Un article est en cours de pr´eparation sur le travail effectu´e dans le chapitre 3. Une version non d´efinitive est incluse ci-dessous.
Mélissa D. Menua,b, Ludovic Petitdemangeb, Sébastien Galtierc
aLaboratoire de Physique des Plasmas, Univ. Paris-Sud, Université Paris-Saclay, École polytechnique, CNRS, Sorbonne Université, Observatoire de Paris, F-91128 Palaiseau Cedex, France
bLERMA, CNRS, Observatoire de Paris, PSL Research University, Sorbonne Université, F-75005 Paris, France
5
cLaboratoire de Physique des Plasmas, Univ. Paris-Sud, Université Paris-Saclay, Institut Universitaire de France, École polytechnique, CNRS, Sorbonne Université, Observatoire de Paris, F-91128 Palaiseau Cedex, France
Abstract
The dynamo eect is the most popular candidate to explain the non primordial magnetic elds of astrophysical objects. Although many systematic studies of parameters have already been made to determine the dierent dynamical regimes explored by direct numerical geodynamo simulations, it is only recently that the regime corresponding to the outer core of the Earth characterized by a balance of forces between the Coriolis and Lorentz forces is accessible numerically. In most previous studies, the Lorentz force played a relatively minor role. For example, they have shown that a purely hydrodynamic parameter (the local Rossby number Ro`) determines the stability domain of dynamos dominated by the axial dipole (dipolar dynamos). In this study, we show that this result cannot hold when the Lorentz force becomes dominant. We model turbulent geodynamo simulations with a strong Lorentz force by varying the important parameters on several orders of magnitude. This method enables us to question previous results and to argue on the applications of numerical dynamos in order to better understand the geodynamo problem. Strong dipolar elds considerably aect the kinetic energy distribution of convective motions which enables the maintenance of this eld con guration. The relative importance of each force depends on the spatial length scale, whereas Ro` is a global output parameter which ignores the spatial dependency. We show that inertia does not induce a dipole collapse as long as the Lorentz and the Coriolis forces remain dominant at large length scales.
Keywords: Dynamo, Magnetohydrodynamics, Planetary interiors 1. Introduction
The mechanism behind planetary magnetic elds, maintained over millions of years despite ohmic dissipation, is one of the most famous scienti c problems. The favorite candidate, the dynamo eect, explains how a magnetic eld can be generated by electromotive forces driven by electrically conductive uid movements in celestial bodies[18, 12]. Dynamo action is an instability by which a conducting uid transfers part of its kinetic energy to magnetic energy. Planetary magnetic elds result from such processes thought to be driven by convection in electrically conducting uid regions. For example, turbulent convective motions in the Earth's outer core are mainly subject to two forces: the Coriolis force resulting from global planetary rotation and the Lorentz force which accounts for the back-reaction of magnetic eld on the ow from which it is generated. In order to improve our understanding about the Earth's magnetic eld, it is of interest to numerically model
∗melissa.menu@lpp.polytechnique.fr
force, the Buoyancy (Archimede) force and the Coriolis force (the so-called MAC-balance). Earth-like elds do not only consist of a simple axial dipole eld which is maintained over time. Paleomagnetic measurements have allowed us to reconstruct the dynamics of the magnetic eld and revealed that the Earth's dipolar eld has reversed its polarity several hundred times during the past 160 million years. These polarity reversals are known to be strongly irregular and chaotic, and such events were observed numerically by Glatzmaier & Roberts [13] for the rst time. Since this rst fully three-dimensional numerical model [e.g. 13], there have been signi cant advances in understanding the geodynamo. Many features of the Earth's magnetic eld have been reproduced numerically [5, 6, 4, 35, 7]. However the realistic physical properties of the Earth's outer core (fast rotation, low dissipation coe cients. . . ) dier by several orders of magnitude from the values acces-sible in direct numerical simulations (DNS). For instance, the Ekman number, ratio of the viscous and Coriolis forces, is approximately E = 10−15in the Earth's outer core whereas E ≥ 10−7can be considered in numerical models (see below for a complete de nition of this dimensionless number). Progress in both numerical methods and parallel machine architecture has made it possible to ex-plore an extensive parameter space in order to deduce the physical ingredients responsible for the dominance of the axial dipole eld [8, 16, 28, 31, 23]. Olson & Christensen [20] have also inferred some of the physical causes associated with eld reversals in planetary interiors from numerical studies. Numerical simulations suggest that reversals may arise from the importance of inertia relative to the Coriolis force in Earth's outer core.
Observations and numerical simulations indicate that rapid global rotation and thus the ordering in uence of the Coriolis force is of major importance for the generation of coherent magnetic elds [34, 14, 3]. Kutzner & Christensen [17] demonstrated the existence of a dipolar and a multipolar dynamo regime, and Christensen & Aubert [8] showed that the transition between these two regimes is governed by a local Rossby number (Ro`), i.e. the in uence of inertia relative to the Coriolis force. Similar results were reported by Sreenivasan & Jones [33], as well. Dipolar models were found for small local Rossby numbers; separated by a fairly sharp regime boundary from multipolar models, where inertia is more important. The models transition from a dipolar morphology to a multipolar state when the local Rossby number reaches a certain value (Ro`> 0.1). However, such transitions have been reported by previous systematic parameter studies such as Christensen & Aubert [8] for models that explore a regime in which the Lorentz force does not play a major role (see for instance King & Buett [15]). Oruba & Dormy [22] have also con rmed the validity of the criteria Ro` < 0.1for dipolar dynamos, but only the data set published by Christensen & Aubert [8] has been used. Here, we study the stability domain of dipolar dynamos in a unexplored regime in which the Lorentz force is dominant.
In DNS, MAC-balance has been obtained either by considering high values of the magnetic Prandtl number P m [9, 11, 23] or by lowering the Ekman number [26, 30, 29]. In the former case, turbulence plays a minor role as the Reynolds number is low. By comparison, in the latter situation, high Reynolds numbers are considered and the numerical cost of such models increases signi cantly. Here, we link these two approaches by increasing the in uence of inertia (via the buoy-ant forcing Ra) in models with high magnetic Prandtl number P m. This method enables us to study the in uence of the Lorentz force on the local Rossby number criteria. Consequently, we study the stability domain of dipolar dynamo simulations in a regime where the Lorentz force is not negligible. After a brief presentation of the model and control parameters used in this study, the validity
characterise a dipolar model are presented, and used to distinguish two branches of dipolar solutions (3.3). An explanation of the dierent behaviours observed for each branch is then proposed (3.4). Finally these results are compared with observations and previous studies (4).
2. Model and input/output parameters
We follow previous extensive parameter studies [8, 20, 28, 24] and use the Boussinesq approxi-mation. The uid freely evolves between an inner and outer spherical shell, respectively at radius riand ro, with no-slip boundary conditions. The aspect ratio is de ned as χ = ri/roand is xed to 0.35. The evolution is constrained by the classical incompressible magneto-hydrodynamic (MHD) equations with rotation (momentum and induction), and an additional equation constraining the temperature evolution. A temperature contrast ∆T is imposed between the inner and outer shells to generate convection and the magnetic eld matches a potential eld outside the uid shell (in-sulating boundaries).
Length is expressed in boundary shell gap units D = ro− ri, time in units of D2/ν, magnetic eld B in units of (ρµηΩ)1/2, pressure in units of ρνΩ and temperature in units of ∆T , where ρ is the density, Ω the rotation rate, ν, µ, η and κ are the kinematic viscosity, magnetic permeabil-ity, magnetic diusivity and thermal diusivity respectively. These units are used to obtain the dimensionless equations : E ∂u ∂t + u· ∇u − ν∇2u =−∇P − 2z × u + Rarr o T + 1 P m(∇ × B) × B , (1) ∂B ∂t =∇ × (u × B) + 1 P m∇2B , (2) ∂T ∂t + u· ∇T = P r1 ∇2T , (3) ∇ · u = 0 , ∇ · B = 0. (4)
The nondimensionalization of these equations reveals commonly used dimensionless numbers: the Ekman number E = ν/ΩD2which represents the ratio of viscous to Coriolis forces, the Rayleigh number Ra = αgo∆T D3/νκ which can be seen as the ratio of buoyancy to viscous force, the magnetic Prandtl number P m = ν/η, and the classical Prandtl number P r = ν/κ which is xed to 1 in our study.
To solve equations (1) to (4) in spherical coordinates, we use the 3D pseudo-spectral code PaRoDy with the dimensionless quantities de ned above as control parameters (see Dormy et al. [10] for more details). All the simulations performed have been started with an initial dipolar magnetic eld. To reach high Rayleigh number values, some simulations have been initialised from the solution of a previous simulation with similar parameter values.
The main global output quantities we will use in this article are the Elsasser number, the Rossby number, and the magnetic Reynolds number. The Elsasser number Λ = B2
rms/2Ωρµηgives information about the importance of the Lorentz force compared to the Coriolis force. The Rossby number Ro = urms/ΩD compares inertia to Coriolis force and the magnetic Reynolds number Rm = urmsD/ηmeasures the ratio of the advection to the diusive term in the induction equation
Moreover we have chosen to use the Rayleigh number normalised by its critical value Rac for convection, whose values has been taken from Christensen & Aubert [8], for ease of reading.
We also compute the local Rossby number Ro` as introduced by Christensen & Aubert [8] to have a better measurement of the balance between inertia and Coriolis force, taking into account a length scale deduced from the mean spherical harmonic degree of the ow ¯`uinstead of the shell gap D : Ro`= Ro Lu with Lu= π¯ `u , where `¯u= P ``hu`· u`i 2Ekin , (5)
with ` the spherical harmonic degree, Ekin= 12R
hu · ui dV and the brackets h·i denotes an average over time, radial and azimuthal directions. The brackets h·iiwill be used to denote an average over the quantity i.
Another characteristic length scale, the kinetic dissipation length scale Lν, can be de ned as: L2ν=h
R u2dV R
(∇ × u)2dVit. (6)
This typical length for dissipation enables us to de ne another local Rossby number RoLν = Ro/Lν. Similarly, a dissipation length scale for the magnetic eld exists L2
η = h
R B2dV R
(∇×B)2dVit. Dormy [9] derived a modi ed Elsasser number Λ0 that is more suitable to measure the relative importance of the Lorentz force compared to the Coriolis force. Indeed, it also takes into account the magnetic dissipation length scale and the magnetic Reynolds number :
Λ0= Λ D
Rm Lη. (7)
In order to distinguish dynamos with a dominant dipolar component from the others, Chris-tensen & Aubert [8] have calculated the relative dipole eld strength fdip, which is the ratio of the mean magnetic dipole component to the others, at the outer shell:
fdip=h(Bl=1· Bl=1)
1/2
P12
l=1(Bl· Bl)1/2iφ,t. (8) Only the twelve rst modes are considered in order to compare simulations with geomagnetic ob-servations. Smaller length scales cannot be observed. A transition from a dipolar state (fdip& 0.5) to a non-dipolar one occurs around a value of the local Rossby number Ro`∼ 0.1. This empirical result, called the local Rossby number criteria, seems quite robust as it is valid for multiple values of E, P m and P r, each of these parameters being varied over at least two orders of magnitude. This feature is interpreted as a boundary between two regimes : dipolar dynamos and multipolar dynamos.
Volume-averaged output parameters enable us to compare models with dierent input parame-ters. However, physical interactions and, in particular, force balance can depend on the length scale in turbulent systems. We calculate the dierent forces as function of the spherical harmonic degree `. To avoid the pressure gradient in uence (or the geostrophic balance), we take the azimuthal average of the φ component of each force Fi as has been done by Sheyko et al. [30]. For sake
dominate, took the absolute value and time averaged, leading to h|Fi(`)|i with Fi(`) =
r < rXo−0.15D r > ri+0.15D
hFiφ(l)iφ,t. (9)
Typical resolutions are 288 points in the radial direction (up to 384 points). The spectral decomposition is truncated at a hundred modes (up to lmax = mmax = 256), in order to observe a drop by a factor 100 or more for the kinetic and the magnetic energy spectra of the spherical harmonic degree l and order m from the maximum to the energy cut-o lmax and mmax. The simulations performed are summarised in Appendix B.
3. Results
3.1. Turbulent dynamos with high Rm and a new de nition for dipolar dynamos
At rst, it is important to notice dierent characteristic behaviours observed on the time evo-lution of output parameters when turbulent dynamos (high buoyant forcing) with Ro` close to the transitional value 0.12 and high Rm (or P m). As a result, a need for a more restricted de nition of a dipolar dynamo appears. By using this new de nition, we will be able to study the stability domain of dipolar dynamos in the next sections.
Considering geodynamo simulations with realistic parameters is still far from being a reality. Except for the magnetic Reynolds number (Rm ∼ 1000), the other parameters dier from realistic ones by several orders of magnitude. Such a value of Rm was considered by recent studies which employed dierent strategies. Dormy [9], Petitdemange [23] and Dormy et al. [11] have shown that the Lorentz force plays a major role when Rm is su ciently high. However, these studies reach high values of Rm = P m Re by considering su ciently high values of P m and low values of the Reynolds number Re = urmsD/ν, meaning that their models cannot be really classi ed as turbulent models. For instance, Dormy [9] considered simulations close to the onset of convection (Ra < 4Rac). Yadav et al. [36], Schaeer et al. [26] and Aubert et al. [1] have also obtained a dominant Lorentz force by considering turbulent models with low P m and low Ekman numbers. In our study, we complete this parameter space and we link the dierent previous studies by considering turbulent models (high values of Ra/Rac) with high P m. To this end, the Rayleigh number has been gradually increased with a xed P m value in order to reach this turbulent regime.
The tilt angle θ, which measures the direction of the magnetic dipole compared to the rotation axis, and the relative dipole eld strength fdip as a function of time are shown for dierent cases (see gure 1) with Ro` higher than 0.12. Figures 1(a) and 1(b) correspond to simulations with the same Ekman number. The buoyant forcing Ra/Rac and the magnetic Prandtl number P m are slightly higher for gure 1(b). On the one hand, large temporal uctuations of θ and fdip are observed for the model with P m = 3 ( gure 1(a)). The axial dipole is not a dominant component at the surface of the numerical domain (low fdip value) and several reversals of this component are obtained (θ oscillations). On the other hand, gure 1(b) with P m = 5 does not present such a behaviour: although a higher Ra/Rac has been employed, the axial dipole dominates at any time with a constant magnetic polarity (no reversal). The latter model can thus be classi ed as a dipolar dynamo. However, it is important to note that Ro`exceeds 0.12.
for a model with dierent parameters: E = 3.10 and P m = 6. After a long period of time (larger than one magnetic diusion time), the dipole collapses and fdip becomes lower than 0.2. At the same time, the polarity of the magnetic dipole measured by θ uctuates around π/2 which indicates that the dominant component of the magnetic dipole is its equatorial dipole component. This situation is only a transient con guration. Then, the axial dipole with an opposite polarity again dominates the other components. This reversal seems to appear between two long periods in which the axial dipole is dominant. It is important to note that large temporal uctuations of the magnetic energy as measured by Λ are observed for this model. In comparison, relatively constant values of Λ are obtained for gures 1(a) and 1(b) (not shown). The condition fdip> 0.5, mentioned by previous studies [8], does not appear as a su cient condition in order to identify dipolar models since it indicates that this dynamo model with unpredictable reversals and important temporal variations of the magnetic eld strength is a dipolar solution. This particular behaviour has been reported before for dynamos with Ro` close to the transitional value 0.12. Other proxies such as the time evolution of the tilt angle must be taken into account.
As mentioned, the magnetic energy as measured by the Elsasser number Λ is another useful piece of information. When a transition from a dipolar con guration to a multipolar one occurs, a decrease of Λ is observed in most of the cases. In order to highlight the evolution of Λ in the parameter space, gure 2 shows the evolution of Λ as a function of Ra/Rac for dierent P m and E. Since we are interested in the relative variations of Λ when a transition occurs, Λ is normalised by its maximum value Λmax = max(Λ(P m)). Figure 2 clearly shows that the dipole slump giving rise to multipolar dynamos (square symbols) is accompanied by a decrease of Λ. This eect (loss of magnetic energy) increases as P m decreases. For example, multipolar dynamos at low P m values have Λ/Λmax lower than 0.25 i.e. the dipole collapse has induced a loss of 75% of the magnetic energy (see yellow markers in gures 2(b) & 2(c)). Increasing P m reduces the dierence in Λ value between dipolar cases and multipolar cases. Indeed, for the highest P m at which a transition is observed (green markers in gure 2), it only represents a drop of about 50% or less. Some symbols marked by a dot will be discussed later in this paper.
Figure 2 shows time averaged quantities. However, such quantities can have large temporal uctuations as shown in gure 1, especially when high P m are considered. For example, the case represented in gure 1(c) has a particular behaviour: fdip temporarily drops below 0.2 around the period t ∼ 12.7 while the tilt angle uctuates around π/2 at the same time. Later fdip increases again to reach its original value (see above). This reversal of the magnetic eld axis attests that this case is not purely dipolar. Looking at the evolution of Λ ( gure 1(d)) con rms that the magnetic eld strength has been aected by this behaviour. This is another indication supporting the fact that some care must be taken in order to characterise the morphology of geodynamo models. In particular, the time evolution of fdip, θ and Λ appear as important proxies.
In order to avoid a deep analysis of our numerical results which explore a new parameter set, it seems crucial to have a proper de nition of dipolar dynamos. Motivated by modelling geomagnetic elds, this term is restricted to dynamos with a dominant dipolar component (fdip > 0.5) at any moment of the time integration. The polarity of the dipole measured by the tilt angle θ has to keep its orientation as well. As mentioned above, the magnetic eld strength measured by Λ is also a good proxy. A dynamo model is called a dipolar dynamo when no important uctuations of Λ are observed during the time integration. This modi ed de nition of dipolar dynamos allows
3.75 3.80 3.85 3.90 3.95 4.00 4.05
Magnetic diffusion time
0.0 0.5 1.0 1.5 2.0 2.5 Tilt angle 0.0 0.2 0.4 0.6 0.8 fdip
(a) Multipolar dynamo at E = 10−4, Ra/Rac= 43.1, P m = 3, Λ = 12.6, Ro`∼ 0.18, fdip∼ 0.25.
4.8 5.0 5.2 5.4
Magnetic diffusion time
0.0 0.5 1.0 1.5 2.0 2.5 Til ta ng le 0.0 0.2 0.4 0.6 0.8 fdip
(b) Dipolar dynamo at E = 10−4, Ra/Rac = 50.2, P m = 5, Λ = 45.9, Ro`∼ 0.18, fdip∼ 0.7.
11.5 12.0 12.5 13.0
Magnetic diffusion time
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Til ta ng le tilt 0.0 0.2 0.4 0.6 0.8 1.0 fdip fdip
(c) Multipolar dynamo at E = 3 × 10−4, Ra/Rac = 24.7, P m = 6, Λ = 21.2, Ro`∼ 0.17, fdip∼ 0.56.
11.5 12.0 12.5 13.0
Magnetic diffusion time
0 10 20 30 40 Elsasser 0.03 0.04 0.05 0.06 0.07 0.08 Ro Rossby
(d) Elsasser (Λ) and Rossby (Ro) number time evolu-tion for the same parameters as gure 1(c).
Figure 1: Panels 1(a), 1(b) & 1(c) : Tilt angle (blue) and fdip(orange) evolution through magnetic diusion time. Panel 1(a) at low P m value denotes a multipolar simulation, whereas panel 1(b) denotes a dipolar dynamo with a low mean value of fdip. Panel 1(d) : Λ (blue) and Ro (orange) evolution through magnetic diusion time. This case is very close to a dipolar state, but sometimes explores another con guration.
101 Ra/Racrit 0.0 0.2 0.4 0.6 0.8 /max Pm = 3 (a) E = 3 × 10−4. 101 Ra/Racrit 0.0 0.2 0.4 0.6 0.8 /max Pm = 2 Pm = 1 (b) E = 1 × 10−4. 101 Ra/Racrit 0.0 0.2 0.4 0.6 0.8 1.0 /max Pm = 2.5 Pm = 1 Pm = 0.25 (c) E = 3 × 10−5. 101 Ra/Racrit 0.0 0.2 0.4 0.6 0.8 1.0 /max Pm = 1 Pm = 0.5 Pm = 0.2 Pm = 0.15 (d) E = 1 × 10−5.
Figure 2: Λ variations with Ra/Racnormalised by the maximum value of each P m branch (colours) for four Ekman