, § 4.3), using $\zeta _m$ from (4.14). Surface plots compare bounded analytical (T) and numerical (N) velocity in a meridional plane (equatorial section of T also shown in panel (b)).

Figure 4. Long description

Panel A: Three line graphs compare theoretical (T) and numerical (N) velocity profiles for polar forcing. The graphs show radial (Vr), latitudinal (Vθ), and azimuthal (-Vφ) components of velocity against the radial distance (r - as). The theoretical profiles are shown in orange and red lines, while the numerical profiles are in black dotted and cyan lines. Insets display velocity distributions in meridional planes. Panel B: Three line graphs compare theoretical (T) and numerical (N) velocity profiles for prograde equatorial forcing. Similar to Panel A, the graphs show radial (Vr), latitudinal (-Vθ), and azimuthal (-Vφ) components of velocity against the radial distance (r - as). The theoretical profiles are shown in orange and red lines, while the numerical profiles are in black dotted lines. Insets display velocity distributions in meridional planes.

For all three modes, we have $\boldsymbol{\nabla} ^2\boldsymbol{U}=\boldsymbol{0}$ , and the inviscid pressure force follows from the linearised unsteady Bernoulli equation. With $\partial _t[\epsilon \varPhi \mathrm{e}^{\textrm{i}(\omega t+m\phi )}] =\mathrm{i}\omega \epsilon \varPhi \mathrm{e}^{\textrm{i}(\omega t+m\phi )}$ , we obtain

(4.4) \begin{equation} -\frac {\boldsymbol{F}_{\!p}}{M_{\!fs}} = C_a^i\,\mathrm{d}_t^2\boldsymbol{\delta } = -\frac {1}{4\pi a_s^3/3} \int _{\tilde S} \mathrm{i}\omega \epsilon \varPhi \mathrm{e}^{\textrm{i}(\omega t+m\phi )} \,\mathrm{d}\boldsymbol{S} = \frac {1}{2}\frac {1+2\tilde a^3}{1-a^3}\, \mathrm{d}_t^2\boldsymbol{\delta } \end{equation}

on any spherical surface $\tilde S$ of radius $\tilde a a_m$ . For the polar mode, the acceleration is axial and reverses during each oscillation, whereas for the equatorial modes, it is centripetal, along $-\hat {\boldsymbol{s}}$ . The inner-sphere force is obtained with $\tilde a=a$ ; the corresponding radial pressure integral at the outer boundary is obtained with $\tilde a=1$ .

Weak rotational effects can be incorporated perturbatively. At leading order for $\gamma \ll 1$ , the rotation-induced azimuthal velocity of the polar-mode potential flow follows from the $\phi$ component of the Navier–Stokes equation, yielding $U_\phi /U_s=\mathrm{i}\gamma$ in cylindrical coordinates $(s,\phi ,z)$ . Perturbing the flow (4.1), the first-order basic flow $\boldsymbol{U}=\boldsymbol{U}_{\!1}$ becomes

(4.5a,b) \begin{equation} \boldsymbol{U}_{\!1}=\boldsymbol{\nabla }\varPhi +U_{1\phi } \hat {\boldsymbol{\phi }} , \quad U_{1\phi }=- \frac {3}{4} \frac {(a_s/r)^3}{1-a^3} \gamma \omega \sin (2 \theta ) \end{equation}

for the polar mode (correcting a typo in equation (10) of BG95).

Figure 4 compares these base flows with XSHELLS (dotted profiles) for strong confinement ( $a=0.7$ ). Dark grey bands near the boundaries indicate regions where viscous effects are significant, with a thickness of $25$ viscous skin depths; these are excluded from the bulk-flow discussion and treated in the boundary-layer analysis. Outside the boundary layers, the irrotational approximation is excellent: the bounded base flow (dark-red lines; (4.2) and (4.3) in panels a,b) accurately reproduces the DNS, remaining accurate for both radial and meridional components even under weak rotation. Conversely, the unbounded flow (orange line) used by BG95 fails to capture the solution when confinement effects become significant for $a\gt 0.3$ . In the lowest panel of figure 4(a), the rotation-perturbed and bounded solution (4.5b ) (dark-red dashed line) agrees well with the DNS for $\gamma =1/3$ (cyan-dotted line).

Since (4.4) applies only to irrotational flows, the rotational correction to the pressure force is evaluated using the kinetic energy approach (e.g. B74, equation (6)). Applying this approach to (4.5) yields the inviscid added-mass coefficient (for the inner sphere)

(4.6) \begin{align} C_a^i=\frac {1}{2} \frac {1+2a^3}{1-a^3} - \frac {3}{10} \frac {\gamma ^2}{1-a^3} + \mathcal{O}(\gamma ^4) \end{align}

for the polar mode. The $\gamma ^2$ correction originates from the centrifugal pressure associated with the azimuthal velocity $U_\phi$ and can equally be derived directly from the pressure field. Equation (4.6) accurately reproduces the rotational dependence of $C_a^i$ at small $\gamma$ , in excellent agreement with the COMSOL DNS (figure 5).

Figure 5.

Inviscid added-mass coefficient $C_a^i$ at the inner (a) and outer (b) boundaries versus $\gamma$ for two radius ratios, with no-slip (NS) and stress-free (SF) boundary conditions at $r=a_s$ (stress-free outer boundary) in COMSOL, at $(L_\nu /(a_m\sqrt {\gamma }))^2=10^{-5}$ and $\epsilon =0.5L_\nu$ . Solid lines in (a,b) correspond to (4.6); dotted lines in (a) follow (A22). In (a) the coloured area and dash–dotted lines represent the spread of the B74 solution (three estimates of $\zeta _m$ ).

4.2. Bulk dissipation of the rotation-perturbed potential flow

In the absence of rotation, the real part of the flow $\boldsymbol{\nabla }\varPhi$ can be used to calculate the viscous dissipation $\mathcal{D}_\nu$ , that reduces here to $\mathcal{D}_\nu = 2 \rho _{\!f} \nu \int _{\partial {\varOmega _{\!f}}} \hat {\boldsymbol{n}} \boldsymbol{\cdot }\boldsymbol{\nabla }( \boldsymbol{V}^2/2) \mathrm{d}S$ (see (2.10) and (2.11)),

(4.7) \begin{eqnarray} \frac { \mathcal{D}_\nu }{P_m } = 12 \pi \frac {1-a^5}{(1-a^3)^2} a_s (\epsilon \omega )^2 (f_s \sin ^2 \omega t + f_c \cos ^2 \omega t ), \end{eqnarray}

with $(f_s,f_c)=(1,0)$ for the polar mode and $(f_s,f_c)=(1,1)$ for the equatorial modes, and where $P_m$ comes from our unit choice (to be replaced by $\rho _{\!f} \nu$ in dimensional units). Considering the base flow $\boldsymbol{U}_{\!1}$ of (4.5), (2.10) and (2.11) give the leading-order rotation-induced correction $(f_s,f_c)=(1,2 \gamma ^2/5)$ for the polar mode. As the $f_c$ correction is of order $\gamma ^2$ , we can wonder if higher-order $\gamma ^2$ corrections of the basic flow could modify this result. Equation (4.7) gives

(4.8) \begin{eqnarray} F_\nu ^d= \frac {6\pi }{\mathcal E} \frac {1-a^5}{(1-a^3)^2} P_m a_s(f_s+f_c)\epsilon \omega , \quad \widehat {\mathcal D}_\nu = \frac {9}{4}\frac {\rho _{\!f}}{\rho _s} \frac {L_\nu ^2}{a_s^2} \frac {1-a^5}{(1-a^3)^2}(f_s+f_c) , \end{eqnarray}

recovering the Levich (Reference Levich1949) drag, given by (A21), and extending it to spherical confinement. We have successfully validated this prediction of the bulk dissipation geometric dependence against COMSOL DNS (figure 22 b in Appendix A.4).

This bulk-flow dissipation can be further enhanced by viscous boundary layers imposed by the boundary conditions. For a rapidly oscillating no-slip inner boundary and a no-slip outer boundary at rest, the boundary-layer contribution is given by (A18); its leading term, proportional to $a_s/L_\nu$ , exceeds the bulk force (4.8) when $a_s/L_\nu \gt 2(1-a^5)/(1+a^4)$ , i.e. for sufficiently thin boundary layers (the $a^4$ term is the outer-boundary-layer contribution). Comparing (4.8) with the $L_\nu$ -independent term of (A18) in the unbounded limit $a=0$ further shows that the viscous layer at the no-slip inner boundary reduces the contribution of this order to the force by a factor $1/2$ .

4.3. Large rotation effects: basic flow and added-mass coefficient

The regime of arbitrary rotation is analytically challenging, notably because inertial modes may be directly forced when $\gamma \gt 1$ . In the planetary-relevant range $\gamma \lt 1$ , the polar mode was analysed by B74, while all three modes were considered by Smylie & McMillan (Reference Smylie and McMillan1998, Reference Smylie and McMillan2000). These studies include confinement by the outer boundary, and Smylie & McMillan (Reference Smylie and McMillan1998, Reference Smylie and McMillan2000) additionally account for the associated Stokes–Ekman layers. Their results relevant to this work are summarised in Appendix A.4. We outline below the formulation of B74 used to analyse rotational effects. Several typographical errors and implicit relations were identified in B74 in the course of our derivation; we provide explicit corrected forms below.

Using cylindrical coordinates and a streamfunction $\psi (s,z)$ , the axisymmetric basic flow $\boldsymbol{U}=\boldsymbol{U}_{B74}$ of B74 is written as $[U_s,U_\phi ,U_z]=[s^{-1}\partial _z\psi ,\,\mathrm{i}\gamma s^{-1}\partial _z\psi ,\,-s^{-1}\partial _s\psi ]$ . Following B74, the streamfunction $\psi (s,z)$ of the polar mode reads

(4.9a,b) \begin{equation} \psi =\frac {k}{2}(1-\varsigma ^2)(\zeta ^2-1)[C_1 \mathcal{H}(\zeta )+C_2 ] , \quad \mathcal{H}(\zeta )= \frac {1}{2} \ln \left ( \frac {\zeta +1}{\zeta -1} \right ) -\frac {\zeta }{\zeta ^2-1}, \end{equation}

and the inviscid added-mass coefficient is

(4.10) \begin{eqnarray} C_a^i=\frac {\mathcal{H}(\zeta _s)-\mathcal{H}(\zeta _m)+\zeta _s^{-1}+\zeta _s(\zeta _s^2-1)^{-1}}{\mathcal{H}(\zeta _m)-\mathcal{H}(\zeta _s)} , \end{eqnarray}

where, correcting a sign error in B74’s expression for $C_2$ (the correct sign is required to satisfy non-penetration at $\zeta = \zeta _m$ ),

(4.11a,b) \begin{equation} C_1=\mathrm{i} \omega k [\mathcal{H}(\zeta _m)-\mathcal{H}(\zeta _s)]^{-1} , \quad C_2= -\mathrm{i} \omega k [1-\mathcal{H}(\zeta _s)/\mathcal{H}(\zeta _m)]^{-1}, \end{equation}

with $\zeta _s=\gamma ^{-1}$ , $k=\gamma a_s (1-\gamma ^2)^{-1/2}$ , and where the ellipsoidal coordinates $(\varsigma ,\zeta )$ are related to $(s,z)$ through $s=k\sqrt {(1-\varsigma ^2)(\zeta ^2-1)}$ and $z=k \varsigma \zeta \sqrt {1-\gamma ^2}$ . Here, $\zeta _s$ and $\zeta _m$ are the values of $\zeta$ at the inner and outer boundaries, respectively. Using these latter equations, four solutions are found for $(\varsigma ,\zeta )$ as functions of $(s,z)$ , and the solution

(4.12a,b,c) \begin{equation} \varsigma = \frac {\varpi _+^{1/2}-\varpi _-^{1/2}}{2k}, \quad \zeta = \frac {2z (1-\gamma ^2)^{-1/2}}{\varpi _+^{1/2}-\varpi _-^{1/2}}, \quad \varpi _\pm = \left (\frac {z}{\sqrt {1-\gamma ^2}} \pm k \right )^2+s^2 \end{equation}

is found to recover the solution (4.2) of the non-rotating limit. The approach of B74 can only approximate the non-penetration condition at the outer boundary. The value of $\zeta =\zeta _m$ at the outer boundary can be bounded from below and above by $\zeta _m^{(i)}$ and $\zeta _m^{(o)}$ , which are given by (correcting a typo in equation (16b) of B74)

(4.13a,b) \begin{equation} \zeta _m^{(i)}=\sqrt {1+(a_m/k)^2} , \quad \zeta _m^{(o)}= \big(1-\gamma ^2 \big)^{-1/2} \, a_m/k , \end{equation}

such that $[\zeta _m^{(i)}/\zeta _m^{(o)}]^2=1-\gamma ^2(1-a^2)$ . For small $\gamma ^2(1-a^2) \ll 1$ , the theory is thus expected to be accurate. An intermediate relevant estimate is given by equation (15) of B74. Integrating the associated cubic polynomial, we obtain

(4.14a,b) \begin{equation} \zeta _m=\zeta _m^{(e)}= \frac {(12\mathcal{A} + 12 \sqrt {\mathcal{A}^2 - 12})^{2/3} + 12}{6(12\mathcal{A} + 12\sqrt {\mathcal{A}^2 - 12})^{1/3}} , \quad \mathcal{A}=9 (a_m/k)^3 \big(1-\gamma ^2 \big)^{-1/2}, \end{equation}

which provides an explicit solution for $\zeta _m$ . In the rapid-oscillation limit, all three estimates of $\zeta _m$ reduce the added-mass coefficient (4.10) to (4.4).

Figure 5(a) compares the B74 prediction with the inviscid added-mass coefficient computed using COMSOL; the DNS (dots) lie within the coloured bounds (4.13). While (4.10) computed with (4.14) and (4.6) coincide for $\gamma \ll 1$ , the former agrees better at moderate confinement ( $a=0.35$ ), whereas (4.6) is more accurate at strong confinement ( $a=0.75$ ) owing to exact enforcement of non-penetration at $r=a_m$ .

Higher-order corrections to the basic flow $\boldsymbol{U}_{\!1}$ of (4.5) can also be derived. In particular, the next-order contribution $\boldsymbol{U}_2$ (in $\gamma ^2$ ; Appendix A.5) is obtained explicitly. Although the three estimates of $\zeta _m$ yield different higher-order cross-terms in $a$ and $\gamma$ , they coincide in the unbounded rotating limit ( $a\ll 1$ ), giving the same basic flow and added-mass coefficient. For example, $\zeta _m=\zeta _m^{(o)}$ yields, at order $\mathcal{O}(\gamma ^4+a^5)$ ,

(4.15) \begin{align} \boldsymbol{U}_2^{a \to 0} &= \boldsymbol{U}_{\!1}+ 3 \omega \left (\frac {\mathrm{i}}{5} f_r \cos \theta \hat {\boldsymbol{r}}+ \mathrm{i} f_\theta \sin \theta \hat {\boldsymbol{\theta }}+\frac {(f_\phi +3a^3) \gamma \sin 2 \theta }{10} \hat {\boldsymbol{\phi }} \right ) \, \frac {a_s^3}{r^3}\gamma ^2 , \\[-12pt]\nonumber \end{align}

(4.16) \begin{align} f_r&=(5 \cos ^2 \theta -3)(a_s^2 r^{-2}-1)(1+a^3) +2a^3(r^3a_s^{-3}-1), \\[-12pt]\nonumber \end{align}

(4.17) \begin{align} f_\theta &= (\cos ^2 \theta -1/5) (3a_s^2 r^{-2}-1)(1+a^3)/4-a^3(2r^3a_s^{-3}+1)/5 , \\[-12pt]\nonumber \end{align}

(4.18) \begin{align} f_\phi &=-[13-15(a_s/r)^2+(35a_s^2/r^2-25) \cos ^2 \theta ](1+a^3)/4 , \end{align}

where $\epsilon \boldsymbol{U}_{\!1}=\epsilon (U_{1\phi } \hat {\boldsymbol{\phi }}+\boldsymbol{\nabla }\varPhi )$ is given by (4.1)−(4.5), and where the unbounded rotation-corrected flow $\boldsymbol{U}_2(a = 0)$ is obtained by putting $a=0$ in (4.15)−(4.18). At $\mathcal{O}(\gamma ^3)$ , this flow confirms the leading-order rotation-induced correction of the viscous force and dissipation proposed in § 4.2, and the $\mathcal{O}(\gamma ^4)$ series expansion of the flow gives $f_s=1+4 \gamma ^4/25$ and $f_c=2 \gamma ^2/5+4\gamma ^4/175$ .

In an unbounded rotating fluid, the added-mass coefficient (4.10) is found to be exactly

(4.19) \begin{eqnarray} C_a^i&=&\frac {(\gamma ^2-1)(2\gamma -\gamma _0)}{2\gamma + \gamma _0(\gamma ^2-1)} , \end{eqnarray}

where $\gamma _0=\ln [(1+\gamma )/(1-\gamma )]$ , providing a compact exact expression for the added-mass coefficient of a sphere oscillating along the rotation axis in an inviscid fluid (see the comparison with Smylie & McMillan (Reference Smylie and McMillan2000) and DNS in Appendix A.4). For weak viscous and rotational effects, a perturbative approach can be used in which the viscous corrections to Stokes’ result (A17) are combined with the rotational terms (from (4.19) in the unbounded limit or from Appendix A.4 in the bounded case). This approach is valid only up to order $\mathcal{O}(\gamma ^2 L_\nu /a_s)$ , since cross-terms may arise at higher order.

Figure 6.

Theoretical (T) basic flow versus XSHELLS (N, black line) for strong rotation ( $\gamma =0.92$ ), at $a=0.35$ (a) and $a=0.05$ (b), polar mode ( $\theta =45^{\circ }$ , $L_\nu =0.002$ ). Potential flow: unbounded case (orange solid line, (4.2) with $a=0$ ), bounded case at $\gamma =0$ (dark-red solid line, (4.2)) and rotation perturbation (azimuthal flow (4.5), dark-red dashed line). The B74 flow (4.9): $\zeta _m$ from (4.14) (green-dashed line), bounds (4.13) shown as a shaded region, and Taylor expansion (4.15) for $a=0$ (blue line). Grey side bands: viscous boundary layers. In (b), surface plots compare the B74 solution $\boldsymbol{U}_{B74}$ , the rotation-perturbed potential flow $\boldsymbol{U}_{\!1}$ (4.5) and its correction (4.15) with the XSHELLS (N) in the meridional plane (zoom to $r=6a_s$ ).

For strong confinement ( $a=0.7$ ) and moderate rotation ( $\gamma =1/3$ ), figure 4 shows that the B74 basic flow (dashed pink line) consistently recovers both the rotation-perturbed solution (4.5) and the XSHELLS DNS. At stronger rotation ( $\gamma \sim 1$ ), considered in figure 6(a) for $a=0.35$ , the B74 solution (dashed line), based on (4.14), provides a markedly improved description of the velocity profiles outside the boundary layers (grey bands) compared with the rotation-perturbed potential flow (4.5) (dark-red line). The influence of $\zeta _m$ is illustrated by the shaded region between the bounds in (4.13); the azimuthal velocity remains only weakly sensitive, with all formulations nearly coincident.

Figure 6(b) presents the same comparison for $a=0.05$ on a logarithmic scale, highlighting the spread of solutions. A horizontal dashed line at $L_\nu$ marks the threshold below which the basic flow becomes comparable to the viscously driven secondary bulk flow $\boldsymbol{\mathcal{U}}$ . The B74 solution again shows the best agreement with the XSHELLS DNS, whereas alternative models, including (4.5) (red line) and (4.15) (blue line), overestimate the velocity, particularly near the inner boundary. Accurate representation of the bulk flow near $r=a_s$ is thus essential, as it sets the boundary-layer solution where dissipation is concentrated.

The meridional fields in figure 6(b), in particular the comparison between $\boldsymbol{U}_{\!1}$ and the XSHELLS (N), provide further evidence of the strong rotational constraint on the bulk flow. This behaviour is captured only by the B74 solution $\boldsymbol{U}_{B74}$ , whereas its weakly bounded Taylor expansion $\boldsymbol{U}_2^{a \to 0}$ , given by (4.15), fails. Even in approximate form, accounting for confinement in B74 is therefore essential. Additional meridional cuts from numerical and analytical models at $\gamma \approx 1$ are provided in supplementary material S4. To our knowledge, this is the first validation and application of the B74 theoretical flow and added-mass coefficient. The hydrodynamic results of this section provide the reference solution for the MHD extensions developed below.

5.1. Radial magnetic field: induction by the hydrodynamic base flow

Our calculations of $b_r$ are performed in an identical way for the three modes. To illustrate the method, we detail explicitly this calculation below in the particular case of a uniform axial field $\boldsymbol{B}_0=[B_0 \cos \theta ,-B_0 \sin \theta ,0]^\top$ , with the rotation-modified polar mode flow given by (4.1)−(4.5). The evolution of $b_r$ in the fluid is then governed by

(5.1) \begin{eqnarray} L_\eta ^2 \partial _r^2 b_r= 2 \mathrm{i} b_r +3 \mathrm{i} \, B_0 \, a_s^3 r^{-4} (3 \cos ^2 \theta -1) (1-a^3)^{-1}, \end{eqnarray}

recalling that the magnetic-field perturbation is $ \epsilon \text{Re} (\boldsymbol{b} )$ . This equation corresponds to (3.11) multiplied by $2\omega ^{-1}$ . In the diffusionless limit $L_\eta =0$ , the solution $b^{(p)}_0$ of (5.1) is readily obtained (Appendix B.1). The induced radial field $b_r=b_r^{(p)}+b_r^{(h)}$ is written as the sum of a particular solution $b_r^{(p)}$ and a homogeneous solution $b_r^{(h)}$ satisfying $ L_\eta ^{2} \partial _r^2 b_r=2 \mathrm{i} b_r$ . The homogeneous solution gives rise to a magnetic skin layer of typical thickness $L_\eta$ , ensuring the electromagnetic interface conditions at $r=a_s$ or $r=a_m$ . In the illustrative case, the solution of (5.1) reads

(5.2) \begin{eqnarray} b_r^{(h)}&=&\tilde {A} \mathrm{e}^{\lambda _{\mathfrak{s}} r}+\tilde {B} \mathrm{e}^{-\lambda _{\mathfrak{s}} r}, \end{eqnarray}

(5.3) \begin{eqnarray} b_r^{(p)} &=& - \frac {B_0 a_s^3 }{2 L_\eta ^4} \frac {3 \cos ^2 \theta -1}{1-a^3} (\mathrm{Ei}_1(\lambda _{\mathfrak{s}} r) \mathrm{e}^{\lambda _{\mathfrak{s}} r}+\mathrm{Ei}_1(-\lambda _{\mathfrak{s}} r) \mathrm{e}^{-\lambda _{\mathfrak{s}} r} + 2 (\lambda _{\mathfrak{s}} r)^{-2}) , \end{eqnarray}

with the (complex) prefactors $(\tilde {A},\tilde {B})$ , the exponential integral $\mathrm{Ei}_1(x)$ and the radial wavenumber $\lambda _{\mathfrak{s}}=(1+\mathrm{i})/L_\eta$ based on the magnetic skin depth $L_\eta$ . Outside the boundary layer, we recover the diffusionless bulk solution (B4) with (5.3) by considering the fifth-order series expansion $\mathrm{Ei}_1(x) \approx \mathrm{e}^{-x} (x^{-1}-x^{-2} +2x^{-3}-6x^{-4})$ for large $x$ .

In the inner and outer solid domains, the particular solution is zero for an imposed uniform axial field. The solution $b_r=b_r^{(h)}$ is thus of the form (5.2), replacing $L_\eta$ by $\breve {\eta }_{\!j}^{1/2} L_\eta$ . We enforce that the boundary-layer solution $b_r^{(h)}$ does not diverge when moving away from the boundary. This imposes the solution $b_r^{(h)}(r \leqslant a_s)$ in the inner solid domain, and the fluid solution $b_r^{(h)}(r \geqslant a_s)$ near $r=a_s$ , to be of the form

(5.4a,b) \begin{equation} b_r(r\leqslant a_s) = \tilde {A}_+ \mathrm{e}^{ \lambda _{\mathfrak{s}} \breve {\eta }_s^{-1/2} \, (r-a_s)} , \quad b_r^{(h)}(r\geqslant a_s) = \tilde {A}_- \mathrm{e}^{-\lambda _{\mathfrak{s}} (r-a_s)} , \end{equation}

with two constants $\tilde {A}_\pm$ . Similarly, the fluid solution $b_r^{(h)}(r \leqslant a_m)$ near $r=a_m$ , and the solution $b_r^{(h)}(r \geqslant a_m)$ in the outer solid domain, must take the form

(5.5a,b) \begin{equation} b_r^{(h)}(r \leqslant a_m) = \tilde {B}_+ \mathrm{e}^{\lambda _{\mathfrak{s}} (r-a_m)} , \quad b_r(r\geqslant a_m) = \tilde {B}_- \mathrm{e}^{-\lambda _{\mathfrak{s}} \breve {\eta }_m^{-1/2} \, (r-a_m)} , \end{equation}

with two constants $\tilde {B}_\pm$ . Using the interface conditions, the four constants can then be analytically obtained (Appendix D.1).

Unless stated otherwise, comparisons with XSHELLS DNS are performed for a confined configuration ( $a = 0.7$ ), identical magnetic diffusivity in the inner sphere and fluid ( $\breve {\eta }_s = 1$ ) and a solid exterior with $\breve {\eta }_m=10$ . Magnetic permeability is uniform across the three domains ( $\breve {\mu }_s=\breve {\mu }_m=1$ ). A uniform axial magnetic field of amplitude $B_0$ is imposed, with $\varLambda _z=B_{0}^2/\omega$ . The fluid magnetic skin depth and magnetic Prandtl number are $L_\eta =1.9\times 10^{-2}$ and $P_m=10^{-1}$ , respectively, ensuring a clear separation of scales. We consider radial profiles at $\theta = 30^\circ$ and $\phi = 0$ . The DNS are shown in the converged regime at $t=(\tau _s+0.35)T_s$ for polar forcing and $t=(\tau _s+0.45)T_s$ for equatorial forcing, where $\tau _s$ is chosen such that $\tau _s T_s\gt 10P_m^{-1}$ , i.e. after more than 10 viscous time scales.

Figure 7 shows excellent agreement between the analytical solution for $b_r$ (dark-red dashed line) and the DNS (orange-dotted line) for both polar (panel a) and equatorial (panel b) modes at weak rotation, with $\gamma =0.1$ and $\gamma =10^{-3}$ , respectively; a similar agreement is obtained in the non-rotating case (not shown). Here and throughout, comparisons are performed using the magnetic perturbation $\Delta \boldsymbol{B}=(\boldsymbol{B}-\boldsymbol{B}_0)\epsilon ^{-1}$ . The interior radial field is accurately captured by the diffusionless correction ((B4); dark-blue dash–dotted line; bottom-left inset). Therefore, it provides an effective particular solution, as shown by the sum of the homogeneous inviscid solution and diffusionless correction (cyan dash–dotted line). The magnetic skin layer extends over a few $L_\eta$ in both the inner sphere ( $\breve {\eta }_s=1$ ) and the fluid, and is thicker in the outer solid ( $\breve {\eta }_m=10$ ). The small discrepancies visible within the fluid boundary layer arise from higher-order radial-field corrections driven by the tangential boundary-layer components. Larger deviations are expected at higher $\gamma$ , where magnetic induction by the rotation-modified basic flow yields a different particular solution.

Figure 7.

Radial part of the magnetic perturbation $\Delta \boldsymbol{B}=(\boldsymbol{B}-\boldsymbol{B}_0)\epsilon ^{-1}$ near the inner (left) and outer (right) boundaries: XSHELLS (orange-dotted line), induced field (dark-red dashed line), induced field with diffusionless correction (cyan dash–dotted line) and diffusionless bulk solution (blue dash–dotted line, (B4)). (a) Polar and (b) equatorial forcing. Parameters: $\varLambda _z=0.667$ , $\varLambda _l=0.5$ , $L_\eta =1.9\times 10^{-2}$ , $P_m=0.1$ , $a=0.7$ , $a_e/a_m=1.1$ , $\breve {\eta }_s=1, \breve {\eta }_m=10$ , $\breve {\mu }_m=\breve {\mu }_s=1$ , $\theta =30^{\circ }$ , $\phi =0$ , polar $\gamma =0.1$ , equatorial $\gamma =10^{-3}$ . Insets: bulk perturbation.

5.2. Tangential components in the inviscid MHD boundary layer

For rapid oscillations ( $\gamma \ll 1$ ), we extend the study to cases where the Coriolis force becomes negligible while the magnetic induction from the rotation-corrected basic flow is not. This occurs when $\gamma ^2$ is small compared with $\varLambda$ , introduced by BG95 to estimate the magnetic-to-inertial force ratio (their equation (17)). For $\gamma ^2 \ll \varLambda$ and an inviscid fluid in a uniform axial field, (3.6) and (3.7) become

(5.6) \begin{eqnarray} B_{0r} \partial _r \boldsymbol{b}_\parallel &=& \mathrm{i} \omega \boldsymbol{u}_\parallel + \boldsymbol{\nabla }_{{\kern-1pt}\parallel} p , \end{eqnarray}

(5.7) \begin{eqnarray} L_\eta ^2 \partial ^2_r \boldsymbol{b}_\parallel &=& 2 \mathrm{i} \boldsymbol{b}_\parallel - \frac {2 B_{0r}}{\omega } \partial _r \boldsymbol{u}_\parallel -\frac {6 B_{0\theta } a_s^3 }{r^4 (1-a^3)} \begin{bmatrix} \mathrm{i} \cos \theta \\[3pt] (1-5 \cos ^2 \theta ) \gamma / 2 \end{bmatrix} \!, \end{eqnarray}

using the rotation-modified polar base flow (4.1)−(4.5) and magnetic skin depth $L_\eta$ . From (5.7), one may test whether $B_{0r}\partial _r\boldsymbol{U}$ dominates the forcing $\mathcal{T}$ in (3.8). Neglecting rotation and considering the boundary-layer tangential components only, we obtain $\mathcal{T}=-\boldsymbol{B}_0(\partial _r U_r)$ since $B_{0r}\partial _r\boldsymbol{U}=-\boldsymbol{B}_0(\partial _r U_r)/2$ is cancelled by $r^{-1}(B_{0r}\boldsymbol{U}-U_r\boldsymbol{B}_0)=\boldsymbol{B}_0(\partial _r U_r)/2$ .

Although a diffusionless solution of (5.6) and (5.7) can be derived for weak Lorentz forces (Appendix B.1), the general solution is written as $[\boldsymbol{u}_\parallel ,\boldsymbol{b}_\parallel ]=[\boldsymbol{u}_\parallel ^{(p)},\boldsymbol{b}_\parallel ^{(p)}]+[\boldsymbol{u}_\parallel ^{(h)},\boldsymbol{b}_\parallel ^{(h)}]$ , i.e. the sum of a particular solution and the homogeneous solution. The latter, obtained by discarding the last term in (5.7), is required to satisfy the boundary conditions. Integration of (5.6) and (5.7) then yields lengthy expressions for $[\boldsymbol{u}_\parallel ^{(p)},\boldsymbol{b}_\parallel ^{(p)}]$ in the inviscid fluid, e.g.

(5.8) \begin{eqnarray} b_\theta ^{(p)} = \frac {L_\eta ^2r^{-2}(\varLambda _l+\mathrm{i})+ (\mathrm{i} \pi -2\, \mathrm{Shi}(\lambda r)) \mathrm{e}^{\lambda r}-(\mathrm{e}^{\lambda r}+\mathrm{e}^{-\lambda r}) \, \mathrm{Ei}_1(-\lambda r)}{B_{0r}^{-1} a_s^{-3} (1-a^3)(1-\mathrm{i} \varLambda _l)^2 L_\eta ^4} \, \sin \theta ,\end{eqnarray}

where the radial wavenumber $\lambda =\lambda _{\mathfrak{s}}(1-\mathrm{i} \varLambda _l)^{-1/2}$ reduces to the magnetic skin-layer wavenumber $\lambda _{\mathfrak{s}}$ for $\varLambda _l \ll 1$ . Here $\mathrm{Shi}(x)=\int _0^x t^{-1}\sinh (t)\,\mathrm{d}t$ denotes the hyperbolic sine integral and $\varLambda _l=B_{0r}^2/\omega =\sigma _{\!f} \tilde {B}_{0r}^2/( \rho _{\!f} \omega _{\textit{s}})$ is the local parameter based on $B_{0r}$ (dimensional counterpart $\tilde {B}_{0r}$ ). This parameter also appears in the BG95 model, where it governs the transition from a skin-depth regime to one dominated by Alfvén-wave radiation. For $\varLambda _l \ll 1$ , the diffusionless approximation $[\boldsymbol{u}^{(p)},\boldsymbol{b}^{(p)}]\approx [\boldsymbol{u}^{(p)}_0,\boldsymbol{b}^{(p)}_0]$ follows as simple analytical expressions for weak Lorentz forces (Appendix B).

From the homogeneous part of (5.6) and (5.7), the solutions $\boldsymbol{b}_\parallel ^{(h)}$ and $\boldsymbol{u}_\parallel ^{(h)}$ in the fluid are

(5.9a,b,c) \begin{equation} \boldsymbol{b}_\parallel ^{(h)}=\begin{bmatrix} A_\pm \\[3pt] B_\pm \end{bmatrix} \mathrm{e}^{\pm \lambda \tilde {r}_\pm } , \, \boldsymbol{u}_\parallel ^{(h)}= \pm \frac { \omega \varLambda _l \lambda }{2 \mathrm{i} B_{0r}} L_\eta ^2 \boldsymbol{b}_\parallel ^{(h)} , \, \boldsymbol{b}_\parallel ^{(s)}= \begin{bmatrix} C_\pm \\[3pt] D_\pm \end{bmatrix} \mathrm{e}^{\mp \breve {\eta }^{-1/2} \, \lambda _{\mathfrak{s}} \tilde {r}_\pm } , \end{equation}

discarding exponentially growing terms, where $\boldsymbol{b}_\parallel ^{(s)}$ is the solution in the solids. At $r\ge a_s$ , the fluid solution is governed by $(A_-,B_-)$ with $\tilde r_-=r-a_s$ , while it is described by $(A_+,B_+)$ at $r\le a_m$ , with $\tilde r_+=r-a_m$ . In the inner and outer solids, $\boldsymbol{b}_\parallel ^{(s)}$ involves $(C_-,D_-)$ and $(C_+,D_+)$ , respectively. The boundary-layer thicknesses in the solids are $\breve {\eta }_s^{1/2}\lambda _{\mathfrak{s}}^{-1}$ and $\breve {\eta }_m^{1/2}\lambda _{\mathfrak{s}}^{-1}$ , while $\lambda =\lambda _{\mathfrak{s}}(1-\mathrm{i}\varLambda _l)^{-1/2}$ sets the fluid thickness $\lambda ^{-1}\breve {\eta }^{1/2}$ ; the flow contribution vanishes as $\varLambda _l\to 0$ . For $\varLambda _l\gg 1$ , $\lambda$ becomes $\lambda \approx \sqrt {2}(\varLambda _l^{-3/2}/2+\mathrm{i}\varLambda _l^{-1/2})L_\eta ^{-1}$ , which recovers the Alfvén phase velocity $\omega /\mathrm{Im}(\lambda )=\tilde {B}_{0r}/\sqrt {\rho _{\!f}\mu _{\!f}}=V_A$ and an attenuation length $L_A=\textit{Re}(\lambda )^{-1}=\sqrt {2}L_\eta \varLambda _l^{3/2}\gg L_\eta$ , much larger than the skin depth $L_\eta$ (as noted by BG95). The inviscid MHD solution therefore transitions from a magnetic skin layer at $\varLambda _l\ll 1$ to an attenuated Alfvén-wave regime. Our BLT approach requires small $L_A$ compared with the available propagation distance $D$ , and absence of significant reflection, or boundary-layer interaction, on the opposite boundary (and, thus, absence of global resonance).

The prefactors of (5.9) are analytically obtained from the interface conditions (Appendix D.2); their complete symbolic expressions are available in the accompanying Zenodo repository Personnettaz & Cébron (Reference Personnettaz and Cébron2026). These inviscid MHD boundary-layer solutions (including the particular contribution) are shown as dark-red dashed lines in the following figures (e.g. figures 8 and 9 for velocity and magnetic fields, respectively). The DNS velocity perturbation is computed as a scaled difference with respect to the potential base flow, $\Delta \boldsymbol{V}=\boldsymbol{V}\epsilon ^{-1}-\text{Re}(\boldsymbol{U}_{\!1}\exp {(\mathrm{i}(\omega t+m\phi ))})$ . Without rotation, polar forcing produces vanishing azimuthal components in both theory and DNS, so only the $\theta$ component is considered. The inviscid MHD solution (dark-red dashed line) agrees with DNS (orange-dotted line) in both skin and Alfvén-wave regimes (figure 9 at $\varLambda _l=2$ and figure 14 at $\varLambda _l=0.5$ ), except near interfaces where viscosity dominates within nested boundary layers (see insets), leading to slope changes absent from the inviscid model. In the Alfvén-wave regime, perturbations penetrate much deeper in the fluid (right insets showing the full liquid core) and exhibit wave-like structures (figure 9) compared with those in the skin-layer regime. In all cases, the diffusionless induced correction ( $b^{(p)}_{0}$ , $u^{(p)}_{0}$ ) is an efficient substitute for the inviscid MHD particular solution, as shown by the cyan dash–dotted line.

Larger discrepancies arise for velocity, especially near solid boundaries (figure 8), where viscous effects dominate. For large $\varLambda _l$ , the velocity perturbation follows the inviscid MHD solution outside viscous layers, notably in the Alfvén-wave regime, whereas viscosity leads to marked deviations for small $\varLambda _l$ (see figure S5 of the supplementary material). Profiles are shown at co-latitude $\theta =30^{\circ }$ ; agreement improves toward the poles where local $\varLambda _l$ is larger and deteriorates at the equator where $\varLambda _l=0$ . The above considerations are valid for both polar and equatorial modes, respectively indicated by panels (a) and (b) in the figures discussed.

Figure 8.

Tangential components of flow perturbation $\Delta \boldsymbol{V}=\boldsymbol{V}\epsilon ^{-1}-\textit{Re}(\boldsymbol{U}_{\!1}\exp {(\mathrm{i}(\omega t+m\phi ))})$ near the inner (left) and outer (right) boundaries: XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line) and purely viscous (purple dashed line) solutions. Insets show bulk perturbations, including secondary bulk flows: inviscid MHD (red dashed), viscomagnetic oscillatory (dark-green line) and purely viscous (purple dash–dotted line); viscomagnetic oscillatory boundary layer (thick green line). (a) Polar and (b) equatorial forcing. Parameters: as in figure 7, but $\varLambda _z=2.67$ , $\varLambda _l=2$ , $\gamma =0$ .

Figure 9.

Tangential magnetic-field perturbation near the inner (left) and outer (right) boundaries: XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line). Insets on the left show profiles close to the interfaces. (a) Polar and (b) equatorial forcing. Parameters: as in figure 7, but $\varLambda _z=2.67$ , $\varLambda _l=2$ , $\gamma =0$ .

5.3. Higher-order bulk fields driven by boundary-layer radial flow

The higher-order correction $\mathfrak{u}_r$ is obtained near both boundaries from $\boldsymbol{u}_\parallel$ using the BLT (3.13). Discarding the particular solution and substituting $r\partial _r$ with $a_s \partial _r$ in (3.13) yields, in the fluid at $r=a_s$ , and noting that $\kappa _\lambda =\lambda ^{-1}(r-a_s) \mp \lambda ^{-2} \approx \mp \lambda ^{-2}$ ,

(5.10) \begin{eqnarray} \mathfrak{u}_r &=& - \frac { \omega \varLambda _l}{2 \mathrm{i} B_{0r} } L_\eta ^2 \frac {A_- (\cot \theta -\kappa _\lambda \partial _\theta \lambda ) +\partial _\theta A_- +\mathrm{i} m B_- / \!\sin \theta }{a_s} \, \mathrm{e}^{- \lambda \tilde {r}_- }. \end{eqnarray}

Within the huBL framework, the $\theta$ dependence of $B_{0r}$ , and hence, of $\varLambda _l$ and $\lambda$ , is neglected in the $\theta$ derivatives of (3.13) and (5.10). Using the basic flow (4.1), we obtain

(5.11) \begin{eqnarray} \frac {\mathfrak{u}_r}{\mathrm{e}^{- \lambda (r-a_s)}}&=& \frac {\mathfrak{u}_r^{(o)}}{-a^4 \mathrm{e}^{- \lambda (a_m-r)}}= - \frac {3}{2} \frac { 1-\mathrm{i}}{\breve {c}_{\!j}+\sqrt {1-\mathrm{i} \varLambda _l}} \frac {L_\eta \omega \varLambda _l}{a_s} \frac {\mathfrak{c}}{1-a^3} , \end{eqnarray}

noting that material properties contrast $\breve {c}_{\!j}=\breve {\eta }_{\!j}^{1/2} \breve {\mu }_{\!j}$ , and $\mathfrak{c}=\cos \theta$ for the polar mode and $\mathfrak{c}=\sin \theta$ for the equatorial modes.

Figure 10 compares the numerical radial velocity perturbation, $\Delta V_r=V_r\epsilon ^{-1}-\text{Re}(U_{1r}\exp {(\mathrm{i}(\omega t+m\phi ))})$ (orange-dotted lines), with the higher-order MHD inviscid solution $\mathfrak{u}_r$ (dark-red dashed lines). The asymptotic solution captures the spatial structure of the perturbation, reproducing both the boundary-layer behaviour and the bulk wavelength, but it underestimates the wave amplitude. This limitation is consistent with the comparatively weak radial velocity, which is more sensitive than the tangential components to higher-order effects neglected in the present approximation. As shown below, the inclusion of viscosity significantly improves the agreement between analytical predictions and DNS.

Through the boundary conditions, the radial boundary-layer fluxes (5.11) drive bulk fields $( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{B}})$ , whose Ohmic dissipation is subsequently evaluated using power arguments (§ 5.5). An explicit expression of $(\boldsymbol{\mathcal{U}},\mathcal{P})$ and $(\boldsymbol{\mathcal{U}}^{(o)},\mathcal{P}^{(o)})$ can be obtained in the limit of weak Lorentz forces. By searching for irrotational flows of velocity potentials $(\varPsi ,\varPsi ^{(o)})$ , with

(5.12) \begin{eqnarray} \mathfrak{u}_r+\mathcal{U}_r|_{(r=a_s)}=\mathcal{U}_r|_{(r=a_m)}= \mathfrak{u}_r^{(o)}+\mathcal{U}_r^{(o)}|_{(r=a_m)}=\mathcal{U}_r^{(o)}|_{(r=a_s)} =0 , \end{eqnarray}

the method used in § 4.1 for $\boldsymbol{U}$ shows that the pressure driven by the inner-boundary layer is

(5.13) \begin{eqnarray} \mathcal{P}&=&- \mathrm{i} \omega \varPsi = \frac {3}{4} \frac {1+\mathrm{i}}{(1-a^3)^2} \frac {\omega ^2 \varLambda _l L_\eta a_s^2 (1+2 r^3/a_m^3) }{\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l}} \frac {\cos \theta }{r^2 } , \end{eqnarray}

and the associated bulk secondary flow

(5.14) \begin{equation} \frac {\boldsymbol{\mathcal{U}}}{ \mathfrak{u}_r (r=a_s)} = - \frac {(a_s/r)^3}{1-a^3} \left [\left (1-\frac {r^3}{a_m^3} \right ) \hat {\boldsymbol{r}} + \left (1+2\frac {r^3}{a_m^3} \right ) \frac {\tan (\theta )}{2} \hat {\boldsymbol{\theta }} +3 \mathrm{i} \frac {\gamma }{2} \sin (\theta ) \hat {\boldsymbol{\phi }} \right ] \end{equation}

for the polar mode, using the weak-rotation-induced correction $\mathcal{U}_\phi / \mathcal{U}_s= \mathrm{i} \gamma$ . For the equatorial modes, $\mathcal{P}$ is given by (5.13) where $\cos \theta$ is replaced by $\sin \theta$ , and

(5.15) \begin{equation} \boldsymbol{\mathcal{U}} = - \frac {3}{4} \frac {1-\mathrm{i}}{(1-a^3)^2} \frac {a_s^2 L_\eta \omega \varLambda _l r^{-3}}{\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l}} \left [- 2 \left (1-\frac {r^3}{a_m^3} \right )\sin (\theta ) \hat {\boldsymbol{r}}+\left (1+2\frac {r^3}{a_m^3} \right ) \boldsymbol{t}_\parallel \right ] \!, \quad \end{equation}

with $\boldsymbol{t}_\parallel =\cos (\theta ) \hat {\boldsymbol{\theta }} + \mathrm{i} m \hat {\boldsymbol{\phi }}$ . For all modes, $\mathfrak{u}_r^{(o)}$ drives $(\mathcal{P}^{(o)},\boldsymbol{\mathcal{U}}^{(o)})$ , which can be obtained by multiplying $\mathcal{P}$ and $\boldsymbol{\mathcal{U}}$ by $a^4$ , and by replacing $\breve {c}_s$ and $a_m$ by $\breve {c}_m$ and $a_s$ , respectively.

Figure 10.

Radial flow perturbation $\Delta V_r$ near the inner (left) and outer (right) boundaries: XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line) and purely viscous (purple dashed line). Viscomagnetic oscillatory (dark green line) and purely viscous (purple dash–dotted line) perturbation with secondary bulk flow. (a) Polar and (b) equatorial forcing. Parameters: as in figure 7, but $\varLambda _z=2.67$ , $\varLambda _l=2$ , $\gamma =0$ . Insets show the bulk perturbation.

From the secondary bulk flow $\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)}$ , we can then obtain the bulk magnetic correction $\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{B}}^{(o)}$ from (3.15) by using $\boldsymbol{B}_0=B_0 \hat {\boldsymbol{z}}$ . For the polar mode, we obtain

(5.16) \begin{equation} \frac {\boldsymbol{\mathcal{B}}}{B_0} = \frac {9}{4} \frac {1+\mathrm{i}}{(1-a^3)^2} \frac {\varLambda _l L_\eta a_s^2 }{\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l}} \frac {(3\cos ^2 \theta -1) \hat {\boldsymbol{r}}+\sin ( 2 \theta )\hat {\boldsymbol{\theta }}+ \varXi \hat {\boldsymbol{\phi }} }{r^4} , \end{equation}

with the rotation-induced term $ \varXi =\mathrm{i} \gamma (5 \cos ^2 \theta -1) \sin \theta$ , and we obtain

(5.17) \begin{equation} \frac {\boldsymbol{\mathcal{B}}}{B_0} = \frac {9}{4} \frac {1+\mathrm{i}}{(1-a^3)^2} \frac {\varLambda _l L_\eta a_s^2}{\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l}} \left [ \frac {3 \sin 2\theta }{2 r^4} \hat {\boldsymbol{r}}+\frac {(1-2 \cos ^2 \theta )\hat {\boldsymbol{\theta }}- \mathrm{i} m \cos (\theta ) \hat {\boldsymbol{\phi }}}{r^4} \right ] \end{equation}

for the equatorial modes. The field $\boldsymbol{\mathcal{B}}^{(o)}$ can then be obtained by multiplying (5.16) and (5.17) by $a^4$ , and by replacing $\breve {c}_s$ by $\breve {c}_m$ . Magnetic corrections are thus negligible, preventing a meaningful comparison with DNS. The secondary flow forced by the boundary layer is also negligible, becoming significant in the bulk only when viscosity is included (see below).

As anticipated in § 3.5, the secondary bulk fields $(\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)},\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{B}}^{(o)})$ do not in general satisfy the tangential boundary conditions, whose enforcement determines the final boundary-layer corrections $(\boldsymbol{\mathfrak{u}}_\parallel ,\boldsymbol{\mathfrak{b}}_\parallel )$ . These boundary-layer solutions are of the form (5.9), using the prefactors (D3) where $ \mathfrak{h}=0$ since the boundary velocity has already been accounted for (at previous order). In these expressions, the velocity is only dependent on the bulk flow (and not on the bulk magnetic field). They can thus be used if $\boldsymbol{U}$ is replaced by $\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)}$ , giving

(5.18) \begin{align} \frac {\boldsymbol{\mathfrak{u}}_\parallel }{ \mathrm{e}^{- \lambda (r-a_s)}} &= \frac {3}{4} \frac {(1+\mathrm{i}) \omega \varLambda _l^2}{(1-a^3)^2} \frac {L_\eta \sin ( \theta )}{a_s} \frac {(1+2 a^3+3a^4 ) \hat {\boldsymbol{\theta }}+3 \mathrm{i} \gamma (1+a^4) \cos \theta \hat {\boldsymbol{\phi }}}{(\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l})^2} , \\[-12pt]\nonumber \end{align}

(5.19) \begin{align} \frac {\boldsymbol{\mathfrak{u}}_\parallel ^{(o)}}{a^3 \mathrm{e}^{\lambda (r-a_m)} } &= \frac {3}{4} \frac {(1+\mathrm{i}) \omega \varLambda _l^2}{(1-a^3)^2} \frac {L_\eta \sin ( \theta )}{a_s} \frac {(3+2 a+a^4 ) \hat {\boldsymbol{\theta }}+3 \mathrm{i} \gamma (1+a^4) \cos \theta \hat {\boldsymbol{\phi }}}{(\breve {c}_m+\sqrt {1-\mathrm{i} \varLambda _l})^2} , \end{align}

for the polar mode, while equatorial modes lead to

(5.20) \begin{align} \frac {\boldsymbol{\mathfrak{u}}_\parallel }{ \mathrm{e}^{- \lambda (r-a_s)}} &= -\frac {3}{4} \frac {(1+\mathrm{i}) \omega \varLambda _l^2}{(1-a^3)^2} \frac {L_\eta }{a_s} \frac {(1+2 a^3+3a^4 ) (\cos ( \theta ) \hat {\boldsymbol{\theta }}+ \mathrm{i} m \hat {\boldsymbol{\phi }})}{(\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda _l})^2} , \\[-12pt]\nonumber \end{align}

(5.21) \begin{align} \frac {\boldsymbol{\mathfrak{u}}_\parallel ^{(o)}}{a^3 \mathrm{e}^{\lambda (r-a_m)} } &= - \frac {3}{4} \frac {(1+\mathrm{i}) \omega \varLambda _l^2}{(1-a^3)^2} \frac {L_\eta }{a_s} \frac {(3+2 a+a^4 )( \cos (\theta ) \hat {\boldsymbol{\theta }}+ \mathrm{i} m\hat {\boldsymbol{\phi }})}{(\breve {c}_m+\sqrt {1-\mathrm{i} \varLambda _l})^2} . \end{align}

For all modes, these flows are associated with the boundary-layer magnetic fields $[\boldsymbol{\mathfrak{b}}_\parallel ,\boldsymbol{\mathfrak{b}}_\parallel ^{(o)}]=2 \mathrm{i} B_{0r}(\omega \varLambda _l \lambda L_\eta ^2)^{-1} [-\boldsymbol{\mathfrak{u}}_\parallel ,\boldsymbol{\mathfrak{u}}_\parallel ^{(o)}]$ , given by the induction equation and arising from the interaction of $\boldsymbol{B}_0$ with $\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)}$ . Interactions of $\boldsymbol{\mathcal{B}}$ with $\boldsymbol{U}$ or $\boldsymbol{\mathcal{U}}+\boldsymbol{\mathcal{U}}^{(o)}$ are of order $2$ or higher in $\epsilon$ and are thus neglected. In (5.18)–(5.21), terms in $a^4$ or higher originate from the $\boldsymbol{\mathcal{U}}^{(o)}$ contribution. These corrections are very small, and thus, very difficult to benchmark against the DNS results.

5.4. Magnetic-tension drag from the inviscid boundary layer

Integrating over the inner and outer boundaries yields the magnetic-tension force $\boldsymbol{F}_{\!mt}$ and detuning $\Delta \omega _{mt}$ . Neglecting the particular solution, $\Delta \omega _{mt}$ for the polar mode follows from (D5). For an arbitrary axisymmetric imposed field within the BLT approximation, it depends only on the radial component $B_{0r}$ and reads

(5.22a,b) \begin{equation} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s}= \frac {9}{16} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {\mathcal{J}_s}{1-a^3} , \quad \mathcal{J}_{\!j}=-\int _{0}^\pi \frac { (1+\mathrm{i}) \varLambda _l (\hat {\boldsymbol{z}}\boldsymbol{\cdot }\hat {\boldsymbol{\theta }} ) \sin ^2 \theta }{ \breve {c}_{\!j} +\sqrt {1-\mathrm{i}\varLambda _l} } \mathrm{d} \theta , \end{equation}

with $\hat {\boldsymbol{z}}\boldsymbol{\cdot }\hat {\boldsymbol{\theta }}=-\sin \theta$ from the projection $\boldsymbol{f}_{\!mt}\boldsymbol{\cdot }\hat {\boldsymbol{z}}$ . In the huBL approximation, $\varLambda _l$ is constant, while it varies as $\varLambda _l=\varLambda _z\cos ^2\theta$ for an axial field, with $\varLambda _z=B_{0}^2/\omega$ . An insulating solid corresponds to $\breve {c}_{\!j}\gg 1$ , which reduces the imaginary part of the detuning for all $\varLambda _z$ relative to uniform electrical properties (figure 11). In the opposite limit of negligible $\breve {c}_{\!j}$ , the integral (5.22) saturates to a curve independent of the electrical properties ratio.

Figure 11.

Role of $\breve {c}_j$ and of $\varLambda _z$ on the real (a) and imaginary (b) parts of the proxy $\mathcal{J}_{\!j} \varLambda _z^{-1}$ of the magnetic tension. For $\breve {c}_{\!j}=1$ : (5.24) in black, with its asymptotic formula for $\varLambda _z \ll 1$ (red line) and $\varLambda _z \gg 1$ (orange line), compared with (5.25) with $\varLambda = \varLambda _z / 3$ (grey dashed line) and including the field-geometry correction factor $\mathcal{I}_{mt}^{\,\textit{pol}}=3/5$ (pink dashed line; defined in § 5.4, (5.26)). The dots in (b) use the same correction factor $\mathcal{I}_{mt}^{\,\textit{pol}}=3/5$ . For $\text{Re}({\mathcal{J}_s})$ , the analogous correction is $\mathcal{R}_{mt}^{\textit{pol}}=3/5$ for $\breve {c}_{\!j}\gg 1$ or $\varLambda _z \ll 1$ (triangles) and $\mathcal{R}_{mt}^{\textit{pol}}=\sqrt {3} (\log (4 \varLambda _z) - 2)/ 4$ for $\breve {c}_{\!j} \ll 1$ and $\varLambda _z \gg 1$ (circles).

The detuning associated with the outer-boundary layer is also given by (5.22), replacing $\mathcal{J}_s$ by $a\mathcal{J}_m$ . The real part of this detuning provides the non-dissipative part of the magnetic-tension force on the outer boundary, and its imaginary part is a dissipation source for the inner-sphere dynamics. The total dissipation is thus

(5.23) \begin{eqnarray} \frac { \mathrm{Im} \left (\Delta \omega _{mt}\right )}{\omega } = \mathrm{Im} \left ( \frac {9}{16} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {\mathcal{J}_s+a \mathcal{J}_m}{1-a^3} \right )\!. \end{eqnarray}

For a uniform axial field, $\varLambda _z$ is constant and the integration can be analytically performed for $\breve {c}_{\!j}=1$ . We have $\mathcal{J}_s=\mathcal{J}_m=\mathcal{J}_1$ and, noting that $\chi =\sqrt {1- \mathrm{i} \varLambda _z}$ ,

(5.24) \begin{equation} \mathcal{J}_1 = \frac {(1+\mathrm{i})\sqrt {\mathrm{-i}}}{12 \varLambda _z^{3/2}} \Big[ (16 -6\chi ) \sqrt {-\mathrm{i} \varLambda _z^{3}}+3 \chi \sqrt {\mathrm{i} \varLambda _z} -(3-12 \mathrm{i}\varLambda _z) \tan ^{-1}( \chi ^{-1} \sqrt {\mathrm{i} \varLambda _z} ) \Big] \end{equation}

shown in black in figure 11. We thus have $\mathcal{J}_1 \approx 2 (1+\mathrm{i}) \varLambda _z/15$ for $\varLambda _z \ll 1$ and $\mathcal{J}_1 \approx 4/3+\mathrm{i} \sqrt {\varLambda _z/2}$ for $\varLambda _z \gg 1$ (red and orange lines in figure 11, respectively).

Since evaluating the exact solution (5.23) requires the field geometry, BG95 rather used the approximate huBL approach where $B_{0r}$ is replaced by a typical constant value at the boundary $r=a_s$ . The local parameter $\varLambda _l$ , based on $B_{0r}$ is then replaced by its surface average $\varLambda = \langle \varLambda _l \rangle$ at $r=a_s$ . Note that the huBL result is exact when the radial imposed field is constant at $r=a_s$ . For the polar mode ( $\boldsymbol{f}_{\!mt}$ along $\hat {\boldsymbol{z}}$ ), the estimate $\mathcal{J}_s \approx \mathcal{J}_s^{\,\textit{huBL}}$ reads

(5.25a,b) \begin{equation} \frac {\mathcal{J}_s^{\,\textit{huBL}}}{ \varLambda } = \frac {4}{3} \frac {1+\mathrm{i}}{\breve {c}_s +\sqrt {1-\mathrm{i}\varLambda } } , \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s}= \frac {3 \varLambda }{4} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\mathrm{i}}{ \breve {c}_s +\sqrt {1-\mathrm{i}\varLambda } } \frac {1+\alpha \gamma ^2}{1-a^3} , \end{equation}

where $\alpha =0$ for the flow $\boldsymbol{U}_{\!1}$ given by (4.5), but re-doing the calculation gives $\alpha =-6a^3(1+a) (1+a+a^2)^{-1}/5$ for the flow $\boldsymbol{U}_2$ given by (A28). Since $L_\eta ^2/a_s^2= 2/(P_m\,\mathrm{Wo}^2)$ , the magnetic skin depth is related to the Womersley number and the magnetic Prandtl number.

Equation (5.25) gives the magnetic-tension correction to $C_a$ as $C_a^i=C_a^i(\varLambda =0)-2 (\rho _s/\rho _{\!f})\text{Re}(\Delta \omega _{mt}/\omega )$ in the inviscid limit. Magnetic tension reduces the effective inertia of the inner solid domain, increasing the frequency of free oscillations (BG95). The influence of the particular solution on the magnetic tension in (5.24) or (5.25) can be assessed analytically using the diffusionless solution (B4) for a uniform axial field and weak Lorentz forces. In the BLT limit $L_\eta \ll a_s$ , (5.24) is then multiplied by $1+(1-\mathrm{i}) L_\eta /a_s$ , showing that the particular solution introduces only a higher-order correction of order $(L_\eta /a_s)^2$ .

Equations (D5) and (5.25) extend equations (26), (27) and (32) of BG95 to rotation-perturbed, bounded flows around an inner sphere with magnetic diffusivity and permeability. Equation (5.25) shows that jumps in $\eta$ and $\mu$ are encapsulated through the ratio $\mu \eta ^{1/2}$ across the boundary (unlike when the particular solution is included). In the limit $\mu _s \gg \mu _{\!f}$ of a high-permeability inner sphere, the tangential components of $\boldsymbol{b}$ vanish at the boundary, and hence, $\boldsymbol{f}_{\!mt}=0$ . The insulating limit $\sigma _s \ll \sigma _{\!f}$ also yields vanishing $\boldsymbol{f}_{\!mt}$ , although the BLT approximation breaks down: the BLT approach requires that $\omega a_s^2 \gg \max ( \eta _{\!f},\eta _s)$ to ensure thin boundary layers. In the opposite case $\sigma _s \gg \sigma _{\!f}$ of a highly conducting solid, $\breve {c}_s \to 0$ produces a modified $\boldsymbol{f}_{\!mt}$ relative to the geophysically relevant case $\breve {c}_s \approx 1$ .

For a uniform axial field and varying electromagnetic properties contrast $\breve {c}_s$ , figure 11 shows how the field strength ( $\varLambda _z$ ) controls the magnetic-tension contribution to the added-mass coefficient (panel a, via $\text{Re}(\mathcal{J}_{\!j})$ ) and to dissipation (panel b, via $\mathrm{Im}(\mathcal{J}_{\!j})$ ). For $\breve {c}_{\!j}=1$ , the exact solution (5.24) (solid black lines) is compared with the huBL estimate (5.25) (grey dashed lines), obtained naively using simply the surface average value $\varLambda =\langle \varLambda _l \rangle =\varLambda _z/3$ for a uniform axial field. Although this averaging captures the overall field intensity, it does not reproduce the weaker detuning predicted by the exact integration (5.24). A further field-geometry correction factor can be estimated by considering the ratio of the exact formulation of (5.22) and its huBL approximation. The real and imaginary parts of the huBL result should then be corrected by a factor, given by $\mathcal{R}_{mt}^{\textit{pol}}=\text{Re}({\mathcal{J}_s})/\text{Re}({\mathcal{J}_s^{\textit{huBL}}})$ and $\mathcal{I}_{mt}^{\,\textit{pol}}=\mathrm{Im}({\mathcal{J}_s})/\mathrm{Im}({\mathcal{J}_s^{\textit{huBL}}})$ , respectively, for the polar mode. For this forcing, these definitions naturally give $\mathcal{R}_{mt}^{\textit{pol}}=\mathcal{I}_{mt}^{\,\textit{pol}}=1$ for a constant radial field at $r=a_s$ , while an arbitrary field geometry can be handled through

(5.26) \begin{equation} \mathcal{R}_{mt}^{\textit{pol}}(\varLambda \ll 1) = \mathcal{I}_{mt}^{\,\textit{pol}} (\varLambda \ll 1) = \frac {3}{2} \frac {\int _{0}^\pi \varLambda _l(\theta ) \sin ^3 \theta \mathrm{d} \theta }{\int _{0}^\pi \varLambda _l(\theta ) \sin \theta \mathrm{d} \theta } \end{equation}

for $\varLambda \ll 1$ . For a uniform axial field, $\varLambda _l(\theta )=\varLambda _z\cos ^2(\theta )$ , yielding $\mathcal{R}_{mt}^{\textit{pol}}=\mathcal{I}_{mt}^{\,\textit{pol}}=3/5$ in this limit. As shown in figure 11(b), this single value already reproduces $\mathrm{Im}(\mathcal{J}_s)$ almost exactly over the entire range of $\varLambda _z$ and $\breve {c}_{\!j}$ . Moreover, in the opposite asymptotic limit $\varLambda \gg 1$ , we obtain $\mathcal{I}_{mt}^{\,\textit{pol}}=3\sqrt {3}/8\approx 0.65$ , sufficiently close to $3/5$ to justify adopting the approximation $\mathcal{I}_{mt}^{\,\textit{pol}}\approx 3/5$ throughout the remainder of the paper. Although introduced here to account for the magnetic-tension drag, this correction factor also enters the Ohmic dissipation (§§ 5.5, 6.3) and the comparison with the DNS of P26, making the choice $\mathcal{I}_{mt}^{\,\textit{pol}}\approx 3/5$ a convenient and accurate approximation for a uniform axial field.

The real part varies more (figure 11 a): $\mathcal{R}_{mt}^{\textit{pol}}\approx 3/5$ remains a satisfactory approximation for $\breve {c}_{\!j}=1$ and becomes asymptotically exact for $\breve {c}_{\!j}\gg 1$ or $\varLambda _z\ll 1$ , but is instead well captured by $\mathcal{R}_{mt}^{\textit{pol}}\approx \sqrt {3}\,(\ln (4\varLambda _z)-2)/4$ in the remaining regime $\breve {c}_{\!j}\ll 1$ , $\varLambda _z\gg 1$ .

In the limit $\varLambda \to 0$ , (5.25) shows that the magnetic tension is linear in $\varLambda$ , corresponding to the magnetic skin-layer regime (figure 2 of BG95). For large $\varLambda \gg 1$ , Alfvén-wave propagation dominates and (5.25) reduces to

(5.27) \begin{eqnarray} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s}&\approx & \frac {3}{4} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {(1-\mathrm{i}) \breve {c}_s+ \mathrm{i} \, \sqrt {2 \varLambda }}{1-a^3} , \end{eqnarray}

which, with the field-geometry correction, gives the dimensional leading-order sum of the inviscid pressure force and magnetic-tension force

(5.28) \begin{eqnarray} {\boldsymbol{F}_{\!p}}+ {\boldsymbol{F}_{\!mt}} &=& - M_{\!fs} C_a \mathrm{d}_t {\boldsymbol{V}}_{\!s}- 4 \pi \mathcal{I}_{mt}^{\,\textit{pol}} \sqrt {\frac {\rho _{\!f}}{\mu _{\!f}}} \frac { a_s^2 \tilde {B}_0}{1-a^3} {\boldsymbol{V}}_{\!s}, \end{eqnarray}

with

(5.29) \begin{eqnarray} C_a &=& \frac {1}{2} \frac {1+2a^3}{1-a^3}- \frac {3}{2} \frac {\mathcal{R}_{mt}^{\textit{pol}} \breve {c}_s}{1-a^3}\frac {L_\eta }{a_s} , \end{eqnarray}

where $\boldsymbol{F}_{\!p}$ contributes only through the first term in $C_a$ (added mass of the basic flow), and where magnetic diffusion no longer contributes to the leading-order drag. Instead, the oscillating sphere emits propagating Alfvén waves that transport momentum away through the mechanical and electromagnetic energy flux. Electromagnetic drag therefore remains finite although local Ohmic dissipation becomes asymptotically negligible, making this regime directly analogous to classical wave drag. Note also that (5.28) gives only the magnetic-tension contribution to the drag; the power approach of § 5.5 additionally recovers the pressure-mediated contribution and thereby the complete mechanical power budget. In a finite cavity, the Alfvén-wave attenuation length $L_A$ must satisfy $L_A\ll D$ for these formulae to be valid. Otherwise, reflected waves couple the two boundaries and the local radiative law no longer describes the cavity response.

The coefficient $C_a$ of (5.28), originating from the real part of (5.27), shows how the magnetic tension modifies the added mass, yielding a non-zero $\varLambda _l$ -independent limit for the frequency shift, as noted by BG95 and here extended to arbitrary $\breve {c}_s$ and $a$ . Whereas BG95 identified $\varLambda \approx 1$ as the transition between the magnetic skin layer and Alfvén-wave regimes for $\breve {c}_s=1$ , the criterion is rather found to be $\varLambda \approx 2(\breve {c}_s + 1)^2$ , with an additional transition at $\breve {c}_s=1$ in the magnetic-skin-layer regime (§ S7 and figure S12).

In the non-rotating case, the prefactors (D5) extend to equatorial modes by replacing $\sin \theta$ with $-\cos \theta$ . Restoring the azimuthal dependence $\mathrm{e}^{\textrm{i} m \phi }$ and expressing the force $\boldsymbol{F}_{\!mt}$ in the Cartesian basis, the huBL force components along $\hat {\boldsymbol{x}}$ and $\hat {\boldsymbol{y}}$ are $F_{mtz}$ and $\mathrm{i} m F_{mtz}$ , where $\boldsymbol{F}_{\!mt}=F_{mtz}\hat {\boldsymbol{z}}=A_{mtz} \epsilon \mathrm{e}^{\textrm{i}\omega t} \hat {\boldsymbol{z}}$ is the polar-mode force (such that the force is $A_{mtz} \boldsymbol{\delta }$ ). Magnetic torque vanishes for the equatorial modes and for the rotation-perturbed polar mode.

Considering the dissipative component of (5.28), the large- $\varLambda$ asymptotics recover the classical linear dependence on magnetic-field strength observed, for translating spheres, in the inductionless regime of low magnetic Reynolds number $\omega _s\epsilon _sa_s/\eta _{\!f}\ll 1$ and large interaction parameter $N=\varLambda a_s/\epsilon _s\gg 1$ , as established experimentally by Yonas (Reference Yonas1967), Maxworthy (Reference Maxworthy1968a ) and numerically by Sekhar et al. (Reference Sekhar, Sivakumar and Kumar2005). The same scaling also emerges in the complementary ideal-MHD limit, where momentum is transported away by the pair of Alfvén wings generated by a conducting sphere translating steadily across a uniform axial magnetic field (Drell et al. Reference Drell, Foley and Ruderman1965; Neubauer Reference Neubauer1980). Following the classical hydrodynamic definition of the drag coefficient, we define the magnetic drag coefficient $C_d^m$ by replacing the dynamic pressure $\rho _{\!f} \mathcal{V}^2/2$ with its magnetic counterpart $\tilde {B}_0^2/(2\mu _{\!f})=\rho _{\!f} \tilde {V}_A^2/2$ , such that the (dimensional) drag is written as

(5.30) \begin{equation} C_d^m\,\pi a_s^2\mu _{\!f}^{-1}\tilde {B}_0^2/2=C_d^m \pi a_s^2 \rho _{\!f} \tilde {V}_A^2/2 , \end{equation}

with the sphere cross-section area $\pi a_s^2$ . The usual drag coefficient $C_d$ , based on the dynamic pressure, is then $C_d=C_d^m/A_l^2$ , where the Alfvén number $A_l=\mathcal{V}/\tilde {V}_A$ (also known as Alfvén Mach number) reads here as $A_l=(\epsilon /L_\eta ) (2/\varLambda )^{1/2}\lt 1$ . For sub-Alfvénic motion ( $A_l\lt 1$ ), ideal-MHD Alfvén-wing theories predict the universal scaling $C_d^m/A_l=\mathcal{O}(1)$ , implying the characteristic force scaling $\propto \tilde {B}_0^2A_l\propto \tilde {B}_0$ , although the associated mechanical drag coefficient cannot be inferred uniquely from their equivalent electric-circuit formulation. In particular, equations (54) and (55) of Saur et al. (Reference Saur, Grambusch, Duling, Neubauer and Simon2013) and equation (29) of Bouvier & Cébron (Reference Bouvier and Cébron2015) yield $C_d^m/A_l=4$ , while finite- $A_l$ effects multiply this ratio by $(1+A_l^{2})^{-1/2}$ (Neubauer Reference Neubauer1980; Zarka Reference Zarka2007). By directly evaluating the Maxwell stresses acting on the sphere, (5.28) not only recovers the Alfvén-wing scaling but also yields the explicit prefactor $C_d^m/A_l=8 \mathcal{I}_{mt}^{\,\textit{pol}}/(1-a^3)$ . As for the classical Stokes drag coefficient (Appendix A.2), the total drag force is actually larger due to pressure effects: in the relevant large $\varLambda \gg 1$ limit, the energetic argument developed in § 5.5 shows that this drag must be multiplied by $3 /[2(1-a^3)]$ giving the drag coefficient

(5.31) \begin{equation} C_d=\frac {12}{A_l} \frac {\mathcal{I}_{mt}^{\,\textit{pol}}}{(1-a^3)^2}. \end{equation}

Normalised by the natural energy density of each problem, this asymptotic drag law recovers the mathematical form of the Stokes law (A18), where $A_l$ replaces the Reynolds number. In this sense, Alfvén-wave radiation transports momentum in ideal MHD just as viscous diffusion transports momentum in low-Reynolds-number hydrodynamics. These predictions remain in good agreement with the global ideal-MHD simulations of Corso & Lai (Reference Corso and Lai2026) for a steady, compressible ideal-MHD flow perpendicular to the field past a non-magnetised conducting sphere. They recover the same asymptotic scaling in the incompressible low- $A_l$ regime (their equations (18) and (19)). Their figure 6 at a large value of $\beta =200 \gg 1$ , approaching the incompressible limit, gives $C_A \approx 1.75$ in their notations, which corresponds to an effective coefficient $C_d^m/A_l\simeq 2 \times 1.75/\pi \approx 1.1$ . The different prefactor is unsurprising since their simulations consider a steadily translating obstacle, whereas the present theory is derived for a rapidly oscillating sphere in the thin-boundary-layer regime $L_\eta \ll a_s$ .

5.5. Ohmic dissipation in inviscid MHD boundary layers

The Ohmic dissipation does not arise solely from the magnetic-tension contribution (5.25). In both the solid and fluid domains it can be obtained from the volume integral of the squared radial derivatives of the magnetic perturbation, as specified by (3.19). For the thin inner-boundary layer, the BLT reduces the volume integral to the surface element $a_s^2 \sin(\theta) \mathrm{d}\theta \mathrm{d}\phi$ , retaining only leading-order terms in $L_\eta /a_s \ll 1$ and $\breve {\eta }_s^{1/2} L_\eta /a_s \ll 1$ (§ 3.6). To keep the analytical integration tractable, the huBL framework is used and gives (for the polar mode)

(5.32) \begin{eqnarray} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \geqslant a_s} &=& \frac {9 }{8} \frac { \varLambda }{\vartheta _1 \vartheta _2^{1/4} |\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda }|^{2}} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\gamma ^2/5+\alpha _2 \gamma ^4}{(1-a^3)^2} \end{eqnarray}

in the fluid, noting that $ \vartheta _1= \cos ( \arg [ (1+\mathrm{i})(1-\mathrm{i} \varLambda )^{-1/2} ] )$ and $ \vartheta _2=4(1+\varLambda ^2)$ , and where rotational effects modify the dissipation through the terms in $\gamma$ where $\alpha _2=0$ for the rotation-modified basic flows $\boldsymbol{U}_{\!1}$ (4.5). Within the solid inner sphere, the relative Ohmic contribution is given by (for the polar mode)

(5.33) \begin{eqnarray} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \leqslant a_s} &=& \breve {\mu }_s \vartheta _1 \vartheta _2^{1/4} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \geqslant a_s}\! . \end{eqnarray}

The relative Ohmic contribution $\breve {\mu }_s \vartheta _1 \vartheta _2^{1/4}$ of the inner sphere decreases from $\breve {\mu }_s$ at $\varLambda \ll 1$ , where the flow-induced contribution is negligible, to $0$ for $\varLambda \gg 1$ , where the energy loss mainly arises from Alfvén-wave radiation in the fluid. The outer-boundary contribution follows from (5.32) and (5.33) by multiplying by $a^4$ and replacing the index $s$ with $m$ . Discarding rotation ( $\gamma =0$ ), we simply obtain twice (5.32) and (5.33) for the equatorial modes. In any case, the total dissipation is obtained by summing the inner and outer solid domain contributions to the two fluid ones (at the two boundaries).

To assess the accuracy of the huBL estimate, we can compute numerically the exact dissipation integrals, retaining the angular dependence of the imposed magnetic field rather than replacing $B_{0r}$ by its surface average, and form the ratio with the corresponding huBL counterparts. For a uniform axial field where $\varLambda =\varLambda _z/3$ , we then find slightly different ratios in each domain (§ S3). By analogy with the correction factor $\mathcal{I}_{mt}^{\,\textit{pol}}$ introduced in § 5.4 for magnetic tension, we thus introduce a global correction factor for the total dissipation (neglecting the outer solid contribution), denoted $\mathcal{I}_{\eta }^{\,\textit{pol}}$ for the polar mode and $\mathcal{I}_{\eta }^{\,\textit{eq}}$ for the equatorial modes. For $\breve {c}=1$ and $\gamma =0$ , we obtain $\mathcal{I}_{\eta }^{\,\textit{pol}}= \mathcal{I}_{mt}^{\,\textit{pol}}$ at machine precision and $\mathcal{I}_{\eta }^{\,\textit{eq}}\approx 2 \mathcal{I}_{\eta }^{\,\textit{pol}}$ . While the huBL approach shows that the equatorial-mode Ohmic dissipation is twice the polar one when $B_{0r}$ is constant at $r=a_s$ , this factor thus increases to $2 \mathcal{I}_{\eta }^{\,\textit{eq}}/\mathcal{I}_{\eta }^{\,\textit{pol}}=4$ for a uniform axial field (figure S1). Note also that, while (5.32) and (5.33) account for magnetic tension and pressure, the axial Lorentz force vanishes at leading order for such a field, owing to the cancellation between magnetic pressure and tension (§ S5.1).

Figure 12.

Scaled Ohmic dissipation for $\breve {\eta }_s=\mathcal{I}_{\eta }^{\,\textit{pol}}=1$ , $\alpha _2=0$ . (a) Fluid, from (5.32), and solid, from (5.33), regions. (b) Sum of fluid and solid contributions compared with the imaginary part of (5.25). Top left inset in (a): ratio between the scaled Ohmic dissipations in the solid and fluid for $\breve {\mu }_s=1$ . Top left inset in (b): ratio between the sum of the two contributions and the imaginary part of (5.25); the colour of the curves is $\breve {\eta }_s$ , assuming that $\breve {\mu }_s=1$ .

Figure 12 shows the effect of magnetic-field strength $\varLambda$ and permeability ratio $\breve {\mu }_s$ on the scaled Ohmic dissipation: panel (a) compares fluid and solid contributions and panel (b) contrasts total dissipation with that due to magnetic tension alone. For $\varLambda \gg 1$ , increasing $\varLambda$ enhances fluid dissipation while leaving the solid contribution unchanged, so the fluid term dominates (inset of panel a). In figure 12(b) the total dissipation (markers) is compared with its estimate (solid line) based on the dissipative part of the magnetic tension (as in BG95).

For $\varLambda \ll 1$ , (5.32) and (5.33) converge towards the same huBL estimate

(5.34) \begin{eqnarray} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \geqslant a_s} &=& \breve {\mu }_s^{-1} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \leqslant a_s}= \frac {9 }{8} \frac { \varLambda }{ (\breve {c}_s+1)^{2}} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\gamma ^2/5+\alpha _2 \gamma ^4}{(1-a^3)^2} , \quad \end{eqnarray}

the total dissipation being thus twice (5.34) at $\breve {\mu }_s = 1$ . For $\varLambda \ll 1$ , the fluid behaves as a solid heated by oscillatory currents, allowing (5.34) to be related to the dissipation in a conducting sphere subjected to a field of magnitude $B$ . Beyond the estimate $V_{ec} \omega B^2/2$ , where $V_{ec}$ is the eddy-current volume (e.g. Bouvier & Cébron Reference Bouvier and Cébron2015), an exact analytical result exists for a uniform axial field of magnitude $B_{0}$ (Chyba et al. Reference Chyba, Hand and Thomas2021, who obtained identical shell dissipation for $L_\eta \ll a_s$ ). Matching equations (5) and (7) of Chyba et al. (Reference Chyba, Hand and Thomas2021) also shows a regime transition at $L_\eta /a_s \gt \sqrt [3]{4/45} \approx 1/2$ for $a \ll 1$ .

The opposite limit $\varLambda \gg 1$ gives a $\varLambda$ -independent value in the solid inner domain, with

(5.35) \begin{eqnarray} \breve {\mu }_s^{-1} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \leqslant a_s} &=& \frac {1}{\sqrt {2 \varLambda }} \left . \widehat {\mathcal D}_\eta ^{\,\textit{pol}} \right |_{r \geqslant a_s} = \frac {9}{8} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\gamma ^2/5+\alpha _2 \gamma ^4}{(1-a^3)^2} . \end{eqnarray}

The full dissipation being the sum of (5.32) and (5.33), we retrieve the scalings in $\varLambda$ and $\varLambda ^{1/2}$ for $\mathrm{Im}(\Delta \omega _{mt})$ that we found for $\varLambda \ll 1$ and $\varLambda \gg 1$ , respectively.

The dissipation from the power approach can now be compared with the one based on the magnetic tension. For the polar mode, $\mathcal{I}_{mt}^{\,\textit{pol}} \approx \mathcal{I}_{\eta }^{\,\textit{pol}}$ such that the ratio can be calculated as the sum of the fluid (5.32) and solid (5.33) contributions divided by the imaginary part of (5.25b ). By neglecting outer-boundary dissipation and rotational effect at $ \breve {\eta }_s = 1$ , we obtain a gain of $3/(2(1-a^3))$ , regardless of the values of $ \varLambda$ and $ \breve {\mu }_s$ . This is illustrated by the black line in the top-left inset of figure 12(b), and justifies equation (20) in P26. For varying $\breve {\eta }_s$ (coloured lines in the top-left inset of panel b), the gain becomes $1/(1-a^3)$ multiplied by $3(1+\breve {\mu }_s)/[2(1+\breve {c}_s)]$ for $\varLambda \ll 1$ and $3/2$ for $\varLambda \gg 1$ . For $\breve {\eta}_s=1$ , these ratios coincide, recovering the same ratio known from the hydrodynamic viscous case (including rotation effects) between the total Stokes force and the tangential-stress contribution (see (6.15), (6.16) and Appendix C). In the BG95 case $(\breve {c}_s,a)=(1,0)$ , the disregarded factor $2/3$ between magnetic tension and total Ohmic dissipation is nearly compensated by the disregarded factor $\mathcal{I}_{mt}^{\,\textit{pol}} \approx 3/5$ from the huBL approach (§ 5.4), explaining the good agreement of BG95 with DNS found by P26.

Higher-order bulk-field estimates (§ 5.3) are uncertain for moderate Lorentz forces owing to the irrotational assumption, but allow comparison with the magnetic-tension force. Contributions from $\boldsymbol{\mathcal{B}}+\boldsymbol{\mathcal{B}}^{(o)}$ and $\boldsymbol{\mathfrak{b}}$ are higher order in $L_\eta$ and negligible, whereas the force from $\mathcal{P}+\mathcal{P}^{(o)}$ is comparable. Using the huBL approach for the polar mode gives

(5.36) \begin{eqnarray} \frac {\Delta \omega _{p}}{\Delta \omega _{mt}} = \frac {1}{2} \frac {1+2a^3+3a^4}{1-a^3} , \end{eqnarray}

with $\Delta \omega _{mt}$ from (5.25); the $a^4$ term arises solely from $\mathcal{P}^{(o)}$ . We thus find that the total magnetic force is $3/2$ times larger than the magnetic tension (for $a \ll 1$ ). Although less accurate than the power-based estimate, (5.36) indicates comparable real and imaginary parts, yielding the magnetic correction to $C_a^i$ , which is inaccessible from the power-based approach.

These results complete the theory and bring it into quantitative agreement with the DNS of P26. There, the theoretical dissipation prefactor $d^\star _{\eta ,2}$ was derived within the huBL framework, which neglects the field-geometry correction. Using $\mathcal{I}_{\eta }^{\,\textit{pol}} \approx \mathcal{I}_{\eta }^{\,\textit{eq}}/2 \approx 3/5$ , the improved estimates are now $d^\star _{\eta ,2}=\pi \sqrt {2}\mathcal{I}_{\eta }^{\,\textit{pol}}/4\approx 0.67$ for the polar mode and $d^\star _{\eta ,2}=\pi \sqrt {2}\mathcal{I}_{\eta }^{\,\textit{eq}}/2\approx 2.7$ for the equatorial modes. Relative to the DNS prefactor of the polar mode and the average of the prograde and retrograde equatorial-mode prefactors (thereby removing rotational effects), the theoretical predictions agree to within $9\,\%$ .

More broadly, the DNS of P26 had already validated the asymptotic functional dependence over the geophysically relevant range $P_m\in [10^{-6},10^{-3}]$ , including the previously unreported confinement scaling $(1-a^3)^{-2}$ . The present correction of the analytical prefactors completes this agreement, demonstrating that the asymptotic theory reproduces both the scaling laws and the absolute magnitude of the Ohmic dissipation for all three translational modes. It also clarifies the physical origin of the confinement correction: estimating the dissipation from magnetic tension alone yields a confinement factor proportional to $(1-a^3)^{-1}$ , whereas evaluating the volume-integrated Ohmic dissipation includes pressure effects and produces the stronger $(1-a^3)^{-2}$ dependence observed in the DNS.

5.6. Bulk-diffusive and boundary-layer Ohmic dissipation

When $L_\eta \gtrsim a_s$ , the boundary-layer approximation breaks down and the Ohmic dissipation is instead governed by bulk induction. In this bulk-diffusive regime, relevant to weakly conducting subsurface oceans (figure 1 b), we obtain the fluid dissipation (Appendix B.2)

(5.37a,b) \begin{equation} \widehat {\mathcal D}_\eta |_{r \geqslant a_s}^{\textit{pol}} = \frac {3}{20}\frac {\rho _{\!f}}{\rho _s}\frac {1+\gamma ^2}{1-a^3} \varLambda _z f_\varLambda , \quad \widehat {\mathcal D}_\eta |_{r \geqslant a_s}^{\textit{eq}} = \frac {\rho _{\!f}}{\rho _s}\frac {\varLambda _z}{1-a^3} \!\left (\frac {7}{20}+a^3\right )\!, \end{equation}

for a uniform axial field with $\varLambda _z\ll 1$ , where $f_\varLambda =1$ . For the unbounded, non-rotating polar problem, Motz (Reference Motz1966b ) obtained and experimentally tested the finite-field correction $f_\varLambda =1-1.08\varLambda _z+1.15\varLambda _z^2$ . We quote this correction for comparison, but do not interpret it as a large- $\varLambda _z$ asymptotic continuation of the present confined bulk-diffusive solution. Through its equivalence with the volume Ohmic dissipation (§ 3.6), (5.37a ) gives the exact quasi-static, weak-Lorentz-force contribution for the non-rotating polar mode and extends the magnetic drag of Singh (Reference Singh1965) from an unbounded fluid to confined spherical shells; its rotation-dependent term and the equatorial expression (5.37b ) are estimates obtained by neglecting the associated non-uniform electric potential (Appendix B.2).

Unlike the boundary-layer results, (5.37) are independent of $L_\eta$ . The corresponding dissipation nevertheless retains the linear scaling with $\varLambda _z$ characteristic of the weak-field magnetic skin-layer regime, rather than the $\varLambda _z^{1/2}$ scaling of the strong-field Alfvén-wave regime. For the polar mode, a formal estimate of the crossover between the weak-field magnetic skin-layer and bulk-diffusive regimes is obtained by equating the bulk-diffusive dissipation in (5.37a ) to the total skin-layer dissipation derived from (5.34). Including the field-geometry correction factor $\mathcal{I}_{\eta }^{\,\textit{pol}}$ , this gives $L_\eta /a_s=(1+\breve {c}_s)^2(1-a^3)/(5\mathcal{I}_{\eta }^{\,\textit{pol}})$ for $\gamma =0$ , $f_\varLambda =\breve {\mu }_s=1$ and $\varLambda _z=3\varLambda$ . For $a=0.35$ , $\breve {c}_s=1$ and $\mathcal{I}_{\eta }^{\,\textit{pol}}\approx 3/5$ , this estimate yields $L_\eta /a_s\approx 1.28$ , or equivalently $2a_s^2/L_\eta ^2\approx 1.23$ , close to the numerical crossover $2a_s^2/L_\eta ^2\approx 1$ reported by P26.

6.1. Tangential components in the viscomagnetic oscillatory layers

Viscosity enters through the additional diffusion term $P_m\partial _r^2\boldsymbol{u}_\parallel$ in (5.6). Analytical particular solutions of the three Slichter modes (4.1)−(4.5) remain explicit, although their expressions are too lengthy to be reported here without obscuring the presentation. All analytical expressions omitted from the paper, including these particular solutions, are nevertheless provided in the accompanying Zenodo repository (Personnettaz & Cébron Reference Personnettaz and Cébron2026), allowing direct evaluation, plotting and independent verification.

The solution $\boldsymbol{b}_\parallel ^{(s)}$ in the solid domains remains unchanged (5.9), while the homogeneous fluid solutions at both boundaries take the form

(6.1) \begin{align} \boldsymbol{b}_\parallel ^{(h)} &= (A_{\pm 1}\hat {\boldsymbol{\theta }}+A_{\pm 2} \hat {\boldsymbol{\phi }}) \mathrm{e}^{\pm \lambda _+ \tilde {r}_\pm }+(B_{\pm 1}\hat {\boldsymbol{\theta }}+B_{\pm 2} \hat {\boldsymbol{\phi }}) \mathrm{e}^{\pm \lambda _- \tilde {r}_\pm } , \end{align}

(6.2) \begin{align} B_{0r} \boldsymbol{u}_\parallel ^{(h)} &= \pm (\mathrm{i} \omega \lambda _+^{-1}-\lambda _+ ) (A_{\pm 1}\hat {\boldsymbol{\theta }}+A_{\pm 2} \hat {\boldsymbol{\phi }}) \mathrm{e}^{\pm \lambda _+ \tilde {r}_\pm }\pm (\mathrm{i} \omega \lambda _-^{-1} -\lambda _-) (B_{\pm 1}\hat {\boldsymbol{\theta }}+B_{\pm 2} \hat {\boldsymbol{\phi }})\mathrm{e}^{\pm \lambda _- \tilde {r}_\pm } , \end{align}

since the induction (3.7) gives $ B_{0r} \boldsymbol{u}_\parallel =- \partial _r \boldsymbol{b}_\parallel + \mathrm{i} \omega \int \boldsymbol{b}_\parallel \mathrm{d}r$ . The prefactors with the subscript $_-$ and $\tilde {r}_-$ (respectively $_+$ and $\tilde {r}_+$ ) are associated with the boundary layer at $r \geqslant a_s$ (respectively $r \leqslant a_m$ ). Here, the (complex) radial wavenumber $\lambda _\pm$ is

(6.3) \begin{eqnarray} \lambda _\pm =\frac {1+\mathrm{i}}{L_\eta } \sqrt {\frac {1+P_m -\mathrm{i} \varLambda _l \pm \mathrm{i} \sqrt {[\varLambda _l+\mathrm{i}(1+P_m)]^2+4P_m}}{2 P_m} } , \end{eqnarray}

which can be written symmetrically for the diffusivities (§ S5.2). The non-magnetic purely viscous limit is detailed in Appendix C.

If we consider the steady limit $\omega =0$ , we find that ${\lambda }_+$ vanishes as ${\lambda }_+ =\mathrm{i} L_\eta ^{-1} (2/\varLambda _l)^{1/2} \propto \omega$ and ${\lambda }_-^{-1}=L_\nu /\sqrt {2 \varLambda _l}=\tilde {B}_{0r}^{-1}\sqrt {\rho _{\!f} \nu \eta _{\!f} \mu _{\!f}}$ , with the viscous skin depth $L_\nu =L_\eta P_m^{1/2}$ . In this limit, ${\lambda }_-^{-1}$ is independent of $\omega$ and is exactly the usual Hartmann layer thickness ( $\lambda _- D$ being then the Hartmann number; see, e.g. Dormy & Soward Reference Dormy and Soward2007). Such layers have thus also been termed Stokes–Hartmann layers (e.g. Cuevas et al. Reference Cuevas, Sierra and Avramenko2002).

In the strong-field limit $\varLambda _l\gg 1$ , diffusive effects modify Alfvén-wave propagation (ideally at speed $V_A=\tilde {B}_{0r}/\sqrt {\rho _{\!f}\mu _{\!f}}$ ). We have $\lambda _- \approx (2\varLambda _l/P_m)^{1/2}/L_\eta \to \infty$ , while $\lambda _+$ gives

(6.4) \begin{eqnarray} \frac { \omega }{\mathrm{Im}(\lambda _+)} \approx V_A \left (1+\frac {3(1+P_m^2)+10 P_m}{ 8 \varLambda _l^2}\right ) \!, \end{eqnarray}

which is the diffusion-modified phase velocity, and attenuation length

(6.5) \begin{eqnarray} L_A\equiv \textrm{Re}(\lambda _+)^{-1}\approx 2^{1/2} (1+P_m)^{-1}L_\eta \varLambda _l^{3/2}, \end{eqnarray}

reduced by the factor $1+P_m$ owing to viscous diffusion (compared with BG95). At a given latitude, the finite-cavity requirement $L_A\ll D$ sets an upper bound for $\varLambda _l$ , given by

(6.6) \begin{eqnarray} \varLambda _l \ll (2^{-1/2}(1+P_m)L_\eta ^{-1} D)^{2/3}, \end{eqnarray}

where waves are attenuated before reaching the opposite boundary, and the inner- and outer-boundary-layer constructions remain independent. When $L_A=\mathcal{O}(D)$ , reflected waves couple the two boundaries and may generate global cavity resonances, for which the local radiative drag law no longer describes the finite-cavity response. This possibility was demonstrated experimentally by Motz (Reference Motz1966a ), who generated Alfvén waves in a mercury-filled spherical shell and determined their propagation speed from a standing-wave resonance.

In the geophysically relevant limit $P_m \ll 1$ , (6.3) gives

(6.7a,b) \begin{equation} \lambda _+ \to (1+\mathrm{i})L_\eta ^{-1 } (1-\mathrm{i} \varLambda _l)^{-1/2} =\lambda , \quad \lambda _- \to (1+\mathrm{i}) L_\nu ^{-1} \sqrt {1-\mathrm{i} \varLambda _l} , \end{equation}

where $\lambda _+$ recovers the inviscid MHD boundary-layer wavenumber (§ 5.2) and $\lambda _-^{-1}\propto L_\nu =L_\eta P_m^{1/2}$ vanishes with $P_m$ , reducing to the viscous Stokes layer for $\varLambda _l\ll 1$ . This formulation provides a continuous transition between regimes (figure S3). For an axial field, $\varLambda _l=\varLambda _z\cos ^2\theta$ , the viscomagnetic oscillatory boundary layer reduces to a purely viscous skin layer at the equator. In the opposite limit $P_m\gg 1$ ,

(6.8a,b) \begin{equation} {\lambda }_+ \to \frac {1+\mathrm{i}}{L_\nu } -\frac {(1-\mathrm{i})\varLambda _l}{2L_\nu P_m} +\mathcal{O}(P_m^{-2}), \quad {\lambda }_- \to \frac {1+\mathrm{i}}{L_\eta } +\frac {(1-\mathrm{i})\varLambda _l}{2L_\eta P_m} +\mathcal{O}(P_m^{-2}) , \end{equation}

so that, at leading order, $\lambda _+$ recovers the viscous Stokes boundary layer, while $\lambda _-$ corresponds to the magnetic skin-depth wavenumber in the absence of flow, the fluid behaving as a solid.

Figure 13.

Tangential components of the flow perturbation near the inner and outer boundaries. (a) Polar and (b) equatorial forcing. XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD using diffusionless particular solution (cyan dash–dotted line), viscomagnetic oscillatory (thick green line) and purely viscous (purple dashed line). Insets: left, zoom on $r=a_s$ ; right, bulk profiles versus sum of boundary-layer and secondary flows from the inviscid MHD (dark-red dashed line), viscomagnetic (dark-green line) and purely viscous (purple dash–dotted line). Parameters: as in figure 7, but $\gamma =0$ .

Figure 14.

Tangential components of the magnetic perturbation near the inner and outer boundaries. (a) Polar and (b) equatorial forcing. XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line), and diffusionless bulk solution (blue dash–dotted line, (B4)). Insets show zoomed-in regions near the interfaces. Parameters: as in figure 7, but $\gamma =0$ .

Figure 14. Long description

Panel A: Two line graphs show the tangential components of the magnetic perturbation near the inner and outer boundaries for polar forcing. The horizontal axis represents the normalized distance from the inner sphere (left) and the outer boundary (right), while the vertical axis represents the perturbation. The graphs compare different models: the XSHELLS code (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash-dotted line), viscomagnetic oscillatory (thick green line), purely viscous (purple dash-dotted line), and diffusionless bulk solution (blue dash-dotted line). Insets zoom in on regions near the interfaces. Panel B: Two line graphs show the tangential components of the magnetic perturbation near the inner and outer boundaries for equatorial forcing. The horizontal axis represents the normalized distance from the inner sphere (left) and the outer boundary (right), while the vertical axis represents the perturbation. The graphs compare the same models as in Panel A. Insets zoom in on regions near the interfaces.

The boundary-layer solution (6.1) and (6.2) is completed by determining the prefactors from the boundary conditions (Appendix D.3), whose complete analytical expressions are provided in the accompanying Zenodo repository Personnettaz & Cébron (Reference Personnettaz and Cébron2026). The resulting oscillating viscomagnetic solution (thick dark-green curves in figures 13 and 14) reproduces the DNS throughout the boundary layer. Within one viscous skin depth $L_\nu$ of the solid boundaries, viscous effects dominate and the solution reduces to the purely viscous Stokes layer (Appendix C), in agreement with the DNS (left insets of figure 13); this remains true even under strong magnetic forcing (figure 9, $\varLambda _l=2$ ). However, for $L_\nu \lt (r-a_s)\lt 2L_\eta$ , the purely viscous solution agrees with the DNS only for very weak local magnetic fields (figure S5, $\varLambda _l=5\times 10^{-3}$ ), whereas the viscomagnetic solution remains accurate from the bulk-diffusive regime to Alfvén waves emission (figures 8 and 9).

For $P_m \ll 1$ , magnetic coupling increases the penetration depth of the velocity boundary layer. For the polar mode, the viscomagnetic solution accurately reproduces the meridional velocity and magnetic-field components under weak rotation up to $\gamma =0.1$ (figures S7 and S8). Similar agreement is obtained for equatorial modes (figures S9 and S10), where oscillatory forcing dominates the dynamics and masks rotational effects for $\gamma \lt 0.1$ .

Viscous effects near the boundaries are also visible in radial profiles of the tangential magnetic field (figure 14 and right insets), where extrema are slightly shifted from the interfaces – more strongly at the outer boundary – due to enhanced Ohmic dissipation. In the bulk (right insets of figure 13), the boundary-layer solution (thick light green lines) alone cannot reproduce the velocity perturbation because it vanishes outside the layer; adding the higher-order viscomagnetic correction $\mathcal{U}$ (dark-green line) restores agreement with the DNS (orange-dotted lines). Radial plots also display vertical dotted lines at five times the characteristic penetration lengths $\text{Re}(\lambda _\pm )^{-1}$ defined in (6.3), providing estimates of the viscously dominated region, $5\text{Re}(\lambda _-)^{-1}$ , and of the full velocity or magnetic boundary layer, $5\text{Re}(\lambda _+)^{-1}$ , even in the presence of waves (e.g. figure 13 and its right insets).

6.2. Tangential-stress drag from the viscomagnetic oscillatory layers

The forces $\boldsymbol{F}_{\!mt}$ and $\boldsymbol{F}_{\!v}$ can now be calculated, including the contribution of the particular solution. However, for thin enough boundary layers, the particular solution can be discarded. One can then use the huBL approach to obtain estimates of the forces $\boldsymbol{F}_{\!mt}$ and $\boldsymbol{F}_{\!v}$ . Using $\tilde {\lambda }_\pm =\lambda _\pm L_\eta$ with a constant radial magnetic field at the boundary, the polar mode leads to (for the inner boundary)

(6.9) \begin{align} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s} &=\frac {3}{4} \varLambda \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\mathrm{i}}{\breve {c}_s (2\mathrm{i}+\tilde {\lambda }_+ \tilde {\lambda }_-) +(1+\mathrm{i})(\tilde {\lambda }_++\tilde {\lambda }_-) } \frac {\tilde {\lambda }_+ \tilde {\lambda }_-}{1-a^3} , \\[-12pt]\nonumber\end{align}

(6.10) \begin{align} \left . \frac {\Delta \omega _v}{\omega } \right |_{r=a_s}&=\frac {3}{8} P_m \frac {\rho _{\!f}}{\rho _s} \frac {L_\eta }{a_s} \frac { 2\mathrm{i} (\tilde {\lambda }_+ +\tilde {\lambda }_-)\breve {c}_s+\tilde {\lambda }_+\tilde {\lambda }_-(1+\mathrm{i})+2(\mathrm{i}-1)}{ \breve {c}_s(2 \mathrm{i}+\tilde {\lambda }_+\tilde {\lambda }_-)+(1+\mathrm{i})(\tilde {\lambda }_++\tilde {\lambda }_-)} \frac {\tilde {\lambda }_+\tilde {\lambda }_-}{1-a^3} , \end{align}

Figure 15.

(a) Normalised magnetic-stress detuning assuming that $\breve {c}_s = 1$ ; insets show the ratio of (6.9) to (5.25b ). Dashed and dotted lines denote the low- $P_m$ ((5.25b )) and large- $P_m$ limits (6.14). (b) Viscous-stress detuning, (6.10), normalised by its non-magnetic limit (C5), evaluated at $\breve {c}_s = 10^3$ ; $P_m\ll 1$ (dashed) and $P_m\gg 1$ (dotted) correspond to (6.13) and (6.15). Left ( $P_m = 10^{-3}$ ) and right ( $\varLambda = 10$ ) insets show maps of (6.10), green horizontal line is at $\breve {c}_s = 10^3$ ; the central inset displays the normalised real part of (6.13).

for no-slip boundary conditions, and (noting that $\xi =4+2\mathrm{i}(\tilde {\lambda }_+^2+\tilde {\lambda }_-^2)=-4(1-\mathrm{i} \varLambda )/P_m$ )

(6.11) \begin{eqnarray} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s}&=&\frac {3}{2} \varLambda \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {\mathrm{i} \tilde {\lambda }_+ \tilde {\lambda }_-(\tilde {\lambda }_++\tilde {\lambda }_-)}{ \breve {c}_s (1+\mathrm{i})\tilde {\lambda }_+ \tilde {\lambda }_-(\tilde {\lambda }_++ \tilde {\lambda }_-) +\xi +2\mathrm{i}\tilde {\lambda }_+ \tilde {\lambda }_- } \frac {1}{1-a^3} \end{eqnarray}

for stress-free boundary conditions, shown in figures 15 and S4 as functions of $P_m$ and $\varLambda$ . In the stress-free case, $\tilde {\lambda }_+\tilde {\lambda }_- = 2 \mathrm{i} P_m^{-1/2}$ and the viscous stress at the boundary vanishes, so that $\Delta \omega _v=0$ . As in § 5.2, the total magnetic-tension dissipation includes the outer-boundary contribution, obtained by multiplying the above expressions by $a$ and replacing $(\breve {\mu }_s,\breve {\eta }_s)$ with $(\breve {\mu }_m,\breve {\eta }_m)$ . The total magnetic-tension dissipation from both boundaries then follows as in (5.23). Equations (6.9)−(6.11) should further be multiplied by $1+\alpha \gamma ^2$ , with $\alpha =-6a^3(1+a)(1+a+a^2)^{-1}/5$ , if the flow $\boldsymbol{U}_2$ of (A28) is used instead of $\boldsymbol{U}_{\!1}$ .

Consider first the limit $\breve {c}_s \to \infty$ , reached either for $\sigma _s \ll \sigma _{\!f}$ (insulating inner sphere) or $\mu _s \gg \mu _{\!f}$ (high-permeability inner sphere). Equation (6.9) then gives a vanishing detuning

(6.12) \begin{eqnarray} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s} &=&\frac {3}{4} \frac {\varLambda }{\breve {c}_s} \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\mathrm{i}}{1-a^3} \frac {1 }{1+\sqrt {P_m}} \end{eqnarray}

for a no-slip sphere. For a stress-free boundary, (6.11) leads to (6.12) multiplied by $1+\sqrt {P_m}$ , yielding a $P_m$ -independent detuning.

In the limit $P_m \ll 1$ , the scaled magnetic tension (6.9) and (6.11) reduce to the inviscid case (5.25), and the viscous-stress detuning simplifies to

(6.13) \begin{eqnarray} \left . \frac {\Delta \omega _v}{\omega } \right |_{r=a_s}&\approx &-\frac {3}{4} \frac {\rho _{\!f}}{\rho _s} \frac {L_\nu }{a_s} \frac {1-\mathrm{i}}{1-a^3} \left ( \frac {1+\breve {c}_s\sqrt {1-\mathrm{i} \varLambda } }{\breve {c}_s+\sqrt {1-\mathrm{i} \varLambda }} \right )\!, \end{eqnarray}

as shown by dashed curves in figure 15(b) for $\breve {c}_s=10^3$ (approximately insulating solid sphere). In the limit $\varLambda \ll 1$ and $\gamma \ll 1$ , this gives $\mathrm{Im}(\Delta \omega _{mt})\gt \mathrm{Im}(\Delta \omega _v)$ when $\varLambda \gt (1+\breve {c}_s) \sqrt {P_m}$ .

Equation (6.13) shows that magnetic effects vanish for $\breve {c}_s=1$ , recovering the hydrodynamic result (C5); see the red line at $1$ in figure 15(b). Its expansion for $\breve {c}_s \gg 1$ gives a minimum of $-\text{Re}(\Delta \omega _v)$ at $\varLambda =\breve {c}_s^{2/3}/2$ for $\varLambda \gg 1$ . A minimum appears in the central inset of figure 15(b). In the small- $P_m$ limit ( $10^{-3}$ , top-left inset), the magnetic correction in parentheses matches (6.10) normalised by the purely viscous detuning. This regime is particularly relevant to planetary interiors. In P26 the time-averaged viscous dissipation was shown numerically to agree with the power-based prediction, including confinement effects, and to recover the expected factor of two between the mean viscous dissipation of the polar and equatorial modes (see § 3.6). The top-left inset in figure 15(b) shows the imaginary part of this normalised quantity, indicating that magnetic corrections become significant only for large $\varLambda$ and $\breve {c}_s$ , as for a high-permeability inner sphere (e.g. $\breve {\mu }_s=800$ for an iron sphere in Galinstan) oscillating in a sufficiently strong magnetic field.

In the opposite limit $P_m \gg 1$ , (6.9) and (6.10) give

(6.14) \begin{align} \left . \frac {\Delta \omega _{mt}}{\omega } \right |_{r=a_s} &= \frac {3}{4} \varLambda \frac {\rho _{\!f}}{\rho _s} \frac { L_\eta }{ a_s } \frac {1+\mathrm{i}}{ (\breve {c}_s+1)(1-a^3) \sqrt {P_m}} , \\[-12pt]\nonumber \end{align}

(6.15) \begin{align} \left . \frac {\Delta \omega _v}{\omega } \right |_{r=a_s}&\approx -\frac {3}{4} \frac {\rho _{\!f}}{\rho _s} \frac {L_\nu }{a_s} \left [ \frac {1-\mathrm{i}}{1-a^3} + \frac {1+\mathrm{i}}{1-a^3} \frac {\breve {c}_s-1}{\breve {c}_s+1} (P_m^{-3/2}-P_m^{-1}/2) \varLambda \right ]\!, \end{align}

while (6.11) yields $\sqrt {P_m}$ times (6.14), leading to a $P_m$ -independent detuning. For any $P_m$ , magnetic corrections to $\Delta \omega _v$ vanish when $\breve {c}_s=1$ , in which case (6.10) reduces to the non-magnetic limit given by the first term of (6.15), consistent with (C5) and shown by the red line at 1 in figure 15(b). In this large- $P_m$ limit, $\varLambda =(1+\breve {c}_s)P_m$ marks the transition between magnetic- and viscous-drag-dominated regimes, where the leading-order terms of (6.14) and (6.15) have comparable real and imaginary components.

Rotation is absent from (6.13) since the leading $\gamma$ correction is azimuthal; however, viscous pressure corrections reintroduce rotational effects in the total force $\boldsymbol{F}_\nu$ . As shown in Appendix A.4, the results of Smylie & McMillan (Reference Smylie and McMillan1998, Reference Smylie and McMillan2000) are recovered from (6.15) at order $\gamma$ , with higher-order corrections modifying $\Delta \omega _v$ . Appendix C further recovers the Stokes result and the detuning from a no-slip outer boundary by multiplying (6.13) by $a$ .

Discarding magnetic and rotational effects, the above result extends directly to the equatorial modes. Restoring the azimuthal dependence $\mathrm{e}^{\textrm{i} \phi }$ and expressing the force $\boldsymbol{F}_{\!v}$ in the Cartesian basis, the force components along $\hat {\boldsymbol{x}}$ and $\hat {\boldsymbol{y}}$ are $F_{vz}$ and $\mathrm{i} m F_{vz}$ , where $\boldsymbol{F}_{\!v}=F_{vz} \hat {\boldsymbol{z}}=A_{vz} \epsilon \mathrm{e}^{\textrm{i}\omega t} \hat {\boldsymbol{z}}$ denotes the corresponding polar-mode force (such that the force is $A_{vz} \boldsymbol{\delta }$ ). All the previous results, notably related to $\Delta \omega _{v}$ , thus hold for the equatorial mode, agreeing with Smylie & McMillan (Reference Smylie and McMillan2000), who obtained the same force for the polar and the equatorial modes in the limit of vanishing rotation effects and outer-boundary displacement.

6.3. Viscous and Ohmic dissipation in viscomagnetic oscillatory layers

To obtain the total dissipation, one can now wonder how pressure effects modify the results of § 6.2. Pressure contributions from the bulk flow $\boldsymbol{\mathcal{U}}$ are required to recover the complete viscous force. When only mean dissipation is required, $\widehat {\mathcal D}_\nu$ follows directly from (3.18). Both approaches are detailed in Appendix C, where the leading-order Stokes limits (A17) and (A18) are recovered and extended to weak rotation.

Performing the volume integration of (3.18) at leading order in $L_\nu /a_s \ll 1$ yields the dissipative viscous force at the inner boundary

(6.16) \begin{eqnarray} {F}_\nu ^{d,{pol}} &=& 6 \pi \frac {P_m a_s }{(1-a^3)^2} \frac {a_s}{L_\nu } \left (1+\frac {\gamma ^2}{5} \right ) \epsilon \omega , \quad \widehat {\mathcal D}_\nu ^{\,\textit{pol}} = \frac {9}{8} \frac {\rho _{\!f}}{\rho _s}\frac {L_\nu }{ a_s} \frac {1+\gamma ^2/5}{(1-a^3)^2} , \end{eqnarray}

where $P_m$ arises from the chosen time scale (equivalent to $\nu$ in dimensional form). Equation (6.16) follows from the $\mathcal{O}(\gamma ^4)$ expansion (4.15) of the B74 bulk flow $\boldsymbol{U}$ (Appendix A.5). Equations (6.15) and (6.16) show that the viscous dissipation exceeds the tangential-stress contribution by $3(1+\gamma ^2/5)/[2(1-a^3)]$ ; neglecting hydrodynamic or magnetic pressure therefore omits rotation and geometric effects, modifying the scaling for $a\sim 1$ .

For a no-slip outer boundary, the corresponding force and scaled mean dissipation are obtained by multiplying (6.16) by $a^4$ . The total dissipative force is thus $(1+a^4)$ times (6.16), recovering the leading-order term of (A18) in the rapid-oscillation limit $L_\nu \to 0$ (Appendix C). Comparing with the bulk dissipation (4.8), the boundary-layer dissipation is thus found to dominate when $L_\nu \lt a_s/2$ in the non-rotating unbounded limit $\gamma =a=0$ , similarly to the transition $L_\eta \lt a_s/2$ found in § 5.5 for the Ohmic dissipation. Balancing the two leading terms of the Stokes’ viscous force expansion (A18) gives a similar $a$ -dependent transition that evolves from $L_\nu \approx a_s$ for $a\to 0$ to $L_\nu \approx (a_m-a_s)/2$ for $a \to 1$ .

For the equatorial modes, performing the volume integration of (3.18) at leading order in $L_\nu /a_s \ll 1$ gives (at $r=a_s$ )

(6.17) \begin{eqnarray} {F}_\nu ^{d,{eq}} &=& 6 \pi \frac {P_m a_s }{(1-a^3)^2} \frac {a_s}{L_\nu } \, \epsilon \omega , \quad \widehat {\mathcal D}_\nu ^{\textit{eq}} = \frac {9}{4} \frac {\rho _{\!f}}{\rho _s} \frac {L_\nu }{ a_s} \frac { 1 }{(1-a^3)^2} , \end{eqnarray}

and ${F}_\nu ^{d}$ is thus identical for all three modes for $\gamma =0$ , as confirmed by direct calculation (Appendix C). At $\gamma =0$ , identical dissipative forces therefore imply identical modal damping rates; the factor of two between (6.16) and (6.17) reflects only $\mathcal{E}_{\textit{eq}}/\mathcal{E}_{\textit{pol}}=2$ . Extending the integration to the order $L_\nu ^2/a_s^2 \ll 1$ multiplies (6.16) and (6.17) by $1+L_\nu /a_s$ , in agreement with the Stokes result (A18) for the unbounded polar mode.

Even within the huBL approximation, no closed-form expression can be derived for the total Ohmic dissipation in viscomagnetic oscillatory layers. Nevertheless, the analytical results obtained for inviscid layers in § 5.5, together with the viscous-dissipation results derived above, strongly suggest that the total Ohmic dissipation can be estimated by multiplying (6.9) or (6.11) by the correction factor $3 \mathcal{I}_{mt}^{\,\textit{pol}}/(2(1-a^3))$ .

6.4. Magnetohydrodynamic Stokes–Ekman boundary layer

Extension to rotating viscous flows with finite $\gamma$ is achieved by reintroducing the Coriolis term, yielding MHD Stokes–Ekman boundary layers. As before, the general boundary-layer solution is determined in terms of a given basic flow $\boldsymbol{U}$ and particular solution. Integration of the homogeneous part of (3.6) and (3.7) then gives, at both boundaries,

(6.18) \begin{align} \boldsymbol{b}_\parallel ^{(h)} &= (A_{\pm 1} \mathrm{e}^{\pm {\iota }_+ \tilde {r}_\pm }+A_{\pm 2} \mathrm{e}^{\pm {\kappa }_+ \tilde {r}_\pm }) (\mathrm{i} \hat {\boldsymbol{\theta }} +\hat {\boldsymbol{\phi }})+(B_{\pm 1} \mathrm{e}^{\pm {\iota }_- \tilde {r}_\pm }+B_{\pm 2} \mathrm{e}^{\pm {\kappa }_- \tilde {r}_\pm }) (-\mathrm{i} \hat {\boldsymbol{\theta }} +\hat {\boldsymbol{\phi }}), \end{align}

where $\boldsymbol{u}_\parallel ^{(h)}$ can be retrieved from the induction equation $ B_{0r} \boldsymbol{u}_\parallel ^{(h)}=- \partial _r \boldsymbol{b}_\parallel ^{(h)} + \mathrm{i} \omega \int \boldsymbol{b}_\parallel ^{(h)} \mathrm{d}r$ ,

(6.19) \begin{align} \frac {2 B_{0r}}{\omega }\boldsymbol{u}_\parallel ^{(h)} = &\mp ( ({\iota }_+ L_\eta ^2-2 \mathrm{i} /{\iota }_+) A_{\pm 1} \mathrm{e}^{\pm {\iota }_+ \tilde {r}_\pm } + ({\kappa }_+ L_\eta ^2-2 \mathrm{i} /{\kappa }_+)A_{\pm 2} \mathrm{e}^{\pm {\kappa }_+ \tilde {r}_\pm }) (\mathrm{i} \hat {\boldsymbol{\theta }} +\hat {\boldsymbol{\phi }}) \nonumber \\ & \mp ( ({\iota }_- L_\eta ^2-2 \mathrm{i} /{\iota }_-) B_{\pm 1} \mathrm{e}^{\pm {\iota }_- \tilde {r}_\pm }+ ({\kappa }_- L_\eta ^2-2 \mathrm{i} /{\kappa }_-)B_{\pm 2} \mathrm{e}^{\pm {\kappa }_- \tilde {r}_\pm }) (-\mathrm{i} \hat {\boldsymbol{\theta }} +\hat {\boldsymbol{\phi }}), \end{align}

with the four radial wavenumbers

(6.20) \begin{eqnarray} \underline {\iota }_\pm = \frac {1+ \mathrm{i}}{L_\eta } \sqrt {\frac {1+P_m - \mathrm{i} \varLambda _l \pm ( \gamma \cos \theta + \mathrm{i} \sqrt { \varpi _\pm } )}{2 P_m} } , \end{eqnarray}

(6.21) \begin{eqnarray} \underline {\kappa }_\pm = \frac {1+ \mathrm{i}}{L_\eta } \sqrt { \frac {1+P_m - \mathrm{i} \varLambda _l \pm ( \gamma \cos \theta - \mathrm{i} \sqrt { \varpi _\pm } )}{2 P_m}} , \end{eqnarray}

where

(6.22) \begin{eqnarray} \varpi _\pm =[\varLambda _l+\mathrm{i}(1+P_m)]^2+4P_m \pm 2(P_m+\mathrm{i}\varLambda _l-1)\gamma \cos \theta -\gamma ^2\cos ^2\theta . \end{eqnarray}

Underlined quantities in (6.20) and (6.21) relate to those in (6.18) and (6.19) via $X=\underline {X}\,\mathrm{sign}[\text{Re}(\underline {X})]$ , so that $\text{Re}(X)\gt 0$ and the boundary-layer solution decays away from the boundary. As reviewed by Hide & Stewartson (Reference Hide and Stewartson1972) or Acheson & Hide (Reference Acheson and Hide1973), such MHD viscous layers from combined oscillation and rotation have been studied previously (e.g. Hide & Roberts Reference Hide and Roberts1960). For instance, the four boundary layers (6.20) and (6.21) are given by Gans (Reference Gans1970, Reference Gans1971). These layers have also been examined near electrically conducting solids for MHD inertial waves (Kerswell Reference Kerswell1994) and precession-nutation flows (Mathews & Guo Reference Mathews and Guo2005; Deleplace & Cardin Reference Deleplace and Cardin2006). Asymptotic limits are reviewed in § S5.4 of the supplementary material.

Figure 16.

Azimuthal velocity (a) and magnetic (b) perturbations close to the inner (left) and outer (right) boundaries for the polar mode. XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line), purely viscous (purple dashed line), homogeneous MHD Stokes–Ekman (red-dotted line), MHD Stokes–Ekman with diffusionless correction (red solid line). Insets: left, zoom on $r=a_s$ ; right, bulk profiles versus sum of boundary-layer and secondary flows from the inviscid MHD (dark-red dashed line), viscomagnetic (dark-green line) and purely viscous (purple dash–dotted line), MHD Stokes–Ekman (red solid line) solutions. Parameters: as in figure 7. See supplementary material S7 and S8 for the other components.

The prefactors $(A_{\!j},B_{\!j})$ in (6.18) and (6.19) follow from the boundary conditions (Appendix D.4) and are additionally provided in the accompanying Zenodo repository Personnettaz & Cébron (Reference Personnettaz and Cébron2026). With rotation, polar forcing generates finite azimuthal velocity and magnetic components (figure 16). For weak rotation (e.g. $\gamma =0.1$ ), DNS results are well reproduced by the MHD Stokes–Ekman model associated with the rotation-corrected potential base flow (4.5) (red-dotted lines). Rotation also deepens the penetration of boundary-layer perturbations relative to the purely viscous skin layer (purple dashed lines) and the viscomagnetic oscillatory solution (dark-green line), as seen near the inner boundary in the left insets of figure 16. Adding the diffusionless particular solution to the homogeneous MHD Stokes–Ekman model (solid red lines) yields an accurate prediction of the azimuthal magnetic perturbation (figure 16 b). At $\gamma =0.1$ , the equatorial modes are accurately captured by the viscomagnetic oscillatory model, which reproduces both polar and azimuthal velocity and magnetic perturbations (dark-green lines in S10). At larger $\gamma$ , the model based on the base flow (4.3) fails to reproduce the DNS for equatorial modes; see figure S11 for $\gamma =0.33$ .

Figure 17.

Meridional (left) and azimuthal (right) velocity (a) and magnetic (b) perturbations near $r=a_s$ (polar mode). XSHELLS (orange-dotted line), inviscid MHD (dark-red dashed line), inviscid MHD with diffusionless correction (cyan dash–dotted line), viscomagnetic oscillatory (thick green line), purely viscous (purple dashed line), homogeneous MHD Stokes–Ekman based on $\boldsymbol{U}_{\!1}$ (red-dotted line), MHD Stokes–Ekman based on $\boldsymbol{U}_{\!1}$ with diffusionless correction (red solid line), MHD Stokes–Ekman based on $\boldsymbol{U}_{B74}$ (grey line) and diffusionless bulk solution (blue dash–dotted line, (B4)). Insets show zoomed in regions near the interfaces. Parameters: as in figure 7, but $\gamma =0.92$ .

For the polar mode, the Stokes–Ekman solution based on the rotation-corrected potential base flow (4.5) loses accuracy at larger $\gamma$ , e.g. $\gamma =0.92$ , as shown by the red curves in figure 17. The discrepancy between the rotation-corrected potential base flow and B74 (dark blue line) is of the same order as the boundary-layer perturbation. Using $\boldsymbol{U}_{B74}$ in the MHD Stokes–Ekman solution (grey lines in figure 17) restores accurate estimates of both polar and azimuthal velocity and magnetic perturbations, notably near $r=a_s$ .