Self-similarity of the dipole–multipole transition in rapidly rotating dynamos

Abstract The dipole–multipole transition in rapidly rotating dynamos is investigated through the analysis of forced magnetohydrodynamic waves in an unstably stratified fluid. The focus of this study is on the inertia-free limit applicable to planetary cores, where the Rossby number is small not only on the core depth but also on the length scale of columnar convection. By progressively increasing the buoyant forcing in a linear magnetoconvection model, the slow magnetic–Archimedean–Coriolis (MAC) waves are significantly attenuated so that their kinetic helicity decreases to zero; the fast MAC wave helicity, on the other hand, is practically unaffected. In turn, polarity reversals in low-inertia spherical dynamos are shown to occur when the slow MAC waves disappear under strong forcing. Two dynamically similar regimes are identified – the suppression of slow waves in a strongly forced dynamo and the excitation of slow waves in a moderately forced dynamo starting from a small seed field. While the former regime results in polarity reversals, the latter regime produces the axial dipole from a chaotic multipolar state. For either polarity transition, a local Rayleigh number based on the mean wavenumber of the energy-containing scales bears the same linear relationship with the square of the peak magnetic field measured at the transition. The self-similarity of the dipole–multipole transition can place a constraint on the Rayleigh number for polarity reversals in the Earth.


Introduction
The dynamo operating in the Earth's outer core generates a predominantly North-South dipole magnetic field.Occasionally, the magnetic dipole axis flips its orientation and retains its approximate alignment with the Earth's rotation axis.The last such polarity reversal occurred nearly 0.78 million years ago (Merrill 2011).Geomagnetic excursions, the periods during which the magnetic axis wanders up to 45 • from the rotation axis before returning to its original state, have been more frequent in the Earth's past.As reversals and excursions are likely to result from similar convective states of the core (Gubbins 1999;Valet et al. 2005), it is possible that the geodynamo operates in a strongly driven state below the threshold for reversals.
The first polarity reversals in numerical dynamo models were obtained by Glatzmaier & Roberts (1995a;1995b).Since then, several other studies reported reversals in comparable parameter regimes (Sarson & Jones 1999;Kutzner & Christensen 2002;Wicht & Olson 2004;Olson et al. 2009;Sreenivasan et al. 2014), and it is now well accepted that dynamo reversals occur under strong buoyancy-driven convection.Any explanation for polarity reversals must follow from an explanation for the preference for the axial dipole in planetary dynamos.The kinetic helicity u u u • ζ ζ ζ (where u u u is the velocity and ζ ζ ζ is the vorticity) generated in convection columns that arise in rapid rotation is thought to be essential for dipolar dynamo action (Moffatt 1978;Olson et al. 1999).Consequently, the loss of columnar helicity can lead to collapse of the dipole (e.g.Soderlund et al. 2012).A number of numerical dynamo models show a dipolemultipole transition for Ro ℓ ≈ 0.1, where Ro ℓ is a 'local' Rossby number that measures the ratio of nonlinear inertial to Coriolis forces on the length scale of columnar convection (Christensen & Aubert 2006).Since it has been hypothesized that the columnar vortices in rotating turbulence are formed by the propagation of linear inertial waves (Davidson et al. 2006), the above polarity transition may result from the suppression of inertial waves in the dynamo above a critical value of Ro ℓ (McDermott & Davidson 2019).Multipolar solutions may also be found when the ratio of nonlinear inertia to Lorentz forces exceeds a critical value of O(1) (Tassin et al. 2021;Zaire et al. 2022).That having been said, in the rapidly rotating limit of zero nonlinear inertia, the helicity of columnar vortices aligned with the rotation axis is considerably enhanced by the magnetic field in the dynamo (Sreenivasan & Jones 2011).It is thought that this field-induced helicity is essential for the formation of the axial dipole in planetary dynamos.Under strong buoyancy-driven convection, the magnetically enhanced columnar flow breaks down, likely causing collapse of the axial dipole (Sreenivasan & Jones 2011).Given the spatial inhomogeneity of the magnetic field (e.g.Schaeffer et al. 2017), an analysis of volume-averaged forces in the dynamo cannot possibly reveal how buoyancy offsets the Lorentz-Coriolis force balance in isolated columns.The analysis of helical dynamo waves at progressively increasing forcing can, on the other hand, provide an insight into the role of buoyancy in dipole collapse.This study distinguishes between polarity-reversing and multipolar dynamos in the inertia-free limit, where the Rossby number is small not only on the depth of the planetary core but also on the columnar length scale transverse to the rotation axis.The analysis of dynamo waves in this limit in turn places a constraint on the convective state of the dynamo that admits polarity reversals.
In convectively-driven planetary cores, isolated density disturbances generate fast and slow Magnetic-Archimedean-Coriolis (MAC) waves under the combined influence of background rotation, magnetic field and unstable stratification.The fast waves are inertial waves weakly modified by the magnetic field and buoyancy while the slow waves are magnetostrophic waves produced by localized balances between the magnetic, Coriolis and buoyancy forces (Braginsky 1967;Busse et al. 2007, pp.165-168).MAC waves in the stably stratified regions of the Earth's core, where gravity acts as a restoring force, have been an active area of research (e.g.Nicolas & Buffett 2023) .The present study, on the other hand, focuses on MAC waves in an unstably stratified medium that supports convection in a planetary dynamo.In a rapidly rotating dynamo, the fast MAC waves of frequency ∼ ω C , the frequency of linear inertial waves, exist even in the strong field state where the square of the scaled mean magnetic field, B 2 /2Ω ρ µη = O(1), where Ω is the angular velocity of rotation, ρ is the fluid density, η is the magnetic diffusivity and µ is the magnetic permeability.The intensity of slow MAC wave motions becomes comparable to that of the fast waves when the ratio of Alfvén to inertial wave frequencies, where V M is the Alfvén wave velocity based on the peak magnetic field, δ is the length scale of the buoyancy disturbance and the subscript '0' refers to the initial state of the disturbance as it is released into the flow (Varma & Sreenivasan 2022).Because of the anisotropy of the convection, the instantaneous value of ω M /ω C for parity between fast and slow wave motion would be higher than its initial value, and inferred to be ∼ 0.1 from the spherical shell dynamo models of Varma & Sreenivasan.In the energy-containing scales, given by the range of spherical harmonic degrees lesser than the mean value at energy injection, the absolute kinetic helicity in the nonlinear dynamo is nearly twice that in the equivalent nonmagnetic convection problem.This result implies that the helicity produced by the slow MAC waves in the nonlinear dynamo would be of the same magnitude as that produced by the inertial waves in nonmagnetic convection.Further, we infer that the slow waves are essential for dipole formation since the hydrodynamic The dipole-multipole transition in rapidly rotating dynamos 3 dynamo at the same parameters, where the Lorentz force is zero, does not generate the axial dipole (see also Sreenivasan & Kar 2018).Although the magnetic diffusion frequency ω η is the lowest frequency in the dynamo, small but finite magnetic diffusion can place a lower bound on the length scale that supports slow MAC waves in the energy-containing scales of the dynamo.Linear magnetoconvection analysis of a forced damped system (Sreenivasan & Maurya 2021) indicates that, for the energy of the fast and slow wave motions are comparable.(Here, ω M /ω C ∼ 0.1).Therefore, slow wave motions at length scales δ ∼ 10 km can be influential in the generation of the dipole field.At higher values of the ratio (ω η /ω C ) 0 , the slow wave energy falls below the fast wave energy, implying that scales smaller than ∼ 10 km are rapidly damped by magnetic diffusion.
The role of buoyancy in a rapidly rotating dynamo is not merely in the excitation of MAC waves.In a convective dynamo that evolves from a small seed magnetic field, the slow MAC waves are first excited when |ω M | > |ω A |, where ω A is a measure of the strength of buoyancy in an unstably stratified fluid and has the same magnitude as that of the frequency of internal gravity waves in a stably stratified fluid (Varma & Sreenivasan 2022).As the dynamo field intensity increases from its starting seed value, |ω M | progressively increases and exceeds |ω A |, after which the axial dipole eventually forms from a chaotic multipolar state.Thus, the inequality represents a large region of the parameter space where dipoledominated dynamos exist.If a rotating dynamo is subject to progressively increasing buoyant forcing, the larger self-generated fields would result in progressively larger ω M until a state is reached where |ω A | ∼ |ω M | as the field attains its highest intensity for a given rotation rate.
Here, the slow MAC waves disappear, likely causing collapse of the dipole.This transition from dipole to polarity-reversing states is dynamically similar to the transition from a multipolar state to the dipole that occurs in the growth phase of a dynamo starting from a seed field.Since ω C remains the dominant frequency while forcing is increased, the dynamo reverses polarity in a rotationally dominant regime where slow magnetostrophic waves are suppressed.Although increased forcing may result in enhanced nonlinear inertia in numerical dynamo models, we consider it unlikely that inertia has any role in polarity transitions in the Earth even for the smallest buoyancy disturbances that support MAC waves.For disturbances of size L ⊥ ≈ 15 km transverse to the rotation axis, the actual ratio of nonlinear inertia to Coriolis forces, ). Low-inertia numerical dynamos must have a small Rossby number based on the length scale of convection.The use of magnetic Prandtl number Pm = ν/η ∼ 1-10, where ν is the kinematic viscosity, is useful in realizing strong magnetic fields in numerical simulations (Sreenivasan & Jones 2006;Willis et al. 2007;Teed et al. 2015;Dormy 2016) in the inertia-less regime relevant to rotating planetary cores.Although the choice of a large Pm at moderate Ekman number E = ν/2Ω L 2 ∼ 10 −4 -10 −6 has the unphysical consequence of the viscous dissipation being at least as high as the Ohmic dissipation, the advantage derived in terms of reproducing the MAC force balance in the energycontaining scales is crucial in the understanding of wave motions in both dipole-dominated and reversing regimes.The present study analyses polarity transitions in strongly driven, low-inertia dynamos.
In §2, we consider the evolution of a buoyancy disturbance in an unstably stratified rotating fluid subject to a magnetic field.In Cartesian geometry, the axes parallel to gravity, rotation, and the magnetic field are chosen to be mutually orthogonal, which is referred to as the "equatorial toroidal configuration" by Loper et al. (2003).At times much shorter than the time scale for exponential increase of the perturbation, the relative intensities of the fast and slow MAC wave motions generated by the perturbation are studied.Apart from the dipole-dominated regime given by |ω Since the relative magnitudes of the frequencies in these inequalities do not depend on the precise orientation of the gravity, rotation and magnetic field axes, the Cartesian linear model serves as the basis for the study of the role of wave motions in polarity transitions in nonlinear dynamo models, given in §3.Here, we find that the formation of the axial dipole from a chaotic multipolar state and the collapse of the axial dipole into a polarityreversing state are dynamically similar phenomena, in that they both occur at |ω A /ω M | ∼ 1.While dipole formation requires the excitation of slow MAC waves as a dynamo evolves from a small seed magnetic field, polarity reversals occur when the slow waves are suppressed in a strongly driven dynamo.The self-similarity of the dipole-multipole transition in the inertiafree regime places a constraint on the Rayleigh number for reversals.In §4, we discuss the implications of our results for planetary cores and future work.
2. Evolution of a density disturbance under rapid rotation and a magnetic field

Problem set-up and governing equations
A localized density disturbance ρ ′ that occurs in an unstably stratified rotating fluid layer threaded by a uniform magnetic field is considered.Since ρ ′ is related to a temperature perturbation Θ by ρ ′ = −ραΘ , where ρ is the ambient density and α is the coefficient of thermal expansion, an initial temperature perturbation is chosen in the form where C is a constant and δ is the length scale of the perturbation.Figure 1 shows the initial perturbation which subsequently evolves under gravity g g g = −gê e e y , background rotation Ω Ω Ω = Ω ê e e z and a uniform magnetic field B B B = Bê e e x in Cartesian coordinates (x, y, z).In an otherwise quiescent medium, the initial temperature perturbation (2.
where ν is the kinematic viscosity, κ is the thermal diffusivity, η is the magnetic diffusivity, µ is the magnetic permeability, is the mean temperature gradient in the unstably stratified fluid.

Solutions for the velocity field
Taking the curl of equations (2.2) and the (2.3) and eliminating the electric current density between them, we obtain for the velocity field, (2.6) where V M = B/ √ µρ is the Alfvén velocity.The assumption of an unbounded domain facilitates the use of Fourier transforms along each coordinate direction, and where Application of the Fourier transform (2.7) to the Cartesian components of equation (2.6) gives (2.9) (2.10) (2.11) where we have combined the transformed temperature equation (2.4) with the transform of (2.6).
In equations (2.9)-(2.11), (2.12) represent the squares of the frequencies of linear inertial, buoyancy and Alfvén waves respectively (Braginsky 1967;Busse et al. 2007).In an unstably stratified medium, wherein ω 2 A < 0, |ω A | is a measure of the strength of buoyancy.As the present study focuses on a system where the viscous and thermal diffusion are much smaller than magnetic diffusion, the frequencies ω ν = νk 2 and ω κ = κk 2 in (2.9)-(2.11)are small compared ω η = ηk 2 .In this limit, the solution of the form ûy ∼ e iλ t for the homogeneous equation (2.10) gives the following quintic equation in λ : (2.13) the approximate roots of which were discussed in Sreenivasan & Maurya (2021) for known relative orders of magnitudes of ω M , ω A , ω C and ω η .In the solution for (2.10), (2.14) the coefficients D m are determined using the initial conditions for ûy and its time derivatives ( §2.3).Of the five terms in the expansion on the right-hand side of (2.14), two terms represent oppositely travelling fast MAC waves, two other terms represent oppositely travelling slow MAC waves, and the fifth term represents the overall growth of the velocity perturbation.
By substituting (2.14) in (2.9) and (2.11), the following solutions for (2.9) and (2.11) are obtained: (2.16) In (2.15) and (2.16), ûH x and ûH z are the homogeneous solutions of (2.9) and (2.11) respectively, ûP x and ûP z are the particular solutions, λ P m are the roots of (2.13) and λ H m are the roots of (2.13) The dipole-multipole transition in rapidly rotating dynamos which give the frequencies of the fast and slow Magneto-Coriolis (MC) waves in the absence of buoyancy (Sreenivasan & Narasimhan 2017).The coefficients A m , C m , M m and N m in (2.15) and (2.16) are evaluated as in §2.3 below.
The solutions for the induced magnetic field transforms bx , by and bz are obtained following a similar approach.

Evaluation of spectral coefficients
From (2.14), the initial conditions for ûy and its time derivatives are given by Algebraic simplifications give the right-hand sides of (2.18) in the limit of ν = κ = 0, as follows: (2.19) The coefficients D m may now be obtained using the roots of equation (2.13).For example, we obtain, for the forward-travelling fast and slow wave solutions respectively.The coefficients A m and C m are determined in a similar way.The coefficients M m and N m in equations (2.15) and (2.16) are obtained using the method of undetermined coefficients, as follows: and with (2.24)

Fast and slow MAC waves in unstable stratification
From the solution of the initial value problem, the velocity and induced magnetic field can be obtained at discrete points in time.The analysis of the solutions is limited to times much shorter than the time scale for the exponential increase of the perturbations.When the buoyancy force is small compared with the Lorentz force (|ω A /ω M | ≪ 1), the parameter regime is determined by the Lehnert number Le and the magnetic Ekman number E η , both based on the length scale δ of the initial buoyancy perturbation (2.1).
Figure 2 shows the evolution of the kinetic helicity u u u • ζ ζ ζ at y = 0.The real-space fields are obtained from the transforms û u u and ζ ζ ζ via the inverse Fourier transform (2.8).Here, a truncation value of ±10/δ is used for the three wavenumbers in the integrals since the initial wavenumber k 0 = √ 3/δ (Appendix A).Apart from the segregation of oppositely signed helicity between the two halves about the mid-plane z = 0, the evolution of blobs into columnar structures through the propagation of damped waves is evident.
As we seek to understand the role of MAC waves in the dipolar and multipolar dynamo regimes, we may separate the fast and slow MAC wave parts of the general solution, which is a linear superposition of the two wave solutions (see Sreenivasan & Maurya 2021).For example, where the subscripts f and s in the left-hand sides of (2.26) and (2.27) denote the fast and slow wave parts of the solution.Figure 3(a) shows the variation of fundamental frequencies with Le.
To compute the frequencies, the mean wavenumbers are first calculated through ratios of L 2 norms; e.g.In figure 4, the contours of the fast and slow MAC wave helicities are shown at two times for |ω M /ω C | = 0.13, which lies in the region of slow wave dominance in figure 3(b).The fast waves split in two and propagate rapidly along z.The slow waves do not propagate as far as the fast waves at the same time due to their lower group velocity.Yet, as indicated by the colour bars, the slow waves are markedly more intense than the fast waves.Both fast and slow wave columns propagate along x at the Alfvén velocity ( §2.5 below).
The effect of progressively increasing the buoyant forcing on the fast and slow waves is examined in the following section.

Effect of progressively increasing buoyancy
(2.31) As |ω A | increases relative to |ω M |, the fast MAC waves, given by frequencies λ 1,2 , are nearly unaffected.However, the slow waves, whose real frequencies are approximated by Braginsky 1967), would be significantly attenuated in an unstably stratified fluid with ω 2 A < 0 as |ω A | nears |ω M |.We see below that the decrease of the slow wave frequency translates into the marked decrease of the slow wave helicity relative to the fast wave helicity.
Figure 5(a) indicates that both fast and slow MAC waves propagate along the mean-field direction x such that x/δ = t/t a , where t a is the Alfvén wave travel time.Contrary to that in the inertial-Alfvén wave system discussed in Bardsley & Davidson (2017), the Alfvén waves propagating in the direction orthogonal to the rotation axis are simply the degenerate form of the MAC waves (see also Varma & Sreenivasan 2022).For small |ω A /ω M |, the helicity of slow wave motions is greater than that of fast waves, but as |ω A /ω M | approaches unity, the slow wave helicity weakens considerably and falls below that of the fast wave.The effect of increasing buoyancy forcing on the fast and slow wave helicity is shown graphically in figure 6.The fast waves are practically unaffected by the strength of forcing as their intensity and z propagation rate are nearly invariant for |ω A /ω M | in the range 0.1-1 (figure 6(a-c)).The slow wave helicity,   prompts us to look at the condition for vanishing slow wave helicity through a relation between Ra ℓ and a parameter Λ defined by which measures the initial ratio of the slow MC wave frequency to the magnetic diffusion frequency.Figure 7 shows that the same linear relation between Ra ℓ and Λ , whose values are tabulated in table 1, holds for any E η .Both Ra ℓ and Λ are measurable in dynamo models, the latter being of the same order of magnitude as the Elsasser number -the square of the scaled magnetic field -in many models.If the state of vanishing slow wave helicity is taken as a proxy for polarity transitions, then we may expect a self-similar relationship between Ra ℓ and Λ in low-inertia dynamos, where the nonlinear inertial force is small compared with the Coriolis force.This idea is explored further in §3.

Nonlinear dynamo simulations
We consider a convection-driven dynamo operating in a spherical shell, the boundaries of which correspond to the inner core boundary (ICB) and the core-mantle boundary (CMB).The ratio of inner to outer radius is 0.35.Fluid motion is driven by thermal buoyancy, although our formulation can also study thermochemical buoyancy using the codensity formulation (Braginsky & Roberts 1995).The other body forces acting on the fluid are the electromagnetic Lorentz force and the Coriolis force.Lengths are scaled by the thickness of the spherical shell L and time is scaled by magnetic diffusion time L 2 /η.The velocity u u u and magnetic field B B B are scaled by η/L and (2Ω ρ µη) 1/2 respectively.The temperature is scaled by β L, where β is the radial temperature gradient at the outer boundary.In the Boussinesq approximation, the non-dimensional MHD equations for the velocity, magnetic field and temperature are given by, The modified pressure p * in equation (3.1) is given by p + 1 2 E Pm −1 |u u u| 2 .The dimensionless parameters in the above equations are the Ekman number E = ν/2Ω L 2 , the Prandtl number, Pr = ν/κ, the magnetic Prandtl number, Pm = ν/η and the modified Rayleigh number gαβ L 2 /2Ω κ.Here, g is the gravitational acceleration, ν is the kinematic viscosity, κ is the thermal diffusivity and α is the coefficient of thermal expansion.
The basic-state temperature profile represents a basal heating given by T 0 (r) = r i r o /r, where r i and r o are the inner and outer radii of the spherical shell.The velocity and magnetic fields satisfy the no-slip and electrically insulating conditions respectively at the two boundaries.The inner boundary is isothermal while the outer boundary has constant heat flux.The calculations are performed by a pseudospectral code that uses spherical harmonic expansions in the angular coordinates (θ , φ ) and finite differences in radius r (Willis et al. 2007).
As in recent studies (Varma & Sreenivasan 2022), the dynamo simulations begin from a dipole seed magnetic field of small volume-averaged intensity B = 0.01.The runs are performed for at least 2 magnetic diffusion times, well into the saturated state of the dynamo.The main output parameters of the dynamo simulations, given in table 2, are time-averaged values in the saturated state.For three values of the Ekman number E, a series of simulations at progressively increasing Rayleigh number Ra are performed, spanning the dipole-dominated regime up to the onset of polarity transitions.The mean spherical harmonic degrees for convection and energy injection are defined by where E k (l) is the kinetic energy spectrum and E T (l) is the spectrum obtained from the product of the transform of u r T and its conjugate.The total kinetic and magnetic energies in the saturated dynamo are given by the volume integrals The relative dipole field strength f dip , which is the ratio of the mean dipole field strength to the field strength in harmonic degrees l = 1-12 at the outer boundary (Christensen & Aubert 2006), takes values > 0.5 in all the dipole-dominated runs (table 2.) The distinction between multipolar and polarity-reversing runs, however, is well understood from the evolution of the dipole colatitude, presented in §3.1.
For each E, the value of Pm = Pr is chosen such that the local Rossby number Ro ℓ , which gives the ratio of the inertial to Coriolis forces on the characteristic length scale of convection (Christensen & Aubert 2006) is < 0.1 (table 2).Therefore, our dynamo simulations lie in the rotationally dominant, or low-inertia, regime.While low-Pm simulations starting from a seed field can result in multipolar fields even for Ro ℓ < 0.1 (Petitdemange 2018), our choice of Pm ensures that the low-inertia dynamos are dipole dominated regardless of whether one begins from a seed field or a strong field.The dipole dominance exists for a wide range of forcing, up to Ra/Ra c ∼ 10 3 , where Ra c is the critical Rayleigh number at the onset of nonmagnetic convection.The only exception here is the last run in each Ekman number series (marked by an asterisk in table 2), where a multipolar or polarity-reversing dynamo is obtained depending on the initial field strength (see §3.1 below).
The role of the magnetic field in helicity generation is well understood by comparing a dynamo calculation with its equivalent nonmagnetic convection calculation.For Pm = Pr, the dynamo obtained by solving equations (3.1)-(3.4) is compared with its nonmagnetic counterpart, obtained by solving equations (B 1)-(B 3), Appendix B.

Dipolar, multipolar and polarity-reversing dynamos
Figure 8 shows the magnetic colatitude of the dipole field, θ at the upper boundary obtained from spherical harmonic Gauss coefficients, as follows: where g 0 1 , g 1 1 and h 1 1 are obtained from the Schmidt-normalized expansion for the scalar potential of the field (Glatzmaier 2013, pp.142-143).For E = 6 × 10 −5 and Pm = Pr =5, the evolution of θ is shown for runs at three closely spaced values of Ra in strongly supercritical convection.For Ra = 20000, the run that begins from a dipole seed field enters a chaotic multipolar state and subsequently regains dipolarity (figure 8(a)), a behaviour first noted by Sreenivasan & Kar (2018).For the same Ra, the run that begins from the saturated (strong field of mean square intensity B 2 = O(1)) state of Ra = 18000 gives a stable dipole field throughout (figure 8(d)).Both these runs saturate at nearly identical dipolar states.For Ra = 21000, the run that begins from a seed field is multipolar throughout (figure 8(b)).At the same Ra, the run that begins from a strong field produces occasional polarity reversals separated by well-defined periods of normal and reversed polarity (figure 8(e)), as in earlier reversing simulations (Glatzmaier et al. 1995;Kutzner & Christensen 2002) and experiments (Monchaux et al. 2009).The radial magnetic fields at the outer boundary before and after a reversal are shown in figure 9.For still higher forcing (Ra = 24000), both seed field and strong field runs give multipolar solutions (figure 8(c) & (f)).Therefore, polarity reversals happen in a narrow range of Ra that lies between dipolar and multipolar regimes, possibly due to variations in outer boundary heat flux.
The range of spherical harmonic degrees l l E is of particular interest since kinetic helicity is known to be generated in the nonlinear dynamo in this range of energy-containing scales (see §1).A relative helicity is defined that measures the augmentation of lower-hemisphere helicity in the nonlinear dynamo (magnetic) run relative to that in the equivalent nonmagnetic simulation.
for l l E .Here, the subscripts m and nm denote the magnetic (dynamo) and nonmagnetic values respectively.The variation of H r with time in dynamos at E = 6 × 10 −5 and progressively increasing Ra is given in figure 10.The runs that begin from a seed field and produce an axial dipole show an approximately two-fold increase in helicity (figures 10(a) & (b)).For Ra = 21000, the run that begins from a seed field and produces a multipolar solution shows no noticeable increase in helicity over the nonmagnetic state (figure 10(c)).At the same Ra, the run that begins from a strong field and produces polarity reversals undergoes a decrease in helicity from its initial value to the nonmagnetic value (figure 10(d)).We emphasize that the effect of the magnetic field on helicity generation can be experienced only in the energy-containing range of harmonic degrees l l E .If the entire spectrum is considered (Ranjan et al. 2020), the dipolar dynamo helicity would be lower than the nonmagnetic helicity.
The results in figure 8 and figure 10 prompt us to examine the nature of wave motions in dipolar and reversing dynamos.An important aim of this study is to show through the analysis of MAC waves that the formation of the dipole from a multipolar state and the onset of polarity reversals lie in similar regimes.Such an argument is essential to place a constraint on the Rayleigh number that admits polarity reversals (see §3.3).

The suppression of slow MAC waves in polarity-reversing dynamos
Isolated density disturbances in a rotating stratified fluid layer excite MAC waves whose frequencies depend on the fundamental frequencies ω M , ω A and ω C , representing Alfvén waves, internal gravity waves and linear inertial waves respectively.In unstable stratification that drives planetary core convection, ω 2 A < 0, where |ω A | is simply a measure of the strength of buoyancy.Since the dimensional frequencies ω 2 M , −ω 2 A and ω 2 C in the dynamo are given by (Varma & Sreenivasan 2022) and scaling the frequencies by η/L 2 , we obtain in dimensionless units, where k s , k φ and k z are the radial, azimuthal and axial wavenumbers in cylindrical coordinates (s, φ , z), k φ = m/s, where m is the spherical harmonic degree, and Here, ω A is evaluated on the equatorial plane where the buoyancy force is maximum.The magnetic (Alfvén) wave frequency ω M is based on the three components of the measured magnetic field at the peak-field location.The wavenumber k φ is evaluated at s = 1, approximately mid-radius of the spherical shell.
In the limit of zero magnetic diffusion (ω η = 0), the characteristic equation (2.13) reduces to (3.11) the roots of which are (Sreenivasan & Maurya 2021), λ 1,2 and λ 3,4 give the frequencies of the fast (ω f ) and slow (ω s ) MAC waves respectively.While the fast waves are linear inertial waves weakly modified by the magnetic field and buoyancy, the slow waves are magnetostrophic.
In figure 11, the magnitudes of the fundamental frequencies are shown as a function of the spherical harmonic order m in the saturated state of the dynamo run at E = 6 × 10 −5 , and Pr = Pm = 5.The frequencies are computed from (3.10) using the mean values of the s and z wavenumbers.For example, real space integration over (s, φ ) gives the kinetic energy as a function of z, the Fourier transform of which gives the one-dimensional spectrum û2 (k z ).Subsequently, we obtain The frequency root λ 3 in (3.13), shown by the black lines, represents the slow MAC waves of frequency ω s when the inequality Evidently, the scales of helicity generation in the nonlinear dynamo overlaps with the scales where the slow MAC waves are generated.The range of m over which the above frequency inequality holds narrows down with Ra, and for the polarity-reversing dynamo with Ra = 21000, this inequality does not exist at any m (figure 11(d)).
Figures 11 (a)-(d) also show the spectral distribution of the power supplied to the poloidal part of the axial dipole field B P 10 , given by (e.g.Buffett & Bloxham 2002) where u u u and B B B share the same value of m (Bullard & Gellman 1954).The axial dipole is predominantly generated in the wavenumbers where the MAC waves are generated, except in the dynamo close to reversals (figure 11 c) where the peak of P 10 lies outside the MAC wave window.In the reversing dynamo without the MAC wave window, the power supplied to the dipole is small.In figure 12, the upper panels show the absolute values of the frequencies for a range of spherical harmonic order m at three different times during the evolution of the dynamo from a seed field.The lower panels show the m-distribution of the power supplied to the axial dipole.As the magnetic field increases from a seed, the slow MAC waves of frequency ω s are absent at early times but are subsequently excited in the scales where the inequality Kinetic helicity is generated in these scales, where a peak of the axial dipole power P 10 is also noted.In the growth phase of the dynamo where the field is multipolar, the dominant contribution to P 10 comes from higher wavenumbers of u u u and B B B within the MAC wave window (figure 12 b); here helicity is generated via slow MAC waves at the peak field locations (figures S1 and S2 in Supplementary Material).Once the axial dipole is established, it can hold itself up (Sreenivasan & Jones 2011) by inducing the generation of slow waves (figure 12 c).
In figure 13, the dynamo frequencies are computed from (3.10) using the mean values of the s, φ and z wavenumbers in the saturated state of the dynamo in the range l l E , which represents the energy-containing scales.The mean spherical harmonic order m is evaluated through a weighted average as in (3.14), but over the range of m within l l E .As the field increases from a small seed value in the dipolar dynamo run at Ra = 3000 (figure 13  field is much smaller in the multipolar regime of Ra = 21000 than in the stable dipole regime of Ra = 3000 (figure 13(d)), which suggests that the slow MAC waves have an important role in the formation of the axial dipole.In the run at Ra = 21000 that begins from a strong field and produces polarity reversals, the dipole intensity decreases and eventually reaches the same order of magnitude as that in the multipolar run.This is consistent with the suppression of the slow waves at this Ra, and reflected in the decrease in kinetic helicity to the nonmagnetic value (figure 10(d)).
Figure 14 shows the measurement of wave motion in the saturated state of dynamos at E = 6 × 10 −5 and Pr = Pm = 5 and three values of Ra spanning the dipole-dominated regime and reversals.Contours of uz at cylindrical radius s = 1 are plotted over small time windows in which the ambient magnetic field and wavenumbers are approximately constant.These contours show the propagation paths of the fluctuating z velocity.In line with the discussion so far, the measurement of axial motions is limited to the energy-containing scales l l E , with no restriction on the wavenumber.The measured axial group velocity of the waves, U g,z -obtained from the slope of the black lines in figure 14 -is compared with the estimated fast (U f ) and slow (U s ) group velocities obtained by taking the derivatives of the respective frequencies in (3.12) and (3.13) with respect to k z (table 3).The theoretical frequencies ω f and ω s are estimated using the three components of the magnetic field at the peak-field location and the mean values of k s , k z and m over the range of energy-containing scales, l l E .In the dipole-dominated run at Ra = 6000, the slow and fast waves coexist, but the intensity of slow wave motions measured at the peak-field locations is at least as high as that of the fast wave motions.At Ra = 20000, the increasing occurrence of the fast waves alongside the slow waves is noted from the nearly vertical propagation paths (figure 14(b)).The measured slow wave group velocity at Ra = 20000 is greater than that at Ra = 6000, which reflects the larger self-generated field at the higher D. Majumder, B. Sreenivasan and G. Maurya Ra Ra/Ra c N r l max Rm Ro Table 2: Summary of the main input and output parameters in the dynamo simulations considered in this study.Here, Ra is the modified Rayleigh number, Ra c is the modified critical Rayleigh number for onset of nonmagnetic convection, N r is the number of radial grid points, l max is the maximum spherical harmonic degree, Rm is the magnetic Reynolds number, Ro ℓ is the local Rossby number, l C and l E are the mean spherical harmonic degrees of convection and energy injection respectively (defined in (3.5)), m is the mean spherical harmonic order in the range l l E , ks and kz are the mean s and z wavenumbers in the range l l E , E k and E m are the time-averaged total kinetic and magnetic energies defined in (3.6), f dip is the relative dipole field strength, Ra ℓ is the local Rayleigh number defined in (3.16) and Λ is the peak Elsasser number obtained from the square of the measured peak field at the earliest time of excitation of slow MAC waves in the dynamo run starting from a small seed field.* The last run in each Ekman number series is a polarity-reversing dynamo, for which Λ is the square of the measured peak field when slow MAC waves cease to exist in the run starting from the saturated state of the penultimate run in that series (see also §3.3).Rayleigh number.In the reversing dynamo at Ra = 21000, slow waves are totally absent while the fast waves are abundant (figure 14(c)).
The dominance of the slow MAC waves in the dipole-dominated dynamo and the fast MAC waves in the reversing dynamo is further evident in figure 15, where the fast Fourier transform (FFT) of uz is shown.The flow largely consists of waves of frequency ω ∼ ω s in the dipolar dynamo (figure 15(a)), whereas in the reversing dynamo, waves of much higher frequency ω ∼ ω f are dominant (figure 15(b)).The coloured vertical lines in figures 15(a) and (b) give the estimated magnitudes of ω C and ω M normalized by ω f , where ω M is based on the peak magnetic field.

Self-similarity of the dipole-multipole transition
In the presence of small but finite magnetic diffusion, the slow MAC waves are known to disappear in an unstably stratified medium for |ω A /ω M | ≈ 1.The same condition must hold for  3. and (b) correspond to ω/ω f = 1 and the dashed vertical line in (a) corresponds to ω ⋆ s = ω s /ω f , where ω f and ω s are the estimated fast and slow MAC wave frequencies.The coloured vertical lines give ω C and ω M , both normalized by ω f .The Rayleigh number in the simulation is given above each panel.The other dynamo parameters are E = 6 × 10 −5 and Pm = Pr =5.
the appearance of slow waves in a medium where the magnetic field progressively increases from a small value.In simulations starting from a small seed field, the earliest time of excitation of the slow MAC waves is noted from group velocity measurements at closely spaced times during the growth phase of the dynamo.The peak Elsasser number Λ = B 2 peak at this time is obtained from the three components of the field at the peak-field location, and presented in the last column of table 2. In each of the three dynamo series considered in this study, the last run is a polarityreversing dynamo, for which Λ is the measured peak Elsasser number when slow MAC waves cease to exist in the run starting from the saturated (strong field) state of the penultimate run in that series.Figure 16(a) shows that the variation of Ra with Λ in the three dynamo series is nearly linear.The Rayleigh number corresponding to reversals (at which slow MAC waves disappear) also lies on this line, indicating that the appearance and disappearance of MAC waves are in similar regimes.
Following the analysis in figure 7, where the Rayleigh number based on the length scale of the buoyant perturbation was studied, we define a local Rayleigh number in the dynamo, where m is the mean spherical harmonic order evaluated over m within the energy-containing scales l l E .The behaviour of Ra ℓ , which is defined for the scales where the MAC waves are excited by buoyancy, is approximately self-similar (figure 16 The fact that the linear variation of Ra ℓ with Λ demarcates the boundary between dipolar and multipolar states is evident by traversing the sections 1 and 2 marked on the Ra ℓ line from left to right (figure 16(b)).In practice, this is done by following the evolution of the dynamo from a small seed field.Figure 17(a) shows the section 1 within dashed vertical lines, where |ω M | crosses |ω A |.The variation of the dipole colatitude θ , shown in figure 17(b), indicates a multipolar field until this crossing, and a stable dipole thereafter.While the flow is predominantly made up of fast MAC waves of frequency ω ∼ ω f before the transition, the slow waves of frequency ω ∼ ω s are dominant after the transition (figures 17(c) & (d)).The multipole-dipole transitions are further evident in the contour plots of the radial magnetic field at the outer boundary, given in figure 18 for sections 1 and 2 marked in figure 16(b).

Concluding remarks
The present study investigates the dipole-multipole transition in rotating dynamos through the analysis of MHD wave motions.The limit of small Rossby number, based not only on the planetary core depth but also on the length scale of core convection, is considered.In this inertia-free limit, the dynamo polarity depends on the relative magnitudes of ω M and ω A , which in turn depend on the peak intensity of the self-generated field and the strength of buoyant forcing in the unstably stratified fluid layer.The linear magnetoconvection model shows that the slow MAC (magnetostrophic) waves have greater helicity than the fast MAC waves for the regime thought to be relevant to dipole-dominated dynamos.Although buoyancy-induced helicity can be spatially segregated about the equator by linear inertial waves in the absence of the magnetic field (Davidson & Ranjan 2018), this purely ), with Λ .The values of Ra, Ra ℓ and Λ in the plots are given in table 2. The sections 1 and 2 marked on the self-similar line are analysed further in figures 17 and 18 below.hydrodynamic process is not likely to generate the axial dipole in rotating dynamos, where the helicity of slow MAC waves is at least as high as that of the fast waves of frequency ∼ ω C , which already exist in the seed field state (figure S2 in Supplementary Material).Moreover, the time scale in which the slow waves establish the dipole field can be a significant fraction of the magnetic diffusion time (see figures 10(a) & (b)), whereas the fast waves would have to form the dipole on a much shorter time scale.Since the slow MAC waves are produced by magnetostrophic balances at the peak field locations, the analysis of wave motions with ω M based on the peak field intensity is performed here rather than a scale-dependent analysis of volume-averaged forces (e.g.Schwaiger et al. 2019).The present study shows that the condition of vanishing slow waves defines polarity transitions in rapidly rotating dynamos.The variation of the local Rayleigh number Ra ℓ with the Elsasser number Λ at the transition is very similar in the linear magnetoconvection and dynamo models, which is remarkable given that the Alfvén frequency ω M is determined by the imposed field in the linear model and by the self-generated field in the dynamo model.
While the onset of slow magnetostrophic waves for | is known to produce the axial dipole from a chaotic multipolar state (see figures 17(a,b) & 18), the suppression of slow waves for the same condition (figure 13(c)) leads to polarity reversals in strongly driven dynamos.This study differentiates between polarity-reversing and multipolar solutions, both of which onset at the same Rayleigh number.Polarity reversals result from the suppression of slow waves that already exist in a strong field state of mean square intensity O(1), whereas the multipolar solution forms from a seed field when the forcing is sufficiently strong that the slow waves are never excited.Multipolar solutions are also obtained at stronger forcing, regardless of whether one begins from a seed field or a strong field.
The critical ratios of forces and energies in the dynamo have been instrumental in setting criteria for polarity transitions.Earlier studies (Olson & Christensen 2006;Christensen & Aubert 2006) have proposed a critical value ≈ 0.1 for the local Rossby number Ro ℓ , above which the dipole-multipole transition occurs.However, this condition relies on the finiteness of nonlinear inertia on the scale of the buoyancy perturbations, which we consider unlikely in rapidly rotating planetary cores.While the condition of the energy ratio E k /E m > 1 (table 2) by itself is admissible as a criterion for the dipole-multipole transition (Kutzner & Christensen 2002;Tassin et al. 2021), the increase of this ratio is likely a consequence of the transition rather than a cause for it.Polarity reversals are never obtained for E k /E m < 1, so it is not certain that the reversals in the present study are Earth-like.
The self-similarity of polarity reversals in the inertia-free regime can place a useful constraint on the Rayleigh number that admits reversals in the Earth.For Ra ℓ ∼ 10 4 , the classical Rayleigh number R = Ra/E ∼ 10 17 , taking a turbulent viscosity ν ≈ η and a plausible ratio of core depth to the convective length scale, L/δ ∼ 10 2 .That having been said, the Earth's core is thought to convect in response to large lateral variations in lower-mantle heat flux (e.g.Olson et al. 2015;Mound et al. 2019), which can potentially reduce the value of Ra for reversals.Our understanding of the convective regime for polarity reversals is not complete, but the idea that polarity transitions are self-similar would eventually lead to improved parameter constraints for reversals.

Figure 1 :
Figure 1: The initial state of a density perturbation ρ ′ that evolves in an unstably stratified fluid subject to a uniform magnetic field B B B = Bê e e x , background rotation Ω Ω Ω = Ω ê e e z and gravity g g g = −gê e e y in Cartesian coordinates (x, y, z).
1) gives rise to a velocity field u u u, which in turn interacts with B B B to generate the induced magnetic field b b b.The initial velocity u u u 0 and induced field b b b 0 are both zero.In the Boussinesq approximation, the following magnetohydrodynamic (MHD) equations give the evolution of u u u, b b b and Θ :
.28) which are based on the velocity field.The values of |ω M /ω C |, given in brackets below the horizontal axis of figure 3(b), are systematically higher than that of Le, which is essentially the initial value of this ratio.The enhanced instantaneous value of |ω M /ω C | is due to the anisotropy of the columnar flow, and would not be evident if Ω Ω Ω were aligned with B B B(Sreenivasan & Maurya 2021).For E η = 2 × 10 −5 , all calculations for Le > 2 × 10 −3 satisfy the inequality|ω C | > |ω M | > |ω A | > |ω η |,thought to be essential for axial dipole formation in convective dynamos.Figure3(b)shows the variation of the dimensionless helicity h * of the fast and slow MAC waves, obtained by summing the helicity at all points in (x, z) for z < 0 and y = 0 and then normalizing this value by the nonmagnetic helicity.For Le > 2 × 10 −3 (|ω M /ω C | > 0.045), the slow wave helicity increases dramatically, and for Le ∼ 10 −2 (|ω M /ω C | ∼ 0.1), the slow wave helicity is greater than the fast wave helicity.This result is consistent with the higher intensity of the slow wave motions relative to that of the fast wave motions for |ω M /ω C | ∼ 0.1, inferred from the fast Fourier transform (FFT) spectra in dynamo simulations(Varma & Sreenivasan 2022).
D.Majumder, B. Sreenivasan and G. Maurya Figure 3: (a) Variation of absolute values of frequencies with Le.(b) Variation with Le of the total kinetic helicity in the region z < 0 normalized by the nonmagnetic helicity.The values of |ω M /ω C | that correspond to the values of Le are given in brackets below the horizontal axis.All calculations are performed for E η = 2 × 10 −5 .The slow wave frequency, ω s takes non-zero values for Le > 2 × 10 −3 , when |ω M | > |ω A |.

Figure 4 :
Figure 4: Helicity on the section y = 0 of fast (a, b) and slow (c,d) MAC waves at two times, t/t η = 2.5 × 10 −3 (left panels) and t/t η = 1 × 10 −2 (right panels).Here, E η = 2 × 10 −5 and Le = 0.03 (|ω M /ω C | = 0.13).on the other hand, is substantially weakened as |ω A /ω M | increases in the same range (figure 6(d-f)).For |ω A /ω M | ≈ 1, a state is reached where the slow wave helicity is nearly zero.The induced magnetic field b z also weakens considerably with increasing |ω A /ω M | (figure 6(g-i)), which indicates that only the slow waves have a direct bearing on field generation.Figure 5(b) shows the normalized slow wave helicity against a 'local' Rayleigh number based on the length scale of the initial perturbation,

Figure 5 :
Figure 5: (a) Variation of the fast wave (dashed line) and slow wave (solid line) helicity for z < 0, normalized by the non-magnetic helicity, with x/δ and |ω A /ω M | at different times (measured in units of the Alfvén wave travel time t a ).Here, E η = 2 × 10 −5 and Le = 0.03.(b) Variation of the normalized slow wave helicity with the local Rayleigh number Ra ℓ (defined in (2.33)) and |ω A /ω M | for different Le.

Figure 8 :
Figure 8: Evolution of dipole colatitude with magnetic diffusion time for three closely spaced values of Ra in strongly driven dynamo simulations.The top panels (a)-(c) are for simulations starting from a seed magnetic field while the bottom panels (d)-(f) are for simulations starting from a strong magnetic field.For Ra = 21000, the superscript † denotes a seed field start whereas the superscript ‡ denotes a strong field start.The other dynamo parameters are E = 6 × 10 −5 , Pm = Pr = 5.

Figure 9 :Figure 10 :
Figure 9: The contours of the radial magnetic field at the outer boundary for Ra = 21000 at magnetic diffusion times (a) t d = 1.1 and (b) t d = 2.3 shown in figure 8(e).The other dynamo parameters are E = 6 × 10 −5 , Pm = Pr = 5.(a) (b)

Figure 11 :
Figure 11: The upper panels of (a)-(d) show the absolute values of wave frequencies plotted for the saturated state of the dynamo.The dashed vertical lines show the upper boundary of the range of wavenumbers m for which the the helicity of the dynamo run is greater than that of the equivalent nonmagnetic run.The lower panels of (a)-(d) show the spectral distribution of the power supplied to the axial dipole, defined in (3.15).The modified Rayleigh number Ra of the model is given above each panel.The other dynamo parameters are E = 6 × 10 −5 , and Pm = Pr = 5.
.14) A similar approach gives ks .The computed frequencies in figure 11(a)-(c), shown for dynamos with Ra = 6000-18000, satisfy the inequality |ω C | > |ω M | > |ω A | in a range of the spherical harmonic order m.The dashed vertical lines show the value of m below which the helicity in the nonlinear dynamo is greater than that in the nonmagnetic run at the same parameters.
(a)), slow MAC waves of frequency ω s are first excited when |ω M | > |ω A |.The formation of the axial dipole from a chaotic field, marked by the dotted vertical line, follows slow wave excitation.In the run at Ra = 21000 that begins from a seed field and produces a multipolar solution, |ω M | remains lower than |ω A | throughout (figure 13(b)), so the slow waves are never excited.From the variation of the frequencies with increasing strength of forcing, given in figure 13(c), we note that |ω M | falls below |ω A | at Ra ≈ 21000, which indicates that polarity reversals would indeed onset in the regime |ω M | ≈ |ω A | when slow MAC waves disappear.The volume-averaged mean square value of the axial dipole

Figure 12 :
Figure 12: The upper panels of (a)-(c) show the absolute values of wave frequencies plotted for three different times in the growth phase of a dynamo starting from a seed field.The shaded grey area shows the range of scales where the helicity of the dynamo run is greater than that of the equivalent nonmagnetic run.The lower panels of (a)-(c) show the spectral distribution of the power supplied to the axial dipole, defined in equation (3.15).The dynamo parameters are Ra = 18000, E = 6 × 10 −5 , and Pm = Pr = 5.

Figure 13 :
Figure 13: (a) & (b) Absolute values of the measured frequencies ω M , ω A , ω C and ω s plotted against time (measured in units of the magnetic diffusion time, t d ).These simulations study the evolution of the dynamo starting from a small seed magnetic field.The modified Rayleigh number Ra is shown above each panel.The dotted vertical line in (a) marks the time of formation of the axial dipole from a multipolar field.(c) Frequencies in the saturated dynamo shown against Ra.The dashed vertical line here gives the value of Ra at which the slow MAC wave frequency ω s goes to zero.(d) The Elsasser number of the axial dipole field component, based on its root mean square value, for Ra = 3000, Ra = 21000 † (seed field start) and Ra = 21000 ‡ (strong field start).The dynamo parameters are E = 6 × 10 −5 , Pm = Pr = 5.The colour codes in (a) are also used in (b) and (c).

Figure 14 :
Figure 14: Contour plots of ∂ u z /∂t at cylindrical radius s = 1 for l l E and small intervals of time in the saturated state of three dynamo simulations.The parallel black lines indicate the predominant direction of travel of the waves and their slope gives the group velocity U g,z .The Rayleigh number Ra of the simulation is given above each panel.The other dynamo parameters are E = 6 × 10 −5 , Pr = Pm = 5.The estimated group velocity of the fast and slow MAC waves (U f and U s respectively) and U g,z are given in table 3.
Figure 15: (a) FFT spectrum of ∂ u z /∂t at cylindrical radius s = 1 for the scales l l E .The spectra are computed at discrete φ points and then averaged azimuthally.The solid vertical line in (a) and (b) correspond to ω/ω f = 1 and the dashed vertical line in (a) corresponds to ω ⋆ s = ω s /ω f , where ω f and ω s are the estimated fast and slow MAC wave frequencies.The coloured vertical lines give ω C and ω M , both normalized by ω f .The Rayleigh number in the simulation is given above each panel.The other dynamo parameters are E = 6 × 10 −5 and Pm = Pr =5.
(b)).The values of Ra ℓ at the onset of polarity reversals, where the slow MAC waves disappear, also lie on the same selfsimilar branch.While the conventional Rayleigh numbers Ra at the onset of reversals in the three dynamo series lie far apart (see the filled symbols in figure16(a)), the respective local Rayleigh numbers Ra ℓ are remarkably close, and ∼ 10 4 (see the filled symbols in figure16(b) and table 2).The magnetic Ekman number based on m in the three dynamo series takes values of E η ∼ 10 −5 for a wide range of Ra, which indicates an energy-containing length scale ∼ 10 km for Earth (see §1).
Figure16: (a) Variation of the modified Rayleigh number Ra with the peak Elsasser number Λ (square of the peak magnetic field) at both excitation and suppression of slow MAC waves.The hollow symbols represent the states where slow waves are first excited as the dynamos evolve from a small seed field; the filled symbols represent the states where slow waves are suppressed in the polarity-reversing dynamos.The parameters of the three dynamo series and their symbolic representations are as follows: E = 3 × 10 −4 , Pm = Pr = 20 (circles), E = 6 × 10 −5 , Pm = Pr = 5 (triangles), E = 1.2 × 10 −5 , Pm = Pr = 1 (diamonds).(b) Variation of the local Rayleigh number Ra ℓ , defined in (3.16), with Λ .The values of Ra, Ra ℓ and Λ in the plots are given in table 2. The sections 1 and 2 marked on the self-similar line are analysed further in figures 17 and 18 below.

Figure 17 :
Figure 17: (a) Evolution the dynamo frequencies in a simulation beginning from a small seed magnetic field at E = 1.2 × 10 −5 , Pm = Pr = 1, Ra = 4000.The two dashed vertical lines at t d = 0.55 and t d = 0.66 represent the end points of section 1 marked in figure 16(b) with peak Elsasser numbers Λ = 32 and 56 respectively.(b) Evolution of the dipole colatitude in the above simulation.(c) & (d) FFT spectra of ∂ u z /∂t at cylindrical radius s = 1 for the scales l l E at Λ = 32 and 56 respectively.The spectra are computed at discrete φ points and then averaged azimuthally.In (d), ω * s = ω s /ω f , where ω f and ω s are the estimated fast and slow MAC wave frequencies.

Table 1 :
Values of Ra ℓ , defined in (2.33), at different Λ , defined in (2.34), for suppression of slow MAC waves in the linear magnetoconvection calculations.The dimensionless parameters Le and E η are defined in (2.25a,b).Figure 7: Variation of the local Rayleigh number Ra ℓ , defined in (2.33), shown against Λ , defined in (2.34), for the state of approximately zero slow MAC wave helicity.Three values of E η are considered -the diamonds represent