Earth and Planetary Science Letters Characterising Jupiter’s dynamo radius using its magnetic energy spectrum

Jupiter’s magnetic ﬁeld is generated by the convection of liquid metallic hydrogen in its interior. The transition from molecular hydrogen to metallic hydrogen as temperature and pressure increase is believed to be a smooth one. As a result, the electrical conductivity in Jupiter varies continuously from being negligible at the surface to a large value in the deeper region. Thus, unlike the Earth where the upper boundary of the dynamo—the dynamo radius—is deﬁnitively located at the core-mantle boundary, it is not clear at what depth dynamo action becomes signiﬁcant in Jupiter. In this paper, using a numerical model of the Jovian dynamo, we examine the magnetic energy spectrum at different depth and identify a dynamo radius below which (and away from the deep inner core) the shape of the magnetic energy spectrum becomes invariant. We ﬁnd that this shift in the behaviour of the magnetic energy spectrum signiﬁes a change in the dynamics of the system as electric current becomes important. Traditionally, a characteristic radius derived from the Lowes–Mauersberger spectrum—the Lowes radius—gives a good estimate to the Earth’s core-mantle boundary. We argue that in our model, the Lowes radius provides a lower bound to the dynamo radius. We also compare the Lowes–Mauersberger spectrum in our model to that obtained from recent Juno observations. The Lowes radius derived from the Juno data is signiﬁcantly lower than that obtained from our models. The existence of a stably stratiﬁed region in the neighbourhood of the transition zone might provide an explanation of this result.


Introduction
Jupiter has the strongest magnetic field among the planets in the Solar System. The magnitude of its surface magnetic field is about ten times larger than that of the Earth. Jupiter and the Earth both have dipole-dominated magnetic fields, with the dipolar axis inclined at about 10 • to the rotation axis. However, the recent NASA Juno mission  revealed that Jupiter's magnetic field has its non-dipolar part mostly confined to the northern hemisphere Jones, 2018), unlike the Earth's field which shows no such preference. The intricate magnetic field of Jupiter is believed to be generated by the convective stirring of liquid metallic hydrogen in the planet's interior. An important and long-standing question is at what depth does such dynamo action begin. The dynamo radius-the location of the top of the dynamo region-is an important factor in understanding the interaction between the interior magnetohydrodynamics and the atmospheric flow in the outer layer (Cao and Stevenson, 2017). Knowledge of the dynamo radius also provides constraints * Corresponding author.
E-mail address: y.tsang@leeds.ac.uk (Y.-K. Tsang). that help to improve the estimation of electrical properties inside Jupiter and will subsequently lead to better modelling of the Jovian magnetic field. The dynamo radius also determines where internal torsional oscillations in Jupiter are reflected as they propagate outwards (Hori et al., 2019).
As existing technology does not allow us to take direct measurement inside Jupiter, or for that matter, inside the Earth, we have to deduce the dynamo radius of a planet from measurements made near its surface. In the case of the Earth, where the dynamo radius is at the core-mantle boundary, Lowes (1974) introduced a strategy by considering the average magnetic energy over a spherical surface of radius r, E B (r) = 1 2μ 0 1 4π |B(r, θ, φ)| 2 sin θ dθ dφ. (1) Here B is the magnetic field, (r, θ, φ) Connerney et al. (2018). A linear fit to log 10 R l (r J ) for 2 ≤ l ≤ 10 gives the Lowes radius r lowes = 0.845r J . Changing the fitting range to 5 ≤ l ≤ 10 results in r lowes = 0.796r J . Connerney et al. (2018) suggest the data is compatible with r lowes = 0.87r J . Here, r J = 6.9894 × 10 7 m. (b) The non-zonal part of the JRM09 data is compared with the full spectrum. Both spectra are normalised by the value of R 1 of the full spectrum. Note that the non-zonal data gives a much closer fit to a straight line in the range 6 ≤ l ≤ 10 than the full data and a linear fit in this range gives r lowes = 0.828r J . (c) Comparison of normalised Lowes spectrum R l /R 1 at r = r J from the Juno data JRM09 and our Jupiter dynamo model at Pm = 10 and Pm = 3.
where a is a reference radius often taken to be the mean planetary radius. P m l are the Schmidt semi-normalised associated Legendre polynomials. The Gauss coefficients g m l and h m l are determined from magnetic field measurement at the surface. Using the expression (2) in (1) yields is the Lowes spectrum, or sometimes the Lowes-Mauersberger spectrum (Mauersberger, 1956). It follows that The downward continuation relation (5) gives the Lowes spectrum R l (r) at some depth r in terms of R l (a) at the surface. It relies crucially on B being purely potential.
To estimate the depth of the dynamo region, we need one further assumption. It has been argued that the large-scale part of R l (a) mainly originates from the Earth's outer core and turbulence there results in a uniform distribution of magnetic energy over different scales l. In particular, at some depth r lowes near the coremantle boundary, R l (r lowes ) is independent of l. This 'white source hypothesis' (Backus et al., 1996), together with (5) implies the linear relation log 10 R l (a) ∼ −β(a)l (6) for the large scales with β(a) satisfying r lowes = 10 −β(a)/2 · a.
Thus, (7) gives the Lowes radius r lowes in terms of the spectral slope β which can be determined solely from magnetic measurement at the surface. The Lowes radius provides an estimate to the location of the Earth's core-mantle boundary that agrees reasonably with seismic measurement. Langlais et al. (2014) found r lowes = 3294.5 km compared to the seismically determined 3481.7 km. Langlais et al. (2014) also found that omitting the m = 0 axisymmetric components in (4), so that only the non-zonal components are used, reduced the scatter of the spectrum and led to a remarkably accurate agreement between r lowes and the Earth's seismic core radius. Compared to the Earth, magnetic field measurements for Jupiter are less extensive. The data available before the Juno mission only allowed for the calculation of the Lowes spectrum up to l = 4 (Connerney et al., 1998) or l = 7 (Ridley and Holme, 2016) depending on the modelling methodology. This has changed since the Juno spacecraft arrived at Jupiter. While the spacecraft is taking more measurements as it continues to orbit Jupiter, Connerney et al. (2018) computed g m l and h m l up to l = 10 from the data collected during eight of the first nine flybys. Using these Gauss coefficients, Fig. 1(a) shows the Lowes spectrum at r = r J where, following French et al. (2012), we take r J = 6.9894 × 10 7 m to be the mean radius of Jupiter. Note that Connerney et al. (2018) used the equatorial radius, 7.1492 × 10 7 m, and we have corrected for this difference in Fig. 1 and Table 1. In sharp contrast to its Earth counterpart (Backus et al., 1996), the Jovian Lowes spectrum R l (r J ) does not show a clean exponential decay. In fact R l (r J ) remains almost constant for 2 ≤ l ≤ 5 before decaying at larger l.
Consequently, routinely applying Lowes' procedure gives different values of r lowes depending on the range of l used in the linear fit, as shown in Fig. 1(a). However, using the non-zonal components only on the JRM09 data, see equation (8), as suggested by Langlais et al. (2014) for the Earth, leads to a better linear fit for l ≥ 6, see Fig. 1(b). The best-fit value of r lowes = 0.828r J . This improvement may arise because the higher order non-axisymmetric field components arise more directly from the non-axisymmetric convection and hence are more randomly distributed than the full spectrum components. A more fundamental issue here concerns the interpretation of r lowes for Jupiter. The interior structure of Jupiter is very different from that of the Earth. Theoretical and experimental studies suggest that the phase transition from molecular to degenerate metallic hydrogen (Wigner and Huntington, 1935) along a Jupiter adiabat is continuous (Wicht et al., 2018;Helled, 2018). As a result, the electrical conductivity σ (r) of the hydrogen-helium mixture in Jupiter varies smoothly with the radial distance (Weir et al., 1996;Liu et al., 2008). For example, Fig. 2 shows the profile σ (r) obtained from an ab initio simulation by French et al. (2012). Therefore unlike the Earth, where the flow and the Lorentz force acting on the flow are both confined within the same region, Jupiter's dynamo is coupled to an outer layer of fluid flow that is free from magnetic effects. Such coupling, with a transition layer in between, is not well understood. It is not clear how large the current-free region where the downward continuation operation (5) is justified actually is. There is also the question about the validity of the white source hypothesis. In fact, how do we characterise the extent of the dynamo region for a continuously varying electrical conductivity profile? Is there a sensible way to define a dynamo radius for Jupiter? In this paper, we examine these issues by considering the magnetic energy spectrum F l (r), to be defined in section 3, in a numerical model of Jupiter. The magnetic energy spectrum F l (r) essentially represents the distribution of magnetic energy over different spherical harmonic degrees l at depth r. The Lowes spectrum R l (r) in (4) is a special case of F l (r) under the condition j = 0 (which is only true near the planetary surface). The change in behaviour of F l (r) along r indicates varying dynamics in different regions. Comparing F l (r) to R l (r) gives further insights into the different physics in these regions.
In the next section, we describe our model for Jupiter's dynamo. In section 3, we first introduce the magnetic energy spectrum and discuss how its behaviour changes with depth. We then show that a dynamo radius can be identified from a transition in the spectral slope. In section 4, we look at the relationship between the dynamo radius and the Lowes radius. We then finish with a discussion on the differences between results from our model and observation.

A model of Jupiter's dynamo
Numerical models of Jupiter's dynamo have recently been developed to study the magnetic field and internal flow of the giant planet (Jones et al., 2011;Jones, 2014;Gastine et al., 2014). The model used in the present study is developed and described in detail by Jones (2014), though here the range of parameters has been extended to get further into the strong-field dynamo regime (Dormy, 2016). We briefly summarise it here.

Anelastic spherical dynamo
We consider the convection of an electrically conducting fluid in a rotating spherical shell of inner radius r in = 0.092r J and outer radius r out = 0.959r J . The heat flux is modelled by an entropy flux proportional to the local entropy gradient with constant diffusivity κ S . The other physical parameters are the angular speed , the constant kinematic viscosity ν and the magnetic diffusivity η(r) which varies with the radial distance r. The dynamical variables of velocity, magnetic field, entropy, density and pressure (u, B, S, ρ, p) are governed by the non-dimensional equations, together with the equation of state In deriving the model, the Lantz and Fan (1999) formulation of the anelastic approximation (Braginsky and Roberts, 2007) has been employed about a spherically symmetric, hydrostatic and adiabatic basic state (ρ, p, S ). This simplifies the system to involve the dynamics of a single thermodynamical variable S = S −S. In (10a), ẑ and r are unit vectors along the rotation axis and the radial direction respectively, = p /ρ + is a generalised pressure combining disturbance pressure p = p −p and disturbance gravitational potential . In (11), ḡ is the gravitational acceleration due to ρ. The system is forced by a constant entropy source H S modelling the secular cooling of the planet. The dissipative terms are: Let T * , ρ * and η * be the dimensional values of T , ρ and η, respectively, at the midpoint of the shell and S be the entropy drop across the thickness L = r out − r in of the shell. Our equations are non-dimensionalised using the unit of length L, time L 2 /η * and magnetic field √ ρ * μ 0 η * , where μ 0 is the permeability of free space. The dimensionless numbers in (10) are defined as: The boundary condition for u is no-slip at r in and stress-free at r out . At both the inner and outer boundaries, it is electrically insulating and S is fixed at a constant value. The initial conditions are u = 0 with a small perturbation in B and S. The spectrum of the initial magnetic perturbation is narrow-banded with 8 ≤ l ≤ 10 and thus has no dipole component.

Hyperbolic electrical conductivity profile
In applying the anelastic convective system described above to model the Jovian dynamo, we need to provide an equilibrium state and a conductivity profile that represent the thermodynamic and transport properties inside Jupiter. French et al. (2012) have calculated the material properties of a hydrogen-helium-water mixture under Jupiter-like condition using density functional theory. Here, we use the same equilibrium density ρ(r) and temperature T (r) profile as in Jones (2014) which are smooth interpolations to the J11-8a data in French et al. (2012). For the magnetic diffusivity η(r), we consider the following hyperbolic model: with five parameters. The values used for these parameters are the same as in Jones (2014) which give a good fit to the J11-8a data. Fig. 2 shows the corresponding electrical conductivity σ (r) = 1/μ 0 η. In the interior, σ (r) decreases roughly linearly and reaches about one-fifth of its maximum value at r = 0.8r J . This is unlike some of the previous studies (Gastine et al., 2014;Glatzmaier, 2018) in which the electrical conductivity is taken to be constant below a certain depth. Dietrich and Jones (2018) studied a wide range of profiles for σ (r) by varying c 2 in (14) and found a diversity of magnetic field morphologies.

Simulation parameters and electric current profile
For the rest of this paper, the following parameters are kept fixed: r in /r out = 0.0963, Ra = 2 × 10 7 , Ek = 1.5 × 10 −5 and Pr = 0.1. Jupiter is believed to have a strong-field dynamo, i.e., its magnetic field strongly influences the flow. Thus, we are interested in cases of large magnetic Prandtl number Pm which produce a strong-field dynamo (Dormy, 2016). Specifically, we investigate the effects of Pm by comparing simulations with Pm = 10 and Pm = 3. The case of Pm = 3 has previously been studied in detail by Jones (2014). Results here are presented in dimensionless units unless otherwise stated. Fig. 3 shows snapshots of magnetic and velocity fields from the Pm = 10 simulation. The equatorial zonal jet and the dipolar nature of the radial magnetic field, both near the surface, are obvious.
With the conductivity σ (r) increasing sharply with depth, we are interested in the radial dependence of the dynamo action. We first examine the average electric current j rms at a depth r, defined as, Above, j = ∇ × B, · t indicates time average over a statistical steady state and the integral is over a spherical surface of radius r. Fig. 4 shows that j rms (r) behaves very similarly for both Pm = 3 and Pm = 10. In the interior where strong magnetic field is being generated, j rms (r) increases slightly with r and peaks at around r = 0.7r J even though σ (r) is monotonically decreasing in r. Approaching the surface, j rms (r) follows the trend of σ (r) and drops quickly and smoothly to negligible value, indicating the cessation of dynamo action. While the variation of j rms (r) with r certainly signifies different dynamics at different depth, Fig. 4 does not locate a characteristic depth that represents the top of the dynamo region. It is not obvious from the profile of j rms (r) at what depth the electric current becomes large enough to generate a significant magnetic field. In the next section, we show that a dynamo radius can be identified using the magnetic energy spectrum.

Magnetic energy spectrum
We again consider the average magnetic energy on a spherical surface given by (1). Unlike in the derivation of the Lowes spectrum R l where the magnetic field is assumed to be potential, here we make no such assumption and expand B in terms of a set of

Table 1
For simulations at two different Pm and Juno observation (with the two fitting ranges in Fig. 1(a)): α(r out ) is the spectral slope of F l (= R l at r out ) measured at the outer boundary r out and α dyn is measured inside the upper dynamo region as discussed below (21). r lowes is the Lowes radius in (7) and r dyn is the dynamo radius in (22) The expansion coefficients are generally function of r and t. Substituting (16) into (1), we get where the magnetic energy spectrum is We are mainly interested in the time-averaged spectrum F l (r) = F l (r, t) t , with the time-averaging done after the system has reached a statistical stationary state. Roughly, F l (r) can be interpreted as the average magnetic energy per spherical harmonic degree l (Maus, 2008). Note that F l is calculated from the full field B.
On the other hand, R l is calculated from the scalar potential in (2). Nevertheless, in a current-free region, the two are identical. Hence,

F l (r) at the planetary surface
We first examine F l (r) at the outer boundary r = r out of the spherical shell. Fig. 5 shows the time evolution of a few modes of F l (r out , t) from the Pm = 10 simulation. The development of the dipolar l = 1 mode can clearly be seen as it outgrows all other modes.
Since the electric current is negligible at r out , F l (r out ) = R l (r out ) in our simulations, see for example Fig. 6 for the case of Pm = 10. Fig. 1(c) plots F l from the Pm = 10 and Pm = 3 simulations as well as R l calculated from the Juno data JRM09. All three spectra have been continued to r = r J using (5) and normalised. The key qualitative difference between the spectra from our two simulations is  (17), and Lowes spectrum R l (r) (circles) at different depth r for the Pm = 10 simulation. R l (r) are obtained from the time-averaged R l (r out ) using (5).
that F l (r J ) for Pm = 3 displays a clear exponential decay for all l (excluding l = 1) while for Pm = 10, the spectrum is roughly flat at small l and only starts to decay exponentially for l 5. In this respect, the Pm = 10 spectrum is similar to the Juno spectrum. However, the decay rate of F l for Pm = 3 is faster and slightly closer to the Juno observed value. Fitting the range 5 ≤ l ≤ 40 yields the values of r lowes shown in Table 1 for Pm = 10 and Pm = 3.

F l (r) at different depth r
We now look at the magnetic energy spectrum in the interior of the spherical shell. We focus on the case of Pm = 10. The solid lines in Fig. 6 show F l (r) at different depth r. The key feature is the transition of F l (r) through three different stages as r varies. We have seen in the previous section the steep exponential decay of F l with l (for l 5) at the surface. As we move below the surface, Fig. 6 shows F l (r) maintains such exponential decay, log 10 F l (r) ∼ −α(r)l, but the spectral slope α(r) decreases rapidly with r inside the layer of 0.91r J r ≤ r out and F l (r) has become rather shallow at 0.91r J . As we delve further into the interior, quite remarkably, F l (r) remains shallow as its shape, and hence α(r), becomes more or less invariant over the substantial region of 0.55r J r 0.91r J . This clearly indicates a shift in the dynamics near 0.91r J . Finally, in the deep interior and close to the core, F l (r) decays superexponentially and the magnetic field is dominated by large scales. Boussinesq geodynamo models which compute the magnetic energy spectrum inside the core also show a spectrum that decays exponentially with l for l ≥ 5, e.g. Christensen et al. (1999).

A dynamo radius
The significance of the change in behaviour of F l (r) becomes clear when we compare F l (r) to the Lowes spectrum R l (r) at the same depth. Recall that R l (r out ) = F l (r out ) at the surface. We now downward continue R l (r out ) using (5) (with a = r out ) to obtain R l (r) at different r, which we plot as circles in Fig. 6. Near the surface, F l (r) and R l (r) are essentially indistinguishable because electric current is negligible there. As r decreases and reaches some depth r dyn , F l (r) starts to deviate from R l (r). The main observation in Fig. 6 is that the shape of F l (r) becomes independent of r, as discussed in the previous section, at essentially the same depth r dyn . This implies that r dyn is the boundary below which electric current becomes important and the dynamics of the system is altered. We therefore identify r dyn as the top of the dynamo region, or the dynamo radius. Fig. 7 vividly illustrates the discussion in the previous paragraph by plotting α(r) together with the Lowes spectral slope β(r), defined analogously to (6), as a function of r. From (5), we have where β(r out ) = α(r out ). Note how α(r) diverges from β(r) and levels off to the value α dyn at r = r dyn . The sharpness of the transition allows for a meaningful definition of r dyn . Fig. 7 also provides a quantitative way to determine r dyn . Fitting a horizontal line through 0.7r J < r < 0.86r J yields the value of α dyn = 0.024. We then obtain the dynamo radius r dyn from (21) using the relation β(r dyn ) = α dyn .
This gives r dyn = 0.907r J for the Pm = 10 simulation. The values of the various spectral slopes and characteristic radii for Pm = 10 are summarised in Table 1. Spacecraft missions can only measure the spectral slope at the planetary surface, from which r lowes is calculated using the white source assumption discussed in section 1. The question is then how well can r lowes predict the actual dynamo radius r dyn . In our simulations, r lowes is where the dashed line in Fig. 7 intersects the horizontal axis, i.e. β(r lowes ) = 0 in (21). This is because by definition, the downward continued R l (r) becomes flat at r = r lowes , at least for the range of l where R l (r out ) is fitted to obtain β(r out ). It is clear from Fig. 7 that generally r lowes ≤ r dyn .
Comparing the values shown in Table 1, we see that for Pm = 10, r lowes is about 3% less than r dyn . The difference stems from α dyn being fairly small but not exactly zero. Comparing F l to R l near r lowes in Fig. 6 again shows the white source assumption is only approximate and F l (r) never becomes exactly flat. Nonetheless, helped by the steep decrease of β(r) with r shown in Fig. 7, we still have a close agreement between r lowes and r dyn . Fig. 3(b) shows a snapshot of the radial magnetic field at r = 0.8r J where α(0.8r J ) ≈ α dyn .   Fig. 8(a). The l = 1 mode has the largest magnitude at all r except for a region around 0.8r J where it is overtaken by l = 2, see also Fig. 3(b). Interestingly, this is also the region in which j rms (r) peaks and α(r) remains virtually constant at α dyn . We also see that all non-dipolar modes are strongly damped just above the dynamo radius r dyn leading to the dipoledominated field observed at the surface. Fig. 8(b) zooms into a region about r = r dyn and shows, for the large-scale modes, how F l (r) deviates from R l (r) as one moves from the current-free outer layer into the dynamo region.

Effects of Pm
We now compare the simulation at Pm = 10 to the one at Pm = 3. We have already discussed the differences in the magnetic energy spectrum at the surface in section 3.1. The slightly steeper F l (r out ) for Pm = 3 means the magnetic field has less small scales.
Generally, results from the two simulations are qualitatively similar. The Pm = 10 simulation has stronger magnetic fields, so the Elsasser number, which is proportional to the root-mean-squared magnetic field over the domain, is about an order of magnitude larger than that of the Pm = 3 simulation, closer to the high value expected in Jupiter. In the Pm = 3 simulation, it was noted in Jones (2014) that the Lorentz force in the interior mostly suppresses the internal differential rotation in the metallic hydrogen region.
At Pm = 10 the stronger magnetic field means the internal differential rotation is even more strongly suppressed, see Fig. 3(c), so that strong zonal flow occurs only in the molecular region near the equator. However, although the zonal flow is confined to the molecular region, its magnitude is about double that of the Pm = 3 simulation. The convective velocity in the Pm = 10 simulation is also about double that in the Pm = 3 simulation, so the magnetic Reynolds number is about twice that of the Pm = 3 simulation. While it is not computationally possible to achieve the parameters believed to operate in Jupiter, the increase in Pm has moved the simulation results in the direction of more realistic values. Fig. 4 shows the electric current is roughly three times smaller in the Pm = 3 case. The spectral slopes for Pm = 3 display the same trend as in Fig. 7. The values of α(r out ) and α dyn together with that of r lowes and r dyn are given in Table 1. A steeper spectrum at the surface means a bigger α(r out ). At the same time, we see that α dyn also increases. The net result is that the dynamo radius r dyn is only marginally less than that of Pm = 10. On the other hand, r lowes drops more significantly which makes r lowes less accurate as a predictor of r dyn . These results suggest that for a dynamo with a larger Pm, the magnetic energy spectrum F l (r) in the upper part of the dynamo region is closer to being 'white'. As a consequence, the Lowes radius gives a better prediction to the dynamo radius.

Discussion and conclusion
The electrical conductivity in Jupiter varies from being negligible at the surface to a very high value in the interior. It thus raises the question about the depth at which dynamo action starts. In this paper, we consider the magnetic energy spectrum F l (r) at depth r in a numerical model of Jupiter's dynamo. For l 5, the magnetic energy spectrum decays exponentially with l, log 10 F l (r) ∼ −α(r)l. We find that a sharp transition in α(r) can be used to identify a dynamo radius r dyn and this dynamo radius can be reasonably predicted by the Lowes radius r lowes as discussed in section 3.3. The situation is in fact rather simple as illustrated in Fig. 7. The two characteristic radii r dyn and r lowes are controlled by two spectral slopes: α(r out ) which is observable at the surface and α dyn which measures the deviation from the white source hypothesis near the top of the dynamo region. Varying α dyn moves the horizontal dot-dashed line in Fig. 7 up and down while changing α(r out ) shifts the dashed curve left and right. These determine the location of r dyn and r lowes as well as the relative distance between them. Notice that the dashed curve, given by (21), is essentially a straight line for r out − r r out , We find that in our two simulations at Pm = 10 and Pm = 3, α(r out ) and α dyn change in such a way that leaves the dynamo radius fairly insensitive to Pm. Incidentally, at r = r dyn , the electrical conductivity σ (r) has dropped by two orders of magnitude from its maximum at the inner boundary. Fig. 9 plots as a function of depth three different magnetic Reynolds numbers, each based on a different velocity scale, for our two Jovian dynamo simulations, reverting to dimensional units. Here, u rms , u r,rms and u φ,rms are the root-mean-squared values of the total, radial and zonal velocity, respectively, over a spherical surface of radius r. All three magnetic Reynolds numbers vary weakly in the interior and then decrease sharply to a negligible value near the surface. We also see that Rm ≈ Rm φ for r r dyn indicating the zonal flow becomes dominant. This is consistent with the depth of the equatorial zonal jet estimated roughly from figures such as Fig. 3. The values of Rm at r dyn and the depth r Rm at which Rm = 1 are given in Table 1. These values suggest that r Rm determined by the criterion Rm = 1 is generally larger We have estimated the Lowes radius from the Juno data in Fig. 1. We find that it has fairly large uncertainties depending on the range of l used in the linear fitting. However, if the zonal components are omitted we find a closer linear fit for l ≥ 6 as did Langlais et al. (2014) for the geomagnetic data. The situation in Jupiter could be similar to our simulation at Pm = 10 where a clean exponential decay in the spectrum emerges only at larger l. This is in contrast to the case of Pm = 3, which has weaker flow and magnetic field. We anticipate further data collected by the ongoing and future flybys will extend the range of the Lowes spectrum in Fig. 1 and hence provide a more reliable estimate of r lowes .
Despite uncertainties in the data, Table 1 shows that r lowes from the Juno observation is clearly smaller than in both of our simulations. Nevertheless, its implication on the location of the dynamo radius is not clear. Whether r lowes gives a good estimate on r dyn relies on the white source hypothesis, which may or may not be valid in Jupiter. However, the Pm = 10 simulation, which we believe is closer to Jupiter conditions than the Pm = 3 simulation, has a smaller α dyn than the Pm = 3 simulation, suggesting that Jupiter's magnetic field might be close to white near the top of the dynamo region. In our simulations, a steeper spectrum observed at the surface tends to be accompanied by steeper spectra in the interior and consequently r dyn could be shallower than r lowes by a fair amount. The present results suggest that the true dynamo radius likely lies above the Lowes radius. Irrespective of its relation to the dynamo radius, the Lowes radius is a property of the magnetic field at the surface of Jupiter. The smaller r lowes of the Juno data stems from a steeper spectrum at the planetary surface, implying Jupiter's magnetic field has less small scales than that in our model. This is slightly surprising as the flow is believed to be more vigorous in Jupiter than in our simulations, because the simulations have enhanced diffusion coefficients to maintain numerical stability. We should point out again that r lowes for the simulations in Table 1 are derived from timeaveraged spectra while the Juno observation essentially provides only a snapshot of Jupiter's magnetic field. Fig. 10 plots the instantaneous Lowes radius r lowes (t) obtained from the time-dependent spectrum F l (r out , t) in the Pm = 10 simulation. The case of Pm = 3 shows similar spread about the mean value. We argue that the difference between simulations and observation is significant even when statistical fluctuation is taken into account.
The difference in r lowes between our numerical model and observation raises several questions and suggests possible avenues for future research. The results presented here are specific to the electrical conductivity profile σ (r) in (14) which is based on data from theoretical ab initio calculation. Could the deeper Lowes radius in observation mean the actual electrical conductivity inside Jupiter is smaller than predicted? It is worth studying how the magnetic energy spectrum responds to perturbations in σ (r). On a related note, the magnetic Reynolds number Rm in our simulations is of the order 10 3 at its maximum, much lower than an estimated value of 10 6 in Jupiter (Jones, 2014). The puzzle here is that increasing Rm in the model will likely increase r lowes rather than reduce it and thus move it further away from the Juno value. While current computing resources prevent us from reaching a much larger Rm, investigating the trend of the dynamics in the neighbourhood of a smaller attainable Rm, possibly by changing σ (r), could provide valuable insights.
In our simulations, the system is forced by the constant entropy source H S in (10c) and we employ a constant entropy boundary condition. Could our numerical setup tend to produce extra small scales that lead to the shallower spectra shown in Fig. 1(c)? In geodynamo simulations, boundary conditions can significantly affect the dynamics (Sakuraba and Roberts, 2009;Dharmaraj and Stanley, 2012). It is important to assess the robustness of the present results and examine their dependence on boundary conditions and the form of forcing.
The formation of a stably stratified layer just under the molecular layer due to 'helium rain' (Stevenson and Salpeter, 1977) has been proposed to explain the near-axisymmetric magnetic field of Saturn (Stevenson, 1982;Dougherty et al., 2018). Although helium rain is more probable to occur in Saturn, it cannot be ruled out for Jupiter (Wahl et al., 2017;Debras and Chabrier, 2019). It would be interesting to see the effects of such a stable layer on the Lowes radius as it displaces the dynamo action deeper into the interior. This is perhaps the most natural way to explain the surprisingly low value of r lowes in the Juno data.