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Edited by: Peter E. Driscoll, Carnegie Institution for Science (CIS), United States

Reviewed by: Binod Sreenivasan, Indian Institute of Science (IISc), India; Kenneth Philip Kodama, Lehigh University, United States

This article was submitted to Geomagnetism and Paleomagnetism, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Convection in the Earth's outer core is driven by buoyancy sources of both thermal and compositional origin. The thermal and compositional molecular diffusivities differ by several orders of magnitude, which can affect the dynamics in various ways. So far, the large majority of numerical simulations have been performed within the codensity framework that consists in combining temperature and composition, assuming artificially enhanced diffusivities for both variables. In this study, we use a particle-in-cell method implemented in a 3D dynamo code to conduct a first qualitative exploration of pure compositional convection in a rotating spherical shell. We focus on the end-member case of infinite Schmidt number by totally neglecting the compositional diffusivity. We show that compositional convection has a very rich physics that deserves several more focused and quantitative studies. We also report, for the first time in numerical simulations, the self-consistent formation of a chemically stratified layer at the top of the shell caused by the accumulation of chemical plumes and blobs emitted at the bottom boundary. When applied to likely numbers for the Earth's core, some (possibly simplistic) physical considerations suggest that a stratified layer formed in such a scenario would be probably weakly stratified and may be compatible with magnetic observations.

Convection in the Earth's outer core is presently driven by buoyancy sources of both thermal and compositional origin. The thermal source results from the combination of the heat extracted by the mantle, the latent heat released at the bottom of the fluid core by the crystallization of the inner core and possibly the decay of radioactive elements. The chemical source is the consequence of the crystallization of the inner core which releases light elements into the fluid core (Braginsky, _{2} (Hirose et al., ^{T} and κ^{ξ}, and for the kinematic viscosity ν of liquid iron in core conditions, with typical ranges: κ^{T} ~ 5 × 10^{−6} − 10^{−5} m^{2}s^{−1} (Poirier, ^{ξ} ~ 5 × 10^{−9} − 2 × 10^{−8} m^{2}s^{−1} (Dobson, ^{−7} − 5 × 10^{−6} m^{2}s^{−1} (de Wijs et al., ^{T}, ^{ξ}, and ^{T}/κ^{ξ} =

the Schmidt number

Classically, the diffusivity difference is ignored in numerical simulations. The thermal and compositional fields are combined into one single variable, the so-called “codensity” (Braginsky and Roberts,

where κ^{turb} is a homogeneous and isotropic “turbulent” or “eddy” diffusivity. Since turbulent mixing would affect all quantities in a similar way, the Prandtl, Schmidt, and Lewis numbers should be of order one:

This approximation is convenient to implement in numerical codes but relies on debatable approximations. First, the turbulent regime of the core is not known, though under the effect of rotation it is probably anisotropic (Braginsky and Roberts, ^{−6}, where

In addition to the high value of the Schmidt number, another fundamental specificity of compositional convection arises from the complexity of the chemical processes occurring at the “mushy layer” that may exist at the interface between the fluid outer core and the crystallizing inner core. Experiments conducted on model systems involving the freezing of ammonium chloride in solution (Chen and Chen,

A second interesting phenomenon envisioned by Moffatt and Loper (

Here, we propose to explore pure chemical convection at high Schmidt number by means of numerical simulations and test whether the accumulation of chemically buoyant parcels of fluid can lead to the formation of a stably stratified layer at the top of the core. Focusing on compositional effects only allows us to gain insight into the specific phenomena involved, independently of an additional forcing of thermal buoyancy. However, though moderately large Prandtl numbers can be reached with the classical codes (for instance, Simitev and Busse, ^{ξ} = 0 and that our approach thus differs from that of Zhang and Busse (

This paper is organized as follows. In section 2, the mathematical model and the numerical method are described. The results of a series of numerical simulations of pure compositional convection are then presented. The chemical structures and their dynamics are first studied in section 3. In section 4 we show that, when no light elements are allowed to cross the core-mantle boundary, a chemically stratified layer systematically forms at the top of the spherical shell in all simulations. The potential implications for the Earth's outer core are discussed in section 5. The final section 6 gives a general conclusion.

The fluid considered is a mixture of a heavy component, representing iron and nickel, and a light constituent, whose compositional mass fraction is denoted by ξ. As thermal effects are neglected here, the density ρ can be modeled as a function of the compositional mass fraction only,

where ξ_{0} and ρ_{0} are the reference compositional mass fraction and density, respectively, and β denotes the coefficient of chemical expansivity.

In addition to the mass conservation,

we solve for the momentum conservation, expressed in the form of the well-known Navier-Stokes equation, for a Newtonian rotating fluid in the Boussinesq approximation,

where _{o} the gravitational acceleration at the top of the shell at radius _{o}. The Lorentz force is neglected. We adopt the viscous time scale ^{2}/ν in which the thickness of the shell _{o} − _{i} is taken as the typical length scale, _{i} being the radius of the inner boundary. Hence, using (ρ_{0}νΩ) as a scaling for pressure, one can write the dimensionless momentum equation,

in which

and _{ξ} a “modified” compositional Rayleigh (or Roberts-Rayleigh) number,

where ^{*}/^{*} depends on the prescribed boundary conditions which are discussed in section 2.2. Note that, in the infinite Prandtl number case, the classical Rayleigh number constructed using the time scale

cannot be used as it is infinite, which means that even a very small value of _{ξ} would lead to buoyancy-driven fluid motion and the notion of super-criticality is no longer relevant in this case.

In the Earth's fluid core, chemicals can be transported by advection and diffusion. The corresponding equation yields

in which σ^{ξ} is a numerical sink/source term ensuring that the mean composition remains constant throughout the simulation (see section 2.2 for details). The equivalent dimensionless equation reads

In the end-member case ^{ξ} = 0) considered here, Equation (12) becomes a first-order hyperbolic equation:

We adopt non-penetration and no-slip conditions at both boundaries which implies

The first-order nature of Equation (13) combined with the condition (14) on the velocity implies that no chemical “signal” imposed at a boundary can propagate inside the shell, neither by advection nor diffusion. The value of the composition at a boundary has therefore no effect on the compositional field inside the shell and will in turn not be affected by the latter. Consequently, once the initial condition has been specified (for instance ξ = 0 everywhere), there is

The choice of an adequate function for

The constants in the exponential function are empirically set to σ = ^{1/3}. These values will be justified later in section 3.1. The scaling factor σ_{0} is computed to satisfy Equation (15). Mathematically, this formulation is equivalent to solving the transport equation

and imposing no boundary conditions for the composition (at least, “mathematically” speaking). The injection of light elements occurring in the ICB region is accounted for by the local source term ^{ξ} in the entire shell so that the mean composition remains constant with time:

Using the corresponding compositional buoyancy flow injected at the inner boundary,

one can derive the following scale for composition:

The expression of the modified compositional Rayleigh number then becomes

Illustration of the shape of the source term

Though moderate Schmidt numbers (< 50) may be numerically accessible with most classic dynamo codes, the infinite Schmidt number case requires a specific numerical method. Indeed, the time integration schemes classically used in the spherical dynamo codes (Crank-Nicolson for diffusion and Adams-Bashforth for the non-linear terms) are conditionally stable and produce spurious oscillations when the chemical diffusivity is set to zero. Other numerical methods such as TVD schemes have been specifically designed for hyperbolic equations but still introduce some amount of numerical diffusion and/or “clipping” effects (see Munz,

In this work, we use a recently developed version of the dynamo code PARODY-JA (Aubert et al., ^{ξ} = 0). For one simulation (case 9^{*}), the homogeneous sink term σ^{ξ} in Equation (17) is replaced by a sink term near the top boundary equivalent to _{c}-periodic along the longitude (see

Summary of the parameters used for the different simulations.

_{r}, ℓ_{max}) |
_{p} (× 10^{6}) |
_{c} |
|||
---|---|---|---|---|---|

1 | (150,200) | 200 | 1 | 10^{−3} |
5 × 10^{2} |

2 | (150,200) | 200 | 1 | 10^{−3} |
10^{4} |

3 | (150,200) | 50 | 4 | 10^{−3} |
10^{5} |

4 | (150,200) | 50 | 4 | 10^{−3} |
5 × 10^{5} |

5 | (150,200) | 50 | 4 | 10^{−3} |
10^{6} |

6 | (200,270) | 110 | 6 | 3 × 10^{−4} |
10^{5} |

7 | (200,270) | 110 | 6 | 3 × 10^{−4} |
10^{6} |

8 | (250,360) | 250 | 8 | 10^{−4} |
10^{5} |

9^{*} |
(250,360) | 250 | 8 | 10^{−4} |
10^{3} |

10 | (300,540) | 500 | 12 | 3 × 10^{−5} |
2 × 10^{6} |

11 | (350,720) | 800 | 16 | 10^{−5} |
10^{6} |

_{r} is the number of radial layers. ℓ_{max} is the maximal degree of spherical harmonics, equal to the maximal order m_{max} in all simulations. N_{p} is the number of particles, m_{c} the symmetry in longitude (1 for full spherical shell, higher value for 2π/m_{c}-symmetric solutions). In all simulations, the grid aspect ratio is fixed and equal to r_{i}/r_{o} = 0.35. For case 9^{*}, no volumetric sink term is introduced in the shell. It is replaced by a top flow, imposed in the form of a local sink term that balances the bottom flow

The continuous injection of light elements creates a lighter layer immediately above the bottom boundary. This configuration is prone to the Rayleigh-Taylor instability which, after some time, causes the layer to destabilize, forming thin rising filamentary chemical plumes with a “sheet-like” or “curtain” shape elongated in the direction of the rotation axis owing to the Taylor-Proudman constraint (see

_{0} centered on a chemical “curtain.”

When they rise, the first chemical plumes carry away light elements and make the bottom light layer more heterogeneous. As a consequence, the following generations of compositional “sheets” assume a more complex and discontinuous structure since light elements are not always locally available to supply the rising curtains (see

We measured the mean thickness δ of these sheet-like or filamentary rising plumes for five different values of the Ekman number in the range 10^{−5} − 10^{−3}. We estimated δ for the first generation of plumes by plotting zonal profiles of the composition in the equatorial plane, at a short distance above the bottom light layer. For each “peak” corresponding to a plume in the compositional profile, we define the width of the plume as the distance over which the composition is positive. This is facilitated by the fact that, at the beginning of the simulation, the background composition is homogeneous and negative due to the volumetric sink term σ^{ξ}. The results are displayed on

which is the same exponent as the onset of classical thermal convection in a rotating spherical shell (Jones et al.,

Mean thickness of the chemical plumes measured in the equatorial plane for different values of the Ekman number.

The exact value of the pre-factor in the scaling law (22) may be slightly sensitive to the parameters σ and

_{ξ} denotes the fraction of chemical forcing.

When rotational effects remain moderate, chemical plumes rise almost radially (see

Azimuthal average of the chemical field for cases 3 (^{−3}, ^{−3} and ^{−4}. The color scales have been deliberately modified and saturated in order to better visualize the structure of the compositional field in the convecting region.

Azimuthal section of the velocity perpendicular to (Vs) and parallel to (Vz) the rotation axis, for case 8 at time

^{−4}.

Due to the absence of chemical diffusion, rising chemical plumes can be affected by secondary instabilities, some of which are usually not observed for thermal plumes (Kerr and Mériaux,

Several laboratory experiments have been dedicated to the study of the dynamics of buoyant plumes in a rotating environment. These were conducted in rotating cylinders in which the rotation axis is parallel to the gravity vector, a situation that is analogous to polar regions in the Earth's outer core. When discharging a single point turbulent plume at the top of a water-filled rotating tank, Frank et al. (_{0} the source buoyancy flux. The vertical extent of the precession region and the time necessary for the development of precession were found to decrease with the Rossby number. Helical motions were also reported for turbulent thermal plumes in laboratory experiments of rotating convection modeling the tangent cylinder region (Aurnou et al.,

We observed similar behaviors for chemical plumes in our simulations. On ^{−4} and ^{−3}, the pronounced plume at the north pole shows some helical undulation before a chemical blob detaches and continues to rise, a phenomenon similar to that reported by Claßen et al. (^{−5} and ^{−4} (see _{δ} ~

We also observed other “blob-instabilities” similar to that reported by Claßen et al. (

Equatorial snapshot of the compositional field for case 9^{*}.

Though plumes have a rather laminar behavior in our simulations (which operate at low Reynolds numbers),

^{*} after the initial bottom light layer starts destabilizing at the beginning of the simulation.

The efficiency of plumes mixing caused by the entrainment of heavier fluid differs between the rotating and non-rotating cases. Many studies have pointed out that turbulent mixing and lateral entrainment of fluid are inhibited in rotating flows (Helfrich and Battisti,

In all our simulations, a fraction of the rising chemical plumes and blobs reach the top of the spherical shell where they accumulate to form a stably stratified layer. This layer is clearly visible on equatorial sections of the chemical field (_{b} above which the mean Brunt-Väisälä frequency

where

in which _{b}, defined by Equation (24), is indicated by dashed lines on ^{2} ratio, respectively. Similar shapes are observed for these profiles in the other simulations. _{b} which require time averaging. In addition, the definition of _{b} remains somehow arbitrary since there is dynamically no neat boundary delimiting the convective and stratified regions. Indeed, the low values of the

^{2} ratio at time

Furthermore, the mass fraction of light elements at the very top of the stratified layer also increases from the equator to the poles. We make the hypothesis that this is due to the different mixing efficiency of the plumes inside and outside the TC. In polar regions, plumes rise almost vertically and experience less mixing and dilution than in the equatorial plane. Consequently, they reach the stratified region with a higher mass anomaly than equatorial plumes which experience more fragmentation and mixing.

After a first phase of rapid growth of the layer caused by the destabilization of the initial bottom light layer, the thickness of the layer [based on definition (24)] increases very slowly. ^{−3} and only 20% for case 6 with ^{−4} and comparable Rayleigh numbers (see

This situation bears some interesting resemblance to the “filling-box” models studied experimentally by Baines and Turner (

In the idealized “filling-box” model described above, parametrization of the turbulent plume allows to derive analytically the time evolution of the density profile and the position of the first front (Baines and Turner,

The compositional profiles shown on

Determining the implication of the results shown above for the Earth is a delicate task that is impeded by several factors. First, assessing the “plausibility” of a given scenario for the formation of a stably stratified layer below the CMB is complicated by the apparent inconsistency between studies based on magnetic and seismic observations (see ^{−5} s^{−1}) layer can explain both the 60-year period in the geomagnetic secular variations together with fluctuations in the dipole field and, possibly, length of the day. Dynamo simulations suggest that the morphology of the magnetic field becomes too little Earth-like when the stratified layer is thicker than 400 km (Olson et al., ^{−3} s^{−1}) (Helffrich and Kaneshima,

List of the main studies that proposed some constraints on the stratified layer at the top of the Earth's core based on either magnetic or seismic observations.

^{-1}) |
|||||
---|---|---|---|---|---|

Lay and Young, |
SKS-SKKS | Yes | 50–100 | − | 1–2% Slower P-waves. |

Tanaka, |
SmKS | Maybe | 140 | − | 0.8% Slower P-waves, Possible contamination of the lowermost mantle. |

Gubbins, |
Magnetic | Maybe | ≲ 100 | − | Upwelling necessary near the top of the core. |

Helffrich and Kaneshima, |
SmKS | Yes | 300 | ≲ 10^{−3} ( |
≲ 0.3% Slower P-waves. Total light-element enrichment is up to 5 wt% at the top of the core. |

Alexandrakis and Eaton, |
SmKS | No | − | − | Different analysis of SmKS waves. |

Kaneshima and Helffrich, |
SmKS | Yes | 300 | − | ≲ 0.45% Slower P-waves. |

Buffett, |
Magnetic | Maybe | 140 | ≲ 7.4 × 10^{−5} s^{−1} |
Stratification is compatible with the 60-year period in the geomagnetic secular variation and fluctuations in the dipole field. |

Kaneshima and Matsuzawa, |
SmKS | Yes | 300 | − | ≲ 0.45% Slower P-waves. |

Lesur et al., |
Magnetic | Maybe | − | − | Small poloidal flow required below the CMB. |

Buffett et al., |
Magnetic | Maybe | 130–140 | ≲ 5.4 − 6.1 × 10^{−5} s^{−1} |
Time-dependent zonal flow at the top of the core, fluctuations of the dipole field, and length of day can be explained by a stratified layer. |

Kaneshima, |
SmKS | Yes | Up to 450 | − | ≲ 0.45% Slower P-waves. |

Irving et al., |
Normal modes | No | − | Close to neutral | Stratification not required to fit observations. |

The presence of a magnetic field can influence the dynamics of buoyant parcels of fluid in a non-trivial way (Moffatt and Loper,

On the other hand, the method of light elements injection in our simulations may not adequately reflect the complex chemical processes occurring in the inner core boundary region. Indeed, injecting light elements via a homogeneous source term above the bottom boundary tends to generate plumes with a sheet-like structure instead of the cylindrical shape that would be expected if plumes erupt from isolated chimneys. A step forward toward a more realistic modeling of the mushy layer could consist of injecting light elements within randomly distributed “patches” mimicking local chimneys, although this brings extra complexities since the number of mush chimneys in the ICB conditions is not constrained. Furthermore, the scaling law (22) may not be relevant to predict the size of chemical plumes in core conditions if these plumes erupt from localized chimneys formed in the mushy layer. The initial width of the plumes would in that case likely be controlled by the dynamics of the mushy layer rather than by the Rayleigh-Taylor destabilization of a homogeneous light layer. Unfortunately, the size of the chimneys formed in the mush is presently unknown in core conditions. An alternative scenario is the presence of a slurry layer above the inner core boundary (Loper and Roberts,

Another crucial question is whether chemical plumes would experience a complete turbulent mixing in core conditions. Laboratory experiments by Jellinek et al. (

Assuming the scaling law (22) can be used as a good proxy for the size of chemical plumes in core conditions, we can estimate

Taking ^{−15} and ^{8} for the core, one obtains _{δ} ~ 200, which suggests that plumes would be totally mixed during their ascension in the core, a phenomenon that cannot be captured in our simulations that operate at low global-scale Reynolds numbers (< 100) and therefore, even lower plumes' Reynolds numbers. However, as rotation is known to partially inhibit lateral entrainment of fluid and subsequent mixing (Helfrich and Battisti,

Taking ^{−6} in core conditions, we get _{δ} ~ 0.5 which suggests that rotational effects may still be influencial for chemical plumes. Future investigations will be necessary to quantify the mixing of chemical plumes and the role played by rotation in more turbulent flow regimes.

One may also question the relevance of the infinite Schmidt number approximation used in the simulations. The dynamics of a rising chemical plume is susceptible to be affected by chemical diffusion if the latter acts at a length scale comparable with the size of the plume (or larger) during its typical rising time. Using the scaling law (22) for the average width of the plumes, one can estimate the typical time τ necessary for chemical diffusion to operate at the scale of a chemical plume or blob of size δ,

where τ_{visc} denotes the viscous time ^{2}/ν. For example, ^{−4} and _{visc}. This time must be compared with the typical time it takes for a chemical plume or blob to reach the top boundary, which likely depends on the convective regime. For instance, in case 8 with ^{−4} and _{visc}. For a finite value of

For intermediate Schmidt numbers, chemical diffusion would therefore not be negligible for the dynamics of chemical plumes. A higher value

Despite the numerous limitations of the present study listed above that prevent any direct extrapolation of our results to the Earth's core, some physical considerations can be invoked to constrain the properties of a chemically stratified layer built by the accumulation of light chemical blobs emitted at the ICB. In such a scenario, the total density anomaly Δρ_{L} across the stably stratified layer is bounded by that Δρ of the chemical blobs erupting from the mushy layer (Δρ_{L} ≤ Δρ). The density anomaly Δρ of such chemical blobs is not known and may have varied throughout the history of the inner core. Here we try to obtain an order of magnitude for this density anomaly using very simple hypotheses and calculations. Following Moffatt (_{s} ~ 1.05ρ, where ρ_{s} is the density of the inner core and ρ that of the bulk fluid core, one can write

where ^{−11} m s^{−1} (Labrosse, ^{−5} s^{−1} and ^{−2} leads to

The parameter ^{−3} − 10^{−2} in some laboratory experiments (Copley et al., ^{−4} m s^{−1} below the CMB (Finlay and Amit, ^{−3} m s^{−1} (Christensen and Aubert, ^{−2} m s^{−1} for the chemical blobs gives

^{−7}ρ over a stratified layer of thickness

For instance, ^{−2} and ^{−6} s^{−1}, which is just slightly smaller than the estimation required by Buffett (^{−5} s^{−1} compatible with magnetic observations would require either a thinner stratified layer (^{−6}. This density ratio ratio would then give ^{−5} and ^{−1}. It should be noted that the mean Brunt–Väisälä frequency of the layer estimated using Equation (33) constitutes an upper bound corresponding to the ideal case in which there is no dilution of the chemical blobs along their path to the top of the core (Δρ_{L} ~ Δρ). Provided the force balance proposed by Moffatt (^{−3} s^{−1}) called for by some seismic studies can be achieved only by invoking unrealistically high blobs velocities (^{2} m s^{−1}). One may however criticize the simplicity of this assumption by arguing that the properties of the blobs erupting from the mush chimneys would rather be controlled by the dynamics of the mushy layer. Though it seems more likely that a stratified layer at the top of the core formed by the accumulation of chemical blobs emitted at the ICB would be weakly stratified (most likely ^{−5} s^{−1}) and, very likely, seismically invisible, we cannot exclude stronger stratifications depending on the precise processes occurring in the ICB region.

Mean Brunt-Väisälä frequency (orange) and ^{−7}.

Should the accumulation of chemical blobs alone be unable to explain the presence of a thick stably stratified layer at the top of the core, it might still play an important role. Gubbins and Davies (

Using a new version of the PARODY code including a particle-in-cell method, we ran a first series of exploratory simulations of pure chemical convection at infinite Schmidt number in a rotating spherical shell. In these simulations, light elements are injected within a thin layer above the bottom boundary. This preliminary work remains descriptive and qualitative, but reveals that chemical convection in the Earth's outer core is a rich and vast topic that warrants several more focused and quantitative studies. We showed that, principally as a result of their lower diffusivity, chemical plumes have a complex dynamics that may be influenced by several distinct instabilities in the different regions of the shell. We observed secondary instabilities such as undulations, plumes precession, fragmentation into chemical “blobs” that comply qualitatively well with some previous predictions and laboratory experiments. Plumes may also experience jet-instability and start widening and entraining ambient fluid. A particularly interesting result is the systematic formation of a chemically stratified layer at the top of the shell in all simulations. The layer has a latitudinal topography and slowly grows owing to the accumulation of incompletely mixed chemical plumes and blobs emitted at the bottom boundary.

The existence of a stably stratified layer at the top of the core has been proposed for several decades based on seismic and magnetic observations (Lay and Young,

As our simulations operate with control parameters that are very remote from core conditions and miss some important ingredients like thermal convection, magnetic field and turbulence, it is not clear whether the results shown in this study would hold when approaching more realistic conditions. Future studies should aim at incorporating these essential ingredients and systematically exploring the parameter space in order to understand more quantitatively the physics of compositional convection and make predictions for the Earth. Our preliminary study suggests that a chemically stratified layer formed by the accumulation of chemically buoyant parcels of fluid released at the bottom of the core would probably be weakly stratified and thus, likely seismically invisible. However, this relies on a force balance assumption that may be irrelevant. Constraints on the stratification that can be produced by such a scenario may rather come from the precise dynamics of the mushy layer which is presently poorly understood and deserves future investigations.

MB ran the simulations, analyzed the data, and wrote the manuscript. GC, SL, and JW significantly contributed to the physical reflection, the interpretation of the results, and to the elaboration of the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank David Gubbins, Binod Sreenivasan, and Kenneth P. Kodama for their valuable comments which were very helpful to improve the manuscript. We also thank Philippe Cardin and Hagay Amit for fruitful discussions on this topic.