The underexplored frontier of ice giant dynamos

Abstract

The Voyager 2 flybys of Uranus and Neptune revealed the first multipolar planetary magnetic fields and highlighted how much we have yet to learn about ice giant planets. In this review, we summarize observations of Uranus’ and Neptune’s magnetic fields and place them in the context of other planetary dynamos. The ingredients for dynamo action in general, and for the ice giants in particular, are discussed, as are the factors thought to control magnetic field strength and morphology. These ideas are then applied to Uranus and Neptune, where we show that no models are yet able to fully explain their observed magnetic fields. We then propose future directions for missions, modelling, experiments and theory necessary to answer outstanding questions about the dynamos of ice giant planets, both within our solar system and beyond.

This article is part of a discussion meeting issue ‘Future exploration of ice giant systems’.

1. Introduction

As the solar system has been explored with spacecraft missions, our understanding of planetary magnetic fields has developed from the Earth as an archetypal example to learning about the large breadth of behaviours exhibited from Mercury to Neptune. Figure 1 shows the surface radial magnetic fields of planets with intrinsic fields at present day. The Earth’s magnetic field is dominated by the axial dipole component with a roughly 10° offset from the rotation axis. Mercury’s magnetic field is similar in that the dominant field component is the axial dipole, but the dipole is offset approximately 500 km northward of the planet centre and aligned with the rotation axis to less than 1° (e.g. [6,7]). Considering the gas giants, Jupiter’s magnetic field is dipole-dominated with an approximately 10° tilt between the magnetic and rotation axes. The surface field is also more spatially complex than those of the terrestrial planets, with a high-intensity band in the northern hemisphere, a strong flux patch near the equator and fewer non-dipolar structures in the southern hemisphere (e.g. [2]). Saturn is again dipole-dominated, with a nearly axisymmetric field like Mercury. Equatorial asymmetries do exist, however, with the magnetic equator located approximately 2800 km (approx. 0.05 Saturnian radii) northward of the planetary equator (e.g. [8]).

Figure 1.

Radial magnetic field at the surfaces of (a) Mercury, (b) Earth, (c) Jupiter, (d) Saturn, (e) Uranus and (f ) Neptune. Datataken from [1] for Mercury (with spectral resolution of spherical harmonic degree l ≤ 5), IGRF-13 coefficients are used for Earth and are available on the IAGA Division V-MOD geomagnetic field modelling website (l ≤ 13), [2] for Jupiter (l ≤ 10), [3] for Saturn (l ≤ 11), [4] for Uranus (l ≤ 4) and [5] for Neptune (l ≤ 3). The colours represent field intensity, where purple (green) indicates outward (inward) directed field. A mollweide projection is used in which the horizontal lines indicate constant latitude. (Online version in colour.)

Dramatically different magnetic fields are found at Uranus and Neptune (e.g. [5,9,10]). The fields are somewhat poorly constrained since observations are limited to single flybys of the Voyager 2 spacecraft that passed within 4.2 R_U of Uranus in 1986 and 1.18 R_N of Neptune in 1989 [11,12], but are better resolved near closest approach, with the different flyby distances at Uranus and Neptune resulting in different sensitivities to higher spherical harmonic degree and order magnetic field components. At Uranus, auroras at the magnetic field line footprints provide an additional constraint. By searching for magnetic field models that agree with both magnetometer data and auroral emissions, multipole moments up to l = 4 can be estimated for Uranus [4] compared to l = 3 for Neptune. Alternatively, a regularized inversion method has been used to further constrain smaller scale features, resolving some higher degree and order field components out to l = 16 for both Uranus and Neptune [5]. All of these observational models show that the magnetic fields are not dipole-dominated nor are they axially aligned. The dipole component is tilted by 59° at Uranus and 47° at Neptune. It may further be expected that the planetary magnetic fields of Uranus and Neptune are likely to be even more complex than existing observations reveal. In addition, the total magnetic field strength of Uranus and Neptune may not be well represented by the observed field. This is because even at the surface, the dipole is not dominant and so it is possible that small-scale unresolved field components are much stronger than the dipole field.

The maps in figure 1 represent snapshots of the planetary magnetic fields in time. However, magnetic fields evolve temporally, resulting in what is known as secular variation (SV) of the field. Evolution of Earth’s and Jupiter’s magnetic fields vary on decadal timescales (e.g. [13–16]). By contrast, no detectable SV has been measured at Saturn over a 30-year period [17] and SV is only inferred at Mercury through a comparison of the present-day intrinsic magnetic field and the remanent crustal magnetization [18,19]. The single flybys of the Voyager 2 spacecraft past Uranus and Neptune do not provide any knowledge of their magnetic SV but could be used to model SV when compared with future measurements.

Planetary magnetic fields fall into three broad categories. Mercury and Saturn have strikingly axisymmetric fields, with varying degrees of equatorial asymmetry between the northern and southern hemispheres, and slow SV. Earth and Jupiter have dipole-dominated fields with approximately 10° tilts, prominent regions of enhanced intensities and measured SV. Uranus and Neptune have multipolar (i.e. non-dipole dominated) surface fields with comparable intensities and no clear symmetries along any axis. The singular set of field measurements for each of Uranus and Neptune prevents the determination of SV. These observations lead to fundamental questions about planetary magnetic field generation: What processes control the magnetic field morphology, strength and temporal evolution? What aspects of the planetary interiors are responsible for the variations we see across the terrestrial, gas giant and ice giant planets as well as within each of these classes?

The paper is organized as follows: §2 reviews the fundamentals of planetary magnetic field generation, and §3 summarizes our current understanding of the factors controlling magnetic field strength and morphology. Hypotheses to explain Uranus’ and Neptune’s unique magnetic fields are discussed in §4. Section 5 discusses future directions to advance our understanding of ice giant dynamos.

2. Magnetic field generation in the ice giants

(a). Fundamentals

Planetary magnetic fields are generated through self-sustained dynamo action, where motions in an electrically conducting fluid region of a planet act to maintain a magnetic field against decay from ohmic dissipation. This leads to several necessary conditions that must be met for a planet to sustain a dynamo. The most obvious is that the planet must have a fluid region that is both in motion and has an appreciable electrical conductivity.

Here, we focus on some of the basics of planetary dynamo theory necessary for our discussion of ice giant dynamos. For a more complete overview of dynamo theory, see [20,21]. The fundamental equation for dynamo action is known as the magnetic induction equation (MIE). It is derived by combining Maxwell’s equations, neglecting the displacement current, with Ohm’s Law in order to form a time evolution equation for magnetic field. For the special case of uniform electrical conductivity, the MIE becomes

$$\frac{\mathrm{\partial }\mathit{B}}{\mathrm{\partial }t}=\mathrm{\nabla }\times (\mathit{u}\times \mathit{B})+\eta {\mathrm{\nabla }}^2\mathit{B},$$ 2.1

where B is magnetic field, u is velocity, t is time and η = (σμ₀)⁻¹ is magnetic diffusivity (which is inversely proportional to electrical conductivity σ and magnetic permeability μ₀).

Physical insight into dynamo action can be found by examining each term in the MIE. The left side of the equation represents how the magnetic field changes in time. The first term on the right side represents magnetic induction, i.e. how interactions between fluid motions and magnetic fields generate new magnetic field. This term can be seen as a source/sink term for the time evolution of the magnetic field. The second term on the right side represents ohmic diffusion. A dynamo is, therefore, a balance between induction and diffusion processes. In addition, note that the requirement that a dynamo needs fluid motions in an electrical conductor is a direct consequence of the MIE. If there are no fluid motions, then u = 0 and we lose the only potential source term for generation of magnetic field. If the fluid is an electrical insulator, then σ = 0 and derivation of the MIE results in Laplace’s equation ${\mathrm{\nabla }}^2\mathit{B}=0$.

A characteristic measure of dynamo action is given by comparing the magnitudes of the induction and diffusion terms on the right side of the MIE using representative estimates for the variables in these terms. Taking the ratio of the induction and diffusion terms gives a non-dimensional parameter known as the magnetic Reynolds number: Rm = UL/η, where U is a characteristic speed and L is a characteristic length-scale. A dynamo must maintain Rm above a critical value in order for the dynamo to be sustained. Detailed mathematical treatments lead to lower bounds on Rm_c of ∼π or ∼π² depending on what choices are made for characteristic velocities. Numerical dynamo simulations typically find this value is $R{m}_c\gtrsim 10$.

Attaining a super-critical Rm is necessary, but not sufficient, for maintaining a dynamo. Various anti-dynamo theorems (e.g. [22]) demonstrate that fluid motions must maintain particular geometries. For example, the motions must be three-dimensional. This means that some of the fundamental fluid motions we expect in planets, such as solid body rotation and zonal flows, cannot independently maintain a dynamo because they do not possess a radial component. Convective motions due to buoyancy differences in the fluid can produce these necessary radial motions. Convection may be thermal in origin if the temperature gradient in a region of the planetary interior is super-adiabatic, and/or it can be compositional in origin if a multi-component fluid is not miscible and is dynamically unstable.

(b). Ice giant interiors

Uranus’ and Neptune’s magnetic fields tell us that there are regions inside the planets where electrically conductive fluids are undergoing dynamo-favourable motions. However, the lack of strong constraints on the ice giants’ interior structure, composition and thermal profiles limits our ability to determine exactly where inside Uranus and Neptune the dynamos are generated.

A recent review on the constraints and inferred interior structures of the ice giants can be found in [23]. As a basic model, we can consider the ice giants as consisting of three layers: a relatively thin hydrogen–helium rich envelope, an ice-rich mantle and a small rock-rich core, where the ‘-rich’ intends to imply that small fractions of the other main components are likely present in each of the layers. However, high-pressure experiments and ab initio simulations (e.g. [24,25]) demonstrate that the ice giant interiors are likely much more complicated than this simple three-layer structure. Icy materials such as water, ammonia and methane (and their mixtures) have complex phase diagrams that we are only beginning to understand. Figure 2 demonstrates how complex the interiors may actually be.

Figure 2.

Three-layer internal structure models for Uranus. (a) ‘Hot’ Uranus model with an imposed thermal boundary layer (TBL) that fits gravity and luminosity data. (b) ‘Icy’ Uranus model without a TBL using updated equations of state that fits gravity but not luminosity data. Adapted from [26,27]. (Online version in colour.)

So how does one determine where the dynamo is generated in the ice giants? With present data and constraints, we are forced to use a combination of elimination and plausibility to answer this question. The hydrogen/helium envelope does not reach pressures high enough to attain strong electrical conductivity. The rocky core, although it may contain all the necessary ingredients for dynamo action, is likely too deep to explain the multipolar nature of Uranus’ and Neptune’s observed surface magnetic fields since the power in smaller-scale field components decays much more strongly with distance than the larger-scale (dipole) component. Eliminating those two layers leaves the ice-rich mantle layer. Considering water as an example of the ices making up the majority of Uranus and Neptune, the phase diagram suggests that appreciable conductivity can be reached at depths of approximately $0.2-0.3$ planetary radii below their surfaces. At these depths, the conductivity is ionic in origin, and although ionic conductivity may be several orders of magnitude lower than the metallic conductivity of iron in the core, estimates of Rm for the ice giants suggest that the conductivity is large enough for the ice-rich mantles to be super-critical to dynamo action. These regions are also expected to be unstable to thermochemical convection, with thermal contributions due to secular cooling, radioactive heating and potentially latent heat release associated with phase transitions, and compositional contributions due, for example, to the formation of lighter elements associated with chemical reactions (e.g. [28]). Double-diffusive convection may also arise from an unstable thermal gradient in the presence of stable compositional stratification (e.g. [29]).

3. Lessons from dynamo theory and experiments

A variety of tools are used to investigate the characteristics of planetary magnetic fields and the fluid flows responsible for their generation, including theoretical arguments, numerical models and laboratory experiments. These tools can be combined to develop scaling laws that allow a dynamo property, such as the strength and dipolarity of the magnetic field, to be estimated using fundamental properties of the planet (e.g.thickness of the dynamo region, rotation rate, fluid properties). Scaling laws also serve to relate numerical models and laboratory experiments to planetary interiors, whose extreme conditions cannot be replicated. While these models may not be tailored to the ice giants explicitly, they provide a broad framework to better understand the physical processes that may be expected. Brief descriptions are given below and we refer the reader to other review papers for more details (e.g. [30–33]).

(a). Field strength

Numerous scaling laws have been proposed to explain the characteristic magnetic field strength inside the dynamo region. Early studies suggested that the magnetic field strength would be set by the assumption of magnetostrophic balance between the Lorentz and Coriolis forces such that the dimensionless Elsasser number, Λ = σB²/(ρΩ), where B is characteristic magnetic field strength, ρ is density and Ω is rotation rate, is on the order of unity (e.g. [34]). While this scaling is consistent with estimates for Earth and Jupiter [35], it does not work well in numerical dynamo models (e.g. [30,36,37]) nor for Mercury or the ice giants.

Alternatively, scaling laws can be derived from the requirement that a dynamo must be thermodynamically consistent such that ohmic dissipation cannot exceed the energy available to drive the dynamo [38,39]. This approach with the assumption that the characteristic flow speed follows a mixing length scaling (obtained by balancing the nonlinear inertia and buoyancy forces) yields

$$\frac{B^2}{2{\mu }_0}=c{f}_{\text{ohm}}{\rho }^{1/3}{\left(\frac{q_c{R}_c}{H_T}\right)}^{2/3},$$ 3.1

with constant prefactor c that has been shown empirically to be 0.63 [30], the fraction of ohmic to total dissipation f_ohm (typically approx. 0.5 in dynamo models, but approx. 1 may be more appropriate in planetary cores [31]), convective heat flux q_c, dynamo region radius R_c, temperature scale height H_T = C_p/(αg), specific heat capacity C_p, thermal expansion coefficient α and gravitational acceleration g. This result, importantly, has no dependence on the rotation rate nor any diffusivity values. Assuming that magnetic energy depends only on convective power following turbulence theory, this expression can also be derived from first principles (i.e. no assumption for a velocity scaling) [33,40].

Figure 3 shows how magnetic field strength scales with power for a collection of numerical dynamo models that span a wide range of parameter space (that do not overlap with planetary core estimates due to technological limitations [22]). With this power-based scaling law, the models roughly collapse onto a single power law and have a best fit exponent of 0.31, in reasonable agreement with the theoretical expectation of one-third in equation (3.1). Moreover, $|B|\propto q_c^{1/3}$ holds approximately for both dipolar and multipolar dynamos (e.g. [42,43]), both stress-free and no-slip mechanical boundary conditions (e.g. [44]), both fixed temperature and fixed flux thermal boundary conditions (e.g. [45]), both Boussinesq and anelastic models (e.g. [43,46]), and different inner core sizes (e.g. [45]). Thus, equation (3.1) appears to be a robust result that is supported by both theory and dynamo models.

Figure 3.

Magnetic field strengths for (a) numerical dynamo models in dimensionless units and (b) planetary interiors in dimensional units. In (a), black (blue) rimmed symbols denote thermal (compositional) convection cases from [30], red (green) rimmed symbols denote ‘coupled Earth’ (standard) cases from [41]. Symbol shape denotes the Ekman number, E (see legend). The grey scale denotes the magnetic Prandtl number, Pm, where white (black)indicates Pm ≥ 10 (Pm ≤ 0.1). Estimates for the geodynamo are in the green shaded box. Le = B/((ρμ₀)^1/2ΩL) is the Lehnert number, or the ratio of the periods of inertial and Alfvén waves. E = ν/(ΩL²) is the ratio of rotational to viscous timescales (ν is kinematic viscosity). Pm = ν/η is the ratio of magnetic to viscous diffusion timescales. In (b), field strength inside the dynamo region (at the planetary surface) is given on the left (right) y-axis. The x-axis includes an efficiency factor F that accounts for radial variations in fluid properties [39]. The diagonal black line denotes the predicted behaviour following equation (3.1). E denotes Earth, J denotes Jupiter, S denotes Saturn, S${\hspace{0.17em}}^{\ast }$ denotes a deep-seated Saturnian dynamo, U denotes Uranus and N denotes Neptune. Adapted from [32,33]. (Online version in colour.)

In order to apply this scaling to planets, several additional steps are required. First, the depth to the top of the dynamo region must be assumed since the dipole field strength decreases with radius as r³ and decreases even more rapidly for higher degree components. Second, the ratio of the mean field strength in the dynamo region to the field strength at the top of the dynamo region must be assumed since only the large-scale radial component is observable; dynamo models suggest that the dynamo surface field tends to be a factor of three to four times smaller than the internal field [30]. Third, the convective heat flux must be assumed, but is often poorly constrained. Figure 3b compares the magnetic field strength predictions of equation (3.1) against estimates for the geodynamo and giant planets. The predictions work well for Earth and Jupiter, fall within the ranges of uncertainty for Uranus and Neptune, and do not match for Saturn unless a deep-seated dynamo is assumed. Thus, the scaling is also moderately consistent with planetary estimates.

(b). Field morphology

Numerical dynamo models are the primary tool for studying magnetic field morphology and have revealed numerous ways to generate multipolar magnetic fields. Magnetic field morphology is often characterized using dipolarity, f_dip, which measures the strength of the dipole component to the total field strength (e.g. [47]) or to the combined field strength in spherical harmonic degrees 1 to 12 (e.g. [30,48]) on the outer boundary of the dynamo. For axial dipolarity, $f_{\text{dip}}^{m=0}$, the dipole component is limited to only the axisymmetric m = 0 contribution (e.g. [48]). Early studies focused on the geodynamo typically assumed a Boussinesq fluid and considered a thick shell geometry, no-slip mechanical boundary conditions and fixed temperature thermal boundary conditions such that buoyancy is concentrated along the inner boundary (termed ‘standard’ models) (e.g. [36,38,49,50]). Near the onset of convection, stable dipole-dominated magnetic fields are obtained and the solutions transition to multipolar dynamos that evolve more rapidly in time as convective supercriticality is increased.

Nonlinear inertia is found to play a ubiquitous role in the transition from dipole-dominated to multipolar magnetic fields (e.g. [36,38,42,51,52]). Most frequently, inertia is compared against the Coriolis force through a local Rossby number: Ro_ℓ = U/(Ωℓ), where ℓ is the typical convective length scale. Figure 4a shows that dipole-dominated solutions are obtained when $R{o}_{\ell }\lesssim 0.12$ in standard cases with χ = r_i/r_o = 0.35 (ratio of inner to outer shell radii) for a wide range of input parameters, with multipolar dynamos occurring for larger Ro_ℓ values. The behaviour becomes richer when stress-free boundary conditions and internal heating are considered. In particular, bistable solutions emerge where both dipolar and multipolar fields are obtained for the same input parameters and boundary conditions depending on the initial conditions [44,53–55]. This bistability is associated with the development of zonal winds, where dipolar solutions have weaker zonal winds and multipolar solutions are associated with stronger winds. The range of magnetic field morphologies also becomes more exotic, including quadrupolar, oscillating and hemispheric dynamos [53,56–59]. The dipolar-multipolar transition becomes less distinct when convection is driven by internal heating, but still roughly follows Ro_ℓ ∼ 0.1 [42,45,55].

Figure 4.

(a) Dipolarity (based on magnetic field strength) versus the local Rossby number for standard dynamo models with a range of Ekman numbers (see legend). Interior colour denotes Pm: blue Pm < 1, white Pm = 1 and red Pm > 1. Interior symbolsdenote the Prandtl number, Pr = ν/κ (ratio of thermal to viscous diffusion times, κ is thermal diffusivity): empty Pr = 1, cross Pr > 1, circle Pr < 1. Most cases have an electrically insulating inner core. Vertical line at Ro_ℓ = 0.12 indicates the approximate transition between dipolar and multipolar dynamos. Adapted from [30]. (b) Dipolarity (based on magnetic energy) versus the local Rossby number for anelastic dynamo models with a range of density stratifications (see legend). Hollow (filled) symbols denote χ = 0.2 (χ = 0.6). Symbol size indicates magnetic field strength. Boundary conditions are fixed temperature and either no slip at the inner boundary and free slip at the outer boundary (mixed) or stress-free at both boundaries. The inner core is electrically insulating. Vertical lines indicate the approximate transitions between dipolar and multipolar dynamos for χ = 0.2 (short dash) and χ = 0.6 (long dash). For (b,c), E = 10⁻⁴, Pm = 2 and Pr = 1. Adapted from [47]. (c) Axial dipolarity (based on magnetic energy) versus density stratification for anelastic dynamo models with χ = 0.2 and the (dimensionless) radial electrical conductivity profiles shown in (d). Boundary conditions are fixed temperature and mixed mechanical, and the inner core is electrically diffusive. Error bars correspond to standard deviations of the time series of each case. Adapted from [48]. (Online version in colour.)

Looking towards giant planets and stars, anelastic effects due to background density stratification as measured by the number of density scale heights, N_ρ = ln(ρ_i/ρ_o), where ρ_i and ρ_o are densities at the inner and outer boundaries, respectively, are shown in figure 4b. As for Boussinesq cases (N_ρ = 0), bistability is evident for stress-free boundary conditions [46,47], which manifests as low dipolarity solutions within the low Ro_ℓ region. Excluding these bistable cases, the Ro_ℓ ∼ 0.1 transition criterion still appears to hold. Differences in shell thickness have a secondary influence, with the transition decreasing to Ro_ℓ ∼ 0.08 for thicker shells (χ = 0.2) and increasing to Ro_ℓ ∼ 0.15 for thinner shells (χ = 0.6). The addition of strong density stratification, however, can inhibit the generation of dipolar magnetic fields due to the concentration of convective features in the lower density region near the outer boundary [46,47]. No dipolar solutions are found for N_ρ > 2 by [47], although they can be recovered by increasing the magnetic Prandtl number (i.e. electrical conductivity) [46].

When electrical conductivity variations with radius are also incorporated, the behaviour becomes even more complex and the Ro_ℓ ∼ 0.1 condition breaks down [48,60–63]. Figure 4c shows how the electrical conductivity profile (figure 4d) modulates the magnetic field morphology as a function of density stratification. For small conductivity variations (χ_m > 0.9), dipolar solutions are found only for small density stratifications (N_ρ < 2). For larger conductivity stratifications, the opposite behaviour is found, where large density variations are required for dipolar solutions. This occurs because zonal winds become concentrated near the outer boundary, where interaction with the magnetic field is diminished.

Properties of the inner core, such as its electrical conductivity and relative size, may also influence the solutions (e.g. [64,65]) [cf. [66,67]]. If electrically conducting and sufficiently large, the inner core can help maintain dipolar dynamos and decrease their reversal frequencies (e.g. [68,69]). Conversely, an insulating inner core or deep stably stratified layer below the convecting fluid can prohibit anchoring of the dipole such that multipolar dynamos are preferred [68–70]. A shallow stable layer above the convecting fluid tends to have the opposite effect and enhance the axisymmetry of the solutions [71–75], although a disruption of the dipole is also possible [76].

In summary, there are many paths to multipolar magnetic fields, such as strong inertial effects, bistability, strong density stratification and deep stable layers. Application of these results to the ice giants is reviewed below.

4. Proposed explanations for ice giant dynamos

Several hypotheses were proposed to explain the unique magnetic field configurations of Uranus and Neptune revealed by the Voyager 2 encounters. After the Uranus flyby, an ongoing reversal and the planet’s large obliquity was hypothesized to be responsible [77], but fell out of favour when the Neptune flyby also revealed a multipolar magnetic field despite a smaller obliquity. An in-progress reversal of the magnetic field was also proposed after the Uranus encounter [78], but after observations of Neptune’s field, it was recognized that this was a statistically improbable scenario if a reversal frequency similar to that of the geomagnetic field is assumed. The large dipole tilts were additionally suggested to be related to a small number of large-scale convective cells in the interior [9,10], and the lack of magnetostrophic balance between the Lorentz and Coriolis forces was suggested to explain why the weak ice giant fields differ from those of Earth and the gas giants [5]. These arguments do not explain, however, why multipolar fields would be preferred. Deep, thin shell dynamos have also been proposed (potentially related to a metalized carbon layer below the ice layer) [79], but multipolar field morphologies at the surface are difficult to achieve with this mechanism due to the rapid attenuation of smaller-scale fields with distance from the dynamo source region. Thus, none of these early explanations are entirely satisfactory.

Shallow thin shell dynamos overlying a region of stable stratification were suggested by [80] to explain the low luminosities of Neptune and especially Uranus. Building upon this idea, Stanley and Bloxham [68,70] carried out a suite of numerical dynamo simulations to test this hypothesis. Figures 5 and 6 compare dynamo simulations with solid inner cores of different sizes (indicated by χ) and with deep stable layers of different thicknesses (indicated by χ_s = r_s/r_o, where r_s is the outer radius of the stable layer). Models with solid inner cores were only found to produce multipolar magnetic fields if the inner core is more magnetically diffusive (i.e. less electrically conducting) than the convecting fluid. This is shown qualitatively in figure 5c with η_io = η_i/η_o = 100 compared to a, b and d with η_io = 1, and quantitatively in figure 6 (top row, η_io ≥ 10²). By contrast, deep stable layers lead to ice giant-like magnetic fields in all models except for when the convecting layer is very thin (figure 5h–j; figure 6, middle row, χ_s ≤ 0.7). All models have field strengths of Λ ∼ 1, which is much larger than the top of the dynamo region estimate of Λ ∼ 10⁻⁴ for the ice giants. The multipolar solutions are also found to evolve on the advective time scale. The radial magnetic field snapshots shown in figure 5 are therefore not indicative of the fields at all times, although the distribution of scales and symmetries in the snapshots is similar to that of Uranus’ and Neptune’s observed fields, as reflected in the time-averaged magnetic spectra (figure 6a–d).

Figure 5.

Magnetic and velocity fields in models with (a–g) solid innercores of different sizes given by χ = r_i/r_o and (h–n) deep stable layers of different sizes given by χ_s = r_s/r_o. (a–d,h–k) Radial magnetic fields at the top of the dynamo source region are instantaneous in time; colours denote field directions. (e,l) Zonal flow snapshot averaged over all longitudes in the fluid layer corresponding to the models in c and j; orange (blue) denotes prograde (retrograde) flow. (f ,m) Meridional circulation flow vectors averaged over all longitudes. (g,n) Axial vorticity, ${\omega }_z=\mathrm{\nabla }\times \mathbf{\text{u}}\cdot \widehat{\mathbf{\text{z}}}$, in the equatorial plane; red (blue) denotes cyclonic (anticyclonic) circulation. Input parameters are E = 4 × 10⁻⁵ (1 − χ)⁻², Pr = 1, Pm = 1 and Ra varied such that all models produce similar magnetic field intensities of $\mathit{\Lambda }=\mathcal{O}(1)$. Boundary conditions are fixed flux and stress-free, the inner core/fluid has the same electrical conductivity as the outer fluid layer or is several orders of magnitude less. Adapted from [68]. (Online version in colour.)

Figure 6.

Magnetic power spectra from dynamo models with (a,b) solid inner cores with different radius ratios χ, inner/outer core magnetic diffusivity ratios η_io and outer core convective vigour (adapted from [68]), (c,d) stably stratified inner regions with different thicknesses χ_s and outer core convective vigour (adapted from [68]), and (e,f ) strongly forced convection with thick and thin shell geometries (after [81]). Spectra in (a–d) are averaged in time and upward continued to the surface assuming the top of the dynamo region is located at 0.7 planetary radii. Spectra in (e,f ) are time-averaged and taken at the outer boundary. H09 refers to [4], HB96 refers to [5]. (Online version in colour.)

Considering the models with χ = 0.7 (figure 5c) and χ_s = 0.7 (figure 5j), convection is characterized by axial ‘Taylor columns’ that are aligned with the rotation axis (figure 5g,n). The axisymmetric zonal (east–west) flows in both models are organized into a prograde (eastward) jet far from the rotation axis with retrograde (westward) flow at smaller cylindrical radii (figure 5e,l). It should be noted that these models only simulated the dynamo regions of the ice giants, i.e. they did not include the low-conductivity regions such as the molecular water region and gas-rich envelope. It is, therefore, not straightforward to compare the zonal winds in these models with those observed at the surfaces of Uranus and Neptune. It is nonetheless interesting to note that both the model and the planets’ zonal winds are organized into a three jet structure, albeit with opposite directions [82]. The axisymmetric meridional circulations have less defined structures (figure 5f ,m).

These models provide two possible mechanisms for generating ice giant-like magnetic fields: a solid inner core that is less electrically conductive than the overlying fluid shell or a deep stably stratified layer beneath the convecting fluid shell. Although the properties of superionic ice are not well known (figure 2), it is believed to behave as a solid [83] and to have an electrical conductivity exceeding that of ionic water [25], arguing against the first option. While the presence of a superionic ice layer would also argue against the second option, properties of the interior are sufficiently poorly constrained that stable stratification in the deep interior cannot be ruled out [23]. If a thick stably stratified layer exists deep in the interior, the convecting region would be relatively thin to provide a short length scale that is more conducive to non-dipolar magnetic fields.

An alternative hypothesis is that the ice giants have multipolar dynamos because convective turbulence in their interiors is weakly constrained by rotation (i.e. inertial effects are strong), in contrast to rotationally constrained convection in the deep interiors of the gas giants that drives dipole-dominated dynamos [81,84,85]. Figures 6 and 7 compare dynamo simulations with thick (χ = 0.35) and thin (χ = 0.75) shell geometries that are strongly driven to produce turbulent convection that is not organized into columnar structures (figure 7e,j). In both cases, the resulting magnetic fields are small scale and vary strongly with time. When limited to the largest spatial scales commensurate with observations (l ≤ 3, figure 7b,g), multiple flux patches with different polarities are evident in both the northern and southern hemispheres with field strengths on the order of Λ ∼ 10⁻³, approaching agreement with the observations. Time-averaged magnetic power spectra indicate that the multipolar nature of the dynamo is persistent, but does not capture the m = 1 peak of both planets well (figure 6, bottom row).

Figure 7.

Magnetic and velocity fields in models with (a–e) thickshells (χ = 0.35) and (f –j) thin shells (χ = 0.75). (a,b,f ,g) Radial magnetic fields near the top of the dynamo region are instantaneous in time; colours denote field directions. (a,f ) At full spatial resolutions; (b,g) limited to spherical harmonic degrees l ≤ 3. (c,h) Zonal flow averaged over time and all longitudes in the fluid layer; red (blue) denotes prograde (retrograde) flow. (d,i) Meridional circulations averaged over time and all longitudes; red (blue) denotes clockwise (counterclockwise) flows. (e,j) Axial vorticity isosurfaces in the bulk fluid; the yellow sphere represents the solid inner core. Red (blue) denotes cyclonic (anticyclonic) ω_z values. Input parameters are E = 3 × 10⁻⁴, Pr = 1, Pm = 1 and Ra = 2.22 × 10⁷. Boundary conditions are isothermal and stress-free, and the outer fluid and solid inner core have the same electrical conductivity. Adapted from [81]. (Online version in colour.)

The models also produce retrograde equatorial jets that are accompanied by overturning circulations with upwelling at low latitudes and downwelling near the tangent cylinder (i.e. axial cylinder that intersects the inner boundary of the shell at the equator) (figure 7c,d,h–i). Additional meridional circulations with polar upwellings develop if the spherical shell is thick. Strong prograde jets develop at smaller cylindrical radii in the thick shell, while the magnetic field decelerates the zonal flows in the thin shell dynamo case. These winds can, therefore, be similar qualitatively to those observed on Uranus and Neptune [82].

While this model seems promising, it is not yet clear if convection weakly constrained by rotation is likely in the ice giant interiors (see [81,85] for further discussion on convective regime estimates of the giant planets). This question is especially acute for Uranus, where the internal energy appears to be small [86]. These models also do not include background density stratification nor variation in electrical conductivity with depth, so the outer molecular envelope is only included in an overly simplified manner.

Bistability has also been suggested as a possible explanation for why Uranus and Neptune have multipolar dynamos while Jupiter and Saturn have dipole-dominated magnetic fields [44,47], which would require small local Rossby numbers for all four of the giant planets [42]. The interplay between electrical conductivity and density stratification has not yet been investigated explicitly in the context of Uranus and Neptune [cf. [87]]. Three-layer internal structure models suggest N_ρ ∼ 1.5 for the water layer of Neptune ($0.30-0.85\hspace{0.17em}{R}_N$) and N_ρ ∼ 1.3 for Uranus ($0.18-0.78\hspace{0.17em}{R}_U$) [88]. Models with gradual compositional changes can have larger values of N_ρ ∼ 2.4 for Uranus and N_ρ ∼ 1.9 for Neptune considering the region extending from the centre to 0.85 planetary radii [89]. The electrical conductivity profile is not well constrained, but is expected to change by roughly an order of magnitude across the ionic water layer and more rapidly across the molecular envelope [90,91]. Thus, both density and electrical conductivity stratifications appear to be moderate at depth, so this potential explanation requires further study.

In summary, numerous hypotheses have been proposed to explain the multipolar magnetic fields of Uranus and Neptune. However, in order to determine which—if any—of these explanations is correct, additional work is needed in regards to observational constraints as well as dynamo and rotating convection behaviour as described in the next section.

5. Future directions

The planetary magnetic fields of our solar system show remarkable variations that reveal secrets about their deep interiors. The major revelations of the structure and dynamics of the interiors of Jupiter and Saturn from Juno and the Cassini Grand Finale, respectively, place our ignorance of ice giant interiors in stark contrast (e.g. [3,92]). Many questions remain about Uranus’ and Neptune’s interiors and magnetic fields, such as:

• 1.What are the detailed configurations of their magnetic fields? Has SV occurred since the Voyager 2 flybys?
• 2.How deep do the atmospheric circulations observed on the surface extend into the interior and do they interact with the dynamo?
• 3.What is the internal density and compositional distribution? Do layers of stable stratification and/or double diffusion exist?
• 4.How do the thermodynamic and transport properties of the planets vary with radius and with time as the planet evolves?
• 5.What processes generate the dynamo? What are the characteristics of zonal winds, meridional circulations and turbulent convective flows in the deep interior?
• 6.What are the dynamo characteristics of ice giant exoplanets? How do they compare to Uranus and Neptune, as well as to gas giant exoplanets and super-Earths?

The first three questions, in particular, illustrate the need for a new mission to Uranus and/or Neptune (e.g. [93]). New magnetic field measurements close to the planet at a variety of latitudes and longitudes would allow characterization of the ice giants’ higher degree magnetic field structure, ideally to better than spherical harmonic degree 10 for comparison with Earth, Jupiter and Saturn. This field determination could be further improved through imaging of auroral and satellite footprints that provide additional constraints (e.g. [4]). The internal magnetic field may have undergone temporal change since the Voyager 2 epoch so new observations, even from a flyby, would provide constraints on SV and potentially identify changes in the locations of flux patches that are indicative of zonal and/or meridional winds in the deep interior (e.g. [15,16]). The depth of zonal wind penetration into the planets as well as their radial density distributions may be established through measurements of the gravity field. Remote sensing would allow inference of deep meridional circulations, elemental abundances in the atmosphere and internal heat flow. Observations of planetary oscillations and/or ring seismology might further constrain interior flows and identify layers of convection versus stable stratification (e.g. [94,95]). In situ measurements of winds, temperatures and composition by a probe would provide ground truth for interpretations of the potential field and remote sensing observations.

The fourth question relies on advancements in our understanding of planetary materials at high pressures and temperatures, especially for mixtures and multi-phase phenomena, coupled with interior structure models (e.g. [23]). Equations of state are a necessary and critical ingredient of internal structure and thermal evolution models that illustrate where the dynamo regions are likely located within the planets and how they might have evolved over time. The mode(s) and efficiency of heat transfer depend on thermodynamic properties, such as specific heat capacity and thermal expansion coefficient, while the dynamics of convecting regions additionally depends on diffusivities (thermal, kinematic and magnetic). Mean values of these properties are useful to estimate the dimensionless parameters that control convection and dynamo action (e.g. E, Pr, Pm, Rm), and their variations with depth may also modulate magnetic field generation as described in §3.

Continued numerical modelling and laboratory experiments of convection and dynamo action are necessary for the fifth question, encompassing several directions. As new hypotheses for magnetic field generation progress with advancements in ice giant internal structures and physical properties, dynamo models provide a means to test these ideas. Dynamo model coupling with atmospheric dynamics and radiative transfer models would also be an interesting future direction. These models would make detailed predictions about magnetic fields, flow fields and heat transfer that could serve to guide instrument and mission concept designs. Additionally, it is important to better understand and test convective regime transitions and their influence on magnetic field generation to answer questions like the feasibility of the ice giants being in the weakly rotating convective regime. As a last example, more realistic parameters (e.g. E, Pm) should be investigated as computing capabilities continue to improve with time since viscous effects may be (artificially) important in existing models [36] and novel behaviours are being found as new regions of parameter space are explored (e.g. [96]).

Comparative planetology is another powerful tool to understand ice giant dynamos, per the sixth question. Exoplanets are routinely being discovered, with sub-Neptunes being among the most prominent [97]. Magnetic fields have been predicted theoretically for exoplanets (e.g. [98]) and telescopic observations now enable their field strengths to be estimated [99,100]. Determining the strengths and morphologies (i.e. dipole-dominated or multipolar) of these planets and their zonal wind profiles will provide critical data points to assess dynamo generation hypotheses more broadly [101, e.g.]. As we learn more about ice giant planets around other stars, we will also learn more about Uranus and Neptune closer to home.

Data accessibility

This article does not contain any additional data.

Authors' contributions

Both authors contributed to the writing of the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the NASA Solar System Workings Program(grant nos. NNX15AL56G (K.M.S.) and 80NSSC20K1046 (S.S.)).