Magnetohydrodynamic Turbulence in a Spherical Shell, Part 2: Emergent Magnetic Field from a Turbulent Geodynamo

Abstract

Using established results, we examine how a turbulent magnetic field in an outer core emerges and manifests itself as the geomagnetic field. Two basic results are demonstrated: First, how the stationary interior magnetic dipole components gain fluctuating parts, leading to polar wander of the geomagnetic dipole. Second, how the relation between the interior dipole energy ${\mathcal{E}}_D$ and magnetic helicity ${\mathcal{H}}_M$, i.e., ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$, permits us to estimate the value of ${\mathcal{H}}_M$ in the outer core from the strength of the geomagnetic dipole field. We also discuss how MHD turbulence with magnetic helicity may be seen as the essential engine of the geodynamo.

1. Introduction

In a recent paper [1], the measured surface geomagnetic field was not explicitly connected to the turbulent magnetic field within the outer core. That is what we will do in this paper: examine how the turbulent magnetic field emerges and manifests itself as the geomagnetic field. This will allow us to demonstrate two things: first, how the stationary interior magnetic dipole component gains a fluctuating part, and second, how the relation between interior dipole energy ${\mathcal{E}}_D$ and magnetic helicity ${\mathcal{H}}_M$, i.e., ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$ (developed previously [2,3]) permits us to estimate the value of ${\mathcal{H}}_M$ in the outer core from the geomagnetic dipole field strength. Following this, we will discuss how helical MHD turbulence may be seen as the essential engine of the geodynamo.

A recent article [1] discussed the relation that Dirichlet and Neumann boundary conditions (b.c.s) had with Galerkin expansions of the magnetic field and velocity of a magnetofluid contained in a spherical shell. These Galerkin expansions, based on spherical Bessel and Neumann functions of radius, along with spherical harmonic functions of colatitude and azimuth, have been investigated by many authors [4,5,6,7,8]. Models based on these expansions, along with the statistical mechanics of magnetohydrodynamic (MHD) turbulence [9,10,11], are essential for understanding the origin of the geodynamo and its dominant dipole component [2,8].

We chose Dirichlet b.c.s in [1] for several reasons: one, because numerical simulations have shown that the interior magnetic field is predominantly transverse as it approaches a boundary, then exhibits a kink to develop a non-zero normal component [12,13]; two, satellite observations lead to profiles of mid-mantle electrical conductivity [14,15]; and three, the numerical requirement that a large electrical conductivity is necessary just above the core–mantle boundary (CMB) to ensure realistic behavior, including a reversal, of the geomagnetic field [12,13]. Recognizing the importance of mantle electrical conductivity allowed for connecting the geomagnetic field measured on the Earth’s surface [16] to one at the CMB that appeared more realistic than not doing so [17].

Thus, we also apply Dirichlet b.c.s here. At any rate, the actual structure of the outer core and nearby mantle is not well known, although significant progress has been made in this regard [18]. There are other phenomena, in addition to outer core turbulence, that affect geomagnetism: differential rotation, precession, variability of heat flow, etc.; these will be discussed in detail in Section 4. If these other phenomena become important, they could move turbulence in the outer core away from a stationary, equilibrium state and destabilize the geodynamo, leading to possible reversals or excursions of the dipole field. However, if the system is in a stationary state, one describable in terms of equilibrium statistical mechanics [2], then MHD turbulence is of primary importance. In such a state, MHD turbulence may be viewed as the essential engine of the geodynamo.

The basic model that has been developed to explain the origin and dynamics of the geomagnetic dipole field has four parts: first, use has been made of the aforementioned Galerkin expansions; second, the principles of statistical mechanics [19,20,21] have been applied to ideal MHD turbulence [9,10,11,22,23]; third, statistical results have been validated in real (forced and dissipative) cases using Fourier-method numerical simulations [3,24,25]; and fourth, a recognition of the effect that mantle electrical conductivity can have on boundary conditions [1,17]).

Here, we build on the results presented in [1]. This will allow for a determination of how a stationary interior magnetic dipole can gain fluctuations, leading to polar wander of the geomagnetic dipole. Then, the relation between interior dipole energy ${\mathcal{E}}_D$ and magnetic helicity ${\mathcal{H}}_M$, i.e., ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$, will permit us to estimate the value of ${\mathcal{H}}_M$ in the outer core from the strength of the observed geomagnetic dipole field. To begin, we will briefly review the basic results that were presented in [1] and that are necessary for developing the results of this paper.

2. Magnetic Field Inside a Spherical Shell with Dirichlet Boundary Conditions

For clarity, we present some results that have appeared previously [1] and are included here for ease of reference during the analytical developments in Section 3.

The magnetic field $\mathbf{b}$ and other divergenceless vector fields in a spherical geometry are typically represented by a ‘poloidal–toroidal decomposition’ [26]:

$$\begin{array}{ccc}\hfill \mathbf{b}(\mathbf{x},t)& =& \nabla \times \left[\mathbf{r}B\right(r,\theta ,\phi ,t\left)\right]+\nabla \times (\nabla \times [\mathbf{r}A(r,\theta ,\phi ,t)\left]\right).\hfill \end{array}$$

(1)

Commensurate with the physical dimensions of the Earth [27], for which the inner-core boundary (ICB) is $R_{ICB}=1220$ km and the CMB is $R_{CMB}=3480$ km, we use $R_{ICB}$ as a standard of distance, so that the non-dimensional coordinates are r, $\theta$ and $\phi$, where $1\le r\le r_{\mathrm{o}}=2.85$, $0\le \theta \le \pi$ and $0\le \phi \le 2\pi$ for the spherical shell representing the outer core.

2.1. Galerkin Expansion Basis Functions

We want to expand B and A in (1) in terms of a complete set of orthogonal basis functions, called a Galerkin expansion:

$$\begin{array}{ccc}\hfill B(r,\theta ,\varphi ,t)=\sum _{l,m,n}{b}_{lmn}\left(t\right){\widehat{g}}_l\left(k_{ln}r\right){\mathrm{Y}}_{lm}(\theta ,\phi ),& & A(r,\theta ,\varphi ,t)=\sum _{l,m,n}{a}_{lmn}\left(t\right){\widehat{g}}_l\left(k_{ln}r\right){\mathrm{Y}}_{lm}(\theta ,\phi ).\hfill \end{array}$$

(2)

Above, the ${\mathrm{Y}}_{lm}(\theta ,\phi )$ are spherical harmonics and the ${\widehat{g}}_l\left(k_{ln}r\right)$ are normalized linear combinations of spherical Bessel functions of the first kind, $j_l\left(z\right)$, and the second kind, $n_l\left(z\right)$ (called spherical Neumann functions), where $z=k_{ln}r$. The exact form of the ${\widehat{g}}_l\left(k_{ln}r\right)$ will be given below and the wavenumbers $k_{ln}$ will be determined by applying Dirichlet b.c.s. An essential property of the functions in (2) is that they satisfy Helmholtz’s equation:

$${\nabla }^2{\widehat{g}}_l\left(k_{ln}r\right){\mathrm{Y}}_{lm}(\theta ,\phi )=-k_{ln}^2{\widehat{g}}_l\left(k_{ln}r\right){\mathrm{Y}}_{lm}(\theta ,\phi ).$$

(3)

The Galerkin method requires that when (2) is put into (1), individual terms in the resulting expansions each satisfy the boundary conditions (b.c.s). Placing (2) into (1) produces the Galerkin poloidal–toroidal expansions

$$\begin{array}{ccc}\hfill \mathbf{b}(\mathbf{x},t)& =& \sum _{l,m,n}\left[b_{lmn}\left(t\right){\mathbf{T}}_{lmn}\left(\mathbf{x}\right)+a_{lmn}\left(t\right){\mathbf{P}}_{lmn}\left(\mathbf{x}\right)\right],\hfill \end{array}$$

(4)

$${\mathbf{T}}_{lmn}\left(\mathbf{x}\right)=c_{lmn}{\widehat{g}}_l\left(k_{ln}r\right)\mathbf{r}\times \nabla {\mathrm{Y}}_{lm}(\theta ,\phi ),$$

(5)

$$\nabla \times {\mathbf{T}}_{lmn}\left(\mathbf{x}\right)={\mathbf{P}}_{lmn}\left(\mathbf{x}\right),\nabla \times {\mathbf{P}}_{lmn}\left(\mathbf{x}\right)=k_{ln}^2{\mathbf{T}}_{lmn}\left(\mathbf{x}\right),$$

(6)

$$b_{l,-m,n}\left(t\right)={(-1)}^m{b}_{lmn}^{\ast }\left(t\right),a_{l,-m,n}\left(t\right)={(-1)}^m{a}_{lmn}^{\ast }\left(t\right).$$

(7)

In (5), $c_{lmn}$ is a normalizing constant, to be defined presently, and the overall minus sign, which occurred because $\nabla \times \mathbf{r}{\mathrm{Y}}_{lm}=-\mathbf{r}\times \nabla {\mathrm{Y}}_{lm}$, has been dropped; also, * denotes complex conjugation.

The expansion (4) has zero divergence since $\nabla ·{\mathbf{T}}_{lmn}\left(\mathbf{x}\right)=\nabla ·{\mathbf{P}}_{lmn}\left(\mathbf{x}\right)=0$, while (7) is required by the realness of $\mathbf{b}$.

Definitions of the toroidal and poloidal vector basis functions ${\mathbf{T}}_{lmn}\left(\mathbf{x}\right)$ and ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$ in terms of vector spherical harmonics are given below. As discussed previously [1], ${\mathbf{T}}_{lmn}\left(\mathbf{x}\right)$ and ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$ form an orthogonal set for the range $1\le r\le r_{\mathrm{o}}$, where each member satisfies the b.c.s (thus, a Galerkin expansion); the wavenumbers $k_{ln}$ are numerical constants with the summation indices $l,m$ and n and range over prescribed, finite sets of integer values. Therefore, we have two sets of complex coefficients, $b_{lmn}$ and $a_{lmn}$, representing the magnetic field.

In terms of the wavenumbers $k_{ln}$, the vector functions ${\mathbf{T}}_{lmn}\left(\mathbf{x}\right)$ and ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$ are defined as follows:

$$\begin{array}{ccc}\hfill {\mathbf{T}}_{lmn}\left(\mathbf{x}\right)& =& {\widehat{g}}_l\left(k_{ln}r\right){\widehat{\mathbf{\Phi }}}_{lm}(\theta ,\phi ),1\le l\le L,-l\le m\le l,1\le n\le N,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\mathbf{P}}_{lmn}\left(\mathbf{x}\right)& =& -{\displaystyle \frac{1}{r}}\left[\sqrt{l(l+1)}{\widehat{g}}_l\left(k_{ln}r\right){\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )+{\displaystyle \frac{d}{dr}}\left[r{\widehat{g}}_l\left(k_{ln}r\right)\right]{\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi )\right].\hfill \end{array}$$

(9)

The vector functions ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )$, ${\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi )$ and ${\widehat{\mathbf{\Phi }}}_{lm}(\theta ,\phi )$ are the normalized, orthonormal vector spherical harmonics of [28] which are based on the spherical harmonics ${\mathrm{Y}}_{l,m}(\theta ,\phi )$ defined in [29]:

$$\begin{array}{ccc}\hfill {\mathrm{Y}}_{l,-m}(\theta ,\phi )& =& {(-1)}^m{\mathrm{Y}}_{lm}^{\ast }(\theta ,\phi ),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )& =& \widehat{\mathbf{r}}{\mathrm{Y}}_{lm}(\theta ,\phi ),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi )& =& {\displaystyle \frac{r}{\sqrt{l(l+1)}}}\nabla {\mathrm{Y}}_{lm}(\theta ,\phi ),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\widehat{\mathbf{\Phi }}}_{lm}(\theta ,\phi )& =& \widehat{\mathbf{r}}\times {\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi ).\hfill \end{array}$$

(13)

The ${\widehat{g}}_l\left(k_{ln}r\right)$ in (8) and (9) are functions of the radius and are an orthonormal set [1] for $n=1,2,\dots ,N$ separately at each value of l for $1\le r\le r_{\mathrm{o}}$; the ${\widehat{g}}_l\left(k_{ln}r\right)$ are linear combinations of spherical Bessel and Neumann functions, as will be explicitly shown in the next section.

Expressions (8)–(13) do appear elsewhere [1] but are reproduced here for ease of reference as they need to be referred to in order to show the efficacy of the Dirichlet b.c.s in matching up with external fields. In particular, the Dirichlet condition (14), to be discussed presently, ensures that the toroidal functions (8) all go to zero on the boundaries, while the poloidal parts (9) only have a transverse part proportional to (12), which allows the outer core magnetic field to be connected immediately to the lower mantle magnetic potential field, without requiring the existence of an induced surface electric current due to the toroidal parts (8), as would be the case for Neumann b.c.s.

2.2. Dirichlet Conditions on Both Boundaries

As discussed in earlier work [1,17], Dirichlet b.c.s seem most appropriate, along with mantle electrical conductivity, as these allow the geomagnetic field to be mapped down to a realistic turbulent magnetic spectrum at the CMB. These b.c.s also seem pertinent to the ICB as the inner core has very high electrical conductivity. Dirichlet b.c.s are satisified when the radial expansion functions ${\widehat{g}}_l\left(k_{ln}r\right)$ satisfy

$${\widehat{g}}_l\left(k_{ln}{r}_{\mathrm{o}}\right)={\widehat{g}}_l\left(k_{ln}\right)=0.$$

(14)

A suitable representation [1,8], in terms of spherical Bessel function $j_l\left(z\right)$ and spherical Neuman functions $n_l\left(z\right)$ has been found to be

$$\begin{array}{ccc}\hfill {\widehat{g}}_l\left(k_{ln}r\right)=N_{ln}^{-1}{g}_l\left(k_{ln}r\right),& & g_l\left(k_{ln}r\right)=n_l\left(k_{ln}{r}_{\mathrm{o}}\right)j_l\left(k_{ln}r\right)-j_l\left(k_{ln}{r}_{\mathrm{o}}\right)n_l\left(k_{ln}r\right)\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill N_{ln}^2& =& {\displaystyle \frac{1}{2k_{ln}^4{r}_{\mathrm{o}}^2}}\left(1-r_{\mathrm{o}}{\left[{\displaystyle \frac{n_l\left(k_{ln}{r}_{\mathrm{o}}\right)}{n_l\left(k_{ln}\right)}}\right]}^2\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill g_l\left(k_{ln}\right)& =& 0,n=1,2,\dots ,N;l=1,2,\dots .\hfill \end{array}$$

(17)

By definition, $g_l\left(k_{ln}{r}_{\mathrm{o}}\right)\equiv 0$, and requiring that (17) holds determines $k_{ln}$, which must satisfy

$$g_l\left(k_{ln}\right)=n_l\left(k_{ln}{r}_{\mathrm{o}}\right)j_l\left(k_{ln}\right)-j_l\left(k_{ln}{r}_{\mathrm{o}}\right)n_l\left(k_{ln}\right)=0.$$

(18)

These results, combined with with (12) and (13), indicate that, in (5), $c_{lmn}={\left(\sqrt{l(l+1)}{N}_{ln}\right)}^{-1}$. The wavenumbers $k_{ln}$ in (18) obviously depend on $r_{\mathrm{o}}\equiv 2.85$, as is appropriate for the CMB. The $k_{ln}$ can be found numerically [30]; they are given in

Table 1. Values of $k_{ln}$, $g_l\left(k_{ln}{r}_{\mathrm{o}}\right)=g_l\left(k_{ln}\right)=0$, with $l,n=1,\dots ,10$, for Dirichlet boundary conditions.

n =	1	2	3	4	5	6	7	8	9	10
$l=1$	1.8638	3.4929	5.1612	6.8434	8.5316	10.2231	11.9165	13.6110	15.3063	17.0022
2	2.1497	3.6788	5.2927	6.9440	8.6129	10.2912	11.9750	13.6623	15.3520	17.0433
3	2.5042	3.9411	5.4851	7.0931	8.7339	10.3928	12.0625	13.7391	15.4204	17.1050
4	2.8910	4.2636	5.7330	7.2884	8.8935	10.5273	12.1785	13.8410	15.5112	17.1869
5	3.2898	4.6294	6.0300	7.5271	9.0903	10.6938	12.3225	13.9678	15.6244	17.2890
6	3.6911	5.0227	6.3684	7.8062	9.3227	10.8915	12.4940	14.1190	15.7595	17.4111
7	4.0909	5.4310	6.7396	8.1218	9.5890	11.1194	12.6923	14.2942	15.9162	17.5528
8	4.4882	5.8457	7.1347	8.4695	9.8870	11.3762	12.9167	14.4929	16.0943	17.7140
9	4.8828	6.2618	7.5452	8.8441	10.2145	11.6608	13.1664	14.7145	16.2932	17.8943
10	5.2749	6.6767	7.9642	9.2399	10.5685	11.9718	13.4405	14.9586	16.5127	18.0935

for $l,n=1,\dots ,10$.

Since $\widehat{\mathbf{r}}·\mathbf{b}=0$ at $r=r_{\mathrm{o}}$ and 1 for Dirichlet b.c.s, $\mathbf{b}$ in the outer core has been connected to the geomagnetic field by observing that some electrical conductivity exists in the mantle [14,15]. This connection was achieved by replacing volumetric mantle electrical conductivity by an electrically conductive spherical surface placed located between the Earth’s surface and the CMB [17]. When the Earth’s surface geomagnetic field, i.e., the International Geomagnetic Reference Field (IGRF) [16], is then mapped down to the CMB, the resulting spectrum on the CMB matches much more closely with a turbulent MHD spectrum than otherwise [1,17]. Thus, qualitatively, validity is given to using Dirichlet b.c.s on the CMB, as well as our specific model of mantle electrical conductivity. As for a b.c. on the ICB, high electrical conductivity of the inner core may be expected to exclude, to a good approximation, magnetic fields from the inner core, so that a Dirichlet b.c. is also approriate on the ICB.

3. The Magnetic Field

Again, for clarity, we present some previously reported results [17] that are included here for ease of reference during the analytical developments in Section 3.4, Section 3.5 and Section 3.6.

3.1. Magnetic Field and Electric Current Inside the Spherical Shell

The magnetic field $\mathbf{b}$ is given by (4) and we present it again here:

$$\begin{array}{ccc}\hfill \mathbf{b}(\mathbf{x},t)& =& \sum _{l,m,n}\left[b_{lmn}\left(t\right){\mathbf{T}}_{lmn}\left(\mathbf{x}\right)+a_{lmn}\left(t\right){\mathbf{P}}_{lmn}\left(\mathbf{x}\right)\right],\hfill \end{array}$$

(19)

Taking the curl of $\mathbf{b}$ and using (5) and (6) gives us the electric current density $\mathbf{j}$:

$$\begin{array}{ccc}\hfill \mathbf{j}(\mathbf{x},t)=\nabla \times \mathbf{b}(\mathbf{x},t)& =& \sum _{l,m,n}\left[k_{ln}^2{a}_{lmn}\left(t\right){\mathbf{T}}_{lmn}\left(\mathbf{x}\right)+b_{lmn}\left(t\right){\mathbf{P}}_{lmn}\left(\mathbf{x}\right)\right].\hfill \end{array}$$

(20)

At the CMB and ICB, $r_b=r_{\mathrm{o}}$ and $r_b=1$, respectively, and the normal components of $\mathbf{b}$ and $\mathbf{j}$ are

$$\widehat{\mathbf{r}}·{\left[\begin{array}{c}\mathbf{b}\\ \mathbf{j}\end{array}\right]}_{r_{\mathrm{o}},1}=-\sum _{l,m,n}{\displaystyle \frac{\sqrt{l(l+1)}}{r_{\mathrm{o}}}}\left[\begin{array}{c}{a}_{lmn}\\ b_{lmn}\end{array}\right]{\widehat{g}}_l\left(k_{ln}{r}_b\right){\mathrm{Y}}_{lm}(\theta ,\phi ).$$

(21)

For Dirichlet b.c.s, ${\widehat{g}}_l\left(k_{ln}{r}_b\right)=0$, so that $\widehat{\mathbf{r}}·\mathbf{b}=\widehat{\mathbf{r}}·\mathbf{j}=0$. In the model we have adopted, $\widehat{\mathbf{r}}\times \mathbf{b}$ is continuous at $r_b=r_{\mathrm{o}}$ but discontinuous at $r_b=1$, while $\widehat{\mathbf{r}}\times \mathbf{j}$ is discontinuous at both $r_b=r_{\mathrm{o}}$ and $r_b=1$. Our reasoning follows from our electrical conductivity model for the mantle [1,17] and from the presumed expulsion of magnetic flux and electric current from the highly conductive inner core.

3.2. Magnetic Field Outside the Spherical Shell

Here, we review our model of the electrically conductive mantle [1,17].

On the Earth’s surface, $r_e=5.22$, and assuming electrical conductivity is negligible, a magnetic potential field expansion is appropriate:

$$\mathbf{B}=-\nabla r_e\sum _{l,m}{B}_{lm}\left(t\right){\left({\displaystyle \frac{r_e}{r}}\right)}^{l+1}{\mathrm{Y}}_{lm}(\theta ,\phi ).$$

(22)

Here, we use coefficients $B_{lm}$ that are linearly related to the Schmidt quasi-normalized [31] Gauss coefficients $g_{lm}$ and $h_{lm}$ used to define the International Geomagnetic Reference Field (IGRF) [16]. At $r=r_{\mathrm{o}}$, $\mathbf{B}$ has the normal component

$$\widehat{\mathbf{r}}{·\mathbf{B}|}_{r_{\mathrm{o}}}=\sum _{l,m}(l+1)B_{lm}^{\mathrm{o}}\left(t\right){\mathrm{Y}}_{lm}(\theta ,\phi ),B_{lm}^{\mathrm{o}}\left(t\right)=B_{lm}\left(t\right){\left({\displaystyle \frac{r_e}{r_{\mathrm{o}}}}\right)}^{l+2},B_{lm}=\sqrt{{\displaystyle \frac{4\pi c_m}{2l+1}}}\left(g_{lm}-i{h}_{lm}\right).$$

(23)

Here, $g_{lm}=g_{l,-m}$, $h_{lm}=-h_{l,-m}$ and $c_m=\frac{1}{2}$ for $m\ne 0$, while $c_0=1$.

The Mauersberger–Lowes spectrum $R_l\left(r_e\right)$ of the IGRF [16,31] is

$${\displaystyle \frac{1}{4\pi }}{\int }_0^{2\pi }{\int }_0^{\pi }{\left|\mathbf{B}\right|}^{2}sin\theta d\theta d\varphi =\sum _{l=1}^L{R}_l\left(r\right),$$

(24)

$$R_l\left(r\right)={\left({\displaystyle \frac{r_e}{r}}\right)}^{2l+4}{R}_l\left(r_e\right)R_l\left(r_e\right)=(l+1)\sum _{m=0}^l\left[{\left(g_{lm}\right)}^2+{\left(h_{lm}\right)}^2\right].$$

(25)

If this is mapped down to the CMB, then $R_l\left(r_{\mathrm{o}}\right)={\left(r_e/r_{\mathrm{o}}\right)}^{2l+4}{R}_l\left(r_e\right)$ results, i.e., a magnetic power spectrum at $r=r_{\mathrm{o}}$ that is much too flat for a non-electrically conductive mantle [17].

The mid-mantle does, however, have significant electrical conductivity $\sigma \left(r\right)$, as measured by [14,15]. Between the mid-mantle and the CMB, there was no data, so we modeled $\sigma \left(r\right)$ from the observed ${\sigma }_s=$ 4 S/m at $r_s=5371.2$ km (1000 km deep) and increasing exponentially down to the CMB [17]. Specifically,

$$\begin{array}{c}\hfill \sigma \left(r\right)={\sigma }_{s}exp\left[\alpha \left(r_s-r\right)\right],r_{\mathrm{o}}\le r\le r_s,\alpha ={\displaystyle \frac{log\left({\sigma }_{\mathrm{o}}/{\sigma }_s\right)}{r_s-r_{\mathrm{o}}}},{\sigma }_{\mathrm{o}}=\sigma \left(r_{\mathrm{o}}\right),\end{array}$$

(26)

$$\begin{array}{c}\hfill {\sigma }_c={\int }_{r_{\mathrm{o}}}^{r_s}\sigma \left(r\right)dr={\displaystyle \frac{{\sigma }_{\mathrm{o}}-{\sigma }_s}{\alpha }},r_c={\int }_{r_{\mathrm{o}}}^{r_s}r\sigma \left(r\right)dr=r_{\mathrm{o}}+{\displaystyle \frac{1}{\alpha }}\left[1-{\displaystyle \frac{{\sigma }_s}{{\sigma }_c}}(r_s-r_{\mathrm{o}})\right].\end{array}$$

(27)

The integration of $\sigma \left(r\right)$ over $r_{\mathrm{o}}\le r\le r_s$ gives an effective conductance ${\sigma }_c$ (S) residing on a spherical surface at $r=r_c$, where $r_{\mathrm{o}}<r_c<r_s$.

The exponential profile $\sigma \left(r\right)$ in (26) was chosen because, although the conductivity at $r=r_s$ was measured as ${\sigma }_s=$ 4 S/m, it was found that a high conductivity just above the CMB (equivalent to outer core conductivity) was needed to give realistic results in a numerical simulation of the geodynamo [12,13]. In lieu of actual knowledge of conductivity below the mid-mantle, an exponential profile was adopted as it gave a smooth variation from a low value to a high value of conductivity (though the high values of ${\sigma }_{\mathrm{o}}$ in [17] were much less than the expected very high conductivity of the outer core [27]).

In this model, between the CMB and the effective surface current at $r=r_c$ is a magnetic potential field of the form

$$\begin{array}{c}\hfill \mathit{\beta }=-\nabla r_e\sum _{l,m}{f}_l\left(r\right)H_{lm}\left(t\right){\mathrm{Y}}_{lm}(\theta ,\phi ),\end{array}$$

(28)

$$\begin{array}{c}\hfill f_l\left(r\right)={\left({\displaystyle \frac{r_e}{r}}\right)}^{l+1}+{\displaystyle \frac{l+1}{l}}{\left({\displaystyle \frac{r_e}{r_{\mathrm{o}}}}\right)}^{2l+1}{\left({\displaystyle \frac{r}{r_e}}\right)}^l.\end{array}$$

(29)

The functional form of $f_l\left(r\right)$ comes from the Case DD requirement that $\widehat{\mathbf{r}}·\mathit{\beta }=0$ at $r=r_{\mathrm{o}}$. Using (22) and (28) and requiring that $\widehat{\mathbf{r}}·\left(\mathbf{B}-\mathit{\beta }\right)=0$ at $r=r_c$ gives

$$\begin{array}{c}\hfill B_{lm}=\left[1-{\left({\displaystyle \frac{r_c}{r_{\mathrm{o}}}}\right)}^{2l+1}\right]H_{lm}.\end{array}$$

(30)

Thus, the magnetic power spectrum at the CMB becomes

$${\displaystyle \frac{1}{4\pi }}{\int }_0^{2\pi }{\int }_0^{\pi }{\left|\mathit{\beta }\right|}^{2}sin\theta d\theta d\varphi =\sum _{l=1}^L{R}_l^c\left(r_{\mathrm{o}}\right),$$

(31)

$$R_l^c\left(r_{\mathrm{o}}\right)={\left({\displaystyle \frac{r_e}{r_{\mathrm{o}}}}\right)}^{2l+4}{F}_l\left(r_c\right)R_l\left(r_e\right)F_l\left(r_c\right)={\displaystyle \frac{2l+1}{l}}{\left[{\left({\displaystyle \frac{r_c}{r_{\mathrm{o}}}}\right)}^{2l+1}-1\right]}^{-2}.$$

(32)

Comparing (25) with (32), we see that once mantle conductivity is taken into account, a factor $F_l\left(r_c\right)$ is gained which was found to correct the flatness problem. In [17], a number of different values for $\sigma \left(r_{\mathrm{o}}\right)$ were examined and each produced a corresponding value of ${\sigma }_c$ and $r_c$; all of these gave a similar correction. Figure 1 of [1] illustrates this, where representative values from [17] are used with ${\sigma }_{\mathrm{o}}={10}^3$ S/m at $r_{\mathrm{o}}=2.85$, so that ${\sigma }_c=341$ kS and $r_c=3.13$ (corresponding to a dimensional $R_c=3815$ km).

Now, Figure 1 of [1] also includes bounds on Kolmogorov spectra that are predicted for a turbulent fluid [32]. These inertial range energy spectra vary as $k^{-5/3}$, with k being the associated wavenumber. Comparing Kolmogorov spectra to geomagnetic spectra presents a challenge, as the IGRF power spectrum is given in terms of spherical harmonic order l, but each l in a Galerkin expansion of the outer core magnetic field has many different wavenumbers associated with it, as seen in Table 1. As representative sets of wavenumbers for each $l=1,2,\dots ,13$, we used both $k_l^{\left(1\right)}=k_{l1}$ and $k_l^{\left(2\right)}=k_{ll}$ from Table 1. The position of these Kolmogorov spectra is arbritrary in Figure 1 of [1], but placed so as to give easy comparison. Qualitatively, Figure 1 of [1] and Figure 2 of [17] are important because they show that the IGRF spectrum maps down to the CMB more closely to what is expected if mantle electrical conductivity is included than otherwise.

3.3. Connecting the Geomagnetic Field to the Outer Core Poloidal Field

The chosen b.c. $\widehat{\mathbf{r}}·\mathbf{b}=0$ at $r=r_{\mathrm{o}}$ is satisfied by the outer core magnetic field $\mathbf{b}$ given in (4) due to the Dirichlet condition (14), i.e., ${\widehat{g}}_l\left(k_{ln}{r}_{\mathrm{o}}\right)=0$, and because of the three vector spherical harmonics (11)–(13), ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )$, appearing in expression (9) for ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$, has only a radial component. Thus, using the explicit forms (9) and (11) for ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$ and ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )$, respectively, we have

$$\widehat{\mathbf{r}}·\mathbf{b}(\mathbf{x},t)=\sum _{l,m,n}{a}_{lmn}\left(t\right)\widehat{\mathbf{r}}·{\mathbf{P}}_{lmn}\left(\mathbf{x}\right)=-{\displaystyle \frac{1}{r}}\sum _{l,m,n}\sqrt{l(l+1)}{a}_{lmn}\left(t\right){\widehat{g}}_l\left(k_{ln}r\right){\mathrm{Y}}_{lm}(\theta ,\phi ).$$

(33)

Since ${\widehat{g}}_l\left(k_{ln}{r}_{\mathrm{o}}\right)=0$, then $\widehat{\mathbf{r}}{·\mathbf{b}|}_{r_{\mathrm{o}}}=0$.

The intermediate mantle field $\mathit{\beta }$ given by (28) also satisifes $\widehat{\mathbf{r}}·\mathit{\beta }=0$ at $r=r_{\mathrm{o}}$ because the radial function $f_l\left(r\right)$ appearing in (28) satisfies $f_l^{\prime }\left(r_{\mathrm{o}}\right)=0$ via its definition (29). Therefore, the radial components of $\mathbf{b}$ and $\mathit{\beta }$ are continuous across the boundary $r=r_{\mathrm{o}}$ because both are zero there. Now, we examine the transverse components of $\mathbf{b}$ and $\mathit{\beta }$ at $r=r_{\mathrm{o}}$, as these must also be equivalent.

The expression (9) for ${\mathbf{P}}_{lmn}\left(\mathbf{x}\right)$ shows it contains only ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )$ and ${\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi )$. Using (9), along with the definition (11) for ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )$, i.e., ${\widehat{\mathbf{Y}}}_{lm}(\theta ,\phi )=\widehat{\mathbf{r}}{\mathrm{Y}}_{lm}(\theta ,\phi )$, and definition (12) for ${\widehat{\mathbf{\Psi }}}_{lm}(\theta ,\phi )\sim \nabla {\mathrm{Y}}_{lm}(\theta ,\phi )$, leads to

$$\begin{array}{ccc}\hfill \widehat{\mathbf{r}}{\times \mathbf{b}(\mathbf{x},t)|}_{r_{\mathrm{o}}}& =& \sum _{l,m,n}{a}_{lmn}\left(t\right)\widehat{\mathbf{r}}\times {\mathbf{P}}_{lmn}{\left(\mathbf{x}\right)|}_{r_{\mathrm{o}}}\hfill \\ & =& -\sum _{l,m,n}{\displaystyle \frac{a_{lmn}\left(t\right)}{\sqrt{l(l+1)}}}{k}_{ln}{\widehat{g}}_l^{\prime }\left(k_{ln}{r}_{\mathrm{o}}\right)\mathbf{r}\times \nabla {\mathrm{Y}}_{lm}(\theta ,\phi ).\hfill \end{array}$$

(34)

This follows because ${\widehat{g}}_l\left(k_{ln}{r}_{\mathrm{o}}\right)=0$ but $d{\widehat{g}}_l\left(k_{ln}r\right){/dr|}_{r_{\mathrm{o}}}=k_{ln}{\widehat{g}}_l^{\prime }\left(k_{ln}{r}_{\mathrm{o}}\right)\ne 0$.

Using the expression (28) for $\mathit{\beta }$, we find the cross product with $\widehat{\mathbf{r}}$ to be

$$\widehat{\mathbf{r}}{\times \mathit{\beta }|}_{r_{\mathrm{o}}}=-{\displaystyle \frac{r_e}{r_{\mathrm{o}}}}\sum _{l,m}{f}_l\left(r_{\mathrm{o}}\right)H_{lm}\left(t\right)\mathbf{r}\times \nabla {\mathrm{Y}}_{lm}(\theta ,\phi ).$$

(35)

Now, we use (34) and (35), along with the continuity requirement $\widehat{\mathbf{r}}{\times (\mathit{\beta }-\mathbf{b})|}_{r_{\mathrm{o}}}=0$, to connect the coefficients of $\mathit{\beta }$ and $\mathbf{b}$. The result is

$$f_l\left(r_{\mathrm{o}}\right){\displaystyle \frac{r_e}{r_{\mathrm{o}}}}{H}_{lm}=\sum _n{\displaystyle \frac{k_{ln}}{\sqrt{l(l+1)}}}{\widehat{g}}_l^{\prime }\left(k_{ln}{r}_{\mathrm{o}}\right)a_{lmn}.$$

(36)

On the left side of (36), we can use the definition (29) of $f_l\left(r\right)$ to express the factor in front of $H_{lm}$ as

$${\displaystyle \frac{r_e}{r_{\mathrm{o}}}}{f}_l\left(r_{\mathrm{o}}\right)={\displaystyle \frac{2l+1}{l}}{\left({\displaystyle \frac{r_e}{r_{\mathrm{o}}}}\right)}^{l+2}.$$

(37)

Next, from the form of $g_l\left(k_{ln}r\right)$ given in (15), along with a standard formula of spherical Bessel functions (shown in Prob. 11.7.6 of [33]), it follows that $g_l^{\prime }\left(k_{ln}{r}_{\mathrm{o}}\right)=-{\left(k_{ln}{r}_{\mathrm{o}}\right)}^{-2}$. Using this, and since ${\widehat{g}}_l\left(k_{ln}r\right)=N_{ln}^{-1}{g}_l\left(k_{ln}r\right)$, as defined in (15), we have

$${\widehat{g}}_l^{\prime }\left(k_{ln}{r}_{\mathrm{o}}\right)=-{\displaystyle \frac{1}{N_{ln}{\left(k_{ln}{r}_{\mathrm{o}}\right)}^2}}=-{\displaystyle \frac{\sqrt{2}}{r_{\mathrm{o}}\sqrt{1-r_{\mathrm{o}}{\left[\frac{n_l\left(k_{ln}{r}_{\mathrm{o}}\right)}{n_l\left(k_{ln}\right)}\right]}^2}}}.$$

(38)

Finally, we link $H_{lm}$ in (36) to the external geomagnetic field coefficients $B_{lm}$ using (30) to get

$$B_{lm}=\left[1-{\left({\displaystyle \frac{r_c}{r_{\mathrm{o}}}}\right)}^{2l+1}\right]H_{lm}=\sum _n{Q}_{ln}{k}_{ln}{a}_{lmn}.$$

(39)

where the factor $Q_{ln}$ is

$$Q_{ln}={\displaystyle \frac{P_l}{r_{\mathrm{o}}\sqrt{1-r_{\mathrm{o}}{\left[\frac{n_l\left(k_{ln}{r}_{\mathrm{o}}\right)}{n_l\left(k_{ln}\right)}\right]}^2}}},P_l={\displaystyle \frac{1}{2l+1}}\sqrt{{\displaystyle \frac{2l}{l+1}}}\left[{\left({\displaystyle \frac{r_c}{r_{\mathrm{o}}}}\right)}^{2l+1}-1\right]{\left({\displaystyle \frac{r_{\mathrm{o}}}{r_e}}\right)}^{l+2}.$$

(40)

The cofactor $Q_{ln}$ can be evaluated using the values of the $k_{ln}$ given in Table 1, along with the values $r_e=5.22$, $r_c=3.13$ and $r_{\mathrm{o}}=2.85$; the results are shown in Figure 1.

In the expression for $Q_{ln}$ given in (40), the only easily adjustable parameter is $r_c$ since $r_e$ is fixed and $r_{\mathrm{o}}$ is used to generate the wavenumbers $k_{ln}$ given in Table 1; if $r_c=4.82$, then the lines in Figure 1 shift upwards so that $Q_{11}=1$. The values of $Q_{ln}$ are dependent on the conductivity profile of the mantle, which was modeled in [17], where it was noted that the conductivity of the mantle begins at $r_c=4.40$. Thus, $r_c=4.82$ is too large, and $Q_{11}=1$ is also too large as it would imply that the dipole field strength leaving the core matched the surface value. Instead, Figure 1 tells us that the dipole field strength in the core is about 33 times that of the surface, if we accept $r_c=3.13$ as appropriate. This gives the predicition of a dipole field within the outer core of at least 20 gauss, as the $|k_{ln}{a}_{lmn}|$ in (39) are the poloidal field strengths and the toroidal field strengths $|b_{lmn}|$ are expected to have the same magnitude [3].

3.4. The Dipole Field and Its Turbulent Fluctuations

In the theory of ideal MHD turbulence [2], the dipole components $a_{1m1}$ and $b_{1m1}$ at the smallest wavenumber $k_{11}$ are seen to be very large and effectively constant (particularly for $m=0$) such that the stationary part of dipole energy is ${\mathcal{E}}_D=k_{11}\left|{\mathcal{H}}_M\right|$, where ${\mathcal{H}}_M$ is, again, the magnetic helicity in the spherical shell. If we consider the dipole components $B_{1m}$ in (39), we see

$$\begin{array}{c}\hfill B_{1m}=Q_{11}{k}_{ln}{a}_{1m1}+\sum _{n>1}{Q}_{ln}{k}_{ln}{a}_{lmn}.\end{array}$$

(41)

Thus, even though $a_{1m1}$, particularly the $m=0$ component, may have become stationary, the addition of the other, non-stationary terms for $n>1$ will cause unavoidable fluctuations in the geomagnetic dipole field strength. Furthermore, the nonlinear interaction of $a_{1m1}$ with all the other modes $a_{lmn}$ and $b_{lmn}$ in the turbulent magnetofluid will also lead to fluctuations in the geomagnetic dipole field strength, which, if large enough, lead to excursions and possibly reversals. The mode $a_{101}$ creates the rotation-axis-aligned geomagnetic dipole and has an observed [16] and predicted value [8] many orders of magnitude larger than any of the other modes and its fluctuations may be thought of as a kind of Brownian motion.

3.5. Estimating Volume-Averaged Magnetic Helicity in the Outer Core

The relation between magnetic dipole energy ${\mathcal{E}}_D$ and magnetic helicity ${\mathcal{H}}_M$, both per unit volume, is ${\mathcal{E}}_D=k_{11}{\mathcal{H}}_M$. Taking a time average ${〈\cdots 〉}_t$, in the usual sense, of the squared magnitude of (41) and summing over m, where $m=-1,0,1$ for $l=1$, assuming statistical independence of the $a_{1mn}$ for $n>1$, leads to a connection between geomagnetic dipole energy density $E_D$ and outer core dipole energy density ${\mathcal{E}}_D={\left(2{\mu }_{\mathrm{o}}\right)}^{-1}{\sum }_{m=-1}^1\left[|k_{11}{a}_{1m1}{|}^2+{\left|b_{1m1}\right|}^2\right]={\mu }_{\mathrm{o}}^{-1}{\sum }_{m=-1}^1{\left|k_{11}{a}_{1m1}\right|}^2$, since the statistical prediction given in [2] is $〈|k_{1n}{a}_{1mn}{|}^2〉=〈|b_{1mn}{|}^2〉$,

$$\begin{array}{cc}\hfill E_D={\displaystyle \frac{1}{2{\mu }_{\mathrm{o}}}}\sum _{m=-1}^1{\left|B_{1m}\right|}^2& {\displaystyle =\frac{1}{{\mu }_{\mathrm{o}}}}{Q}_{11}^2\sum _{m=-1}^1{〈|k_{11}{a}_{1m1}{|}^2〉}_t+{\displaystyle \frac{1}{{\mu }_{\mathrm{o}}}}\sum _{n>1}{Q}_{1n}^2\sum _{m=-1}^1{〈|k_{ln}{a}_{1mn}{|}^2〉}_t\hfill \\ & =Q_{11}^2{\mathcal{E}}_D+{\displaystyle \frac{1}{{\mu }_{\mathrm{o}}}}\sum _{m=-1}^1\sum _{n>1}{Q}_{1n}^2{〈|k_{ln}{a}_{1mn}{|}^2〉}_t.\hfill \end{array}$$

(42)

Dimensionally, $k_{11}\to K_{11}=k_{11}/R_{ICB}=1.55\times {10}^{-6}$ m⁻¹; thus, ${\mathcal{E}}_D=E_D/Q_{11}^2=1.74\times {10}^5$ J/m³. Using the IGRF values of $g_{lm}$ and $h_{lm}$ [16], along with (23), leads to the following estimate of the outer core volume-averaged $H_M$:

$$H_M\approx E_D/\left(K_{11}{Q}_{11}^2\right)=1.12\times {10}^{11}{\mathrm{J}/\mathrm{m}}^2.$$

(43)

The large numerical value is due to the denominator $K_{11}{Q}_{11}^2=1.47\times {10}^{-14}$.

3.6. Magnetic Energy Spectrum and IGRF Power Spectrum

Here, we take a slight diversion from the main thrust of this paper. We note, for clarity, that the IGRF power spectrum [16] is not the energy spectrum of the geomagnetic field, as the former is found by integrating over the surface $r=r_e$ and the latter by integrating over the volume $r\ge r_{\mathrm{o}}$. The integral over the volume is split into two parts, volume $V_1$: $r\ge r_c$ and volume $V_2$: $r_c\ge r\ge r_{\mathrm{o}}$; thus, the magnetic energy above the outer core is

$$E_M=\frac{1}{2}{\int }_{V_1}{\left|\mathbf{B}\right|}^{2}dV+\frac{1}{2}{\int }_{V_2}{\left|\mathit{\beta }\right|}^{2}dV.$$

(44)

(We omit the factor $1/{\mu }_{\mathrm{o}}$ as this is omitted in the definition of the IGRF power spectrum.)

Now, using the Gauss divergence theorem on both parts of (44), along with the definitions of $\mathbf{B}$ and $\mathit{\beta }$, given by (22) and (28), along with (23), (29) and (30), gives us

$$E_M=2\pi r_e^3\sum _{l=1}^L\sum _{m=0}^l{\displaystyle \frac{l+1}{l}}{\left({\displaystyle \frac{r_e}{r_c}}\right)}^{2l+1}{\displaystyle \frac{g_{lm}^2+h_{lm}^2}{\left[1-{\left(r_{\mathrm{o}}/r_c\right)}^{2l+1}\right]}}.$$

(45)

If we divide this by the Earth’s volume $4\pi r_e^3$, we obtain the magnetic energy density spectrum ${\widehat{E}}_M$ of

$${\widehat{E}}_M=\sum _{l=1}^L{R}_l^M,R_l^M=C_l{\displaystyle \frac{l+1}{l}}\sum _{m=0}^l\left(g_{lm}^2+h_{lm}^2\right),C_l={\displaystyle \frac{3{\left(r_e/r_c\right)}^{2l+1}}{2\left[1-{\left(r_{\mathrm{o}}/r_c\right)}^{2l+1}\right]}}.$$

(46)

It is, however, the IGRF power spectrum [16] that allows us to connect the coefficients of the Earth’s surface magnetic field to the Galerkin expansion coefficients in (19) and to estimate outer core magnetic field strength and magnetic helicity.

4. Discussion

The geodynamo and its origin in core dynamics have been of long-standing interest, and although the Earth’s core is hidden from direct observation, much has been learned about its qualitative features and estimates have been made about its physical parameters [27,34]. In order to understand the geodynamo, numerical simulations have been an important tool [35,36,37,38,39]. Rotating MHD [40] and turbulence [41] are essential features of the Earth’s outer core that must, of course, be taken into account. Since computer capabilities have always had and seemingly always will have limited resolution, theoretical studies have been necessary and complimentary, particularly in determining the relevance of MHD turbulence (and especially magnetic helicity) to understanding the geodynamo [8].

Of particular importance in understanding MHD turbulence is statistical mechanics, which predicts that, in the limit of no dissipation, the stationary part of the turbulent dipole energy ${\mathcal{E}}_D$ is related to magnetic helicity ${\mathcal{H}}_M$ (taken to be positive) by ${\mathcal{E}}_D=k_{11}{\mathcal{H}}_M$ in a spherical shell [2,8]; this relation has also been seen to hold in numerical simulations of forced and dissipative MHD turbulence [2,24,25]. As discussed in this paper, the presence of all of the smaller modes causes the dipole moment vector defined by the coefficients $a_{1,m,1}$ and $b_{1,m,1}$, $m=0,\pm 1$, to fluctuate, much like a pollen grain in a fluid undergoing Brownian motion, the difference being that the Earth’s rotation keeps the dipole moment vector more or less aligned with the rotation axis. Here, we have detailed how, using Dirichlet b.c.s, the coefficients of the outer core magnetic field emerge and are connected to the IGRF magnetic field coefficients. This connection explains the polar wander exhibited by the geomagnetic dipole upon which the turbulent fluctuations are manifested.

Reversals and excursions are not included in the explanation, the reason being that the theory presented here applies to a turbulent magnetofluid in equilibrium, which is maintained by a stationary, sufficiently long-term forcing mechanism that includes helicity injection. Reversals and excursions are presumably due to significant and perhaps abrupt changes in the mechanisms that energize MHD turbulence and inject magnetic helicity into the liquid core. Here, we have assumed that magnetofluid turbulence in the outer core is fully developed and is a large dimensional dynamical system; however, if the system is near the onset of convection, a situation where only a few dynamical modes are important, it has been suggested that reversals may be a manifestation of chaotic dynamics [42]. Again, we deal with fully developed turbulence here, which is generally thought to be caused by inner core growth (releasing the heat of fusion) with possible contributions from heating due to radioactivity and compositional buoyancy, all combining to maintain a strongly convecting, turbulent magnetofluid in the outer core [43].

There have been various theories for the creation of the geodynamo: Precession alone had once been posited as a primary cause [44], while for other authors, a combination of convection and precession has been seen as essential for creating a geodynamo [45]. Core heat flow [26] might be introduced through radioactive decay [46] with intermittent activity disrupting stationarity [47]; heat flow may also be moderated by a convecting inner core [48]. Prior to the existence of an inner core, tidal forces and precession due to lunar and solar influences have also been seen as possible drivers of the geodynamo [49]. Additionally, reversals and excursions may occur due to changes in heat flow, either in total amount [47] or through variations in mantle convection [50]. Planetary waves [51] or super-rotation coupled with a differential inner core growth rate [52,53] are alternative mechanisms for reversals and excursions. Changes in any of these mechanisms could upset the stationarity of MHD turbulence and, if their influence is strong enough, diminish dipole strength and cause a reversal or noticeable large excursion.

Our view is that the fundamental engine of the geodynamo is MHD turbulence, per se, an engine fueled by heat flow. Furthermore, the fundamental equation ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$ of a turbulent magnetofluid in equilibrium tells us that magnetic helicity ${\mathcal{H}}_M$ in the outer core is essential in explaining the Earth’s dominant geomagnetic dipole field. Obviously, if ${\mathcal{H}}_M$ were to vary, then so would the dipole field. Thus, producing and maintaining a sufficient level of magnetic helicity in the core is a necessary part of the geodynamo. Production can occur through the injection of helicity into the outer core by one or more of the mechanisms mentioned above, e.g., differential rotation of the inner core, for which there is evidence [54,55,56], or by the creation of helical structures (convective rolls) within the outer core [57,58]. In addition to injecting magnetic helicity directly in to the outer core, these mechanisms could also inject kinetic helicity, as it has been seen numerically that injected kinetic helicity transforms into magnetic helicity in MHD turbulence [2,24,25].

Lastly, since internal dipole energy and magnetic helicity are connected by the relation ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$, we are able to map the IGRF dipole field coefficients $g_{1m}$ and $h_{1m}$ from the Earth’s surface down to the the coefficients $a_{1,m,1}$ and $b_{1,m,1}$, $m=0,\pm 1$ in the outer core, and thereby to estimate a value for ${\mathcal{H}}_M$. Of course, this value is approximate because the theory leading to this result is itself based on a minimal but essential set of assumptions. The strength of the theory is that it is based on a relatively simple and straighforward model system, consisting of a rotating spherical shell containing a turbulent magnetofluid, possessing magnetic helicity, in equilibrium. Mechanisms such as those mentioned above can affect dynamo action and stability. However, a prolonged period of stationarity would imply that these mechanisms (1) maintain a sufficient level of magnetic helicity and (2) have become synergistic with turbulent equilibrium in the massive liquid metal outer core, resulting in a relatively stable geomagnetic dipole field. Reversals and excursions occur when this equilibrium is disrupted.

5. Conclusions

Here, we have discussed how the stationary interior magnetic dipole component gains a fluctuating part and also how the relation between interior dipole energy ${\mathcal{E}}_D$ and magnetic helicity ${\mathcal{H}}_M$, i.e., ${\mathcal{E}}_D=k_{min}{\mathcal{H}}_M$ (developed previously [2,3]), permits us to estimate the value of ${\mathcal{H}}_M$ in the outer core from the geomagnetic dipole field strength. The results presented herein, as well as previous ones, all lead to one major, qualitative conclusion: when the outer core is in an energetic equilibrium state, helical MHD turbulence may be considered to be the essential engine of the geodynamo.