An accelerating high-latitude jet in Earth’s core

Observations of the change in Earth’s magnetic ﬁeld, the secular variation, provide information on the motion of liquid metal within the core that is responsible for its generation. The very latest high-resolution observations from ESA’s Swarm satellite mission show intense ﬁeld change at high-latitude localised in a distinctive circular daisy-chain conﬁgu-ration centred on the north geographic pole. Here we explain this feature with a localised, non-axisymmetric, westwards jet of 420 km width on the tangent cylinder, the cylinder of ﬂuid within the core that is aligned with the rotation axis and tangent to the solid inner core. We ﬁnd that the jet has increased in magnitude by a factor of three over the period 2000– 2016 to about 40 km/yr, and is now much stronger than typical large-scale ﬂows inferred for the core. The current accelerating phase may be a part of a longer term ﬂuctuation of the jet causing both eastwards and westwards movement of magnetic features over historical periods, and may contribute to recent changes in torsional wave activity and the rotation direction of the inner core.

geographic pole at about latitude 70 • . The same structure of SV is present in many other observation-based models of the geomagnetic secular variation covering the past 17 years (see methods).
The distinctive pattern of SV is caused by a westward motion of the flux lobes [12,7], situated in the regions between A (80 • E) and B (120 • E), between C (170 • E) and D (220 • E), and a weaker patch between E (280 • E) and F (310 • E): see also the supplementary movies S1 and S2 of the time-evolution of the main field and its SV. As the patches of negative radial flux move,  Fig. 1), which shows a uniformly low secular variation. In 2001, the high latitude SV patches were recognisable in their present form but relatively weak (about 10-15 µTyr −1 in magnitude; see figure S1).
Between 2004 and 2016 these SV patches, required by the observational data, have notably increased in magnitude. This change is robustly seen not only in CHAOS-6, but also across a wide range of other field models, built using different data sets and employing different modelling assumptions (see methods). The only plausible explanation for the signal appears to be that it is a signature of rapidly changing magnetohydrodynamic processes taking place in the Earth's core. It is too localized to be due to remote magnetospheric sources, and the fact it is seen only in the northern hemisphere and not the southern hemisphere makes an explanation in terms of magnetosphere-ionosphere coupling or related ionospheric currents unlikely. Furthermore, this SV signal is clearly related to the evolution of the core-field's high latitude flux lobes close to the tangent cylinder region (see movie S1) and thus is most likely of internal origin.
The majority of the intense SV signal is contained within spherical harmonic degrees 11-13.

Localised flow close to the tangent cylinder
The latitudinal position and extent of the northern polar SV patches suggest an intimate link to dynamics close to the tangent cylinder, whose intersection with the CMB is at latitude ±69 • .
The tangent cylinder likely separates regions of quite distinct physical processes [13,8,14], a probable cause of which is that magnetic forces fail to satisfy a certain continuity condition [15,16]. Any mismatch in the radial flux of core-fluid would lead to a local convergence or divergence of flow, which, assuming incompressibility, would drive a lateral flow in order to redistribute fluid. In the rapidly rotating regime of Earth's core, this lateral flow takes the form of an equatorially-symmetric jet [16] which is predominantly in the azimuthal direction and localised to the tangent cylinder. For the axisymmetric component of the jet the need to satisfy Taylor's constraint [17] adds additional complications. Additional evidence for such localised equatorially-symmetric jets comes from numerical dynamo models at very low viscosity [18,19]. We therefore propose that the flow local to the tangent cylinder is a member of the well studied class of (equatorially-symmetric) quasi-geostrophic flows.
In a full-sphere geometry (neglecting the existence of the inner core), an incompressible quasi-geostrophic flow, u, can be written in terms of a stream function Ψ in the following form [20] u s = 1 Hs where H = √ 1 − s 2 , (s, φ, z) are cylindrical coordinates and where we have non-dimensionalised length by the radius of the core (3480 km). Provided the boundary condition ∂Ψ/∂φ = 0 at s = 1 is fulfilled, this flow also satisfies the impenetrable condition everywhere on the CMB.
In order to define a localised flow, here we define Ψ by the real part of the modal sum where M is the maximum wavenumber, a m are complex coefficients to be determined, β denotes s−r i δ where r i is the non-dimensional inner-core radius 1221/3480 and δ is the prescribed jet width. The constants c m are determined through imposition of the boundary condition on each Φ m . Each mode then has an azimuthal component proportional to Φ m (s) which is localised to the tangent cylinder. Note that we included a factor √ 1 − s 2 , which multiplies the exponential in the definition of Φ m , in order that Ψ could be determined analytically.

Observationally-constrained high-latitude flow
We now investigate whether the localised SV on the CMB close to the tangent cylinder can be accounted for by such a jet. The equation that describes the rate of change of the radial field (B r ) on the core-surface, under the assumption of frozen-flux (i.e. neglecting magnetic diffusion on account of its time scale being much longer than the interannual variations that we seek to explain), is the followingḂ where u H denotes the horizontal flow [21].
where SV obs is the radial component of secular variation from the observational model, and SV syn is the synthetic SV as determined from the interaction of the flow with the background field. The majority of our results are computed using the model CHAOS-6 which describes the main field at the CMB reliably to degree 13 and its associated SV to degree 16: we truncate also SV syn to degree 16. Since the target residual R N +S is quadratic in the unknown coefficients its global minimum is straightforward to find; for each choice of M and δ a best-fitting model can be produced.  [9], and three times the typical rms speed of 13 km yr −1 from globalscale core-flow inversions [21]. This jet is confined in longitude and so it is neither circumpolar nor zonal, despite being predominantly azimuthal. A similar equatorially-symmetric westwards flow close to the tangent cylinder has been identified in previous direct core-flow inversions e.g. [22,23], and is sometimes interpreted as an integral part of a planetary-scale gyre.

Symmetry of the flow
Although we have demonstrated that a simple non-axisymmetric and equatorially-symmetric jet fits the high-latitude signal in both the northern and southern hemispheres, we now assess evidence for other flows that might fit the signal equally well. Table 1  from all these jets are about four times smaller than that generated by zero flow (row 8).
We next consider how well axisymmetric flows can fit the data (rows 4-5), by considering not only our jet model (with M = 0) but also that of an axisymmetric polar-vortex of the form u φ = as + bs 3 + cs 5 , similar to that proposed in earlier studies [8,24], where the constants a, b, c were fit by minimising the same target residual as before. Both axisymmetric models fit the data far less well than our preferred non-axisymmetric model, principally because they predict large SV under Greenland (caused by the westwards advection of a nearby flux lobe) which is absent from the observed SV signal. Thus the higher level of detail now available in the tangent cylinder region appears to favour non-axisymmetric over axisymmetric flow structures.
Lastly, we assess whether or not the SV signal provides evidence for a jet that is equatorially-   Table 1: A comparison of residuals produced by a variety of flow models: our preferred model is shown in row 1. The model type and maximum azimuthal wavenumber M is given in column 1; the models are fitted using the target residual defined over either the northern polar region (N), the southern polar region (S) or both (N+S, see equation (2)), as shown in column 2. The integrated SV residual R calculated over these three regions is given in columns 3-5, expressed in units of (µT /yr) 2 . All jets have optimal width δ = 0.12 correct to 2 d.p. The last row shows the residuals assuming zero flow everywhere. asymmetric (i.e. different in the northern and southern hemispheres), by restricting the target residual to include only one hemispheric polar region. Row 6 shows the fit of an M = 1 (equatorially-symmetric) jet using only the signal from the north. The residual is only slightly lower in the northern polar region than our preferred model of row 1, and only slightly greater in the southern polar region, demonstrating that the southern signal lies predominantly in the null-space of the inversion. Performing a similar exercise and fitting a jet using only signal from the south leads to a residual in the south being comparable to the residual with no flow at all. Therefore, although the constraints from the northern SV signal are strong and require a jet, by contrast the southern SV signal by itself provides essentially no constraint on the flow and serves neither to support, nor disprove, equatorial symmetry. Nevertheless, overall the data do not rule out our theoretically-preferred equatorially-symmetric model [16].
At first glance, it may appear surprising that our proposed equatorially-symmetric jet, although strong close to the southern geographic pole, remains consistent with the small SV observed in this region (Fig 1). To explain this, it is useful to note that if the flow is dominated by azimuthal advection rather than upwelling, then ∇ H · u H ≈ 0 (which is the case for the optimised jets), and (1) simplifies toḂ whereφ is the unit vector in the azimuthal direction. In this simple case, the secular variation is then simply longitudinal advection of the azimuthal derivative of the radial field. The importance of this effect will become clear from figure 4, which shows a superposition of the jet structure with the azimuthal derivative of the radial field in both the northern and southern hemispheres. In the north, stronger azimuthal gradients in the radial field are advected producing the stronger patches of SV. In the south, the gradient of the radial field is relatively weak and thus advection by a strong flow produces little SV.

An accelerating jet
We are also able to explain the increasing magnitude of the high-latitude northern SV signal by considering a jet with the same M = 1 structure which we now allow to vary in time. The westward movement of the northern flux lobes bears some resemblance to the westwardmoving flux patches on the equator [9]. However, the equatorial patches are not accelerating and likely have a different explanation from those at high-latitude, either steady advection [25,19] or wave motion [26].

Implications for core dynamics
Because the jet may ultimately result from an imbalance in fluid transport across the tangent cylinder, changes in its magnitude may reflect alterations in the dynamics on either, or both, sides of the tangent cylinder, on decadal timescales. Because inertia and viscosity are so small in the core, this would either have to come about through changes in the internal structure of the magnetic field or through changes in the fluid buoyancy distribution. Decadal changes in composition or temperature driving changes in buoyancy may be possible during an intense upwelling event. However, changes in the interior magnetic field seem to be a more likely explanation as it is well known that the surface field, at least, changes on such timescales.
Indeed, a recent model [19] has shown that altering the l = 3, 4 harmonics of a magnetic field (mimicking changes over centennial timescales) can not only alter the torque on the inner core, but cause significant change to the tangent cylinder jet structure and direction. Higher harmonics are expected to change faster, hence it is likely that decadal changes in the structure of the jet can be driven through internal changes in the magnetic field.
The strength and magnitude of the jet is sensitive to the sign and magnitude of the force imbalance and not the forces themselves: consequently subtle changes in the structure of the magnetic field could be enough to cause large fluctuations in the jet and its associated SV. If so, the present dynamics on the tangent cylinder is likely to episodically repeat and reverse, as the internal field changes over time. Evidence of such a westward-eastward wobble of the Canadian flux lobe [12] can be seen in the historical model gufm1 (see movie S3), which shows eastward motion during 1730-1800, and westward motion during 1900-1960; there are also indications of east-west oscillations of the flux lobes on longer (centennial) timescales [27]. We suggest that we are currently observing the accelerating phase of such a wobble. This adds to the evidence for distinctive SV at high-latitude [10,11], although the visibility of the jet within the SV relies on there being an appropriate structure of radial field to advect. Because the jet flow is dominantly in the azimuthal direction, it would likely not be affected by outer-core stratification [28].
Changes in the magnitude of the jet will also have repercussions for the dynamics deep within the core. Large-scale changes in axisymmetric core-flow affect the net angular momentum of the core and therefore of the mantle; however the jets we computed have a significant non-axisymmetric component and therefore do not have a simple signature in change in length of day. Nevertheless, the jet will now be supplying a westwards-directed force on the inner core due to electromagnetic coupling. Interestingly, the acceleration of the jet that we find from 2004 onwards is coincident with an abrupt alteration in the rotation direction of the inner core from eastwards to westwards [29] that has been inferred at about the same time. Lastly, mounting evidence [30,31] suggests that torsional waves may be launched from the tangent cylinder, perhaps by the dynamics associated with the cylindrical jet [16]. An accelerating phase of the jet is consistent with independent studies [32] showing an increase in torsional-wave magnitudes over the last decade.

Methods
The CHAOS-6 model This model provides an excellent global description of recent secular variation, fitting ground observatories to a Huber-weighted rms level of 3.1 nT/yr for the eastward components and 3.8 and 3.7 nT/yr for the vertical and southward components. It has secular variation at the core surface which is stable out to at least spherical harmonic degree 16 [1]. In this article, we focus on CHAOS-6 in preference to other available models because of its continuous treatment of ground and satellite data since the start of the modern era of satellite geomagnetism in 1999, its focus on high resolution secular variation, and the generally good agreement of the CHAOS family of models with other geomagnetic reference models [33].

Robustness of the observation
In addition to CHAOS-6, the same structure of intense high-latitude SV is present in many other observation-based models of the geomagnetic secular variation covering the past 17 years.

Robustness of jet acceleration
Here we compare CHAOS-6 to a variety of other families of observation-based geomagnetic field models, to show strong evidence for an increase in the jet magnitude over the past two decades. The families of models we compare are: • The GRIMM series from Lesur and co-workers [35,36], from which we use a recent version, GRIMM-3. A similar algorithm using only data from Swarm was used by GFZ to produce the Swarm Level 2 Data Product: Dedicated Inversion [38]. For both models we used the SV to degree 14.
• The Comprehensive Inversion (CM) family, of which the latest published version is CM5 [34]. The Swarm Level 2 Data Product: Comprehensive Inversion (CI) [38] has been produced using a similar method but using only Swarm data. For both models we used the SV to degree 13.
• The Swarm Initial Field Model (SIFM) using data only from Swarm at the single epoch of 2014.5 [37]; we used the SV to degree 11.
• The POMME model series [39], of which the latest is POMME-10. We used the SV to • A lower resolution CHAOS-type model built using only data from Ørsted and ground observatories. We used the SV to degree 15.
Assuming a jet structure of δ = 0.12 and M = 1 (as determined from CHAOS-6) we computed the best-fit jet over a set of discrete points in time (typically every 0.1 years) from 1999 onwards. For all models the main field was truncated at degree 13, but the SV truncation was chosen for each model individually in order to ensure that the SV power spectra at the CMB was not diverging.
The results are summarised in figure S2. The POMME-10 model shows the same three-fold increase in jet velocity since 2002 as CHAOS-6, although the jet strength from POMME-10 is notably more variable, likely due to its piecewise linear temporal parameterization. The strengthening jet is also supported, although to a lesser extent, by the CHAOS-type model using only data from the Ørsted satellite and ground observatories. This means that both the intensifying SV polar signal and the evidence of a strengthening jet are not a consequence of the descent of the CHAMP satellite or the inclusion of recent data from Swarm. The very recent high jet strength is also supported by the single data point of SIFM by an even larger flow velocity than that inferred from CHAOS-6. The GRIMM/GFZ family, for which there is a gap in temporal coverage between 2009 and 2014.5, also shows the increase in jet velocity, although more modest than CHAOS-6. The CM5/CI family shows a constant maximum jet velocity until 2012 of about 27 km/yr, but when restricted to data from Swarm the jet velocity jumps to a value above 40 km/yr, which is comparable to that inferred from CHAOS-6. Thus this family of field model does, overall, still support a strengthening jet.
The fact that CM5 (shown 2001-2012) shows no evidence itself for a strengthening jet seems to be related to its relatively strong regularization of secular acceleration, which will therefore also result in lower accelerations of any fitted flow. This effect is explored in figure   S3, which shows the SV power spectra for CHAOS-6 alongside the GRIMM/GFZ and CM5/CI families of models. Because the majority of the intense SV patches are present in degrees 11-13, it is important that the modeling procedure allows power in this range to change. It is notable that CHAOS-6 shows the greatest temporal variability in power at high degree. The GRIMM/GFZ family shows less but still significant variability, and is largely in agreement with CHAOS-6. The CM5/CI family in contrast shows very little temporal variability in power from degree 6 upwards, as the lines defining the different epochs almost overplot. Therefore it is perhaps no surprise that the jet, when fitted to this relatively temporally-restricted SV, shows little variability. The independently computed CI model (using only Swarm data) shows a significant change in the shape of the SV spectra at high degree.
The families of models compared here use a range of data selection and processing methods.
The fact that they all agree on an acceleration of the jet gives us confidence in our interpretation.

Code availability
The code used to generate the results shown can be obtained from the corresponding author upon request (PWL).

Data availability
Both CHAOS-6 and the SIFM geomagnetic field models can be accessed via the URL Information about how to access Swarm L2 products (including the field models we used) can be found at https://earth.esa.int/web/guest/swarm/data-access.  Figure S2: Comparison of maximum westward speeds from the simple jet structure (assuming M = 1 and δ = 0.12) fit to a variety of observational models, whose truncations are described in the methods section of the main text.  Figure S3: Secular variation power spectra for three different families of models. Both CHAOS-6 and GRIMM/GFZ show time-variation in power at high degree, whereas CM5 shows little variation above degree 6. Thus the strong temporal regularisation used in CM5 does not permit the intensification of the polar SV patches (mainly in degrees 11-13) related to jet acceleration seen in other models.