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# A dynamo cascade interpretation of the geomagnetic dipole decrease

Hagay Amit, Peter Olson

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Hagay Amit, Peter Olson. A dynamo cascade interpretation of the geomagnetic dipole decrease. Geophysical Journal International, 2010, 10.1111/j.1365-246X.2010.04596.x. hal-02360071

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# A dynamo cascade interpretation of the geomagnetic dipole decrease

Hagay Amit<fnref n="1" /> and Peter Olson<fnref n="2" />

Accepted 2010 March 11. Received 2010 March 11; in original form 2009 November 13

## SUMMARY

We propose a spectral transfer model for the secular variation of the geomagnetic core field to explain the simultaneous decrease in dipole field intensity and the increase in non-dipole field intensity from 1840 to the present in terms of a dynamo cascade process. The main assumption of this model is that magnetic energy is transferred between adjacent spherical harmonic degrees in the Mauersberger-Lowes spectrum of the geomagnetic field. The key parameters are a set of coefficients $\gamma_n$ that indicate the rate and direction of magnetic energy transfer through the spectrum. Applying the spectral transfer model to the historical period, we find that the quadrupole family of the core field can be characterized by a persistent inverse magnetic energy cascade from higher towards lower spherical harmonics. In the dipole family of the core field, we find cascade behaviour generally from lower to higher spherical harmonics, consistent with axial dipole decrease, but with a high level of time variability that correlates with variations in the dipole family intensity. During time intervals when the dipole family intensity rapidly decreases, energy appears to cascade towards higher spherical harmonics, beyond the limit of the observable part of the core field spectrum. During time intervals when the dipole family intensity is nearly constant, a more limited forward cascade appears to trap energy at intermediate spherical harmonics. Similar fluctuations in the rate and direction of spectral transfer are also seen in the Mauersberger-Lowes spectrum of a numerical dynamo model during a dipole decrease event that led to a polarity excursion. We discuss the possibility of this scenario for the current geomagnetic dipole decrease.

**Key words:** Dynamo: theories and simulations; Geomagnetic excursions; Rapid time variations.

## 1 INTRODUCTION

During the historical period, the shape of the low degree part of the geomagnetic field spectrum has rapidly changed with time. Since the advent of geomagnetic field intensity measurements about 170 yr ago, the geomagnetic dipole moment has been losing intensity, while at the same time the non-dipole part of the core field has been gaining intensity. According to the historical *gufm1* field model (Jackson *et al.* 2000), the geomagnetic dipole moment has dropped from $8.50 \times 10^{22} \text{ Am}^2$ in 1840 to $7.84 \times 10^{22} \text{ Am}^2$ in 1990, and similarly, its dominant axial component has decreased from $8.33 \times 10^{22} \text{ Am}^2$ in 1840 to $7.70 \times 10^{22} \text{ Am}^2$ in 1990. The rate of change of the axial dipole moment in 1990 was $-4.38 \times 10^4 \text{ Am}^2 \text{ yr}^{-1}$, compared to the average rate of $-4.22 \times 10^4 \text{ Am}^2 \text{ yr}^{-1}$ over the 150-yr period. Recent geomagnetic field models based on satellite data show that the historical dipole decrease continues to the present-day. According to the *xCHAOS* field model (Olsen & Mandea 2008), the 2009 dipole moment was $7.75 \times 10^{22} \text{ Am}^2$, its axial component was $7.63 \times 10^{22} \text{ Am}^2$, and its current rate of change is $-4.92 \times 10^4 \text{ Am}^2 \text{ yr}^{-1}$, indicating that the dipole decrease has recently accelerated.

Fig. 1 shows time-series of the rms field intensity, rms dipole intensity, rms axial dipole intensity, and rms non-dipole field intensity on the core-mantle boundary (CMB) from 1840 to present. The period 1840–1990 is covered by model *gufm1* from which we expanded until spherical harmonic degree $n_{max} = 8$, while the period 1999–2009 is represented by model *xCHAOS* until $n_{max} = 14$. The mismatch between the rms field intensities in the two models reflects different regularization assumptions and different levels of harmonic truncation due to the greater quality and time resolution of the modern data. While the dipole field decreases monotonically in both models, the non-dipole field has the opposite trend, substantially increasing with time. Overall, the rms intensity (Verosub & Cox 1971; Benton & Alldredge 1987) and the average absolute value of the radial component (Benton & Voorhies 1987) of the core field have changed relatively little during the historical period; if anything, they have increased very slightly, although this may be an artefact of improved resolution of the core field with time.

Were the current trends to persist, the axial dipole would hypothetically vanish in about 1600 yr time. However, the dipole variation has not been uniform in the past. Evidence suggests that the

<footnote n="1">¹CNRS, Université de Nantes, Nantes Atlantiques Universités, UMR CNRS 6112, Laboratoire de Planétologie et de Géodynamique, 2 rue de la Houssinière, F-44000 Nantes, France. E-mail: Hagay.Amit@univ-nantes.fr</footnote>

<footnote n="2">²Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218, USA</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Time-series of the rms core field intensities: (a) total field $B_{rms}$; (b) dipole $B_{rms}(n = 1)$ and axial dipole $B_{rms}(n = 1, m = 0)$; and (c) non-dipole $B_{rms}(n > 1)$, all from model *gufm1* (Jackson *et al.* 2000) to spherical harmonic degree $n_{max} = 8$ for the period 1840–1990, and similarly (d)-(f) from model *xCHAOS* (Olsen & Mandea 2008) to $n_{max} = 14$ for the period 1999–2009. All intensities are in mT at the core–mantle boundary. In (b) and (e) the solid curve is the full dipole intensity and the dashed curve is the axial dipole intensity. $1 \text{ Am}^2$ is equivalent to $3.87 \cdot 10^{-25} \text{ mT}$ for the geomagnetic $g_1^0$ Gauss coefficient, or to $2.74 \cdot 10^{-24} \text{ mT}$ for the corresponding rms field.</figcaption></figure>

dipole decrease during the 250 yr prior to 1840 was significantly slower than today (Gubbins *et al.* 2006), possibly remaining constant (Finlay 2008) during that time. According to the archaeomagnetic core field model *CALS7K.2* (Korte & Constable 2005), the current episode of dipole decrease began within the past 1 kyr and followed a period of about 2 kyr when the dipole intensity fluctuated. Less is known about the non-dipole core field during that time, but the irregular dipole variations over the past 3 kyr, and moreover, the opposing trends of the dipole and non-dipole parts over the past 150 yr, are the primary evidence that the shape of the geomagnetic spectrum undergoes rapid changes.

There are a variety of dynamo mechanisms that could drive the dipole and non-dipole fields in opposite directions. One possibility is that magnetic diffusion is preferentially weakening the dipole field by decay of its higher modes (Moffatt 1978) or by expulsion of reversed dipolar field from the core (Bloxham 1986), while at the same time, dynamo processes are strengthening the non-dipole field. Another possibility is that advection by fluid motions in the outer core is distorting the pattern of the field on the CMB, and thereby directly transferring magnetic energy from the dipole part to the non-dipole part of the core field. The rate of dipole decrease is variable, but on average it is about one order of magnitude faster than predicted by free decay (Olson & Amit 2006), so it is likely that the trends shown in Fig. 1 are a consequence of both advective (Roberts & Scott 1965) and diffusive (Gubbins 1987) processes operating simultaneously.

Importantly however, the effects of advection and diffusion may not operate equally over the whole spherical harmonic spectrum of the core field. Because the magnetic Reynolds number of the core is large (e.g. Bloxham & Jackson 1991), it is expected that advection effects play a proportionally larger role in the geomagnetic secular variation (SV) on the largest scales (i.e. for the lowest spherical harmonics), whereas diffusion plays a proportionally larger role on the smaller scales (i.e. for higher spherical harmonics).

In this paper, we propose a simple cascade model that connects the dipole moment decrease to the time evolution of the energy spectrum of the core field. The model assumes that magnetic energy SV is dominated by transfer between adjacent spherical harmonic degrees by fluid motions in the outer core. Our model is motivated by the trends evident in Fig. 1 and supported by results of numerical dynamo simulations, which show qualitatively similar spectral transfer effects during times of rapid dipole collapse, events that sometimes precede polarity reversals and excursions. We apply our cascade model to estimate the rates of spectral transfer in the core during the present-day dipole decrease.

## 2 GEODYNAMO CASCADE MODEL

The average magnetic energy density of the core field on the CMB is proportional to $R$, the mean squared field intensity at the core radius $r = c$. The spectrum of $R$, called the Mauersberger–Lowes
spectrum, is conventionally written as

$$R = \sum_{n=1}^{n_{\max}} R_n, \qquad (1)$$

where $n$ is spherical harmonic degree and $n_{\max}$ is the truncation, usually $n_{\max} \simeq 14$ for the core field. The Mauersberger-Lowes spectrum at the CMB is one of the primary constraints on the dynamo process in the core (Dormy et al. 2000). The terms $R_n$ at the CMB can be expressed as sums of squares of the Gauss coefficients of the core field, according to (Lowes 1974)

$$R_n = (n + 1) \left(\frac{a}{c}\right)^{2n+4} \sum_{m=0}^{n} \left( (g_n^m)^2 + (h_n^m)^2 \right), \qquad (2)$$

where $a$ is Earth's radius and $g_n^m$ and $h_n^m$ are the Gauss coefficients of the core field in mT at spherical harmonic degree and order $n$ and $m$, respectively. The mean squared field intensity can also be defined for the equatorially antisymmetric part of the core field termed the dipole family by summing (2) over odd values of $n-m$, and for the equatorially symmetric part of the core field termed the quadrupole family by summing over even values of $n-m$ (Roberts 1971). In this paper, we denote spectra with these symmetries using superscripts $a$ and $s$, for example $R_n^a$ and $R_n^s$, respectively.

Using (2), the time variation of the spectrum of $R$ is written as (Cain et al. 1989)

$$\dot{R}_n = 2(n+1) \left(\frac{a}{c}\right)^{2n+4} \sum_{m=0}^{n} \left( (g_n^m g_n^{\prime m} + h_n^m \dot{h}_n^{\prime m}) \right), \qquad (3)$$

where the dot denotes time derivative. Like the Mauersberger-Lowes spectrum, (3) can be split into its equatorially antisymmetric and symmetric parts, denoted here by $\dot{R}_n^a$ and $\dot{R}_n^s$, respectively. Note that (3) which has units of $mT^2yr^{-1}$ differs from the definition of the spectrum of the SV field itself at the CMB, which has units of $mT^2yr^{-2}$ and is given by (Alldredge 1984; McLeod 1996; Voorhies 2004)

$$S_n = (n + 1) \left(\frac{a}{c}\right)^{2n+4} \sum_{m=0}^{n} \left( (g_n^{\prime m})^2 + (\dot{h}_n^{\prime m})^2 \right). \qquad (4)$$

Temporal changes in the shape of the Mauersberger-Lowes spectrum are reflected in non-uniform time variations in the individual terms $\dot{R}_n$, $\dot{R}_n^a$, $\dot{R}_n^s$, $S_n$, $S_n^a$ or $S_n^s$.

A cascade model for the evolution of $R_n$ can be derived starting from the radial component of the magnetic induction equation at the top of the core. Assuming incompressible flow, the radial component of the induction equation at the CMB is

$$\frac{\partial B_r}{\partial t} + \mathbf{u}_h \cdot \nabla B_r + B_r \nabla_h \cdot \mathbf{u}_h = \eta \hat{r} \cdot \nabla^2 \mathbf{B}, \qquad (5)$$

where $B_r$ is the radial component of the magnetic field on the CMB, $t$ is time, $\mathbf{u}_h$ is the free stream fluid velocity in the outer core below the CMB, $r$ is the radial coordinate, $\hat{r}$ is its unit vector, $\eta$ is the magnetic diffusivity of the outer core, $\mathbf{B}$ the magnetic field vector on the CMB and the subscript $h$ denotes coordinates tangent to the CMB. The first term on the left-hand side of (5) is the SV, the second and third terms on the left-hand side represent advection of magnetic field by the flow, and the term on the right-hand side represents magnetic diffusion. Units of all these variables are given in Table 1.

Multiplying (5) by $B_r$ and averaging over the CMB gives

$$\frac{\partial}{\partial t} \langle B_r^2 \rangle = - \langle \mathbf{u}_h \cdot \nabla B_r^2 \rangle - 2 \langle B_r^2 \nabla_h \cdot \mathbf{u}_h \rangle + 2\eta \langle B_r \hat{r} \cdot \nabla^2 \mathbf{B} \rangle, \qquad (6)$$

where $\langle \rangle$ denotes CMB surface average. The left hand side of (6) is the time derivative of the mean square radial field. Using the relationship

$$\langle B_r^2 \rangle = \sum_{n=1}^{n_{\max}} \frac{n+1}{2n+1} R_n, \qquad (7)$$

(6) can be written symbolically for each spherical harmonic degree as

$$\dot{R}_n = T_n - D_n \qquad (8)$$

where

$$T_n = - \frac{2n+1}{n+1} \left( \langle \mathbf{u}_h \cdot \nabla B_r^2 \rangle_n + 2 \langle B_r^2 \nabla_h \cdot \mathbf{u}_h \rangle_n \right) \qquad (9)$$

is the transfer spectrum of the core field and

$$D_n = -2 \frac{2n+1}{n+1} \eta \langle B_r \hat{r} \cdot \nabla^2 \mathbf{B} \rangle_n \qquad (10)$$

is its diffusion spectrum, both at harmonic degree $n$. In the following subsections we develop parametrized representations for both the diffusion spectrum $D_n$ and the transfer spectrum $T_n$.

## 2.1 Diffusion spectrum

The exact form of magnetic diffusion in the core cannot be deduced from the geomagnetic field at the CMB, and therefore the diffusion terms in (5) and (6) cannot be inferred from observations. However, we can estimate a lower limit on the diffusion spectrum $D_n$ from the rate of free decay of the core field at each harmonic degree $n$, as follows. The radial field on the CMB is given in terms of the Gauss coefficients by

$$B_r = \sum_{n=1}^{n_{\max}} \sum_{m=0}^{n} B_{r,n}^m = \sum_{n=1}^{n_{\max}} \sum_{m=0}^{n} \left( \left(\frac{a}{c}\right)^{n+2} (n+1) P_n^m(\cos\theta) \times (g_n^m \cos m\phi + h_n^m \sin m\phi), \qquad (11)$$

where $B_{r,n}^m$ is the radial field contribution of degree $n$ and order $m$, $P_n^m$ are the Legendre functions, $\theta$ is colatitude and $\phi$ is longitude. Focusing on free decay, the diffusion term in (5) can be rewritten using (11) as

$$\eta \hat{r} \cdot \nabla^2 \mathbf{B} = - \frac{\eta}{c^2} \sum_{n=1}^{n_{\max}} \sum_{m=0}^{n} P_n^{j2} B_{r,n}^m, \qquad (12)$$

where $P_n^j$ denote the free decay eigenvalues of each mode $j$. The eigenvalues $P_n^j$ correspond to the roots of the Bessel function $J_{(n-1/2)}(p_n') = 0$ (see Moffatt 1978), the smallest root at a given $n$ corresponding to the fundamental mode $p_n$, and the larger roots corresponding to the higher modes, which have more radial structure in the core and therefore diffuse faster. Multiplying both sides of (12) by (11), summing over $m$, and averaging the resulting expression at the CMB, we obtain the following representation of the diffusion spectrum at harmonic degree $n$:

$$D_n = 2 \frac{\eta}{c^2} P_n^2 R_n, \qquad (13)$$

where we choose for simplicity the fundamental mode $p_n$. Although (13) is not a complete model for magnetic diffusion, it serves to define the fundamental timescales for free decay at each spherical harmonic

$$\tau_n^d = \frac{c^2}{2\eta p_n^2}. \qquad (14)$$
<table><caption>**Table 1.** Summary of variables and their dimensions. MKS units are: T is Tesla, m is metres, s is seconds, and Γ is a any tracer such as temperature or chemical concentration. In the text we sometimes use other units, e.g. mT, km, yr. Note that all variables are also defined for the equatorially asymmetric dipole family and the equatorially symmetric quadrupole family by the superscripts *a* and *s*, respectively.</caption><thead><tr><th>Variable</th><th>Symbol</th><th>Dimension</th></tr></thead><tbody><tr><td>Spherical radial, colatitude and longitude coordinates</td><td>r, θ and φ</td><td>m and radians</td></tr><tr><td>Cartesian wavenumber</td><td>k</td><td>m<sup>-1</sup></td></tr><tr><td>Spherical harmonic degree and order</td><td>n and m</td><td>—</td></tr><tr><td>Legendre function</td><td>P<sub>n</sub><sup>m</sup></td><td>—</td></tr><tr><td>Earth's surface radius</td><td>a</td><td>m</td></tr><tr><td>Earth's core radius</td><td>c</td><td>m</td></tr><tr><td>Gauss coefficients</td><td>g<sub>n</sub><sup>m</sup> and h<sub>n</sub><sup>m</sup></td><td>T</td></tr><tr><td>Magnetic energy spectrum</td><td>R<sub>n</sub></td><td>T<sup>2</sup></td></tr><tr><td>SV of Magnetic energy spectrum</td><td>R&#770;<sub>n</sub></td><td>T<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>SV spectrum</td><td>S<sub>n</sub></td><td>T<sup>2</sup> s<sup>-2</sup></td></tr><tr><td>Radial magnetic field</td><td>B<sub>r</sub></td><td>T</td></tr><tr><td>Radial magnetic field of harmonic n and m</td><td>B<sub>r</sub><sup>m</sup></td><td>T</td></tr><tr><td>Rms magnetic field</td><td>B<sub>rms</sub></td><td>T</td></tr><tr><td>Magnetic field vector</td><td>B</td><td>T</td></tr><tr><td>Time</td><td>t</td><td>s</td></tr><tr><td>Tangential velocity vector</td><td>u<sub>h</sub></td><td>m s<sup>-1</sup></td></tr><tr><td>Magnetic diffusivity</td><td>η</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Free decay eigenvalues</td><td>P<sub>n</sub><sup>j</sup></td><td>—</td></tr><tr><td>Fundamental free decay mode</td><td>p<sub>n</sub></td><td>—</td></tr><tr><td>Fundamental free decay timescale</td><td>τ<sub>n</sub><sup>d</sup></td><td>s</td></tr><tr><td>Cartesian scalar variable</td><td>Γ<sub>k</sub></td><td>Γ</td></tr><tr><td>Cartesian scalar energy spectrum</td><td>R<sub>k</sub></td><td>Γ</td></tr><tr><td>SV of Cartesian scalar variable</td><td>Γ&#770;<sub>k</sub></td><td>Γ s<sup>-1</sup></td></tr><tr><td>Cartesian transfer spectrum</td><td>T<sub>k</sub></td><td>Γ s<sup>-1</sup></td></tr><tr><td>Diffusivity of Cartesian scalar variable</td><td>κ</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Cartesian flux spectrum</td><td>F<sub>k</sub></td><td>Γ ms<sup>-1</sup></td></tr><tr><td>Cartesian spectral transfer rate</td><td>γ<sub>k</sub></td><td>m s<sup>-1</sup></td></tr><tr><td>Spherical transfer spectrum</td><td>T<sub>n</sub></td><td>T<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Spherical diffusion spectrum</td><td>D<sub>n</sub></td><td>T<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Spherical flux spectrum</td><td>F<sub>n</sub></td><td>T<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Spherical spectral transfer rate</td><td>γ<sub>n</sub></td><td>s<sup>-1</sup></td></tr><tr><td>SV timescale based on S<sub>n</sub></td><td>τ<sub>n</sub></td><td>s</td></tr><tr><td>SV timescale based on R&#770;<sub>n</sub></td><td>τ<sub>n</sub><sup>*</sup></td><td>s</td></tr></tbody></table>

Slow spectral variations correspond to $\dot{R}_n$-values small compared to fundamental mode $D_n$-values, and conversely, rapid spectral variations correspond to $\dot{R}_n$-values large compared to the same $D_n$-values. Our cascade model is intended for the fast spectral change regime.

## 2.2 Transfer spectrum

We model the transfer spectrum $T_n$ of the core field as a cascade process, on the basis of the similarity between (8) and the equation governing spectral evolution of a scalar variable in a turbulent flow. In the classical Kolmogorov description of hydrodynamic turbulence, energy is exchanged between similar size eddies without major jumps from one scale to another (Frisch 1995). This Kolmogorov type of energy transfer is usually referred to as *local*, and the term *cascade* is often used to describe the transfer process. In this paper, we use the term *local* for the transfer of magnetic energy between adjacent harmonics in spectral space. A *forward* cascade denotes energy transfer from large to small scales, or in our case, from lower to higher spherical harmonics. Conversely, an *inverse* cascade denotes energy transfer from higher to lower harmonics.

Analyses of numerical simulations of magnetohydrodynamic (MHD) turbulence (Alexakis et al. 2007) associate these two types of spectral transfer with different terms in the induction equation. Local magnetic energy transfer is associated with advection of magnetic field by the flow (first term on the right-hand side of 9), whereas non-local transfer is associated with magnetic field stretching (second term on the right-hand side of 9). For non-rotating MHD turbulence, Alexakis et al. (2007) find that the non-local transfer of magnetic energy due to stretching actually dominates over local magnetic energy transfer by direct energy advection. In addition, recent core flow models argue that non-local transfer, in the form of interactions between small-scale field and large-scale flow or small-scale flow and large-scale field leading to small-scale SV, dominates their model misfit (Eymin & Hulot 2005; Pais & Jault 2008). However, we argue that local spectral transfer is the dominant process that shapes the core field spectrum, because so much of the core field SV can be explained as frozen-flux advection by an evolving large-scale dominantly toroidal flow (e.g. Jackson 1997; Amit & Olson 2006) with the non-local energy transfer from smaller scales being of secondary importance.

In terms of Cartesian wavenumbers $k$, the spectral evolution equation of a scalar variable subject to advection and diffusion is
commonly written as (Hill 1978)

$$ \dot{\Gamma}_k = T_k - 2\kappa k^2 \Gamma_k, \qquad (15) $$

where $\Gamma_k$ is the wavenumber spectrum of the scalar variable, $T_k$ is the transfer spectrum, and $\kappa$ is the diffusivity of the scalar variable. Local cascade models relate $T_k$ to a quantity called the flux spectrum $F_k$ via

$$ T_k = -\frac{\partial}{\partial k} F_k. \qquad (16) $$

Physically, this spectrum $F_k$ represents the flux of magnetic energy through wavenumber space. It is often assumed that the flux spectrum $F_k$ is linked to the energy spectrum $R_k$ via

$$ F_k = \gamma_k R_k, \qquad (17) $$

where $\gamma_k$ is the spectral transfer rate in units of velocity (Corrsin 1961, 1964). In general, all quantities in (17) are functions of both wavenumber and time. However, under conditions of statistical equilibrium, the time dependence of $\gamma_k$ in (17) is assumed to vanish, and in this situation it is often found that $\gamma_k$ can be approximated as a power-law function of $k$. For example, a standard form of $\gamma_k$ for the spectrum of a passive scalar in homogeneous isotropic turbulence is (Hill 1978; Frisch 1995)

$$ \gamma_k \propto k^{5/3}. \qquad (18) $$

At the top of Earth's core the flow is neither isotropic nor homogeneous (due to the rotational effect and mantle control), so although the 5/3 Kolmogorov exponent may not hold, it has been proposed that a general power law of the form (18) is applicable there (Voorhies 2004).

The preceding discussion applies to continuous wavenumber spectra. For spherical harmonic spectra, there is a close analogy between (8) and (15) in the asymptotic limit of large $n$, for which $kc = \sqrt{n(n+1)}$ becomes a good approximation (Backus et al. 1996). However, the correspondence is not exact because $k$ is continuous while $n$ is discrete (see Appendix B4 of Voorhies 2004). Accordingly, in place of (16), we hypothesize that the spectral transfer and spectral flux are related by the following difference formula

$$ T_n = F_{n-1} - F_n. \qquad (19) $$

Note that the dependence of $T$ on $F$ in (19) involves only nearest-neighbour spectral terms, consistent with our local spectral transfer assumption. Substituting (19) into (8) then yields the following recursion formula for the flux spectrum:

$$ F_n = F_{n-1} - \dot{R}_n - D_n \qquad (20) $$

with the initial value $F_0 = 0$ (corresponding to zero geomagnetic monopole). We further hypothesize that, by analogy with (17) the flux spectrum $F_n$ and Mauersberger-Lowes spectrum $R_n$ are related by

$$ F_n = \gamma_n R_n, \qquad (21) $$

where $\gamma_n$ represents the local rate of transfer through the Mauersberger-Lowes spectrum in units of reciprocal time (Table 1). According to (21), $\gamma_n$ is just given by $F_n/R_n$. However, to avoid singularities where $R_n$ is small, we obtain smoother and more stable results if we calculate $\gamma_n$ as

$$ \gamma_n = \frac{F_n + F_{n+1}}{R_n + R_{n+1}} \qquad (22) $$

in which case $\gamma_n$ in our model represents the rate of transfer from spherical harmonic degree $n$ to degree $n+1$. With this definition,

positive values of $\gamma_n$ correspond to a forward cascade of magnetic energy through the Mauersberger-Lowes spectrum (from $n$ to $n+1$), whereas negative values of $\gamma_n$ correspond to an inverse magnetic energy cascade (from $n+1$ to $n$).

Although our model assumes only local spectral transfer, it is important to point out that non-local spectral transfer may be important in the geodynamo, particularly as it contributes to the dipole moment. Non-local spectral transfer is sometimes observed in non-magnetic turbulence, both in experiments (Wiltse & Glezer 1993) and in numerical simulations (Alexakis et al. 2005). Spectral transfer is well established in MHD turbulence (Maron & Goldreich 2001; Biskamp 2003; Müller & Grappin 2005; Boldyrev 2006). Numerical simulations show that the spectral transfer typically consists of two components, one local and Kolmogorov-like, the other non-local, in which the magnetic energy jumps discontinuously from small-scales to large-scales (Yousef et al. 2007). Theoretically, this type of non-local transfer is implicit in the $\alpha$-dynamo effect (Moffatt 1978; Biskamp 2003). It is also linked to magnetic helicity, which is thought to be a vital ingredient in the generation of large-scale magnetic fields from small-scale turbulent flows in astrophysical bodies (Pouquet et al. 1976; Cho et al. 2002).

## 2.3 Analytical tests of local spectral transfer

The relationship between the transfer spectrum and the flux spectrum (16) holds if the spectral transfer is local, but may give spurious results in situations where the spectral transfer is non-local. In addition, our finite difference (19) and smoothing (22) approximations might constitute additional sources of error in some situations. Here we use analytical examples in which we specify the spectral transfer, in order to show where our model assumptions work and where they breakdown.

The first examples are of purely local spectral transfer, so these serve to test our method of calculating $\gamma_n$. We specify the actual local spectral transfer rate $\dot{\gamma}_n$ as the net magnetic energy transfer from degree $n$ to degree $n+1$ in the Mauersberger-Lowes spectrum, normalized by $R_n$, and located in spectral space at $n$. For simplicity in this section we adopt the frozen-flux limit, so the diffusion term $D_n$ is ignored. Accordingly, the relationship between the actual local spectral transfer rate $\dot{\gamma}_n$, the Mauersberger-Lowes spectrum $R_n$ and its SV $\dot{R}_n$ is

$$ \dot{R}_n = \dot{\gamma}_{n-1} R_{n-1} - \dot{\gamma}_n R_n. \qquad (23) $$

Also note that in the frozen-flux limit our model $F_n$ (20) yields

$$ F_n = - \sum_{q=1}^{n} \dot{R}_q. \qquad (24) $$

Consider a decreasing dipole field with $\dot{R}_1 < 0$ in which the dipole energy is transferred to successively higher harmonics at a uniform rate, while the non-dipole part of the Mauersberger-Lowes spectrum is constant in time. The situation is sketched in Fig. 2(a). Further assume in this first example that the Mauersberger-Lowes spectrum is flat, that is, $R_n = R_1$. Then according to (23), $\dot{\gamma}_n = -\dot{R}_1/R_1$ is the actual spectral transfer rate through the Mauersberger-Lowes spectrum. For comparison, our model gives $F_n = -\dot{R}_1$ according to (24), and according to (22) $\gamma_n = -\dot{R}_1/R_1$, identical to the actual transfer rate.

If the field spectrum is not flat, then the spectral transfer rates in our model will not be identical to the actual ones. However, the two will be very close if the Mauersberger-Lowes spectrum has a continuous variation with $n$. Consider for instance the same transfer
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) Schematic illustration of a uniform, local, forward energy cascade. (b) Normalized spectral transfer rates versus spherical harmonic degree *n* with $R_n = R_1/n$: actual values $-(R_1/\dot{R}_1)\tilde{\gamma}_n$ (25) denoted by asterisks versus the model $-(R_1/\dot{R}_1)\gamma_n$ (26) denoted by triangles. The positions of the model $\gamma_n$ have been shifted by one half unit of *n* to represent transfer between harmonics *n* and *n* + 1.</figcaption></figure>

from the dipole field as in the previous example, but in this case assume $R_n = R_1/n$ for the Mauersberger-Lowes spectrum. Then according to (23)

$$ \tilde{\gamma}_n = -n \frac{\dot{R}_1}{R_1}, \qquad (25) $$

whereas according to (22) and (24)

$$ \gamma_n = -n \frac{\dot{R}_1}{R_1} \frac{2n + 2}{2n + 1}. \qquad (26) $$

As shown in Fig. 2(b), the difference between our model rates (26) and the actual rates (25) are minor, particularly with increasing *n*.

A second example again assumes local spectral transfer but allows for time variation in the non-dipole as well as the dipole parts of the Mauersberger-Lowes spectrum. Consider as before a decreasing dipole field with $\dot{R}_1 < 0$, but in this case the forward cascade is non-uniform, and transfers successively smaller amounts of magnetic energy towards successively higher degrees *n* (Fig. 3a). This results in an increasing non-dipole spectrum in which

$$ \dot{R}_{n>1} = -\alpha \dot{R}_1, \qquad (27) $$

where $\alpha$ is the fractional difference in the energy transferred from one harmonic to the next higher. Although this is just a slight extension of the first examples, it is an idealization of the historical behaviour of the core field. Assuming as in the previous case that $R_n = R_1/n$, (23) and (27) give for the actual transfer rate

$$ \tilde{\gamma}_n = -n \frac{\dot{R}_1}{R_1} (1 - (n - 1)\alpha), \qquad (28) $$

whereas according to (22), (24) and (27), the model values are

$$ \gamma_n = -n \frac{\dot{R}_1}{R_1} \frac{n+1}{2n+1} (2 - (2n-1)\alpha). \qquad (29) $$

Figs 3(b)–(e) compare between (28) and (29) for several values of the parameter $\alpha$. As in the previous examples, the agreement

is excellent. According to (28) and (29), the cascade is forward ($\gamma > 0$) up to $n = N$, where

$$ N = \frac{\alpha + \epsilon}{\epsilon\alpha}, \qquad (30) $$

where $\epsilon = 1$ for the actual transfer and $\epsilon = 2$ for our model. For $n > N$, the $\gamma$-coefficients change sign. In these situations, our model interpretation is that magnetic energy cascading up the spectrum becomes absorbed (trapped) in the spectral band $2 \le n \le N$.

For the geodynamo, spectral transfer of magnetic energy is more complex than these simple examples, involving a combination of local, non-local, forward and inverse cascades (Fig. 4a). To illustrate how our local spectral transfer model treats non-local spectral transfer, consider the situation shown in Figs 4(b) and (c). In these cases, energy from the dipole is transferred directly to some higher harmonic $n'$ beyond (Fig. 4b) or within (Fig. 4c) the observable core field spectrum, while all other harmonics experience zero net transfer. In this case our model incorrectly predicts a continuous forward cascade from $n = 1$ to $n = n' - 1$ (as in e.g. (26) and Fig. 2), while the actual spectral transfer is non-local, that is, $\tilde{\gamma}_n = 0$. Although this example demonstrates the pitfalls in distinguishing between local and non-local transfer effects, it also suggests there are warning signs, such as abrupt changes in the inferred spectral transfer rate, which indicate where the transfer is non-local.

Even though it is not possible to uniquely distinguish local from non-local spectral transfer, there are some situations that strongly favour the local transfer interpretation. For example, consider the case in which a local peak in the Mauersberger-Lowes spectrum is observed to propagate towards higher degrees over time. As illustrated in Figs 5(a) and (b), the energy peak centred at $n'$ at time $t_0$ propagates to degree $n' + 1$ at time $t_0 + \delta t$. The amplitude of the peak is $\delta R$ relative to the background level $R_1$ of the spectrum. In this case, the actual spectral transfer rates in the neighbourhood of
<figure><img src="image_5.png" /><figcaption>Figure 3. As in Fig. 2 for a non-uniform, local, forward energy cascade (28)–(29). In (b)–(e) the spectral transfer rates are given for various $\alpha$ values.</figcaption></figure>

the peak are

$$ 
\begin{aligned}
\tilde{\gamma}_{n'-1} &= \tilde{\gamma}_{n'+1} = \frac{1}{\delta t} \frac{\delta R}{2R_1 + \delta R} \\
\tilde{\gamma}_{n'} &= \frac{1}{\delta t} \frac{\delta R}{R_1 + \delta R},
\end{aligned}
\qquad (31) 
$$

whereas our model rates in the same neighbourhood are

$$ 
\begin{aligned}
\gamma_{n'-2} &= \gamma_{n'+1} = \frac{1}{\delta t} \frac{\delta R}{4R_1 + \delta R} \\
\gamma_{n'-1} &= \gamma_n = \frac{1}{\delta t} \frac{3\delta R}{4R_1 + 3\delta R}.
\end{aligned}
\qquad (32) 
$$

Again our model captures well the actual rates, independently of the peak-to-background ratio (Figs 5c and d).

In Fig. 6(c) a travelling peak that propagates from higher towards lower degrees in the quadrupole family of the Mauersberger-Lowes spectrum is evident in the geomagnetic core field. For this structure, the peak-to-background amplitude is $(R_1 + \delta R)/R_1 \sim 4$, comparable to the propagating example used in Fig. 5(d). Because our model is shown to recover the propagation direction, location in spectral space and speed, we suggest that the evolution of this particular

structure in Fig. 6(c) is evidence for local energy transfer in the geomagnetic field.

## 2.4 Spectral transfer rates and secular variation timescales

The traditional measures of core field variations are the SV timescales $\tau_n$, which in our notation are defined as

$$ \tau_n = \sqrt{\frac{R_n}{S_n}} \qquad (33) $$

(Hulot & LeMouël 1994; McLeod 1996; Hongre et al. 1998; Christensen & Tilgner 2004; Holme & Olsen 2006; Gillet et al. 2009). The magnitude of $\tau_n$ may vary significantly from one epoch to another (Hulot & LeMouël 1994). Voorhies (2004) argued that the timescale

$$ \tau_n^* = \frac{R_n}{|\dot{R}_n|} \qquad (34) $$

based on the SV of the spectrum $\dot{R}_n$ is more relevant than $\tau_n$ which is based on the spectrum of SV $S_n$. Physical interpretation of (33)
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Schematic illustrations of non-local transfer. (a) A general mixed state of local, non-local, forward and inverse cascades. (b) Non-local transfer from the dipole to a degree higher than the observable spectrum. (c) Non-local transfer from the dipole to a high degree within the observable spectrum.</figcaption></figure>

and (34) depends strongly, although implicitly, on the dominant processes that control the SV at each spherical harmonic degree. At low magnetic Reynolds number conditions for example, (33) represents free decay timescales for each harmonic of the field, whereas at high magnetic Reynolds number conditions, (33) represents a combination of spectral transfer by advection and diffusion (Liu & Olson 2009). Hulot & LeMouël (1994) determined $τ_n$ for the non-axial dipole field to be on the order of several centuries, with decreasing values for higher degrees. The axial dipole timescale is longer than the timescales of other harmonics (Lowes 1974; Alldredge 1984). Hulot & LeMouël (1994) argued that diffusive effects dominate the axial dipole behaviour, whereas advection dominates the other timescales. They also showed that, apart from the axial dipole term, the dipole and quadrupole families have similar $τ_n$-values. Christensen & Tilgner (2004) found they could match core field SV timescales for $n > 2$ with the $τ_n$-values from a numerical dynamo model with a large magnetic Reynolds number, $Rm \simeq 500$. Spectral transfer processes certainly play an important role in this regime of $Rm$. Another piece of supporting evidence are the very short timescales at the harmonics on the edge of the core field at $n \simeq 14$. Assuming power law fit for $τ_n$ in the range $n = 3-12$, satellite geomagnetic field models predict $τ_{14} \simeq 20$ yr for the highest degrees in the core field (Holme & Olsen 2006; Gillet et al. 2009). Another short timescale relevant to the geomagnetic field is the dynamo memory, the time required for the perturbed and unperturbed dynamos to diverge. Hulot et al. (2010) inferred this is about one century for the geodynamo.

Our spectral transfer rates $γ_n$ conceptually differ from both the SV timescales $τ_n$ and from $τ_n^*$ in several respects. Most importantly, $γ_n$ can be positive or negative, its sign indicating the energy cascade direction. A more subtle difference is that $1/τ_n$ and $1/τ_n^*$ are related to the rate of field change at a particular harmonic degree, whereas $γ_n$ refers to the rate at which energy transfers by one harmonic degree. In spite of these differences, we show in the next section that there are similarities between $1/τ_n$ and the magnitude of $γ_n$ in the historical field behaviour.

## 3 SPECTRAL CASCADE AND GEOMAGNETIC DIPOLE DECREASE

We use the historical field model *gufm1* on the CMB for the period 1840–1990 (Jackson *et al.* 2000) and the field model *xCHAOS* on the CMB for the period 1999–2009 (Olsen & Mandea 2008) to interpret the geomagnetic dipole decrease shown in Fig. 1 in terms of a spectral cascade. For each core field model, we first calculate $R_n$ and its time derivative $\dot{R}_n$ on the CMB. To avoid artefacts due to temporal variations in the resolution of *gufm1*, we truncate this field model at degree and order $n_{max} = m_{max} = 8$, whereas we truncate the *xCHAOS* field model at $n_{max} = m_{max} = 14$. Fig. 6(a) shows the Mauersberger–Lowes spectrum of the historical geomagnetic field on the CMB as a function of time, and Figs 6(b) and (c) show the corresponding Mauersberger–Lowes spectra for the dipole and quadrupole families, respectively. The same spectra for the
<figure><img src="image_7.png" /><figcaption>**Figure 5.** (a) The Mauersberger-Lowes spectra of a peak travelling between times $t_0$ to $t_0 + \delta t$ from $n'$ to $n' + 1$, and (b) a schematic illustration of the energy cascade. The background flat spectrum is $R_1$ and the peak is elevated by $\delta R$. Normalized spectral transfer rates versus spherical harmonic degree $n$ for a peak-to-background ratios of $(R_1 + \delta R)/R_1 - 3$ (c) and $(R_1 + \delta R)/R_1 = 4$ (d): actual values $\delta t \tilde{\gamma}_n$ (31) denoted by asterisks versus the model $\delta t \gamma_n$ (32) denoted by triangles. The positions of the model $\gamma_n$ have been shifted by one half unit of $n$ to represent transfer between harmonics $n$ and $n + 1$.</figcaption></figure>

1999–2009 core field are shown in Figs 6(d)–(f). These contour plots provide some qualitative evidence for geomagnetic energy cascade, in the form of migration of local maxima and minima from one $n$-value to its neighbours over time, for example, in the dipole family spectrum from $n = 3$ to 4 and from $n = 5$ to 6 (Fig. 6b), and more pronouncedly in the quadrupole family spectrum from $n = 5$ through 4 to 3 (Fig. 6c).

The next step is to calculate the flux spectra $F_n$ from (20) using $\dot{R}_n$ from Fig. 6 and diffusion spectra $D_n$ from (13), in which we use the fundamental mode eigenvalues for $p_n$ and assume $\eta = 2 \text{ m}^2 \text{ s}^{-1}$ for the magnetic diffusivity of the outer core (well within the range of estimated values, see Poirier 2000). Table 2 lists the first 14 fundamental free decay modes and the corresponding free decay timescales. Fig. 7 shows the time dependence of $F_n$ obtained this way. The black curves are derived from $\dot{R}_n$ and $D_n$ at $n = 1$, whereas the coloured $F_n$ curves are derived from the $\dot{R}_n$ and $D_n$ degrees from 1 to $n$. In general, we find that $D_n$ increases with $n$, but in nearly all

cases $|D_n| \ll |\dot{R}_n|$, that is, the impact of fundamental mode diffusion on the flux spectrum of the core field is of secondary importance.

To illustrate how the recursion formula (20) works, consider Fig. 7(a). The black curve is the most positive because the dipole decreases, while the other curves are in general successively smaller due to the pervasive increase of most non-dipole components, in agreement with uniform flux convergence in the non-dipole part of the spectrum and consistent with local transfer. Crossing of individual $F_n$ curves is produced by instantaneous decrease of some non-dipole harmonics, and this may indicate deviations from purely local transfer.

The final step is to compute the spectral transfer rates $\gamma_n$ from $F_n$ and $R_n$ using (22). In Fig. 8, we compare the time average of the absolute spectral transfer rates $|\gamma_n|$ with the time average $1/\tau_n$ defined in (33). The positions of the spectral transfer rates have been shifted by one half unit of $n$ in Fig. 8 (and in subsequent figures) in order to emphasize that they represent transfer between a pair of
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The Mauersberger-Lowes spectra of the geomagnetic field at the core-mantle boundary as a function of time. Spectral values are defined only at integer spherical harmonic degrees *n*. For the period 1840–1990 from core field model *gufm1* (Jackson *et al.* 2000) with *n*<sub>max</sub> = 8: (a) full field; (b) dipole family; (c) quadrupole family. For the period 1999–2009 from core field model *xCHAOS* (Olsen & Mandea 2008) with *n*<sub>max</sub> = 14: (d) full field; (e) dipole family; (f) quadrupole family. The log-scale ranges of the spectra in nT<sup>2</sup> are given in the colour bars.</figcaption></figure>

<table><caption>**Table 2.** Fundamental free decay modes $p_n$, and free decay timescales $\tau_n^d$ in years based on magnetic diffusivity of $\eta = 2 \text{ m}^2\text{s}^{-1}$, for spherical harmonic degrees $n = 1-14$.</caption><thead><tr><th>Degree</th><th>p<sub>n</sub></th><th>τ<sub>n</sub><sup>d</sup> [yr]</th></tr></thead><tbody><tr><td>1</td><td>π</td><td>9733</td></tr><tr><td>2</td><td>4.493</td><td>4758</td></tr><tr><td>3</td><td>5.763</td><td>2892</td></tr><tr><td>4</td><td>6.988</td><td>1967</td></tr><tr><td>5</td><td>8.183</td><td>1435</td></tr><tr><td>6</td><td>9.356</td><td>1097</td></tr><tr><td>7</td><td>10.513</td><td>869</td></tr><tr><td>8</td><td>11.657</td><td>707</td></tr><tr><td>9</td><td>12.790</td><td>587</td></tr><tr><td>10</td><td>13.916</td><td>496</td></tr><tr><td>11</td><td>15.033</td><td>425</td></tr><tr><td>12</td><td>16.145</td><td>369</td></tr><tr><td>13</td><td>17.250</td><td>323</td></tr><tr><td>14</td><td>18.351</td><td>285</td></tr></tbody></table>

neighbouring harmonics. The overall magnitudes of the two rates are comparable at low degrees in the total field and in both the dipole and quadrupole families. Also, their trends are similar up to spherical harmonic degree $n = 8$ in the total field, $n = 7$ in

the dipole family, and $n = 6$ in the quadrupole family, although there are substantial differences at higher degrees. The absolute spectral transfer rates in Fig. 8 tend to be less than the corresponding SV rates $1/\tau_n$ for the total field because $\gamma_n$ can have either sign, whereas $\tau_n$ is not. The dipole family $\gamma_n^a$ are dominantly positive, whereas the quadrupole family $\gamma_n^s$ are dominantly negative, so the absolute rates for the total field are smaller than those for each family. The generally monotonic variations of $\gamma_n$ with $n$ in Fig. 8 is suggestive of the power-law spectral transfer rates commonly found in turbulence measurements (Corrsin 1961, 1964; Hill 1978). This behaviour is consistent with (although does not prove) our interpretation of local, nearest-neighbour energy transfer through the geomagnetic spectrum.

## 4 DIPOLE COLLAPSE AND SPECTRAL TRANSFER IN A NUMERICAL DYNAMO

Numerical dynamos show energy cascade behaviour, especially during dipole collapse events that precede polarity reversals or excursions (Olson *et al.* 2009). Although these events involve both magnetic advection and diffusion, even at very high magnetic Reynolds numbers (Liu & Olson 2009), there are time intervals when the net effects of diffusion on the low harmonic degree part of the magnetic field are negligibly small. During these time intervals, the mean square field intensity on the outer boundary of the
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Flux spectra of the core field $F_n$ in nT$^2$ yr$^{-1}$ as a function of spherical harmonic degree $n$ and time.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Time average of the absolute value of the spectral transfer rate $\gamma_n$ (triangles) compared with time average of the SV rate $1/\tau_n$ (squares), both in yr$^{-1}$, versus spherical harmonic degree $n$, for core field models *gufm1* between 1840–1990 (a)-(c) and *xCHAOS* between 1999–2000 (d)-(f).</figcaption></figure>
dynamo remains practically constant, the dipole intensity rapidly decreases while the non-dipole field compensates by increasing intensity.

Fig. 9 shows an example of magnetic energy cascade during a dipole collapse event from a numerical dynamo model. The control parameters of this model are Ekman number $E = 0.006$, Prandtl number $Pr = 1$, magnetic Prandtl number $Pm = 20$, and a com- positional Rayleigh number $Ra$ about 10 times supercritical, with no-slip mechanical boundary conditions, zero light element flux at the outer boundary, and fixed light element concentration at the in- ner core boundary (for parameter definitions and more details see Olson 2007). The choice of $Pr = 1$ can be justified by the Prandtl turbulence hypothesis, while the choice of large $Pm$ is aimed to produce a large magnetic Reynolds number and a chaotic dynamo. Although the control parameters in this specific dynamo model are very far from being Earth-like, especially the large Ekman number,

this model is suitable for our purposes because it has a dipole- dominated magnetic field during stable polarity periods, with rapid dipole collapse events that sometimes result in polarity reversals (Olson 2007; Olson et al. 2009), a combination that is hard to find in numerical dynamos (Kutzner & Christensen 2002). The dipole collapse event shown here culminated in a polarity excursion, in which the dipole became very weak for about one half of a free decay time then re-strengthened on return to its original polarity. Magnetic field variables in Fig. 9 are scaled by $(\rho\Omega/\sigma)^{1/2}$, where $\sigma$, $\rho$, $\Omega$ are electrical conductivity, density, and angular velocity of rotation, respectively, and time is scaled by the most fundamental (longest) dipole free decay time, starting from an arbitrarily selected time before the main dipole collapse. Over the whole time interval the average dipole decrease rate is just 1.5 times the dipole free decay rate, however over some shorter periods, equivalent to about 150 yr, the dipole decrease rate reaches 6.5 times dipole free decay

<figure><img src="image_11.png" /><figcaption>**Figure 9.** (a) The Mauersberger–Lowes spectrum of the total field at the outer boundary of the dynamo model as a function of time during a dipole decrease event. Spectral values are defined only at integer spherical harmonic degrees $n$. The linear-scale range of the dimensionless spectrum is given in the colour bar. (b) Time variation of the dimensionless rms total field intensity $B_{rms}$ (solid), the rms non-dipole field intensity truncated at $n = 9 B_{rms}(10 > n > 1)$ (dotted), and the rms dipole field intensity (dashed) $B_{rms}(n = 1)$ on the outer boundary. Dimensionless time is in dipole free decay units. Blue vertical lines correspond to the three times indicated in Fig. 10, and the three periods are marked by the red arrow lines.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Time averages of dimensionless flux spectra $F_n$ (top) and dimensionless spectral transfer rates $\gamma_n$ (bottom) both scaled by the dipole free decay rate, versus spherical harmonic degree $n$ during three time intervals of the dipole collapse event shown in Fig. 9. (a) time interval 0.05–0.28; (b) time interval 0.05–0.17; (c) time interval 0.17–0.28.</figcaption></figure>

rate. For comparison, the present geomagnetic dipole decrease rate is about 11 times dipole free decay rate (Olson & Amit 2006).

Fig. 9(a) shows the variation of the Mauersberger–Lowes spectrum as a function of time at the outer boundary during the collapse. Although the model is taken to $n_{max} = 32$, we consider only harmonics up to $n = 9$. Fig. 9(b) shows the variation of the total field intensity, the non-dipole intensity truncated at $n = 9$, and the dipole intensity, all at the outer boundary and non-dimensionalized with the same parameters as $R_n$. Over the time interval between 0.05 and 0.28 (which corresponds to about 4.5 kyrs for the core), the dipole intensity decreases substantially but the total field intensity net change from beginning to end is negligible. Meanwhile, the non-dipole intensity increases and magnetic energy in the Mauersberger–Lowes spectrum (Fig. 9a) appears at progressively higher harmonic degrees with time, as in an ideal forward cascade.

Fig. 10 shows time averages of the dimensionless flux spectrum $F_n$ (top) and the dimensionless spectral transfer rate $\gamma_n$ (bottom) over three time intervals from the dynamo model, calculated according to (20) and (22), respectively, using the dimensionless Mauersberger–Lowes spectrum $R_n$ shown in Fig. 9 and values of $D_n$ from the dynamo model. Fig. 10(a) corresponds to the time interval 0.05–0.28, which covers the main dipole collapse event. Fig. 10(b) corresponds to the time interval 0.05–0.17, while Fig. 10(c) corresponds to the time interval 0.17–0.28. Over each of these time intervals the net change in total field intensity is negligible, and as expected, the net effect of radial diffusion is also negligible.

Fig. 10(a) shows that, on average the flux spectrum $F_n$ and the transfer rates $\gamma_n$ are all positive and are of the same order of magnitude, corresponding to a nearly uniform, forward energy cascade throughout the low degree Mauersberger–Lowes spectrum. It is worth noting that the odd $F_n$ are systematically larger than the even $F_n$, a consequence of $\dot{R}_n < 0$ for the odd whereas $\dot{R}_n > 0$ for the even harmonics. This behaviour may suggest that a quasi-local

transfer in steps of two harmonics is significant in the evolution of the spectrum in this numerical dynamo model.

However, there are substantial short term variations in both $F_n$ and $\gamma_n$ within the collapse event, as the difference between Figs 10(b) and (c) reveals. Both of these sub-intervals begin with a dipole intensity fall, but in the first interval the non-dipole exhibits zero net intensity change, whereas in the second interval the non-dipole intensity has a net increase (see Fig. 9b). Fig. 10(b) shows that $F_n$ and $\gamma_n$ are larger than average and increase strongly with harmonic degree $n$ when the dipole collapse is particularly rapid, consistent with the accelerated dipole decrease in a strong forward cascade during this time. In contrast, Fig. 10(c) shows that $F_n$ and $\gamma_n$ are small or negative and decrease with harmonic degree $n$, consistent with inverse cascade at high harmonics and energy trapping at intermediate harmonics during this time. It is perhaps unexpected that the cascade process is so strongly variable in time, even within a single dipole collapse event. In the next section, we show that comparable short-term variability is seen in the historical geomagnetic field.

## 5 GEOMAGNETIC DIPOLE INTENSITY STEPS

As noted before, in our local transfer model the spectral rate $\gamma_n$ may be positive or negative, depending on whether the energy cascade is forward or inverse. During the historical era the dipole family is generally characterized by positive $\gamma_n^a$, corresponding to forward energy cascade from low to high harmonic degrees. In contrast, the quadrupole family exhibits negative $\gamma_n^s$ and decreasing with $n$ (Fig. 11, which is a signed version of Fig. 8c), corresponding to inverse energy cascade. The latter behaviour persists over the entire historical period, with only minor variations from one snapshot to another. It is perhaps counter-intuitive that $\gamma_1^s$ is negative, suggesting
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Time average (1840–1990) quadrupole family spectral transfer rate $\gamma_n^s$ (triangles) compared with the corresponding SV rates $1/\tau_n^s$ (squares), both in yr$^{-1}$, versus spherical harmonic degree $n$, from core field model *gufm1*.</figcaption></figure>

an energy flux to the equatorial dipole, despite the decrease in $R_1^s$ over the historical period (Amit & Olson 2008). The reason is that over most of the historical period the equatorial dipole varies much more slowly compared to the other harmonics so free decay becomes comparable to and even larger than its SV, i.e. $|\dot{R}_1^s| < |D_1^s|$. Since 1960, however, the equatorial dipole decrease has accelerated, causing a poleward motion of the dipole tilt (Amit & Olson 2008), and in that period $F_1^s$ became positive and large. The interpretation of the equatorial dipole time-evolution prior to 1960 in terms of our flux transfer method is therefore probably non-robust.

The numerical dynamo model results in the previous section demonstrate that $\gamma_n$ may vary substantially over short time intervals within a longer dipole decrease event. Interpreted as a cascade process, changes in $\gamma_n$ correspond to changes in the cascade direction, and can decelerate or accelerate a dipole decrease event. Evidence for these types of fluctuations can be seen in the historical core field. Fig. 12 shows the rms intensity of the dipole family $B_{\text{rms}}^a$ on the CMB during the historical period. We focus on two time intervals, 1895–1935 labeled (a) when $B_{\text{rms}}^a$ was nearly constant in time, and 1955–1990 labeled (b) when $B_{\text{rms}}^a$ decreased precipitously. The time averages of the dipole family spectral transfer rates $\gamma_n^a$ over these two intervals are shown in Fig. 13.

The transition from positive to negative $\gamma_n^a$ with increasing $n$ during 1895–1935 (Fig. 13a) is indicative of magnetic energy forward

cascade from the dipole and magnetic energy inverse cascade from $n=8$, resulting in magnetic energy trapping at intermediate harmonic degrees. This is consistent with simultaneous axial dipole decrease and constant $B_{\text{rms}}^a$ (Fig. 12) in which the energy cascading from the dipole remains within the observable part of the core field spectrum. Note the similarity between $\gamma_n^a$ in Fig. 13(a) to the spectral transfer rate in the numerical dynamo dipole collapse during its slower stage (Fig. 10c). Similarly, the increase in $\gamma_n^a$ with increasing $n$ for 1955–1990 (Fig. 13b) indicates a forward energy cascade, which simultaneously accounts for the decrease of the axial dipole and the sharp decrease in $B_{\text{rms}}^a$ (Fig. 12), allowing energy to leave the observable part of the core field spectrum via a uniformly forward cascade. Here again there is close resemblance to $\gamma_n$ in the numerical dynamo during its time of most rapid dipole decrease (Fig. 10b).

Analysis of the high resolution satellite field model xCHAOS reveals that the difference among the individual $F_n$ curves increases significantly within one decade, demonstrating the rapid changes in the geodynamo (Olsen & Mandea 2008). Comparing the large-scale spectral transfer rates for 1999–2009 (Fig. 8e) with the corresponding time average for the preceding period 1955–1990 (Fig. 13b) shows in both cases positive trends, suggesting forward cascade at low harmonic degrees, although at lower rates than in the historical field. The higher degrees revealed using the modern field

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Dipole family rms intensity $B_{\text{rms}}^a$ in mT for 1840–1990 from core field model *gufm1*. The two shaded time intervals (a) and (b) correspond to the time averages shown in Fig. 13.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Dipole family spectral transfer rates $\gamma_n^a$ (triangles) compared with corresponding SV rates $1/\tau_n^a$ (squares), both in $\text{yr}^{-1}$, versus spherical harmonic degree $n$ averaged over the two time intervals (a) 1895–1935; and (b) 1955–1990, shown in Fig. 12.</figcaption></figure>

model xCHAOS exhibit negative $\gamma_n^a$ at higher harmonic degrees (not shown), suggesting that magnetic energy is trapped at $n \sim 9$. For the quadrupole family, the monotonic decrease of $\gamma_n^s$ with $n$ during the historical era (Fig. 11) transforms into a more complicated $n$-dependence in the modern era (not shown), although for most degrees the rates are again negative, indicative of inverse magnetic energy cascade in the quadrupole family.

## 6 RELATION WITH PREVIOUS GEOMAGNETIC DIPOLE MOMENT ANALYSES

Core flow models inferred from inversions of geomagnetic SV data were used to study dipole moment variations. Most core flow models neglect magnetic diffusion, under the so-called frozen-flux approximation (Roberts & Scott 1965), assuming that the magnetic Reynolds number in the core is large. Olson & Amit (2006) introduced a new theory that combines core flow and magnetic field models to map the advective contributions to the axial dipole SV. According to their formulation, axial dipole moment advective sources and sinks form where meridional flow interacts with radial magnetic field. Applying their derivation for the geomagnetic field, they identified regions below the CMB where advection weakens the dipole moment. They concluded that the geomagnetic axial dipole moment decrease is caused by the combined effects of intensification and poleward migration of reversed dipolar field as well as

normal dipolar field equatorward motion (see also Voorhies 1991). Amit & Olson (2008) applied the same theory to localize advective sources and sinks of equatorial dipole moment that are responsible for dipole tilt variations. They found that the recent dipole tilt decrease event in the past 40 yr is caused by two equatorial dipole advective sinks where positive/negative radial magnetic field is transported westward/northward towards/away from the negative equatorial pole, respectively.

In fact, the core flow models of Jackson (1997) systematically underestimate axial dipole SV, which he hypothesized is an artefact of the tangential geostrophy constraint. In contrast, Whaler & Davis (1997) argued that this underestimation is due to unmodelled effects of magnetic diffusion. Core flow models constructed assuming steady flow in a drifting reference frame (Voorhies & Backus 1985) were found to improve the fit to the axial dipole SV (Holme & Whaler 2001). Increasing evidence for magnetic diffusion motivated incorporation of diffusion effects in core flow inversion schemes. The difficulty of accounting for diffusion is that the variation of the field with depth is unknown, so radial diffusion SV cannot be imaged from observations. Voorhies (1993) argued that diffusive field expulsion and fluid upwelling will have the same kinematic effect in terms of the observed SV. Holme & Olsen (2006) estimated diffusive SV using free decay modes. Their largest estimates were assigned for the lowest degrees, most notably the dipole. These free decay models were used as error estimates in their core flow inversions. Surprisingly, the resulting flows were nearly unchanged compared to the flows inverted using conventional error estimates,
perhaps indicating that the flows cannot account for diffusive ef-
fects (Holme 2007). A different approach was taken by Amit &
Christensen (2008). They found in numerical dynamos that radial
and tangential diffusion effects are well-correlated. They then used
the pattern of tangential diffusion from observations and an extrap-
olated magnitude to account for radial diffusion in order to model
full magnetic diffusion. Comparison of their diffusive and frozen-
flux inverted core flows suggests that accounting for radial magnetic
diffusion may in some locations affect significantly images of the
flow at the top of Earth's core.

Our cascade model may be viewed as a complimentary approach
to core flow inversions. Our calculated transfer rates cannot substi-
tute for core flow models; these rates provide only a 1-D description
of magnetic field advection in spectral space, while core flows pro-
vide a 2-D image on the CMB. However, the process of core flow
inversions suffer from numerous assumptions, making the results
of theses models open to diverse interpretations (for a review see
Holme 2007). Our study may provide a framework for testing in-
verted core flow models. By substituting a spectral description of
the velocity field at the top of the core into (9) the transfer spec-
trum T_n is found, from which the spectral flux F_n can be calculated
(19), and finally the spectral transfer rate γ_n is obtained using (22).
Comparison of γ_n inferred from an inverted core flow model with
our direct calculation from the Mauersberger-Lowes spectra may
connect inverted core flow models with the Mauersberger-Lowes
spectra and highlight the ability of these models to capture the
spectral transfer phenomenon in general, and the dipole decrease
in particular. Such a comparison, though beyond the scope of this
study, is a worth-while future prospect.

7 CONCLUDING REMARKS

We interpret the changes in the low degree Mauersberger–Lowes
geomagnetic spectrum at the CMB since 1840 using a magnetic
energy cascade model. Our model is motivated by the simultaneous
dipole field decrease and the non-dipole field increase over the
past 170 yr. We quantify the evolution of the Mauersberger–Lowes
spectrum by the spectral transfer rate γ<sub>n</sub>, which indicates the rates of
energy transfer between neighbouring spherical harmonic degrees.
In our local transfer model, positive γ<sub>n</sub> corresponds to forward
cascade of energy through the spectrum (from n to n + 1), negative
γ<sub>n</sub> corresponds to inverse energy cascade (from n + 1 to n).

The values of our inferred spectral transfer rates are generally
consistent with the traditional SV rates 1/τn, and also consistent
with the independently estimated geodynamo memory time (Hulot
et al. 2010). However, we find that the transfer rates themselves fluc-
tuate significantly, sometimes changing sign, particularly within the
dipole family. Between 1955 and 1990 for example, the rms dipole
family intensity B<sup>a</sup><sub>rms</sub> decreases rapidly and γ<sup>a</sup><sub>n</sub> increases with n,
implying a forward cascade with energy transfer out of the core
field spectrum towards n > 8. In contrast, between 1895–1935 B<sup>a</sup><sub>rms</sub>
is nearly constant while γ<sup>a</sup><sub>n</sub> decreases with n, becoming negative
at n > 6, implying energy trapping at intermediate n-values in the
core field spectrum. A numerical dynamo model shows comparable
behaviour during a prolonged dipole decrease event. For the geo-
magnetic quadrupole family, we find γ<sup>s</sup><sub>n</sub> is negative and decreases
with n, suggesting an inverse cascade.

In spite of these general similarities to a cascade process, we ac-
knowledge that caution should be exercised in applying this model
to the geomagnetic field. The central assumption of our model, local
transfer through the magnetic energy spectrum, is not a universal

property in MHD systems. Our free decay model of magnetic dif-
fusion serves as an energy sink only, while diffusive sources in
the form of magnetic field expulsion are not modelled. In addition,
there is conflicting evidence in the geomagnetic field behaviour (see
Fig. 7 and its discussion in Section 3).

Our analytical examples show that our model can capture the cas-
cade phenomenon provided that local energy transfer is dominant,
but it can equally interpret non-local transfer as local. Although we
point to some geomagnetic examples (e.g. Figs 2–5), further iden-
tification of local spectral transfer in core field is needed, in order
to substantiate our model. Extended, high resolution records of the
core field expected from future satellite missions such as SWARM
could make this possible.

Finally we address the important issue of whether the current
geomagnetic dipole decrease will result in a reversal. The similarity
between the forward energy cascade in the dipole collapse event that
preceded a reversal in the numerical dynamo model and the forward
energy cascade in the dipole family of the geomagnetic field may
be viewed as evidence for a beginning of a reversal. However, we
have shown that the time-variability of the spectral transfer rates
is very strong. It is therefore probably impossible to know, based
on the relatively short 150-yr period of available high resolution
geomagnetic field observations, whether these cascade trends will
prevail and lead to a reversal or not. Improvement in the quality
and spatial resolution of archaeomagnetic field models may allow
examining the persistence of the forward cascade trends inferred
from the historical field over millennial timescales.

ACKNOWLEDGMENTS

We thank F.J. Lowes and C.V. Voorhies for their very detailed re-
views that significantly improved the manuscript. We also thank
R. Holme for his useful comments. PO gratefully acknowledges
support from NSF Geophysics Grant EAR0604974.

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