Nonlinear MHD Rossby wave interactions and persistent geomagnetic field structures

Abstract

The geomagnetic field presents several stationary features that are thought to be linked to inhomogeneities at the core–mantle boundary. Particularly important stationary structures of the geomagnetic field are the flux lobes, which appear in pairs in mid- to high mid- to high latitudes. A recently discovered stratified layer at the top of the Earth’s core poses important constraints on the dynamics at this layer and on the interaction of the core dynamics and the base of the mantle. In this article, we introduce the linear and nonlinear theories of magnetic Rossby waves in a thin shell at the top of the Earth’s core. We study the nonlinear interaction of these waves in the presence of prescribed forcings at the base of the mantle of both a thermal and a topographic nature. We show that the combined effects of forcing and nonlinear interaction can lead the wave phases to be locked around a particular geographical longitude, generating a quasi- stationary flow pattern with a significant meridional component. The solutions of the system are shown to be analogous to atmospheric blocking phenomena. Therefore, we argue that persistent and long-lived structures of the geomagnetic field, such as the geomagnetic lobes, might be associated with a blocking at the top of the Earth’s core due to nonlinear stationary waves.

1. Introduction

The geomagnetic field is characterized by several asymmetries that can be evidenced by analysing either its long-term averages or local differences in its secular variations. These asymmetries or persistent structures must in some way reflect a lack of homogeneity in the dynamo process in the Earth’s core. Below we list a few of the most noticeable non-homogeneous/persistent structures of the geomagnetic field.

One of the most remarkable persistent structures of the geomagnetic field is the so-called geomagnetic flux lobes, which are found in reconstructions of the geomagnetic field at the core–mantle boundary [1]. These are persistent quasi-stationary structures that appear in two pairs: one southward of the Americas and the other in the western Pacific, with both of the pairs exhibiting one lobe in high latitudes of each hemisphere. Unlike many other signatures of the geomagnetic field that exhibit intrinsic westward/eastward drifts, these structures are characterized by patches of intense magnetic flux fluctuations around a mean position. Given their persistent behaviour, the geomagnetic lobes appear in time-averaged maps of the geomagnetic field evaluated with archeomagnetic data [2], and can also be detected through lava flow recordings [3].

Another remarkable asymmetric signature of the geomagnetic field refers to the so-called dipole asymmetries. In fact, there are indications that the spatial distribution of the non-axial component of the geomagnetic dipole is no longer symmetric, for several studies have pointed out the existence of preferential paths for geomagnetic dipole reversals. These preferential paths are longitudinal strips in which the occurrence probability of a polarity transition is increased, namely the one located roughly in the same longitudes as the American continent and the other near Australia and the eastern Pacific (see, for instance, [4,5]).

Apart from the two aforementioned examples of geomagnetic field persistent anomalies, it is observed that the secular variation in the geomagnetic field exhibits an asymmetry between the Eastern and Western Hemispheres, with the Pacific sector being characterized by lower secular variation than the Atlantic one. This asymmetric feature was first noted in palaeomagnetic recordings from Hawaiian lava flows [6] and has been labelled the Pacific low secular variation.

Phenomena associated with heterogeneous conditions in the lower mantle are natural candidates to explain the existence of these asymmetric and persistent features of the geomagnetic field, in particular the temperature variations. This statement is corroborated by several studies of numerical geodynamo modelling with heterogeneous core–mantle boundary conditions [7–9]. Besides the lateral variations in temperature at the core–mantle boundary, there is also the under-explored hypothesis of the topography at the base of the mantle playing a role in these persistent geomagnetic field structures. As evidence for this hypothesis, we note that topographic anomalies at the mantle base roughly coincide with the longitudes of the geomagnetic field lobes and with preferential paths of the geomagnetic field reversals, as can be observed in core–mantle boundary topography reconstructions [10].

On the other hand, in the geophysical fluid dynamics literature both topographic and thermal anomalies are well-known forcing mechanisms for large-scale wave disturbances (see, for instance, [11,12] and references therein). In particular, some papers in the atmospheric fluid dynamics literature have pointed out the role of topography and/or parametric thermal forcings in coupling different wave modes [12,13]. In this context, the combined mode coupling effects of such parametric forcings and the advective nonlinearity can yield chaotic transitions in the phase-space dynamics that yield persistent flow patterns that are characterized by predominantly meridional flow [14,15]. These patterns are labelled in the atmospheric dynamics literature as blocking states. Recently, Ciro et al. [15] showed that the emergence of such blocking states does not rely on complex spectral networks that require a regular truncation of the spectral equations with a large number of modes. Rather, they demonstrated that the dynamics of a single interacting wave triplet, which constitutes the most elementary form of the nonlinearity, supports the existence of a non-drifting wave-pattern solution that acquires a finite measure in the phase space when the triad equations are perturbed by a single topography wave mode.

Since the wave modes that are believed to exist in the Earth’s core are generalizations of the hydrodynamic eigenmodes of the atmosphere and ocean by including the magnetic field effects [16,17], it would be natural to extend the existing wave interaction theory of geophysical fluid dynamics to such magnetohydrodynamic (MHD) modes. Therefore, the scope of this article is to propose an MHD generalization of the nonlinear theory of Rossby–Haurwitz waves for the context of the Earth’s core, in the presence of both thermal and topographic forcings. We adopt here the same reductionist philosophy of the aforementioned theoretical works of the geophysical fluid dynamics literature in that we isolate the essential physics in the simplest possible setting, rather than trying to reproduce particular features in a complete MHD numerical model. In this reductionist context, as in [15], we shall analyse the dynamics of a single interacting wave triad, since it contains the essential features of the nonlinearity of the full model equations, and the thermal and topographic forcings will be assumed to bear a single wave-mode structure.

Therefore, let us first consider the simplest model to describe planetary MHD rotational waves of the Earth’s core that may be directly affected by mantle–core thermal anomalies. In this context, although the Earth’s core is a thick layer, several studies have pointed out the existence of a stratified sublayer at its top that results from the accumulation of light elements in this region [18–20], because as the metallic alloy that constitutes the Earth’s core cools down it separates into liquid iron plus the light element that is pushed to the top of the core by buoyancy forces. Very little is known about the structure and the thickness of this layer, although it is believed to be of the order of a few hundred kilometres, which corresponds to around 5–10% of the radius of the Earth’s outer core. This relatively thin layer could have important impacts for the wave dynamics in the Earth’s core; for instance, the possibility of trapped waves in the equatorial region [21]. Consequently, as the fluid depth in this case is much smaller than the characteristic wavelength of the disturbances we are interested here, we adopt the MHD generalization of a model describing a barotropic quasi-geostrophic flow in the presence of topography and a prescribed divergence forcing mimicking the effects of thermal anomalies.

Therefore, this article is organized as follows. We present the linear theory of MHD Rossby–Haurwitz waves in a shallow shell in §2. In §3, we revisit the nonlinear interaction theory of MHD Rossby–Haurwitz waves on the sphere developed by [22] in the context of the solar tachocline. Section 4 demonstrates the role of topography in coupling two different wave modes. In §4, we also obtain a low-order theoretical model to study the persistent structures of the geomagnetic field by augmenting the nonlinear interaction theory presented in §3 to include a prescribed thermal forcing and a parametric forcing associated with the effect of topography. The results of the numerical integration of the low-order nonlinear model developed in §4 are presented in §5 for the case where both the thermal forcing and topography exhibit the spatial structure of the single spherical harmonic $Y_2^2$. We show that, for certain forcing parameters, the primarily forced mode becomes locked, in the sense that its phase vacillates around an average value instead of propagating around the globe. In view of this result, in §6, we show that a quasi-stationary streamline pattern dominated by a meridional component can constitute a transport barrier for the radial component of the magnetic field, making the radial magnetic field structures become trapped in a localized geographical region in a similar fashion to the blocking phenomenon in the Earth’s atmosphere. Consequently, the arguments in §6 suggest that a blocked state in the flow at the core–mantle boundary is a plausible candidate for explaining the geomagnetic flux lobes. The main conclusions are summarized in §7.

2. MHD Rossby waves: linear theory

MHD Rossby waves arise in conducting fluids (plasmas) subjected to the action of the Coriolis and Lorenz forces such as in planets and stars. Recently, they have received special attention from the solar physics community, as can be seen in [23,24]. In particular, in thin shells they can be found as eigensolutions of the shallow-water MHD equations [25], which also support the existence of magneto-inertio-gravity waves. In order to simplify our analysis and make it analytically tractable in spherical coordinates, we shall employ a simpler system, the quasi-geostrophic MHD equations derived in [26] (see also [27] for the derivation of these equations as a distinguished limit of the rotating MHD shallow-water equations). This model has the advantage of supporting only Rossby waves, along with being probably a good model for the weak field regime explored here. For a more complete analysis of the linear eigenmodes of the MHD shallow-water equations on a sphere, see [28]. In a compact form, the barotropic version of the quasi-geostrophic MHD equations can be written as

$$\frac{\mathrm{\partial }q}{\mathrm{\partial }t}+J(\psi ,q)=\frac{1}{{\mu }_0\rho }J(A,j)$$ 2.1

and

$$\frac{\mathrm{\partial }A}{\mathrm{\partial }t}+J(\psi ,A)=0,$$ 2.2

where

$$q={\mathrm{\nabla }}^2\psi -F\psi +2\mathrm{\Omega }sin\theta +Fh$$ 2.3

is the potential vorticity, ψ is the streamfunction, h(ϕ, θ) represents the topography of the core–mantle boundary, with (θ, ϕ) representing the regular spherical coordinate system (θ being the latitude and ϕ the longitude), and the magnetic current is given by $j={\mathrm{\nabla }}^{2}A$, where A is the magnetic potential; μ₀ and ρ are, respectively, the magnetic permeability and the density of the medium, and F is the inverse of the Rossby deformation radius squared. In the equations above, the bilinear operator J is the Jacobian operator defined by

$$J(\hspace{0.17em}f,g)={\mathrm{\nabla }}^{\perp }f\bullet \mathrm{\nabla }g,$$ 2.4

with ${\mathrm{\nabla }}^{\perp }$ being the perpendicular two-dimensional gradient operator; for any two differentiable functions $f,g:{\mathbb{R}}^2\to \mathbb{R}$, J will assume different expressions depending on the chosen system of coordinates.

(a). β-plane approximation

For pedagogical reasons, and in order to have some understanding of the time scales involved, we first derive the dispersion relation of MHD Rossby waves in Cartesian β-plane coordinates. The β-plane approximation is widely used in the literature of atmosphere and ocean dynamics [29] and consists of considering a linear approximation of the Coriolis parameter with respect to a reference latitude; this is done by a Taylor expansion of the rotation parameter,  f = f₀ + β(y − y₀) + O(y − y₀)², and neglecting the O(y − y₀)² terms, where f₀ = f(y₀) = 2Ωsinθ₀, $\beta =\text{d}f/\text{d}y{|}_{y=y_0}=2\mathrm{\Omega }\cos {\theta }_0/R$, with Ω = 2π/day being the Earth’s rotation rate, θ₀ the reference latitude, R the radius of the spherical shell and y − y₀ the meridional displacement. The Jacobian operator in Cartesian coordinates (x, y) is written as

$$J(\hspace{0.17em}f,g)={\mathrm{\nabla }}^{\perp }f\bullet \mathrm{\nabla }g=(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\frac{\mathrm{\partial }g}{\mathrm{\partial }y}-\frac{\mathrm{\partial }g}{\mathrm{\partial }x}\frac{\mathrm{\partial }f}{\mathrm{\partial }y}).$$ 2.5

In order to derive the dispersion relation of the eigenmodes in the present model, we need to choose a reference state that is characterized by a resting velocity field, $\overrightarrow{u}={\mathrm{\nabla }}^{\perp }\psi =\overrightarrow{0}$, and a constant background magnetic field in the zonal direction, $\overrightarrow{B}=(B_0,0)$. The resulting linearized equations, in the absence of topographic variations, h(ϕ, θ) ≡ 0, can therefore be written in operator form as follows:

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[\begin{array}{c}{\mathrm{\nabla }}^2\psi \\ A\end{array}\right]=\left[\begin{array}{cc}\beta {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}& {\displaystyle \frac{B_0}{{\mu }_0\rho }}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}({\mathrm{\nabla }}^2+F)\\ B_0{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}& 0\end{array}\right]\left[\begin{array}{c}\psi \\ A\end{array}\right].$$ 2.6

In the equation above, the linear differential operator is skew-Hermitian and therefore has imaginary eigenvalues, so the solutions of (2.6) are purely oscillatory. Thus, in order to find these solutions, we consider the following ansatz:

$$\left[\begin{array}{c}\psi \\ A\end{array}\right]=\mathrm{\Lambda }{e}^{\text{i}kx+\text{i}ly-\text{i}\omega t}\overrightarrow{R},$$ 2.7

where Λ is an arbitrary constant, $\overrightarrow{k}=(k,l)$ is the vector wavenumber, $\omega =\omega (\overrightarrow{k})$ is the eigenfrequency and $\overrightarrow{R}$ is the right eigenvector of the symbol of the differential operator, ${\mathcal{L}}^{\ast }$, that results from the substitutions ${\mathrm{\partial }}_x\mapsto ik$ and ${\mathrm{\partial }}_y\mapsto il$, namely

$${\mathcal{L}}^{\ast }=\text{i}\left[\begin{array}{cc}\beta k& {\displaystyle \frac{B_0}{{\mu }_0\rho }}k(k^2+l^2+F)\\ B_{0}k& 0\end{array}\right].$$ 2.8

Therefore, the eigenfrequencies $\omega =\omega (\overrightarrow{k})$ are obtained as the roots of the characteristic polynomial, i.e.

$$\text{det}({\mathcal{L}}^{\ast }-\text{i}\omega I_d)=0,$$ 2.9

where I_d represents the identity matrix. The two possible solutions of the characteristic equation above are given by

$${\omega }_{\pm }=\frac{-\beta k\pm \sqrt{{\beta }^2+4v_a^2{(k^2+l^2+F)}^2}k}{2(k^2+l^2+F)},$$ 2.10a

with the branches ω₊ and ω₋ representing the slow and fast Rossby modes, respectively, where $v_a=B_0/\sqrt{{\mu }_0\rho }$ is the Alfvén wave phase speed. In order to show that this derivation is consistent, note that by taking v_a = 0 (B₀ = 0) the dispersion relation reduces to

$$\omega =\frac{-\beta k}{k^2+l^2+F},$$ 2.11

which constitutes the dispersion relation of classical hydrodynamic Rossby waves found in textbooks on atmospheric/ocean dynamics (e.g. [30]). On the other hand, neglecting the effects of rotation by setting β = 0, one obtains

$$\omega =\pm v_{a}k,$$ 2.12

which is the dispersion relation of Alfvén waves [31]. Therefore, as expected, the MHD Rossby wave is a hybrid of the hydrodynamic Rossby mode with the Alfvén wave. In figure 1, we present the dispersion curves of both branches, for the values of β = 4.28 × 10⁻¹¹ m⁻¹ s⁻¹ and B₀ = 4 mT. As can be noted in figure 1, for the parameters of the Earth’s core, the MHD Rossby waves exhibit a large time-scale separation between the slow and fast branches. Indeed, the typical time scale for a fast Rossby mode oscillation is several days to months, while the typical time scale of the slow oscillations is thousands of years. Hide [16] used a similar model in a thick shell to derive the dispersion relation of planetary waves that exhibit a columnar flow pattern. Owing to their characteristic frequencies and westward propagation, it was suggested that these waves can contribute to the westward drift of the geomagnetic field. On the other hand, the normal modes analysed here are of a different nature and are likely to exist in the thin shell at the top of the Earth’s core. According to equation (2.10), their propagation can be either retrograde (in the case of the fast branch) or prograde (in the case of the slow branch). As a consequence, the rotational waves studied here might in principle contribute to different aspects of the observed secular variation of the geomagnetic field.

Figure 1.

Dispersion relation of MHD Rossby modes on the β-plane approximation for the slow modes (a) and fast modes (b). (Online version in colour.)

(b). Spherical coordinates: MHD Rossby–Haurwitz waves

The Jacobian operator in spherical coordinates is written as

$$J(\hspace{0.17em}f,g)=\frac{1}{a^2\cos \theta }(\frac{\mathrm{\partial }f}{\mathrm{\partial }\varphi }\frac{\mathrm{\partial }g}{\mathrm{\partial }\theta }-\frac{\mathrm{\partial }g}{\mathrm{\partial }\varphi }\frac{\mathrm{\partial }f}{\mathrm{\partial }\theta })={\mathrm{\nabla }}^{\perp }f\bullet \mathrm{\nabla }g,$$ 2.13

for any two differentiable functions f and g, where a is the radius of the spherical shell, Ω is the rotation rate of the Earth and

$${\mathrm{\nabla }}^{\perp }=\frac{1}{a}(\frac{1}{cos\theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }{e}_{\theta }-\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }{e}_{\varphi })$$

is the two-dimensional perpendicular gradient operator in the spherical coordinate system.

In this way, linearizing equations (2.1) and (2.2), in the absence of topographic variations, around a basic state characterized by $\overline{\psi }\equiv 0$, $\overline{q}=2\mathrm{\Omega }\sin \theta$ and $\overline{A}$ being chosen to represent a global toroidal field ${\overline{B}}_0\cos \theta$ in the zonal direction, the governing equations for the perturbations now read

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\left[\begin{array}{c}{\mathrm{\nabla }}^2\psi \\ A\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{2}{a^2}}\mathrm{\Omega }{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }}& {\displaystyle \frac{B_0}{a{\mu }_0\rho }}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }}({\displaystyle \frac{1}{a^2}}-{\mathrm{\nabla }}^2)\\ {\displaystyle \frac{B_0}{a}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }}& 0\end{array}\right]\left[\begin{array}{c}\psi \\ A\end{array}\right].$$ 2.14

The particular form of the background toroidal field mentioned above is chosen for two reasons. First, since it is difficult to infer the structure of the toroidal magnetic field inside the Earth’s core, it seems reasonable to assume a simple pattern with the strongest toroidal fields near the Equator. The second reason is for mathematical convenience, since this choice of the background state makes the linear differential operator separable in spherical coordinates [25], which means that the corresponding eigenfrequencies can be found analytically, as will be shown below. In fact, equation (2.14) can be solved by spherical harmonics (see [25] for a similar treatise). Briefly, we assume a solution obeying the ansatz

$$\left[\begin{array}{c}\psi \\ A\end{array}\right]=\mathrm{\Lambda }{Y}_n^m(\theta ,\varphi )\hspace{0.17em}{\text{e}}^{-\text{i}\omega t}\overrightarrow{R}=\mathrm{\Lambda }{N}_n^m{P}_n^m(\sin \theta )\hspace{0.17em}{\text{e}}^{\text{i}m\varphi -\text{i}\omega t}\overrightarrow{R},$$ 2.15

where $Y_n^m$ are the eigenfunctions of the Laplace operator, that is,

$${\mathrm{\nabla }}^2{Y}_n^m=-\frac{n(n+1)}{a^2}{Y}_n^m,$$ 2.16

with Legendre functions satisfying the following orthogonality relation:

$${\int }_{-1}^1{P}_{n_1}^{m1}{P}_{n_2}^{m_2}\hspace{0.17em}\text{d}z=\frac{2(n_1+m_1)!}{(n_1-m_1)!(2n_1+1)}{\delta }_{n_1{n}_2},$$ 2.17

where ${\delta }_{n_1{n}_2}=1$ if n₁ = n₂ and 0 otherwise. The normalization constant $N_n^m$ is given by

$$N_n^m={\left(\frac{(n-|m|)!(2n+1)}{(n+|m|)!}\right)}^{1/2}.$$

In (2.15), Λ is an arbitrary constant and $\overrightarrow{R}$ is the eigenvector of the matrix

$${\mathcal{L}}^{\ast }=\left[\begin{array}{cc}-2\mathrm{\Omega }{\displaystyle \frac{m}{n(n+1)}}& {\displaystyle \frac{B_{0}m}{{\mu }_0\rho }}({\displaystyle \frac{1}{a^2}}-{\displaystyle \frac{2}{n(n+1)}})\\ {\displaystyle \frac{B_{0}m}{a}}& 0\end{array}\right]$$ 2.18

associated with the eigenvalue ω, given by

$$\overrightarrow{R}(\omega (m,n))=\left[\begin{array}{c}{\displaystyle \frac{\omega (m,n)}{m}}\\ {\displaystyle \frac{B_0}{a}}\end{array}\right].$$ 2.19

The eigenvectors associated with the eigenvalue problem analysed here have an important physical interpretation, as they inform how the energy associated with the different wave modes is partitioned between kinetic and magnetic energies.

In fact, plugging the ansatz (2.15) into the linearized equations (2.14) leads to the following dispersion relation:

$$\omega =\frac{1}{2}(-\frac{2\mathrm{\Omega }m}{n(n+1)}\pm \sqrt[2]{{\left(\frac{2\mathrm{\Omega }m}{n(n+1)}\right)}^2+\frac{4B_0^2{m}^2}{{\mu }_0\rho a^2}(1-\frac{2}{n(n+1)})}).$$ 2.20

The dispersion relation above reduces to ordinary hydrodynamic Rossby–Haurwitz waves for B₀ = 0. Note also that when n = 1 the magnetic part of the dispersion relation cancels out and again there is only one branch corresponding to hydrodynamical contributions. This agrees with the dispersion relation obtained in [22,25].

3. Nonlinear interaction of MHD Rossby waves on the sphere

So far we have discussed the linear dynamics of MHD Rossby waves. The linear theory is valid in the limit of infinitesimal amplitudes and short periods of time, in the sense that the smaller the amplitudes of the waves, the closer to observations the linear theoretical estimates should be. Also, it is important to note that the validity of the linear theory is limited by the length of the time interval on which the system evolves: as time passes the influence of the nonlinear terms of the governing equations will start to become important, with energy initially concentrated in a particular mode being transferred to other modes and eventually generating ‘new waves’. The study of the nonlinearity can be done in a first approximation through the so-called weakly nonlinear theory in which the amplitudes of the waves are still infinitesimal; in this case, it can be shown that the waves interact with each other only in sets of resonant modes [32]. When the wave amplitudes are large enough, the weakly nonlinear theory breaks down, and several effects can take place modifying the dynamics of the waves; most notably, the effects of wave phases on the wave amplitudes begin to be important [33].¹ The study of nonlinear theory for MHD Rossby waves was introduced in [22,34] in the context of the solar tachocline. Here, we extend this theory to include the effects of both topographic variations and thermal forcing in a thin fluid layer. As we shall see later, the effect of a non-homogeneous topography is to add a linear variable coefficient in the perturbation equations (2.14). It turns out that the presence of such a linear variable coefficient allows the wave modes to interact in duets, similarly, for instance, to the parametric resonance in harmonic oscillators [35]. In the context of atmospheric and oceanic dynamics, wave interaction through topography has been extensively studied; see, for instance, [13,36]. In particular, the interaction of Rossby waves through topography seems to have an important role in the dynamics of atmospheric blocking [14,15]. Additionally, as will be further discussed later, the effect of a thermal forcing in the present model can be represented by a non-homogeneous term on the r.h.s. of the vorticity balance equation (2.1).

Therefore, first we review in the present section the theoretical framework of nonlinear MHD Rossby waves in spherical coordinates, following the approach employed in [22,37]. Then, in the following sections, we shall augment the nonlinear interaction theory of MHD Rossby modes by including the effects of topography and thermal forcings.

To analyse the nonlinear dynamics of the MHD Rossby waves introduced in the previous section, we restore the nonlinear terms disregarded in (2.14). In this case, equation (2.14) can be written as

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[\begin{array}{c}{\mathrm{\nabla }}^2\psi \\ A\end{array}\right]=\mathcal{L}\left[\begin{array}{c}\psi \\ A\end{array}\right]+\mathcal{B}(\left[\begin{array}{c}\psi \\ A\end{array}\right],\left[\begin{array}{c}\psi \\ A\end{array}\right]).$$ 3.1

In the equation above, the linear operator $\mathcal{L}$ is the same as in (2.14), and $\mathcal{B}$ is the bilinear operator containing the nonlinear terms of the perturbation equations, namely

$$\mathcal{B}(\left[\begin{array}{c}{\psi }_1\\ A_1\end{array}\right],\left[\begin{array}{c}{\psi }_2\\ A_2\end{array}\right])=\left[\begin{array}{c}-\mathcal{J}({\psi }_1,{\mathrm{\nabla }}^2{\psi }_2)+{\displaystyle \frac{1}{{\mu }_0\rho }}\mathcal{J}(A_1,{\mathrm{\nabla }}^2{A}_2)\\ \mathcal{J}({\psi }_1,A_2)\end{array}\right].$$ 3.2

Therefore, to obtain the spectral equations for the nonlinear evolution of the waves, we consider the following ansatz:

$$\begin{array}{rl}\left[\begin{array}{c}\psi \\ A\end{array}\right]& \hspace{0.17em}=\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n\phantom{0}}^n\sum _{j=0}^{1}a{\mathrm{\Lambda }}_{m,n}^{{\alpha }_j}(t)Y_n^m(\varphi ,\theta ){\overrightarrow{R}}_{m,n}^{{\alpha }_j}\\ & \hspace{0.17em}=\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n\phantom{0}}^n\sum _{j=0}^{1}a{\mathrm{\Lambda }}_{m,n}^{{\alpha }_j}(t)N_n^m{P}_n^m(\sin \theta )\hspace{0.17em}{\text{e}}^{im\varphi }{\overrightarrow{R}}_{m,n}^{{\alpha }_j},\end{array}$$ 3.3

where α_j ∈ { − , + } denotes the mode type, fast or slow, and ${\mathrm{\Lambda }}_{m,n}^{{\alpha }_j}$ now refers to the complex-valued mode amplitudes, satisfying ${\mathrm{\Lambda }}_{-m,n}^{{\alpha }_j}={({\mathrm{\Lambda }}_{m,n}^{{\alpha }_j})}^{\ast }$, with the asterisk denoting the complex conjugate. By inserting the above expression into the dynamical equations (3.1), we obtain

$$\frac{\text{d}{\mathrm{\Lambda }}_{k_1}^{{\alpha }_1}}{\text{d}t}=\text{i}{\omega }_{k_1}^{{\alpha }_1}{\mathrm{\Lambda }}_{k_1}^{{\alpha }_1}+\sum _{k_2,k_3,{\alpha }_2,{\alpha }_3}{C}_{k_1,k_2,k_3}^{{\alpha }_1,{\alpha }_2,{\alpha }_3}{\mathrm{\Lambda }}_{k_2}^{{\alpha }_2}{\mathrm{\Lambda }}_{k_3}^{{\alpha }_3},$$ 3.4

where k₁ = (m₁, n₁), k₂ = (m₂, n₂), k₃ = (m₃, n₃) indicate three arbitrary spherical harmonics and $C_{k_1,k_2,k_3}^{{\alpha }_1,{\alpha }_2,{\alpha }_3}$ refers to the nonlinear interaction coefficients, given by

$$\begin{array}{rl}{C}_{k_1,k_2,k_3}^{{\alpha }_1,{\alpha }_2,{\alpha }_3}& \hspace{0.17em}=\frac{{\int }_{S_2}⟨\mathcal{B}({\overrightarrow{R}}_{k_2}^{{\alpha }_2}{Y}_{n_2}^{m_2},{\overrightarrow{R}}_{k_3}^{{\alpha }_3}{Y}_{n_3}^{m_3})+\mathcal{B}({\overrightarrow{R}}_{k_3}^{{\alpha }_3}{Y}_{n_3}^{m_3},{\overrightarrow{R}}_{k_2}^{{\alpha }_2}{Y}_{n_2}^{m_2}),{\overrightarrow{R}}_{k_1}^{{\alpha }_1}{Y}_{n_1}^{m_1}⟩\text{d}s}{{\int }_{S_2}⟨{\overrightarrow{R}}_{k_1}^{{\alpha }_1}{Y}_{n_1}^{m_1},{\overrightarrow{R}}_{k_1}^{{\alpha }_1}{Y}_{n_1}^{m_1}⟩\text{d}s}\\ & \hspace{0.17em}=\frac{-\text{i}}{2a(n_1(n_1+1))|{\overrightarrow{R}}_{k_1}^{{\alpha }_1}{|}^2}{K}_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}[I_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}+L_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}].\end{array}$$ 3.5

In the equation above, 〈.,.〉 indicates the following inner product:

$$⟨\overrightarrow{X},\overrightarrow{Y}⟩=X_1^{\ast }{Y}_1+\frac{1}{{\mu }_0\rho }{X}_2^{\ast }{Y}_2,$$ 3.6

for two arbitrary vectors $\overrightarrow{X}$, $\overrightarrow{Y}\in {\mathbb{C}}^2$, and the coefficients $I_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}$, $L_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}$ and $K_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}$ are given by

$$\begin{array}{rl}& I_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}=[(n_k(n_k+1))-(n_l(n_l+1))],\end{array}$$ 3.7a

$$\begin{array}{rl}& L_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}={\left(\frac{V_A}{a}\right)}^2(\frac{{\omega }_2}{m_2}-\frac{{\omega }_3}{m_3}),\end{array}$$ 3.7b

$$\begin{array}{rl}& K_{n_1{n}_2{n}_3}^{m_1{m}_2{m}_3}=N_{n_1}^{m_1}{N}_{n_2}^{m_2}{N}_{n_3}^{m_3}{\int }_{-1}^1{P}_{n_1}^{m1}(m_2{P}_{n_2}^{m_2}\frac{\text{d}{P}_{n_3}^{m_3}}{\text{d}z}-m_3{P}_{n_3}^{m_3}\frac{\text{d}{P}_{n_2}^{m_2}}{\text{d}z})\text{d}z,\end{array}$$ 3.7c

where z = sinθ. For a more detailed derivation of the interaction coefficients displayed above, see [22]. In order for the coefficients to be non-zero, the mode indices must satisfy the Elsasser selection rules [38]

$$\begin{array}{rl}& m_2+m_3=m_1,\\ & \hspace{0.17em}{(m_1)}^2+{(m_3)}^2\ne 0,\\ & n_2{n}_1{n}_3\ne 0,\\ & n_2+n_1+n_3\text{ is odd},\\ & \hspace{0.17em}(n_2^2-|m_2{|}^2)+(n_3^2-|m_3{|}^2)>0,\\ & \hspace{0.17em}|n_2-n_3|<n_1<n_2+n_3,\\ & \hspace{0.17em}(m_2,n_2)\ne (m_1,n_1),(m_3,n_3)\ne (m_1,n_1).\end{array}$$ 3.8

Let us now truncate the expansion (3.3) to consider only three wave modes satisfying the selection rules above. In this case of a single interacting wave triplet, equation (3.4) now reads

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_1}{\text{d}t}=\text{i}{\omega }_1{\mathrm{\Lambda }}_1+C_{1,2,3}{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3,\end{array}$$ 3.9a

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_2}{\text{d}t}=\text{i}{\omega }_2{\mathrm{\Lambda }}_2+C_{2,3,1}{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_3^{\ast },\end{array}$$ 3.9b

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_3}{\text{d}t}=\text{i}{\omega }_3{\mathrm{\Lambda }}_3+C_{3,1,2}{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2^{\ast },\end{array}$$ 3.9c

where in the equations above we have used a simplified notation for the mode indices in which each mode j, j = 1, 2, 3, is characterized by a spherical harmonic (m_j, n_j) and a wave type labelled by α_j. The total energy conservation of model equations (3.1) implies that for any interacting triad the coupling coefficients must satisfy the following constraint (see appendix A for details):

$$E_1{C}_{1,2,3}-E_2{C}_{2,1,3}-E_3{C}_{3,1,2}=0,$$ 3.10

with $E_j=n_j(n_j+1)|{\overrightarrow{R}}_j{|}^2$, j = 1, 2, 3, corresponding to the intrinsic energy norm of a particular eigenmode. Since the energies E_j are positive definite, condition (3.10) implies that |C_1,2,3| > |C_2(3),1,3(2)|. This means that mode 1 always receives energy from or supplies energy to the other two modes of the triplet. In the literature of plasma physics, this mode is labelled the ‘pump mode’ or ‘pump wave’ [39].

4. Effects of topography and non-homogeneous thermal forcing on Rossby wave interaction

(a). Linear wave interaction through topography

Here, we show how topography can trigger the interaction between two MHD Rossby waves. The hydrodynamical counterpart to this mechanism was studied by [13,40]. The basic idea here is that the topography will act as a zero-frequency mode with a given wavenumber (i.e. topography with a single wavenumber), allowing two waves to interact given that their wavenumbers k₁ and k₂ are compatible with the wavenumber of the topography, k_T. In Cartesian coordinates, this compatibility consists of the sum of their wavenumbers being zero (k_T − k₁ − k₂ = 0). In the spherical coordinate system, the wavenumbers must satisfy the Elsasser relations (3.8) presented in the previous section. In general, the topography function h(θ, ϕ) can be expanded in a spherical harmonic series, which will lead to many duo interactions between different modes. However, for simplicity, we consider a single wave duet in the presence of a single mode topography

$$\begin{array}{rl}& {\psi }_j(\theta ,\varphi )={\mathrm{\Lambda }}_j(t)Y_{n_j}^{m_j}(\varphi ,\theta )\frac{\omega (m_j,n_j)}{m_j},\end{array}$$ 4.1a

$$\begin{array}{rl}& A_j(\theta ,\varphi )={\mathrm{\Lambda }}_j(t)Y_{n_j}^{m_j}(\varphi ,\theta )\frac{B_0}{a},\end{array}$$ 4.1b

$$\begin{array}{rl}& h(\theta ,\varphi )=h_{n_T}^{m_T}{Y}_{n_T}^{m_T}(\theta ,\varphi ).\end{array}$$ 4.1c

In the equations above, j = 1, 2, and the topography wavenumber is represented by k_T = (n_T, m_T). Thus, inserting the anstaz (4.1) into the linearized version of equations (2.1) and (2.2) in the presence of single-mode topography yields

$$\frac{\text{d}{\mathrm{\Lambda }}_1}{\text{d}t}=\text{i}{\omega }_1{\mathrm{\Lambda }}_1+H_1{h}_{n_T}^{m_T}{\mathrm{\Lambda }}_2^{\ast }$$ 4.2a

and

$$\frac{\text{d}{\mathrm{\Lambda }}_2}{\text{d}t}=\text{i}{\omega }_2{\mathrm{\Lambda }}_2+H_2{h}_{n_T}^{m_T}{\mathrm{\Lambda }}_1^{\ast },$$ 4.2b

where H_j, j = 1, 2, refers to the linear coupling constants of modes 1 and 2 through the topography mode (n_T, m_T), given by

$$H_2=-{\int }_{S_2}J({\psi }_1,Y_{n_T}^{m_T}){\psi }_2\hspace{0.17em}\text{d}s=-{\int }_0^{2\pi }{\int }_{-(\pi /2)}^{\pi /2}J({\psi }_1,Y_{n_T}^{m_T}){\psi }_2{a}^2\cos \theta \hspace{0.17em}\text{d}\theta \text{d}\varphi$$ 4.3a

and

$$H_1=-{\int }_{S_2}J({\psi }_2,Y_{n_T}^{m_T}){\psi }_1\hspace{0.17em}\text{d}s=-{\int }_0^{2\pi }{\int }_{-(\pi /2)}^{\pi /2}J({\psi }_2,Y_{n_T}^{m_T}){\psi }_1{a}^2\cos \theta \hspace{0.17em}\text{d}\theta \text{d}\varphi .$$ 4.3b

Integrating by parts the equations above, it is possible to show that H₁ = −H₂. Consequently, the mode amplitudes Λ₁₍₂₎(t) are bounded in time, being described by trigonometric functions, with the frequency of energy exchange between modes 1 and 2 depending linearly on the amplitude of the topography mode.

(b). Effect of a non-homogeneous thermal forcing

Now that we have shown the role of both nonlinearity and topography in coupling different wave modes of (2.14), here we augment the wave interaction theories described by (3.9) and (4.2) to include a prescribed thermal forcing and dissipation. Indeed, the core–mantle boundary is known to be highly inhomogeneous, so lateral differences in the heat flow through this layer are likely to drive lateral pressure differences in the fluid at the stratified layer at the top of the core [41,42]. Since a negative temperature anomaly at the top of the Earth’s core results in downdraft motion and a positive temperature anomaly results in upward motion, it is reasonable to represent a thermal forcing in the barotropic quasi-geostrophic MHD model as a prescribed divergence field in the vorticity balance (2.1). In other words, we replace the term proportional to $\mathrm{\nabla }\cdot \mathbf{\text{u}}$ in the vorticity budget equation by a prescribed term. This approach of considering a prescribed field of the linear divergence term of the vorticity balance as a Rossby wave source was adopted in [22,32] in the context of MHD Rossby waves at the solar tachocline. Also, this approach is common in the atmospheric dynamics literature to study the remote impacts of thermal forcings in the large-scale atmospheric circulation (see, for instance, [43] and references therein). Therefore, including a prescribed divergence forcing in the potential vorticity equation (2.1) with h ≠ 0, it follows that the forced-dissipative version of the perturbation equation (3.1) can be written as

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[\begin{array}{c}{\mathrm{\nabla }}^2\psi \\ A\end{array}\right]=\mathcal{L}\left[\begin{array}{c}\psi \\ A\end{array}\right]+\mathcal{B}(\left[\begin{array}{c}\psi \\ A\end{array}\right],\left[\begin{array}{c}\psi \\ A\end{array}\right])+{\mathbf{\text{F}}}_{\mathbf{\text{c}}}+\mathbf{\text{F}}\left[\begin{array}{c}\psi \\ A\end{array}\right]+\mathcal{D}\left(\right[\begin{array}{c}\psi \\ A\end{array}\left]\right),$$ 4.4

where the operators $\mathcal{L}$ and $\mathcal{B}$ are the same as those presented in §§2 and 3, and the vectors F_c and F are given by

$${\mathbf{\text{F}}}_c=\left[\begin{array}{c}-2\mathrm{\Omega }sin\theta D_u(\theta ,\varphi ,t)\\ 0\end{array}\right]$$ 4.5

and

$$\mathbf{\text{F}}=\left[\begin{array}{c}-J(\psi ,Fh)\\ 0\end{array}\right],$$ 4.6

with D_u(θ, ϕ, t) representing the horizontal divergence of the velocity field. In (4.4), $\mathcal{D}$ refers to the dissipation operator

$$\mathcal{D}=\left[\begin{array}{cc}\nu {\mathrm{\nabla }}^2({\mathrm{\nabla }}^2)& 0\\ 0& \eta {\mathrm{\nabla }}^2\end{array}\right],$$ 4.7

where the parameters ν and η are the kinematic and magnetic viscosities, respectively.

Given that the spatial structure of both thermal and topographic anomalies at the core–mantle boundary is believed to be predominantly described by the spherical harmonic (2,2) [10,44], we consider a triad of MHD Rossby modes in which the forcing term F_c projects only onto the spherical harmonic $Y_2^2$. In addition, we assume a single-mode topography with the same spatial structure. In this context, substituting the truncated version of expansion (3.3) for a single interacting triad, as well as the single-mode topography expansion (4.1c), into (4.4) yields

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_1}{\text{d}t}=\text{i}{\omega }_1{\mathrm{\Lambda }}_1+f_1+C_{1,2,3}{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3-d_1{\mathrm{\Lambda }}_1,\end{array}$$ 4.8a

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_2}{\text{d}t}=\text{i}{\omega }_2{\mathrm{\Lambda }}_2+C_{2,3,1}{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_3^{\ast }+H_2{h}_{n_T}^{m_T}{\mathrm{\Lambda }}_3^{\ast }-d_2{\mathrm{\Lambda }}_2,\end{array}$$ 4.8b

$$\begin{array}{rl}& \frac{\text{d}{\mathrm{\Lambda }}_3}{\text{d}t}=\text{i}{\omega }_3{\mathrm{\Lambda }}_3+C_{3,2,1}{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2^{\ast }+H_3{h}_{n_T}^{m_T}{\mathrm{\Lambda }}_2^{\ast }-d_3{\mathrm{\Lambda }}_3.\end{array}$$ 4.8c

In the equations above, H_j, j = 2, 3, refers to the topographic coupling constants given by (4.3), f₁ is the projection of the divergence forcing onto mode 1 and d_i, i = 1, 2, 3, represents the spectral damping coefficients, given by

$$d_i=n_i(n_i+1)⟨{\overrightarrow{R}}_i,\left[\begin{array}{cc}\nu & 0\\ 0& \hspace{0.17em}\eta \end{array}\right]{\overrightarrow{R}}_i⟩.$$ 4.9

In (4.8), besides the divergence forcing D_u(θ, ϕ, t) projecting only onto the mode 1 spatial structure, we have also assumed that the time variation of this forcing resonates with mode 1. The effect of such an assumption is to yield a linear growth of the amplitude of this mode. Note also that a topography function with the same spatial structure as the divergence forcing promotes a linear coupling between the two remaining modes of the triad (modes 2 and 3).

5. Numerical results for a forcing with the $Y_2^2$ spatial structure

As previously discussed, thermal anomalies on the core–mantle boundary are believed to exhibit a spatial structure that is essentially described by the spherical harmonic (2, 2) [44]. This is also the case for the topographic anomalies according to [10]. Consequently, we seek triads of wave modes in which mode 1 has the (2, 2) wavenumber. The following is a list of all possible pairs of spherical harmonics satisfying the selection rules (3.8) with the harmonic (2, 2) in a box in the wavenumber space (|m| ≤ 10, |n| ≤ 10): {(1, 3), (3, 4)}, {(1, 4), (3, 5)}, {(2, 4), (4, 5)}, {(1, 5), (3, 6)}, {(2, 5), (4, 6)}, {(3, 5), (5, 6)}, {(1, 6), (3, 7)}, {(2, 6), (4, 7)}, {(3, 6), (5, 7)}, {(4, 6), (6, 7)}, {(1, 7), (3, 8)}, {(2, 7), (4, 8)}, {(3, 7), (5, 8)}, {(4, 7), (6, 8)}, {(5, 7), (7, 8)}, {(1, 8), (3, 9)}, {(2, 8), (4, 9)}, {(3, 8), (5, 9)}, {(4, 8), (6, 9)}, {(5, 8), (7, 9)}, {(6, 8), (8, 9)}, {(1, 9), (3, 10)}, {(2, 9), (4, 10)}, {(3, 9), (5, 10)}, {(4, 9), (6, 10)}, {(5, 9), (7, 10)}, {(6, 9), (8, 10)}, {(7, 9), (9, 10)}.

Besides mode 1 having the same spatial structure as the thermal and orographic forcings (i.e. with the (2, 2) wavenumber), we are also interested in the case in which this mode, after being excited by the thermal forcing, is unstable to the other modes of the triad, efficiently transferring energy to them. This requirement is achieved if mode 1 is the pump mode of the triad, that is, it has the nonlinear interaction coefficient with the largest absolute value. As discussed in §3, in this case, mode 1 always receives energy from or supplies energy to these remaining modes of the triad. On the other hand, as discussed in §2, there is a large separation between the time–frequency scales associated with the two branches of the MHD Rossby wave dispersion relation in the present model. As a consequence of this gap in the time–frequency spectrum of MHD Rossby modes, the only possibilities of mode 1 being the pump mode of the triad are: (i) mode 1 is a fast mode and modes 2 and 3 belong to the slow branch and (ii) all three modes are of the fast type. Furthermore, the large time-scale separation between the two branches implies a large mismatch among the eigenfrequencies of the triad members in the former case, leading to inefficient energy exchanges among the wave modes in a similar fashion to the interaction between Rossby and inertial gravity waves in the atmosphere [45]. Therefore, we focus here on the latter case in which all three modes belong to the fast branch. A representative example of such interacting triads is illustrated in table 1. In this triad, modes 1, 2 and 3 are characterized by the spherical harmonics (2,2), (2,4) and (4,5), respectively, all of them belonging to the fast branch. Although the frequencies of these modes are probably too fast (time scales of weeks) to account for observed features of the geomagnetic field, the reduced system (4.8) can develop long-lived, persistent structures that might be relevant for the understanding of the geomagnetic field and of the flow at the top of the Earth’s core, as will be demonstrated below.

Table 1.

Parameters of the nonlinear triad connected by a (2,2) mode.

mode	wavenumber	type	frequency (Hz)	coupling coefficient	damping coefficient
1	(2,2)	fast	−0.000048	10.30 i	3.53 × 10−8
2	(2,4)	fast	−0.000015	+7.41 i	1.06 × 10−8
3	(4,5)	fast	−0.000022	+2.88 i	7.05 × 10−9

A result of numerical integration of system (4.8) is displayed in figures 2–6. In order to set the initial conditions for the spectral amplitudes, we need to have an estimated value for the velocity field near the core–mantle boundary. For this purpose, following [46] and [47], we have used the estimated order of magnitude for the velocity field of 3 × 10⁻⁵ m s⁻¹ to 7.5 × 10⁻⁴ m s⁻¹. Given that for a single wave field the approximate relationship between the amplitude of the wave and the magnitude of the velocity field associated with this wave is given by $||\overrightarrow{u}||\approx |a.A\omega /m|$, the resulting order of magnitude for the spectral amplitudes is 10⁻⁶.

Figure 2.

Solution for the spectral amplitudes of the selected triad displayed in table 1 in the conservative case (no forcing, topography and dissipation). The blue curve refers to mode 1, the yellow to mode 2 and the green curve to mode 3. (Online version in colour.)

Figure 6.

Stream function of a snapshot of the flow pattern associated with the solution displayed in figures 3 and 4 superposed with a zonal flow associated with a combination of the spherical harmonics $Y_2^0$ and $Y_4^0$. Normalized colour legend ( × 1.2 × 10⁻¹ m² s⁻¹) denotes the value of the stream function. (Online version in colour.)

For the damping coefficients, we have used the value of ν = 10 m² s⁻¹ for the kinematic viscosity, which is compatible with [48], and the value of η = 1.6 m² s⁻¹ for the magnetic viscosity, following [46]. These are not well-constrained parameters, in particular the kinematic viscosity. This may have some implications for the dynamics of our reduced model (4.8). A possible consequence is regarding the size of the attractor associated with the dynamical system (4.8), which may be rather dependent upon the viscosity parameters. This dependence can be demonstrated by constructing a Lyapunov function to be contained inside a closed surface of dimension N − 1 in the N-dimensional phase space (with N = 6 for system (4.8)), in which all vectors are pointing inside the surface [49].

Figure 2 shows the time evolution of the mode amplitudes $|A_i|=\sqrt{A_i{A}_i^{\ast }}$, i = 1, 2, 3, for the conservative case, i.e. with the parameters f₁, $h_{n_T}^{m_T}$, ν and η being set to zero. In this integration displayed in figure 2, the role of the pump mode, mode 1, in either receiving energy from or supplying energy to modes 2 and 3 is clear. In this case, all the wave modes of the triplet do propagate freely.

By contrast, when the thermal and topographic forcings are considered, the propagation of the first mode is affected in such a way that it no longer propagates freely. Rather, the phase of this mode oscillates around a mean value and does not complete a cycle around the globe. This is illustrated in figures 3 and 4, which show the integration of system (4.8) in the forced-dissipative case. Figure 3 displays the real and imaginary parts of the spectral amplitude of each mode, while figure 4 illustrates the projection of the trajectory in the Re[A₁] × Im[A₁] plane. From figure 4, it is possible to note that the trajectory does not cross the Im[A₁] = 0 axis, meaning that the phase of the first mode no longer completes a [0, 2π] cycle.

Figure 3.

Solution for the spectral amplitudes of the selected triad displayed in figure 2 with the inclusion of thermal forcing, topography and dissipation, with both forcing and topography projecting onto mode 1. (Online version in colour.)

Figure 4.

Phase plot of the forced mode, ReA₁ × ImA₁, referring to the same numerical solution as figure 3. (Online version in colour.)

In figure 5, we show a snapshot of the streamfunction associated with the solution displayed in figures 3 and 4. The fact that mode 1 is directly excited by the thermal forcing considered here implies that, in the non-conservative case, it remains the mode with the largest amplitude of the triad. Consequently, the flow pattern illustrated in figure 5 is dominated by the dynamical fields associated with this mode, and thus exhibits the spatial structure of the spherical harmonic (2,2).

Figure 5.

Stream function of a snapshot of the flow pattern associated with the solution displayed in figures 3 and 4. Normalized colour legend (×2 × 10⁻² m² s⁻¹) denotes the value of the stream function. (Online version in colour.)

Since the flow at the top of the Earth’s core is known to exhibit a significant zonal flow, as is also supposedly the case in any planetary and stellar flow, we have combined the flow pattern generated by the waves composing the triad displayed in table 1 with a zonal flow characterized by a combination of the spherical harmonics $Y_2^0$ and $Y_4^0$. We choose the amplitude of the spherical harmonic coefficients in order to match the order of magnitude of the mean flow observed in [47]. This results in a flow pattern with two pairs of vortex structures with closed streamlines in mid- to high latitudes, as illustrated in figure 6. This combination is reminiscent of the atmospheric blocking phenomenon. In addition, the spatial structure exhibited in figure 6 is also similar to the geometrical structure of the geomagnetic flux lobes, appearing in pairs in antipodal longitudes. Both the latitudinal position and size of the ‘blocked cells’ are dependent on the coefficients of the mean flow components $Y_2^0$ and $Y_4^0$.

6. Discussion

So far we have shown that the combined effects of spatially heterogeneous thermal and topographic forcings and the nonlinear interaction can affect a fast MHD Rossby wave mode with a $Y_2^2$ spatial structure in such a way that it locks the phase of this mode. In this case, the wave no longer propagates freely; instead, it oscillates around a particular latitude value. This situation is completely analogous to the phenomenon of atmospheric blocking in the Earth’s atmosphere, whereby wind patterns become predominately meridional and correlated with topography or with a non-homogeneous thermal forcing [14]. A blocked atmospheric flow becomes a transport barrier for aerosols, humidity and smaller scale waves embedded in the flow, so that blocking can be responsible for long periods of drought in certain regions (see, for instance, [50]).

We argue here that the phenomenon of geomagnetic lobes can be analogous to the atmospheric blocking. Indeed, we have demonstrated that the flow pattern generated by our low-order spectral model describing the MHD Rossby wave interaction in the presence of both thermal and topographic forcings is characterized by closed streamlines. On the other hand, the radial component of the geomagnetic field, B_r, is basically described by a transport equation with a source term

$$\frac{\mathrm{\partial }{B}_r}{\mathrm{\partial }t}+{\mathbf{\text{u}}}_{\mathbf{\text{H}}}.\mathrm{\nabla }{B}_r=-B_r\text{div}\hspace{0.17em}{\mathbf{\text{u}}}_{\mathbf{\text{H}}},$$ 6.1

where u_H is the horizontal velocity field at the top of the core and $\mathrm{\nabla }$ and div represent the two-dimensional gradient and divergent operators acting on the plane, respectively. The flow pattern associated with the barotropic Rossby waves presented here is non-divergent so that we may denote by u_G the component of u_H associated with the waves studied here (or the geostrophic component). By contrast, the ageostrophic, significantly divergent, component of the flow will be denoted by uA. Since divu_G = 0, it follows that ${\mathbf{\text{u}}}_{\mathbf{\text{G}}}={\mathrm{\nabla }}^{\perp }\psi$. Consequently, for large-scale, predominantly geostrophic flows, we have

$$\frac{\mathrm{\partial }{B}_r}{\mathrm{\partial }t}+{\mathrm{\nabla }}^{\perp }\psi .\mathrm{\nabla }{B}_r\approx 0.$$ 6.2

Furthermore, let us define a quasi-stationary field as a magnetic field structure such that the average value of its derivative over a sufficiently long time and large spatial scale approaches zero

$$⟨\frac{\mathrm{\partial }{B}_r}{\mathrm{\partial }t}{⟩}_{T,L}\approx 0.$$ 6.3

From this assumption, we conclude that

$${⟨{\mathrm{\nabla }}^{\perp }\psi .\mathrm{\nabla }{B}_r⟩}_{T,L}\approx 0.$$ 6.4

In other words, in order for a localized, persistent geomagnetic field structure to exist, its contour lines should be significantly correlated with the streamlines of the flow. Taking as a reference the flow pattern illustrated in figure 6, obtained as a combination of the meridional flow associated with the stationary wave and the background zonal flow, we see that these requirements are fulfilled. An anomaly of a radial magnetic field generated by a localized convergence of the flow will not drift away if the surrounding streamlines are closed.

A second possibility is that there exist oscillatory lobes with a horizontal structure with a smaller spatial scale than the blocked cell. This may happen, for instance, if a horizontal divergence is associated with the projection of a columnar flow in the core–mantle boundary [41]. If this divergence takes place in a geographical location where the streamline at the core–mantle boundary is closed, then the corresponding B_r anomaly will be trapped by the closed streamlines, which in turn will constitute a transport barrier. In this case, the radial magnetic field will oscillate inside the blocked region, but will not drift away freely around the globe.

The discussion presented here suggests that a blocked state in the flow at the core–mantle boundary is a plausible candidate for explaining the geomagnetic flux lobes.

7. Conclusion

Here, we have presented the theory of linear and nonlinear MHD Rossby waves in a stratified layer at the top of the Earth’s core, and we have shown how a spatially varying thermal forcing and topography at the core–mantle boundary can impact the dynamics of these waves. In summary:

• —We derived the dispersion relation for fast and slow Rossby waves in a barotropic quasi-geostrophic model at the top of the Earth’s core. We found that, for the fast branch, the eigenfrequencies are associated with periods of the order of weeks/months, which are probably too fast to be observed on Earth. The slow modes, on the other hand, have typical time scales of thousands of years. The eigenvector structures of the fast modes are dominated by their velocity field component with negligible horizontal magnetic field perturbations. On the other hand, the slow modes are dominated by their magnetic field component, with negligible effects of the velocity field.
• —Equations for the interaction of different MHD Rossby waves were derived. The time scale of energy transfer between these waves varies, depending on the energy in each mode, but a typical value would be at the typical time scale for a strong nonlinearity regime and one order of magnitude slower for weak interactions.
• —We have shown that the combined effects of nonlinearity, forcing on a selected mode and parametric interaction through topography can make the excited mode quasi-stationary. Instead of propagating as a free wave, this mode oscillates around a particular value of the phase (or longitude). The energy of the system is therefore dominated by the forced mode, which holds a larger amplitude than the other modes.
• —We have proposed that the phenomenon of geomagnetic flux lobes, localized stationary structures of the geomagnetic field, is analogous to the phenomenon of atmospheric blocking. A quasi-stationary streamline pattern dominated by a meridional component will constitute a transport barrier for the radial component of the magnetic field. Radial magnetic field structures will become trapped in a localized geographical region. The situation is similar to blocked atmospheric states that prevent aerosol and humidity transport in the zonal direction, causing extended periods of drought in certain regions.

We drew our conclusions from a simple nonlinear model with only a single interacting wave triad. It would be important to extend this theory to more complete numerical models with thermal forcing and topography, along with including a more realistic forcing structure involving a larger number of modes. An interesting question is whether chaotic transitions from atmospheric blocked to zonal states have their counterpart in the geomagnetic field. In such a scenario the predominantly meridional flow would break down into a zonal one, allowing localized structures of the radial geomagnetic field to drift away with the flow. In this case, geomagnetic flux lobes would possibly be absent in the observational records. Constable et al. [1] have indicated that the flux lobes might be absent in the records for certain periods. Thus, if their findings reflect the true behaviour of the geomagnetic field, and are not just an artefact of insufficient data or coarse temporal resolution, then they could be an indication that blocking transitions also occur in the Earth’s core.

Appendix A. Conserved quantities and their constraints for triad interactions

Following the work of [51], the quantities that play a relevant role in the three-wave interactions of MHD Rossby waves are the ones that are quadratic in the perturbations. The total energy corresponding to the perturbation equations (3.1) is given by

$$E=\frac{1}{2}\int |{\mathrm{\nabla }}^{\perp }\psi {|}^2+\frac{1}{{\mu }_0\rho }|{\mathrm{\nabla }}^{\perp }A{|}^2\hspace{0.17em}\text{d}\mathbf{\text{x}},$$ A 1

which is a conserved quantity of these equations ($\dot{E}=0$). The exactly quadratic nature of the total energy implies that, for any arbitrary truncation of the expansion (3.3), the truncated energy will be conserved. In this way, consider the arbitrary truncated expansion of the solution in terms of the eigenmodes of the linear problem,

$${\left[\begin{array}{c}\psi \\ A\end{array}\right]}_{\mathrm{\Gamma }}=\sum _{(k,\alpha )\in \mathrm{\Gamma }}a{\mathrm{\Lambda }}_k^{\alpha }(t)Y_{n_k}^{m_k}{\overrightarrow{R}}_k^{\alpha }=\sum _{(k,\alpha )\in \mathrm{\Gamma }}a{\mathrm{\Lambda }}_k^{\alpha }(t)N_{n_k}^{m_k}{P}_{n_k}^{m_k}\hspace{0.17em}{\text{e}}^{i{m}_k\varphi }{\overrightarrow{R}}_k^{\alpha },$$ A 2

where $\mathrm{\Gamma }\subset \mathbb{Z}\times \mathbb{N}\times \{-,+\}$ is the set of wavenumber pairs (m, n) and wave types that characterizes the truncation. Then, inserting the expansion above into (1) yields

$$\frac{\text{d}}{\text{d}t}{E}_{\mathrm{\Gamma }}=\frac{\text{d}}{\text{d}t}[\sum _{(k,\alpha )\in \mathrm{\Gamma }}{n}_k(n_k+1)|{\mathrm{\Lambda }}_k^{\alpha }{|}^2(|R_k^{\alpha }(1){|}^2+\frac{1}{{\mu }_0\rho }|R_k^{\alpha }(2){|}^2)]=0,$$ A 3

where $R_k^{\alpha }(1)$ and $R_k^{\alpha }(2)$ refer to the components of the eigenvector ${\overrightarrow{R}}_k^{\alpha }$. It is also important to note that the total energy has a diagonalized representation in terms of the eigenmodes. In particular, the total energy is conserved for any set of three waves whose wavenumbers satisfy the Elsasser selection rules (3.8). Thus, let us consider such an arbitrary wave triad whose spectral amplitudes are governed by equations (3.9). From these equations, it follows that

$$\begin{array}{rl}& \frac{\text{d}|{\mathrm{\Lambda }}_1{|}^2}{\text{d}t}=2i{C}_{1,2,3}Im({\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2^{\ast }{\mathrm{\Lambda }}_3^{\ast }),\end{array}$$ A 4a

$$\begin{array}{rl}& \frac{\text{d}|{\mathrm{\Lambda }}_2{|}^2}{\text{d}t}=2i{C}_{2,1,3}Im({\mathrm{\Lambda }}_1^{\ast }{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3),\end{array}$$ A 4b

$$\begin{array}{rl}& \frac{\text{d}|{\mathrm{\Lambda }}_3{|}^2}{\text{d}t}=2i{C}_{3,1,2}Im({\mathrm{\Lambda }}_1^{\ast }{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3).\end{array}$$ A 4c

From the equations above it follows that

$$\frac{\text{d}{E}_T}{\text{d}t}=0\hspace{0.17em}⟺\hspace{0.17em}{E}_1{C}_{1,2,3}-E_2{C}_{2,1,3}-E_3{C}_{3,1,2}=0,$$ A 5

where $E_j=n_j(n_j+1)(|R_{(j,\alpha )}(1){|}^2+(1/{\mu }_0\rho )|R_{(j,\alpha )}(2){|}^2)=n_j(n_j+1)|{\overrightarrow{R}}_j{|}^2$, j = 1, 2, 3, refers to the intrinsic energy norm of the eigenmodes. Therefore, in order for the total energy to be conserved in an interacting wave triad, the mode whose coupling coefficient has the largest absolute value (mode 1) of the triad must either provide energy to or receive energy from the other waves.

Apart from the total energy, the other quadratic invariant of the perturbation equations (3.1) refers to the total enstrophy conservation. The total enstrophy of equations (3.1) is given by

$${ϵ}_T=\frac{1}{2}\int |{\mathrm{\nabla }}^2\psi {|}^2+\frac{1}{{\mu }_0\rho }|{\mathrm{\nabla }}^{2}A{|}^2\hspace{0.17em}\text{d}\mathbf{\text{x}}.$$ A 6

Similarly to total energy conservation, the total enstrophy conservation implies the following constraint for the coupling coefficients of an interacting wave triad:

$$E_1{n}_1(n_1+1)C_{1,2,3}-E_2{n}_2(n_2+1)C_{2,1,3}-E_3{n}_3(n_3+1)C_{3,1,2}=0.$$ A 7

Equation (7), together with (5), implies that the pump mode of an interacting triad (say, mode 1) is the one having the intermediate total wavenumber, that is,

$$|C_{1,2,3}|>|C_{2(3),1,3(2)}|\hspace{0.17em}⟹\hspace{0.17em}{n}_{2(3)}(n_{2(3)}+1)<n_1(n_1+1)<n_{3(2)}(n_{3(2)}+1).$$

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Authors' contributions

B.R. designed the study and wrote the text. C.F.M.R. discussed the results and helped with the writing.

Competing interests

We declare we have no competing interest.

Funding

The research presented here was financially supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) through grant nos. 2017/23417-5 and 2015/50686-1. The work of C.F.M.R. is also supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.