Fast Quasi-Geostrophic Magneto-Coriolis Modes in the Earth's core
mmanuscript submitted to
Geophysical Research Letters
Fast Quasi-Geostrophic Magneto-Coriolis Modes in theEarth’s core
F. Gerick , , D. Jault , J. Noir CNRS, ISTerre, University of Grenoble Alpes, Grenoble, France Institute of Geophysics, ETH Zurich, Zurich, Switzerland
Key Points: • Magneto-Coriolis modes of periods close to torsional Alfv´en modes could bepresent in Earth’s core model without stratification. • The magnetic field changes of such modes show properties similar to geomagneticobservations, with fast changes localized near the equator. • Our model could allow core-flow inversions from geomagnetic field data to theflow and simultaneously the magnetic field within the core.
Corresponding author: Felix Gerick, [email protected] –1– a r X i v : . [ phy s i c s . g e o - ph ] F e b anuscript submitted to Geophysical Research Letters
Abstract
Fast changes of Earth’s magnetic field could be explained by inviscid and diffusion-lessquasi-geostrophic (QG) Magneto-Coriolis modes. We present a hybrid QG model withcolumnar flows and three-dimensional magnetic fields and find modes with periods ofa few years at parameters relevant to Earth’s core. For the simple poloidal magneticfield that we consider here they show a localization of kinetic and magnetic energyin the equatorial region. This concentration of energy near the equator and the highfrequency make them a plausible mechanism to explain similar features observed inrecent geomagnetic field observations. Our model potentially opens a way to probethe otherwise inaccessible magnetic field structure in the Earth’s outer core.
Hide (1966) proposed that temporal changes of Earth’s magnetic field, calledsecular variations (SV), could originate from linear modes present in the Earth’s liquidouter core. These modes are separated into modes dominated by a balance of magnetic,Coriolis and pressure forces, known as Magneto-Coriolis modes (MCM), and modesdominated by inertial, Coriolis and pressure forces. The latter are often referred toas quasi-geostrophic (QG) inertial modes, or Rossby modes (RM). Torsional Alfv´enmodes (TM), consisting of geostrophic motions (Braginsky, 1970), complete the set ofincompressible and diffusion-less magnetohydrodynamic (MHD) modes. They obey abalance between inertia and the magnetic force. In this study, we use a reduced model,based on the QG assumption for the velocity and a three-dimensional (3-D) magneticfield that is compatible with an insulating mantle, to investigate the SV associatedwith such QG modes, with a focus on MCM, in the Earth’s core.Monitoring the radial magnetic field component at the core-mantle boundary(CMB) is the main way of probing flows in the liquid outer core of Earth. A largenumber of studies are concerned with the inversion of the downward projected geo-magnetic field to flows in the outer core, a process referred to as core-flow inversion(see Holme, 2015, for a review). The most commonly applied core-flow inversions arebased on geostrophic flows tangential to the CMB (Le Mou¨el, 1984; Chulliat & Hulot,2000). These inversions give the flow field local to the CMB. Several works have theninferred core dynamics from the inverted flow field at the CMB. Zatman and Bloxham(1997) and more recently Gillet et al. (2010) have inverted these surface flows to themean radial magnetic field component within Earth’s core through TM. Buffett (2014)correlated Magneto-Archimedes-Coriolis (MAC) waves in a stably-stratified layer atthe top of the outer core with the inferred surface flows.Our model potentially serves as a new forward model to invert geomagnetic fieldobservations. In this approach a reduced set of MHD equations is solved in the bulkof the fluid. It is based on the QG assumption for the velocity, where a balancebetween Coriolis and pressure gradient forces is dominant, while allowing linear axialdependence of the flow field. This assumption is appropriate to investigate fluids underrapid rotation at time scales much larger than the rotation period. Different studieshave shown that a large part of the inferred surface core flow is equatorially symmetricand may account for a large part of the observed secular variations (Gillet et al.,2009, 2011). Additionally, recent 3-D high resolution numerical simulations revealed alargely columnar flow structure, in agreement with the QG assumption (e.g. Schaefferet al., 2017). To establish and maintain such a columnar flow within the Earth’score different mechanisms have been proposed, e.g. 3-D inertial-Alfv´en waves thattransport energy along the rotation axis (Bardsley & Davidson, 2016). Such built upof columns occurs on diurnal periods and are thus not captured in the QG model, wherea columnar structure is assumed to be already established. Previously, consistentlyderived QG models that include a magnetic field were limited to magnetic fields that –2–anuscript submitted to
Geophysical Research Letters treated the CMB as a perfectly conducting boundary (Busse, 1976; Canet et al., 2014;Labb´e et al., 2015; Gerick et al., 2020). For such magnetic fields the radial componentat the boundary must vanish, rendering them unsuitable to associate core flows withmagnetic field imprints at the CMB. There have been approaches to combine a QGmodel with a magnetic field that has a non-zero radial magnetic field component atthe surface, but they rely on the neglect of surface terms in the induction equationthat is difficult to justify (Canet et al., 2009; Maffei & Jackson, 2017). Here wepresent a hybrid model that combines QG velocities with a 3-D insulating magneticfield. We follow the approach presented in Gerick et al. (2020) with a new basisfor the magnetic field that satisfies the insulating boundary condition at the CMB.Both the QG velocity and the magnetic field basis vectors are expressed in Cartesianpolynomials. This methodology has been fruitful to model modes and instabilitiesin rapidly rotating ellipsoids (Vantieghem, 2014; Vidal & C´ebron, 2017; Vidal et al.,2019, 2020). This Cartesian presentation of the basis vectors is particularly useful forthe Galerkin approach used here, due to the easy integration of Cartesian monomialsover the volume (Lebovitz, 1989). We derive a basis for the magnetic field in Cartesianpolynomials, that exploits the properties of spherical harmonics.
The equations governing the incompressible flow u and the magnetic field B in arapidly rotating planetary core of volume V , here assumed to be a full sphere withoutan inner core, are given in non-dimensionalized form by ∂ u ∂t + ( u · ∇ ) u = − z × u − ∇ p + PmLu ∇ u + ( ∇ × B ) × B , (1a) ∂ B ∂t = ∇ × ( u × B ) + 1Lu ∇ B . (1b)The non-dimensional Lehnert, Lundquist and magnetic Prandtl number are given byLe = B Ω R √ µ ρ , Lu = R B η √ µ ρ , Pm = νη , (2)with Ω = Ω z the rotation vector, ρ the fluid density, p the reduced pressure, ν the kinematic viscosity, µ the permeability of vacuum, η the magnetic diffusivity, R the core radius and B the characteristic strength of the magnetic field. Thecharacteristic time scale is the Alfv´en time scale T A = R /u A , where u A = B / √ ρµ is the characteristic Alfv´en velocity. Equations (1) are subject to the non-slip boundarycondition u = and the continuity of the magnetic field across the boundary [ B ] = ,where [ · ] denotes a jump.For parameters relevant for Earth’s core, Le ∼ − , Lu ∼ and Pm ∼ − (Wijs et al., 1998; Gillet et al., 2010; Pozzo et al., 2014). Thus, if we additionallyconsider time scales on the order of T A , it is appropriate to neglect viscous and diffusiveeffects in the bulk. In the next step, since we are interested in the linear response of thesystem, the velocity and magnetic field are perturbed around a background state withno motion and steady magnetic field B . In the Earth’s core, the characteristic meanvelocity field is thought to be negligible compared to the Alfv´en wave velocity (Gilletet al., 2015; B¨arenzung et al., 2018). Hence, the equations describing the evolution ofthe velocity and magnetic perturbations [˜ u , ˜ B ] are given by ∂ ˜ u ∂t + 2Le z × ˜ u = − ∇ p + ( ∇ × B ) × ˜ B + ( ∇ × ˜ B ) × B , (3a) ∂ ˜ B ∂t = ∇ × (˜ u × B ) . (3b) –3–anuscript submitted to Geophysical Research Letters
In the limit Pm → u · n = 0, with n the vector normal to the boundary, and themagnetic boundary condition is not modified (Stewartson, 1957; Hide & Stewartson,1972). Previous studies allowed for a jump in the tangential component of the magneticfield across a diffusive boundary layer (Braginsky, 1970; Jault & Finlay, 2015). Here,we assume that the motions that we are investigating are able to eliminate any currentlayer on the fluid surface. Assuming that the equatorial components of the velocity are independent of thecoordinate z along the rotation axis; the non-penetration boundary condition, u · n = 0on the core-mantle boundary ∂ V holds and the flow is incompressible, ∇ · u = 0, thequasi-geostrophic (QG) velocity takes the form (Amit & Olson, 2004; Schaeffer &Cardin, 2005; Bardsley, 2018) u = ∇ ψ × ∇ (cid:16) zh (cid:17) , (4)with h the half height of the fluid column and ψ a scalar stream function dependingonly on the horizontal coordinates.In Cartesian coordinates the stream function can be expressed as (Maffei &Jackson, 2016; Gerick et al., 2020) ψ i = h Π i , (5)with Π i being a monomial in the equatorial Cartesian coordinates x and y of degree N , so that i ∈ [0 , N ] with N = N ( N + 1) /
2. The QG basis vectors u i are given by u i = h ∇ Π i × z + 3Π i ∇ G × z − z ∇ Π i × ∇ G, (6)with ∇ G = h ∇ h = − x x − y y . In this section we present a set of basis vectors for the 3-D magnetic field, sat-isfying insulating boundary conditions at the CMB. Unlike in classical geodynamosimulations, where the boundary condition is enforced at each time step of the for-ward iteration, the boundary condition is included in the basis elements (Zhang &Fearn, 1995; Li et al., 2010; Chen et al., 2018). The detailed derivation of such a basisis given in Appendix A.We write the magnetic field B in the toroidal-poloidal expansion, so that B = B t + B p = ∇ × T r + ∇ × ∇ × P r . (7)The toroidal and poloidal scalars are written for each spherical harmonic degree l , order m and radial degree n , so that T lmn = (1 − r ) r n R ml , (8) P lmn = − r n +1) R ml n + 1)(2( l + n ) + 3) . (9)with R ml = r l Y ml ( θ, φ ) the solid spherical harmonics. We have | m | ≤ l and l ∈ [1 , N ], n ∈ [0 , ( N − l ) / (cid:5) for the toroidal basis and l ∈ [0 , N − n ∈ [0 , ( N + 1 − l ) / − (cid:5) for the poloidal basis, resulting in a total of N = N ( N + 1)(2 N + 7) basis vectors.The toroidal part of B is given by B t,lmn = ∇ × T lmn r . (10) –4–anuscript submitted to Geophysical Research Letters
The poloidal magnetic field has to satisfy the continuity at the core-mantle boundary ∂ V , so that ∇ Φ i + B p = ∇ Φ e , with Φ i and Φ e the interior and exterior potential field,respectively. The exterior potential field must vanish at infinity, if the source of themagnetic field lies within the interior. The poloidal basis vectors are thus given by B p,lmn = ∇ × ∇ × P lmn r + ∇ Φ ilmn , (11)with Φ ilmn = − ( l +1)(2 l +1)(2 n +2) r l Y ml .Together with the toroidal component (10), this basis can be transformed toCartesian coordinates by representing the spherical harmonics in terms of unit Carte-sian coordinates, as presented in Appendix B. We choose the Schmidt-semi normal-ization for the spherical harmonics, but any other may be chosen, as we normalize thebasis vectors afterwards, so that R V B i · B i d V = 1. We introduce a hybrid quasi-geostrophic (QG) model with QG velocities and 3-Dmagnetic field , following Gerick et al. (2020). The linearized momentum equation(3a) and induction equation (3b) are projected onto a QG basis u of the form of(4) and a 3-D magnetic field basis B of the form (10) and (11), respectively. Thismethod is essentially a variational approach, which consists in finding solutions [˜ u , ˜ B ]satisfying Z V u · ∂ ˜ u ∂t d V = − Z V u · (cid:18) z × ˜ u + ∇ p (cid:19) d V + Z V u · (cid:16) ( ∇ × B ) × ˜ B + ( ∇ × ˜ B ) × B (cid:17) d V ∀ u (12a) Z V B · ∂ B ∂t d V = Z V B · ∇ × (˜ u × B ) d V ∀ B (12b)This set of equations may be reduced to a scalar evolution equation for the streamfunction ψ accompanied by the 3-D induction equation, as shown in equation (48) ofGerick et al. (2020). This hybrid model has been verified against a fully 3-D model atmoderate polynomial truncation (see also Gerick et al., 2020).When replacing the test functions u in (12a) by the subset of purely geostrophicvelocities u G = u G ( s ) φ we obtain the equation for the diffusion-less torsional Alfv´enmodes (TM) initially discovered by Braginsky (1970). The one-dimensional (1-D) TMequation is written s h ∂ ξ∂t = ∂∂s (cid:18) s hv A ∂ξ∂s (cid:19) (13)with ξ = u G ( s ) /s and the mean squared cylindrical Alfv´en velocity v A ( s ) = 14 πsh I Z ( B · s ) s d z d φ. (14)For more details on the derivation we refer the reader to Jault (2003). Equation(13) diverges near the equator as s →
1, but solutions exist, if v A (1) = 0 (Maffei& Jackson, 2016). Since we can compute solutions to the 1-D equation for a givenbackground magnetic field satisfying these conditions, TM suit well as a benchmarkof our hybrid QG model capable of capturing TM. Lehnert (1954) introduced two distinct families of MHD modes as solutions tothe linearized MHD equations (3), namely slow MCM and fast, slightly modified RM. –5–anuscript submitted to
Geophysical Research Letters
The phase velocity is prograde for slightly modified RM and retrograde for MCM.Malkus (1967) has shown that for an idealized magnetic field B ,M = s φ of uniformcurrent density analytical solutions exists for these two mode families. The dispersionrelations are given by (Labb´e et al., 2015) ω ± n,m = 12Le λ n,m ± (cid:18) m ( m − λ n,m ) λ n,m (cid:19) / ! , (15)with ω + n,m , ω − n,m and λ n,m the frequencies of the slightly modified RM, MCM andhydrodynamic (HD) inertial modes of azimuthal wave number m and radial scale n , respectively. The dispersion relation shows that the difference between ω + n,m and λ n,m / Le is small, if Le (cid:28)
1. The frequencies λ n,m are scaled by the rotation frequencyand can be obtained as solutions to a univariate polynomial in the sphere (Zhang etal., 2001). An approximate value for the equatorially symmetric inertial modes is givenby (Zhang et al., 2001) λ n,m ≈ − m + 2 (cid:18) m ( m + 2) n (2 n + 2 m + 1) (cid:19) / − ! . (16)As the magnetic field perturbations in the Malkus field satisfy the perfectly con-ducting boundary condition, with B · n = 0 at ∂ V , the solutions cannot be associatedwith the SV at the CMB and a more suitable background magnetic field needs to beintroduced. The velocity and magnetic field perturbations are assumed to be periodic in time,i.e. ˜ u ( r , t ) = ˜ u ( r ) exp( iωt ) , (17a)˜ B ( r , t ) = ˜ B ( r ) exp( iωt ) . (17b)Enumerating the QG velocity basis ˜ u i , with i = 1 , ..., N and magnetic field basis ˜ B i ,with i = 1 , ..., N , the projections (12) discretize toi ω Mx = Dx , (18)with x = (ˆ α j , ζ j ) ∈ C N + N , and M , D ∈ R N + N × N + N of the form M = (cid:18) U ij B ij (cid:19) , D = (cid:18) C ij L ij V ij (cid:19) . (19)Here, the coefficient matrices U ij , C ij ∈ R N × N , L ij ∈ R N × N , B ij ∈ R N × N and V ij ∈ R N × N correspond to the inertial acceleration, Coriolis force, Lorentz force, timechange of magnetic field and magnetic induction respectively. This form is referred toas a generalized eigen problem solvable for eigen pairs ( ω k , x k ). A large part of M and D is zero, which allows us to use large polynomial degrees whilst keeping computationalefforts small. To avoid numerical inaccuracies we use quadruple precision numbers (seesupplementary material for more details). Our model extends the code available at https://github.com/fgerick/Mire.jl by the magnetic field basis introduced in 2.3. We choose a poloidal magnetic field of low polynomial degree, given by B = ∇ × ∇ × ( P + P ) r + ∇ (cid:0) α Φ i + α Φ i (cid:1) = 1 c x + xz + 2 y + 2 z − / − xy + yz − x − xz − y − z + 5 / , (20) –6–anuscript submitted to Geophysical Research Letters − − | ω | ω − − | ω | − − E k i n / E m ag RMTMMCM
Figure 1.
An overlap of the density of eigen solution spectrums at different truncation9 ≤ N ≤
29 and Le = 10 − (left). Ratio of kinetic to magnetic energy at N = 29 (right). with the constant c = 5 q , so that R V B · B d V = 1. This field has a non-zeromean radial magnetic field at the equator. Its mean squared radial Alfv´en velocityprofile is given by v A ( s ) = − s − s + 12811840 , (21)which decays slowly enough to ensure that TM are captured with low polynomialdegrees. The Lehnert number is chosen to be Le = 10 − , corresponding to a meanmagnetic field strength of about 3 mT within Earth’s core (Gillet et al., 2010). The density of the eigen solution spectrum is shown in Figure 1 (left) for differentdegrees of truncation up to N = 29. The band limitation of the spectrum is easilyseen on the fast end of the spectrum with a similar upper end of frequencies for alltruncation degrees. The fastest mode in the spectrum approximately corresponds to | ω +1 , | ≈ . − (or 0 . | ω | <
2Ω for inertial modes (Greenspan, 1968).For MCM the slowest frequency is affected by the truncation. This is due to the factthat the convergence of magnetic modes depends on the truncation, as the Lorentzforce alters the polynomial degree of the modes, unlike the Coriolis operator (Iverset al., 2015). For a truncation N ≤
13 TM are separated from the MCM and thefast modes. At larger N some MCM are present also in the frequency range of TM.At low truncation the classification of MCM, RM and TM is straightforward by thedifference in frequencies. At higher truncation we classify the modes by their kineticand magnetic energies.The kinetic and magnetic energies are respectively given by E kin = 12 Z V u · u d V, (22) E mag = 12 Z V B · B d V. (23)At the degree of N = 29 both RM and MCM reach frequencies around theTM frequency range, but their energy ratio E kin /E mag is still different from unity(see Figure 1, right). At the considered polynomial degrees more MCM have periodscomparable to those of TM than RM. This bias can be explained by the periods ofMalkus modes as a function of n and m (compare middle Figure 2 in Labb´e et al., –7–anuscript submitted to Geophysical Research Letters . . . . . . s − . − . . . . . . . ξ ( s ) ω =2 . ω =4 . ω =6 . ω =7 . ω =9 . ω =10 . Figure 2.
The six largest scale TM of the hybrid model at N = 29 and Le = 10 − (solid col-ors) and of the 1-D equation (dashed black). n , not of m (unless m (cid:29) m and n leads to an increase of themode period towards unity. This explains why at a certain truncation level the fastestMCM is closer to the periods of TM than the slowest RM. Compared to the Malkusfield, MCM (RM) spread out to higher (lower) frequencies and higher (lower) energyratio. Given the profile of v A ( s ) we compute the TM by integrating (13) using finitedifferencing. The 1-D equation implies that ∂ξ/∂s = 0 at s = 1 and it is automaticallysatisfied in our solver, if v A (1) = 0. The selection of TM in the spectrum of modesof the hybrid solution is done by considering the frequency range indicated by the1-D solutions and by a unit ratio of kinetic to magnetic energy of the eigen solutions.The comparison between the 1-D solutions (dashed black) and the hybrid model (solidcolors) is shown in Figure 2. The six largest scale TM calculated by the hybrid modelare in excellent agreement with the 1-D solutions. The frequencies obtained fromthe two models have a relative difference of O (10 − ). We see that for both models ∂ξ/∂s = 0 at s = 1, as expected. For the hybrid model the resolution of this boundarycondition depends on the radial wave number of the TM in s , the spatial heterogeneityof B and the polynomial truncation. At a polynomial truncation of N = 29 at leastthe six largest scale TM are well resolved by the basis. The spatial structure of themagnetic field component of the TM depends only on the structure of B and thecomplexity of the TM in s (not presented here, see e.g. Cox et al., 2016). For the range of polynomial degrees studied here, RM are only slightly influencedby the presence of the magnetic field. Their spatial structure and frequency remaincomparable to that of the RM in the purely HD case. We compared the frequencies ofsome of the largest scale RM to the frequencies of the RM when including magneticforces with the Malkus field B ,M = s φ and (20). The relative differences between –8–anuscript submitted to Geophysical Research Letters
Table 1.
Non-dimensional frequency ω , dimensionalized period T in years (for | B | = 3 mT),and ratio of kinematic to magnetic energy E kin /E mag of the six slowest TM (displayed in Figure2), the three fastest RM and MCM (displayed in Figure 3). Type ω T [yr] E kin /E mag TM 2.67 10.3 0.86TM 4.39 6.2 0.94TM 6.05 4.5 0.97TM 7.68 3.6 0.98TM 9.30 2.9 0.99TM 10.92 2.5 0.99MCM1 1.72 15.91 0.02MCM2 2.66 10.28 0.03MCM3 5.95 4.60 0.13RM1 112.25 0.24 49.57RM2 118.09 0.23 12.26RM3 136.79 0.20 23.03the frequencies of the three models are below 10 − for all modes up to a truncationof N = 29. This difference can increase at N >
29, as may be anticipated by thedispersion relation (15). The spatial structure also remains mostly unchanged andtheir phase velocity is prograde, as observed in the HD case. We show the velocityand the radial magnetic field at the surface of three RM in Figure 3a. The slowestmodes are associated with m = 1 and increasingly large N (see the top mode). Thiscan be seen also by a careful analysis of the approximated dispersion relation (16).We found even slower RM in our model, but we display three modes that are wellconverged from degree N = 29 to N = 35 (see details in supplementary material). Inboth the HD and MHD case an increase of the velocity amplitudes near the equatoris observed (compare with RM in Kloss & Finlay, 2019). Their ratio of kinetic tomagnetic energy is O (10), suggesting that their surface magnetic field perturbationmight be observable in the future. MCM are strongly influenced by the background magnetic field. They are noteasily compared between different magnetic fields, e.g. between the idealized Malkusfield and the magnetic field B . Instead we focus on the MCM of relatively high fre-quency, that are of particular interest here. From Figure 1 we see that some MCMevolve on time scales similar to those of TM. We select three MCM (MCM1–3) withdimensionalized periods of a few years (see exact figures in Table 1), that show a poly-nomial complexity below that of the truncation degree and a converged structure (seesupplementary material). The spatial structure of these selected MCM is presented inFigure 3b, showing that they have a large complexity along the cylindrical radius anda relatively small azimuthal wave number. The short length scale in cylindrical radiusis even more evident in the radial profile of the azimuthal velocity in the equatorialplane, shown in Figure 3c. Similarly to the slow RM, the fast MCM concentrate theirkinetic and magnetic energy near the equator. All MCM observed here travel retro-grade, compared to the prograde direction of RM, as predicted by Hide (1966). Foran azimuthal wave number m = 2 in the equatorial band, the magnetic field pertur-bations of the three displayed MCM have a phase velocity of ωm − u A ≈ − –9–anuscript submitted to Geophysical Research Letters a)RM1 RM2 RM3b)MCM1 MCM2 MCM3c) . . . . . . s − . . . . u φ MCM1MCM2MCM3
Figure 3. a) Core surface flows (top row) and associated radial magnetic field perturbationat the surface (bottom row) of the three fastest, converged, RM (RM1–3). b) Core surface flows(top row) and associated radial magnetic field perturbation at the surface (bottom row) of threeselected MCM (MCM1–3) in the TM frequency range. Colors indicate the azimuthal velocitymagnitude with blue being prograde and red being retrograde and inward (blue) and outward(orange) magnetic flux. The arrows indicate tracers of the surface velocity. c) Cylindrical-radialprofile of the azimuthal velocity in the equatorial plane at φ = 0 of MCM1–3. The modes arecomputed for a maximum polynomial degree N = 35. The frequencies and periods of the modesare given in Table 1. –10–anuscript submitted to Geophysical Research Letters
We have shown, for Le = 10 − and a mean magnetic field strength in the coreinterior of about 3 mT, that changes of the magnetic field on periods as short as afew years could be explained by MCM. Since the periods of these fast MCM are onlya few years they may be associated with the periodic secular acceleration impulsesinferred from recent satellite observations (Chulliat et al., 2015; Kloss & Finlay, 2019;Chi-Dur´an et al., 2020). These observations have been interpreted as the signatureof MCM in the presence of a strong azimuthal magnetic field (Hori et al., 2015) or ofMAC waves in a stratified layer at the top of the core (Knezek & Buffett, 2018; Buf-fett & Matsui, 2019). We find there is no need to introduce a magnetic field strongerthan inferred from TM or a stratified layer to account for fast wave propagation inthe equatorial region of the core. A key result of our study is the presence of largehorizontal scales of B r at the core surface next to the equator associated with MCM,while the mode structure itself remains small scale in the cylindrical radial direction.Such large magnetic features near the equator should be captured by satellite observa-tions. Meanwhile, Aubert and Finlay (2019) have linked the secular variation impulsesto so-called QG Alfv´en waves arising near strongly heterogeneous magnetic fields ofbuoyant plumes in their numerical simulations. Whether or not our fast MCM arein agreement with their explanation remains to be investigated. Our model could beused to invert geomagnetic observations for such a possible excitation mechanism, asdescribed by Buffett et al. (2009) for TM.Fast MCM show a concentration of energy near the equator, similar to the slowestRM. Equatorially trapped waves have been much discussed either from observations(Chulliat et al., 2015) or from physical models (Bergman, 1993). The surface core flowcalculations of Gillet et al. (2019) also show the largest core flow acceleration patternin an equatorial belt below 10 ◦ of latitude. Concentration of energy of the modes couldbe favored by the weaker intensity of B in the equatorial region, which Knezek andBuffett (2018); Buffett and Matsui (2019) also found to be important for the focusingof MAC waves in the equatorial region of a stratified layer. A systematic study overa wider range of magnetic field geometries would be needed to make this statementquantitative.Bergman (1993) and Buffett and Matsui (2019) have shown that equatoriallytrapped MAC modes are strongly affected by damping. We haven’t included diffusionin our study and an investigation into how a diffusive layer at the top of the core mayinfluence the observed fast MCM is necessary, even though they are of large spatialscale at the equator. The new basis presented here potentially allows us to includemagnetic diffusion, at a substantial computational cost.Previously, dynamics in the bulk of the core have been linked to inverted surfaceflows, but not directly to the observed changes in the magnetic field. Being ableto associate at once MCM, as well as TM, to magnetic field changes occurring withperiods of 10 years or less and yet with large horizontal scale in an equatorial band atthe CMB opens new perspectives for data assimilation and analyses of the dynamicsoccurring in the Earth’s outer core. References
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Let us write the current density j = ∇ × B in the toroidal-poloidal expansion,so that –14–anuscript submitted to Geophysical Research Letters j = ∇ × Q r + ∇ × ∇ × S r . (A1)We can also write the magnetic field in the toroidal-poloidal expansion, with B = ∇ × T r + ∇ × ∇ × P r . (A2)It follows (Backus et al., 1996) that S = T, (A3) ∇ P = − Q. (A4)We construct the toroidal and poloidal scalars for the basis of j following thevelocity basis introduced by Ivers et al. (2015), so that Q lmn = r n R ml , (A5) S lmn = (1 − r ) r n R ml , (A6)with R ml = r l Y ml ( θ, φ ) the solid spherical harmonics. We have | m | ≤ l and l ∈ [1 , N ], n ∈ [0 , ( N − l ) / (cid:5) for the toroidal basis and l ∈ [0 , N − n ∈ [0 , ( N + 1 − l ) / − (cid:5) forthe poloidal basis. These N = N ( N + 1)(2 N + 7) elements form a complete basisfor the current density in the set of polynomial vector fields of degree N in the volume V (Ivers et al., 2015). Then, the toroidal part of B is directly given by B t,lmn = ∇ × S lmn r . (A7)For the poloidal part B p,lmn = ∇ × ∇ × P lmn r , (A8)we need to solve the Poisson equation ∇ P lmn = − Q lmn . (A9)We can use a slightly modified version of equation (3.1.9) in Backus et al. (1996) ∇ (cid:18) r n +1) R ml n + 1)(2( l + n ) + 3) (cid:19) = r n R ml , (A10)to find that P lmn = − r n +1) R ml n + 1)(2( l + n ) + 3) . (A11)It remains to ensure that the internal magnetic field can be matched to a potentialfield at the boundary. The poloidal component has to satisfy ∇ Φ i + B p = ∇ Φ e at ∂ V , (A12)with Φ i and Φ e the interior and exterior potential field, respectively. The exteriorpotential field must vanish at infinity, if the source of the magnetic field lies within theinterior. We can solve for Φ i and Φ e by considering the radial component of (A12)and the horizontal component of (A12) ∂ r Φ i − ∂ r Φ e = − B p,r , (A13a) ∇ H Φ i − ∇ H Φ e = − B p,H (A13b)Using the properties of the spherical harmonics, we find that B p,r = 1 r L P, B p,H = ∇ H (cid:18) ∂∂r ( rP ) (cid:19) , (A14) –15–anuscript submitted to Geophysical Research Letters with L P = ∂ r ( r ∂P/∂r ) − r ∇ P and the system simplifies to ∂ r Φ i − ∂ r Φ e = − L P, (A15a)Φ i − Φ e = − ∂ r ( rP ) . (A15b)Since this is linearly independent of l , m , and n we can consider this system for each B p,lmn individually, so that ∂ r Φ ilmn − ∂ r Φ elmn = − L P lmn , (A16a)Φ ilmn − Φ elmn = − ∂ r ( rP lmn ) . (A16b)Since L Y ml = l ( l + 1) Y lm and ∂ r ( rP lmn ) = 2( n + 1) + l + 12( n + 1)(2( l + n ) + 3) r n +1)+ l Y ml , (A17)we need to search for potentials of the formΦ ilmn = α lmn r l Y ml , (A18a)Φ elmn = β lmn r − ( l +1) Y ml . (A18b)The system to be solved for each l, n is then (cid:18) l l + 11 − (cid:19) (cid:18) α lmn β lmn (cid:19) = − α (cid:18) l ( l + 1)2 n + l + 3 (cid:19) , (A19)with α = 2( n + 1)(2( l + n ) + 3), for each l, n . The solutions to (A19) are α lmn = − ( l + 1)(2 l + 1)(2 n + 2) , (A20a) β lmn = l (2 l + 1)(2 l + 2 n + 3) , (A20b)so that the poloidal basis vectors are given by B p,lmn = ∇ × ∇ × P lmn r + ∇ Φ ilmn . (A21)Li et al. (2010) presented a similar basis to express the magnetic field for an in-sulating mantle, which involves slightly more complicated expressions for the poloidaland toroidal scalars with Jacobi polynomials. We have not proven any weighted or-thogonal inner products of the poloidal and toroidal scalars, but our basis vectors showthe same orthogonality for the unweighted inner product between vectors of differentharmonic order and degree as does the basis of Li et al. (2010). Orthogonal bases,based on Jacobi polynomials and spherical harmonics, have been presented in Chenet al. (2018); Li et al. (2018). These bases are desirable for reducing computationalefforts, but without an orthogonal QG basis, no such computational advantage is givenfor the hybrid model presented here. Appendix B Spherical Harmonics in Cartesian Coordinates
The unnormalized spherical harmonics are defined as˜ Y ml ( θ, φ ) = e imφ P ml (cos θ ) , (B1)with the so-called associated Legendre functions P ml ( x ) = 12 l l ! (cid:0) − x (cid:1) m/ ∂ m ∂x m P l ( x ) (B2) –16–anuscript submitted to Geophysical Research Letters with P l ( x ) = ∂ l ∂x l (cid:0) x − (cid:1) l . (B3)The unit Cartesian coordinates are given byˆ x = x/r = cos φ sin θ, (B4)ˆ y = y/r = sin φ sin θ, (B5)ˆ z = z/r = cos θ. (B6)We can rewrite (B2), so that P ml (cos θ ) = P ml (ˆ z ) = 12 l l ! (sin θ ) m ∂ m ∂ ˆ z m P l (ˆ z ) . (B7)Using the trigonometric identitiessin( mφ ) = X k odd ( − k − (cid:18) mk (cid:19) (cos φ ) m − k (sin φ ) k , (B8)cos( mφ ) = X k even ( − k (cid:18) mk (cid:19) (cos φ ) m − k (sin φ ) k , (B9)we are able to rewrite the spherical harmonics in terms of the unit Cartesian coordi-nates, so that ˜ Y ml ( θ, φ ) = ˜ Y ml (ˆ x, ˆ y, ˆ z )= 12 l l ! X k odd ( − k − (cid:18) mk (cid:19) ˆ x m − k ˆ y k + i X k even ( − k (cid:18) mk (cid:19) ˆ x m − k ˆ y k ! ∂ m ∂ ˆ z m P l (ˆ z ) . (B10)The solid spherical harmonics are˜ R ml ( x, y, z ) = r l ˜ Y ml (ˆ x, ˆ y, ˆ z ) , (B11)which is polynomial in x , y and z . Acknowledgments
The authors like to thank Henri-Claude Nataf and two anonymous reviewers for theirhelp in improving this work. FG was partly funded by Labex OSUG@2020 (ANR10LABX56). Support is acknowledged from the European Space Agency through con-tract 4000127193/19/NL/IA. This work has been carried out with financial supportfrom CNES (Centre National d’´Etudes Spatiales, France). JN was partly funded bySNF Grant https://dx.doi.org/10.5281/zenodo.4008396 . –17– EOPHYSICAL RESEARCH LETTERS
Supporting Information for ”Fast Quasi-GeostrophicMagneto-Coriolis Modes in the Earth’s core”
F. Gerick , , D. Jault , J. Noir CNRS, ISTerre, University of Grenoble Alpes, Grenoble, France Institute of Geophysics, ETH Zurich, Zurich, Switzerland
Solving the Generalized Eigen Problem
The generalized eigen problem i ω Mx = Dx , (1)is transformed to a standard eigenvalue problem by inverting M , so that λ x = Ax , (2)where λ = i ω and A = M − D . In practice this inverse is not calculated explicitly, as M − is not necessarily sparse and its calculation costly. Instead we factorize M to be able tosolve the linear problem Ax = b for x . The standard eigenvalue problem (2) can thenbe solved iteratively for the largest eigen pairs ( λ k , x k ) using the Krylov-Schur methodimplemented in the Julia programming language. The advantage of this implementationis an easy adaptation to higher accuracy floating point numbers. In our model we observe February 24, 2021, 4:11am a r X i v : . [ phy s i c s . g e o - ph ] F e b - 2 : an increase in the real part of the numerical eigen solutions for increasing polynomialdegree due to numerical inaccuracies, demonstrated in Figure S1 for the Malkus field. Wecompensate this by using quadruple precision (128bit) floating point numbers throughoutall of our computations, which are able to ensure stable enough solutions. Due to thehybrid model a large portion of the eigen solutions are degenerate modes of zero frequencyso that we choose to calculate only the non-zero modes.Another difficulty, present for magnetic fields that are not linear in x , y and z , is theconvergence of the modes. In this case spurious modes are always possible and theycan perturb the physically relevant modes significantly. We have thus to make sure thatthe calculated modes are converged. To do so, first, the frequency shouldn’t changesignificantly between different truncation degrees. Further, the velocity and magnetic fieldmust also not change more than a given threshold. We impose that the mode calculatedat a degree N , u N , must be able to be matched to a mode calculated at a higher degree N + 2, u N +2 , with a correlation R V u N · u N +2 d V R V u N · u N d V > − (cid:15), (3)with (cid:15) the allowed error.For the RM we choose (cid:15) = 0 .
01 and from a degree N = 29 to N = 35 the three modespresented in Figure 3a and S2a are the fastest modes that satisfy this constraint. For theMCM the threshold is lowered to (cid:15) = 0 .
05. This way we ensure that for N = 29 , , , February 24, 2021, 4:11am
X - 3 dominant wave number and equatorial focusing, are preserved throughout different degreesof truncation, making us confident that these modes are indeed sufficiently converged.
February 24, 2021, 4:11am - 4 : | Im( λ k ) | + (cid:15) − − − − − − − (cid:15) | R e ( λ k ) | + (cid:15) m a x ( d e g ( u k )) Figure S1.
Comparison of eigen values to the Malkus problem using a 64bit accurate solverand a 128bit accurate solver for a truncation of N = 20. We focus here on the frequency regimeof the fast RM. The solutions of the 64bit solver are colored by the polynomial degree of theirpeak amplitude basis vector. The solutions to the 128bit solver are shown as orange crosses andlie on the line of Re( λ k ) + (cid:15) ≈ (cid:15) ≈ . × − , the 64bit floating point error. February 24, 2021, 4:11am
X - 5 a)RM1 RM2 RM3b)MCM1 MCM2 MCM3
Figure S2.
Same as Figure 3 of the article with truncation degree N = 29.= 29.