## Abstract

A novel theory for the origin of the westward drift of the Earth’s magnetic field is proposed, based upon the propagation of hydrodynamic Rossby waves in the liquid outer core. These waves have the obscure property that their crests always progress eastwards—but, for a certain subset, energy can nevertheless be transmitted westwards. In fact, this subset corresponds to sheet-like flow structures, extended in both the axial and radial directions, which are likely to be preferentially excited by convective upwellings in the Earth’s rapidly rotating outer core. To enable their analysis, the *quasi-geostrophic* (QG) approximation is employed, which assumes horizontal motions to be independent of distance along the rotation axis, yet accounts for variations in the container height (i.e. the slope of the core–mantle boundary). By projecting the momentum equation onto flows of a QG form, a general equation governing their evolution is derived, which is then adapted for the treatment of two initial value problems—in both Cartesian and spherical geometries—which demonstrate the preference for westward energy propagation by the waves in question. The merits of this mechanism as an explanation for westward drift are discussed.

## 1. Introduction

Since its discovery over three centuries ago by Halley [1], the westward drift of the Earth’s magnetic field has remained an intriguing problem in geophysics. Why should it be that many features of the observable geomagnetic field have systematically tracked west throughout 400 years of measurements? The first detailed analysis of this phenomenon was performed by Bullard *et al.* [2], to be followed by many others [3] seeking to elucidate its behaviour through interrogation of both historical data and recent satellite measurements.

The allure of westward drift can perhaps be put down to its ubiquity. Regardless of whether one looks at maps of declination at the Earth’s surface or radial field at the core–mantle boundary, westward drift arises as a robust feature of geomagnetic secular variation. The apparent indifference to both spatial scale [4] and epoch [5] suggests that the westward drift is an artefact of fundamental core dynamics, and therefore its explanation may open a window onto the Earth’s deep interior.

While the westward drift spans many temporal and spatial scales, it also appears to be geographically localized, with secular variation being limited to low-latitude (equatorial) regions [5–7], and dominated by the motion of intense flux spots with a tendency to pair north and south of the Equator [8]. For unknown reasons, the Pacific hemisphere appears to have been relatively quiet in the modern era [9], with a weaker field magnitude and lack of any convincing secular variation patterns. The dominant contribution to westward drift—centred over the Equator in the Atlantic hemisphere—was found by Finlay & Jackson [5] to be at a rate of 0.27° yr^{−1}, or 17 km yr^{−1} at the core–mantle boundary. It is worth noting, however, that these observations face an unfortunate constraint; owing to the interference of crustal magnetism, their resolution is limited to spherical harmonic degrees below 13—meaning there is a dearth of information at all but the largest scales of magnetic field [10]. This is a cause for concern, especially since spectra of the observable secular variation show its power increasing with harmonic degree [4], suggesting its origin is to be found at the invisible small scales. Therefore, theoretical models of the small-scale dynamics may prove useful tools for explaining the westward drift.

The liquid outer core, approximately occupying the spherical shell 1231–3485 km from the Earth’s centre, is the cradle of our planet’s magnetic field. Its periphery—the core–mantle boundary—may be considered an impermeable, electrically insulating solid on the time scales of interest. Its internal boundary with the solid, conducting inner core will be neglected in this study, as it is thought the geodynamo operates chiefly outside of the tangent cylinder (an imaginary surface aligned with the rotation axis and circumscribing the inner core). The molten iron which makes up the outer core has a kinematic viscosity which is probably not dissimilar to that of water [11], and therefore may be taken as inviscid over the large length scales considered. The force balance is instead thought to be dominated by the effect of the rapid background rotation of the Earth at an angular velocity *Ω* of approximately 2*π* radians per day. It is well known that rapidly rotating fluids have a tendency to evolve in a manner which is effectively two dimensional, being independent of distance along the rotation axis [12,13], and therefore simplified models which presuppose this disposition are often employed in their study.

Motion in the outer core is thought to be stirred by vigorous convection, with thermally or compositionally buoyant material pushing radially outwards from the hot inner core. The convection is strongly forced, meaning the distribution of density anomaly within the core is likely to be chaotic and span a vast range of scales. This raises questions for both the geodynamo as a whole and the westward drift; how does the organized dipolar field structure emerge from this stochastic forcing, and how can it also produce the systematic drift observed at large scales?

At present, there exist two main schools of thought on the answer to this final question. Arguably the most popular model, due to Pais & Jault [14], invokes a large-scale eccentric gyre—or westward-directed jet—which advects the mean magnetic field. An alternative hypothesis [15–18] rests upon certain magnetohydrodynamic modes with an invariably westward phase velocity. Without remark upon the merits of either of the above models, a third possibility is here put forward, underpinned by the hydrodynamic Rossby waves produced when a rapidly rotating fluid is forced to deviate from two-dimensionality by the presence of the container walls (in this case, the core–mantle boundary). We introduce the quasi-geostrophic (QG) theory of these waves in §2, deriving their governing equation for a general container geometry, and dispersive properties in a canonical example. The waves are linked to westward drift in §3, supported by a simplistic initial value problem. A similar problem is approached in §4, only in a spherical geometry much more reminiscent of the Earth. The discussion of §5 appraises the value of Rossby waves as a possible source of the observed westward drift.

## 2. Theory of quasi-geostrophic Rossby waves

Consider the Earth’s outer core to be an inviscid, incompressible fluid in a state of rapid bulk rotation at an angular velocity ** Ω**=

*Ω*

*e*_{z}, where

*e*_{z}is the unit vector in the axial direction of either a Cartesian (

*x*,

*y*,

*z*) or cylindrical polar (

*s*,

*ϕ*,

*z*) co-ordinate system. In either case, a subscript ⊥ denotes the component of a vector perpendicular to

*e*_{z}. In a reference frame rotating at

**, the Eulerian fluid velocity is**

*Ω***(**

*u***,**

*x**t*). The core–mantle boundary is represented by symmetric, impermeable surfaces at

*z*=±

*h*(

*x*_{⊥}); for a spherical geometry of unit radius, one would have

### (a) Kinematics and the quasi-geostrophic approximation

In a bid to simplify the analysis, the so-called *QG approximation* [19–21] is made. This is in deference to the fact that the rapid background rotation forces the fluid to seek steady states which are independent of the axial co-ordinate *z* (geostrophic). The presence of the boundaries at ±*h* introduces small departures from geostrophy which cause these states to evolve on a time scale much longer than the rotation period; such motions might be called QG, though the definition of the phrase is somewhat imprecise. Here, the term QG is used in a strict sense: as a label for the assumption that the velocity components perpendicular to the rotation axis (*u*_{⊥}) are independent of the axial co-ordinate, an approach which has seen much success in modelling of outer core convection [22–24]. This is despite the fact that the assumption is only strictly valid when the boundary slope is small, a condition clearly violated in the equatorial regions of the Earth’s spherical core [21,25]. Moreover, the approximation remains reasonable even in the presence of a background magnetic field [26].

The velocity field ** u** in this formulation is subject to three kinematic conditions:

(i) incompressibility, ∇⋅

=0;*u*(ii) non-penetration at the upper and lower boundaries, (∇

*h*∓*e*_{z})⋅|*u*_{±h}=0;(iii) the QG approximation,

*u*_{⊥}=*u*_{⊥}(*x*_{⊥},*t*).

It can be shown that a representation of the form
*χ*(*x*_{⊥},*t*) neatly encapsulating the evolution of the vector field ** u**(

**,**

*x**t*) through a scalar function of just two spatial co-ordinates. Note that this form is a generalization of that introduced by Schaeffer & Cardin [20], which itself improves upon the classical perturbation expansion approach (as discussed in [23]). Restricting solutions to the form (2.1) offers a drastic simplification of the analysis, while providing a useful tool with which to probe the physics of axially elongated structures in the core of the Earth.

### (b) Dynamics and governing equation

A governing equation for the streamfunction *χ* in this QG approach is now derived. In a reference frame rotating at the bulk angular velocity ** Ω**, conservation of momentum for an inviscid, incompressible fluid may be written

*Π*. In the limit of small Rossby number (

**=∇×**

*ω***. Evidently, steady solutions must be independent of distance along the rotation axis (i.e. geostrophic)—this is the Taylor–Proudman theorem [12,13]. However, in order to satisfy non-penetration at**

*u**z*=±

*h*, QG solutions must possess a weak

*z*-dependence, and therefore can exhibit unsteadiness. To derive an equation for the evolution of a QG flow, one could simply substitute the representation (2.1) into the axial component of the vorticity equation (2.3), a procedure commonly employed in the literature (e.g. [21,22]). However, as pointed out in Labbé

*et al.*[27], a more efficacious approach is to instead project the momentum equation (2.2) onto flows of the QG form (2.1), thereby obtaining a reduced model which better approximates the dynamics. This has been verified analytically in a full sphere by Maffei

*et al.*[25], who found remarkably good agreement with the fully three-dimensional solutions of Zhang

*et al.*[28].

We therefore proceed by following the derivation of Labbé *et al.* [27], generalizing their results in a sphere to a more arbitrary geometry. Consider a control volume V of fluid of constant horizontal cross section A, bounded at the top and bottom by the caps *z*=±*h*(*x*_{⊥}). The boundaries of V and A are denoted *u*^{′}=∇*χ*^{′}×∇(*z*/*h*), which by construction satisfies *χ*^{′} guarantees streamlines of *u*^{′} cannot pass through *z*=−*h* to *z*=*h* can be completed, thereby projecting the equations onto the horizontal plane A:
** n** is the in-plane unit outward normal to

*χ*

^{′}, the streamfunction

*χ*must obey the governing equation

*z*-component of (2.3)); for moderate values of ∇

*h*, however, this difference becomes significant.

### (c) Quasi-geostrophic Rossby waves

The governing equation (2.15) can support oscillatory solutions known as QG Rossby waves, in analogy to their atmospheric counterparts [15,29], discussed in detail by Vallis [30], for example. The theory of QG Rossby waves in the Earth’s interior mirrors this classical analysis—to extract their archetypal form, select Cartesian co-ordinates (*x*,*y*,*z*) and a linear height profile *h*(*y*)=*H*+*h*^{′}*y* for positive constants *H* and *h*^{′}, the domain height and slope, respectively. This aims to capture the slope of the core–mantle boundary at zero order, with the *x*-axis oriented east and the *y*-axis radially inwards. Furthermore, the slope is for the moment assumed small (in comparison with the aspect ratio of the QG structures), so (2.15) may be written in the linearized form
*χ*(*x*,*y*,*t*)∝*exp*{*i*(*k*_{⊥}⋅*x*_{⊥}−*ϖ*(*k*_{⊥})*t*)} with frequency *ϖ* and wavevector *α* is the angle between the wavevector and the *x*-axis and *k*_{⊥}=|*k*_{⊥}|. This yields the dispersion relationship for QG Rossby waves
*e*_{k⊥} for the unit vector in the direction of *k*_{⊥} and *e*_{α} for the unit vector in the direction of increasing *α* (figure 2), the phase velocity corresponding to (2.17) can be expressed as
*x*-direction is always positive, meaning wave crests invariably progress eastwards. However, the same is not true for the wave energy, which instead propagates at the group velocity, given by the gradient in *k*_{⊥}-space of the frequency,
*c*_{g} is independent of *α*, so on the velocity diagram the vector *c*_{g} is the diameter of a circle of radius *c*_{p}+*c*_{g})⋅*c*_{p}=0, so *c*_{p} is a chord of the same circle terminating at the base of *c*_{g}. As *c*_{p} is always in the positive *x*-direction, the circle must lie to the right of the origin as shown.

Using figure 2, it is possible to probe the effect of varying the wavevector orientation *α* for a given *k*_{⊥} (i.e. a specified horizontal length scale). Consider only positive frequencies (−*π*/2<*α*≤*π*/2), for which figure 2 makes sense. The phase velocity vector *c*_{p} is constrained to move along the dashed circle, whereas the group velocity vector *c*_{g} starts where *c*_{p} finishes and is necessarily a diameter of the same circle. Although the phase velocity always has a positive *x*-component, the group velocity shows no such preference. In fact, the *x*-component of *c*_{g} is negative (westward) for |*α*|<*π*/4 and positive (eastward) for |*α*|>*π*/4. When *α*=0, *c*_{p} and *c*_{g} are exactly opposite, so an observer following a wave group moving westward would see wave crests heading in the opposite direction at twice the speed of the group. When *α*=*π*/2 the group velocity remains finite (and due east) despite the fact the waves have no phase velocity. Waves with *α*=*π*/4 propagate directly outwards and waves with *α*=−*π*/4 inwards.

The dependence of wave velocity on *k*_{⊥} is comparatively trivial; waves with longer wavelengths (smaller *k*_{⊥}) travel faster. As

## 3. Quasi-geostrophic Rossby waves and the westward drift

A possible explanation for the westward drift of the Earth’s magnetic field at the core–mantle boundary based upon this classical theory of hydrodynamic QG Rossby waves is now offered. Suppose, not unreasonably, that the fluid outer core is stirred by gravitating buoyant anomalies, which constitute localized disturbances to the system in the form of convective upwellings or plumes. Consider one such disturbance introduced at a location outside of the tangent cylinder (i.e. not directly north or south of the solid inner core). The disturbance will in general be three dimensional, but the velocity field it instigates will rapidly become elongated along the rotation axis through the action of inertial waves [33], and therefore QG after a short transient period. It is thus useful to consider a thought experiment posed as an initial value problem in which a localized QG velocity field is specified as an initial condition, and the flow allowed to evolve as an assemblage of QG Rossby waves, operating on a time scale much longer than the rotation period.

A generic initial condition will excite a broad spectrum of waves—that is to say, many different choices of wavevector *k*_{⊥}—which will all spread from the source according to their individual dispersive properties. A disturbance of characteristic size ℓ will have a spectrum peaked around a wavevector magnitude *k*_{⊥} of order ℓ^{−1}, but will in general excite wavevectors of all possible orientations *α*. Therefore, consider the dependence of the group velocity on wavevector orientation by referring back to figure 2. For wavevectors with *α*≈±*π*/2, corresponding to structures elongated in the east–west direction, the group velocity is east; for wavevectors with *α*≈0, i.e. structures elongated in the radial direction, the group velocity is west. QG Rossby waves, therefore, disperse in a manner which segregates different spatial structures from an arbitrary initial disturbance, with east–west extended features heading east and radially-extended features west.

However, it is unreasonable to assume that the excitation of these waves in the core of the Earth is arbitrary. For motions continually stirred by vigorous convection, one might expect the proliferation of sinuous radial plumes, emanating from the inner core and being much longer than they are wide. Such structures commonly arise in numerical and experimental studies of core dynamics (e.g. [34,35]). Owing to the constraint imposed by the rapid background rotation, such plumes would also be elongated in the axial direction, forming a series of radial sheets which are likely to be well represented by the QG approximation [36]. In the context of our thought experiment, a radial sheet (extended in the *y*-direction) will possess much more energy in wavevectors pointing east–west (*α*≈0) than radially (*α*≈±*π*/2). When the solution to such an initial value problem is evolved, the abundance of wavevectors with *α*≈0 will dominate the picture. As the group velocity for these solutions is in the negative *x*-direction, a radially-extended disturbance will preferentially transmit energy to the west, making this class of QG Rossby waves an intriguing candidate for the mechanism underlying westward drift.

### (a) Demonstration through a model problem

To support the arguments made so far, consider a simplistic model problem which demonstrates the ability of QG Rossby waves to segregate different spatial structures. The canonical equation (2.16) is solved in a domain which is infinite in *x* and *y*, starting from some initial condition *χ*(*x*_{⊥},*t*=0)=*χ*_{init}(*x*_{⊥}), by taking a two-dimensional spatial Fourier transform. Emphasis is placed upon the significance of the choice of initial condition, which is constrained to be of the form
*χ* are equivalent to streamlines in the equatorial plane, this corresponds to a columnar vortex of extent ℓ_{x} in the *x*-direction and ℓ_{y} in the *y*-direction. Using *k*_{⊥} and *α* have the same interpretations as in the dispersion relation (2.17). This expression is evaluated numerically for three different choices of the initial condition’s aspect ratio ℓ_{x}/ℓ_{y} (figure 3). Figure 3*b* shows the case of an axisymmetric initial condition, which excites waves of all orientations *α* equally and therefore shows no preferential direction for energy transport, although the partition of different spatial structures is visible. Figure 3*a* shows the case of a radially-extended initial condition, four times longer in the direction of the slope than it is wide; the bias towards westward-propagating waves is self-evident. For completeness, figure 3*c* features an initial condition which is elongated in the east–west direction, which exhibits a strong preference for eastward propagation. It is worth remarking that the aspect ratio need not be extreme for this effect to be apparent; it is very clear-cut here even for a moderate value of 4 : 1.

## 4. A model problem: westward-propagating waves in a sphere

It has been established so far that, in the case of a gentle slope, the linearized equation (2.16) supports wave motions which partition different spatial structures—crucially, with flows elongated in the direction of the slope going west. However, it is not obvious that the same will necessarily hold true for a more complicated geometry with an appreciable slope, such as that presented by the core–mantle boundary. Therefore, consider an initial value problem similar to that of the previous section, but in an enclosed spherical geometry reminiscent of the Earth’s core. A full sphere—deficient of the solid inner core—is used in order to simplify the analysis. First, mode shapes and frequencies are derived in this geometry, following Maffei *et al.* [25], before the calculated modes are used to solve an illustrative initial value problem.

Despite the spherical geometry, cylindrical polar coordinates (*s*,*ϕ*,*z*)—with *s* being the radial location, *ϕ* the azimuthal angle and *z* the distance along the rotation axis—are adopted in order to usefully apply the QG approximation. Taking lengths in units of outer core radii, the governing equation is then (2.15) with the axisymmetric height profile *χ*/∂*t*=∂*χ*/∂*ϕ*=0 will be a particular integral of this equation. This corresponds to the steady, strictly geostrophic motion of coaxial cylinders (** u**=

*u*

_{ϕ}(

*s*)

*e*_{ϕ}); hence, the axisymmetric component of any initial condition will not evolve, and one need only solve (4.1) for the non-axisymmetric portion of the flow. This can be done by seeking normal mode solutions of the form

*m*and modal frequency

*ϖ*, with

*s*=0) and the equatorial boundary (

*s*=1). Regularity at the origin requires

*χ*is a streamfunction, its constant value at the outer boundary can be chosen, so is set at zero; more specifically, we must have

*u*

_{s}=(1/

*hs*)(∂

*χ*/∂

*ϕ*) to be zero and

*u*

_{ϕ}=−(1/

*h*)(∂

*ϕ*/∂

*s*) to be finite at the outer boundary. The solution to the eigenvalue problem posed by (4.3) and these boundary conditions is given in Maffei

*et al.*[25]; the mode shapes are of the form

*n*≥1 is the radial mode number, equal to the number of turning points of

*m*≥1, the frequency

*x*-direction. Just as in that problem, this does not preclude the possibility that the energy from a localized disturbance can nevertheless propagate west, as demonstrated below.

The general solution to (4.1) (setting aside the axisymmetric particular integral for a moment) can be written as an infinite sum of the above modes,
*χ*_{init}(*s*,*ϕ*). In fact, it can be seen from the self-adjoint equation (4.3) that the radial mode shapes (4.5) are orthogonal with respect to the function *s*/*h*^{3}, i.e.
*m* and *n*.

### (a) Choice of initial condition

If the simple Cartesian cartoon discussed in §3 is to be believed, the choice of initial condition, and therefore distribution of energy in *k*_{⊥}-space (or, equivalently, between modes), will have a profound effect on the direction of net energy propagation. In fact, those modes for which the frequency *m* will be associated with westward propagation of energy; in the Cartesian case, *c*_{g,x} is negative for ∂*ϖ*/∂*k*_{x}<0, and analogously retrograde group velocity is seen in the sphere for *h*^{3}*s* ensuring the boundary conditions are satisfied from the outset. It is broadly unimportant exactly what form the initial condition takes, however— it is the general distribution of energy between modes which will dictate the solution’s character.

There are three controlling lengths in the initial condition (4.12); the radial and azimuthal extents of the vortex ℓ_{s} and ℓ_{ϕ}, and the radial location of the vortex centre *s*_{0}. In search of a westward bias to energy propagation, the parameters ℓ_{s}=0.1, ℓ_{ϕ}=0.01 and *s*_{0}=0.7 are selected, giving a slender radially-extended structure near the middle of the outer core region 0.35<*s*<1 (though there is no inner core boundary in this calculation). The solution is expressed as a finite sum of modes (i.e. a truncated version of (4.7)) by evaluating the integrals (4.9) for the coefficients *m*≤200 and *n*≤50 are used.

### (b) Westward bias to energy propagation in a sphere

Figure 4 shows the solution to the initial value problem evaluated at *Ωt*=2×10^{4} (*t*=8.7 *yr*). Streamlines in the equatorial plane are produced by plotting contours of *χ*, with the black contours corresponding to the initial value (4.12). Although this solution is many times more complicated than the Cartesian problem of figure 3, there remains a striking preference for wave propagation to the west of the initial disturbance. This is despite the fact that each individual eigenmode has an eastward phase velocity; it is the superposition of modes which creates the visible westward bias. Wave motion appears to be confined to a circular band near the initial radial location *s*_{0}, with little activity very close to the outer boundary or near the rotation axis. The dashed line is at *s*=0.35, where the inner core would be if it were included in the model; thankfully, the vast majority of activity occurs outside of this region.

The preference for westward propagation may be understood in exactly the same way as the Cartesian problem of §3, but with the small slope assumption now relaxed. The majority of the energy from the radially-extended initial condition is contained in modes with large azimuthal wavenumbers, which conspire to produce a westward group velocity despite individually having eastward phase velocities. To demonstrate that the former prevails, consider the distribution of energy in the sphere as a function of longitude and time. Namely, the meridionally averaged specific kinetic energy,
*ϕ* at a few choice times (figure 5). The energy, initially localized around *ϕ*=0, is almost all at negative *ϕ* when *Ωt*=2×10^{3}, and is still sharply peaked as the waves have had little time to disperse. Come *Ωt*=2×10^{4}, which corresponds to figure 4, the energy is much more dispersed but retains its westward bias; of course, reflections and circumnavigations mean a little energy does end up to the east of the initial disturbance.

## 5. Discussion

The model problems above successfully demonstrate the possibility for westward transport of energy by hydrodynamic QG Rossby waves, but are intended as a proof-of-concept rather than an accurate representation of core dynamics. Clearly, the flow in the Earth’s outer core is not the solution to an initial value problem, but rather the result of continual convective stirring; the interplay between buoyancy and velocity fields will introduce complexity beyond the scope of this study [24]. However, the present theory demonstrates that westward propagation requires only the prevalence of radial plumes, which are likely to be robustly generated by the buoyant upwellings associated with strongly forced convection.

Indeed, discussions of vigorously forced convection are intrinsically linked to the relevant smallest length scale in the core of the Earth, which is itself pertinent to dynamo action [38]. The thickness of the radial plumes will have a strong bearing on the propagation speed of their associated wave packets, as the magnitude of the group velocity (2.19) is proportional to the square of the wavelength, meaning narrower structures propagate much slower. The most strongly forced simulations to date (e.g. [34]) show structures more slender than those considered in the model problem of figure 4, and it is not unreasonable to suspect that the true length scale is even smaller. In fact, it is possible to infer this length scale under the assumption that QG Rossby waves are responsible for the westward drift. From the expression for group velocity (2.19), the speed of a wavepacket at a certain cylindrical radius *s* is given by
*c*_{g}|/*s*; equating this to the observed drift rate of the magnetic field *D* gives an expression for the dominant wavelength of the packet,
*s*∼2000 km gives the estimate *λ*∼18 km, a conceivable value for the prevalent scales in the Earth’s core, but one which should be treated with caution. Firstly, it is within touching distance of the Rhines length *λ* is greater, with additional factors—interactions with the buoyancy and magnetic fields, large values of boundary slope or departures from quasi-geostrophy—acting to slow the wave groups down. Unfortunately, the machinery required to investigate these nonlinear phenomena lies beyond our present scope.

Nevertheless, it seems remiss that so far no consideration has been given to the magnetic field, despite the fact that the observed westward drift of its large-scale features is the motivation for this study. It is therefore necessary to ask what could link hydrodynamic QG Rossby waves to the apparent motion of the spherical radial magnetic field *B*_{r} at the core–mantle boundary. Since the drift is observed to be mainly in the equatorial regions [8], motion of the *cylindrical* radial field *B*_{s}, which will be approximately equal to *B*_{r} at low latitudes, is discussed instead. To a first approximation, magnetic field lines may be thought of as material curves, pinned into the fluid at all points [29,39], so there are essentially two ways of modifying the radial magnetic field: advection and stretching of an existing *B*_{s} by a mean flow, or rotation of the other components (*B*_{ϕ},*B*_{z}) by transverse gradients in radial velocity (∂_{ϕ}*u*_{s},∂_{z}*u*_{s} respectively); both mechanisms are discussed in the context of westward drift by Finlay [40] and Aubert *et al.* [41]. Westward-propagating QG Rossby waves, which necessarily consist of radially-extended sheet-like structures, have small azimuthal velocities so are unlikely to advect *B*_{s} strongly enough to account for the westward drift. The radial velocity is much greater, but has small derivatives in *s* and *z* so stretching of an existing *B*_{s} or shearing of *B*_{z} are both unlikely mechanisms. The best candidate for generation of *B*_{s} is therefore shearing of *B*_{ϕ} by azimuthal gradients in *u*_{s}, which are large for the slender radial jets. Furthermore, the azimuthal magnetic field is likely to be relatively strong within the core [42], and largest at mid- to low latitudes—which could explain the equatorial bias to the observed drift, since a low-latitude *B*_{ϕ} swept out by a QG radial jet would produce a radial field anomaly at the core–mantle boundary in the vicinity of the Equator.

At first glance, this argument appears to suffer from the deficiency that the manipulation of *B*_{s} occurs on the small scale of the wavelength *λ*, whereas the observed drift occurs in magnetic field features hundreds of kilometres across. However, the observations themselves are hampered by a lack of spatial resolution, so small-scale features simply are not visible, even though they may in fact contain a significant portion of the energy [4]. The observations instead feature large, westward-moving patches [8] which one might compare to wave groups, with the small-scale details (wave crests and troughs) within each patch unavailable. It is therefore to be expected that, if the present theory were to explain the observations, large magnetic field features would appear to be advected at the group (rather than phase) velocity of QG Rossby wave packets.

The feedback of the magnetic field on the dynamics through the action of the Lorentz force has been ignored in this study. Indeed, for highly simplified field configurations it has been shown that its inclusion introduces additional oscillations known as *slow magnetic Rossby waves* which themselves have been suggested as a possible source of magnetic field drift since their phase velocity is always westward [15–18]. However, these slow solutions coexist with others known as *fast magnetic Rossby waves*, which are little more than a weakly modified version of the hydrodynamic solutions discussed at length above. The perturbation to the magnetic field does not strongly influence the dynamics of these waves, and so they remain an equally viable source of westward drift, with the magnetic field approximating a passive tracer at leading order.

Moreover, the fact that the dynamics of QG Rossby waves are independent of the magnetic field configuration and magnitude is a strength of the present theory. Slow magnetic Rossby modes have to date only been demonstrated for simple choices of background field [21,27], and it is therefore unclear whether such solutions are meaningful in a geophysical context. Conversely, the fast (i.e. hydrodynamic) solutions are likely to persist regardless of the magnetic field structure, meaning they are an almost unavoidable feature of QG flows in the Earth’s outer core. This robustness tallies with the observation of westward drift as a systematic component of the geomagnetic secular variation, and the fact that the waves operate on a scale much smaller than the observed field features may explain the broad scale-independence of the observed drift rate. These advantages, along with those discussed above, lend credibility to the theory presented here—that hydrodynamic QG Rossby waves with radially-extended structures may underpin the westward drift present in geomagnetic secular variation records.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

The author is funded by an EPSRC Doctoral Training Award. The majority of this work was completed during an academic visit to ETH Zürich, funded by a David Crighton fellowship from the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK.

## Acknowledgements

I am grateful to Andrew Jackson and the EPM group at ETH Zürich for hosting me for a two-month academic visit in autumn 2017. Helpful suggestion for improvements to this manuscript were provided by Jack Atkinson, Peter Davidson, Andrew Jackson, Ben McDermott and two anonymous reviewers.

- Received February 19, 2018.
- Accepted April 20, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.