2.1. A simple example: Kolmogorov’s universal turbulence

We first illustrate the construction rules of 𝜏–ℓ diagrams with the simple example of Kolmogorov’s universal turbulence [Kolmogorov 1941]. Although this is not the kind of turbulence we expect in planetary cores, we pick a range of length-scales and timescales typical of the Earth core. Figure 1 is a log–log plot of timescales spanning a range from 10 s to 32,000 years versus length-scales from 1 cm to R_o = 3480 km, the radius of the core.

3.1. Physical phenomena and their 𝜏–ℓ expressions

Table 1 gives the expressions of major 𝜏(ℓ) times pertaining to planetary cores. 𝜏_𝜈(ℓ) and 𝜏_u(ℓ) times have been introduced in Section 2.1. Convection adds thermal diffusion and buoyancy scales. Rotation, spherical boundaries, and magnetic field contribute key additional timescales.

We discuss the origin and meaning of these various 𝜏(ℓ) scales, and illustrate the 𝜏–ℓ template they provide in Earth’s core example (Figure 2), using its properties presented in Section 3.3. We first ignore the magnetic field and build the template of Figure 2a.

3.2. Dimensionless numbers

In order to connect to the huge literature pertaining to geophysical and astrophysical fluid dynamics, it is important to relate our 𝜏–ℓ regime diagrams to widely used dimensionless numbers. Dimensionless numbers provide the minimum number of parameters needed to describe a physical system. They permit a comparison of widely different systems that yield the same dimensionless numbers. These numbers are dimensionless combinations of properties and field variables that appear when the equations governing the dynamics of the system under study are made dimensionless by normalizing their various terms by “typical scales”.

For example, Reynolds number for a system of size L will be written: ℜ = U L∕𝜈, where U is a typical fluid velocity, and 𝜈 kinematic viscosity. Usually, it is when this dimensionless number is of order 1 that a change of regime occurs. In this example: a change between a regime where momentum diffusion dominates over advection when ℜ < 1 to one where advection dominates for ℜ > 1.

Most dimensionless numbers can be written as the ratio of two times. In our approach, we define length-scale dependent dimensionless numbers, constructed as the ratios of the timescales of the relevant physical phenomena at that length-scale. We thus define ℓ-scale Reynolds number as: ℜ(ℓ) =𝜏 _𝜈(ℓ)∕𝜏_u(ℓ), where 𝜏_𝜈(ℓ) is momentum diffusion timescale at length-scale ℓ, while 𝜏_u(ℓ) is the overturn time of a vortex of radius ℓ. Table 2 gives the expressions of ℓ-scale dimensionless numbers pertaining to planetary liquid core dynamics.

In 𝜏–ℓ regime diagrams, the intersection of the 𝜏_x(ℓ) and 𝜏_y(ℓ) lines of physical phenomena x and y occurs where ℓ-scale dimensionless number Z(ℓ) =𝜏 _y(ℓ)∕𝜏_x(ℓ) equals 1. Each such intersection marks a change in the system’s dynamic regime.

In Figures 2a and 2b, we have labeled several line intersections, where specific dimensionless numbers equal 1. The intersections of line t_𝛺 and the 𝜈, 𝜅 and 𝜂 lines indicate where corresponding ℓ-scale Ekman numbers equal 1 in the 𝜏–ℓ plane. The intersection of the Alfvén and magnetic diffusion lines defines where the ℓ-scale Lundquist number equals 1, marking a change from propagating Alfvén waves at larger scales to damped waves at smaller scales. Similarly, the intersection of the Alfvén and t_𝛺 lines yields 𝜆(ℓ) ∼ 1, where 𝜆 is the Lehnert number [Lehnert 1954, Jault 2008]. System rotation favors quasi-geostrophic Alfvén waves at timescales above this intersection.

More dimensionless numbers, such as ℜ, Rm, Ra, Ro, Ro_ff, and 𝛬_d, will appear when we plot lines 𝜏_𝜌(ℓ), 𝜏_u(ℓ) and 𝜏_b(ℓ) of the system’s density, velocity and magnetic fields for the different turbulence scenarios we will explore.

3.3. A word on Earth’s core properties

Core properties used to build the 𝜏–ℓ diagrams of Figure 2 are listed in Table 3. Most are taken from Peter Olson’s review in Treatise on Geophysics [Olson 2015]. Some of them are known with great precision (to about 1‰ for core radius R_o and liquid core mass M_o), but others are poorly constrained (to about 1 or 2 orders of magnitude for viscosity 𝜈 and compositional diffusivity 𝜒). In addition, most physical properties are expected to vary with radius. None of these (important) subtleties are taken into account in the “fuzzy” approach we advocate for building 𝜏–ℓ regime diagrams. Note that we systematically drop all numerical prefactors, including 2π.

As noted in Section 1, the power available to drive the dynamics of the system under study is a key ingredient. It largely controls the different turbulence regimes the system will experience. Thermal evolution of the Earth has received considerable attention (see Nimmo [2015], Landeau et al. [2022], Driscoll and Davies [2023] for reviews). It is now well established that the dynamics of Earth’s core today is powered by its slow cooling, enhanced by the resulting growth of the solid inner core. As iron-nickel alloy crystallizes at its surface, it releases latent heat and light elements that drive convection and power the geodynamo.

Despite uncertainties on isentropic heat flux, the available convective power is found to be in the range 0.8–5 × 10¹² W for present-day core [Nimmo 2015, Landeau et al. 2022]. We adopt value 𝓟 diss = 3 TW. This value is pointed by a blue arrow on the Rossby line in Figure 2a, and by a red arrow on the magnetic dissipation line in Figure 2b.

The last two rows in Table 3 are not used to build 𝜏–ℓ diagrams, but instead to test their relevance. Large-scale vortex turnover time t u obs ≃ 3 0 0 years is retrieved from core flow inversions of magnetic secular variation [e.g., Pais and Jault 2008]. Large-scale Alfvén wave propagation time t b obs ≃ 4 years is deduced from the discovery and analysis of “torsional oscillations” in Earth’s core [Gillet et al. 2010].

It is also observed that the Lowes–Mauersberger spectrum of magnetic energy is flat at the core-mantle boundary up to harmonic degree 10 [e.g., Langlais et al. 2014]. This means that energy is independent of length-scale in this scale-range, which translates into a 𝜏_b(ℓ) ∝ ℓ^3∕2 trend at large-scale (see Appendix A.2.1). Similarly, core flow inversions favor an almost flat harmonic spectrum of kinetic energy up to degree 10 [Roberts and King 2013, Aubert 2013, Gillet et al. 2015, Baerenzung et al. 2016]. These trends are sketched by colored disks labeled t u obs and t b obs in Figure 2.

4.1. 𝜏–ℓ expression of force balances in rotating convection

Let us start from the Navier–Stokes equation for deviations from hydrostatic equilibrium in an incompressible fluid under the Boussinesq approximation:

We already recalled in Section 3.1.4 that Equation (15) reduces to geostrophic equilibrium (8) when Coriolis acceleration strongly dominates, yielding Proudman–Taylor constraint (9). However, this equation is a diagnostic equation that can’t be used on its own, as gets clear when Equation (15) is curled to obtain the vorticity equation (see Jones [2015] for a more complete treatment). Reintroducing other accelerations and forces in the vorticity equation can lead to two different situations: (i) Proudman–Taylor constraint is broken and we get a three-term balance between Coriolis, inertia (or viscosity) and buoyancy; (ii) Coriolis acceleration is still dominant, and flow is quasi-geostrophic at leading order, with a small velocity gradient along the spin axis (𝜴⋅𝛻)u, scaling as 1∕ℓ_∥, where ℓ_∥∼ R_o [Julien et al. 2012].

4.2. 𝜏–ℓ diagram of a remarkable numerical simulation

We focus here on rapid rotation regimes (small Rossby number), for which the leading order balance is quasi-geostrophic. Important results have been obtained for this regime by Guervilly et al. [2019], who performed numerical simulations of thermal convection at low Prandtl number (Pr = 10⁻² and 10⁻¹) in a sphere, at Ekman numbers down to Ek(R_o) = 10⁻⁸ in 3D, and to Ek(R_o) = 10⁻¹¹ in quasi-geostrophic 2D. In this section, we build the 𝜏–ℓ diagram of their most extreme 3D simulation, at Ek(R_o) = 10⁻⁸and Pr = 10⁻², and discuss its implications. Figure 3a displays the 𝜏–ℓ diagram we obtain.

We first build its “template” as in Figure 2a, using radius R_o and t_𝛺 (spin rate’s inverse) of the actual Earth’s core as length-scale and timescale, respectively, in order to compare with the Earth. Input dimensionless parameters Ek and Pr of the simulation provide values needed to build lines 𝜏_𝜈(ℓ) = ℓ²∕𝜈 and 𝜏_𝜅(ℓ) = ℓ²∕𝜅. As in Figure 2a, power dissipation markers are drawn along the Rossby and viscous lines, using outer core mass M_o. The simulation provides the power dissipated by viscosity in the bulk, pointed by a blue arrow on the viscous line, and the slightly larger viscous dissipation in the Ekman boundary layer, pointed by a blue arrow on the Rossby line.

We now turn to extracting lines 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) from the simulation. Line 𝜏_u(ℓ), labeled u, is obtained from the conversion of the volumetric average of a snapshot’s azimuthal order m-kinetic energy spectrum, following Equation (A57) of Appendix A.2.3. Note that line 𝜏_u(ℓ) stays above the Rossby line at all length-scales, implying that the flow is quasi-geostrophic, as expected from the high degree of z-invariance observed in this simulation [Guervilly et al. 2019]. Remember that, in this case, length-scale ℓ has to be understood as the flow’s length-scale in the equatorial plane. Flow becomes anisotropic at large length-scale, as shown by the additional 𝜏_{u r}(ℓ) line, labeled u_r, of radial velocities. The azimuthal over radial velocity ratio increases with ℓ. Line 𝜏_𝜌(ℓ), labeled 𝜌, is obtained in a similar way, using the conversion rule given by Equation (A59) of Appendix A.2.3, with gravity given by g = Ra ( R o ) R o 𝜅 𝜈 ∕ R o 4 and Ra(R_o) = 2.5 × 10¹⁰.

Both 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) lines display a sharp timescale minimum, very close to the Rossby line, defining length-scale ℓ_⊥. This simulation thus nicely illustrates the QG-CIA force balance, with 𝜏_Rossby(ℓ_⊥) ∼𝜏_u(ℓ_⊥) ∼𝜏_𝜌(ℓ_⊥), as advocated by Guervilly et al. [2019]. Note that the same force balance appears to apply for ℓ < ℓ_⊥, as envisioned by Rhines [1975]. Length-scale ℓ_⊥ coincides with the length-scale given by power dissipation occurring in Ekman layers (blue arrow pinned to the Rossby line). As expected from Equations (13) and (5), viscous dissipation is mostly due to flow at the minimum 𝜏_u(ℓ) time.

4.3. 𝜏–ℓ diagram for a non-magnetic Earth core

We now have all elements to start building a 𝜏–ℓ scenario for rotating convection in a non-magnetic Earth’s core, which we present in Figure 3b. The goal is to create and draw realistic 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) lines over the background “template” of Figure 2a. We assume that viscous dissipation mainly occurs in Ekman layers. Applying Equation (13), we obtain time 𝜏 ∗ = [ ( M o R o ∕ 𝓟 diss ) 𝜈 ∕ 𝛺 3 ] 1 ∕ 4, which yields the dissipated power 𝓟 diss that we estimate for the Earth’s core (see Section 3.3), marked by a blue arrow in Figure 3b. We further formulate the ansatz of a QG-CIA force balance at the “optimum” length-scale ℓ_∗, which thus lies on the Rossby line. We plot point (ℓ_∗, 𝜏_∗) in Figure 3b.

Inspired by Figure 3a, and by Rhines’ arguments, we infer that flow obeys the QG-CIA balance for all length-scales ℓ < ℓ_∗. We thus plot 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) lines along the Rossby line all the way to their intersection with dissipation 𝜏_𝜈(ℓ) and 𝜏_𝜅(ℓ) lines, respectively. It remains to draw lines 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) for ℓ larger than ℓ_⊥. Flow becomes anisotropic for ℓ > ℓ_⊥, with radial velocities decreasing as ℓ increases, while azimuthal velocities increase with ℓ. For ℓ > ℓ_⊥ we thus loosely prescribe 𝜏_{u r}(ℓ) ∝ ℓ² (for a spectral energy density E(k) ∝ k), 𝜏_{u a z}(ℓ) ≃𝜏_u(ℓ) ∝ ℓ (for E(k) ∝ k⁻¹), and 𝜏_𝜌(ℓ) ∝ ℓ^1∕2 (for a k⁻²-spectrum).

Reading the 𝜏–ℓ diagram of Figure 3b, we see that core flow in a non-magnetic Earth would be quasi-geostrophic at all scales, with azimuthal velocities reaching 3 m⋅s⁻¹, much larger than present-day core flow velocities represented by its t u obs value and trend. The radius of dominant columnar vortices would be around 200 km. Ekman layer viscous dissipation would dominate over bulk viscous dissipation by many orders of magnitude.

5.1. 𝜏–ℓ expression of force balances in rotating convective dynamos

Following the approach of Section 4.1, we add the Lorentz force in Navier–Stokes’ equation:

5.2. 𝜏–ℓ diagram of a remarkable numerical simulation

Let us start by building and discussing the 𝜏–ℓ regime diagram (Figure 4a) of one of the most extreme dynamo simulations available today: the S2 DNS of Schaeffer et al. [2017].

We first build its “template” as in Figure 2b, using radius R_o and t_𝛺 of the actual Earth’s core as length-scale and timescale, respectively, in order to compare with the Earth. Input dimensionless parameters of the simulation (Ek(R_o − R_i) = 10⁻⁷, Pr = 1, Pm = 0.1) provide values needed to build lines 𝜏_𝜈(ℓ), 𝜏_𝜅(ℓ) and 𝜏_𝜂(ℓ). As in Figure 2b, power dissipation markers are drawn and labeled along the magnetic diffusion line, and along the Rossby and viscous lines, using outer core mass M_o. Ohmic and viscous dissipations D_𝜂 and D_𝜈 of the simulation are obtained from Table 2 of Schaeffer et al. [2017], and scaled to Earth’s core by: 𝓟 = D 𝜌 𝜈 3 ∕ ( R o − R i ). They are pointed by a red arrow on the magnetic diffusion line, and a blue arrow on the viscous line, respectively.

Next, we turn to extracting lines 𝜏_u(ℓ) and 𝜏_b(ℓ) from the simulation’s spherical harmonic spectra, applying conversion rules of Equations (A52) and (A53) of Appendix A.2.2, respectively. The simulated acceleration of gravity g at the top boundary is obtained from:

Reading the resulting 𝜏–ℓ diagram, we see that: magnetic energy largely dominates over kinetic energy (𝜏_b(R_o)≪𝜏_u(R_o)); Ohmic dissipation dominates over viscous dissipation (compare dissipation powers indicated by arrows pinned to lines 𝜏_𝜂(ℓ) and 𝜏_𝜈(ℓ), respectively); flow should be largely quasi-geostrophic, since the 𝜏_u(ℓ) line stays above the Rossby line down to dissipation length-scales. We also observe that both 𝜏_u(ℓ) and 𝜏_b(ℓ) lines have slopes steeper than 1 at large length-scale, while the slope of the 𝜏_𝜌(ℓ) line is closer to 1/2. Finally, we observe that a QG-MAC force balance seems approximately satisfied at length-scale ℓ_⊥≃ R_o∕18, marked by a vertical dotted line, at which we observe (𝜏_b∕t_𝛺)(𝜏_b∕𝜏_u) ∼ R_o∕ℓ_⊥ and 𝜏_b(ℓ_⊥) ∼𝜏_𝜌(ℓ_⊥).

5.3. A simple MAC-balance scenario

Figure 4b proposes a first attempt to complete the template of Figure 2b with plausible 𝜏_u(ℓ), 𝜏_b(ℓ) and 𝜏_𝜌(ℓ) lines. As in Section 4.3, our starting point is the available convective power 𝓟 diss ≃ 3 TW. We formulate the ansatz that it is dissipated by Joule heating only, which means that line 𝜏_b(ℓ) should get down to (but not below) time 𝜏 ∗ = M o 𝜂 ∕ 𝓟 diss that yields a dissipation equal to 𝓟 diss, as pointed by the red arrow on line 𝜏_𝜂(ℓ). However, in contrast with the situation of Section 4.3, we cannot attach line 𝜏_b(ℓ) to this point, because we anticipate that the velocity would be too weak at that scale to amplify the magnetic field.

One of the guidelines of our 𝜏–ℓ approach is that regime changes should occur where relevant ℓ-scale dimensionless numbers reach 1, i.e., at the intersection of corresponding 𝜏–ℓ lines. In a dynamo, we expect a regime change for Rm(ℓ) ∼ 1, at the intersection of lines 𝜏_u(ℓ) and 𝜏_𝜂(ℓ), below which magnetic diffusion takes precedence over induction. Furthermore, if the magnetic field is strong, the intensity of turbulence is strongly reduced in that low Rm regime. We thus assume that Ohmic dissipation is maximum at length-scale ℓ_∗ where Rm(ℓ_∗) = 1, implying 𝜏_b(ℓ_∗) ≃𝜏_∗.

This first condition links the velocity field to the magnetic field but is not sufficient to provide ℓ_∗. Another constraint is needed, which we get from a force balance. As a first guess, we request that our system obeys a MAC force balance at length-scale ℓ_∗. Recalling Equation (24), we have:

The next step consists in drawing lines 𝜏_u(ℓ) and 𝜏_b(ℓ) for ℓ < ℓ_∗ and for ℓ > ℓ_∗. For ℓ < ℓ_∗, we note that the regime of MHD turbulence in the presence of a strong magnetic field is characterized by steep energy density spectra: E(k) ∝ k⁻³ and E_m(k) ∝ k⁻⁵ [Alemany et al. 1979]. Converting to 𝜏–ℓ language with Equation (A44), we get 𝜏_u(ℓ) ∝ ℓ⁰ and 𝜏_b(ℓ) ∝ ℓ⁻¹, yielding the slopes drawn in Figure 4b. Adding rotation further reduces the intensity of turbulence [Nataf and Gagnière 2008, Kaplan et al. 2018], but we lack constraints on the resulting energy spectra.

In the dynamo region, for ℓ > ℓ_∗, we assume k⁻¹ energy density spectra for both u and b (i.e., 𝜏(ℓ) ∝ ℓ), implying that line 𝜏_b(ℓ) follows the Alfvén wave line 𝜏_Alfven(ℓ). This choice is mostly for pedagogical reasons as explained below. Finally, we rather arbitrarily assume 𝜏_𝜌(ℓ) ∝ ℓ^1∕2 at all length-scales. This completes the 𝜏–ℓ diagram shown in Figure 4b.

Reading this 𝜏–ℓ diagram, we see that our scenario yields velocity and magnetic amplitudes that are not too far from the observed t u obs and t b obs values. They translate into a magnetic to kinetic energy ratio of about 10⁴, according to Equations (6) and (14). Bulk and boundary viscous dissipations have comparable amplitudes, both many orders of magnitude smaller than Ohmic dissipation, as assumed. The smallest QG vortices (on the Rossby line) are very sluggish, with a turnover time of about 1 year and a radius of 1 km. Magnetic diffusion is largest at a length-scale of about 10 km.

It is interesting to observe that our scenario implies that the Alfvén line intersects line 𝜏_𝜂(ℓ) at time t_𝛺, meaning that all three ℓ-scale dimensionless numbers Ek_𝜂, Lu and 𝜆 equal 1 at this same scale. This implies that 𝜏_b(R_o) can be deduced from the intersection of 𝜏_𝜂(ℓ) and t_𝛺 lines. In other words, even though our scenario has been built to achieve a given Ohmic dissipation 𝓟 diss, the actual value of 𝜏_b(R_o) does not depend on 𝓟 diss. Only the kinetic energy depends on 𝓟 diss, as given by these scaling laws:

5.4. A simple QG-MAC balance scenario

The simple MAC-balance scenario of the previous section meets a problem: line 𝜏_u(ℓ) plots above the Rossby line for all length-scales ℓ > ℓ_∗, and all these scales have 𝛬_d(ℓ) < 1. This means that flow should be quasi-geostrophic in this scale range, even though the magnetic field is strong. Leading-order force balance should therefore be quasi-geostrophic, with convective motions forming columnar vortices parallel to the spin axis. Figure 5a shows a first plausible alternative scenario, built as follows.

As in Section 5.3 we first obtain the minimum magnetic time 𝜏_b, which is set to 𝜏 ∗ = M o 𝜂 ∕ 𝓟 diss, marked by a red arrow on line 𝜏_𝜂(ℓ), and the corresponding length-scale of maximum dissipation ℓ_∗ is such that Rm(ℓ_∗) = 1. This first condition links the velocity field to the magnetic field but is not sufficient to provide ℓ_∗. Another constraint is needed, which we get from a QG-MAC force balance. We recall that such a balance at length-scale ℓ_⊥ is given by Equation (27):

Graphically, this means that 𝜏_b(ℓ_⊥) plots at mid-distance between 𝜏_u(ℓ_⊥) and 𝜏_Rossby(ℓ_⊥). One then needs to guess at which scale ℓ_⊥ this balance should apply. In contrast with our simple MAC scenario, one cannot choose ℓ_⊥ = ℓ_∗, because this would place 𝜏_u(ℓ_∗) very far down, below the t_𝛺 line, breaking quasi-geostrophy, and yielding 𝓔 k ≫ 𝓔 m in strong disagreement with observations. And we also need to decide how velocity and magnetic fields vary between ℓ_∗ and R_o. As in Section 5.3 we assume 𝜏_u(ℓ) ∝ ℓ and 𝜏_b(ℓ) ∝ ℓ. We then obtain:

Figure 5a displays the 𝜏–ℓ diagram of such a QG-MAC scenario with ℓ_⊥ = R_o∕10. Comparing with Figure 4b, we see that this scenario predicts a larger magnetic over kinetic energy ratio, with 𝜏_u above the Rossby line down to scales of a few hundred meters. Another difference is the level of line 𝜏_𝜌(ℓ).

5.5. A QG-MAC balance scenario à la Aubert

In the previous scenario, choosing ℓ_⊥ = R_o∕10 was borrowed from Aubert et al. [2017] and Aubert [2019], who find it in good agreement with their numerical simulation results. Following Davidson [2013], Aubert [2019] proposes a 𝜏_u(ℓ) scaling for ℓ_∗ < ℓ < ℓ_⊥ that differs from the one we used in Section 5.4. In that interval, Davidson [2013] infers that vorticity is independent of ℓ. This translates into 𝜏_u(ℓ) ∝ ℓ^1∕2 instead of 𝜏_u(ℓ) ∝ ℓ. Using the same scaling for 𝜏_b(ℓ), length-scale ℓ_∗ is then given by:

The corresponding 𝜏–ℓ regime diagram is shown in Figure 5b, where we chose to follow the observed ℓ^3∕2 trend for 𝜏_u and 𝜏_b above ℓ_⊥ (blue and red disks, respectively). This scenario provides an amazing fit to the observed t u obs and t b obs values. Furthermore, we observe that the dynamical Elsasser number 𝛬_d(ℓ) = (t_𝛺∕𝜏_b(ℓ))(𝜏_u(ℓ)∕𝜏_b(ℓ)) remains below 1 at all length-scales ℓ, validating our assumption of leading-order Quasi-Geostrophy.

This scenario yields the following (ugly-looking) scaling laws for the largest scale velocity and magnetic fields:

5.6. The interesting case of Venus

The internal structure of Venus is very poorly known, but we know that it does not generate a detectable magnetic field. This important difference from its sister planet Earth is classically explained by a different thermal history, leading to a hot mantle convecting beneath a rigid lid, preventing core cooling, hence halting the convective engine of the dynamo [Stevenson et al. 1983, Nimmo 2002].

Venus and Earth also differ by their spinning rate: one turn in 243 days instead of one day. This difference is usually considered as unimportant since rotation still appears overwhelming, with an Ekman number Ek(R_o) ∼ 10⁻¹³ [Russell 1980]. However, if we adopt for Venus the same physical properties as for Earth, including its available convective power 𝓟 diss, but update the spinning rate to the one of Venus, we meet a problem, illustrated in Figure 6.

We first observe that Ohmic dissipation time 𝜏 ∗ = M o 𝜂 ∕ 𝓟 diss plots below the t_𝛺 line. We also see that the core of Venus would not be able to dissipate such a power by friction in its Ekman layers, as indicated by the markers along the Rossby line (compare with Figure 3b). Venus does not appear to be a “fast rotator”, and we should not expect the flow to be dominantly Quasi-Geostrophic as in Earth’s core. It might be more similar to the solar dynamo.

We therefore propose a very tentative “devil’s advocate” scenario, in which we assume that dissipation takes place in the bulk, with equipartioned viscous and Ohmic dissipations, yielding the blue arrow on the viscous line, and the red arrow on the magnetic diffusion line. In this region far below the t_𝛺 line, Kolmogorov energy density spectra E(k) ∝ k^−5∕3 seem plausible for both kinetic and magnetic energies, which translate into 𝜏_u(ℓ) ∝ ℓ^2∕3 and 𝜏_b(ℓ) ∝ ℓ^2∕3, as drawn in Figure 6 from their respective dissipation scales up to timescale t_𝛺, which is met at a length-scale ℓ_𝛺 ≃ 5 km.

The force balance we expect there is of type Inertia-Magneto-Archimedean-Coriolis (IMAC), given by:

Our exercise is very formal since we recall that there are good reasons to believe that such a 𝓟 diss convective power is not available for Venus [Stevenson et al. 1983, Nimmo 2002]. Nevertheless, our diversion to Venus questions the notion of “fast rotator” and shows that for a given convective power 𝓟 diss, planet’s spin rate might play a more important role than simply inferred from the low Ekman number it delivers.

6.1. Validity domain of dynamo regimes

One goal of our approach is to better appreciate the validity domain of the various regimes encountered in planetary cores. For example, no dynamo will exist if line 𝜏_u(ℓ) plots too high in the diagram, yielding Rm(R_o) ∼ 1. In Earth’s core, this would only happen for 𝓟 diss values several orders of magnitude smaller than our reference value of 3 TW. Actually, very low values (even negative) are not excluded by some thermal history models, before the birth of the inner core [e.g., Landeau et al. 2022].

At the other end, large 𝓟 diss values can pull the dynamo generation domain (Rm(ℓ) > 1) partly beneath the Rossby line. Complete columnar vortices won’t then have time to form at length-scales below the Rossby line, and we might have a somewhat different turbulence regime. Davidson [2014] proposed an original dynamo scenario that corresponds to such a situation. Noting that inertial waves are strongly helical, and that flow helicity is a key ingredient for dynamo action, he suggests that Earth’s dynamo might operate this way. None of the three scenarios we presented (Figures 4b, 5a, 5b) puts Rm(ℓ) ∼ 1 below the Rossby line. However, it would only take 𝓟 diss ≃ 1 0 TW for the QG-MAC scenario à la Aubert (Figure 5b) to qualify (remember we only consider orders of magnitude).

An even more dramatic regime change should occur when 𝜏_∗ < t_𝛺, as in our devil’s advocate scenario for Venus (Figure 6). Then, the object should not be considered as a “fast rotator” anymore, and the dynamo regime is more of a small-scale type. This occurs when 𝓟 diss > M o 𝜂 𝛺 2. For the Earth’s core, this translates into a power of 10⁴ TW, which was never reached in Earth’s history.

We believe that 𝜏–ℓ diagrams could help inferring relevant dynamo regimes for planets and stars.

6.2. Dynamo scaling laws

Christensen [2010] nicely reviews a number of scaling laws proposed to infer the magnetic field intensity of planets (and stars). Among the nine proposed scaling laws he lists, five relate magnetic intensity to planetary rotation rate 𝛺, with no influence of the available convective power 𝓟 diss, three involve both 𝛺 and 𝓟 diss, and one only 𝓟 diss. The latter one, first proposed by Christensen and Aubert [2006], stems from the analysis of a large corpus of numerical simulations of the geodynamo, backed by an appraisal of the dominant terms in the governing equations. It is the law preferred by Christensen [2010] who shows that it is in good agreement with the measured magnetic intensity of planets and stars.

Translated in 𝜏–ℓ formulation, Christensen’s preferred laws yield:

Davidson [2013] re-examined this question and noted that inertia is playing too strong a role in Christensen and Aubert [2006]’s simulations. He derived a QG-MAC-balance variant and obtained a revised flow velocity law (his Equation (15)), which translates into:

Our discussion on scenario validity domain questions the obtention of universal dynamo scaling laws. However, we have seen that such scaling laws can be easily derived from the 𝜏–ℓ diagram of various scenarios. Each scenario produces 𝜏_u(R_o) and 𝜏_b(R_o) that combine 𝜏_∗ (or T_diss), T_𝜂 and t_𝛺 with various powers. Equations (31) and (32) give the scaling laws for a simple MAC scenario. Although built to dissipate a given convective power 𝓟 diss by Ohmic dissipation, this scenario provides a large-scale magnetic field B that does not depend upon 𝓟 diss, while U is independent of magnetic diffusivity 𝜂. The prediction of our QG-MAC scenario “à la Aubert” is given by Equations (35) and (36) for U and B, respectively.

6.3. Dominant length-scale controversy

Several recent studies have revived a debate on the dominant length-scales to be expected in the Earth’s core [Yan et al. 2021, Bouillaut et al. 2021, Yan and Calkins 2022, Cattaneo and Hughes 2022, Nicoski et al. 2024, Abbate and Aurnou 2023, Hawkins et al. 2023]. Fast rotation inhibits thermal convection, and viscosity is needed to break the Taylor–Proudman constraint [Chandrasekhar 1961]. Studies on the onset of thermal convection in rapidly rotating plane layers [Chandrasekhar 1961], annulus [Busse 1970], and spherical shells or spheres [Busse 1970, Jones et al. 2000, Dormy et al. 2004] therefore all show that, at the onset of convection, the flow consists in columnar vortices aligned with the rotation axis, with a width ℓ_c ∼ R_o[Ek(R_o)]^1∕3 (see Appendix B). In the Earth’s core, such columns would be 30 m in diameter, extending several thousand kilometers in length. At least two effects could strongly widen such columns: (i) the magnetic field [Chandrasekhar 1961, Fearn 1979, Sreenivasan and Jones 2011, Aujogue et al. 2015]; (ii) non-linear flow advection [Rhines 1975, Ingersoll and Pollard 1982, Gillet et al. 2007, Jones 2015, Guervilly et al. 2019].

The effect of the magnetic field can be studied at the onset of convection, but only for given—often simplistic—geometries of the magnetic field that never occur in natural systems. These studies show that convective cells can get as large as the depth of the container R_o when a magnetic field is present. Concerning planetary dynamos, one can argue that such an effect can only be invoked when the dynamo has already produced a strong enough magnetic field. This led Cattaneo and Hughes [2022] to call on the Moon for help, arguing that in an unmagnetized Earth, “convection would consist of hundreds of thousands of extremely thin columns” that would be unable to produce a magnetic field.

This view is clearly challenged by Guervilly et al. [2019] who advocate a dominant convective length-scale around 30 km. Their prediction rests on the extrapolation of a QG-CIA force balance to Earth’s core conditions, and is backed by numerical simulations such as the one we present in Figure 3a. Our QG-CIA scenario, shown in Figure 3b obeys the same rules. It predicts an even larger dominant length-scale (∼200 km) because it is built on the available convective power 𝓟 diss rather than on the observed large-scale velocity, which is reduced by the magnetic field. Interestingly, Figure 3a shows that the simulations of Guervilly et al. [2019] remain very close to the convection onset, which would yield a critical length-scale ℓ_c close to the intersection of the 𝜏_𝜅(ℓ) and 𝜏_Rossby(ℓ) lines (see Appendix B). Yet, they reach a turbulent QG-CIA balance because of the low value of the Prandtl number (Pr = 10⁻²) that allows for large Reynolds numbers near the onset [Aubert et al. 2001, Kaplan et al. 2017].

Our 𝜏–ℓ diagrams clearly emphasize the role of 𝓟 diss in controlling the dominant length-scale, both with and without a magnetic field. This is particularly clear in Figure 3b where the minima of curves 𝜏_u(ℓ) and 𝜏_𝜌(ℓ) would “slide” along the Rossby line following the 𝓟 diss blue arrow. This diagram also suggests that while viscosity in the bulk controls the convective length-scale at the onset (black triangle), it progressively looses its importance as viscous dissipation in the Ekman layers takes over. To achieve a turbulent convection regime where viscosity in the bulk is unimportant, it is crucial to provide another way of dissipating the convective power, like Ekman layer friction or Ohmic dissipation, which are both present in Earth’s core. Because you always need to dissipate the power injected into turbulence, the previous sentence sounds rather obvious. Yet, this might explain why several recent studies find no evidence of a diffusion-free scaling, in which viscosity plays no role. In the numerical simulations of Yan et al. [2021], Yan and Calkins [2022], Nicoski et al. [2024], boundaries are stress-free so that no other energy sink than viscous dissipation in the bulk can equilibrate the convective power.

6.4. Path strategy

Our 𝜏–ℓ approach has much in common with the “path strategy” devised by Julien Aubert and colleagues [Aubert et al. 2017, Aubert and Gillet 2021, Aubert et al. 2022, Aubert 2023]. Their strategy also targets the dynamical regime of a given object, the Earth’s core, and aims at defining a “path” that numerical simulations should follow to reach that regime. As stressed by Dormy [2016], minimizing Pm at a given Ekman number Ek ( R o ) = 𝜈 ∕ 𝛺 R o 2 is not a good strategy, since it increases magnetic Ekman number Ek 𝜂 ( R o ) = 𝜂 ∕ 𝛺 R o 2, while Christensen et al. [2010] show that Ek_𝜂(R_o) < 10⁻⁴ is needed for obtaining Earth-like geodynamo simulations. Hence the need for targeting a more elaborate “distinguished limit” [Dormy 2016].

Aubert and his colleagues stress the importance of respecting the hierarchy of dynamic times, and propose a unidimensional path based on scaling laws obtained by Davidson [2013] for a QG-MAC force balance. Position along the path is measured by an 𝜖 parameter. As exposed in Aubert [2023], the idea is to keep the magnetic diffusion time T_𝜂 =𝜏 _𝜂(R_o) fixed to the Earth value, and to decrease the magnetic Ekman number step by step, by decreasing the rotation time t_𝛺 from values at which appropriate numerical simulations are available (defining 𝜖 = 1) down to the actual value of the Earth (𝜖 = 10⁻⁷). Long-period behaviour of the Earth’s core is well recovered from 𝜖 = 1, while decreasing 𝜖 to Earth’s value allows for the development of short-term dynamics.

Figure 7 shows how 𝜏–ℓ diagrams can help visualize and extend this strategy. We show three diagrams, which correspond to three different values of 𝜖: 1, 10⁻³, and 10⁻⁷. Their “template” are built from the values of the magnetic Ekman and Prandtl numbers listed for “prior PB” in Table 1 of Aubert [2023]. The values of 𝓟 diss (red arrow), t u obs (blue disk), and t b obs (red disk), are from the corresponding simulation outputs given in his Table 2. We then add the 𝜏_𝜌(ℓ), 𝜏_u(ℓ) and 𝜏_b(ℓ) lines that our “à la Aubert” QG-MAC balance scenario (see Section 5.5) predicts for the listed 𝓟 diss. An amazing fit of the “observed” t_u and t_b is obtained, choosing ℓ_⊥ = R_o∕5.

A key contribution of Aubert and colleagues was to determine how the input parameters of numerical simulations should scale with 𝜖 in order to follow the path. This is where the QG-MAC scaling laws of Davidson [2013] come in, and this is also where 𝜏–ℓ diagrams can help. Using the scaling laws obtained for 𝜏_u(R_o) and 𝜏_b(R_o) in Section 5.5, we test two different paths. Note that parameter 𝜖 is defined such that Ek 𝜂 [ 𝜖 ] = 𝜖 Ek 𝜂 [ 1 ], implying t 𝛺 [ 𝜖 ] = 𝜖 t 𝛺 [ 1 ].

Aubert and colleagues chose to keep Rm(R_o) constant along the path, i.e., 𝜏_u(R_o)[𝜖] = C^{t e}. Equation (35) then implies 𝜏 ∗ [ 𝜖 ] ∝ t 𝛺 [ 𝜖 ] ∝ 𝜖 1 ∕ 4, since T_𝜂 does not vary with 𝜖, and assuming that ℓ_⊥∕R_o is constant as well. We thus get 𝓟 diss [ 𝜖 ] ∝ 𝜖 − 1 ∕ 2 since 𝜏 ∗ = M o 𝜂 ∕ 𝓟 diss. Equation (36) then implies 𝜏_b(R_o)[𝜖] ∝𝜖^1∕2. These scalings agree with those of Aubert [2023], as can be seen in Figure 7.

Instead of keeping Rm(R_o) constant along the path, one could choose to keep the dynamo-generating domain above the Rossby line, as we obtain for the Earth. Imposing that Rm(ℓ) = 1 occurs at the intersection of lines 𝜏_𝜂(ℓ) and 𝜏_Rossby(ℓ) yields 𝜏_u(R_o)[𝜖] ∝𝜖^1∕4, 𝜏_b(R_o)[𝜖] ∝𝜖^3∕8, and 𝓟 diss [ 𝜖 ] ∝ 𝜖 − 1 1 ∕ 1 2. It would be interesting to run numerical simulations along such a path.

A.2.1. Lowes–Mauersberger spectrum

In geomagnetism, the variation of magnetic energy with length-scale is usually measured by its Lowes–Mauersberger spectrum [Lowes 1966]. This spectrum is expressed in terms of Gauss coefficients g n m and h n m of scalar magnetic potential V , which defines the internal magnetic field at any radius above the core-mantle boundary when the mantle is considered as an electrical insulator.

Potential V (r,𝜃,𝜑) is then solution of Laplace equation and can be expressed in terms of spherical harmonics as:

Following Langlais et al. [2014], the Lowes–Mauersberger spectrum at any r > R_o is then given by the suite of 𝓛 b ( n ) defined by:

A.2.2. Degree n-spectra from numerical simulations

Numerical simulations of planetary dynamos are most often performed using a pseudo-spectral expansion in spherical harmonics Y n m ( 𝜃 , 𝜑 ) of degree n and order m. Degree-n or order-m spectra are thus readily obtained for both velocity, magnetic and codensity fields. These spectra are usually for u², b² and C² in dimensionless units, such that the sum over all n and m yields 2∕𝜌 times the energy per unit mass of that dimensionless field. Given length-scale L and time-scale T chosen in the simulation, u and b spectra should be multiplied by L²∕T² (assuming that b is expressed in Alfvén wave velocity units).

Given these precisions, the procedure is similar to that exposed in Appendix A.2.1. ℓ(n) is still given by: