Matsushima (2015) gave a method for estimating fluid flow near the core surface with magnetic diffusion in a viscous boundary layer. Here this theory is recalled in brief, and the method is extended to include not only the effect of the Coriolis force for a tangentially geostrophic flow, but also that of the Lorentz force for a tangentially magnetostrophic flow. The radial component, which is denoted by subscript *r*, of the induction equation is given as

$${\dot{B}}_{ri}=\{-\left({{\varvec{V}}}_{i}\cdot \nabla \right){B}_{ri}+\left({{\varvec{B}}}_{i}\cdot \nabla \right){V}_{ri}\}({\delta }_{i1}+{\delta }_{i2})+\frac{\eta }{{r}_{i}}{\nabla }^{2}\left({r}_{i}{B}_{ri}\right)({\delta }_{i0}+{\delta }_{i1}),$$

(1)

where \({\varvec{B}}\) is the magnetic field, \({\varvec{V}}\) the velocity of incompressible core fluid, and \({\delta }_{ij}\) the Kronecker delta. A dot denotes partial differentiation with respect to time, \(t\). The other subscript, \(i\), indicates the depth to be considered: \(i=0\) at the CMB (assumed to be a spherical surface with radius \(r={r}_{0}=\) 3480 km), \(i=1\) inside the boundary layer (\(r={r}_{1}={r}_{0}-{\xi }_{1}\)) at a depth of \({\xi }_{1}\) from the CMB, and \(i=2\) below the boundary layer (\(r={r}_{2}={r}_{0}-{\xi }_{2}\)) at a depth of \({\xi }_{2}\). For \(i=0\), where the core flow relative to a reference frame rotating with the mantle must vanish under the no-slip condition, the first and second terms of the right-hand side of Eq. (1) must also vanish, and only the third term, or magnetic diffusion term, remains; that is, temporal variations of the magnetic field at the CMB arise from the magnetic diffusion only. For \(i=1\), all three right-hand-side terms contribute to temporal variation of the geomagnetic field. For \(i=2\), the magnetic diffusion term is presupposed to be negligible, although the thickness of the magnetic boundary layer would be thicker than that of the viscous boundary layer (e.g., Chulliat and Olsen 2010), because contribution of the motional induction to temporal variations in the magnetic field is likely to be much larger than that of the magnetic diffusion, as in the frozen-flux approximation (e.g., Holme 2015).

Core flow is assumed to be tangentially geostrophic below the boundary layer, while the viscous force is presumed to play an important role inside the boundary layer. Therefore, core flow \({{\varvec{V}}}_{1}\) and \({{\varvec{V}}}_{2}\) should satisfy Eqs. (2a) and (2b), respectively:

$$\hat{{\varvec{r}}}\cdot \nabla \times (-2{\varvec{\Omega}}\times {{\varvec{V}}}_{1}+{\nu }_{\mathrm{edd}}{\nabla }^{2}{{\varvec{V}}}_{1})=0,$$

(2a)

$$\hat{{\varvec{r}}}\cdot \nabla \times (-2{\varvec{\Omega}}\times {{\varvec{V}}}_{2})=0,$$

(2b)

where \({\varvec{\Omega}}\) denotes the angular velocity vector of the mantle, \(\hat{{\varvec{r}}}\) the radial unit vector, and \({\nu }_{\mathrm{edd}}\) the eddy kinematic viscosity. The typical length scale parallel to the boundary layer is likely to be much larger than the thickness of the boundary layer. The horizontal flow, \({\varvec{V}}_{H}\), near the core surface is then expressed as in classical Ekman layer theory (e.g., Pedlosky 1987),

$$\begin{aligned}{\varvec{V}}_{H}&={\overline{\varvec{V}}}_{H}\left\{1-{\mathrm{exp}}\left(-\frac{\xi }{{\delta }_{E}}\right){\mathrm{cos}}\left(\frac{\xi }{{\delta }_{E}}\right)\right\} \\ & \quad +({\mathrm{sgn cos}}\,\theta )\hat{{\varvec{r}}}\times {\overline{\varvec{V}}}_{H}{\mathrm{exp}}\left(-\frac{\xi }{{\delta }_{E}}\right){\mathrm{sin}}\left(\frac{\xi }{{\delta }_{E}}\right),\end{aligned}$$

(3)

where sgn is the signum function, \({\delta }_{E}={\left({\nu }_{\mathrm{edd}}/\Omega |\mathrm{cos}~\theta |\right)}^{1/2}\) with \(\Omega =\left|{\varvec{\Omega}}\right|=7.29\times {10}^{-5} ~\mathrm{rad}~{\mathrm{s}}^{-1}\), and \(\theta\) is the colatitude in the spherical coordinates \(\left(r, \theta , \phi \right).\) The tangentially geostrophic flow, \({\overline{\varvec{V}}}_{H}\), significantly below the viscous boundary layer should satisfy Eq. (4) obtained from Eq. (2b):

$${\nabla}_{H}\cdot (\mathrm{cos}~\theta ~{\overline{\varvec{V}}}_{H})=0,$$

(4)

where \({\nabla }_{H}\) is the horizontal gradient, and \({\overline{V}}_{r}\) is significantly smaller than \(|\overline{\varvec{V}}_{\it H}|\) and can be neglected. For the case of tangentially geostrophic flow, core electrical conductivity\(\sigma\) has an effect on the magnetic diffusion alone, which leads to second partial derivatives of \({B}_{r}\) at \(r={r}_{0}\) with respect to the radius.

To examine the effect of \(\sigma\) on a core flow model, core flow is next assumed to be tangentially magnetostrophic below the boundary layer, which is an Ekman–Hartmann layer in this case. Therefore, core flow \({{\varvec{V}}}_{1}\) and \({{\varvec{V}}}_{2}\) should satisfy Eqs. (5a) and (5b), respectively:

$$\hat{{\varvec{r}}}\cdot \nabla \times (-2{\varvec{\Omega}}\times {{\varvec{V}}}_{1}+{\rho }^{-1}{{\varvec{J}}}_{1}\times {{\varvec{B}}}_{1}+\nu {\nabla }^{2}{{\varvec{V}}}_{1})=0,$$

(5a)

$$\hat{{\varvec{r}}}\cdot \nabla \times (-2{\varvec{\Omega}}\times {{\varvec{V}}}_{2}+{\rho }^{-1}{{\varvec{J}}}_{2}\times {{\varvec{B}}}_{2})=0,$$

(5b)

where \(\rho\) and \({\varvec{J}}\) denote the mass density of the core fluid and the electric current density, respectively. In this study, as mentioned in Appendix, contribution of the electric field to the current density is ignored (e.g., Shimizu 2006), and \({J}_{r}\) is likely to be much smaller than \(|{{\varvec{J}}}_{H}|\) near the mantle, which is assumed to be an electrical insulator (Benton and Muth 1979). The horizontal component of current density is then given as

$${{\varvec{J}}}_{H}=\sigma {\left({\varvec{V}}\times {\varvec{B}}\right)}_{H}\approx \sigma {B}_{r}{{\varvec{V}}}_{H}\times \hat{{\varvec{r}}}.$$

(6)

The horizontal flow, \({{\varvec{V}}}_{H}\), near the core surface is expressed as

$$\begin{aligned}{{\varvec{V}}}_{H}&={\overline{{\varvec{V}}}}_{H}\left\{1-{\mathrm{exp}}\left(-\frac{\xi }{{\delta }_{EH}^{+}}\right)\mathrm{cos}\left(\frac{\xi }{{\delta }_{EH}^{-}}\right)\right\}\\ & \quad +({\mathrm{sgn\ cos}}\,\theta )\hat{\varvec{r}}\times {\overline{{\varvec{V}}}}_{H}\mathrm{exp}\left(-\frac{\xi }{{\delta }_{EH}^{+}}\right)\mathrm{sin}\left(\frac{\xi }{{\delta }_{EH}^{-}}\right),\end{aligned}$$

(7)

and the tangentially magnetostrophic flow, \({\overline{\varvec{V}}}_{H}\), significantly below the viscous boundary layer satisfies

$${\nabla}_{H}\cdot (2\Omega {\mathrm{cos}}\ \theta {\overline{{\varvec{V}}}}_{H}+{\rho }^{-1}\sigma {B}_{r2}^{2}{\overline{\varvec{V}}}_{H}\times {\hat{\varvec{r}}})=0.$$

(8)

Here, \({\delta }_{EH}^{+}\) and \({\delta }_{EH}^{-}\) are given by

$${\delta }_{EH}^{\pm }=\frac{{\delta }_{E}}{{\{{\left(1+{\Lambda }^{2}/4\right)}^{1/2}\pm\Lambda /2\}}^{1/2}}$$

(9)

(double sign correspondence), and

$$\Lambda =\frac{\sigma {B}_{r}^{2}}{\rho\Omega |\mathrm{cos}~\theta |}$$

(10)

is the Elsasser number. For the case of tangentially magnetostrophic flow, core electrical conductivity \(\sigma\) has an effect not only on the magnetic diffusion, but also on the magnetostrophy through the Lorentz force.

The horizontal geostrophic (or magnetostrophic) velocity can be expressed in terms of poloidal and toroidal constituents as

$${\overline{\varvec{V}}}_{H}=r{\nabla }_{H}\overline{U}+\nabla \times ({\varvec{r}}\overline{W}),$$

(11)

$$\overline{U}\left(\theta ,\phi ,t\right)=\sum_{l=1}^{L}\sum_{m=0}^{l}\left\{{\overline{U}}_{l}^{mc}\left(t\right)\mathrm{cos}~{\it m}\phi +{\overline{U}}_{l}^{\it ms}\left(t\right)\mathrm{sin}~m\phi \right\}{P}_{l}^{\it m}(\mathrm{cos}~\theta ),$$

(12a)

$$\overline{W}\left(\theta ,\phi ,t\right)=\sum_{l=1}^{L}\sum_{m=0}^{l}\left\{{\overline{W}}_{l}^{mc}\left(t\right)\mathrm{cos}~m\phi +{\overline{W}}_{l}^{ms}\left(t\right)\mathrm{sin}~m\phi \right\}{P}_{l}^{m}(\mathrm{cos}~\theta ),$$

(12b)

where \({P}_{l}^{m}\) is a Schmidt-normalized associated Legendre function of degree \(l\) and order \(m\), \(L\) is the truncation level, \({\varvec{r}}\) is a position vector, and \(\overline{U}\) and \(\overline{W}\) are poloidal and toroidal scalar functions, respectively.